Subgroups — C₂₃×#25984 #19173610

Order 5888.

876 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3084	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3083	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2897	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2897	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2092	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2063	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2063	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2027	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1866	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1866	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1860	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1860	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1816	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1815	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1792	C₂	1	1	yes	maximal, characteristic
16	C₂×C₂₃×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
19	C₂₃×#544	C₂²	1	1	yes	characteristic, complemented
20	C₂₃×#520	C₂²	1	1	yes	characteristic
21	C₂₃×#520	C₂²	1	1	yes	characteristic
22	C₂₃×#479	—	2	1	no	—
23	C₂₃×#479	—	2	1	no	—
24	C₂₃×#478	—	2	1	no	—
25	C₂₃×#478	—	2	1	no	—
26	C₂₃×#477	—	2	1	no	—
27	C₂₃×#476	—	2	1	no	—
28	C₂₃×#475	—	2	1	no	—
29	C₂₃×#474	—	2	1	no	—
30	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic, complemented
31	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic, complemented
32	C₂₃×#428	C₂²	1	1	yes	characteristic, complemented
33	C₂₃×#428	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic, complemented
35	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic, complemented
36	C₂₃×#424	C₂²	1	1	yes	characteristic, complemented
37	C₂₃×#424	C₂²	1	1	yes	characteristic, complemented
38	C₂×C₂₃×#104	C₂²	1	1	yes	characteristic
39	C₂×C₂₃×#104	C₂²	1	1	yes	characteristic
40	C₂₃×#418	—	2	1	no	—
41	C₂₃×#418	—	2	1	no	—
42	C₂₃×#418	C₂²	1	1	yes	characteristic
43	C₂₃×#418	C₂²	1	1	yes	characteristic
44	C₂₃×#417	—	2	1	no	—
45	C₂₃×#417	—	2	1	no	—
46	C₂₃×#417	C₂²	1	1	yes	characteristic
47	C₂₃×#417	C₂²	1	1	yes	characteristic
48	C₂₃×#417	C₂²	1	1	yes	characteristic
49	C₂₃×#417	C₂²	1	1	yes	characteristic
50	C₂₃×#416	—	2	1	no	—
51	C₂₃×#416	—	2	1	no	—
52	C₂₃×#416	C₂²	1	1	yes	characteristic
53	C₂₃×#416	C₂²	1	1	yes	characteristic
54	C₂×C₂₃×#102	—	2	1	no	—
55	C₂×C₂₃×#102	—	2	1	no	—
56	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
57	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
58	C₂×C₂₃×#100	C₂²	1	1	yes	characteristic
59	C₂₃×#390	C₂²	1	1	yes	characteristic
60	C₂₃×#382	C₂²	1	1	yes	characteristic, complemented
61	C₂₃×#343	C₂²	1	1	yes	characteristic, complemented
62	C₂₃×#343	C₂²	1	1	yes	characteristic, complemented
63	C₂₃×#343	C₂²	1	1	yes	characteristic, complemented
64	C₂₃×#339	C₂²	1	1	yes	characteristic, complemented
65	C₂₃×#339	C₂²	1	1	yes	characteristic, complemented
66	C₂₃×#336	C₂²	1	1	yes	characteristic, complemented
67	C₂₃×H(5,𝔽₂)	—	2	2	no	—
68	C₂₃×H(5,𝔽₂)	—	2	2	no	—
69	C₂₃×H(5,𝔽₂)	—	4	1	no	—
70	C₂₃×H(5,𝔽₂)	Q₈	1	4	yes	complemented
71	C₂×#41×C₂₃	—	2	1	no	—
72	C₂×#41×C₂₃	—	2	1	no	—
73	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
74	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
75	C₂²×D₄×C₂₃	—	2	1	no	—
76	C₂²×D₄×C₂₃	—	2	1	no	—
77	C₂²×D₄×C₂₃	—	4	1	no	—
78	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
79	C₂₃×#131	—	2	2	no	—
80	C₂₃×#131	—	2	2	no	—
81	C₂₃×#131	—	4	1	no	—
82	C₂₃×#131	—	4	1	no	—
83	C₂×M₁₆×C₂₃	—	2	1	no	—
84	C₂×M₁₆×C₂₃	—	2	1	no	—
85	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
86	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
87	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
88	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
89	C₂²×C₈×C₂₃	—	2	1	no	—
90	C₂²×C₈×C₂₃	—	2	1	no	—
91	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
92	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
93	C₂₃×C₄⋊Q₈	—	2	1	no	—
94	C₂₃×C₄⋊Q₈	—	2	1	no	—
95	C₂₃×#125	—	2	1	no	—
96	C₂₃×#125	—	2	1	no	—
97	C₂₃×C₂²⋊Q₈	—	4	1	no	—
98	C₄×Q₈×C₂₃	—	2	1	no	—
99	C₄×Q₈×C₂₃	—	2	1	no	—
100	C₄×D₄×C₂₃	—	2	1	no	—
101	C₄×D₄×C₂₃	—	2	1	no	—
102	C₄×D₄×C₂₃	—	4	1	no	—
103	C₄×D₄×C₂₃	—	4	1	no	—
104	C₂₃×#117	—	2	1	no	—
105	C₂₃×#117	—	2	1	no	—
106	C₂₃×#117	C₂³	1	1	yes	characteristic
107	C₂₃×#117	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
109	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
110	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
113	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
114	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
115	C₂₃×#108	—	4	1	no	—
116	C₂₃×#108	—	4	1	no	—
117	C₂₃×#107	—	2	1	no	—
118	C₂₃×#107	—	2	1	no	—
119	C₂₃×#107	—	2	1	no	—
120	C₂₃×#107	—	2	1	no	—
121	C₂₃×#106	—	2	1	no	—
122	C₂₃×#106	—	2	1	no	—
123	C₂₃×#106	—	2	1	no	—
124	C₂₃×#106	—	2	1	no	—
125	C₂₃×C₄⋊C₈	—	2	1	no	—
126	C₂₃×C₄⋊C₈	—	2	1	no	—
127	C₂₃×C₄⋊C₈	—	2	1	no	—
128	C₂₃×C₄⋊C₈	—	2	1	no	—
129	C₂₃×#104	—	2	2	no	—
130	C₂₃×#104	—	2	2	no	—
131	C₂₃×#103	—	2	1	no	—
132	C₂₃×#103	—	2	1	no	—
133	C₂₃×#103	—	2	1	no	—
134	C₂₃×#103	—	2	1	no	—
135	C₂₃×#103	D₄	1	1	yes	characteristic
136	C₂₃×#103	D₄	1	1	yes	characteristic
137	C₂₃×#103	D₄	1	1	yes	characteristic
138	C₂₃×#103	D₄	1	1	yes	characteristic
139	C₂₃×#102	—	2	1	no	—
140	C₂₃×#102	—	2	1	no	—
141	C₂₃×#102	—	2	1	no	—
142	C₂₃×#102	—	2	1	no	—
143	C₂₃×#102	—	2	1	no	—
144	C₂₃×#102	—	2	1	no	—
145	C₂₃×#102	—	2	1	no	—
146	C₂₃×#102	—	2	1	no	—
147	C₂₃×#102	—	2	1	no	—
148	C₂₃×#102	—	2	1	no	—
149	C₂₃×#102	—	2	1	no	—
150	C₂₃×#102	—	2	1	no	—
151	C₂₃×#102	D₄	1	1	yes	characteristic
152	C₂₃×#102	D₄	1	1	yes	characteristic
153	C₂₃×#102	D₄	1	1	yes	characteristic
154	C₂₃×#102	D₄	1	1	yes	characteristic
155	C₂₃×#100	—	4	1	no	—
156	C₂₃×#95	—	4	1	no	—
157	C₂⁴×C₂₃	—	2	1	no	—
158	C₂⁴×C₂₃	—	4	1	no	—
159	#41×C₂₃	—	2	1	no	—
160	#41×C₂₃	—	2	1	no	—
161	#41×C₂₃	—	2	1	no	—
162	#41×C₂₃	—	2	1	no	—
163	#41×C₂₃	—	2	4	no	—
164	#41×C₂₃	—	2	4	no	—
165	#41×C₂₃	—	4	2	no	—
166	#41×C₂₃	—	4	2	no	—
167	#41×C₂₃	C₂×Q₈	1	2	yes	—
168	#41×C₂₃	C₂×Q₈	1	2	yes	—
169	#41×C₂₃	#41	1	2	yes	—
170	#41×C₂₃	#41	1	2	yes	—
171	C₂×Q₈×C₂₃	—	4	1	no	—
172	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
173	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
174	C₂×D₄×C₂₃	—	2	1	no	—
175	C₂×D₄×C₂₃	—	2	1	no	—
176	C₂×D₄×C₂₃	—	2	4	no	—
177	C₂×D₄×C₂₃	—	2	4	no	—
178	C₂×D₄×C₂₃	—	2	1	no	—
179	C₂×D₄×C₂₃	—	2	1	no	—
180	C₂×D₄×C₂₃	—	2	1	no	—
181	C₂×D₄×C₂₃	—	2	1	no	—
182	C₂×D₄×C₂₃	—	2	1	no	—
183	C₂×D₄×C₂₃	—	4	2	no	—
184	C₂×D₄×C₂₃	—	4	2	no	—
185	C₂×D₄×C₂₃	—	4	1	no	—
186	C₂×D₄×C₂₃	—	4	1	no	—
187	C₂×D₄×C₂₃	—	4	4	no	—
188	C₂×D₄×C₂₃	—	4	4	no	—
189	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
190	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
191	C₂×D₄×C₂₃	C₂×Q₈	1	2	yes	—
192	C₂²×C₄×C₂₃	—	2	1	no	—
193	C₂²×C₄×C₂₃	—	2	1	no	—
194	C₂²×C₄×C₂₃	—	2	1	no	—
195	C₂²×C₄×C₂₃	—	2	1	no	—
196	C₂²×C₄×C₂₃	—	2	1	no	—
197	C₂²×C₄×C₂₃	—	2	1	no	—
198	C₂²×C₄×C₂₃	—	4	1	no	—
199	C₂²×C₄×C₂₃	—	4	1	no	—
200	C₂²×C₄×C₂₃	—	4	1	no	—
201	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
202	M₁₆×C₂₃	—	2	1	no	—
203	M₁₆×C₂₃	—	2	1	no	—
204	M₁₆×C₂₃	—	2	1	no	—
205	M₁₆×C₂₃	—	2	1	no	—
206	M₁₆×C₂₃	—	2	1	no	—
207	M₁₆×C₂₃	—	2	1	no	—
208	M₁₆×C₂₃	—	2	1	no	—
209	M₁₆×C₂₃	—	2	1	no	—
210	M₁₆×C₂₃	—	4	2	no	—
211	M₁₆×C₂₃	—	4	2	no	—
212	C₂×C₈×C₂₃	—	2	1	no	—
213	C₂×C₈×C₂₃	—	2	1	no	—
214	C₂×C₈×C₂₃	—	2	1	no	—
215	C₂×C₈×C₂₃	—	2	1	no	—
216	C₂×C₈×C₂₃	—	2	1	no	—
217	C₂×C₈×C₂₃	—	2	1	no	—
218	C₂×C₈×C₂₃	—	2	1	no	—
219	C₂×C₈×C₂₃	—	2	1	no	—
220	C₂×C₈×C₂₃	—	4	2	no	—
221	C₂×C₈×C₂₃	—	4	2	no	—
222	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
223	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
224	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
225	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
226	C₄⋊C₄×C₂₃	—	2	1	no	—
227	C₄⋊C₄×C₂₃	—	2	1	no	—
228	C₄⋊C₄×C₂₃	—	2	1	no	—
229	C₄⋊C₄×C₂₃	—	2	1	no	—
230	C₄⋊C₄×C₂₃	—	2	1	no	—
231	C₄⋊C₄×C₂₃	—	2	1	no	—
232	C₄⋊C₄×C₂₃	—	4	1	no	—
233	C₄⋊C₄×C₂₃	—	4	1	no	—
234	C₄⋊C₄×C₂₃	—	4	1	no	—
235	C₄⋊C₄×C₂₃	—	4	1	no	—
236	C₄⋊C₄×C₂₃	—	4	1	no	—
237	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
238	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
239	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
240	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
241	C₂²⋊C₄×C₂₃	—	2	1	no	—
242	C₂²⋊C₄×C₂₃	—	2	1	no	—
243	C₂²⋊C₄×C₂₃	—	4	1	no	—
244	C₂²⋊C₄×C₂₃	—	4	1	no	—
245	C₂²⋊C₄×C₂₃	—	8	2	no	—
246	C₄²×C₂₃	—	2	1	no	—
247	C₄²×C₂₃	—	2	1	no	—
248	C₄²×C₂₃	—	2	1	no	—
249	C₄²×C₂₃	—	2	1	no	—
250	C₄²×C₂₃	—	4	1	no	—
251	C₄²×C₂₃	—	4	1	no	—
252	#25984	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
253	C₂³×C₂₃	—	2	1	no	—
254	C₂³×C₂₃	—	2	1	no	—
255	C₂³×C₂₃	—	2	2	no	—
256	C₂³×C₂₃	—	4	4	no	—
257	C₂³×C₂₃	—	4	1	no	—
258	C₂³×C₂₃	—	4	1	no	—
259	C₂³×C₂₃	—	8	1	no	—
260	C₂³×C₂₃	—	8	1	no	—
261	C₂³×C₂₃	—	16	1	no	—
262	C₂³×C₂₃	—	16	1	no	—
263	C₂³×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
264	Q₈×C₂₃	—	2	1	no	—
265	Q₈×C₂₃	—	2	1	no	—
266	Q₈×C₂₃	—	16	1	no	—
267	Q₈×C₂₃	C₂²⋊Q₈	1	2	yes	—
268	Q₈×C₂₃	C₂²⋊Q₈	1	2	yes	—
269	D₄×C₂₃	—	2	2	no	—
270	D₄×C₂₃	—	2	2	no	—
271	D₄×C₂₃	—	2	2	no	—
272	D₄×C₂₃	—	2	2	no	—
273	D₄×C₂₃	—	2	2	no	—
274	D₄×C₂₃	—	2	2	no	—
275	D₄×C₂₃	—	2	1	no	—
276	D₄×C₂₃	—	2	1	no	—
277	D₄×C₂₃	—	4	4	no	—
278	D₄×C₂₃	—	4	4	no	—
279	D₄×C₂₃	—	4	1	no	—
280	D₄×C₂₃	—	4	1	no	—
281	D₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
282	D₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
283	C₂×C₄×C₂₃	—	2	1	no	—
284	C₂×C₄×C₂₃	—	2	1	no	—
285	C₂×C₄×C₂₃	—	2	2	no	—
286	C₂×C₄×C₂₃	—	2	2	no	—
287	C₂×C₄×C₂₃	—	2	2	no	—
288	C₂×C₄×C₂₃	—	2	2	no	—
289	C₂×C₄×C₂₃	—	2	1	no	—
290	C₂×C₄×C₂₃	—	2	1	no	—
291	C₂×C₄×C₂₃	—	2	1	no	—
292	C₂×C₄×C₂₃	—	2	1	no	—
293	C₂×C₄×C₂₃	—	4	4	no	—
294	C₂×C₄×C₂₃	—	4	1	no	—
295	C₂×C₄×C₂₃	—	4	1	no	—
296	C₂×C₄×C₂₃	—	4	1	no	—
297	C₂×C₄×C₂₃	—	4	1	no	—
298	C₂×C₄×C₂₃	—	4	1	no	—
299	C₂×C₄×C₂₃	—	4	1	no	—
300	C₂×C₄×C₂₃	—	8	1	no	—
301	C₂×C₄×C₂₃	—	8	1	no	—
302	C₂×C₄×C₂₃	—	8	1	no	—
303	C₂×C₄×C₂₃	—	16	1	no	—
304	C₂×C₄×C₂₃	—	16	1	no	—
305	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
306	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
307	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
308	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
309	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
310	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
311	C₈×C₂₃	—	2	1	no	—
312	C₈×C₂₃	—	2	1	no	—
313	C₈×C₂₃	—	2	1	no	—
314	C₈×C₂₃	—	2	1	no	—
315	C₈×C₂₃	—	4	1	no	—
316	C₈×C₂₃	—	4	1	no	—
317	C₈×C₂₃	—	4	1	no	—
318	C₈×C₂₃	—	4	1	no	—
319	#3084	C₂×C₂₃	1	1	yes	characteristic, complemented
320	#3083	C₂×C₂₃	1	1	yes	characteristic, complemented
321	#2897	C₂×C₂₃	1	1	yes	characteristic
322	#2897	C₂×C₂₃	1	1	yes	characteristic
323	#2092	C₂×C₂₃	1	1	yes	characteristic, complemented
324	#2063	C₂×C₂₃	1	1	yes	characteristic, complemented
325	#2063	C₂×C₂₃	1	1	yes	characteristic, complemented
326	#2027	C₂×C₂₃	1	1	yes	characteristic, complemented
327	#1866	C₂×C₂₃	1	1	yes	characteristic, complemented
328	#1866	C₂×C₂₃	1	1	yes	characteristic, complemented
329	#1860	C₂×C₂₃	1	1	yes	characteristic, complemented
330	#1860	C₂×C₂₃	1	1	yes	characteristic, complemented
331	#1816	C₂×C₂₃	1	1	yes	characteristic, complemented
332	#1815	C₂×C₂₃	1	1	yes	characteristic, complemented
333	#1792	C₂×C₂₃	1	1	yes	characteristic
334	C₂²×C₂₃	—	2	1	no	—
335	C₂²×C₂₃	—	2	1	no	—
336	C₂²×C₂₃	—	2	2	no	—
337	C₂²×C₂₃	—	2	2	no	—
338	C₂²×C₂₃	—	4	1	no	—
339	C₂²×C₂₃	—	4	1	no	—
340	C₂²×C₂₃	—	4	1	no	—
341	C₂²×C₂₃	—	4	1	no	—
342	C₂²×C₂₃	—	4	2	no	—
343	C₂²×C₂₃	—	8	2	no	—
344	C₂²×C₂₃	—	8	1	no	—
345	C₂²×C₂₃	—	8	1	no	—
346	C₂²×C₂₃	—	16	1	no	—
347	C₂²×C₂₃	—	16	4	no	—
348	C₂²×C₂₃	#543	1	1	yes	characteristic
349	C₂²×C₂₃	D₄²	1	1	yes	characteristic
350	C₂²×C₂₃	#551	1	1	yes	characteristic
351	C₄×C₂₃	—	2	2	no	—
352	C₄×C₂₃	—	2	2	no	—
353	C₄×C₂₃	—	4	1	no	—
354	C₄×C₂₃	—	4	1	no	—
355	C₄×C₂₃	—	4	1	no	—
356	C₄×C₂₃	—	4	1	no	—
357	C₄×C₂₃	—	8	1	no	—
358	C₄×C₂₃	—	8	1	no	—
359	C₄×C₂₃	—	8	1	no	—
360	C₄×C₂₃	—	8	1	no	—
361	C₄×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
362	C₄×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
363	C₄×C₂₃	#546	1	1	yes	characteristic
364	C₄×C₂₃	#546	1	1	yes	characteristic
365	C₂×H(5,𝔽₂)	C₂²×C₂₃	1	1	yes	characteristic
366	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
367	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
368	#544	C₂²×C₂₃	1	1	yes	characteristic, complemented
369	#520	C₂²×C₂₃	1	1	yes	characteristic
370	#520	C₂²×C₂₃	1	1	yes	characteristic
371	#479	—	2	1	no	—
372	#479	—	2	1	no	—
373	#478	—	2	1	no	—
374	#478	—	2	1	no	—
375	#477	—	2	1	no	—
376	#476	—	2	1	no	—
377	#475	—	2	1	no	—
378	#474	—	2	1	no	—
379	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic, complemented
380	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic, complemented
381	#428	C₂²×C₂₃	1	1	yes	characteristic, complemented
382	#428	C₂²×C₂₃	1	1	yes	characteristic, complemented
383	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic, complemented
384	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic, complemented
385	#424	C₂²×C₂₃	1	1	yes	characteristic, complemented
386	#424	C₂²×C₂₃	1	1	yes	characteristic, complemented
387	C₂×#104	C₂²×C₂₃	1	1	yes	characteristic
388	C₂×#104	C₂²×C₂₃	1	1	yes	characteristic
389	#418	—	2	1	no	—
390	#418	—	2	1	no	—
391	#418	C₂²×C₂₃	1	1	yes	characteristic
392	#418	C₂²×C₂₃	1	1	yes	characteristic
393	#417	—	2	1	no	—
394	#417	—	2	1	no	—
395	#417	C₂²×C₂₃	1	1	yes	characteristic
396	#417	C₂²×C₂₃	1	1	yes	characteristic
397	#417	C₂²×C₂₃	1	1	yes	characteristic
398	#417	C₂²×C₂₃	1	1	yes	characteristic
399	#416	—	2	1	no	—
400	#416	—	2	1	no	—
401	#416	C₂²×C₂₃	1	1	yes	characteristic
402	#416	C₂²×C₂₃	1	1	yes	characteristic
403	C₂×#102	—	2	1	no	—
404	C₂×#102	—	2	1	no	—
405	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
406	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
407	C₂×#100	C₂²×C₂₃	1	1	yes	characteristic
408	#390	C₂²×C₂₃	1	1	yes	characteristic
409	#382	C₂²×C₂₃	1	1	yes	characteristic, complemented
410	#343	C₂²×C₂₃	1	1	yes	characteristic, complemented
411	#343	C₂²×C₂₃	1	1	yes	characteristic, complemented
412	#343	C₂²×C₂₃	1	1	yes	characteristic, complemented
413	#339	C₂²×C₂₃	1	1	yes	characteristic, complemented
414	#339	C₂²×C₂₃	1	1	yes	characteristic, complemented
415	#336	C₂²×C₂₃	1	1	yes	characteristic, complemented
416	C₂×C₂₃	—	2	1	no	—
417	C₂×C₂₃	—	2	1	no	—
418	C₂×C₂₃	—	4	2	no	—
419	C₂×C₂₃	—	4	2	no	—
420	C₂×C₂₃	—	8	2	no	—
421	C₂×C₂₃	#2437	1	1	yes	characteristic
422	C₂×C₂₃	#3288	1	1	yes	characteristic
423	C₂×C₂₃	#3290	1	1	yes	characteristic
424	H(5,𝔽₂)	—	2	2	no	—
425	H(5,𝔽₂)	—	2	2	no	—
426	H(5,𝔽₂)	—	4	1	no	—
427	H(5,𝔽₂)	Q₈×C₂₃	1	4	yes	complemented
428	C₂×#41	—	2	1	no	—
429	C₂×#41	—	2	1	no	—
430	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
431	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
432	C₂²×D₄	—	2	1	no	—
433	C₂²×D₄	—	2	1	no	—
434	C₂²×D₄	—	4	1	no	—
435	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
436	#131	—	2	2	no	—
437	#131	—	2	2	no	—
438	#131	—	4	1	no	—
439	#131	—	4	1	no	—
440	C₂×M₁₆	—	2	1	no	—
441	C₂×M₁₆	—	2	1	no	—
442	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
443	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
444	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
445	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
446	C₂²×C₈	—	2	1	no	—
447	C₂²×C₈	—	2	1	no	—
448	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
449	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
450	C₄⋊Q₈	—	2	1	no	—
451	C₄⋊Q₈	—	2	1	no	—
452	#125	—	2	1	no	—
453	#125	—	2	1	no	—
454	C₂²⋊Q₈	—	4	1	no	—
455	C₄×Q₈	—	2	1	no	—
456	C₄×Q₈	—	2	1	no	—
457	C₄×D₄	—	2	1	no	—
458	C₄×D₄	—	2	1	no	—
459	C₄×D₄	—	4	1	no	—
460	C₄×D₄	—	4	1	no	—
461	#117	—	2	1	no	—
462	#117	—	2	1	no	—
463	#117	C₂³×C₂₃	1	1	yes	characteristic
464	#117	C₂³×C₂₃	1	1	yes	characteristic
465	C₂×C₄⋊C₄	—	2	1	no	—
466	C₂×C₄⋊C₄	—	2	1	no	—
467	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
468	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
469	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
470	C₂×C₂²⋊C₄	—	4	1	no	—
471	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
472	#108	—	4	1	no	—
473	#108	—	4	1	no	—
474	#107	—	2	1	no	—
475	#107	—	2	1	no	—
476	#107	—	2	1	no	—
477	#107	—	2	1	no	—
478	#106	—	2	1	no	—
479	#106	—	2	1	no	—
480	#106	—	2	1	no	—
481	#106	—	2	1	no	—
482	C₄⋊C₈	—	2	1	no	—
483	C₄⋊C₈	—	2	1	no	—
484	C₄⋊C₈	—	2	1	no	—
485	C₄⋊C₈	—	2	1	no	—
486	#104	—	2	2	no	—
487	#104	—	2	2	no	—
488	#103	—	2	1	no	—
489	#103	—	2	1	no	—
490	#103	—	2	1	no	—
491	#103	—	2	1	no	—
492	#103	D₄×C₂₃	1	1	yes	characteristic
493	#103	D₄×C₂₃	1	1	yes	characteristic
494	#103	D₄×C₂₃	1	1	yes	characteristic
495	#103	D₄×C₂₃	1	1	yes	characteristic
496	#102	—	2	1	no	—
497	#102	—	2	1	no	—
498	#102	—	2	1	no	—
499	#102	—	2	1	no	—
500	#102	—	2	1	no	—
501	#102	—	2	1	no	—
502	#102	—	2	1	no	—
503	#102	—	2	1	no	—
504	#102	—	2	1	no	—
505	#102	—	2	1	no	—
506	#102	—	2	1	no	—
507	#102	—	2	1	no	—
508	#102	D₄×C₂₃	1	1	yes	characteristic
509	#102	D₄×C₂₃	1	1	yes	characteristic
510	#102	D₄×C₂₃	1	1	yes	characteristic
511	#102	D₄×C₂₃	1	1	yes	characteristic
512	#100	—	4	1	no	—
513	#95	—	4	1	no	—
514	C₂₃	#25984	1	1	yes	characteristic, direct product factor, Sylow
515	C₂⁴	—	2	1	no	—
516	C₂⁴	—	4	1	no	—
517	#41	—	2	1	no	—
518	#41	—	2	1	no	—
519	#41	—	2	1	no	—
520	#41	—	2	1	no	—
521	#41	—	2	4	no	—
522	#41	—	2	4	no	—
523	#41	—	4	2	no	—
524	#41	—	4	2	no	—
525	#41	C₂×Q₈×C₂₃	1	2	yes	—
526	#41	C₂×Q₈×C₂₃	1	2	yes	—
527	#41	#41×C₂₃	1	2	yes	—
528	#41	#41×C₂₃	1	2	yes	—
529	C₂×Q₈	—	4	1	no	—
530	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
531	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
532	C₂×D₄	—	2	1	no	—
533	C₂×D₄	—	2	1	no	—
534	C₂×D₄	—	2	4	no	—
535	C₂×D₄	—	2	4	no	—
536	C₂×D₄	—	2	1	no	—
537	C₂×D₄	—	2	1	no	—
538	C₂×D₄	—	2	1	no	—
539	C₂×D₄	—	2	1	no	—
540	C₂×D₄	—	2	1	no	—
541	C₂×D₄	—	4	2	no	—
542	C₂×D₄	—	4	2	no	—
543	C₂×D₄	—	4	1	no	—
544	C₂×D₄	—	4	1	no	—
545	C₂×D₄	—	4	4	no	—
546	C₂×D₄	—	4	4	no	—
547	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
548	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
549	C₂×D₄	C₂×Q₈×C₂₃	1	2	yes	—
550	C₂²×C₄	—	2	1	no	—
551	C₂²×C₄	—	2	1	no	—
552	C₂²×C₄	—	2	1	no	—
553	C₂²×C₄	—	2	1	no	—
554	C₂²×C₄	—	2	1	no	—
555	C₂²×C₄	—	2	1	no	—
556	C₂²×C₄	—	4	1	no	—
557	C₂²×C₄	—	4	1	no	—
558	C₂²×C₄	—	4	1	no	—
559	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
560	M₁₆	—	2	1	no	—
561	M₁₆	—	2	1	no	—
562	M₁₆	—	2	1	no	—
563	M₁₆	—	2	1	no	—
564	M₁₆	—	2	1	no	—
565	M₁₆	—	2	1	no	—
566	M₁₆	—	2	1	no	—
567	M₁₆	—	2	1	no	—
568	M₁₆	—	4	2	no	—
569	M₁₆	—	4	2	no	—
570	C₂×C₈	—	2	1	no	—
571	C₂×C₈	—	2	1	no	—
572	C₂×C₈	—	2	1	no	—
573	C₂×C₈	—	2	1	no	—
574	C₂×C₈	—	2	1	no	—
575	C₂×C₈	—	2	1	no	—
576	C₂×C₈	—	2	1	no	—
577	C₂×C₈	—	2	1	no	—
578	C₂×C₈	—	4	2	no	—
579	C₂×C₈	—	4	2	no	—
580	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
581	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
582	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
583	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
584	C₄⋊C₄	—	2	1	no	—
585	C₄⋊C₄	—	2	1	no	—
586	C₄⋊C₄	—	2	1	no	—
587	C₄⋊C₄	—	2	1	no	—
588	C₄⋊C₄	—	2	1	no	—
589	C₄⋊C₄	—	2	1	no	—
590	C₄⋊C₄	—	4	1	no	—
591	C₄⋊C₄	—	4	1	no	—
592	C₄⋊C₄	—	4	1	no	—
593	C₄⋊C₄	—	4	1	no	—
594	C₄⋊C₄	—	4	1	no	—
595	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
596	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
597	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
598	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
599	C₂²⋊C₄	—	2	1	no	—
600	C₂²⋊C₄	—	2	1	no	—
601	C₂²⋊C₄	—	4	1	no	—
602	C₂²⋊C₄	—	4	1	no	—
603	C₂²⋊C₄	—	8	2	no	—
604	C₄²	—	2	1	no	—
605	C₄²	—	2	1	no	—
606	C₄²	—	2	1	no	—
607	C₄²	—	2	1	no	—
608	C₄²	—	4	1	no	—
609	C₄²	—	4	1	no	—
610	C₂³	—	2	1	no	—
611	C₂³	—	2	1	no	—
612	C₂³	—	2	2	no	—
613	C₂³	—	4	4	no	—
614	C₂³	—	4	1	no	—
615	C₂³	—	4	1	no	—
616	C₂³	—	8	1	no	—
617	C₂³	—	8	1	no	—
618	C₂³	—	16	1	no	—
619	C₂³	—	16	1	no	—
620	C₂³	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
621	Q₈	—	2	1	no	—
622	Q₈	—	2	1	no	—
623	Q₈	—	16	1	no	—
624	Q₈	C₂₃×C₂²⋊Q₈	1	2	yes	—
625	Q₈	C₂₃×C₂²⋊Q₈	1	2	yes	—
626	D₄	—	2	2	no	—
627	D₄	—	2	2	no	—
628	D₄	—	2	2	no	—
629	D₄	—	2	2	no	—
630	D₄	—	2	2	no	—
631	D₄	—	2	2	no	—
632	D₄	—	2	1	no	—
633	D₄	—	2	1	no	—
634	D₄	—	4	4	no	—
635	D₄	—	4	4	no	—
636	D₄	—	4	1	no	—
637	D₄	—	4	1	no	—
638	D₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
639	D₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
640	C₂×C₄	—	2	1	no	—
641	C₂×C₄	—	2	1	no	—
642	C₂×C₄	—	2	2	no	—
643	C₂×C₄	—	2	2	no	—
644	C₂×C₄	—	2	2	no	—
645	C₂×C₄	—	2	2	no	—
646	C₂×C₄	—	2	1	no	—
647	C₂×C₄	—	2	1	no	—
648	C₂×C₄	—	2	1	no	—
649	C₂×C₄	—	2	1	no	—
650	C₂×C₄	—	4	4	no	—
651	C₂×C₄	—	4	1	no	—
652	C₂×C₄	—	4	1	no	—
653	C₂×C₄	—	4	1	no	—
654	C₂×C₄	—	4	1	no	—
655	C₂×C₄	—	4	1	no	—
656	C₂×C₄	—	4	1	no	—
657	C₂×C₄	—	8	1	no	—
658	C₂×C₄	—	8	1	no	—
659	C₂×C₄	—	8	1	no	—
660	C₂×C₄	—	16	1	no	—
661	C₂×C₄	—	16	1	no	—
662	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
663	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
664	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
665	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
666	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
667	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
668	C₈	—	2	1	no	—
669	C₈	—	2	1	no	—
670	C₈	—	2	1	no	—
671	C₈	—	2	1	no	—
672	C₈	—	4	1	no	—
673	C₈	—	4	1	no	—
674	C₈	—	4	1	no	—
675	C₈	—	4	1	no	—
676	C₂²	—	2	1	no	—
677	C₂²	—	2	1	no	—
678	C₂²	—	2	2	no	—
679	C₂²	—	2	2	no	—
680	C₂²	—	4	1	no	—
681	C₂²	—	4	1	no	—
682	C₂²	—	4	1	no	—
683	C₂²	—	4	1	no	—
684	C₂²	—	4	2	no	—
685	C₂²	—	8	2	no	—
686	C₂²	—	8	1	no	—
687	C₂²	—	8	1	no	—
688	C₂²	—	16	1	no	—
689	C₂²	—	16	4	no	—
690	C₂²	C₂₃×#543	1	1	yes	characteristic
691	C₂²	D₄²×C₂₃	1	1	yes	characteristic
692	C₂²	C₂₃×#551	1	1	yes	characteristic
693	C₄	—	2	2	no	—
694	C₄	—	2	2	no	—
695	C₄	—	4	1	no	—
696	C₄	—	4	1	no	—
697	C₄	—	4	1	no	—
698	C₄	—	4	1	no	—
699	C₄	—	8	1	no	—
700	C₄	—	8	1	no	—
701	C₄	—	8	1	no	—
702	C₄	—	8	1	no	—
703	C₄	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
704	C₄	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
705	C₄	C₂₃×#546	1	1	yes	characteristic
706	C₄	C₂₃×#546	1	1	yes	characteristic
707	C₂	—	2	1	no	—
708	C₂	—	2	1	no	—
709	C₂	—	4	2	no	—
710	C₂	—	4	2	no	—
711	C₂	—	8	2	no	—
712	C₂	C₂₃×#2437	1	1	yes	characteristic
713	C₂	C₂₃×#3288	1	1	yes	characteristic
714	C₂	C₂₃×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂₃×#3084	C₂	semidirect
2. C₂₃×#3083	C₂	semidirect
5. C₂₃×#2092	C₂	semidirect
6. C₂₃×#2063	C₂	semidirect
7. C₂₃×#2063	C₂	semidirect
8. C₂₃×#2027	C₂	semidirect
9. C₂₃×#1866	C₂	semidirect
10. C₂₃×#1866	C₂	semidirect
11. C₂₃×#1860	C₂	semidirect
12. C₂₃×#1860	C₂	semidirect
13. C₂₃×#1816	C₂	semidirect
14. C₂₃×#1815	C₂	semidirect
19. C₂₃×#544	C₂²	semidirect
30. C₂×C₂₃×#108	C₂²	semidirect
31. C₂×C₂₃×#108	C₂²	semidirect
32. C₂₃×#428	C₂²	semidirect
33. C₂₃×#428	C₂²	semidirect
34. C₂×C₂₃×#107	C₂²	semidirect
35. C₂×C₂₃×#106	C₂²	semidirect
36. C₂₃×#424	C₂²	semidirect
37. C₂₃×#424	C₂²	semidirect
60. C₂₃×#382	C₂²	semidirect
61. C₂₃×#343	C₂²	semidirect
62. C₂₃×#343	C₂²	semidirect
63. C₂₃×#343	C₂²	semidirect
64. C₂₃×#339	C₂²	semidirect
65. C₂₃×#339	C₂²	semidirect
66. C₂₃×#336	C₂²	semidirect
70. C₂₃×H(5,𝔽₂)	Q₈	semidirect
252. #25984	C₂₃	direct (N × H)
319. #3084	C₂×C₂₃	semidirect
320. #3083	C₂×C₂₃	semidirect
323. #2092	C₂×C₂₃	semidirect
324. #2063	C₂×C₂₃	semidirect
325. #2063	C₂×C₂₃	semidirect
326. #2027	C₂×C₂₃	semidirect
327. #1866	C₂×C₂₃	semidirect
328. #1866	C₂×C₂₃	semidirect
329. #1860	C₂×C₂₃	semidirect
330. #1860	C₂×C₂₃	semidirect
331. #1816	C₂×C₂₃	semidirect
332. #1815	C₂×C₂₃	semidirect
368. #544	C₂²×C₂₃	semidirect
379. C₂×#108	C₂²×C₂₃	semidirect
380. C₂×#108	C₂²×C₂₃	semidirect
381. #428	C₂²×C₂₃	semidirect
382. #428	C₂²×C₂₃	semidirect
383. C₂×#107	C₂²×C₂₃	semidirect
384. C₂×#106	C₂²×C₂₃	semidirect
385. #424	C₂²×C₂₃	semidirect
386. #424	C₂²×C₂₃	semidirect
409. #382	C₂²×C₂₃	semidirect
410. #343	C₂²×C₂₃	semidirect
411. #343	C₂²×C₂₃	semidirect
412. #343	C₂²×C₂₃	semidirect
413. #339	C₂²×C₂₃	semidirect
414. #339	C₂²×C₂₃	semidirect
415. #336	C₂²×C₂₃	semidirect
427. H(5,𝔽₂)	Q₈×C₂₃	semidirect
514. C₂₃	#25984	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₃×#3084	2
C₂₃×#3083	2
C₂₃×#2897	2
C₂₃×#2092	2
C₂₃×#2063	2
C₂₃×#2027	2
C₂₃×#1866	2
C₂₃×#1860	2
C₂₃×#1816	2
C₂₃×#1815	2
C₂₃×#1792	2
#25984	23

Subgroups — C23×#25984 #19173610

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#25984 #19173610