Subgroups — C₂₃×#18752 #19166378

Order 5888.

1020 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#554	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3206	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3100	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#3037	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1644	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1622	C₂	1	4	yes	maximal, complemented
7	C₂₃×#1548	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1486	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1479	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×#137	—	2	1	no	—
11	C₂₃×#554	—	2	2	no	—
12	C₂₃×#554	—	2	2	no	—
13	C₂₃×#554	—	2	2	no	—
14	C₂₃×#554	C₂²	1	4	yes	—
15	C₂₃×#532	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#125	—	2	1	no	—
18	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
19	C₂×C₂₃×C₂²⋊Q₈	—	2	2	no	—
20	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₂₃	—	2	1	no	—
22	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
24	C₂₃×#492	C₂²	1	2	yes	complemented
25	C₂₃×#487	C₂²	1	2	yes	complemented
26	C₂₃×#480	—	2	1	no	—
27	C₂₃×#474	C₂²	1	2	yes	—
28	C₂₃×#464	—	2	2	no	—
29	C₂₃×#462	C₂²	1	2	yes	complemented
30	C₂₃×#461	—	2	2	no	—
31	C₂₃×#460	C₂²	1	2	yes	complemented
32	C₂₃×#459	C₂²	1	2	yes	—
33	C₂₃×#448	—	2	2	no	—
34	C₂₃×#447	—	2	1	no	—
35	C₄×D₈×C₂₃	—	2	2	no	—
36	C₂₃×#431	C₂²	1	1	yes	characteristic
37	C₂₃×#431	C₂²	1	1	yes	characteristic
38	C₂₃×#423	C₂²	1	1	yes	characteristic
39	C₂₃×#417	—	2	1	no	—
40	C₂₃×#333	C₂²	1	4	yes	complemented
41	C₂₃×Q₈⋊C₈	C₂²	1	4	yes	complemented
42	C₂₃×D₄⋊C₈	C₂²	1	4	yes	—
43	C₂×#41×C₂₃	—	2	1	no	—
44	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
45	C₂²×Q₈×C₂₃	D₄	1	2	yes	—
46	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₂₃	—	2	2	no	—
48	C₂³×C₄×C₂₃	—	2	1	no	—
49	C₂₃×#137	—	4	4	no	—
50	C₂×Q₁₆×C₂₃	—	2	2	no	—
51	C₂×SD₁₆×C₂₃	—	2	2	no	—
52	C₂×SD₁₆×C₂₃	—	2	4	no	—
53	C₂×D₈×C₂₃	—	2	2	no	—
54	C₂×M₁₆×C₂₃	—	2	1	no	—
55	C₂₃×C₄⋊Q₈	—	2	1	no	—
56	C₂₃×C₄⋊Q₈	—	2	1	no	—
57	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
58	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
59	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
60	C₂₃×#127	C₂³	1	1	yes	characteristic
61	C₂₃×#125	—	2	4	no	—
62	C₂₃×#125	—	2	4	no	—
63	C₂₃×#124	—	2	1	no	—
64	C₂₃×C₂²⋊Q₈	—	2	4	no	—
65	C₂₃×C₂²⋊Q₈	—	2	8	no	—
66	C₂₃×C₂²⋊Q₈	—	2	4	no	—
67	C₂₃×C₂²⋊Q₈	—	2	8	no	—
68	C₂₃×#121	—	2	1	no	—
69	C₂₃×#121	—	2	1	no	—
70	C₄×Q₈×C₂₃	—	2	1	no	—
71	C₄×Q₈×C₂₃	—	2	1	no	—
72	C₄×Q₈×C₂₃	D₄	1	2	yes	—
73	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
74	C₄×D₄×C₂₃	—	2	4	no	—
75	C₄×D₄×C₂₃	—	2	4	no	—
76	C₄×D₄×C₂₃	—	2	1	no	—
77	C₄×D₄×C₂₃	—	2	1	no	—
78	C₄×D₄×C₂₃	—	2	1	no	—
79	C₄×D₄×C₂₃	—	2	1	no	—
80	C₄×D₄×C₂₃	D₄	1	2	yes	—
81	C₄×D₄×C₂₃	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
83	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
84	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
85	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
86	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
87	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
88	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
89	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
91	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
92	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
93	C₂×C₄²×C₂₃	—	2	1	no	—
94	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
95	C₂₃×#107	—	2	2	no	—
96	C₂₃×C₄⋊C₈	—	2	1	no	—
97	C₂₃×C₄⋊C₈	—	2	1	no	—
98	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
99	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
100	C₂₃×#103	—	2	4	no	—
101	C₂₃×#103	—	2	2	no	—
102	C₂₃×#102	—	2	2	no	—
103	C₂₃×#102	—	2	2	no	—
104	C₂₃×#102	—	2	2	no	—
105	C₂₃×C₂²⋊C₈	—	2	1	no	—
106	C₂₃×C₂²⋊C₈	—	2	1	no	—
107	C₄×C₈×C₂₃	—	2	1	no	—
108	C₄×C₈×C₂₃	C₂³	1	2	yes	—
109	C₂⁴×C₂₃	—	2	1	no	—
110	#41×C₂₃	—	4	2	no	—
111	#41×C₂₃	—	4	2	no	—
112	C₂×Q₈×C₂₃	—	2	1	no	—
113	C₂×Q₈×C₂₃	—	2	2	no	—
114	C₂×Q₈×C₂₃	—	2	2	no	—
115	C₂×Q₈×C₂₃	—	2	2	no	—
116	C₂×Q₈×C₂₃	—	2	1	no	—
117	C₂×Q₈×C₂₃	—	8	2	no	—
118	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
119	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
120	C₂×D₄×C₂₃	—	2	4	no	—
121	C₂×D₄×C₂₃	—	2	1	no	—
122	C₂×D₄×C₂₃	—	2	1	no	—
123	C₂×D₄×C₂₃	—	2	1	no	—
124	C₂×D₄×C₂₃	—	2	1	no	—
125	C₂×D₄×C₂₃	—	4	1	no	—
126	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₂₃	—	2	1	no	—
128	C₂²×C₄×C₂₃	—	2	1	no	—
129	C₂²×C₄×C₂₃	—	2	1	no	—
130	C₂²×C₄×C₂₃	—	2	2	no	—
131	C₂²×C₄×C₂₃	—	2	8	no	—
132	C₂²×C₄×C₂₃	—	2	2	no	—
133	C₂²×C₄×C₂₃	—	2	2	no	—
134	C₂²×C₄×C₂₃	—	4	2	no	—
135	C₂²×C₄×C₂₃	—	4	2	no	—
136	C₂²×C₄×C₂₃	—	4	1	no	—
137	C₂²×C₄×C₂₃	—	8	4	no	—
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	Q₁₆×C₂₃	—	4	4	no	—
142	SD₁₆×C₂₃	—	4	4	no	—
143	SD₁₆×C₂₃	—	8	4	no	—
144	D₈×C₂₃	—	4	4	no	—
145	M₁₆×C₂₃	—	4	2	no	—
146	C₂×C₈×C₂₃	—	2	2	no	—
147	C₂×C₈×C₂₃	—	2	2	no	—
148	C₂×C₈×C₂₃	—	2	2	no	—
149	C₄⋊C₄×C₂₃	—	2	8	no	—
150	C₄⋊C₄×C₂₃	—	2	8	no	—
151	C₄⋊C₄×C₂₃	—	2	1	no	—
152	C₄⋊C₄×C₂₃	—	2	2	no	—
153	C₄⋊C₄×C₂₃	—	2	2	no	—
154	C₄⋊C₄×C₂₃	—	2	2	no	—
155	C₄⋊C₄×C₂₃	—	2	1	no	—
156	C₄⋊C₄×C₂₃	—	2	2	no	—
157	C₄⋊C₄×C₂₃	—	2	2	no	—
158	C₄⋊C₄×C₂₃	—	2	2	no	—
159	C₄⋊C₄×C₂₃	—	2	1	no	—
160	C₄⋊C₄×C₂₃	—	2	1	no	—
161	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
162	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
163	C₂²⋊C₄×C₂₃	—	2	8	no	—
164	C₂²⋊C₄×C₂₃	—	2	2	no	—
165	C₂²⋊C₄×C₂₃	—	2	2	no	—
166	C₂²⋊C₄×C₂₃	—	4	1	no	—
167	C₂²⋊C₄×C₂₃	—	4	1	no	—
168	C₄²×C₂₃	—	2	2	no	—
169	C₄²×C₂₃	—	2	2	no	—
170	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
171	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
172	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
173	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
174	#18752	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	—	4	2	no	—
178	C₂³×C₂₃	—	4	2	no	—
179	C₂³×C₂₃	—	4	1	no	—
180	C₂³×C₂₃	#120	1	1	yes	characteristic
181	Q₈×C₂₃	—	2	2	no	—
182	Q₈×C₂₃	—	2	2	no	—
183	Q₈×C₂₃	—	4	4	no	—
184	Q₈×C₂₃	—	4	2	no	—
185	Q₈×C₂₃	—	4	1	no	—
186	Q₈×C₂₃	—	4	1	no	—
187	Q₈×C₂₃	—	8	2	no	—
188	D₄×C₂₃	—	2	4	no	—
189	D₄×C₂₃	—	2	2	no	—
190	D₄×C₂₃	—	4	2	no	—
191	D₄×C₂₃	—	4	1	no	—
192	D₄×C₂₃	—	4	1	no	—
193	D₄×C₂₃	—	4	2	no	—
194	D₄×C₂₃	—	8	2	no	—
195	D₄×C₂₃	#137	1	4	yes	complemented
196	C₂×C₄×C₂₃	—	2	2	no	—
197	C₂×C₄×C₂₃	—	2	2	no	—
198	C₂×C₄×C₂₃	—	2	2	no	—
199	C₂×C₄×C₂₃	—	2	2	no	—
200	C₂×C₄×C₂₃	—	2	4	no	—
201	C₂×C₄×C₂₃	—	4	1	no	—
202	C₂×C₄×C₂₃	—	4	2	no	—
203	C₂×C₄×C₂₃	—	4	4	no	—
204	C₂×C₄×C₂₃	—	4	2	no	—
205	C₂×C₄×C₂₃	—	4	2	no	—
206	C₂×C₄×C₂₃	—	4	2	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	—	8	8	no	—
211	C₂×C₄×C₂₃	—	8	2	no	—
212	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
213	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
214	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
215	C₂×C₄×C₂₃	#135	1	2	yes	—
216	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
217	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
220	C₈×C₂₃	—	4	2	no	—
221	C₈×C₂₃	—	4	2	no	—
222	C₈×C₂₃	—	4	2	no	—
223	C₂×#554	C₂×C₂₃	1	1	yes	characteristic, complemented
224	#3206	C₂×C₂₃	1	1	yes	characteristic, complemented
225	#3100	C₂×C₂₃	1	1	yes	characteristic
226	#3037	C₂×C₂₃	1	1	yes	characteristic, complemented
227	#1644	C₂×C₂₃	1	4	yes	complemented
228	#1622	C₂×C₂₃	1	4	yes	complemented
229	#1548	C₂×C₂₃	1	1	yes	characteristic, complemented
230	#1486	C₂×C₂₃	1	1	yes	characteristic, complemented
231	#1479	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	—	4	8	no	—
236	C₂²×C₂₃	—	4	2	no	—
237	C₂²×C₂₃	—	4	2	no	—
238	C₂²×C₂₃	—	8	1	no	—
239	C₂²×C₂₃	—	8	1	no	—
240	C₂²×C₂₃	—	8	1	no	—
241	C₂²×C₂₃	#449	1	2	yes	—
242	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
243	C₄×C₂₃	—	2	1	no	—
244	C₄×C₂₃	—	2	1	no	—
245	C₄×C₂₃	—	2	1	no	—
246	C₄×C₂₃	—	4	2	no	—
247	C₄×C₂₃	—	4	2	no	—
248	C₄×C₂₃	—	4	2	no	—
249	C₄×C₂₃	—	4	4	no	—
250	C₄×C₂₃	—	8	1	no	—
251	C₄×C₂₃	#449	1	2	yes	—
252	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
253	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
254	C₄×C₂₃	C₂×#137	1	2	yes	—
255	C₂×#137	—	2	1	no	—
256	#554	—	2	2	no	—
257	#554	—	2	2	no	—
258	#554	—	2	2	no	—
259	#554	C₂²×C₂₃	1	4	yes	—
260	#532	C₂²×C₂₃	1	1	yes	characteristic
261	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
262	C₂×#125	—	2	1	no	—
263	C₂×C₂²⋊Q₈	—	2	1	no	—
264	C₂×C₂²⋊Q₈	—	2	2	no	—
265	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
266	C₂×C₄×D₄	—	2	1	no	—
267	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
268	C₂²×C₄⋊C₄	—	2	1	no	—
269	#492	C₂²×C₂₃	1	2	yes	complemented
270	#487	C₂²×C₂₃	1	2	yes	complemented
271	#480	—	2	1	no	—
272	#474	C₂²×C₂₃	1	2	yes	—
273	#464	—	2	2	no	—
274	#462	C₂²×C₂₃	1	2	yes	complemented
275	#461	—	2	2	no	—
276	#460	C₂²×C₂₃	1	2	yes	complemented
277	#459	C₂²×C₂₃	1	2	yes	—
278	#448	—	2	2	no	—
279	#447	—	2	1	no	—
280	C₄×D₈	—	2	2	no	—
281	#431	C₂²×C₂₃	1	1	yes	characteristic
282	#431	C₂²×C₂₃	1	1	yes	characteristic
283	#423	C₂²×C₂₃	1	1	yes	characteristic
284	#417	—	2	1	no	—
285	#333	C₂²×C₂₃	1	4	yes	complemented
286	Q₈⋊C₈	C₂²×C₂₃	1	4	yes	complemented
287	D₄⋊C₈	C₂²×C₂₃	1	4	yes	—
288	C₂×C₂₃	—	2	2	no	—
289	C₂×C₂₃	—	4	4	no	—
290	C₂×C₂₃	—	16	1	no	—
291	C₂×C₂₃	C₂×#449	1	1	yes	characteristic
292	C₂×C₂₃	#3004	1	1	yes	characteristic
293	C₂×C₂₃	#3019	1	1	yes	characteristic
294	C₂×#41	—	2	1	no	—
295	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
296	C₂²×Q₈	D₄×C₂₃	1	2	yes	—
297	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
298	C₂³×C₄	—	2	2	no	—
299	C₂³×C₄	—	2	1	no	—
300	#137	—	4	4	no	—
301	C₂×Q₁₆	—	2	2	no	—
302	C₂×SD₁₆	—	2	2	no	—
303	C₂×SD₁₆	—	2	4	no	—
304	C₂×D₈	—	2	2	no	—
305	C₂×M₁₆	—	2	1	no	—
306	C₄⋊Q₈	—	2	1	no	—
307	C₄⋊Q₈	—	2	1	no	—
308	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
309	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
310	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
311	#127	C₂³×C₂₃	1	1	yes	characteristic
312	#125	—	2	4	no	—
313	#125	—	2	4	no	—
314	#124	—	2	1	no	—
315	C₂²⋊Q₈	—	2	4	no	—
316	C₂²⋊Q₈	—	2	8	no	—
317	C₂²⋊Q₈	—	2	4	no	—
318	C₂²⋊Q₈	—	2	8	no	—
319	#121	—	2	1	no	—
320	#121	—	2	1	no	—
321	C₄×Q₈	—	2	1	no	—
322	C₄×Q₈	—	2	1	no	—
323	C₄×Q₈	D₄×C₂₃	1	2	yes	—
324	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
325	C₄×D₄	—	2	4	no	—
326	C₄×D₄	—	2	4	no	—
327	C₄×D₄	—	2	1	no	—
328	C₄×D₄	—	2	1	no	—
329	C₄×D₄	—	2	1	no	—
330	C₄×D₄	—	2	1	no	—
331	C₄×D₄	D₄×C₂₃	1	2	yes	—
332	C₄×D₄	C₂³×C₂₃	1	2	yes	—
333	C₂×C₄⋊C₄	—	2	2	no	—
334	C₂×C₄⋊C₄	—	2	4	no	—
335	C₂×C₄⋊C₄	—	2	2	no	—
336	C₂×C₄⋊C₄	—	2	1	no	—
337	C₂×C₄⋊C₄	—	2	1	no	—
338	C₂×C₄⋊C₄	—	2	1	no	—
339	C₂×C₄⋊C₄	—	2	4	no	—
340	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
341	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
342	C₂×C₂²⋊C₄	—	2	2	no	—
343	C₂×C₂²⋊C₄	—	2	1	no	—
344	C₂×C₄²	—	2	1	no	—
345	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
346	#107	—	2	2	no	—
347	C₄⋊C₈	—	2	1	no	—
348	C₄⋊C₈	—	2	1	no	—
349	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
350	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
351	#103	—	2	4	no	—
352	#103	—	2	2	no	—
353	#102	—	2	2	no	—
354	#102	—	2	2	no	—
355	#102	—	2	2	no	—
356	C₂²⋊C₈	—	2	1	no	—
357	C₂²⋊C₈	—	2	1	no	—
358	C₄×C₈	—	2	1	no	—
359	C₄×C₈	C₂³×C₂₃	1	2	yes	—
360	C₂₃	#18752	1	1	yes	characteristic, direct product factor, Sylow
361	C₂⁴	—	2	1	no	—
362	#41	—	4	2	no	—
363	#41	—	4	2	no	—
364	C₂×Q₈	—	2	1	no	—
365	C₂×Q₈	—	2	2	no	—
366	C₂×Q₈	—	2	2	no	—
367	C₂×Q₈	—	2	2	no	—
368	C₂×Q₈	—	2	1	no	—
369	C₂×Q₈	—	8	2	no	—
370	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
371	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
372	C₂×D₄	—	2	4	no	—
373	C₂×D₄	—	2	1	no	—
374	C₂×D₄	—	2	1	no	—
375	C₂×D₄	—	2	1	no	—
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	—	4	1	no	—
378	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	8	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	4	2	no	—
387	C₂²×C₄	—	4	2	no	—
388	C₂²×C₄	—	4	1	no	—
389	C₂²×C₄	—	8	4	no	—
390	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
391	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
392	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
393	Q₁₆	—	4	4	no	—
394	SD₁₆	—	4	4	no	—
395	SD₁₆	—	8	4	no	—
396	D₈	—	4	4	no	—
397	M₁₆	—	4	2	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	2	2	no	—
401	C₄⋊C₄	—	2	8	no	—
402	C₄⋊C₄	—	2	8	no	—
403	C₄⋊C₄	—	2	1	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	2	1	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	—	2	2	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	1	no	—
412	C₄⋊C₄	—	2	1	no	—
413	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
414	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
415	C₂²⋊C₄	—	2	8	no	—
416	C₂²⋊C₄	—	2	2	no	—
417	C₂²⋊C₄	—	2	2	no	—
418	C₂²⋊C₄	—	4	1	no	—
419	C₂²⋊C₄	—	4	1	no	—
420	C₄²	—	2	2	no	—
421	C₄²	—	2	2	no	—
422	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
423	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
424	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
425	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
426	C₂³	—	2	4	no	—
427	C₂³	—	2	2	no	—
428	C₂³	—	4	2	no	—
429	C₂³	—	4	2	no	—
430	C₂³	—	4	1	no	—
431	C₂³	C₂₃×#120	1	1	yes	characteristic
432	Q₈	—	2	2	no	—
433	Q₈	—	2	2	no	—
434	Q₈	—	4	4	no	—
435	Q₈	—	4	2	no	—
436	Q₈	—	4	1	no	—
437	Q₈	—	4	1	no	—
438	Q₈	—	8	2	no	—
439	D₄	—	2	4	no	—
440	D₄	—	2	2	no	—
441	D₄	—	4	2	no	—
442	D₄	—	4	1	no	—
443	D₄	—	4	1	no	—
444	D₄	—	4	2	no	—
445	D₄	—	8	2	no	—
446	D₄	C₂₃×#137	1	4	yes	complemented
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	4	4	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	4	4	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	8	8	no	—
462	C₂×C₄	—	8	2	no	—
463	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
464	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
465	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
466	C₂×C₄	C₂₃×#135	1	2	yes	—
467	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
468	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
469	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
470	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
471	C₈	—	4	2	no	—
472	C₈	—	4	2	no	—
473	C₈	—	4	2	no	—
474	C₂²	—	2	1	no	—
475	C₂²	—	2	1	no	—
476	C₂²	—	2	4	no	—
477	C₂²	—	4	8	no	—
478	C₂²	—	4	2	no	—
479	C₂²	—	4	2	no	—
480	C₂²	—	8	1	no	—
481	C₂²	—	8	1	no	—
482	C₂²	—	8	1	no	—
483	C₂²	C₂₃×#449	1	2	yes	—
484	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
485	C₄	—	2	1	no	—
486	C₄	—	2	1	no	—
487	C₄	—	2	1	no	—
488	C₄	—	4	2	no	—
489	C₄	—	4	2	no	—
490	C₄	—	4	2	no	—
491	C₄	—	4	4	no	—
492	C₄	—	8	1	no	—
493	C₄	C₂₃×#449	1	2	yes	—
494	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
495	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
496	C₄	C₂×C₂₃×#137	1	2	yes	—
497	C₂	—	2	2	no	—
498	C₂	—	4	4	no	—
499	C₂	—	16	1	no	—
500	C₂	C₂×C₂₃×#449	1	1	yes	characteristic
501	C₂	C₂₃×#3004	1	1	yes	characteristic
502	C₂	C₂₃×#3019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3100	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#554	C₂	semidirect
2. C₂₃×#3206	C₂	semidirect
4. C₂₃×#3037	C₂	semidirect
5. C₂₃×#1644	C₂	semidirect
6. C₂₃×#1622	C₂	semidirect
7. C₂₃×#1548	C₂	semidirect
8. C₂₃×#1486	C₂	semidirect
9. C₂₃×#1479	C₂	semidirect
24. C₂₃×#492	C₂²	semidirect
25. C₂₃×#487	C₂²	semidirect
29. C₂₃×#462	C₂²	semidirect
31. C₂₃×#460	C₂²	semidirect
40. C₂₃×#333	C₂²	semidirect
41. C₂₃×Q₈⋊C₈	C₂²	semidirect
174. #18752	C₂₃	direct (N × H)
195. D₄×C₂₃	#137	semidirect
223. C₂×#554	C₂×C₂₃	semidirect
224. #3206	C₂×C₂₃	semidirect
226. #3037	C₂×C₂₃	semidirect
227. #1644	C₂×C₂₃	semidirect
228. #1622	C₂×C₂₃	semidirect
229. #1548	C₂×C₂₃	semidirect
230. #1486	C₂×C₂₃	semidirect
231. #1479	C₂×C₂₃	semidirect
269. #492	C₂²×C₂₃	semidirect
270. #487	C₂²×C₂₃	semidirect
274. #462	C₂²×C₂₃	semidirect
276. #460	C₂²×C₂₃	semidirect
285. #333	C₂²×C₂₃	semidirect
286. Q₈⋊C₈	C₂²×C₂₃	semidirect
360. C₂₃	#18752	direct (N × H)
446. D₄	C₂₃×#137	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#554	2
C₂₃×#3206	2
C₂₃×#3100	2
C₂₃×#3037	2
C₂₃×#1644	2
C₂₃×#1622	2
C₂₃×#1548	2
C₂₃×#1486	2
C₂₃×#1479	2
#18752	23

Subgroups — C23×#18752 #19166378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18752 #19166378