Subgroups — C₂₃×#18071 #19165697

Order 5888.

646 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3167	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2962	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2935	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1592	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1588	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1585	C₂	1	2	yes	maximal, complemented
7	C₂₃×SD₁₆⋊C₈	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1481	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1480	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#131	—	2	1	no	—
13	C₂₃×#533	C₂²	1	1	yes	characteristic
14	C₂₃×#532	C₂²	1	1	yes	characteristic
15	C₂₃×#498	C₄	1	1	yes	characteristic
16	C₂₃×#492	C₄	1	1	yes	characteristic
17	C₂₃×#487	C₄	1	1	yes	characteristic
18	C₂₃×#486	C₄	1	1	yes	characteristic
19	C₂₃×#465	C₄	1	2	yes	—
20	C₂₃×#465	C₄	1	2	yes	—
21	C₂₃×#446	—	2	2	no	—
22	C₂₃×#446	C₂²	1	2	yes	—
23	C₈×Q₈×C₂₃	C₂²	1	2	yes	—
24	C₄×SD₁₆×C₂₃	C₂²	1	2	yes	—
25	C₄×SD₁₆×C₂₃	C₂²	1	2	yes	—
26	C₂₃×#436	C₂²	1	2	yes	—
27	C₂₃×#435	—	2	2	no	—
28	C₈×D₄×C₂₃	C₂²	1	2	yes	—
29	C₂₃×#431	C₂²	1	1	yes	characteristic
30	C₂₃×#424	—	2	1	no	—
31	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₂₃×#408	—	2	1	no	—
34	C₂₃×#405	—	2	1	no	—
35	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
37	C₂₃×#335	C₂²	1	1	yes	characteristic
38	C₂₃×#335	C₂²	1	1	yes	characteristic
39	C₂₃×#335	C₂²	1	1	yes	characteristic
40	C₂₃×#335	C₂²	1	1	yes	characteristic
41	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
42	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
43	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
44	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
45	C₂₃×#322	C₂²	1	2	yes	—
46	C₂₃×#322	C₂²	1	2	yes	—
47	C₂×#41×C₂₃	—	2	1	no	—
48	C₂×SD₁₆×C₂₃	C₂×C₄	1	2	yes	—
49	C₂×SD₁₆×C₂₃	C₂×C₄	1	2	yes	—
50	C₂₃×#131	—	4	4	no	—
51	C₂×M₁₆×C₂₃	—	2	2	no	—
52	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
53	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₂₃	—	2	1	no	—
55	C₂²×C₈×C₂₃	—	2	1	no	—
56	C₂²×C₈×C₂₃	—	2	2	no	—
57	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
59	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
60	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
61	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
62	C₂₃×#127	C₂×C₄	1	1	yes	characteristic
63	C₂₃×#125	—	2	1	no	—
64	C₂₃×#124	—	2	1	no	—
65	C₂₃×C₂²⋊Q₈	—	2	1	no	—
66	C₂₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
67	C₂₃×#121	—	2	1	no	—
68	C₂₃×#121	C₂×C₄	1	2	yes	—
69	C₄×Q₈×C₂₃	—	2	1	no	—
70	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
71	C₄×D₄×C₂₃	—	2	1	no	—
72	C₄×D₄×C₂₃	C₂³	1	2	yes	—
73	C₂₃×#117	—	2	1	no	—
74	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
75	C₂₃×#106	C₂×C₄	1	2	yes	—
76	C₂₃×#106	C₂×C₄	1	2	yes	—
77	C₂₃×C₄⋊C₈	—	2	2	no	—
78	C₂₃×C₄⋊C₈	—	2	2	no	—
79	C₂₃×C₄⋊C₈	—	2	1	no	—
80	C₂₃×C₄⋊C₈	—	2	1	no	—
81	C₂₃×C₄⋊C₈	—	2	1	no	—
82	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
83	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
84	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
85	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
86	C₂₃×#103	C₂×C₄	1	2	yes	—
87	C₂₃×#103	C₂×C₄	1	2	yes	—
88	C₂₃×#102	C₂×C₄	1	2	yes	—
89	C₂₃×#102	C₂×C₄	1	2	yes	—
90	C₂₃×C₂²⋊C₈	—	2	1	no	—
91	C₂₃×C₂²⋊C₈	—	2	2	no	—
92	C₂₃×C₂²⋊C₈	—	2	2	no	—
93	C₂₃×#97	—	2	2	no	—
94	C₂₃×#97	D₄	1	2	yes	—
95	C₄×C₈×C₂₃	—	2	2	no	—
96	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
97	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
98	C₄×C₈×C₂₃	C₂³	1	2	yes	—
99	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₂₃	C₂³	1	2	yes	—
102	#41×C₂₃	—	4	2	no	—
103	#41×C₂₃	—	4	2	no	—
104	C₂×Q₈×C₂₃	—	2	1	no	—
105	C₂×Q₈×C₂₃	—	2	1	no	—
106	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
107	C₂×D₄×C₂₃	—	2	2	no	—
108	C₂×D₄×C₂₃	—	2	1	no	—
109	C₂×D₄×C₂₃	—	2	1	no	—
110	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₂₃	—	2	2	no	—
112	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
115	SD₁₆×C₂₃	—	2	4	no	—
116	SD₁₆×C₂₃	—	4	2	no	—
117	M₁₆×C₂₃	—	2	2	no	—
118	M₁₆×C₂₃	—	2	2	no	—
119	M₁₆×C₂₃	—	4	4	no	—
120	C₂×C₈×C₂₃	—	2	2	no	—
121	C₂×C₈×C₂₃	—	2	1	no	—
122	C₂×C₈×C₂₃	—	2	1	no	—
123	C₂×C₈×C₂₃	—	2	1	no	—
124	C₂×C₈×C₂₃	—	2	1	no	—
125	C₂×C₈×C₂₃	—	2	2	no	—
126	C₂×C₈×C₂₃	—	4	1	no	—
127	C₂×C₈×C₂₃	—	4	4	no	—
128	C₂×C₈×C₂₃	—	4	2	no	—
129	C₂×C₈×C₂₃	—	4	2	no	—
130	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
135	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
136	C₄⋊C₄×C₂₃	—	2	1	no	—
137	C₄⋊C₄×C₂₃	—	2	1	no	—
138	C₄⋊C₄×C₂₃	—	2	1	no	—
139	C₄⋊C₄×C₂₃	—	2	1	no	—
140	C₄⋊C₄×C₂₃	—	2	2	no	—
141	C₄⋊C₄×C₂₃	—	2	2	no	—
142	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
143	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
144	C₂²⋊C₄×C₂₃	—	2	2	no	—
145	C₂²⋊C₄×C₂₃	—	2	2	no	—
146	C₂²⋊C₄×C₂₃	—	2	2	no	—
147	C₄²×C₂₃	—	2	2	no	—
148	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
149	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
150	C₄²×C₂₃	#41	1	1	yes	characteristic
151	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
152	#18071	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂³×C₂₃	—	2	2	no	—
154	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
155	Q₈×C₂₃	—	2	2	no	—
156	Q₈×C₂₃	—	2	2	no	—
157	Q₈×C₂₃	—	4	2	no	—
158	Q₈×C₂₃	—	8	1	no	—
159	D₄×C₂₃	—	2	2	no	—
160	D₄×C₂₃	—	2	2	no	—
161	D₄×C₂₃	—	4	4	no	—
162	D₄×C₂₃	—	4	2	no	—
163	D₄×C₂₃	—	8	1	no	—
164	C₂×C₄×C₂₃	—	2	1	no	—
165	C₂×C₄×C₂₃	—	2	1	no	—
166	C₂×C₄×C₂₃	—	2	2	no	—
167	C₂×C₄×C₂₃	—	2	2	no	—
168	C₂×C₄×C₂₃	—	2	2	no	—
169	C₂×C₄×C₂₃	—	4	1	no	—
170	C₂×C₄×C₂₃	—	4	2	no	—
171	C₂×C₄×C₂₃	—	4	2	no	—
172	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
173	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
174	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
175	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
176	C₂×C₄×C₂₃	#137	1	2	yes	—
177	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
178	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
180	C₈×C₂₃	—	2	1	no	—
181	C₈×C₂₃	—	2	1	no	—
182	C₈×C₂₃	—	2	1	no	—
183	C₈×C₂₃	—	2	2	no	—
184	C₈×C₂₃	—	2	2	no	—
185	C₈×C₂₃	—	4	2	no	—
186	C₈×C₂₃	—	4	1	no	—
187	C₈×C₂₃	—	4	1	no	—
188	C₈×C₂₃	—	4	1	no	—
189	C₈×C₂₃	—	4	1	no	—
190	C₈×C₂₃	#131	1	2	yes	complemented
191	#3167	C₂×C₂₃	1	1	yes	characteristic
192	#2962	C₂×C₂₃	1	1	yes	characteristic, complemented
193	#2935	C₂×C₂₃	1	1	yes	characteristic
194	#1592	C₂×C₂₃	1	2	yes	complemented
195	#1588	C₂×C₂₃	1	2	yes	complemented
196	#1585	C₂×C₂₃	1	2	yes	complemented
197	SD₁₆⋊C₈	C₂×C₂₃	1	2	yes	complemented
198	C₂×#335	C₂×C₂₃	1	1	yes	characteristic, complemented
199	#1481	C₂×C₂₃	1	1	yes	characteristic, complemented
200	#1480	C₂×C₂₃	1	1	yes	characteristic
201	#1451	C₂×C₂₃	1	1	yes	characteristic, complemented
202	C₂²×C₂₃	—	2	1	no	—
203	C₂²×C₂₃	—	2	1	no	—
204	C₂²×C₂₃	—	4	2	no	—
205	C₂²×C₂₃	—	4	2	no	—
206	C₂²×C₂₃	—	4	2	no	—
207	C₂²×C₂₃	#440	1	2	yes	—
208	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
209	C₄×C₂₃	—	2	1	no	—
210	C₄×C₂₃	—	2	1	no	—
211	C₄×C₂₃	—	2	1	no	—
212	C₄×C₂₃	—	4	2	no	—
213	C₄×C₂₃	—	4	2	no	—
214	C₄×C₂₃	—	4	2	no	—
215	C₄×C₂₃	#440	1	2	yes	—
216	C₄×C₂₃	C₂×#131	1	1	yes	characteristic
217	C₄×C₂₃	#568	1	1	yes	characteristic
218	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
219	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
220	C₂×#131	—	2	1	no	—
221	#533	C₂²×C₂₃	1	1	yes	characteristic
222	#532	C₂²×C₂₃	1	1	yes	characteristic
223	#498	C₄×C₂₃	1	1	yes	characteristic
224	#492	C₄×C₂₃	1	1	yes	characteristic
225	#487	C₄×C₂₃	1	1	yes	characteristic
226	#486	C₄×C₂₃	1	1	yes	characteristic
227	#465	C₄×C₂₃	1	2	yes	—
228	#465	C₄×C₂₃	1	2	yes	—
229	#446	—	2	2	no	—
230	#446	C₂²×C₂₃	1	2	yes	—
231	C₈×Q₈	C₂²×C₂₃	1	2	yes	—
232	C₄×SD₁₆	C₂²×C₂₃	1	2	yes	—
233	C₄×SD₁₆	C₂²×C₂₃	1	2	yes	—
234	#436	C₂²×C₂₃	1	2	yes	—
235	#435	—	2	2	no	—
236	C₈×D₄	C₂²×C₂₃	1	2	yes	—
237	#431	C₂²×C₂₃	1	1	yes	characteristic
238	#424	—	2	1	no	—
239	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
240	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
241	#408	—	2	1	no	—
242	#405	—	2	1	no	—
243	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
244	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
245	#335	C₂²×C₂₃	1	1	yes	characteristic
246	#335	C₂²×C₂₃	1	1	yes	characteristic
247	#335	C₂²×C₂₃	1	1	yes	characteristic
248	#335	C₂²×C₂₃	1	1	yes	characteristic
249	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
250	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
251	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
252	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
253	#322	C₂²×C₂₃	1	2	yes	—
254	#322	C₂²×C₂₃	1	2	yes	—
255	C₂×C₂₃	—	2	2	no	—
256	C₂×C₂₃	—	8	2	no	—
257	C₂×C₂₃	#2935	1	1	yes	characteristic
258	C₂×C₂₃	C₂×#440	1	1	yes	characteristic
259	C₂×C₂₃	#2957	1	1	yes	characteristic
260	C₂×#41	—	2	1	no	—
261	C₂×SD₁₆	C₂×C₄×C₂₃	1	2	yes	—
262	C₂×SD₁₆	C₂×C₄×C₂₃	1	2	yes	—
263	#131	—	4	4	no	—
264	C₂×M₁₆	—	2	2	no	—
265	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
266	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
267	C₂²×C₈	—	2	1	no	—
268	C₂²×C₈	—	2	1	no	—
269	C₂²×C₈	—	2	2	no	—
270	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
271	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
272	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
273	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
274	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
275	#127	C₂×C₄×C₂₃	1	1	yes	characteristic
276	#125	—	2	1	no	—
277	#124	—	2	1	no	—
278	C₂²⋊Q₈	—	2	1	no	—
279	C₂²⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
280	#121	—	2	1	no	—
281	#121	C₂×C₄×C₂₃	1	2	yes	—
282	C₄×Q₈	—	2	1	no	—
283	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
284	C₄×D₄	—	2	1	no	—
285	C₄×D₄	C₂³×C₂₃	1	2	yes	—
286	#117	—	2	1	no	—
287	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
288	#106	C₂×C₄×C₂₃	1	2	yes	—
289	#106	C₂×C₄×C₂₃	1	2	yes	—
290	C₄⋊C₈	—	2	2	no	—
291	C₄⋊C₈	—	2	2	no	—
292	C₄⋊C₈	—	2	1	no	—
293	C₄⋊C₈	—	2	1	no	—
294	C₄⋊C₈	—	2	1	no	—
295	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
296	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
297	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
298	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
299	#103	C₂×C₄×C₂₃	1	2	yes	—
300	#103	C₂×C₄×C₂₃	1	2	yes	—
301	#102	C₂×C₄×C₂₃	1	2	yes	—
302	#102	C₂×C₄×C₂₃	1	2	yes	—
303	C₂²⋊C₈	—	2	1	no	—
304	C₂²⋊C₈	—	2	2	no	—
305	C₂²⋊C₈	—	2	2	no	—
306	#97	—	2	2	no	—
307	#97	D₄×C₂₃	1	2	yes	—
308	C₄×C₈	—	2	2	no	—
309	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
310	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
311	C₄×C₈	C₂³×C₂₃	1	2	yes	—
312	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
313	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
314	C₄×C₈	C₂³×C₂₃	1	2	yes	—
315	C₂₃	#18071	1	1	yes	characteristic, direct product factor, Sylow
316	#41	—	4	2	no	—
317	#41	—	4	2	no	—
318	C₂×Q₈	—	2	1	no	—
319	C₂×Q₈	—	2	1	no	—
320	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
321	C₂×D₄	—	2	2	no	—
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
325	C₂²×C₄	—	2	2	no	—
326	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
327	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
328	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
329	SD₁₆	—	2	4	no	—
330	SD₁₆	—	4	2	no	—
331	M₁₆	—	2	2	no	—
332	M₁₆	—	2	2	no	—
333	M₁₆	—	4	4	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	2	1	no	—
337	C₂×C₈	—	2	1	no	—
338	C₂×C₈	—	2	1	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	4	1	no	—
341	C₂×C₈	—	4	4	no	—
342	C₂×C₈	—	4	2	no	—
343	C₂×C₈	—	4	2	no	—
344	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
345	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
346	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
347	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
348	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
349	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	2	1	no	—
352	C₄⋊C₄	—	2	1	no	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
357	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
358	C₂²⋊C₄	—	2	2	no	—
359	C₂²⋊C₄	—	2	2	no	—
360	C₂²⋊C₄	—	2	2	no	—
361	C₄²	—	2	2	no	—
362	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
363	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
364	C₄²	#41×C₂₃	1	1	yes	characteristic
365	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
366	C₂³	—	2	2	no	—
367	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
368	Q₈	—	2	2	no	—
369	Q₈	—	2	2	no	—
370	Q₈	—	4	2	no	—
371	Q₈	—	8	1	no	—
372	D₄	—	2	2	no	—
373	D₄	—	2	2	no	—
374	D₄	—	4	4	no	—
375	D₄	—	4	2	no	—
376	D₄	—	8	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
386	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
387	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
388	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
389	C₂×C₄	C₂₃×#137	1	2	yes	—
390	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
391	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
392	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
393	C₈	—	2	1	no	—
394	C₈	—	2	1	no	—
395	C₈	—	2	1	no	—
396	C₈	—	2	2	no	—
397	C₈	—	2	2	no	—
398	C₈	—	4	2	no	—
399	C₈	—	4	1	no	—
400	C₈	—	4	1	no	—
401	C₈	—	4	1	no	—
402	C₈	—	4	1	no	—
403	C₈	C₂₃×#131	1	2	yes	complemented
404	C₂²	—	2	1	no	—
405	C₂²	—	2	1	no	—
406	C₂²	—	4	2	no	—
407	C₂²	—	4	2	no	—
408	C₂²	—	4	2	no	—
409	C₂²	C₂₃×#440	1	2	yes	—
410	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
411	C₄	—	2	1	no	—
412	C₄	—	2	1	no	—
413	C₄	—	2	1	no	—
414	C₄	—	4	2	no	—
415	C₄	—	4	2	no	—
416	C₄	—	4	2	no	—
417	C₄	C₂₃×#440	1	2	yes	—
418	C₄	C₂×C₂₃×#131	1	1	yes	characteristic
419	C₄	C₂₃×#568	1	1	yes	characteristic
420	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
421	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
422	C₂	—	2	2	no	—
423	C₂	—	8	2	no	—
424	C₂	C₂₃×#2935	1	1	yes	characteristic
425	C₂	C₂×C₂₃×#440	1	1	yes	characteristic
426	C₂	C₂₃×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂₃×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₄×M₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2962	C₂	semidirect
4. C₂₃×#1592	C₂	semidirect
5. C₂₃×#1588	C₂	semidirect
6. C₂₃×#1585	C₂	semidirect
7. C₂₃×SD₁₆⋊C₈	C₂	semidirect
8. C₂×C₂₃×#335	C₂	semidirect
9. C₂₃×#1481	C₂	semidirect
11. C₂₃×#1451	C₂	semidirect
152. #18071	C₂₃	direct (N × H)
190. C₈×C₂₃	#131	semidirect
192. #2962	C₂×C₂₃	semidirect
194. #1592	C₂×C₂₃	semidirect
195. #1588	C₂×C₂₃	semidirect
196. #1585	C₂×C₂₃	semidirect
197. SD₁₆⋊C₈	C₂×C₂₃	semidirect
198. C₂×#335	C₂×C₂₃	semidirect
199. #1481	C₂×C₂₃	semidirect
201. #1451	C₂×C₂₃	semidirect
315. C₂₃	#18071	direct (N × H)
403. C₈	C₂₃×#131	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3167	2
C₂₃×#2962	2
C₂₃×#2935	2
C₂₃×#1592	2
C₂₃×#1588	2
C₂₃×#1585	2
C₂₃×SD₁₆⋊C₈	2
C₂×C₂₃×#335	2
C₂₃×#1481	2
C₂₃×#1480	2
C₂₃×#1451	2
#18071	23

Subgroups — C23×#18071 #19165697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18071 #19165697