Subgroups — C₂₃×#25985 #19173611

Order 5888.

842 conjugacy classes of subgroups.

The subgroup lattice has 694 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×#2898	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2898	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2097	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2064	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2064	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2028	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×#1793	C₂	1	1	yes	maximal, characteristic
16	C₂×C₂₃×2₋¹⁺⁴	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
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
31	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
32	C₂₃×#428	C₂²	1	1	yes	characteristic
33	C₂₃×#428	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
36	C₂₃×#424	C₂²	1	1	yes	characteristic
37	C₂₃×#424	C₂²	1	1	yes	characteristic
38	C₂×C₂₃×#104	C₂²	1	1	yes	characteristic
39	C₂×C₂₃×#104	C₂²	1	1	yes	characteristic
40	C₂₃×#419	—	2	1	no	—
41	C₂₃×#419	—	2	1	no	—
42	C₂₃×#419	C₂²	1	1	yes	characteristic
43	C₂₃×#419	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×#103	—	2	1	no	—
55	C₂×C₂₃×#103	—	2	1	no	—
56	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
57	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
58	C₂×C₂₃×#101	C₂²	1	1	yes	characteristic
59	C₂₃×#391	C₂²	1	1	yes	characteristic
60	C₂₃×#382	C₂²	1	1	yes	characteristic
61	C₂₃×#343	C₂²	1	1	yes	characteristic
62	C₂₃×#343	C₂²	1	1	yes	characteristic
63	C₂₃×#343	C₂²	1	1	yes	characteristic
64	C₂₃×#339	C₂²	1	1	yes	characteristic
65	C₂₃×#339	C₂²	1	1	yes	characteristic
66	C₂₃×#336	C₂²	1	1	yes	characteristic
67	C₂₃×2₋¹⁺⁴	—	2	2	no	—
68	C₂₃×2₋¹⁺⁴	—	2	2	no	—
69	C₂₃×2₋¹⁺⁴	—	4	1	no	—
70	C₂₃×2₋¹⁺⁴	Q₈	1	4	yes	complemented
71	C₂×#41×C₂₃	—	2	1	no	—
72	C₂×#41×C₂₃	—	2	1	no	—
73	C₂×#41×C₂₃	—	4	1	no	—
74	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
75	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
76	C₂²×Q₈×C₂₃	—	2	1	no	—
77	C₂²×Q₈×C₂₃	—	2	1	no	—
78	C₂²×Q₈×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×C₄²×C₂₃	C₂³	1	1	yes	characteristic
114	C₂₃×#108	—	4	1	no	—
115	C₂₃×#108	—	4	1	no	—
116	C₂₃×#107	—	2	1	no	—
117	C₂₃×#107	—	2	1	no	—
118	C₂₃×#107	—	2	1	no	—
119	C₂₃×#107	—	2	1	no	—
120	C₂₃×#106	—	2	1	no	—
121	C₂₃×#106	—	2	1	no	—
122	C₂₃×#106	—	2	1	no	—
123	C₂₃×#106	—	2	1	no	—
124	C₂₃×C₄⋊C₈	—	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×#104	—	2	2	no	—
129	C₂₃×#104	—	2	2	no	—
130	C₂₃×#103	—	2	1	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	—	2	1	no	—
136	C₂₃×#103	—	2	1	no	—
137	C₂₃×#103	—	2	1	no	—
138	C₂₃×#103	—	2	1	no	—
139	C₂₃×#103	—	2	1	no	—
140	C₂₃×#103	—	2	1	no	—
141	C₂₃×#103	—	2	1	no	—
142	C₂₃×#103	D₄	1	1	yes	characteristic
143	C₂₃×#103	D₄	1	1	yes	characteristic
144	C₂₃×#103	D₄	1	1	yes	characteristic
145	C₂₃×#103	D₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
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×#101	—	4	1	no	—
155	C₂₃×#95	—	4	1	no	—
156	C₂₃×#95	—	4	1	no	—
157	#41×C₂₃	—	2	1	no	—
158	#41×C₂₃	—	2	1	no	—
159	#41×C₂₃	—	2	1	no	—
160	#41×C₂₃	—	2	1	no	—
161	#41×C₂₃	—	2	4	no	—
162	#41×C₂₃	—	2	4	no	—
163	#41×C₂₃	—	4	4	no	—
164	#41×C₂₃	—	4	2	no	—
165	#41×C₂₃	—	4	2	no	—
166	#41×C₂₃	—	4	4	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	—	2	1	no	—
172	C₂×Q₈×C₂₃	—	2	4	no	—
173	C₂×Q₈×C₂₃	—	2	4	no	—
174	C₂×Q₈×C₂₃	—	2	1	no	—
175	C₂×Q₈×C₂₃	—	2	1	no	—
176	C₂×Q₈×C₂₃	—	2	1	no	—
177	C₂×Q₈×C₂₃	—	2	1	no	—
178	C₂×Q₈×C₂₃	—	4	1	no	—
179	C₂×Q₈×C₂₃	—	4	1	no	—
180	C₂×Q₈×C₂₃	—	4	2	no	—
181	C₂×Q₈×C₂₃	—	4	2	no	—
182	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
183	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
184	C₂×Q₈×C₂₃	C₂×Q₈	1	2	yes	—
185	C₂×D₄×C₂₃	—	2	1	no	—
186	C₂×D₄×C₂₃	—	2	1	no	—
187	C₂×D₄×C₂₃	—	4	1	no	—
188	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
189	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
190	C₂²×C₄×C₂₃	—	2	1	no	—
191	C₂²×C₄×C₂₃	—	2	1	no	—
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	—	4	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	C₄²×C₂₃	—	2	1	no	—
246	C₄²×C₂₃	—	2	1	no	—
247	C₄²×C₂₃	—	2	1	no	—
248	C₄²×C₂₃	—	2	1	no	—
249	C₄²×C₂₃	—	4	1	no	—
250	C₄²×C₂₃	—	4	1	no	—
251	#25985	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
252	C₂³×C₂₃	—	2	1	no	—
253	C₂³×C₂₃	—	2	1	no	—
254	C₂³×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
255	Q₈×C₂₃	—	2	1	no	—
256	Q₈×C₂₃	—	2	1	no	—
257	Q₈×C₂₃	—	2	2	no	—
258	Q₈×C₂₃	—	2	2	no	—
259	Q₈×C₂₃	—	4	4	no	—
260	Q₈×C₂₃	—	4	1	no	—
261	Q₈×C₂₃	—	4	1	no	—
262	Q₈×C₂₃	—	16	1	no	—
263	Q₈×C₂₃	C₂²⋊Q₈	1	2	yes	—
264	Q₈×C₂₃	C₂²⋊Q₈	1	2	yes	—
265	D₄×C₂₃	—	2	2	no	—
266	D₄×C₂₃	—	2	2	no	—
267	D₄×C₂₃	—	2	2	no	—
268	D₄×C₂₃	—	2	2	no	—
269	D₄×C₂₃	—	2	1	no	—
270	D₄×C₂₃	—	2	1	no	—
271	D₄×C₂₃	—	4	4	no	—
272	D₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
273	D₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
274	C₂×C₄×C₂₃	—	2	1	no	—
275	C₂×C₄×C₂₃	—	2	1	no	—
276	C₂×C₄×C₂₃	—	2	2	no	—
277	C₂×C₄×C₂₃	—	2	2	no	—
278	C₂×C₄×C₂₃	—	2	2	no	—
279	C₂×C₄×C₂₃	—	2	2	no	—
280	C₂×C₄×C₂₃	—	2	2	no	—
281	C₂×C₄×C₂₃	—	2	1	no	—
282	C₂×C₄×C₂₃	—	2	1	no	—
283	C₂×C₄×C₂₃	—	2	1	no	—
284	C₂×C₄×C₂₃	—	2	1	no	—
285	C₂×C₄×C₂₃	—	4	4	no	—
286	C₂×C₄×C₂₃	—	4	4	no	—
287	C₂×C₄×C₂₃	—	4	1	no	—
288	C₂×C₄×C₂₃	—	4	1	no	—
289	C₂×C₄×C₂₃	—	4	1	no	—
290	C₂×C₄×C₂₃	—	4	1	no	—
291	C₂×C₄×C₂₃	—	4	1	no	—
292	C₂×C₄×C₂₃	—	4	1	no	—
293	C₂×C₄×C₂₃	—	4	1	no	—
294	C₂×C₄×C₂₃	—	4	1	no	—
295	C₂×C₄×C₂₃	—	8	1	no	—
296	C₂×C₄×C₂₃	—	8	1	no	—
297	C₂×C₄×C₂₃	—	8	1	no	—
298	C₂×C₄×C₂₃	—	16	1	no	—
299	C₂×C₄×C₂₃	—	16	1	no	—
300	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
301	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
302	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
303	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
304	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
305	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
306	C₈×C₂₃	—	2	1	no	—
307	C₈×C₂₃	—	2	1	no	—
308	C₈×C₂₃	—	2	1	no	—
309	C₈×C₂₃	—	2	1	no	—
310	C₈×C₂₃	—	4	1	no	—
311	C₈×C₂₃	—	4	1	no	—
312	C₈×C₂₃	—	4	1	no	—
313	C₈×C₂₃	—	4	1	no	—
314	#3084	C₂×C₂₃	1	1	yes	characteristic, complemented
315	#3083	C₂×C₂₃	1	1	yes	characteristic, complemented
316	#2898	C₂×C₂₃	1	1	yes	characteristic
317	#2898	C₂×C₂₃	1	1	yes	characteristic
318	#2097	C₂×C₂₃	1	1	yes	characteristic, complemented
319	#2064	C₂×C₂₃	1	1	yes	characteristic, complemented
320	#2064	C₂×C₂₃	1	1	yes	characteristic, complemented
321	#2028	C₂×C₂₃	1	1	yes	characteristic, complemented
322	#1866	C₂×C₂₃	1	1	yes	characteristic, complemented
323	#1866	C₂×C₂₃	1	1	yes	characteristic, complemented
324	#1860	C₂×C₂₃	1	1	yes	characteristic, complemented
325	#1860	C₂×C₂₃	1	1	yes	characteristic, complemented
326	#1816	C₂×C₂₃	1	1	yes	characteristic, complemented
327	#1815	C₂×C₂₃	1	1	yes	characteristic, complemented
328	#1793	C₂×C₂₃	1	1	yes	characteristic
329	C₂²×C₂₃	—	2	1	no	—
330	C₂²×C₂₃	—	2	1	no	—
331	C₂²×C₂₃	—	2	2	no	—
332	C₂²×C₂₃	—	2	2	no	—
333	C₂²×C₂₃	—	4	1	no	—
334	C₂²×C₂₃	—	4	1	no	—
335	C₂²×C₂₃	—	4	1	no	—
336	C₂²×C₂₃	—	4	1	no	—
337	C₂²×C₂₃	#543	1	1	yes	characteristic
338	C₂²×C₂₃	D₄²	1	1	yes	characteristic
339	C₂²×C₂₃	#551	1	1	yes	characteristic
340	C₄×C₂₃	—	2	2	no	—
341	C₄×C₂₃	—	2	2	no	—
342	C₄×C₂₃	—	4	2	no	—
343	C₄×C₂₃	—	4	1	no	—
344	C₄×C₂₃	—	4	1	no	—
345	C₄×C₂₃	—	4	1	no	—
346	C₄×C₂₃	—	4	1	no	—
347	C₄×C₂₃	—	8	1	no	—
348	C₄×C₂₃	—	8	1	no	—
349	C₄×C₂₃	—	8	1	no	—
350	C₄×C₂₃	—	8	1	no	—
351	C₄×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
352	C₄×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
353	C₄×C₂₃	#546	1	1	yes	characteristic
354	C₄×C₂₃	#546	1	1	yes	characteristic
355	C₂×2₋¹⁺⁴	C₂²×C₂₃	1	1	yes	characteristic
356	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
357	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
358	#544	C₂²×C₂₃	1	1	yes	characteristic
359	#520	C₂²×C₂₃	1	1	yes	characteristic
360	#520	C₂²×C₂₃	1	1	yes	characteristic
361	#479	—	2	1	no	—
362	#479	—	2	1	no	—
363	#478	—	2	1	no	—
364	#478	—	2	1	no	—
365	#477	—	2	1	no	—
366	#476	—	2	1	no	—
367	#475	—	2	1	no	—
368	#474	—	2	1	no	—
369	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
370	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
371	#428	C₂²×C₂₃	1	1	yes	characteristic
372	#428	C₂²×C₂₃	1	1	yes	characteristic
373	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
374	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
375	#424	C₂²×C₂₃	1	1	yes	characteristic
376	#424	C₂²×C₂₃	1	1	yes	characteristic
377	C₂×#104	C₂²×C₂₃	1	1	yes	characteristic
378	C₂×#104	C₂²×C₂₃	1	1	yes	characteristic
379	#419	—	2	1	no	—
380	#419	—	2	1	no	—
381	#419	C₂²×C₂₃	1	1	yes	characteristic
382	#419	C₂²×C₂₃	1	1	yes	characteristic
383	#417	—	2	1	no	—
384	#417	—	2	1	no	—
385	#417	C₂²×C₂₃	1	1	yes	characteristic
386	#417	C₂²×C₂₃	1	1	yes	characteristic
387	#417	C₂²×C₂₃	1	1	yes	characteristic
388	#417	C₂²×C₂₃	1	1	yes	characteristic
389	#416	—	2	1	no	—
390	#416	—	2	1	no	—
391	#416	C₂²×C₂₃	1	1	yes	characteristic
392	#416	C₂²×C₂₃	1	1	yes	characteristic
393	C₂×#103	—	2	1	no	—
394	C₂×#103	—	2	1	no	—
395	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
396	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
397	C₂×#101	C₂²×C₂₃	1	1	yes	characteristic
398	#391	C₂²×C₂₃	1	1	yes	characteristic
399	#382	C₂²×C₂₃	1	1	yes	characteristic
400	#343	C₂²×C₂₃	1	1	yes	characteristic
401	#343	C₂²×C₂₃	1	1	yes	characteristic
402	#343	C₂²×C₂₃	1	1	yes	characteristic
403	#339	C₂²×C₂₃	1	1	yes	characteristic
404	#339	C₂²×C₂₃	1	1	yes	characteristic
405	#336	C₂²×C₂₃	1	1	yes	characteristic
406	C₂×C₂₃	—	2	1	no	—
407	C₂×C₂₃	—	2	1	no	—
408	C₂×C₂₃	—	4	2	no	—
409	C₂×C₂₃	—	4	2	no	—
410	C₂×C₂₃	#2437	1	1	yes	characteristic
411	C₂×C₂₃	#3289	1	1	yes	characteristic
412	C₂×C₂₃	#3290	1	1	yes	characteristic
413	2₋¹⁺⁴	—	2	2	no	—
414	2₋¹⁺⁴	—	2	2	no	—
415	2₋¹⁺⁴	—	4	1	no	—
416	2₋¹⁺⁴	Q₈×C₂₃	1	4	yes	complemented
417	C₂×#41	—	2	1	no	—
418	C₂×#41	—	2	1	no	—
419	C₂×#41	—	4	1	no	—
420	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
421	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
422	C₂²×Q₈	—	2	1	no	—
423	C₂²×Q₈	—	2	1	no	—
424	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
425	#131	—	2	2	no	—
426	#131	—	2	2	no	—
427	#131	—	4	1	no	—
428	#131	—	4	1	no	—
429	C₂×M₁₆	—	2	1	no	—
430	C₂×M₁₆	—	2	1	no	—
431	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
432	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
433	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
434	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
435	C₂²×C₈	—	2	1	no	—
436	C₂²×C₈	—	2	1	no	—
437	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
438	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
439	C₄⋊Q₈	—	2	1	no	—
440	C₄⋊Q₈	—	2	1	no	—
441	#125	—	2	1	no	—
442	#125	—	2	1	no	—
443	C₂²⋊Q₈	—	4	1	no	—
444	C₄×Q₈	—	2	1	no	—
445	C₄×Q₈	—	2	1	no	—
446	C₄×D₄	—	2	1	no	—
447	C₄×D₄	—	2	1	no	—
448	C₄×D₄	—	4	1	no	—
449	C₄×D₄	—	4	1	no	—
450	#117	—	2	1	no	—
451	#117	—	2	1	no	—
452	#117	C₂³×C₂₃	1	1	yes	characteristic
453	#117	C₂³×C₂₃	1	1	yes	characteristic
454	C₂×C₄⋊C₄	—	2	1	no	—
455	C₂×C₄⋊C₄	—	2	1	no	—
456	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
457	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
458	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
459	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
460	#108	—	4	1	no	—
461	#108	—	4	1	no	—
462	#107	—	2	1	no	—
463	#107	—	2	1	no	—
464	#107	—	2	1	no	—
465	#107	—	2	1	no	—
466	#106	—	2	1	no	—
467	#106	—	2	1	no	—
468	#106	—	2	1	no	—
469	#106	—	2	1	no	—
470	C₄⋊C₈	—	2	1	no	—
471	C₄⋊C₈	—	2	1	no	—
472	C₄⋊C₈	—	2	1	no	—
473	C₄⋊C₈	—	2	1	no	—
474	#104	—	2	2	no	—
475	#104	—	2	2	no	—
476	#103	—	2	1	no	—
477	#103	—	2	1	no	—
478	#103	—	2	1	no	—
479	#103	—	2	1	no	—
480	#103	—	2	1	no	—
481	#103	—	2	1	no	—
482	#103	—	2	1	no	—
483	#103	—	2	1	no	—
484	#103	—	2	1	no	—
485	#103	—	2	1	no	—
486	#103	—	2	1	no	—
487	#103	—	2	1	no	—
488	#103	D₄×C₂₃	1	1	yes	characteristic
489	#103	D₄×C₂₃	1	1	yes	characteristic
490	#103	D₄×C₂₃	1	1	yes	characteristic
491	#103	D₄×C₂₃	1	1	yes	characteristic
492	#102	—	2	1	no	—
493	#102	—	2	1	no	—
494	#102	—	2	1	no	—
495	#102	—	2	1	no	—
496	#102	D₄×C₂₃	1	1	yes	characteristic
497	#102	D₄×C₂₃	1	1	yes	characteristic
498	#102	D₄×C₂₃	1	1	yes	characteristic
499	#102	D₄×C₂₃	1	1	yes	characteristic
500	#101	—	4	1	no	—
501	#95	—	4	1	no	—
502	#95	—	4	1	no	—
503	C₂₃	#25985	1	1	yes	characteristic, direct product factor, Sylow
504	#41	—	2	1	no	—
505	#41	—	2	1	no	—
506	#41	—	2	1	no	—
507	#41	—	2	1	no	—
508	#41	—	2	4	no	—
509	#41	—	2	4	no	—
510	#41	—	4	4	no	—
511	#41	—	4	2	no	—
512	#41	—	4	2	no	—
513	#41	—	4	4	no	—
514	#41	C₂×Q₈×C₂₃	1	2	yes	—
515	#41	C₂×Q₈×C₂₃	1	2	yes	—
516	#41	#41×C₂₃	1	2	yes	—
517	#41	#41×C₂₃	1	2	yes	—
518	C₂×Q₈	—	2	1	no	—
519	C₂×Q₈	—	2	4	no	—
520	C₂×Q₈	—	2	4	no	—
521	C₂×Q₈	—	2	1	no	—
522	C₂×Q₈	—	2	1	no	—
523	C₂×Q₈	—	2	1	no	—
524	C₂×Q₈	—	2	1	no	—
525	C₂×Q₈	—	4	1	no	—
526	C₂×Q₈	—	4	1	no	—
527	C₂×Q₈	—	4	2	no	—
528	C₂×Q₈	—	4	2	no	—
529	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
530	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
531	C₂×Q₈	C₂×Q₈×C₂₃	1	2	yes	—
532	C₂×D₄	—	2	1	no	—
533	C₂×D₄	—	2	1	no	—
534	C₂×D₄	—	4	1	no	—
535	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
536	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
537	C₂²×C₄	—	2	1	no	—
538	C₂²×C₄	—	2	1	no	—
539	C₂²×C₄	—	2	1	no	—
540	C₂²×C₄	—	2	1	no	—
541	C₂²×C₄	—	2	1	no	—
542	C₂²×C₄	—	2	1	no	—
543	C₂²×C₄	—	2	1	no	—
544	C₂²×C₄	—	4	1	no	—
545	C₂²×C₄	—	4	1	no	—
546	C₂²×C₄	—	4	1	no	—
547	C₂²×C₄	—	4	1	no	—
548	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
549	M₁₆	—	2	1	no	—
550	M₁₆	—	2	1	no	—
551	M₁₆	—	2	1	no	—
552	M₁₆	—	2	1	no	—
553	M₁₆	—	2	1	no	—
554	M₁₆	—	2	1	no	—
555	M₁₆	—	2	1	no	—
556	M₁₆	—	2	1	no	—
557	M₁₆	—	4	2	no	—
558	M₁₆	—	4	2	no	—
559	C₂×C₈	—	2	1	no	—
560	C₂×C₈	—	2	1	no	—
561	C₂×C₈	—	2	1	no	—
562	C₂×C₈	—	2	1	no	—
563	C₂×C₈	—	2	1	no	—
564	C₂×C₈	—	2	1	no	—
565	C₂×C₈	—	2	1	no	—
566	C₂×C₈	—	2	1	no	—
567	C₂×C₈	—	4	2	no	—
568	C₂×C₈	—	4	2	no	—
569	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
570	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
571	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
572	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
573	C₄⋊C₄	—	2	1	no	—
574	C₄⋊C₄	—	2	1	no	—
575	C₄⋊C₄	—	2	1	no	—
576	C₄⋊C₄	—	2	1	no	—
577	C₄⋊C₄	—	2	1	no	—
578	C₄⋊C₄	—	2	1	no	—
579	C₄⋊C₄	—	4	1	no	—
580	C₄⋊C₄	—	4	1	no	—
581	C₄⋊C₄	—	4	1	no	—
582	C₄⋊C₄	—	4	1	no	—
583	C₄⋊C₄	—	4	1	no	—
584	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
585	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
586	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
587	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
588	C₂²⋊C₄	—	2	1	no	—
589	C₂²⋊C₄	—	2	1	no	—
590	C₂²⋊C₄	—	4	1	no	—
591	C₂²⋊C₄	—	4	1	no	—
592	C₄²	—	2	1	no	—
593	C₄²	—	2	1	no	—
594	C₄²	—	2	1	no	—
595	C₄²	—	2	1	no	—
596	C₄²	—	4	1	no	—
597	C₄²	—	4	1	no	—
598	C₂³	—	2	1	no	—
599	C₂³	—	2	1	no	—
600	C₂³	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
601	Q₈	—	2	1	no	—
602	Q₈	—	2	1	no	—
603	Q₈	—	2	2	no	—
604	Q₈	—	2	2	no	—
605	Q₈	—	4	4	no	—
606	Q₈	—	4	1	no	—
607	Q₈	—	4	1	no	—
608	Q₈	—	16	1	no	—
609	Q₈	C₂₃×C₂²⋊Q₈	1	2	yes	—
610	Q₈	C₂₃×C₂²⋊Q₈	1	2	yes	—
611	D₄	—	2	2	no	—
612	D₄	—	2	2	no	—
613	D₄	—	2	2	no	—
614	D₄	—	2	2	no	—
615	D₄	—	2	1	no	—
616	D₄	—	2	1	no	—
617	D₄	—	4	4	no	—
618	D₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
619	D₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
620	C₂×C₄	—	2	1	no	—
621	C₂×C₄	—	2	1	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	2	2	no	—
624	C₂×C₄	—	2	2	no	—
625	C₂×C₄	—	2	2	no	—
626	C₂×C₄	—	2	2	no	—
627	C₂×C₄	—	2	1	no	—
628	C₂×C₄	—	2	1	no	—
629	C₂×C₄	—	2	1	no	—
630	C₂×C₄	—	2	1	no	—
631	C₂×C₄	—	4	4	no	—
632	C₂×C₄	—	4	4	no	—
633	C₂×C₄	—	4	1	no	—
634	C₂×C₄	—	4	1	no	—
635	C₂×C₄	—	4	1	no	—
636	C₂×C₄	—	4	1	no	—
637	C₂×C₄	—	4	1	no	—
638	C₂×C₄	—	4	1	no	—
639	C₂×C₄	—	4	1	no	—
640	C₂×C₄	—	4	1	no	—
641	C₂×C₄	—	8	1	no	—
642	C₂×C₄	—	8	1	no	—
643	C₂×C₄	—	8	1	no	—
644	C₂×C₄	—	16	1	no	—
645	C₂×C₄	—	16	1	no	—
646	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
647	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
648	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
649	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
650	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
651	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
652	C₈	—	2	1	no	—
653	C₈	—	2	1	no	—
654	C₈	—	2	1	no	—
655	C₈	—	2	1	no	—
656	C₈	—	4	1	no	—
657	C₈	—	4	1	no	—
658	C₈	—	4	1	no	—
659	C₈	—	4	1	no	—
660	C₂²	—	2	1	no	—
661	C₂²	—	2	1	no	—
662	C₂²	—	2	2	no	—
663	C₂²	—	2	2	no	—
664	C₂²	—	4	1	no	—
665	C₂²	—	4	1	no	—
666	C₂²	—	4	1	no	—
667	C₂²	—	4	1	no	—
668	C₂²	C₂₃×#543	1	1	yes	characteristic
669	C₂²	D₄²×C₂₃	1	1	yes	characteristic
670	C₂²	C₂₃×#551	1	1	yes	characteristic
671	C₄	—	2	2	no	—
672	C₄	—	2	2	no	—
673	C₄	—	4	2	no	—
674	C₄	—	4	1	no	—
675	C₄	—	4	1	no	—
676	C₄	—	4	1	no	—
677	C₄	—	4	1	no	—
678	C₄	—	8	1	no	—
679	C₄	—	8	1	no	—
680	C₄	—	8	1	no	—
681	C₄	—	8	1	no	—
682	C₄	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
683	C₄	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
684	C₄	C₂₃×#546	1	1	yes	characteristic
685	C₄	C₂₃×#546	1	1	yes	characteristic
686	C₂	—	2	1	no	—
687	C₂	—	2	1	no	—
688	C₂	—	4	2	no	—
689	C₂	—	4	2	no	—
690	C₂	C₂₃×#2437	1	1	yes	characteristic
691	C₂	C₂₃×#3289	1	1	yes	characteristic
692	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×C₂₃×2₋¹⁺⁴	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×#2097	C₂	semidirect
6. C₂₃×#2064	C₂	semidirect
7. C₂₃×#2064	C₂	semidirect
8. C₂₃×#2028	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
70. C₂₃×2₋¹⁺⁴	Q₈	semidirect
251. #25985	C₂₃	direct (N × H)
314. #3084	C₂×C₂₃	semidirect
315. #3083	C₂×C₂₃	semidirect
318. #2097	C₂×C₂₃	semidirect
319. #2064	C₂×C₂₃	semidirect
320. #2064	C₂×C₂₃	semidirect
321. #2028	C₂×C₂₃	semidirect
322. #1866	C₂×C₂₃	semidirect
323. #1866	C₂×C₂₃	semidirect
324. #1860	C₂×C₂₃	semidirect
325. #1860	C₂×C₂₃	semidirect
326. #1816	C₂×C₂₃	semidirect
327. #1815	C₂×C₂₃	semidirect
416. 2₋¹⁺⁴	Q₈×C₂₃	semidirect
503. C₂₃	#25985	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×#2898	2
C₂₃×#2097	2
C₂₃×#2064	2
C₂₃×#2028	2
C₂₃×#1866	2
C₂₃×#1860	2
C₂₃×#1816	2
C₂₃×#1815	2
C₂₃×#1793	2
#25985	23

Subgroups — C23×#25985 #19173611

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#25985 #19173611