Subgroups — C₂₃×#61585 #19209211

Order 5888.

2378 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3496	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#547	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3330	C₂	1	8	yes	maximal, complemented
4	C₂₃×#3272	C₂	1	8	yes	maximal, complemented
5	C₂₃×#3206	C₂	1	2	yes	maximal, complemented
6	C₂₃×#3193	C₂	1	4	yes	maximal, complemented
7	C₂₃×#3100	C₂	1	2	yes	maximal, complemented
8	C₂₃×#3006	C₂	1	4	yes	maximal, complemented
9	C₂₃×#2975	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×H(5,𝔽₂)	—	2	1	no	—
11	C₂²×#41×C₂₃	C₂²	1	2	yes	complemented
12	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	8	yes	complemented
13	C₂₃×#547	—	2	4	no	—
14	C₂₃×#547	C₂²	1	8	yes	complemented
15	C₂₃×#547	C₂²	1	4	yes	complemented
16	C₂₃×#546	—	2	4	no	—
17	D₄²×C₂₃	C₂²	1	4	yes	complemented
18	C₂₃×#537	—	2	1	no	—
19	C₂₃×#534	—	2	2	no	—
20	C₂₃×#532	C₂²	1	2	yes	—
21	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₂₃×#123	—	2	2	no	—
23	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
24	C₂×C₂₃×#121	—	2	1	no	—
25	C₂×C₂₃×#121	C₂²	1	4	yes	—
26	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
28	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
29	C₂₃×#492	C₂²	1	4	yes	complemented
30	C₂₃×#487	C₂²	1	4	yes	complemented
31	C₂₃×#484	C₂²	1	8	yes	complemented
32	C₂₃×#480	C₂²	1	4	yes	complemented
33	C₂₃×#474	C₂²	1	4	yes	complemented
34	C₂₃×#468	C₂²	1	16	yes	complemented
35	C₂₃×#461	C₂²	1	8	yes	complemented
36	C₂₃×#459	C₂²	1	4	yes	complemented
37	C₂₃×#450	C₂²	1	8	yes	complemented
38	C₂₃×#449	C₂²	1	16	yes	complemented
39	C₂₃×#448	C₂²	1	8	yes	complemented
40	C₂₃×#447	C₂²	1	4	yes	complemented
41	C₄×D₈×C₂₃	C₂²	1	8	yes	complemented
42	C₂₃×#436	C₂²	1	8	yes	complemented
43	C₂₃×#431	C₂²	1	2	yes	complemented
44	C₂₃×#417	C₂²	1	8	yes	complemented
45	C₂₃×#407	C₂²	1	4	yes	complemented
46	C₂₃×H(5,𝔽₂)	—	4	8	no	—
47	C₂×#41×C₂₃	—	2	4	no	—
48	C₂×#41×C₂₃	—	2	4	no	—
49	C₂×#41×C₂₃	—	2	2	no	—
50	C₂×#41×C₂₃	—	4	8	no	—
51	C₂×#41×C₂₃	C₂³	1	8	yes	—
52	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
53	C₂²×D₄×C₂₃	—	2	2	no	—
54	C₂²×D₄×C₂₃	—	2	4	no	—
55	C₂²×D₄×C₂₃	—	2	4	no	—
56	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
57	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
58	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
59	C₂³×C₄×C₂₃	—	2	2	no	—
60	C₂³×C₄×C₂₃	—	2	1	no	—
61	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
62	C₂₃×AGL(1,ℤ/8)	—	4	16	no	—
63	C₂×SD₁₆×C₂₃	C₂³	1	16	yes	—
64	C₂×D₈×C₂₃	C₂³	1	8	yes	—
65	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
66	C₂₃×C₄⋊Q₈	—	2	2	no	—
67	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
68	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
69	C₂₃×#127	C₂³	1	2	yes	—
70	C₂₃×#124	—	2	2	no	—
71	C₂₃×#123	—	2	16	no	—
72	C₂₃×#123	—	2	8	no	—
73	C₂₃×C₂²⋊Q₈	—	2	4	no	—
74	C₂₃×C₂²⋊Q₈	—	2	4	no	—
75	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
76	C₂₃×#121	—	2	8	no	—
77	C₂₃×#121	—	2	4	no	—
78	C₂₃×#121	—	2	4	no	—
79	C₂₃×#121	—	2	2	no	—
80	C₂₃×#121	—	2	4	no	—
81	C₂₃×#121	C₂³	1	8	yes	—
82	C₂₃×#121	C₂³	1	4	yes	—
83	C₂₃×#120	—	2	8	no	—
84	C₄×D₄×C₂₃	—	2	2	no	—
85	C₄×D₄×C₂₃	—	2	2	no	—
86	C₄×D₄×C₂₃	D₄	1	4	yes	—
87	C₄×D₄×C₂₃	C₂³	1	4	yes	—
88	C₄×D₄×C₂₃	C₂³	1	4	yes	—
89	C₄×D₄×C₂₃	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
91	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
92	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
93	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
94	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
96	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
97	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
98	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
99	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
100	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
101	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
102	C₂₃×#107	C₂³	1	8	yes	—
103	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
104	C₂₃×#103	C₂³	1	16	yes	—
105	C₂₃×#102	C₂³	1	8	yes	—
106	C₂₃×#102	C₂³	1	8	yes	—
107	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
108	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
109	C₄×C₈×C₂₃	C₂³	1	4	yes	—
110	C₂⁴×C₂₃	—	2	1	no	—
111	C₂⁴×C₂₃	—	2	4	no	—
112	C₂⁴×C₂₃	C₂×D₄	1	2	yes	—
113	#41×C₂₃	—	2	16	no	—
114	#41×C₂₃	—	4	8	no	—
115	#41×C₂₃	—	4	16	no	—
116	#41×C₂₃	—	4	16	no	—
117	#41×C₂₃	—	4	8	no	—
118	C₂×Q₈×C₂₃	—	2	2	no	—
119	C₂×Q₈×C₂₃	—	2	2	no	—
120	C₂×Q₈×C₂₃	—	4	4	no	—
121	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
122	C₂×D₄×C₂₃	—	2	4	no	—
123	C₂×D₄×C₂₃	—	2	8	no	—
124	C₂×D₄×C₂₃	—	2	4	no	—
125	C₂×D₄×C₂₃	—	2	8	no	—
126	C₂×D₄×C₂₃	—	2	4	no	—
127	C₂×D₄×C₂₃	—	2	1	no	—
128	C₂×D₄×C₂₃	—	2	2	no	—
129	C₂×D₄×C₂₃	—	2	2	no	—
130	C₂×D₄×C₂₃	—	2	4	no	—
131	C₂×D₄×C₂₃	—	4	8	no	—
132	C₂×D₄×C₂₃	—	4	16	no	—
133	C₂×D₄×C₂₃	—	4	8	no	—
134	C₂×D₄×C₂₃	—	4	16	no	—
135	C₂×D₄×C₂₃	C₂×D₄	1	4	yes	—
136	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
137	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
138	C₂²×C₄×C₂₃	—	2	1	no	—
139	C₂²×C₄×C₂₃	—	2	2	no	—
140	C₂²×C₄×C₂₃	—	2	4	no	—
141	C₂²×C₄×C₂₃	—	2	4	no	—
142	C₂²×C₄×C₂₃	—	2	8	no	—
143	C₂²×C₄×C₂₃	—	2	4	no	—
144	C₂²×C₄×C₂₃	—	4	4	no	—
145	C₂²×C₄×C₂₃	—	4	8	no	—
146	C₂²×C₄×C₂₃	—	4	4	no	—
147	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
149	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
150	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
152	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
153	SD₁₆×C₂₃	—	4	16	no	—
154	D₈×C₂₃	—	2	16	no	—
155	M₁₆×C₂₃	—	4	8	no	—
156	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
157	C₄⋊C₄×C₂₃	—	2	4	no	—
158	C₄⋊C₄×C₂₃	—	2	8	no	—
159	C₄⋊C₄×C₂₃	—	2	4	no	—
160	C₄⋊C₄×C₂₃	—	2	1	no	—
161	C₄⋊C₄×C₂₃	—	2	4	no	—
162	C₄⋊C₄×C₂₃	—	2	2	no	—
163	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
164	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
165	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
166	C₂²⋊C₄×C₂₃	—	2	8	no	—
167	C₂²⋊C₄×C₂₃	—	2	4	no	—
168	C₂²⋊C₄×C₂₃	—	2	4	no	—
169	C₂²⋊C₄×C₂₃	—	2	8	no	—
170	C₂²⋊C₄×C₂₃	—	2	4	no	—
171	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
172	C₄²×C₂₃	C₂×D₄	1	2	yes	—
173	C₄²×C₂₃	C₂⁴	1	2	yes	—
174	#61585	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
175	C₂³×C₂₃	—	2	4	no	—
176	C₂³×C₂₃	—	2	2	no	—
177	C₂³×C₂₃	—	2	4	no	—
178	C₂³×C₂₃	—	2	4	no	—
179	C₂³×C₂₃	—	4	2	no	—
180	C₂³×C₂₃	—	4	2	no	—
181	C₂³×C₂₃	—	4	2	no	—
182	C₂³×C₂₃	—	4	8	no	—
183	C₂³×C₂₃	—	4	2	no	—
184	C₂³×C₂₃	—	8	4	no	—
185	C₂³×C₂₃	—	8	4	no	—
186	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
187	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
188	Q₈×C₂₃	—	2	8	no	—
189	Q₈×C₂₃	—	4	4	no	—
190	Q₈×C₂₃	—	4	4	no	—
191	D₄×C₂₃	—	2	4	no	—
192	D₄×C₂₃	—	2	8	no	—
193	D₄×C₂₃	—	2	2	no	—
194	D₄×C₂₃	—	2	8	no	—
195	D₄×C₂₃	—	4	16	no	—
196	D₄×C₂₃	—	4	16	no	—
197	D₄×C₂₃	—	4	8	no	—
198	D₄×C₂₃	—	4	4	no	—
199	D₄×C₂₃	—	8	2	no	—
200	D₄×C₂₃	—	8	4	no	—
201	D₄×C₂₃	H(5,𝔽₂)	1	4	yes	complemented
202	C₂×C₄×C₂₃	—	2	8	no	—
203	C₂×C₄×C₂₃	—	2	2	no	—
204	C₂×C₄×C₂₃	—	2	2	no	—
205	C₂×C₄×C₂₃	—	2	4	no	—
206	C₂×C₄×C₂₃	—	2	4	no	—
207	C₂×C₄×C₂₃	—	4	2	no	—
208	C₂×C₄×C₂₃	—	4	4	no	—
209	C₂×C₄×C₂₃	—	4	1	no	—
210	C₂×C₄×C₂₃	—	4	16	no	—
211	C₂×C₄×C₂₃	—	4	4	no	—
212	C₂×C₄×C₂₃	—	4	8	no	—
213	C₂×C₄×C₂₃	—	4	2	no	—
214	C₂×C₄×C₂₃	—	4	8	no	—
215	C₂×C₄×C₂₃	—	4	8	no	—
216	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	complemented
217	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
219	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
220	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
221	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
222	C₈×C₂₃	—	2	8	no	—
223	#3496	C₂×C₂₃	1	1	yes	characteristic, complemented
224	C₂×#547	C₂×C₂₃	1	1	yes	characteristic, complemented
225	#3330	C₂×C₂₃	1	8	yes	complemented
226	#3272	C₂×C₂₃	1	8	yes	complemented
227	#3206	C₂×C₂₃	1	2	yes	complemented
228	#3193	C₂×C₂₃	1	4	yes	complemented
229	#3100	C₂×C₂₃	1	2	yes	complemented
230	#3006	C₂×C₂₃	1	4	yes	complemented
231	#2975	C₂×C₂₃	1	1	yes	characteristic, complemented
232	C₂²×C₂₃	—	2	1	no	—
233	C₂²×C₂₃	—	2	1	no	—
234	C₂²×C₂₃	—	2	4	no	—
235	C₂²×C₂₃	—	2	4	no	—
236	C₂²×C₂₃	—	2	4	no	—
237	C₂²×C₂₃	—	2	4	no	—
238	C₂²×C₂₃	—	4	8	no	—
239	C₂²×C₂₃	—	4	2	no	—
240	C₂²×C₂₃	—	4	2	no	—
241	C₂²×C₂₃	—	4	4	no	—
242	C₂²×C₂₃	—	4	4	no	—
243	C₂²×C₂₃	—	4	4	no	—
244	C₂²×C₂₃	—	8	4	no	—
245	C₂²×C₂₃	—	16	8	no	—
246	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
247	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
248	C₄×C₂₃	—	2	2	no	—
249	C₄×C₂₃	—	2	4	no	—
250	C₄×C₂₃	—	2	1	no	—
251	C₄×C₂₃	—	4	4	no	—
252	C₄×C₂₃	—	4	2	no	—
253	C₄×C₂₃	—	4	4	no	—
254	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
255	C₄×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
256	C₂×H(5,𝔽₂)	—	2	1	no	—
257	C₂²×#41	C₂²×C₂₃	1	2	yes	complemented
258	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	8	yes	complemented
259	#547	—	2	4	no	—
260	#547	C₂²×C₂₃	1	8	yes	complemented
261	#547	C₂²×C₂₃	1	4	yes	complemented
262	#546	—	2	4	no	—
263	D₄²	C₂²×C₂₃	1	4	yes	complemented
264	#537	—	2	1	no	—
265	#534	—	2	2	no	—
266	#532	C₂²×C₂₃	1	2	yes	—
267	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
268	C₂×#123	—	2	2	no	—
269	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
270	C₂×#121	—	2	1	no	—
271	C₂×#121	C₂²×C₂₃	1	4	yes	—
272	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
273	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
274	C₂²×C₄⋊C₄	—	2	1	no	—
275	#492	C₂²×C₂₃	1	4	yes	complemented
276	#487	C₂²×C₂₃	1	4	yes	complemented
277	#484	C₂²×C₂₃	1	8	yes	complemented
278	#480	C₂²×C₂₃	1	4	yes	complemented
279	#474	C₂²×C₂₃	1	4	yes	complemented
280	#468	C₂²×C₂₃	1	16	yes	complemented
281	#461	C₂²×C₂₃	1	8	yes	complemented
282	#459	C₂²×C₂₃	1	4	yes	complemented
283	#450	C₂²×C₂₃	1	8	yes	complemented
284	#449	C₂²×C₂₃	1	16	yes	complemented
285	#448	C₂²×C₂₃	1	8	yes	complemented
286	#447	C₂²×C₂₃	1	4	yes	complemented
287	C₄×D₈	C₂²×C₂₃	1	8	yes	complemented
288	#436	C₂²×C₂₃	1	8	yes	complemented
289	#431	C₂²×C₂₃	1	2	yes	complemented
290	#417	C₂²×C₂₃	1	8	yes	complemented
291	#407	C₂²×C₂₃	1	4	yes	complemented
292	C₂×C₂₃	—	2	2	no	—
293	C₂×C₂₃	—	4	4	no	—
294	C₂×C₂₃	—	4	4	no	—
295	C₂×C₂₃	—	8	4	no	—
296	C₂×C₂₃	#3484	1	1	yes	characteristic
297	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
298	C₂×C₂₃	#3586	1	1	yes	characteristic
299	H(5,𝔽₂)	—	4	8	no	—
300	C₂×#41	—	2	4	no	—
301	C₂×#41	—	2	2	no	—
302	C₂×#41	—	2	4	no	—
303	C₂×#41	—	4	8	no	—
304	C₂×#41	C₂³×C₂₃	1	8	yes	—
305	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
306	C₂²×D₄	—	2	2	no	—
307	C₂²×D₄	—	2	4	no	—
308	C₂²×D₄	—	2	4	no	—
309	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
310	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
311	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
312	C₂³×C₄	—	2	1	no	—
313	C₂³×C₄	—	2	2	no	—
314	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
315	AGL(1,ℤ/8)	—	4	16	no	—
316	C₂×SD₁₆	C₂³×C₂₃	1	16	yes	—
317	C₂×D₈	C₂³×C₂₃	1	8	yes	—
318	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
319	C₄⋊Q₈	—	2	2	no	—
320	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
321	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
322	#127	C₂³×C₂₃	1	2	yes	—
323	#124	—	2	2	no	—
324	#123	—	2	16	no	—
325	#123	—	2	8	no	—
326	C₂²⋊Q₈	—	2	4	no	—
327	C₂²⋊Q₈	—	2	4	no	—
328	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
329	#121	—	2	8	no	—
330	#121	—	2	4	no	—
331	#121	—	2	4	no	—
332	#121	—	2	2	no	—
333	#121	—	2	4	no	—
334	#121	C₂³×C₂₃	1	4	yes	—
335	#121	C₂³×C₂₃	1	8	yes	—
336	#120	—	2	8	no	—
337	C₄×D₄	—	2	2	no	—
338	C₄×D₄	—	2	2	no	—
339	C₄×D₄	D₄×C₂₃	1	4	yes	—
340	C₄×D₄	C₂³×C₂₃	1	4	yes	—
341	C₄×D₄	C₂³×C₂₃	1	4	yes	—
342	C₄×D₄	C₂³×C₂₃	1	4	yes	—
343	C₂×C₄⋊C₄	—	2	1	no	—
344	C₂×C₄⋊C₄	—	2	2	no	—
345	C₂×C₄⋊C₄	—	2	1	no	—
346	C₂×C₄⋊C₄	—	2	8	no	—
347	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
348	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
349	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
350	C₂×C₂²⋊C₄	—	2	2	no	—
351	C₂×C₂²⋊C₄	—	2	1	no	—
352	C₂×C₂²⋊C₄	—	2	4	no	—
353	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
354	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
355	#107	C₂³×C₂₃	1	8	yes	—
356	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
357	#103	C₂³×C₂₃	1	16	yes	—
358	#102	C₂³×C₂₃	1	8	yes	—
359	#102	C₂³×C₂₃	1	8	yes	—
360	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
361	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
362	C₄×C₈	C₂³×C₂₃	1	4	yes	—
363	C₂₃	#61585	1	1	yes	characteristic, direct product factor, Sylow
364	C₂⁴	—	2	1	no	—
365	C₂⁴	—	2	4	no	—
366	C₂⁴	C₂×D₄×C₂₃	1	2	yes	—
367	#41	—	2	16	no	—
368	#41	—	4	8	no	—
369	#41	—	4	16	no	—
370	#41	—	4	16	no	—
371	#41	—	4	8	no	—
372	C₂×Q₈	—	2	2	no	—
373	C₂×Q₈	—	2	2	no	—
374	C₂×Q₈	—	4	4	no	—
375	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
376	C₂×D₄	—	2	4	no	—
377	C₂×D₄	—	2	8	no	—
378	C₂×D₄	—	2	4	no	—
379	C₂×D₄	—	2	8	no	—
380	C₂×D₄	—	2	4	no	—
381	C₂×D₄	—	2	1	no	—
382	C₂×D₄	—	2	2	no	—
383	C₂×D₄	—	2	2	no	—
384	C₂×D₄	—	2	4	no	—
385	C₂×D₄	—	4	8	no	—
386	C₂×D₄	—	4	16	no	—
387	C₂×D₄	—	4	8	no	—
388	C₂×D₄	—	4	16	no	—
389	C₂×D₄	C₂×D₄×C₂₃	1	4	yes	—
390	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
391	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	2	4	no	—
395	C₂²×C₄	—	2	4	no	—
396	C₂²×C₄	—	2	8	no	—
397	C₂²×C₄	—	2	4	no	—
398	C₂²×C₄	—	4	4	no	—
399	C₂²×C₄	—	4	8	no	—
400	C₂²×C₄	—	4	4	no	—
401	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
402	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
403	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
404	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
405	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
406	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
407	SD₁₆	—	4	16	no	—
408	D₈	—	2	16	no	—
409	M₁₆	—	4	8	no	—
410	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
411	C₄⋊C₄	—	2	4	no	—
412	C₄⋊C₄	—	2	8	no	—
413	C₄⋊C₄	—	2	4	no	—
414	C₄⋊C₄	—	2	1	no	—
415	C₄⋊C₄	—	2	4	no	—
416	C₄⋊C₄	—	2	2	no	—
417	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
418	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
419	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
420	C₂²⋊C₄	—	2	8	no	—
421	C₂²⋊C₄	—	2	4	no	—
422	C₂²⋊C₄	—	2	4	no	—
423	C₂²⋊C₄	—	2	8	no	—
424	C₂²⋊C₄	—	2	4	no	—
425	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
426	C₄²	C₂×D₄×C₂₃	1	2	yes	—
427	C₄²	C₂⁴×C₂₃	1	2	yes	—
428	C₂³	—	2	4	no	—
429	C₂³	—	2	2	no	—
430	C₂³	—	2	4	no	—
431	C₂³	—	2	4	no	—
432	C₂³	—	4	2	no	—
433	C₂³	—	4	2	no	—
434	C₂³	—	4	2	no	—
435	C₂³	—	4	8	no	—
436	C₂³	—	4	2	no	—
437	C₂³	—	8	4	no	—
438	C₂³	—	8	4	no	—
439	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
440	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
441	Q₈	—	2	8	no	—
442	Q₈	—	4	4	no	—
443	Q₈	—	4	4	no	—
444	D₄	—	2	4	no	—
445	D₄	—	2	8	no	—
446	D₄	—	2	2	no	—
447	D₄	—	2	8	no	—
448	D₄	—	4	16	no	—
449	D₄	—	4	16	no	—
450	D₄	—	4	8	no	—
451	D₄	—	4	4	no	—
452	D₄	—	8	4	no	—
453	D₄	—	8	2	no	—
454	D₄	C₂₃×H(5,𝔽₂)	1	4	yes	complemented
455	C₂×C₄	—	2	8	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	4	4	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	16	no	—
464	C₂×C₄	—	4	4	no	—
465	C₂×C₄	—	4	8	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	4	8	no	—
468	C₂×C₄	—	4	8	no	—
469	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	complemented
470	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
471	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
472	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
473	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
474	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
475	C₈	—	2	8	no	—
476	C₂²	—	2	1	no	—
477	C₂²	—	2	1	no	—
478	C₂²	—	2	4	no	—
479	C₂²	—	2	4	no	—
480	C₂²	—	2	4	no	—
481	C₂²	—	2	4	no	—
482	C₂²	—	4	8	no	—
483	C₂²	—	4	2	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	4	no	—
486	C₂²	—	4	4	no	—
487	C₂²	—	4	4	no	—
488	C₂²	—	8	4	no	—
489	C₂²	—	16	8	no	—
490	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
491	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
492	C₄	—	2	4	no	—
493	C₄	—	2	1	no	—
494	C₄	—	2	2	no	—
495	C₄	—	4	4	no	—
496	C₄	—	4	4	no	—
497	C₄	—	4	2	no	—
498	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	4	yes	—
499	C₄	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
500	C₂	—	2	2	no	—
501	C₂	—	4	4	no	—
502	C₂	—	4	4	no	—
503	C₂	—	8	4	no	—
504	C₂	C₂₃×#3484	1	1	yes	characteristic
505	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
506	C₂	C₂₃×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄×D₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3496	C₂	semidirect
2. C₂×C₂₃×#547	C₂	semidirect
3. C₂₃×#3330	C₂	semidirect
4. C₂₃×#3272	C₂	semidirect
5. C₂₃×#3206	C₂	semidirect
6. C₂₃×#3193	C₂	semidirect
7. C₂₃×#3100	C₂	semidirect
8. C₂₃×#3006	C₂	semidirect
9. C₂₃×#2975	C₂	semidirect
11. C₂²×#41×C₂₃	C₂²	semidirect
12. C₂×C₂₃×AGL(1,ℤ/8)	C₂²	semidirect
14. C₂₃×#547	C₂²	semidirect
15. C₂₃×#547	C₂²	semidirect
17. D₄²×C₂₃	C₂²	semidirect
21. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
23. C₂×C₂₃×C₂²⋊Q₈	C₂²	semidirect
27. C₂×C₄×D₄×C₂₃	C₂²	semidirect
29. C₂₃×#492	C₂²	semidirect
30. C₂₃×#487	C₂²	semidirect
31. C₂₃×#484	C₂²	semidirect
32. C₂₃×#480	C₂²	semidirect
33. C₂₃×#474	C₂²	semidirect
34. C₂₃×#468	C₂²	semidirect
35. C₂₃×#461	C₂²	semidirect
36. C₂₃×#459	C₂²	semidirect
37. C₂₃×#450	C₂²	semidirect
38. C₂₃×#449	C₂²	semidirect
39. C₂₃×#448	C₂²	semidirect
40. C₂₃×#447	C₂²	semidirect
41. C₄×D₈×C₂₃	C₂²	semidirect
42. C₂₃×#436	C₂²	semidirect
43. C₂₃×#431	C₂²	semidirect
44. C₂₃×#417	C₂²	semidirect
45. C₂₃×#407	C₂²	semidirect
174. #61585	C₂₃	direct (N × H)
201. D₄×C₂₃	H(5,𝔽₂)	semidirect
216. C₂×C₄×C₂₃	AGL(1,ℤ/8)	semidirect
223. #3496	C₂×C₂₃	semidirect
224. C₂×#547	C₂×C₂₃	semidirect
225. #3330	C₂×C₂₃	semidirect
226. #3272	C₂×C₂₃	semidirect
227. #3206	C₂×C₂₃	semidirect
228. #3193	C₂×C₂₃	semidirect
229. #3100	C₂×C₂₃	semidirect
230. #3006	C₂×C₂₃	semidirect
231. #2975	C₂×C₂₃	semidirect
257. C₂²×#41	C₂²×C₂₃	semidirect
258. C₂×AGL(1,ℤ/8)	C₂²×C₂₃	semidirect
260. #547	C₂²×C₂₃	semidirect
261. #547	C₂²×C₂₃	semidirect
263. D₄²	C₂²×C₂₃	semidirect
267. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
269. C₂×C₂²⋊Q₈	C₂²×C₂₃	semidirect
273. C₂×C₄×D₄	C₂²×C₂₃	semidirect
275. #492	C₂²×C₂₃	semidirect
276. #487	C₂²×C₂₃	semidirect
277. #484	C₂²×C₂₃	semidirect
278. #480	C₂²×C₂₃	semidirect
279. #474	C₂²×C₂₃	semidirect
280. #468	C₂²×C₂₃	semidirect
281. #461	C₂²×C₂₃	semidirect
282. #459	C₂²×C₂₃	semidirect
283. #450	C₂²×C₂₃	semidirect
284. #449	C₂²×C₂₃	semidirect
285. #448	C₂²×C₂₃	semidirect
286. #447	C₂²×C₂₃	semidirect
287. C₄×D₈	C₂²×C₂₃	semidirect
288. #436	C₂²×C₂₃	semidirect
289. #431	C₂²×C₂₃	semidirect
290. #417	C₂²×C₂₃	semidirect
291. #407	C₂²×C₂₃	semidirect
363. C₂₃	#61585	direct (N × H)
454. D₄	C₂₃×H(5,𝔽₂)	semidirect
469. C₂×C₄	C₂₃×AGL(1,ℤ/8)	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3496	2
C₂×C₂₃×#547	2
C₂₃×#3330	2
C₂₃×#3272	2
C₂₃×#3206	2
C₂₃×#3193	2
C₂₃×#3100	2
C₂₃×#3006	2
C₂₃×#2975	2
#61585	23

Subgroups — C23×#61585 #19209211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#61585 #19209211