Subgroups — C₂₃×#32834 #19180460

Order 5888.

720 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2868	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#2860	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2099	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2093	C₂	1	2	yes	maximal
5	C₂₃×#2083	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2071	C₂	1	2	yes	maximal
7	C₂₃×#2057	C₂	1	2	yes	maximal, complemented
8	C₂₃×#2023	C₂	1	2	yes	maximal
9	C₂₃×#1839	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
11	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
12	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₂×C₂₃×#126	—	2	1	no	—
14	C₂×C₂₃×#107	C₂²	1	4	yes	—
15	C₂×C₂₃×#102	C₂²	1	4	yes	—
16	C₂₃×#390	C₂²	1	1	yes	characteristic
17	C₂₃×#390	C₂²	1	1	yes	characteristic
18	C₂₃×#390	C₂²	1	1	yes	characteristic
19	C₂₃×#389	C₂²	1	1	yes	characteristic
20	C₂₃×#389	C₂²	1	1	yes	characteristic
21	C₂₃×#389	C₂²	1	1	yes	characteristic
22	C₂₃×#388	—	2	1	no	—
23	C₂₃×#388	—	2	1	no	—
24	C₂₃×#388	—	2	1	no	—
25	C₂₃×#384	—	2	1	no	—
26	C₂₃×#384	—	2	1	no	—
27	C₂₃×#384	C₂²	1	2	yes	—
28	C₂₃×#384	C₂²	1	2	yes	—
29	C₄³×C₂₃	C₂²	1	1	yes	characteristic
30	C₂₃×#340	C₂²	1	4	yes	—
31	C₂₃×#340	C₂²	1	4	yes	—
32	C₂₃×#336	C₂²	1	2	yes	—
33	C₂₃×#336	C₂²	1	2	yes	—
34	C₂₃×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₂₃	—	2	2	no	—
36	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
38	C₂₃×C₄⋊Q₈	—	4	4	no	—
39	C₂₃×C₄⋊Q₈	—	4	4	no	—
40	C₂₃×C₄⋊Q₈	—	4	4	no	—
41	C₂₃×#126	—	4	4	no	—
42	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
47	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
48	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
49	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
50	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
53	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
54	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
55	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
56	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
57	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
58	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
59	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
60	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
61	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
62	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
64	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
67	C₂₃×#107	—	4	4	no	—
68	C₂₃×#102	—	4	4	no	—
69	C₂₃×#95	—	2	1	no	—
70	C₂₃×#95	—	2	1	no	—
71	C₂₃×#95	—	2	1	no	—
72	C₂⁴×C₂₃	—	2	1	no	—
73	C₂×Q₈×C₂₃	—	4	4	no	—
74	C₂×Q₈×C₂₃	—	4	4	no	—
75	C₂×Q₈×C₂₃	—	4	4	no	—
76	C₂×D₄×C₂₃	—	2	2	no	—
77	C₂×D₄×C₂₃	—	2	2	no	—
78	C₂×D₄×C₂₃	—	2	2	no	—
79	C₂²×C₄×C₂₃	—	2	1	no	—
80	C₂²×C₄×C₂₃	—	2	1	no	—
81	C₂²×C₄×C₂₃	—	2	1	no	—
82	C₂²×C₄×C₂₃	—	2	4	no	—
83	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
90	C₂×C₈×C₂₃	—	2	4	no	—
91	C₂×C₈×C₂₃	—	2	4	no	—
92	C₂×C₈×C₂₃	—	2	4	no	—
93	C₄⋊C₄×C₂₃	—	2	4	no	—
94	C₄⋊C₄×C₂₃	—	2	2	no	—
95	C₄⋊C₄×C₂₃	—	2	4	no	—
96	C₄⋊C₄×C₂₃	—	2	2	no	—
97	C₄⋊C₄×C₂₃	—	2	2	no	—
98	C₄⋊C₄×C₂₃	—	4	4	no	—
99	C₄⋊C₄×C₂₃	—	4	4	no	—
100	C₄⋊C₄×C₂₃	—	4	2	no	—
101	C₄⋊C₄×C₂₃	—	4	2	no	—
102	C₄⋊C₄×C₂₃	—	4	2	no	—
103	C₄⋊C₄×C₂₃	—	4	4	no	—
104	C₄⋊C₄×C₂₃	—	4	4	no	—
105	C₄⋊C₄×C₂₃	—	4	4	no	—
106	C₂²⋊C₄×C₂₃	—	4	2	no	—
107	C₂²⋊C₄×C₂₃	—	4	2	no	—
108	C₂²⋊C₄×C₂₃	—	4	2	no	—
109	C₂²⋊C₄×C₂₃	—	4	2	no	—
110	C₂²⋊C₄×C₂₃	—	4	2	no	—
111	C₂²⋊C₄×C₂₃	—	4	2	no	—
112	C₄²×C₂₃	—	2	2	no	—
113	C₄²×C₂₃	—	2	2	no	—
114	C₄²×C₂₃	—	2	2	no	—
115	C₄²×C₂₃	—	2	2	no	—
116	C₄²×C₂₃	—	2	2	no	—
117	C₄²×C₂₃	—	2	2	no	—
118	C₄²×C₂₃	D₈	1	4	yes	—
119	#32834	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂³×C₂₃	—	4	1	no	—
121	C₂³×C₂₃	—	4	1	no	—
122	C₂³×C₂₃	—	4	1	no	—
123	C₂³×C₂₃	—	4	1	no	—
124	C₂³×C₂₃	—	4	1	no	—
125	C₂³×C₂₃	—	4	1	no	—
126	C₂³×C₂₃	—	4	1	no	—
127	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
128	Q₈×C₂₃	—	8	8	no	—
129	D₄×C₂₃	—	4	4	no	—
130	C₂×C₄×C₂₃	—	2	1	no	—
131	C₂×C₄×C₂₃	—	2	1	no	—
132	C₂×C₄×C₂₃	—	2	1	no	—
133	C₂×C₄×C₂₃	—	2	1	no	—
134	C₂×C₄×C₂₃	—	2	1	no	—
135	C₂×C₄×C₂₃	—	2	1	no	—
136	C₂×C₄×C₂₃	—	2	1	no	—
137	C₂×C₄×C₂₃	—	2	1	no	—
138	C₂×C₄×C₂₃	—	2	1	no	—
139	C₂×C₄×C₂₃	—	2	1	no	—
140	C₂×C₄×C₂₃	—	2	1	no	—
141	C₂×C₄×C₂₃	—	2	1	no	—
142	C₂×C₄×C₂₃	—	4	4	no	—
143	C₂×C₄×C₂₃	—	4	1	no	—
144	C₂×C₄×C₂₃	—	4	1	no	—
145	C₂×C₄×C₂₃	—	4	1	no	—
146	C₂×C₄×C₂₃	—	4	1	no	—
147	C₂×C₄×C₂₃	—	4	1	no	—
148	C₂×C₄×C₂₃	—	4	1	no	—
149	C₂×C₄×C₂₃	—	4	4	no	—
150	C₂×C₄×C₂₃	—	4	1	no	—
151	C₂×C₄×C₂₃	—	4	4	no	—
152	C₂×C₄×C₂₃	—	4	1	no	—
153	C₂×C₄×C₂₃	—	4	1	no	—
154	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
155	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
156	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
157	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
158	C₂×C₄×C₂₃	#137	1	2	yes	—
159	C₂×C₄×C₂₃	#137	1	2	yes	—
160	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
161	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
162	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
163	C₈×C₂₃	—	4	4	no	—
164	#2868	C₂×C₂₃	1	1	yes	characteristic, complemented
165	#2860	C₂×C₂₃	1	1	yes	characteristic
166	#2099	C₂×C₂₃	1	2	yes	complemented
167	#2093	C₂×C₂₃	1	2	yes	—
168	#2083	C₂×C₂₃	1	2	yes	complemented
169	#2071	C₂×C₂₃	1	2	yes	—
170	#2057	C₂×C₂₃	1	2	yes	complemented
171	#2023	C₂×C₂₃	1	2	yes	—
172	#1839	C₂×C₂₃	1	1	yes	characteristic, complemented
173	C₂²×C₂₃	—	8	1	no	—
174	C₂²×C₂₃	—	8	1	no	—
175	C₂²×C₂₃	—	8	1	no	—
176	C₂²×C₂₃	—	8	1	no	—
177	C₂²×C₂₃	—	8	1	no	—
178	C₂²×C₂₃	—	8	1	no	—
179	C₂²×C₂₃	—	8	1	no	—
180	C₂²×C₂₃	#535	1	1	yes	characteristic
181	C₂²×C₂₃	#536	1	1	yes	characteristic
182	C₂²×C₂₃	#536	1	1	yes	characteristic
183	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
184	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
185	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
186	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
187	C₄×C₂₃	—	2	2	no	—
188	C₄×C₂₃	—	2	2	no	—
189	C₄×C₂₃	—	2	2	no	—
190	C₄×C₂₃	—	2	2	no	—
191	C₄×C₂₃	—	2	2	no	—
192	C₄×C₂₃	—	2	2	no	—
193	C₄×C₂₃	—	8	1	no	—
194	C₄×C₂₃	—	8	4	no	—
195	C₄×C₂₃	—	8	1	no	—
196	C₄×C₂₃	—	8	1	no	—
197	C₄×C₂₃	#563	1	4	yes	—
198	C₂×C₄⋊Q₈	—	2	1	no	—
199	C₂×C₄⋊Q₈	—	2	1	no	—
200	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
201	C₂×#126	—	2	1	no	—
202	C₂×#107	C₂²×C₂₃	1	4	yes	—
203	C₂×#102	C₂²×C₂₃	1	4	yes	—
204	#390	C₂²×C₂₃	1	1	yes	characteristic
205	#390	C₂²×C₂₃	1	1	yes	characteristic
206	#390	C₂²×C₂₃	1	1	yes	characteristic
207	#389	C₂²×C₂₃	1	1	yes	characteristic
208	#389	C₂²×C₂₃	1	1	yes	characteristic
209	#389	C₂²×C₂₃	1	1	yes	characteristic
210	#388	—	2	1	no	—
211	#388	—	2	1	no	—
212	#388	—	2	1	no	—
213	#384	—	2	1	no	—
214	#384	—	2	1	no	—
215	#384	C₂²×C₂₃	1	2	yes	—
216	#384	C₂²×C₂₃	1	2	yes	—
217	C₄³	C₂²×C₂₃	1	1	yes	characteristic
218	#340	C₂²×C₂₃	1	4	yes	—
219	#340	C₂²×C₂₃	1	4	yes	—
220	#336	C₂²×C₂₃	1	2	yes	—
221	#336	C₂²×C₂₃	1	2	yes	—
222	#336	C₂²×C₂₃	1	2	yes	—
223	C₂×C₂₃	—	16	1	no	—
224	C₂×C₂₃	#2821	1	1	yes	characteristic
225	C₂×C₂₃	#3205	1	1	yes	characteristic
226	C₂×C₂₃	#3218	1	1	yes	characteristic
227	C₂×C₂₃	#3226	1	1	yes	characteristic
228	C₂×C₂₃	#3226	1	1	yes	characteristic
229	C₂×C₂₃	#3238	1	1	yes	characteristic
230	C₂×C₂₃	#3239	1	1	yes	characteristic
231	C₂²×Q₈	—	2	2	no	—
232	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
233	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
234	C₄⋊Q₈	—	4	4	no	—
235	C₄⋊Q₈	—	4	4	no	—
236	C₄⋊Q₈	—	4	4	no	—
237	#126	—	4	4	no	—
238	C₂×C₄⋊C₄	—	2	1	no	—
239	C₂×C₄⋊C₄	—	2	1	no	—
240	C₂×C₄⋊C₄	—	2	1	no	—
241	C₂×C₄⋊C₄	—	2	2	no	—
242	C₂×C₄⋊C₄	—	2	2	no	—
243	C₂×C₄⋊C₄	—	2	2	no	—
244	C₂×C₄⋊C₄	—	2	2	no	—
245	C₂×C₄⋊C₄	—	2	2	no	—
246	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
247	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
248	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
249	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
250	C₂×C₂²⋊C₄	—	2	1	no	—
251	C₂×C₂²⋊C₄	—	2	1	no	—
252	C₂×C₂²⋊C₄	—	2	1	no	—
253	C₂×C₂²⋊C₄	—	2	1	no	—
254	C₂×C₂²⋊C₄	—	2	1	no	—
255	C₂×C₂²⋊C₄	—	2	1	no	—
256	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
257	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
258	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
259	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
260	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
261	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
262	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
263	#107	—	4	4	no	—
264	#102	—	4	4	no	—
265	#95	—	2	1	no	—
266	#95	—	2	1	no	—
267	#95	—	2	1	no	—
268	C₂₃	#32834	1	1	yes	characteristic, direct product factor, Sylow
269	C₂⁴	—	2	1	no	—
270	C₂×Q₈	—	4	4	no	—
271	C₂×Q₈	—	4	4	no	—
272	C₂×Q₈	—	4	4	no	—
273	C₂×D₄	—	2	2	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂²×C₄	—	2	1	no	—
277	C₂²×C₄	—	2	1	no	—
278	C₂²×C₄	—	2	1	no	—
279	C₂²×C₄	—	2	4	no	—
280	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
281	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
282	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
283	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
284	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
285	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
286	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
287	C₂×C₈	—	2	4	no	—
288	C₂×C₈	—	2	4	no	—
289	C₂×C₈	—	2	4	no	—
290	C₄⋊C₄	—	2	4	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	4	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	4	4	no	—
296	C₄⋊C₄	—	4	4	no	—
297	C₄⋊C₄	—	4	2	no	—
298	C₄⋊C₄	—	4	2	no	—
299	C₄⋊C₄	—	4	2	no	—
300	C₄⋊C₄	—	4	4	no	—
301	C₄⋊C₄	—	4	4	no	—
302	C₄⋊C₄	—	4	4	no	—
303	C₂²⋊C₄	—	4	2	no	—
304	C₂²⋊C₄	—	4	2	no	—
305	C₂²⋊C₄	—	4	2	no	—
306	C₂²⋊C₄	—	4	2	no	—
307	C₂²⋊C₄	—	4	2	no	—
308	C₂²⋊C₄	—	4	2	no	—
309	C₄²	—	2	2	no	—
310	C₄²	—	2	2	no	—
311	C₄²	—	2	2	no	—
312	C₄²	—	2	2	no	—
313	C₄²	—	2	2	no	—
314	C₄²	—	2	2	no	—
315	C₄²	D₈×C₂₃	1	4	yes	—
316	C₂³	—	4	1	no	—
317	C₂³	—	4	1	no	—
318	C₂³	—	4	1	no	—
319	C₂³	—	4	1	no	—
320	C₂³	—	4	1	no	—
321	C₂³	—	4	1	no	—
322	C₂³	—	4	1	no	—
323	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
324	Q₈	—	8	8	no	—
325	D₄	—	4	4	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	4	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	4	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	4	4	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
351	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
352	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
353	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
354	C₂×C₄	C₂₃×#137	1	2	yes	—
355	C₂×C₄	C₂₃×#137	1	2	yes	—
356	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
357	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
358	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
359	C₈	—	4	4	no	—
360	C₂²	—	8	1	no	—
361	C₂²	—	8	1	no	—
362	C₂²	—	8	1	no	—
363	C₂²	—	8	1	no	—
364	C₂²	—	8	1	no	—
365	C₂²	—	8	1	no	—
366	C₂²	—	8	1	no	—
367	C₂²	C₂₃×#535	1	1	yes	characteristic
368	C₂²	C₂₃×#536	1	1	yes	characteristic
369	C₂²	C₂₃×#536	1	1	yes	characteristic
370	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
371	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
372	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
373	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
374	C₄	—	2	2	no	—
375	C₄	—	2	2	no	—
376	C₄	—	2	2	no	—
377	C₄	—	2	2	no	—
378	C₄	—	2	2	no	—
379	C₄	—	2	2	no	—
380	C₄	—	8	4	no	—
381	C₄	—	8	1	no	—
382	C₄	—	8	1	no	—
383	C₄	—	8	1	no	—
384	C₄	C₂₃×#563	1	4	yes	—
385	C₂	—	16	1	no	—
386	C₂	C₂₃×#2821	1	1	yes	characteristic
387	C₂	C₂₃×#3205	1	1	yes	characteristic
388	C₂	C₂₃×#3218	1	1	yes	characteristic
389	C₂	C₂₃×#3226	1	1	yes	characteristic
390	C₂	C₂₃×#3226	1	1	yes	characteristic
391	C₂	C₂₃×#3238	1	1	yes	characteristic
392	C₂	C₂₃×#3239	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 ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2868	C₂	semidirect
3. C₂₃×#2099	C₂	semidirect
5. C₂₃×#2083	C₂	semidirect
7. C₂₃×#2057	C₂	semidirect
9. C₂₃×#1839	C₂	semidirect
119. #32834	C₂₃	direct (N × H)
164. #2868	C₂×C₂₃	semidirect
166. #2099	C₂×C₂₃	semidirect
168. #2083	C₂×C₂₃	semidirect
170. #2057	C₂×C₂₃	semidirect
172. #1839	C₂×C₂₃	semidirect
268. C₂₃	#32834	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×#2868	2
C₂₃×#2860	2
C₂₃×#2099	2
C₂₃×#2093	2
C₂₃×#2083	2
C₂₃×#2071	2
C₂₃×#2057	2
C₂₃×#2023	2
C₂₃×#1839	2
#32834	23

Subgroups — C23×#32834 #19180460

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#32834 #19180460