Subgroups — C₂₃×#60712 #19208338

Order 5888.

1950 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3509	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3508	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3182	C₂	1	4	yes	maximal, complemented
4	C₂₃×#3174	C₂	1	8	yes	maximal, complemented
5	C₂₃×#3167	C₂	1	4	yes	maximal, complemented
6	C₂₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#3139	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#3138	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#3057	C₂	1	4	yes	maximal, complemented
11	C₂₃×#2949	C₂	1	4	yes	maximal, complemented
12	C₄×C₂₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×SD₁₆×C₂₃	C₂²	1	4	yes	—
14	C₂×C₂₃×#131	C₂²	1	2	yes	—
15	C₂₃×#559	—	2	2	no	—
16	C₂₃×#557	—	2	2	no	—
17	C₂₃×#556	—	2	2	no	—
18	C₂₃×#552	—	2	4	no	—
19	C₂₃×#550	—	2	2	no	—
20	C₂₃×#546	—	2	4	no	—
21	C₂₃×#541	C₂²	1	4	yes	complemented
22	C₂₃×#540	C₂²	1	4	yes	—
23	C₂₃×#536	C₂²	1	2	yes	—
24	C₂₃×#535	C₂²	1	2	yes	—
25	C₂₃×#533	C₂²	1	1	yes	characteristic
26	C₂₃×#533	C₂²	1	1	yes	characteristic
27	C₂₃×#533	C₂²	1	4	yes	complemented
28	C₂₃×#532	C₂²	1	4	yes	—
29	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
31	C₂₃×#519	C₂²	1	2	yes	—
32	C₂₃×#518	C₂²	1	2	yes	—
33	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
34	C₂₃×#500	C₂²	1	2	yes	complemented
35	C₂₃×#499	C₂²	1	4	yes	complemented
36	C₂₃×#498	C₂²	1	2	yes	complemented
37	C₂₃×#492	C₂²	1	2	yes	—
38	C₂₃×#492	C₂²	1	4	yes	—
39	C₂₃×#492	C₂²	1	2	yes	—
40	C₂₃×#491	C₂²	1	4	yes	complemented
41	C₂₃×#490	C₂²	1	4	yes	—
42	C₂₃×#487	C₂²	1	4	yes	complemented
43	C₂₃×#486	C₂²	1	4	yes	—
44	C₂₃×#468	C₂²	1	16	yes	complemented
45	C₂₃×#465	C₂²	1	16	yes	complemented
46	C₂₃×#440	C₂²	1	16	yes	complemented
47	C₄×SD₁₆×C₂₃	C₂²	1	16	yes	complemented
48	C₂₃×#427	C₂²	1	4	yes	—
49	C₂₃×#419	C₂²	1	4	yes	—
50	C₂₃×#418	C₂²	1	4	yes	—
51	C₂₃×#405	C₂²	1	2	yes	—
52	C₂₃×#405	C₂²	1	4	yes	complemented
53	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
54	C₄×M₁₆×C₂₃	C₂²	1	2	yes	complemented
55	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
56	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	complemented
57	C₂×#41×C₂₃	—	2	4	no	—
58	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
59	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
60	C₂²×D₄×C₂₃	—	2	2	no	—
61	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
62	C₂×SD₁₆×C₂₃	—	2	16	no	—
63	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
64	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
65	C₂₃×#131	—	2	8	no	—
66	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
67	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
68	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
69	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
70	C₂₃×C₄⋊Q₈	—	2	2	no	—
71	C₂₃×C₄⋊Q₈	—	2	2	no	—
72	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
73	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
74	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
75	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
76	C₂₃×#127	—	2	2	no	—
77	C₂₃×#127	—	2	2	no	—
78	C₂₃×#127	C₂³	1	2	yes	—
79	C₂₃×#127	C₂³	1	2	yes	—
80	C₂₃×#126	—	2	8	no	—
81	C₂₃×#125	—	2	8	no	—
82	C₂₃×#125	—	2	4	no	—
83	C₂₃×#125	C₂³	1	2	yes	—
84	C₂₃×#125	C₂³	1	2	yes	—
85	C₂₃×#124	—	2	2	no	—
86	C₂₃×#124	—	2	2	no	—
87	C₂₃×#124	—	2	4	no	—
88	C₂₃×#123	—	2	4	no	—
89	C₂₃×#123	—	2	8	no	—
90	C₂₃×C₂²⋊Q₈	—	2	2	no	—
91	C₂₃×C₂²⋊Q₈	—	2	4	no	—
92	C₂₃×C₂²⋊Q₈	—	2	2	no	—
93	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
94	C₂₃×#121	—	2	4	no	—
95	C₂₃×#121	—	2	8	no	—
96	C₂₃×#121	C₂³	1	8	yes	—
97	C₂₃×#120	—	2	4	no	—
98	C₄×Q₈×C₂₃	—	2	4	no	—
99	C₄×Q₈×C₂₃	—	2	2	no	—
100	C₄×Q₈×C₂₃	—	2	4	no	—
101	C₄×Q₈×C₂₃	—	2	2	no	—
102	C₄×Q₈×C₂₃	D₄	1	2	yes	—
103	C₄×Q₈×C₂₃	C₂³	1	8	yes	—
104	C₄×D₄×C₂₃	—	2	4	no	—
105	C₄×D₄×C₂₃	—	2	4	no	—
106	C₄×D₄×C₂₃	D₄	1	2	yes	—
107	C₄×D₄×C₂₃	D₄	1	4	yes	—
108	C₄×D₄×C₂₃	C₂³	1	8	yes	—
109	C₂₃×#117	—	2	4	no	—
110	C₂₃×#117	—	2	4	no	—
111	C₂₃×#117	C₂³	1	2	yes	—
112	C₂₃×#117	C₂³	1	2	yes	—
113	C₂₃×#117	C₂³	1	1	yes	characteristic
114	C₂₃×#117	C₂³	1	2	yes	—
115	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
116	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
117	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
119	C₂₃×#107	C₂³	1	8	yes	—
120	C₂₃×#106	C₂³	1	8	yes	—
121	C₂₃×#103	C₂³	1	16	yes	—
122	C₂₃×#102	C₂³	1	16	yes	—
123	C₂₃×#97	C₂³	1	2	yes	—
124	C₂₃×#97	C₂³	1	4	yes	—
125	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
126	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
127	C₄×C₈×C₂₃	C₂³	1	2	yes	—
128	C₄×C₈×C₂₃	C₂³	1	2	yes	—
129	C₄×C₈×C₂₃	C₂³	1	4	yes	—
130	C₂⁴×C₂₃	—	2	2	no	—
131	#41×C₂₃	—	2	4	no	—
132	#41×C₂₃	—	8	8	no	—
133	C₂×Q₈×C₂₃	—	2	2	no	—
134	C₂×Q₈×C₂₃	—	2	2	no	—
135	C₂×Q₈×C₂₃	—	2	2	no	—
136	C₂×Q₈×C₂₃	—	2	8	no	—
137	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
138	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
139	C₂×D₄×C₂₃	—	2	8	no	—
140	C₂×D₄×C₂₃	—	2	2	no	—
141	C₂×D₄×C₂₃	—	2	4	no	—
142	C₂×D₄×C₂₃	—	2	4	no	—
143	C₂×D₄×C₂₃	—	2	8	no	—
144	C₂×D₄×C₂₃	—	8	4	no	—
145	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
146	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
147	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₂₃	—	2	2	no	—
149	C₂²×C₄×C₂₃	—	2	8	no	—
150	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
151	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
152	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
153	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
154	SD₁₆×C₂₃	—	2	16	no	—
155	SD₁₆×C₂₃	—	2	16	no	—
156	M₁₆×C₂₃	—	2	4	no	—
157	M₁₆×C₂₃	—	2	8	no	—
158	C₂×C₈×C₂₃	—	2	8	no	—
159	C₂×C₈×C₂₃	#41	1	8	yes	complemented
160	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
161	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
162	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
163	C₄⋊C₄×C₂₃	—	2	4	no	—
164	C₄⋊C₄×C₂₃	—	2	4	no	—
165	C₄⋊C₄×C₂₃	—	2	2	no	—
166	C₄⋊C₄×C₂₃	—	2	8	no	—
167	C₄⋊C₄×C₂₃	—	2	2	no	—
168	C₄⋊C₄×C₂₃	—	2	4	no	—
169	C₄⋊C₄×C₂₃	—	2	4	no	—
170	C₄⋊C₄×C₂₃	—	2	8	no	—
171	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
172	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
173	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
174	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
175	C₂²⋊C₄×C₂₃	—	2	2	no	—
176	C₂²⋊C₄×C₂₃	—	2	2	no	—
177	C₂²⋊C₄×C₂₃	—	2	8	no	—
178	C₂²⋊C₄×C₂₃	—	2	8	no	—
179	C₂²⋊C₄×C₂₃	—	2	8	no	—
180	C₂²⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
181	C₂²⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
182	C₄²×C₂₃	—	2	8	no	—
183	C₄²×C₂₃	—	2	2	no	—
184	C₄²×C₂₃	—	2	2	no	—
185	C₄²×C₂₃	C₂×D₄	1	4	yes	—
186	C₄²×C₂₃	C₂×D₄	1	2	yes	—
187	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
188	C₄²×C₂₃	C₂⁴	1	2	yes	—
189	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
190	#60712	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
191	C₂³×C₂₃	—	2	4	no	—
192	C₂³×C₂₃	—	4	4	no	—
193	C₂³×C₂₃	—	8	4	no	—
194	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
195	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
196	Q₈×C₂₃	—	2	2	no	—
197	Q₈×C₂₃	—	2	8	no	—
198	Q₈×C₂₃	—	4	4	no	—
199	Q₈×C₂₃	—	4	4	no	—
200	Q₈×C₂₃	—	4	4	no	—
201	D₄×C₂₃	—	2	4	no	—
202	D₄×C₂₃	—	2	2	no	—
203	D₄×C₂₃	—	2	8	no	—
204	D₄×C₂₃	—	4	4	no	—
205	D₄×C₂₃	—	4	8	no	—
206	D₄×C₂₃	—	8	4	no	—
207	D₄×C₂₃	—	8	8	no	—
208	C₂×C₄×C₂₃	—	2	4	no	—
209	C₂×C₄×C₂₃	—	2	8	no	—
210	C₂×C₄×C₂₃	—	2	4	no	—
211	C₂×C₄×C₂₃	—	2	4	no	—
212	C₂×C₄×C₂₃	—	2	4	no	—
213	C₂×C₄×C₂₃	—	2	4	no	—
214	C₂×C₄×C₂₃	—	4	4	no	—
215	C₂×C₄×C₂₃	—	8	8	no	—
216	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
217	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
218	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
219	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
221	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
222	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
223	C₈×C₂₃	—	2	4	no	—
224	C₈×C₂₃	C₂×#41	1	4	yes	—
225	C₈×C₂₃	C₂×#41	1	4	yes	—
226	#3509	C₂×C₂₃	1	1	yes	characteristic, complemented
227	#3508	C₂×C₂₃	1	1	yes	characteristic
228	#3182	C₂×C₂₃	1	4	yes	complemented
229	#3174	C₂×C₂₃	1	8	yes	complemented
230	#3167	C₂×C₂₃	1	4	yes	complemented
231	#3160	C₂×C₂₃	1	1	yes	characteristic, complemented
232	C₂×#492	C₂×C₂₃	1	1	yes	characteristic, complemented
233	#3139	C₂×C₂₃	1	1	yes	characteristic, complemented
234	#3138	C₂×C₂₃	1	1	yes	characteristic, complemented
235	#3057	C₂×C₂₃	1	4	yes	complemented
236	#2949	C₂×C₂₃	1	4	yes	complemented
237	C₄×#131	C₂×C₂₃	1	1	yes	characteristic, complemented
238	C₂²×C₂₃	—	2	2	no	—
239	C₂²×C₂₃	—	2	2	no	—
240	C₂²×C₂₃	—	2	4	no	—
241	C₂²×C₂₃	—	4	4	no	—
242	C₂²×C₂₃	—	4	8	no	—
243	C₂²×C₂₃	—	16	4	no	—
244	C₂²×C₂₃	#532	1	2	yes	—
245	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
246	C₄×C₂₃	—	2	2	no	—
247	C₄×C₂₃	—	2	4	no	—
248	C₄×C₂₃	—	2	2	no	—
249	C₄×C₂₃	—	4	4	no	—
250	C₄×C₂₃	—	4	4	no	—
251	C₄×C₂₃	—	4	4	no	—
252	C₄×C₂₃	#532	1	2	yes	—
253	C₄×C₂₃	#577	1	4	yes	—
254	C₄×C₂₃	C₂²×#41	1	2	yes	—
255	C₂²×SD₁₆	C₂²×C₂₃	1	4	yes	—
256	C₂×#131	C₂²×C₂₃	1	2	yes	—
257	#559	—	2	2	no	—
258	#557	—	2	2	no	—
259	#556	—	2	2	no	—
260	#552	—	2	4	no	—
261	#550	—	2	2	no	—
262	#546	—	2	4	no	—
263	#541	C₂²×C₂₃	1	4	yes	complemented
264	#540	C₂²×C₂₃	1	4	yes	—
265	#536	C₂²×C₂₃	1	2	yes	—
266	#535	C₂²×C₂₃	1	2	yes	—
267	#533	C₂²×C₂₃	1	1	yes	characteristic
268	#533	C₂²×C₂₃	1	1	yes	characteristic
269	#533	C₂²×C₂₃	1	4	yes	complemented
270	#532	C₂²×C₂₃	1	4	yes	—
271	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
272	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
273	#519	C₂²×C₂₃	1	2	yes	—
274	#518	C₂²×C₂₃	1	2	yes	—
275	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
276	#500	C₂²×C₂₃	1	2	yes	complemented
277	#499	C₂²×C₂₃	1	4	yes	complemented
278	#498	C₂²×C₂₃	1	2	yes	complemented
279	#492	C₂²×C₂₃	1	2	yes	—
280	#492	C₂²×C₂₃	1	4	yes	—
281	#492	C₂²×C₂₃	1	2	yes	—
282	#491	C₂²×C₂₃	1	4	yes	complemented
283	#490	C₂²×C₂₃	1	4	yes	—
284	#487	C₂²×C₂₃	1	4	yes	complemented
285	#486	C₂²×C₂₃	1	4	yes	—
286	#468	C₂²×C₂₃	1	16	yes	complemented
287	#465	C₂²×C₂₃	1	16	yes	complemented
288	#440	C₂²×C₂₃	1	16	yes	complemented
289	C₄×SD₁₆	C₂²×C₂₃	1	16	yes	complemented
290	#427	C₂²×C₂₃	1	4	yes	—
291	#419	C₂²×C₂₃	1	4	yes	—
292	#418	C₂²×C₂₃	1	4	yes	—
293	#405	C₂²×C₂₃	1	2	yes	—
294	#405	C₂²×C₂₃	1	4	yes	complemented
295	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
296	C₄×M₁₆	C₂²×C₂₃	1	2	yes	complemented
297	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
298	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	complemented
299	C₂×C₂₃	—	2	2	no	—
300	C₂×C₂₃	—	4	2	no	—
301	C₂×C₂₃	—	8	4	no	—
302	C₂×C₂₃	C₂×#532	1	1	yes	characteristic
303	C₂×C₂₃	C₂×#577	1	2	yes	—
304	C₂×#41	—	2	4	no	—
305	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
306	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
307	C₂²×D₄	—	2	2	no	—
308	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
309	C₂×SD₁₆	—	2	16	no	—
310	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
311	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
312	#131	—	2	8	no	—
313	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
314	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
315	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
316	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
317	C₄⋊Q₈	—	2	2	no	—
318	C₄⋊Q₈	—	2	2	no	—
319	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
320	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
321	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
322	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
323	#127	—	2	2	no	—
324	#127	—	2	2	no	—
325	#127	C₂³×C₂₃	1	2	yes	—
326	#127	C₂³×C₂₃	1	2	yes	—
327	#126	—	2	8	no	—
328	#125	—	2	8	no	—
329	#125	—	2	4	no	—
330	#125	C₂³×C₂₃	1	2	yes	—
331	#125	C₂³×C₂₃	1	2	yes	—
332	#124	—	2	2	no	—
333	#124	—	2	2	no	—
334	#124	—	2	4	no	—
335	#123	—	2	4	no	—
336	#123	—	2	8	no	—
337	C₂²⋊Q₈	—	2	2	no	—
338	C₂²⋊Q₈	—	2	2	no	—
339	C₂²⋊Q₈	—	2	4	no	—
340	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
341	#121	—	2	4	no	—
342	#121	—	2	8	no	—
343	#121	C₂³×C₂₃	1	8	yes	—
344	#120	—	2	4	no	—
345	C₄×Q₈	—	2	4	no	—
346	C₄×Q₈	—	2	2	no	—
347	C₄×Q₈	—	2	2	no	—
348	C₄×Q₈	—	2	4	no	—
349	C₄×Q₈	D₄×C₂₃	1	2	yes	—
350	C₄×Q₈	C₂³×C₂₃	1	8	yes	—
351	C₄×D₄	—	2	4	no	—
352	C₄×D₄	—	2	4	no	—
353	C₄×D₄	D₄×C₂₃	1	4	yes	—
354	C₄×D₄	D₄×C₂₃	1	2	yes	—
355	C₄×D₄	C₂³×C₂₃	1	8	yes	—
356	#117	—	2	4	no	—
357	#117	—	2	4	no	—
358	#117	C₂³×C₂₃	1	2	yes	—
359	#117	C₂³×C₂₃	1	2	yes	—
360	#117	C₂³×C₂₃	1	1	yes	characteristic
361	#117	C₂³×C₂₃	1	2	yes	—
362	C₂×C₄⋊C₄	—	2	2	no	—
363	C₂×C₂²⋊C₄	—	2	4	no	—
364	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
365	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
366	#107	C₂³×C₂₃	1	8	yes	—
367	#106	C₂³×C₂₃	1	8	yes	—
368	#103	C₂³×C₂₃	1	16	yes	—
369	#102	C₂³×C₂₃	1	16	yes	—
370	#97	C₂³×C₂₃	1	2	yes	—
371	#97	C₂³×C₂₃	1	4	yes	—
372	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
373	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
374	C₄×C₈	C₂³×C₂₃	1	2	yes	—
375	C₄×C₈	C₂³×C₂₃	1	2	yes	—
376	C₄×C₈	C₂³×C₂₃	1	4	yes	—
377	C₂₃	#60712	1	1	yes	characteristic, direct product factor, Sylow
378	C₂⁴	—	2	2	no	—
379	#41	—	2	4	no	—
380	#41	—	8	8	no	—
381	C₂×Q₈	—	2	2	no	—
382	C₂×Q₈	—	2	2	no	—
383	C₂×Q₈	—	2	2	no	—
384	C₂×Q₈	—	2	8	no	—
385	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
386	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
387	C₂×D₄	—	2	8	no	—
388	C₂×D₄	—	2	2	no	—
389	C₂×D₄	—	2	4	no	—
390	C₂×D₄	—	2	4	no	—
391	C₂×D₄	—	2	8	no	—
392	C₂×D₄	—	8	4	no	—
393	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
394	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
395	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	2	8	no	—
398	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
399	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
400	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
401	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
402	SD₁₆	—	2	16	no	—
403	SD₁₆	—	2	16	no	—
404	M₁₆	—	2	4	no	—
405	M₁₆	—	2	8	no	—
406	C₂×C₈	—	2	8	no	—
407	C₂×C₈	#41×C₂₃	1	8	yes	complemented
408	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
409	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
410	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
411	C₄⋊C₄	—	2	4	no	—
412	C₄⋊C₄	—	2	4	no	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	2	8	no	—
415	C₄⋊C₄	—	2	2	no	—
416	C₄⋊C₄	—	2	4	no	—
417	C₄⋊C₄	—	2	4	no	—
418	C₄⋊C₄	—	2	8	no	—
419	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
420	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
421	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
422	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
423	C₂²⋊C₄	—	2	2	no	—
424	C₂²⋊C₄	—	2	2	no	—
425	C₂²⋊C₄	—	2	8	no	—
426	C₂²⋊C₄	—	2	8	no	—
427	C₂²⋊C₄	—	2	8	no	—
428	C₂²⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
429	C₂²⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
430	C₄²	—	2	8	no	—
431	C₄²	—	2	2	no	—
432	C₄²	—	2	2	no	—
433	C₄²	C₂×D₄×C₂₃	1	4	yes	—
434	C₄²	C₂×D₄×C₂₃	1	2	yes	—
435	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
436	C₄²	C₂⁴×C₂₃	1	2	yes	—
437	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
438	C₂³	—	2	4	no	—
439	C₂³	—	4	4	no	—
440	C₂³	—	8	4	no	—
441	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
442	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
443	Q₈	—	2	2	no	—
444	Q₈	—	2	8	no	—
445	Q₈	—	4	4	no	—
446	Q₈	—	4	4	no	—
447	Q₈	—	4	4	no	—
448	D₄	—	2	4	no	—
449	D₄	—	2	8	no	—
450	D₄	—	2	2	no	—
451	D₄	—	4	8	no	—
452	D₄	—	4	4	no	—
453	D₄	—	8	4	no	—
454	D₄	—	8	8	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	8	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	4	no	—
461	C₂×C₄	—	4	4	no	—
462	C₂×C₄	—	8	8	no	—
463	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
464	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
465	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
466	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
467	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
468	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
469	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
470	C₈	—	2	4	no	—
471	C₈	C₂×#41×C₂₃	1	4	yes	—
472	C₈	C₂×#41×C₂₃	1	4	yes	—
473	C₂²	—	2	2	no	—
474	C₂²	—	2	2	no	—
475	C₂²	—	2	4	no	—
476	C₂²	—	4	4	no	—
477	C₂²	—	4	8	no	—
478	C₂²	—	16	4	no	—
479	C₂²	C₂₃×#532	1	2	yes	—
480	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
481	C₄	—	2	2	no	—
482	C₄	—	2	4	no	—
483	C₄	—	2	2	no	—
484	C₄	—	4	4	no	—
485	C₄	—	4	4	no	—
486	C₄	—	4	4	no	—
487	C₄	C₂₃×#532	1	2	yes	—
488	C₄	C₂₃×#577	1	4	yes	—
489	C₄	C₂²×#41×C₂₃	1	2	yes	—
490	C₂	—	2	2	no	—
491	C₂	—	4	2	no	—
492	C₂	—	8	4	no	—
493	C₂	C₂×C₂₃×#532	1	1	yes	characteristic
494	C₂	C₂×C₂₃×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂₃×#3508	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3509	C₂	semidirect
3. C₂₃×#3182	C₂	semidirect
4. C₂₃×#3174	C₂	semidirect
5. C₂₃×#3167	C₂	semidirect
6. C₂₃×#3160	C₂	semidirect
7. C₂×C₂₃×#492	C₂	semidirect
8. C₂₃×#3139	C₂	semidirect
9. C₂₃×#3138	C₂	semidirect
10. C₂₃×#3057	C₂	semidirect
11. C₂₃×#2949	C₂	semidirect
12. C₄×C₂₃×#131	C₂	semidirect
21. C₂₃×#541	C₂²	semidirect
27. C₂₃×#533	C₂²	semidirect
34. C₂₃×#500	C₂²	semidirect
35. C₂₃×#499	C₂²	semidirect
36. C₂₃×#498	C₂²	semidirect
40. C₂₃×#491	C₂²	semidirect
42. C₂₃×#487	C₂²	semidirect
44. C₂₃×#468	C₂²	semidirect
45. C₂₃×#465	C₂²	semidirect
46. C₂₃×#440	C₂²	semidirect
47. C₄×SD₁₆×C₂₃	C₂²	semidirect
52. C₂₃×#405	C₂²	semidirect
54. C₄×M₁₆×C₂₃	C₂²	semidirect
56. C₂×C₄×C₈×C₂₃	C₂²	semidirect
159. C₂×C₈×C₂₃	#41	semidirect
190. #60712	C₂₃	direct (N × H)
226. #3509	C₂×C₂₃	semidirect
228. #3182	C₂×C₂₃	semidirect
229. #3174	C₂×C₂₃	semidirect
230. #3167	C₂×C₂₃	semidirect
231. #3160	C₂×C₂₃	semidirect
232. C₂×#492	C₂×C₂₃	semidirect
233. #3139	C₂×C₂₃	semidirect
234. #3138	C₂×C₂₃	semidirect
235. #3057	C₂×C₂₃	semidirect
236. #2949	C₂×C₂₃	semidirect
237. C₄×#131	C₂×C₂₃	semidirect
263. #541	C₂²×C₂₃	semidirect
269. #533	C₂²×C₂₃	semidirect
276. #500	C₂²×C₂₃	semidirect
277. #499	C₂²×C₂₃	semidirect
278. #498	C₂²×C₂₃	semidirect
282. #491	C₂²×C₂₃	semidirect
284. #487	C₂²×C₂₃	semidirect
286. #468	C₂²×C₂₃	semidirect
287. #465	C₂²×C₂₃	semidirect
288. #440	C₂²×C₂₃	semidirect
289. C₄×SD₁₆	C₂²×C₂₃	semidirect
294. #405	C₂²×C₂₃	semidirect
296. C₄×M₁₆	C₂²×C₂₃	semidirect
298. C₂×C₄×C₈	C₂²×C₂₃	semidirect
377. C₂₃	#60712	direct (N × H)
407. C₂×C₈	#41×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3509	2
C₂₃×#3508	2
C₂₃×#3182	2
C₂₃×#3174	2
C₂₃×#3167	2
C₂₃×#3160	2
C₂×C₂₃×#492	2
C₂₃×#3139	2
C₂₃×#3138	2
C₂₃×#3057	2
C₂₃×#2949	2
C₄×C₂₃×#131	2
#60712	23

Subgroups — C23×#60712 #19208338

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60712 #19208338