Subgroups — C₂₃×#27026 #19174652

Order 5888.

726 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2452	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2370	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2065	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2059	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2058	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2025	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1988	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1943	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1926	C₂	1	1	yes	maximal, characteristic
10	C₂₃×#1925	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#1920	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1920	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1917	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1864	C₂	1	1	yes	maximal, characteristic
15	C₂₃×#1762	C₂	1	1	yes	maximal, characteristic
16	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
19	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#106	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×#102	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
30	C₂₃×#396	—	2	1	no	—
31	C₂₃×#393	—	2	1	no	—
32	C₂₃×#390	C₂²	1	1	yes	characteristic
33	C₂₃×#390	C₂²	1	1	yes	characteristic
34	C₂₃×#389	—	2	1	no	—
35	C₂₃×#389	C₂²	1	1	yes	characteristic
36	C₂₃×#389	C₂²	1	1	yes	characteristic
37	C₂₃×#389	C₂²	1	1	yes	characteristic
38	C₂₃×#389	C₂²	1	1	yes	characteristic
39	C₂₃×#387	—	2	1	no	—
40	C₂₃×#387	—	2	1	no	—
41	C₂₃×#385	—	2	1	no	—
42	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
43	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
44	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
45	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
46	C₄×C₄⋊C₄×C₂₃	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×#336	C₂²	1	1	yes	characteristic
56	C₂₃×#336	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	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₂₃	—	2	1	no	—
62	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
66	C₂₃×C₄⋊Q₈	—	2	2	no	—
67	C₂₃×C₄⋊Q₈	—	4	1	no	—
68	C₂₃×#125	—	2	2	no	—
69	C₂₃×#125	—	2	2	no	—
70	C₂₃×C₂²⋊Q₈	—	4	4	no	—
71	C₄×D₄×C₂₃	—	2	2	no	—
72	C₄×D₄×C₂₃	—	4	1	no	—
73	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
74	C₂×C₄⋊C₄×C₂₃	—	2	1	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	D₄	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₂₃	D₄	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	—	2	1	no	—
93	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
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	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
100	C₂₃×#107	—	2	2	no	—
101	C₂₃×#106	—	2	2	no	—
102	C₂₃×#106	—	2	2	no	—
103	C₂₃×#106	—	4	1	no	—
104	C₂₃×C₄⋊C₈	—	4	1	no	—
105	C₂₃×#102	—	2	2	no	—
106	C₂₃×#102	—	2	2	no	—
107	C₂₃×#102	—	4	1	no	—
108	C₂₃×#102	Q₈	1	4	yes	complemented
109	C₂₃×#97	—	4	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	—	2	1	no	—
117	C₂₃×#95	D₄	1	1	yes	characteristic
118	C₂₃×#95	D₄	1	1	yes	characteristic
119	C₂⁴×C₂₃	—	2	1	no	—
120	C₂×Q₈×C₂₃	—	4	2	no	—
121	C₂×Q₈×C₂₃	—	4	2	no	—
122	C₂×Q₈×C₂₃	—	8	1	no	—
123	C₂×D₄×C₂₃	—	2	1	no	—
124	C₂×D₄×C₂₃	—	2	1	no	—
125	C₂×D₄×C₂₃	—	2	1	no	—
126	C₂×D₄×C₂₃	—	2	1	no	—
127	C₂×D₄×C₂₃	—	2	1	no	—
128	C₂×D₄×C₂₃	C₂×Q₈	1	2	yes	—
129	C₂²×C₄×C₂₃	—	2	1	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	—	2	1	no	—
136	C₂²×C₄×C₂₃	—	4	2	no	—
137	C₂²×C₄×C₂₃	—	4	2	no	—
138	C₂²×C₄×C₂₃	—	8	1	no	—
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×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
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	C₂×Q₈	1	2	yes	—
158	C₄⋊C₄×C₂₃	—	2	4	no	—
159	C₄⋊C₄×C₂₃	—	2	4	no	—
160	C₄⋊C₄×C₂₃	—	2	2	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	—	4	1	no	—
172	C₄⋊C₄×C₂₃	—	4	2	no	—
173	C₄⋊C₄×C₂₃	—	4	2	no	—
174	C₄⋊C₄×C₂₃	—	4	2	no	—
175	C₄⋊C₄×C₂₃	—	4	2	no	—
176	C₄⋊C₄×C₂₃	—	8	1	no	—
177	C₄⋊C₄×C₂₃	C₂×Q₈	1	2	yes	—
178	C₄⋊C₄×C₂₃	#41	1	2	yes	—
179	C₄⋊C₄×C₂₃	#41	1	2	yes	—
180	C₄⋊C₄×C₂₃	#41	1	2	yes	—
181	C₂²⋊C₄×C₂₃	—	4	2	no	—
182	C₂²⋊C₄×C₂₃	—	4	2	no	—
183	C₂²⋊C₄×C₂₃	—	4	2	no	—
184	C₂²⋊C₄×C₂₃	—	4	2	no	—
185	C₂²⋊C₄×C₂₃	—	8	1	no	—
186	C₄²×C₂₃	—	2	1	no	—
187	C₄²×C₂₃	—	2	1	no	—
188	C₄²×C₂₃	—	2	1	no	—
189	C₄²×C₂₃	—	2	1	no	—
190	C₄²×C₂₃	—	2	1	no	—
191	C₄²×C₂₃	—	2	1	no	—
192	C₄²×C₂₃	—	4	2	no	—
193	C₄²×C₂₃	—	8	1	no	—
194	C₄²×C₂₃	—	8	1	no	—
195	#27026	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
196	C₂³×C₂₃	—	2	2	no	—
197	C₂³×C₂₃	—	4	1	no	—
198	C₂³×C₂₃	—	4	1	no	—
199	C₂³×C₂₃	—	4	1	no	—
200	C₂³×C₂₃	—	4	1	no	—
201	C₂³×C₂₃	—	4	1	no	—
202	C₂³×C₂₃	—	4	1	no	—
203	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
204	Q₈×C₂₃	—	8	4	no	—
205	D₄×C₂₃	—	2	2	no	—
206	D₄×C₂₃	—	2	2	no	—
207	D₄×C₂₃	—	4	1	no	—
208	D₄×C₂₃	—	4	1	no	—
209	C₂×C₄×C₂₃	—	2	1	no	—
210	C₂×C₄×C₂₃	—	2	1	no	—
211	C₂×C₄×C₂₃	—	2	1	no	—
212	C₂×C₄×C₂₃	—	2	1	no	—
213	C₂×C₄×C₂₃	—	2	2	no	—
214	C₂×C₄×C₂₃	—	2	2	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	2	no	—
219	C₂×C₄×C₂₃	—	2	2	no	—
220	C₂×C₄×C₂₃	—	2	1	no	—
221	C₂×C₄×C₂₃	—	2	1	no	—
222	C₂×C₄×C₂₃	—	2	1	no	—
223	C₂×C₄×C₂₃	—	2	1	no	—
224	C₂×C₄×C₂₃	—	2	1	no	—
225	C₂×C₄×C₂₃	—	2	1	no	—
226	C₂×C₄×C₂₃	—	2	1	no	—
227	C₂×C₄×C₂₃	—	2	1	no	—
228	C₂×C₄×C₂₃	—	2	1	no	—
229	C₂×C₄×C₂₃	—	2	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	—	4	1	no	—
244	C₂×C₄×C₂₃	—	4	1	no	—
245	C₂×C₄×C₂₃	—	4	1	no	—
246	C₂×C₄×C₂₃	—	4	1	no	—
247	C₂×C₄×C₂₃	—	8	4	no	—
248	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
249	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
250	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
251	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
252	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
253	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
254	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
255	C₈×C₂₃	—	2	2	no	—
256	C₈×C₂₃	—	4	1	no	—
257	C₈×C₂₃	—	4	1	no	—
258	C₈×C₂₃	—	4	1	no	—
259	#2452	C₂×C₂₃	1	1	yes	characteristic
260	#2370	C₂×C₂₃	1	1	yes	characteristic, complemented
261	#2065	C₂×C₂₃	1	1	yes	characteristic
262	#2059	C₂×C₂₃	1	1	yes	characteristic, complemented
263	#2058	C₂×C₂₃	1	1	yes	characteristic, complemented
264	#2025	C₂×C₂₃	1	1	yes	characteristic
265	#1988	C₂×C₂₃	1	1	yes	characteristic, complemented
266	#1943	C₂×C₂₃	1	1	yes	characteristic, complemented
267	#1926	C₂×C₂₃	1	1	yes	characteristic
268	#1925	C₂×C₂₃	1	1	yes	characteristic
269	#1920	C₂×C₂₃	1	1	yes	characteristic, complemented
270	#1920	C₂×C₂₃	1	1	yes	characteristic, complemented
271	#1917	C₂×C₂₃	1	1	yes	characteristic, complemented
272	#1864	C₂×C₂₃	1	1	yes	characteristic
273	#1762	C₂×C₂₃	1	1	yes	characteristic
274	C₂²×C₂₃	—	4	2	no	—
275	C₂²×C₂₃	—	4	2	no	—
276	C₂²×C₂₃	—	4	2	no	—
277	C₂²×C₂₃	—	8	1	no	—
278	C₂²×C₂₃	—	8	1	no	—
279	C₂²×C₂₃	—	8	1	no	—
280	C₂²×C₂₃	—	8	1	no	—
281	C₂²×C₂₃	#525	1	1	yes	characteristic
282	C₂²×C₂₃	#546	1	1	yes	characteristic
283	C₂²×C₂₃	D₄×Q₈	1	1	yes	characteristic
284	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
285	C₂²×C₂₃	#575	1	1	yes	characteristic
286	C₂²×C₂₃	#577	1	1	yes	characteristic
287	C₂²×C₂₃	#577	1	1	yes	characteristic
288	C₄×C₂₃	—	2	2	no	—
289	C₄×C₂₃	—	4	1	no	—
290	C₄×C₂₃	—	4	2	no	—
291	C₄×C₂₃	—	4	1	no	—
292	C₄×C₂₃	—	4	2	no	—
293	C₄×C₂₃	—	4	2	no	—
294	C₄×C₂₃	—	4	1	no	—
295	C₄×C₂₃	—	4	2	no	—
296	C₄×C₂₃	—	4	1	no	—
297	C₄×C₂₃	—	4	1	no	—
298	C₄×C₂₃	—	8	1	no	—
299	C₄×C₂₃	—	8	1	no	—
300	C₄×C₂₃	—	8	1	no	—
301	C₄×C₂₃	#533	1	1	yes	characteristic
302	C₄×C₂₃	#542	1	1	yes	characteristic
303	C₄×C₂₃	#554	1	1	yes	characteristic
304	C₄×C₂₃	#556	1	1	yes	characteristic
305	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
306	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
307	C₂×C₂²⋊Q₈	—	2	1	no	—
308	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
309	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
310	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
311	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
312	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
313	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
314	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
315	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
316	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
317	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
318	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
319	#396	—	2	1	no	—
320	#393	—	2	1	no	—
321	#390	C₂²×C₂₃	1	1	yes	characteristic
322	#390	C₂²×C₂₃	1	1	yes	characteristic
323	#389	—	2	1	no	—
324	#389	C₂²×C₂₃	1	1	yes	characteristic
325	#389	C₂²×C₂₃	1	1	yes	characteristic
326	#389	C₂²×C₂₃	1	1	yes	characteristic
327	#389	C₂²×C₂₃	1	1	yes	characteristic
328	#387	—	2	1	no	—
329	#387	—	2	1	no	—
330	#385	—	2	1	no	—
331	C₄×C₄⋊C₄	—	2	1	no	—
332	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
333	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
334	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
335	C₄×C₄⋊C₄	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	#340	C₂²×C₂₃	1	1	yes	characteristic
344	#336	C₂²×C₂₃	1	1	yes	characteristic
345	#336	C₂²×C₂₃	1	1	yes	characteristic
346	#336	C₂²×C₂₃	1	1	yes	characteristic
347	#336	C₂²×C₂₃	1	1	yes	characteristic
348	C₂×C₂₃	—	8	2	no	—
349	C₂×C₂₃	#2465	1	1	yes	characteristic
350	C₂×C₂₃	#3115	1	1	yes	characteristic
351	C₂×C₂₃	#3125	1	1	yes	characteristic
352	C₂×C₂₃	#3310	1	1	yes	characteristic
353	C₂×C₂₃	#3326	1	1	yes	characteristic
354	C₂×C₂₃	#3384	1	1	yes	characteristic
355	C₂×C₂₃	#3388	1	1	yes	characteristic
356	C₂²×Q₈	—	2	1	no	—
357	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
358	C₂³×C₄	—	2	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	#125	—	2	2	no	—
366	#125	—	2	2	no	—
367	C₂²⋊Q₈	—	4	4	no	—
368	C₄×D₄	—	2	2	no	—
369	C₄×D₄	—	4	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	—	2	1	no	—
375	C₂×C₄⋊C₄	—	2	1	no	—
376	C₂×C₄⋊C₄	—	2	1	no	—
377	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
378	C₂×C₄⋊C₄	D₄×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×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
384	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
385	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
386	C₂×C₂²⋊C₄	—	2	1	no	—
387	C₂×C₂²⋊C₄	—	2	1	no	—
388	C₂×C₂²⋊C₄	—	2	1	no	—
389	C₂×C₂²⋊C₄	—	2	1	no	—
390	C₂×C₂²⋊C₄	—	2	1	no	—
391	C₂×C₄²	—	2	1	no	—
392	C₂×C₄²	—	2	1	no	—
393	C₂×C₄²	—	2	1	no	—
394	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
395	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
396	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
397	#107	—	2	2	no	—
398	#106	—	2	2	no	—
399	#106	—	2	2	no	—
400	#106	—	4	1	no	—
401	C₄⋊C₈	—	4	1	no	—
402	#102	—	2	2	no	—
403	#102	—	2	2	no	—
404	#102	—	4	1	no	—
405	#102	Q₈×C₂₃	1	4	yes	complemented
406	#97	—	4	1	no	—
407	#95	—	2	1	no	—
408	#95	—	2	1	no	—
409	#95	—	2	1	no	—
410	#95	—	2	1	no	—
411	#95	—	2	1	no	—
412	#95	—	2	1	no	—
413	#95	—	2	1	no	—
414	#95	D₄×C₂₃	1	1	yes	characteristic
415	#95	D₄×C₂₃	1	1	yes	characteristic
416	C₂₃	#27026	1	1	yes	characteristic, direct product factor, Sylow
417	C₂⁴	—	2	1	no	—
418	C₂×Q₈	—	4	2	no	—
419	C₂×Q₈	—	4	2	no	—
420	C₂×Q₈	—	8	1	no	—
421	C₂×D₄	—	2	1	no	—
422	C₂×D₄	—	2	1	no	—
423	C₂×D₄	—	2	1	no	—
424	C₂×D₄	—	2	1	no	—
425	C₂×D₄	—	2	1	no	—
426	C₂×D₄	C₂×Q₈×C₂₃	1	2	yes	—
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	—	2	1	no	—
429	C₂²×C₄	—	2	1	no	—
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	—	2	1	no	—
432	C₂²×C₄	—	2	1	no	—
433	C₂²×C₄	—	2	1	no	—
434	C₂²×C₄	—	4	2	no	—
435	C₂²×C₄	—	4	2	no	—
436	C₂²×C₄	—	8	1	no	—
437	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
438	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
439	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
440	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
441	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
442	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
443	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
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	—	2	1	no	—
453	C₂×C₈	—	2	1	no	—
454	C₂×C₈	—	2	1	no	—
455	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
456	C₄⋊C₄	—	2	4	no	—
457	C₄⋊C₄	—	2	4	no	—
458	C₄⋊C₄	—	2	2	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	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	4	1	no	—
470	C₄⋊C₄	—	4	2	no	—
471	C₄⋊C₄	—	4	2	no	—
472	C₄⋊C₄	—	4	2	no	—
473	C₄⋊C₄	—	4	2	no	—
474	C₄⋊C₄	—	8	1	no	—
475	C₄⋊C₄	C₂×Q₈×C₂₃	1	2	yes	—
476	C₄⋊C₄	#41×C₂₃	1	2	yes	—
477	C₄⋊C₄	#41×C₂₃	1	2	yes	—
478	C₄⋊C₄	#41×C₂₃	1	2	yes	—
479	C₂²⋊C₄	—	4	2	no	—
480	C₂²⋊C₄	—	4	2	no	—
481	C₂²⋊C₄	—	4	2	no	—
482	C₂²⋊C₄	—	4	2	no	—
483	C₂²⋊C₄	—	8	1	no	—
484	C₄²	—	2	1	no	—
485	C₄²	—	2	1	no	—
486	C₄²	—	2	1	no	—
487	C₄²	—	2	1	no	—
488	C₄²	—	2	1	no	—
489	C₄²	—	2	1	no	—
490	C₄²	—	4	2	no	—
491	C₄²	—	8	1	no	—
492	C₄²	—	8	1	no	—
493	C₂³	—	2	2	no	—
494	C₂³	—	4	1	no	—
495	C₂³	—	4	1	no	—
496	C₂³	—	4	1	no	—
497	C₂³	—	4	1	no	—
498	C₂³	—	4	1	no	—
499	C₂³	—	4	1	no	—
500	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
501	Q₈	—	8	4	no	—
502	D₄	—	2	2	no	—
503	D₄	—	2	2	no	—
504	D₄	—	4	1	no	—
505	D₄	—	4	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	2	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	—	4	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	8	4	no	—
545	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
546	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
547	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
548	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
549	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
550	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
551	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
552	C₈	—	2	2	no	—
553	C₈	—	4	1	no	—
554	C₈	—	4	1	no	—
555	C₈	—	4	1	no	—
556	C₂²	—	4	2	no	—
557	C₂²	—	4	2	no	—
558	C₂²	—	4	2	no	—
559	C₂²	—	8	1	no	—
560	C₂²	—	8	1	no	—
561	C₂²	—	8	1	no	—
562	C₂²	—	8	1	no	—
563	C₂²	C₂₃×#525	1	1	yes	characteristic
564	C₂²	C₂₃×#546	1	1	yes	characteristic
565	C₂²	D₄×Q₈×C₂₃	1	1	yes	characteristic
566	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
567	C₂²	C₂₃×#575	1	1	yes	characteristic
568	C₂²	C₂₃×#577	1	1	yes	characteristic
569	C₂²	C₂₃×#577	1	1	yes	characteristic
570	C₄	—	2	2	no	—
571	C₄	—	4	1	no	—
572	C₄	—	4	2	no	—
573	C₄	—	4	1	no	—
574	C₄	—	4	1	no	—
575	C₄	—	4	1	no	—
576	C₄	—	4	2	no	—
577	C₄	—	4	1	no	—
578	C₄	—	4	2	no	—
579	C₄	—	4	2	no	—
580	C₄	—	8	1	no	—
581	C₄	—	8	1	no	—
582	C₄	—	8	1	no	—
583	C₄	C₂₃×#533	1	1	yes	characteristic
584	C₄	C₂₃×#542	1	1	yes	characteristic
585	C₄	C₂₃×#554	1	1	yes	characteristic
586	C₄	C₂₃×#556	1	1	yes	characteristic
587	C₂	—	8	2	no	—
588	C₂	C₂₃×#2465	1	1	yes	characteristic
589	C₂	C₂₃×#3115	1	1	yes	characteristic
590	C₂	C₂₃×#3125	1	1	yes	characteristic
591	C₂	C₂₃×#3310	1	1	yes	characteristic
592	C₂	C₂₃×#3326	1	1	yes	characteristic
593	C₂	C₂₃×#3384	1	1	yes	characteristic
594	C₂	C₂₃×#3388	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2370	C₂	semidirect
4. C₂₃×#2059	C₂	semidirect
5. C₂₃×#2058	C₂	semidirect
7. C₂₃×#1988	C₂	semidirect
8. C₂₃×#1943	C₂	semidirect
11. C₂₃×#1920	C₂	semidirect
12. C₂₃×#1920	C₂	semidirect
13. C₂₃×#1917	C₂	semidirect
108. C₂₃×#102	Q₈	semidirect
195. #27026	C₂₃	direct (N × H)
260. #2370	C₂×C₂₃	semidirect
262. #2059	C₂×C₂₃	semidirect
263. #2058	C₂×C₂₃	semidirect
265. #1988	C₂×C₂₃	semidirect
266. #1943	C₂×C₂₃	semidirect
269. #1920	C₂×C₂₃	semidirect
270. #1920	C₂×C₂₃	semidirect
271. #1917	C₂×C₂₃	semidirect
405. #102	Q₈×C₂₃	semidirect
416. C₂₃	#27026	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×#2452	2
C₂₃×#2370	2
C₂₃×#2065	2
C₂₃×#2059	2
C₂₃×#2058	2
C₂₃×#2025	2
C₂₃×#1988	2
C₂₃×#1943	2
C₂₃×#1926	2
C₂₃×#1925	2
C₂₃×#1920	2
C₂₃×#1917	2
C₂₃×#1864	2
C₂₃×#1762	2
#27026	23

Subgroups — C23×#27026 #19174652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#27026 #19174652