Subgroups — C₂₃×#26008 #19173634

Order 5888.

712 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2439	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2345	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2101	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2087	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2086	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2058	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1988	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1949	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1921	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1920	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1854	C₂	1	1	yes	maximal, characteristic
12	C₂₃×#1851	C₂	1	1	yes	maximal, characteristic
13	C₂₃×#1823	C₂	1	1	yes	maximal, characteristic
14	C₂₃×#1809	C₂	1	1	yes	maximal, characteristic
15	C₂₃×#1736	C₂	1	1	yes	maximal, characteristic
16	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
27	C₂₃×#396	—	2	1	no	—
28	C₂₃×#393	—	2	1	no	—
29	C₂₃×#391	—	2	1	no	—
30	C₂₃×#389	C₂²	1	1	yes	characteristic
31	C₂₃×#389	C₂²	1	1	yes	characteristic
32	C₂₃×#389	C₂²	1	1	yes	characteristic
33	C₂₃×#389	C₂²	1	1	yes	characteristic
34	C₂₃×#387	—	2	1	no	—
35	C₂₃×#387	—	2	1	no	—
36	C₂₃×#387	—	2	1	no	—
37	C₂₃×#382	C₂²	1	1	yes	characteristic
38	C₂₃×#380	—	2	1	no	—
39	C₂₃×#380	C₂²	1	1	yes	characteristic
40	C₂₃×#380	C₂²	1	1	yes	characteristic
41	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
42	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
43	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
44	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
45	C₂₃×#340	C₂²	1	1	yes	characteristic
46	C₂₃×#340	C₂²	1	1	yes	characteristic
47	C₂₃×#340	C₂²	1	1	yes	characteristic
48	C₂₃×#340	C₂²	1	1	yes	characteristic
49	C₂₃×#340	C₂²	1	1	yes	characteristic
50	C₂₃×#340	C₂²	1	1	yes	characteristic
51	C₂₃×#340	C₂²	1	1	yes	characteristic
52	C₂₃×#340	C₂²	1	1	yes	characteristic
53	C₂₃×#340	C₂²	1	1	yes	characteristic
54	C₂₃×#340	C₂²	1	1	yes	characteristic
55	C₂₃×#340	C₂²	1	1	yes	characteristic
56	C₂₃×#340	C₂²	1	1	yes	characteristic
57	C₂₃×#336	C₂²	1	1	yes	characteristic
58	C₂₃×#336	C₂²	1	1	yes	characteristic
59	C₂²×Q₈×C₂₃	—	2	1	no	—
60	C₂³×C₄×C₂₃	—	2	1	no	—
61	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
62	C₂×M₁₆×C₂₃	—	2	2	no	—
63	C₂×M₁₆×C₂₃	—	2	2	no	—
64	C₂×M₁₆×C₂₃	—	4	1	no	—
65	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
69	C₂₃×C₄⋊Q₈	—	2	2	no	—
70	C₂₃×C₄⋊Q₈	—	4	1	no	—
71	C₂₃×#117	—	2	2	no	—
72	C₂₃×#117	—	4	1	no	—
73	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
74	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
75	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
76	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
77	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
78	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
79	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
80	C₂×C₄⋊C₄×C₂₃	Q₈	1	4	yes	complemented
81	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
89	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
90	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
91	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
92	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
93	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
94	C₂×C₄²×C₂₃	—	2	1	no	—
95	C₂×C₄²×C₂₃	—	2	1	no	—
96	C₂×C₄²×C₂₃	—	2	1	no	—
97	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
99	C₂₃×#107	—	2	2	no	—
100	C₂₃×#107	—	4	1	no	—
101	C₂₃×#106	—	2	2	no	—
102	C₂₃×#106	—	4	1	no	—
103	C₂₃×C₄⋊C₈	—	2	2	no	—
104	C₂₃×C₄⋊C₈	—	4	1	no	—
105	C₂₃×C₂²⋊C₈	—	2	2	no	—
106	C₂₃×#95	—	2	1	no	—
107	C₂₃×#95	—	2	1	no	—
108	C₂₃×#95	—	2	1	no	—
109	C₂₃×#95	—	2	1	no	—
110	C₂₃×#95	—	2	1	no	—
111	C₂₃×#95	—	2	1	no	—
112	C₂₃×#95	—	2	1	no	—
113	C₂₃×#95	—	2	1	no	—
114	C₂₃×#95	—	2	1	no	—
115	C₂₃×#95	—	2	1	no	—
116	C₂₃×#95	D₄	1	1	yes	characteristic
117	C₂₃×#95	D₄	1	1	yes	characteristic
118	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
119	C₂×Q₈×C₂₃	—	4	2	no	—
120	C₂×Q₈×C₂₃	—	4	2	no	—
121	C₂×Q₈×C₂₃	—	8	1	no	—
122	C₂²×C₄×C₂₃	—	2	1	no	—
123	C₂²×C₄×C₂₃	—	2	1	no	—
124	C₂²×C₄×C₂₃	—	2	1	no	—
125	C₂²×C₄×C₂₃	—	2	1	no	—
126	C₂²×C₄×C₂₃	—	2	1	no	—
127	C₂²×C₄×C₂₃	—	2	1	no	—
128	C₂²×C₄×C₂₃	—	2	1	no	—
129	C₂²×C₄×C₂₃	—	2	4	no	—
130	C₂²×C₄×C₂₃	—	2	1	no	—
131	C₂²×C₄×C₂₃	—	2	1	no	—
132	C₂²×C₄×C₂₃	—	2	1	no	—
133	C₂²×C₄×C₂₃	—	2	1	no	—
134	C₂²×C₄×C₂₃	—	2	1	no	—
135	C₂²×C₄×C₂₃	—	8	1	no	—
136	C₂²×C₄×C₂₃	—	8	1	no	—
137	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
143	C₂²×C₄×C₂₃	#41	1	2	yes	—
144	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
145	M₁₆×C₂₃	—	4	4	no	—
146	C₂×C₈×C₂₃	—	2	1	no	—
147	C₂×C₈×C₂₃	—	2	1	no	—
148	C₂×C₈×C₂₃	—	2	1	no	—
149	C₂×C₈×C₂₃	—	2	1	no	—
150	C₂×C₈×C₂₃	—	2	1	no	—
151	C₂×C₈×C₂₃	—	2	1	no	—
152	C₂×C₈×C₂₃	—	2	1	no	—
153	C₂×C₈×C₂₃	—	2	1	no	—
154	C₂×C₈×C₂₃	—	2	1	no	—
155	C₂×C₈×C₂₃	—	2	1	no	—
156	C₂×C₈×C₂₃	—	2	1	no	—
157	C₂×C₈×C₂₃	#41	1	2	yes	—
158	C₄⋊C₄×C₂₃	—	2	4	no	—
159	C₄⋊C₄×C₂₃	—	2	1	no	—
160	C₄⋊C₄×C₂₃	—	2	1	no	—
161	C₄⋊C₄×C₂₃	—	2	1	no	—
162	C₄⋊C₄×C₂₃	—	2	1	no	—
163	C₄⋊C₄×C₂₃	—	2	1	no	—
164	C₄⋊C₄×C₂₃	—	2	1	no	—
165	C₄⋊C₄×C₂₃	—	2	1	no	—
166	C₄⋊C₄×C₂₃	—	2	1	no	—
167	C₄⋊C₄×C₂₃	—	2	1	no	—
168	C₄⋊C₄×C₂₃	—	2	1	no	—
169	C₄⋊C₄×C₂₃	—	2	1	no	—
170	C₄⋊C₄×C₂₃	—	2	1	no	—
171	C₄⋊C₄×C₂₃	—	2	1	no	—
172	C₄⋊C₄×C₂₃	—	4	2	no	—
173	C₄⋊C₄×C₂₃	—	4	2	no	—
174	C₄⋊C₄×C₂₃	—	8	1	no	—
175	C₄⋊C₄×C₂₃	—	8	1	no	—
176	C₄⋊C₄×C₂₃	—	8	1	no	—
177	C₄⋊C₄×C₂₃	C₂×Q₈	1	2	yes	—
178	C₄⋊C₄×C₂₃	C₂×Q₈	1	2	yes	—
179	C₄⋊C₄×C₂₃	#41	1	2	yes	—
180	C₂²⋊C₄×C₂₃	—	2	2	no	—
181	C₂²⋊C₄×C₂₃	—	4	1	no	—
182	C₂²⋊C₄×C₂₃	—	4	2	no	—
183	C₂²⋊C₄×C₂₃	—	4	2	no	—
184	C₂²⋊C₄×C₂₃	—	8	1	no	—
185	C₄²×C₂₃	—	2	1	no	—
186	C₄²×C₂₃	—	2	1	no	—
187	C₄²×C₂₃	—	2	1	no	—
188	C₄²×C₂₃	—	2	1	no	—
189	C₄²×C₂₃	—	4	2	no	—
190	C₄²×C₂₃	—	8	1	no	—
191	C₄²×C₂₃	—	8	1	no	—
192	#26008	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
193	C₂³×C₂₃	—	2	1	no	—
194	C₂³×C₂₃	—	2	1	no	—
195	C₂³×C₂₃	—	2	1	no	—
196	C₂³×C₂₃	—	2	1	no	—
197	C₂³×C₂₃	—	2	1	no	—
198	C₂³×C₂₃	—	2	1	no	—
199	C₂³×C₂₃	C₂²⋊Q₈	1	2	yes	—
200	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
201	Q₈×C₂₃	—	8	4	no	—
202	C₂×C₄×C₂₃	—	2	1	no	—
203	C₂×C₄×C₂₃	—	2	1	no	—
204	C₂×C₄×C₂₃	—	2	1	no	—
205	C₂×C₄×C₂₃	—	2	1	no	—
206	C₂×C₄×C₂₃	—	2	2	no	—
207	C₂×C₄×C₂₃	—	2	2	no	—
208	C₂×C₄×C₂₃	—	2	2	no	—
209	C₂×C₄×C₂₃	—	2	2	no	—
210	C₂×C₄×C₂₃	—	2	2	no	—
211	C₂×C₄×C₂₃	—	2	2	no	—
212	C₂×C₄×C₂₃	—	2	1	no	—
213	C₂×C₄×C₂₃	—	2	2	no	—
214	C₂×C₄×C₂₃	—	2	1	no	—
215	C₂×C₄×C₂₃	—	2	1	no	—
216	C₂×C₄×C₂₃	—	2	1	no	—
217	C₂×C₄×C₂₃	—	2	1	no	—
218	C₂×C₄×C₂₃	—	2	1	no	—
219	C₂×C₄×C₂₃	—	2	1	no	—
220	C₂×C₄×C₂₃	—	2	1	no	—
221	C₂×C₄×C₂₃	—	2	1	no	—
222	C₂×C₄×C₂₃	—	4	1	no	—
223	C₂×C₄×C₂₃	—	4	1	no	—
224	C₂×C₄×C₂₃	—	4	1	no	—
225	C₂×C₄×C₂₃	—	4	1	no	—
226	C₂×C₄×C₂₃	—	4	1	no	—
227	C₂×C₄×C₂₃	—	4	1	no	—
228	C₂×C₄×C₂₃	—	4	1	no	—
229	C₂×C₄×C₂₃	—	4	1	no	—
230	C₂×C₄×C₂₃	—	4	1	no	—
231	C₂×C₄×C₂₃	—	4	1	no	—
232	C₂×C₄×C₂₃	—	4	1	no	—
233	C₂×C₄×C₂₃	—	4	1	no	—
234	C₂×C₄×C₂₃	—	4	1	no	—
235	C₂×C₄×C₂₃	—	4	1	no	—
236	C₂×C₄×C₂₃	—	4	1	no	—
237	C₂×C₄×C₂₃	—	4	1	no	—
238	C₂×C₄×C₂₃	—	4	1	no	—
239	C₂×C₄×C₂₃	—	4	1	no	—
240	C₂×C₄×C₂₃	—	4	1	no	—
241	C₂×C₄×C₂₃	—	4	1	no	—
242	C₂×C₄×C₂₃	—	4	1	no	—
243	C₂×C₄×C₂₃	—	8	4	no	—
244	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
245	C₂×C₄×C₂₃	#137	1	2	yes	—
246	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
247	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
248	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
249	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
250	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
251	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
252	C₈×C₂₃	—	2	2	no	—
253	C₈×C₂₃	—	4	1	no	—
254	C₈×C₂₃	—	4	1	no	—
255	C₈×C₂₃	—	4	1	no	—
256	#2439	C₂×C₂₃	1	1	yes	characteristic
257	#2345	C₂×C₂₃	1	1	yes	characteristic
258	#2101	C₂×C₂₃	1	1	yes	characteristic, complemented
259	#2087	C₂×C₂₃	1	1	yes	characteristic, complemented
260	#2086	C₂×C₂₃	1	1	yes	characteristic, complemented
261	#2058	C₂×C₂₃	1	1	yes	characteristic, complemented
262	#1988	C₂×C₂₃	1	1	yes	characteristic, complemented
263	#1949	C₂×C₂₃	1	1	yes	characteristic, complemented
264	#1921	C₂×C₂₃	1	1	yes	characteristic, complemented
265	#1920	C₂×C₂₃	1	1	yes	characteristic, complemented
266	#1854	C₂×C₂₃	1	1	yes	characteristic
267	#1851	C₂×C₂₃	1	1	yes	characteristic
268	#1823	C₂×C₂₃	1	1	yes	characteristic
269	#1809	C₂×C₂₃	1	1	yes	characteristic
270	#1736	C₂×C₂₃	1	1	yes	characteristic
271	C₂²×C₂₃	—	2	2	no	—
272	C₂²×C₂₃	—	2	2	no	—
273	C₂²×C₂₃	—	2	2	no	—
274	C₂²×C₂₃	—	4	1	no	—
275	C₂²×C₂₃	—	4	1	no	—
276	C₂²×C₂₃	—	4	1	no	—
277	C₂²×C₂₃	—	4	1	no	—
278	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
279	C₂²×C₂₃	#546	1	1	yes	characteristic
280	C₂²×C₂₃	#547	1	1	yes	characteristic
281	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
282	C₂²×C₂₃	#575	1	1	yes	characteristic
283	C₂²×C₂₃	#576	1	1	yes	characteristic
284	C₂²×C₂₃	#577	1	1	yes	characteristic
285	C₄×C₂₃	—	2	2	no	—
286	C₄×C₂₃	—	2	2	no	—
287	C₄×C₂₃	—	4	2	no	—
288	C₄×C₂₃	—	4	2	no	—
289	C₄×C₂₃	—	4	1	no	—
290	C₄×C₂₃	—	4	1	no	—
291	C₄×C₂₃	—	4	2	no	—
292	C₄×C₂₃	—	4	1	no	—
293	C₄×C₂₃	—	8	1	no	—
294	C₄×C₂₃	—	8	1	no	—
295	C₄×C₂₃	—	8	1	no	—
296	C₄×C₂₃	—	8	1	no	—
297	C₄×C₂₃	—	8	1	no	—
298	C₄×C₂₃	#533	1	1	yes	characteristic
299	C₄×C₂₃	#544	1	1	yes	characteristic
300	C₄×C₂₃	#551	1	1	yes	characteristic
301	C₄×C₂₃	#552	1	1	yes	characteristic
302	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
303	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
304	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
305	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
306	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
307	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
308	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
309	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
310	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
311	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
312	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
313	#396	—	2	1	no	—
314	#393	—	2	1	no	—
315	#391	—	2	1	no	—
316	#389	C₂²×C₂₃	1	1	yes	characteristic
317	#389	C₂²×C₂₃	1	1	yes	characteristic
318	#389	C₂²×C₂₃	1	1	yes	characteristic
319	#389	C₂²×C₂₃	1	1	yes	characteristic
320	#387	—	2	1	no	—
321	#387	—	2	1	no	—
322	#387	—	2	1	no	—
323	#382	C₂²×C₂₃	1	1	yes	characteristic
324	#380	—	2	1	no	—
325	#380	C₂²×C₂₃	1	1	yes	characteristic
326	#380	C₂²×C₂₃	1	1	yes	characteristic
327	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
328	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
329	C₄×C₂²⋊C₄	—	2	1	no	—
330	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
331	#340	C₂²×C₂₃	1	1	yes	characteristic
332	#340	C₂²×C₂₃	1	1	yes	characteristic
333	#340	C₂²×C₂₃	1	1	yes	characteristic
334	#340	C₂²×C₂₃	1	1	yes	characteristic
335	#340	C₂²×C₂₃	1	1	yes	characteristic
336	#340	C₂²×C₂₃	1	1	yes	characteristic
337	#340	C₂²×C₂₃	1	1	yes	characteristic
338	#340	C₂²×C₂₃	1	1	yes	characteristic
339	#340	C₂²×C₂₃	1	1	yes	characteristic
340	#340	C₂²×C₂₃	1	1	yes	characteristic
341	#340	C₂²×C₂₃	1	1	yes	characteristic
342	#340	C₂²×C₂₃	1	1	yes	characteristic
343	#336	C₂²×C₂₃	1	1	yes	characteristic
344	#336	C₂²×C₂₃	1	1	yes	characteristic
345	C₂×C₂₃	—	4	2	no	—
346	C₂×C₂₃	#2438	1	1	yes	characteristic
347	C₂×C₂₃	#3078	1	1	yes	characteristic
348	C₂×C₂₃	#3081	1	1	yes	characteristic
349	C₂×C₂₃	#3314	1	1	yes	characteristic
350	C₂×C₂₃	#3326	1	1	yes	characteristic
351	C₂×C₂₃	#3347	1	1	yes	characteristic
352	C₂×C₂₃	#3356	1	1	yes	characteristic
353	C₂²×Q₈	—	2	1	no	—
354	C₂³×C₄	—	2	1	no	—
355	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
356	C₂×M₁₆	—	2	2	no	—
357	C₂×M₁₆	—	2	2	no	—
358	C₂×M₁₆	—	4	1	no	—
359	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
360	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
361	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
362	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
363	C₄⋊Q₈	—	2	2	no	—
364	C₄⋊Q₈	—	4	1	no	—
365	#117	—	2	2	no	—
366	#117	—	4	1	no	—
367	C₂×C₄⋊C₄	—	2	1	no	—
368	C₂×C₄⋊C₄	—	2	2	no	—
369	C₂×C₄⋊C₄	—	2	1	no	—
370	C₂×C₄⋊C₄	—	2	1	no	—
371	C₂×C₄⋊C₄	—	2	1	no	—
372	C₂×C₄⋊C₄	—	2	1	no	—
373	C₂×C₄⋊C₄	—	2	1	no	—
374	C₂×C₄⋊C₄	Q₈×C₂₃	1	4	yes	complemented
375	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
376	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
377	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
378	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
379	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
380	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
381	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
382	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
383	C₂×C₂²⋊C₄	—	2	1	no	—
384	C₂×C₂²⋊C₄	—	2	1	no	—
385	C₂×C₂²⋊C₄	—	2	1	no	—
386	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
387	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
388	C₂×C₄²	—	2	1	no	—
389	C₂×C₄²	—	2	1	no	—
390	C₂×C₄²	—	2	1	no	—
391	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
392	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
393	#107	—	2	2	no	—
394	#107	—	4	1	no	—
395	#106	—	2	2	no	—
396	#106	—	4	1	no	—
397	C₄⋊C₈	—	2	2	no	—
398	C₄⋊C₈	—	4	1	no	—
399	C₂²⋊C₈	—	2	2	no	—
400	#95	—	2	1	no	—
401	#95	—	2	1	no	—
402	#95	—	2	1	no	—
403	#95	—	2	1	no	—
404	#95	—	2	1	no	—
405	#95	—	2	1	no	—
406	#95	—	2	1	no	—
407	#95	—	2	1	no	—
408	#95	—	2	1	no	—
409	#95	—	2	1	no	—
410	#95	D₄×C₂₃	1	1	yes	characteristic
411	#95	D₄×C₂₃	1	1	yes	characteristic
412	C₂₃	#26008	1	1	yes	characteristic, direct product factor, Sylow
413	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
414	C₂×Q₈	—	4	2	no	—
415	C₂×Q₈	—	4	2	no	—
416	C₂×Q₈	—	8	1	no	—
417	C₂²×C₄	—	2	1	no	—
418	C₂²×C₄	—	2	1	no	—
419	C₂²×C₄	—	2	1	no	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	—	2	1	no	—
424	C₂²×C₄	—	2	4	no	—
425	C₂²×C₄	—	2	1	no	—
426	C₂²×C₄	—	2	1	no	—
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	—	2	1	no	—
429	C₂²×C₄	—	2	1	no	—
430	C₂²×C₄	—	8	1	no	—
431	C₂²×C₄	—	8	1	no	—
432	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
433	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
434	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
435	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
436	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
437	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
438	C₂²×C₄	#41×C₂₃	1	2	yes	—
439	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
440	M₁₆	—	4	4	no	—
441	C₂×C₈	—	2	1	no	—
442	C₂×C₈	—	2	1	no	—
443	C₂×C₈	—	2	1	no	—
444	C₂×C₈	—	2	1	no	—
445	C₂×C₈	—	2	1	no	—
446	C₂×C₈	—	2	1	no	—
447	C₂×C₈	—	2	1	no	—
448	C₂×C₈	—	2	1	no	—
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	#41×C₂₃	1	2	yes	—
453	C₄⋊C₄	—	2	4	no	—
454	C₄⋊C₄	—	2	1	no	—
455	C₄⋊C₄	—	2	1	no	—
456	C₄⋊C₄	—	2	1	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	2	1	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	4	2	no	—
468	C₄⋊C₄	—	4	2	no	—
469	C₄⋊C₄	—	8	1	no	—
470	C₄⋊C₄	—	8	1	no	—
471	C₄⋊C₄	—	8	1	no	—
472	C₄⋊C₄	C₂×Q₈×C₂₃	1	2	yes	—
473	C₄⋊C₄	C₂×Q₈×C₂₃	1	2	yes	—
474	C₄⋊C₄	#41×C₂₃	1	2	yes	—
475	C₂²⋊C₄	—	2	2	no	—
476	C₂²⋊C₄	—	4	1	no	—
477	C₂²⋊C₄	—	4	2	no	—
478	C₂²⋊C₄	—	4	2	no	—
479	C₂²⋊C₄	—	8	1	no	—
480	C₄²	—	2	1	no	—
481	C₄²	—	2	1	no	—
482	C₄²	—	2	1	no	—
483	C₄²	—	2	1	no	—
484	C₄²	—	4	2	no	—
485	C₄²	—	8	1	no	—
486	C₄²	—	8	1	no	—
487	C₂³	—	2	1	no	—
488	C₂³	—	2	1	no	—
489	C₂³	—	2	1	no	—
490	C₂³	—	2	1	no	—
491	C₂³	—	2	1	no	—
492	C₂³	—	2	1	no	—
493	C₂³	C₂₃×C₂²⋊Q₈	1	2	yes	—
494	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
495	Q₈	—	8	4	no	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	4	1	no	—
517	C₂×C₄	—	4	1	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	4	1	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	4	1	no	—
522	C₂×C₄	—	4	1	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	4	1	no	—
525	C₂×C₄	—	4	1	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	1	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	4	1	no	—
531	C₂×C₄	—	4	1	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	8	4	no	—
538	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
539	C₂×C₄	C₂₃×#137	1	2	yes	—
540	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
541	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
542	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
543	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
544	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
545	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
546	C₈	—	2	2	no	—
547	C₈	—	4	1	no	—
548	C₈	—	4	1	no	—
549	C₈	—	4	1	no	—
550	C₂²	—	2	2	no	—
551	C₂²	—	2	2	no	—
552	C₂²	—	2	2	no	—
553	C₂²	—	4	1	no	—
554	C₂²	—	4	1	no	—
555	C₂²	—	4	1	no	—
556	C₂²	—	4	1	no	—
557	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
558	C₂²	C₂₃×#546	1	1	yes	characteristic
559	C₂²	C₂₃×#547	1	1	yes	characteristic
560	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
561	C₂²	C₂₃×#575	1	1	yes	characteristic
562	C₂²	C₂₃×#576	1	1	yes	characteristic
563	C₂²	C₂₃×#577	1	1	yes	characteristic
564	C₄	—	2	2	no	—
565	C₄	—	2	2	no	—
566	C₄	—	4	2	no	—
567	C₄	—	4	1	no	—
568	C₄	—	4	2	no	—
569	C₄	—	4	2	no	—
570	C₄	—	4	1	no	—
571	C₄	—	4	1	no	—
572	C₄	—	8	1	no	—
573	C₄	—	8	1	no	—
574	C₄	—	8	1	no	—
575	C₄	—	8	1	no	—
576	C₄	—	8	1	no	—
577	C₄	C₂₃×#533	1	1	yes	characteristic
578	C₄	C₂₃×#544	1	1	yes	characteristic
579	C₄	C₂₃×#551	1	1	yes	characteristic
580	C₄	C₂₃×#552	1	1	yes	characteristic
581	C₂	—	4	2	no	—
582	C₂	C₂₃×#2438	1	1	yes	characteristic
583	C₂	C₂₃×#3078	1	1	yes	characteristic
584	C₂	C₂₃×#3081	1	1	yes	characteristic
585	C₂	C₂₃×#3314	1	1	yes	characteristic
586	C₂	C₂₃×#3326	1	1	yes	characteristic
587	C₂	C₂₃×#3347	1	1	yes	characteristic
588	C₂	C₂₃×#3356	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄×C₂²⋊C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2101	C₂	semidirect
4. C₂₃×#2087	C₂	semidirect
5. C₂₃×#2086	C₂	semidirect
6. C₂₃×#2058	C₂	semidirect
7. C₂₃×#1988	C₂	semidirect
8. C₂₃×#1949	C₂	semidirect
9. C₂₃×#1921	C₂	semidirect
10. C₂₃×#1920	C₂	semidirect
80. C₂×C₄⋊C₄×C₂₃	Q₈	semidirect
192. #26008	C₂₃	direct (N × H)
258. #2101	C₂×C₂₃	semidirect
259. #2087	C₂×C₂₃	semidirect
260. #2086	C₂×C₂₃	semidirect
261. #2058	C₂×C₂₃	semidirect
262. #1988	C₂×C₂₃	semidirect
263. #1949	C₂×C₂₃	semidirect
264. #1921	C₂×C₂₃	semidirect
265. #1920	C₂×C₂₃	semidirect
374. C₂×C₄⋊C₄	Q₈×C₂₃	semidirect
412. C₂₃	#26008	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×#2439	2
C₂₃×#2345	2
C₂₃×#2101	2
C₂₃×#2087	2
C₂₃×#2086	2
C₂₃×#2058	2
C₂₃×#1988	2
C₂₃×#1949	2
C₂₃×#1921	2
C₂₃×#1920	2
C₂₃×#1854	2
C₂₃×#1851	2
C₂₃×#1823	2
C₂₃×#1809	2
C₂₃×#1736	2
#26008	23

Subgroups — C23×#26008 #19173634

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#26008 #19173634