Subgroups — C₂₃×#26400 #19174026

Order 5888.

740 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3083	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3081	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2946	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2913	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2093	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2089	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2063	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2060	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1952	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1922	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1907	C₂	1	1	yes	maximal, characteristic
12	C₂₃×#1883	C₂	1	1	yes	maximal, characteristic
13	C₂₃×#1866	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1859	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1856	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
18	C₂₃×#544	C₂²	1	1	yes	characteristic
19	C₂₃×#520	C₂²	1	1	yes	characteristic
20	C₂₃×#520	C₂²	1	1	yes	characteristic
21	C₂₃×#518	C₂²	1	1	yes	characteristic
22	C₂₃×#479	—	2	1	no	—
23	C₂₃×#478	—	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×#474	—	2	1	no	—
29	C₂₃×#474	—	2	1	no	—
30	C₂₃×#442	—	2	1	no	—
31	C₂₃×#440	—	2	1	no	—
32	C₄×SD₁₆×C₂₃	—	2	1	no	—
33	C₄×D₈×C₂₃	—	2	1	no	—
34	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
35	C₂₃×#428	—	2	1	no	—
36	C₂₃×#428	—	2	1	no	—
37	C₂₃×#428	C₂²	1	1	yes	characteristic
38	C₂₃×#428	C₂²	1	1	yes	characteristic
39	C₂₃×#428	C₂²	1	1	yes	characteristic
40	C₂₃×#428	C₂²	1	1	yes	characteristic
41	C₂₃×#428	C₂²	1	1	yes	characteristic
42	C₂₃×#427	—	2	1	no	—
43	C₂₃×#427	C₂²	1	1	yes	characteristic
44	C₂×C₂₃×#107	—	2	1	no	—
45	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
46	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
47	C₂₃×#424	C₂²	1	1	yes	characteristic
48	C₂₃×#424	C₂²	1	1	yes	characteristic
49	C₂×C₂₃×#104	C₂²	1	1	yes	characteristic
50	C₂₃×#418	C₂²	1	1	yes	characteristic
51	C₂₃×#417	C₂²	1	1	yes	characteristic
52	C₂₃×#416	C₂²	1	1	yes	characteristic
53	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
54	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
55	C₂×C₂₃×#100	C₂²	1	1	yes	characteristic
56	C₂₃×#405	C₂²	1	1	yes	characteristic
57	C₂₃×#390	C₂²	1	1	yes	characteristic
58	C₂₃×#384	C₂²	1	1	yes	characteristic
59	C₂₃×#343	C₂²	1	1	yes	characteristic
60	C₂₃×#343	C₂²	1	1	yes	characteristic
61	C₂₃×#343	C₂²	1	1	yes	characteristic
62	C₂₃×#341	C₂²	1	1	yes	characteristic
63	C₂₃×#339	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	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
68	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₂₃×AGL(1,ℤ/8)	—	2	2	no	—
70	C₂₃×AGL(1,ℤ/8)	Q₈	1	4	yes	complemented
71	C₂×SD₁₆×C₂₃	—	2	1	no	—
72	C₂×D₈×C₂₃	—	2	1	no	—
73	C₂₃×#131	—	2	2	no	—
74	C₂₃×#131	—	2	2	no	—
75	C₂×M₁₆×C₂₃	—	2	1	no	—
76	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
77	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
78	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
79	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₂₃	—	2	1	no	—
81	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
83	C₂₃×C₄⋊Q₈	—	2	1	no	—
84	C₂₃×C₄⋊Q₈	—	2	1	no	—
85	C₂₃×#125	—	2	1	no	—
86	C₂₃×#125	—	2	1	no	—
87	C₂₃×C₂²⋊Q₈	—	4	1	no	—
88	C₄×Q₈×C₂₃	—	2	1	no	—
89	C₄×Q₈×C₂₃	—	2	1	no	—
90	C₄×D₄×C₂₃	—	2	1	no	—
91	C₄×D₄×C₂₃	—	2	1	no	—
92	C₄×D₄×C₂₃	—	2	1	no	—
93	C₄×D₄×C₂₃	—	2	1	no	—
94	C₄×D₄×C₂₃	—	2	1	no	—
95	C₄×D₄×C₂₃	—	2	1	no	—
96	C₄×D₄×C₂₃	—	4	1	no	—
97	C₄×D₄×C₂₃	—	4	1	no	—
98	C₂₃×#117	—	2	1	no	—
99	C₂₃×#117	—	2	1	no	—
100	C₂₃×#117	C₂³	1	1	yes	characteristic
101	C₂₃×#117	C₂³	1	1	yes	characteristic
102	C₂₃×#117	C₂³	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
104	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
105	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
106	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
109	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
110	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
111	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
112	C₂₃×#108	—	2	2	no	—
113	C₂₃×#107	—	2	1	no	—
114	C₂₃×#107	—	2	1	no	—
115	C₂₃×#107	—	2	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×#107	—	2	1	no	—
121	C₂₃×#107	—	2	1	no	—
122	C₂₃×#107	D₄	1	1	yes	characteristic
123	C₂₃×#107	D₄	1	1	yes	characteristic
124	C₂₃×#106	—	2	1	no	—
125	C₂₃×#106	—	2	1	no	—
126	C₂₃×#106	—	2	1	no	—
127	C₂₃×#106	—	2	1	no	—
128	C₂₃×#106	—	2	1	no	—
129	C₂₃×#106	D₄	1	1	yes	characteristic
130	C₂₃×#106	D₄	1	1	yes	characteristic
131	C₂₃×C₄⋊C₈	—	2	1	no	—
132	C₂₃×C₄⋊C₈	—	2	1	no	—
133	C₂₃×C₄⋊C₈	—	2	1	no	—
134	C₂₃×C₄⋊C₈	—	2	1	no	—
135	C₂₃×#104	—	2	2	no	—
136	C₂₃×#103	—	2	1	no	—
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	D₄	1	1	yes	characteristic
147	C₂₃×#102	D₄	1	1	yes	characteristic
148	C₂₃×#100	—	2	2	no	—
149	C₂₃×#97	—	2	1	no	—
150	C₄×C₈×C₂₃	—	2	1	no	—
151	C₂⁴×C₂₃	—	2	1	no	—
152	#41×C₂₃	—	2	1	no	—
153	#41×C₂₃	—	2	1	no	—
154	#41×C₂₃	C₂×Q₈	1	2	yes	—
155	#41×C₂₃	#41	1	2	yes	—
156	C₂×Q₈×C₂₃	—	4	1	no	—
157	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
158	C₂×D₄×C₂₃	—	2	1	no	—
159	C₂×D₄×C₂₃	—	2	1	no	—
160	C₂×D₄×C₂₃	—	2	1	no	—
161	C₂×D₄×C₂₃	—	2	1	no	—
162	C₂×D₄×C₂₃	—	2	1	no	—
163	C₂×D₄×C₂₃	—	2	1	no	—
164	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
165	C₂×D₄×C₂₃	C₂×Q₈	1	2	yes	—
166	C₂²×C₄×C₂₃	—	2	1	no	—
167	C₂²×C₄×C₂₃	—	2	1	no	—
168	C₂²×C₄×C₂₃	—	2	1	no	—
169	C₂²×C₄×C₂₃	—	2	1	no	—
170	C₂²×C₄×C₂₃	—	2	1	no	—
171	C₂²×C₄×C₂₃	—	4	1	no	—
172	C₂²×C₄×C₂₃	—	4	1	no	—
173	C₂²×C₄×C₂₃	—	4	1	no	—
174	C₂²×C₄×C₂₃	—	4	1	no	—
175	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
176	SD₁₆×C₂₃	—	2	4	no	—
177	D₈×C₂₃	—	2	4	no	—
178	M₁₆×C₂₃	—	2	4	no	—
179	M₁₆×C₂₃	—	2	1	no	—
180	M₁₆×C₂₃	—	2	1	no	—
181	M₁₆×C₂₃	—	2	1	no	—
182	M₁₆×C₂₃	—	2	1	no	—
183	M₁₆×C₂₃	—	2	1	no	—
184	M₁₆×C₂₃	—	2	1	no	—
185	M₁₆×C₂₃	C₂×Q₈	1	2	yes	—
186	M₁₆×C₂₃	#41	1	2	yes	—
187	C₂×C₈×C₂₃	—	2	4	no	—
188	C₂×C₈×C₂₃	—	2	1	no	—
189	C₂×C₈×C₂₃	—	2	1	no	—
190	C₂×C₈×C₂₃	—	2	1	no	—
191	C₂×C₈×C₂₃	—	2	1	no	—
192	C₂×C₈×C₂₃	—	2	1	no	—
193	C₂×C₈×C₂₃	—	2	1	no	—
194	C₂×C₈×C₂₃	—	2	1	no	—
195	C₂×C₈×C₂₃	—	2	1	no	—
196	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
197	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
198	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
199	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
200	C₄⋊C₄×C₂₃	—	2	1	no	—
201	C₄⋊C₄×C₂₃	—	2	1	no	—
202	C₄⋊C₄×C₂₃	—	2	1	no	—
203	C₄⋊C₄×C₂₃	—	2	1	no	—
204	C₄⋊C₄×C₂₃	—	2	1	no	—
205	C₄⋊C₄×C₂₃	—	2	1	no	—
206	C₄⋊C₄×C₂₃	—	4	1	no	—
207	C₄⋊C₄×C₂₃	—	4	1	no	—
208	C₄⋊C₄×C₂₃	—	4	1	no	—
209	C₄⋊C₄×C₂₃	—	4	1	no	—
210	C₄⋊C₄×C₂₃	—	4	1	no	—
211	C₄⋊C₄×C₂₃	—	8	2	no	—
212	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
213	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
214	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
215	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
216	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
217	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
218	C₂²⋊C₄×C₂₃	—	2	1	no	—
219	C₂²⋊C₄×C₂₃	—	2	1	no	—
220	C₂²⋊C₄×C₂₃	—	2	1	no	—
221	C₂²⋊C₄×C₂₃	—	4	1	no	—
222	C₂²⋊C₄×C₂₃	—	4	1	no	—
223	C₂²⋊C₄×C₂₃	—	4	1	no	—
224	C₂²⋊C₄×C₂₃	—	4	1	no	—
225	C₂²⋊C₄×C₂₃	—	8	2	no	—
226	C₄²×C₂₃	—	2	1	no	—
227	C₄²×C₂₃	—	2	1	no	—
228	C₄²×C₂₃	—	2	1	no	—
229	C₄²×C₂₃	—	2	1	no	—
230	C₄²×C₂₃	—	2	1	no	—
231	C₄²×C₂₃	—	4	1	no	—
232	C₄²×C₂₃	—	4	1	no	—
233	#26400	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
234	C₂³×C₂₃	—	2	1	no	—
235	C₂³×C₂₃	—	2	2	no	—
236	C₂³×C₂₃	—	4	1	no	—
237	C₂³×C₂₃	—	4	1	no	—
238	C₂³×C₂₃	—	8	1	no	—
239	C₂³×C₂₃	—	8	1	no	—
240	C₂³×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
241	Q₈×C₂₃	—	2	1	no	—
242	Q₈×C₂₃	—	16	1	no	—
243	Q₈×C₂₃	C₂²⋊Q₈	1	2	yes	—
244	D₄×C₂₃	—	2	2	no	—
245	D₄×C₂₃	—	2	2	no	—
246	D₄×C₂₃	—	2	2	no	—
247	D₄×C₂₃	—	2	1	no	—
248	D₄×C₂₃	—	2	2	no	—
249	D₄×C₂₃	—	4	1	no	—
250	D₄×C₂₃	—	4	1	no	—
251	D₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
252	C₂×C₄×C₂₃	—	2	1	no	—
253	C₂×C₄×C₂₃	—	2	2	no	—
254	C₂×C₄×C₂₃	—	2	2	no	—
255	C₂×C₄×C₂₃	—	2	2	no	—
256	C₂×C₄×C₂₃	—	2	1	no	—
257	C₂×C₄×C₂₃	—	2	1	no	—
258	C₂×C₄×C₂₃	—	2	1	no	—
259	C₂×C₄×C₂₃	—	2	1	no	—
260	C₂×C₄×C₂₃	—	2	1	no	—
261	C₂×C₄×C₂₃	—	2	1	no	—
262	C₂×C₄×C₂₃	—	4	1	no	—
263	C₂×C₄×C₂₃	—	4	1	no	—
264	C₂×C₄×C₂₃	—	4	1	no	—
265	C₂×C₄×C₂₃	—	4	1	no	—
266	C₂×C₄×C₂₃	—	4	1	no	—
267	C₂×C₄×C₂₃	—	8	1	no	—
268	C₂×C₄×C₂₃	—	8	1	no	—
269	C₂×C₄×C₂₃	—	8	1	no	—
270	C₂×C₄×C₂₃	—	16	1	no	—
271	C₂×C₄×C₂₃	—	16	1	no	—
272	C₂×C₄×C₂₃	—	16	1	no	—
273	C₂×C₄×C₂₃	—	16	1	no	—
274	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
275	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
276	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
277	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
278	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
279	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
280	C₈×C₂₃	—	2	2	no	—
281	C₈×C₂₃	—	2	1	no	—
282	C₈×C₂₃	—	2	2	no	—
283	C₈×C₂₃	—	2	1	no	—
284	C₈×C₂₃	—	2	1	no	—
285	C₈×C₂₃	—	2	1	no	—
286	C₈×C₂₃	—	4	1	no	—
287	C₈×C₂₃	—	4	1	no	—
288	#3083	C₂×C₂₃	1	1	yes	characteristic, complemented
289	#3081	C₂×C₂₃	1	1	yes	characteristic, complemented
290	#2946	C₂×C₂₃	1	1	yes	characteristic
291	#2913	C₂×C₂₃	1	1	yes	characteristic, complemented
292	#2093	C₂×C₂₃	1	1	yes	characteristic, complemented
293	#2089	C₂×C₂₃	1	1	yes	characteristic, complemented
294	#2063	C₂×C₂₃	1	1	yes	characteristic, complemented
295	#2060	C₂×C₂₃	1	1	yes	characteristic, complemented
296	#1952	C₂×C₂₃	1	1	yes	characteristic, complemented
297	#1922	C₂×C₂₃	1	1	yes	characteristic, complemented
298	#1907	C₂×C₂₃	1	1	yes	characteristic
299	#1883	C₂×C₂₃	1	1	yes	characteristic
300	#1866	C₂×C₂₃	1	1	yes	characteristic, complemented
301	#1859	C₂×C₂₃	1	1	yes	characteristic, complemented
302	#1856	C₂×C₂₃	1	1	yes	characteristic, complemented
303	C₂²×C₂₃	—	2	1	no	—
304	C₂²×C₂₃	—	2	1	no	—
305	C₂²×C₂₃	—	2	2	no	—
306	C₂²×C₂₃	—	4	1	no	—
307	C₂²×C₂₃	—	4	1	no	—
308	C₂²×C₂₃	—	4	2	no	—
309	C₂²×C₂₃	—	8	2	no	—
310	C₂²×C₂₃	—	8	1	no	—
311	C₂²×C₂₃	—	8	1	no	—
312	C₂²×C₂₃	—	16	1	no	—
313	C₂²×C₂₃	#539	1	1	yes	characteristic
314	C₂²×C₂₃	D₄²	1	1	yes	characteristic
315	C₂²×C₂₃	#554	1	1	yes	characteristic
316	C₄×C₂₃	—	2	2	no	—
317	C₄×C₂₃	—	4	2	no	—
318	C₄×C₂₃	—	4	1	no	—
319	C₄×C₂₃	—	4	1	no	—
320	C₄×C₂₃	—	4	1	no	—
321	C₄×C₂₃	—	4	1	no	—
322	C₄×C₂₃	—	4	1	no	—
323	C₄×C₂₃	—	4	1	no	—
324	C₄×C₂₃	—	8	1	no	—
325	C₄×C₂₃	—	8	1	no	—
326	C₄×C₂₃	—	8	1	no	—
327	C₄×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
328	C₄×C₂₃	#525	1	1	yes	characteristic
329	C₄×C₂₃	#547	1	1	yes	characteristic
330	C₄×C₂₃	D₄×Q₈	1	1	yes	characteristic
331	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	1	yes	characteristic
332	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
333	#544	C₂²×C₂₃	1	1	yes	characteristic
334	#520	C₂²×C₂₃	1	1	yes	characteristic
335	#520	C₂²×C₂₃	1	1	yes	characteristic
336	#518	C₂²×C₂₃	1	1	yes	characteristic
337	#479	—	2	1	no	—
338	#478	—	2	1	no	—
339	#478	—	2	1	no	—
340	#478	—	2	1	no	—
341	#477	—	2	1	no	—
342	#476	—	2	1	no	—
343	#474	—	2	1	no	—
344	#474	—	2	1	no	—
345	#442	—	2	1	no	—
346	#440	—	2	1	no	—
347	C₄×SD₁₆	—	2	1	no	—
348	C₄×D₈	—	2	1	no	—
349	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
350	#428	—	2	1	no	—
351	#428	—	2	1	no	—
352	#428	C₂²×C₂₃	1	1	yes	characteristic
353	#428	C₂²×C₂₃	1	1	yes	characteristic
354	#428	C₂²×C₂₃	1	1	yes	characteristic
355	#428	C₂²×C₂₃	1	1	yes	characteristic
356	#428	C₂²×C₂₃	1	1	yes	characteristic
357	#427	—	2	1	no	—
358	#427	C₂²×C₂₃	1	1	yes	characteristic
359	C₂×#107	—	2	1	no	—
360	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
361	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
362	#424	C₂²×C₂₃	1	1	yes	characteristic
363	#424	C₂²×C₂₃	1	1	yes	characteristic
364	C₂×#104	C₂²×C₂₃	1	1	yes	characteristic
365	#418	C₂²×C₂₃	1	1	yes	characteristic
366	#417	C₂²×C₂₃	1	1	yes	characteristic
367	#416	C₂²×C₂₃	1	1	yes	characteristic
368	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
369	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
370	C₂×#100	C₂²×C₂₃	1	1	yes	characteristic
371	#405	C₂²×C₂₃	1	1	yes	characteristic
372	#390	C₂²×C₂₃	1	1	yes	characteristic
373	#384	C₂²×C₂₃	1	1	yes	characteristic
374	#343	C₂²×C₂₃	1	1	yes	characteristic
375	#343	C₂²×C₂₃	1	1	yes	characteristic
376	#343	C₂²×C₂₃	1	1	yes	characteristic
377	#341	C₂²×C₂₃	1	1	yes	characteristic
378	#339	C₂²×C₂₃	1	1	yes	characteristic
379	#339	C₂²×C₂₃	1	1	yes	characteristic
380	#339	C₂²×C₂₃	1	1	yes	characteristic
381	#336	C₂²×C₂₃	1	1	yes	characteristic
382	C₂×C₂₃	—	2	1	no	—
383	C₂×C₂₃	—	2	1	no	—
384	C₂×C₂₃	—	4	2	no	—
385	C₂×C₂₃	—	8	2	no	—
386	C₂×C₂₃	#2449	1	1	yes	characteristic
387	C₂×C₂₃	#3289	1	1	yes	characteristic
388	C₂×C₂₃	#3292	1	1	yes	characteristic
389	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
390	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
391	AGL(1,ℤ/8)	—	2	2	no	—
392	AGL(1,ℤ/8)	Q₈×C₂₃	1	4	yes	complemented
393	C₂×SD₁₆	—	2	1	no	—
394	C₂×D₈	—	2	1	no	—
395	#131	—	2	2	no	—
396	#131	—	2	2	no	—
397	C₂×M₁₆	—	2	1	no	—
398	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
399	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
400	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
401	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
402	C₂²×C₈	—	2	1	no	—
403	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
404	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
405	C₄⋊Q₈	—	2	1	no	—
406	C₄⋊Q₈	—	2	1	no	—
407	#125	—	2	1	no	—
408	#125	—	2	1	no	—
409	C₂²⋊Q₈	—	4	1	no	—
410	C₄×Q₈	—	2	1	no	—
411	C₄×Q₈	—	2	1	no	—
412	C₄×D₄	—	2	1	no	—
413	C₄×D₄	—	2	1	no	—
414	C₄×D₄	—	2	1	no	—
415	C₄×D₄	—	2	1	no	—
416	C₄×D₄	—	2	1	no	—
417	C₄×D₄	—	2	1	no	—
418	C₄×D₄	—	4	1	no	—
419	C₄×D₄	—	4	1	no	—
420	#117	—	2	1	no	—
421	#117	—	2	1	no	—
422	#117	C₂³×C₂₃	1	1	yes	characteristic
423	#117	C₂³×C₂₃	1	1	yes	characteristic
424	#117	C₂³×C₂₃	1	1	yes	characteristic
425	C₂×C₄⋊C₄	—	2	1	no	—
426	C₂×C₄⋊C₄	—	2	1	no	—
427	C₂×C₄⋊C₄	—	4	1	no	—
428	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
429	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
430	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
431	C₂×C₂²⋊C₄	—	4	1	no	—
432	C₂×C₂²⋊C₄	—	4	1	no	—
433	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
434	#108	—	2	2	no	—
435	#107	—	2	1	no	—
436	#107	—	2	1	no	—
437	#107	—	2	1	no	—
438	#107	—	2	1	no	—
439	#107	—	2	1	no	—
440	#107	—	2	1	no	—
441	#107	—	2	1	no	—
442	#107	—	2	1	no	—
443	#107	—	2	1	no	—
444	#107	D₄×C₂₃	1	1	yes	characteristic
445	#107	D₄×C₂₃	1	1	yes	characteristic
446	#106	—	2	1	no	—
447	#106	—	2	1	no	—
448	#106	—	2	1	no	—
449	#106	—	2	1	no	—
450	#106	—	2	1	no	—
451	#106	D₄×C₂₃	1	1	yes	characteristic
452	#106	D₄×C₂₃	1	1	yes	characteristic
453	C₄⋊C₈	—	2	1	no	—
454	C₄⋊C₈	—	2	1	no	—
455	C₄⋊C₈	—	2	1	no	—
456	C₄⋊C₈	—	2	1	no	—
457	#104	—	2	2	no	—
458	#103	—	2	1	no	—
459	#103	D₄×C₂₃	1	1	yes	characteristic
460	#103	D₄×C₂₃	1	1	yes	characteristic
461	#102	—	2	1	no	—
462	#102	—	2	1	no	—
463	#102	—	2	1	no	—
464	#102	—	2	1	no	—
465	#102	—	2	1	no	—
466	#102	—	2	1	no	—
467	#102	—	2	1	no	—
468	#102	D₄×C₂₃	1	1	yes	characteristic
469	#102	D₄×C₂₃	1	1	yes	characteristic
470	#100	—	2	2	no	—
471	#97	—	2	1	no	—
472	C₄×C₈	—	2	1	no	—
473	C₂₃	#26400	1	1	yes	characteristic, direct product factor, Sylow
474	C₂⁴	—	2	1	no	—
475	#41	—	2	1	no	—
476	#41	—	2	1	no	—
477	#41	C₂×Q₈×C₂₃	1	2	yes	—
478	#41	#41×C₂₃	1	2	yes	—
479	C₂×Q₈	—	4	1	no	—
480	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
481	C₂×D₄	—	2	1	no	—
482	C₂×D₄	—	2	1	no	—
483	C₂×D₄	—	2	1	no	—
484	C₂×D₄	—	2	1	no	—
485	C₂×D₄	—	2	1	no	—
486	C₂×D₄	—	2	1	no	—
487	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
488	C₂×D₄	C₂×Q₈×C₂₃	1	2	yes	—
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	2	1	no	—
493	C₂²×C₄	—	2	1	no	—
494	C₂²×C₄	—	4	1	no	—
495	C₂²×C₄	—	4	1	no	—
496	C₂²×C₄	—	4	1	no	—
497	C₂²×C₄	—	4	1	no	—
498	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
499	SD₁₆	—	2	4	no	—
500	D₈	—	2	4	no	—
501	M₁₆	—	2	4	no	—
502	M₁₆	—	2	1	no	—
503	M₁₆	—	2	1	no	—
504	M₁₆	—	2	1	no	—
505	M₁₆	—	2	1	no	—
506	M₁₆	—	2	1	no	—
507	M₁₆	—	2	1	no	—
508	M₁₆	C₂×Q₈×C₂₃	1	2	yes	—
509	M₁₆	#41×C₂₃	1	2	yes	—
510	C₂×C₈	—	2	4	no	—
511	C₂×C₈	—	2	1	no	—
512	C₂×C₈	—	2	1	no	—
513	C₂×C₈	—	2	1	no	—
514	C₂×C₈	—	2	1	no	—
515	C₂×C₈	—	2	1	no	—
516	C₂×C₈	—	2	1	no	—
517	C₂×C₈	—	2	1	no	—
518	C₂×C₈	—	2	1	no	—
519	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
520	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
521	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
522	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
523	C₄⋊C₄	—	2	1	no	—
524	C₄⋊C₄	—	2	1	no	—
525	C₄⋊C₄	—	2	1	no	—
526	C₄⋊C₄	—	2	1	no	—
527	C₄⋊C₄	—	2	1	no	—
528	C₄⋊C₄	—	2	1	no	—
529	C₄⋊C₄	—	4	1	no	—
530	C₄⋊C₄	—	4	1	no	—
531	C₄⋊C₄	—	4	1	no	—
532	C₄⋊C₄	—	4	1	no	—
533	C₄⋊C₄	—	4	1	no	—
534	C₄⋊C₄	—	8	2	no	—
535	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
536	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
537	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
538	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
539	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
540	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
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	—	8	2	no	—
549	C₄²	—	2	1	no	—
550	C₄²	—	2	1	no	—
551	C₄²	—	2	1	no	—
552	C₄²	—	2	1	no	—
553	C₄²	—	2	1	no	—
554	C₄²	—	4	1	no	—
555	C₄²	—	4	1	no	—
556	C₂³	—	2	1	no	—
557	C₂³	—	2	2	no	—
558	C₂³	—	4	1	no	—
559	C₂³	—	4	1	no	—
560	C₂³	—	8	1	no	—
561	C₂³	—	8	1	no	—
562	C₂³	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
563	Q₈	—	2	1	no	—
564	Q₈	—	16	1	no	—
565	Q₈	C₂₃×C₂²⋊Q₈	1	2	yes	—
566	D₄	—	2	2	no	—
567	D₄	—	2	2	no	—
568	D₄	—	2	1	no	—
569	D₄	—	2	2	no	—
570	D₄	—	2	2	no	—
571	D₄	—	4	1	no	—
572	D₄	—	4	1	no	—
573	D₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	1	no	—
586	C₂×C₄	—	4	1	no	—
587	C₂×C₄	—	4	1	no	—
588	C₂×C₄	—	4	1	no	—
589	C₂×C₄	—	8	1	no	—
590	C₂×C₄	—	8	1	no	—
591	C₂×C₄	—	8	1	no	—
592	C₂×C₄	—	16	1	no	—
593	C₂×C₄	—	16	1	no	—
594	C₂×C₄	—	16	1	no	—
595	C₂×C₄	—	16	1	no	—
596	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
597	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
598	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
599	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
600	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
601	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
602	C₈	—	2	1	no	—
603	C₈	—	2	1	no	—
604	C₈	—	2	2	no	—
605	C₈	—	2	1	no	—
606	C₈	—	2	2	no	—
607	C₈	—	2	1	no	—
608	C₈	—	4	1	no	—
609	C₈	—	4	1	no	—
610	C₂²	—	2	1	no	—
611	C₂²	—	2	1	no	—
612	C₂²	—	2	2	no	—
613	C₂²	—	4	1	no	—
614	C₂²	—	4	1	no	—
615	C₂²	—	4	2	no	—
616	C₂²	—	8	2	no	—
617	C₂²	—	8	1	no	—
618	C₂²	—	8	1	no	—
619	C₂²	—	16	1	no	—
620	C₂²	C₂₃×#539	1	1	yes	characteristic
621	C₂²	D₄²×C₂₃	1	1	yes	characteristic
622	C₂²	C₂₃×#554	1	1	yes	characteristic
623	C₄	—	2	2	no	—
624	C₄	—	4	2	no	—
625	C₄	—	4	1	no	—
626	C₄	—	4	1	no	—
627	C₄	—	4	1	no	—
628	C₄	—	4	1	no	—
629	C₄	—	4	1	no	—
630	C₄	—	4	1	no	—
631	C₄	—	8	1	no	—
632	C₄	—	8	1	no	—
633	C₄	—	8	1	no	—
634	C₄	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
635	C₄	C₂₃×#525	1	1	yes	characteristic
636	C₄	C₂₃×#547	1	1	yes	characteristic
637	C₄	D₄×Q₈×C₂₃	1	1	yes	characteristic
638	C₂	—	2	1	no	—
639	C₂	—	2	1	no	—
640	C₂	—	4	2	no	—
641	C₂	—	8	2	no	—
642	C₂	C₂₃×#2449	1	1	yes	characteristic
643	C₂	C₂₃×#3289	1	1	yes	characteristic
644	C₂	C₂₃×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×AGL(1,ℤ/8)	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×#3083	C₂	semidirect
2. C₂₃×#3081	C₂	semidirect
4. C₂₃×#2913	C₂	semidirect
5. C₂₃×#2093	C₂	semidirect
6. C₂₃×#2089	C₂	semidirect
7. C₂₃×#2063	C₂	semidirect
8. C₂₃×#2060	C₂	semidirect
9. C₂₃×#1952	C₂	semidirect
10. C₂₃×#1922	C₂	semidirect
13. C₂₃×#1866	C₂	semidirect
14. C₂₃×#1859	C₂	semidirect
15. C₂₃×#1856	C₂	semidirect
70. C₂₃×AGL(1,ℤ/8)	Q₈	semidirect
233. #26400	C₂₃	direct (N × H)
288. #3083	C₂×C₂₃	semidirect
289. #3081	C₂×C₂₃	semidirect
291. #2913	C₂×C₂₃	semidirect
292. #2093	C₂×C₂₃	semidirect
293. #2089	C₂×C₂₃	semidirect
294. #2063	C₂×C₂₃	semidirect
295. #2060	C₂×C₂₃	semidirect
296. #1952	C₂×C₂₃	semidirect
297. #1922	C₂×C₂₃	semidirect
300. #1866	C₂×C₂₃	semidirect
301. #1859	C₂×C₂₃	semidirect
302. #1856	C₂×C₂₃	semidirect
392. AGL(1,ℤ/8)	Q₈×C₂₃	semidirect
473. C₂₃	#26400	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×#3083	2
C₂₃×#3081	2
C₂₃×#2946	2
C₂₃×#2913	2
C₂₃×#2093	2
C₂₃×#2089	2
C₂₃×#2063	2
C₂₃×#2060	2
C₂₃×#1952	2
C₂₃×#1922	2
C₂₃×#1907	2
C₂₃×#1883	2
C₂₃×#1866	2
C₂₃×#1859	2
C₂₃×#1856	2
#26400	23

Subgroups — C23×#26400 #19174026

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#26400 #19174026