Subgroups — C₂₃×#9101 #19156727

Order 5888.

630 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2972	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#2328	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1478	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1460	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1274	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#127	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₈×D₄×C₂₃	—	2	4	no	—
10	C₂₃×#431	C₂²	1	2	yes	—
11	C₂₃×#431	C₂²	1	2	yes	—
12	C₂×C₂₃×C₂²⋊C₈	—	2	2	no	—
13	C₂₃×#390	C₄	1	1	yes	characteristic
14	C₂₃×#390	C₄	1	1	yes	characteristic
15	C₂₃×#386	—	2	2	no	—
16	C₂₃×#386	—	2	2	no	—
17	C₂₃×#382	C₄	1	1	yes	characteristic
18	C₂₃×#376	C₂²	1	1	yes	characteristic
19	C₂₃×D₄⋊C₈	—	2	4	no	—
20	C₂₃×#323	—	2	2	no	—
21	C₂²×D₄×C₂₃	—	2	2	no	—
22	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₂₃	—	2	1	no	—
25	C₂³×C₄×C₂₃	D₄	1	2	yes	—
26	C₂²×C₈×C₂₃	—	2	4	no	—
27	C₂₃×#127	—	2	2	no	—
28	C₂₃×#127	C₈	1	4	yes	complemented
29	C₄×D₄×C₂₃	—	2	2	no	—
30	C₄×D₄×C₂₃	—	2	2	no	—
31	C₄×D₄×C₂₃	—	2	2	no	—
32	C₄×D₄×C₂₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
36	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
37	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
38	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
39	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
42	C₂₃×C₄⋊C₈	—	2	2	no	—
43	C₂₃×C₄⋊C₈	—	2	2	no	—
44	C₂₃×C₂²⋊C₈	—	2	4	no	—
45	C₂₃×C₂²⋊C₈	—	2	2	no	—
46	C₂₃×C₂²⋊C₈	—	2	2	no	—
47	C₄×C₈×C₂₃	—	2	2	no	—
48	C₄×C₈×C₂₃	D₄	1	4	yes	complemented
49	C₂₃×#95	—	2	2	no	—
50	C₂₃×#95	—	2	1	no	—
51	C₂₃×#95	—	2	1	no	—
52	C₂⁴×C₂₃	—	2	1	no	—
53	C₂⁴×C₂₃	C₂²⋊C₄	1	2	yes	—
54	C₂×D₄×C₂₃	—	2	8	no	—
55	C₂×D₄×C₂₃	—	2	2	no	—
56	C₂×D₄×C₂₃	—	2	2	no	—
57	C₂×D₄×C₂₃	—	2	1	no	—
58	C₂×D₄×C₂₃	—	2	2	no	—
59	C₂×D₄×C₂₃	—	2	2	no	—
60	C₂×D₄×C₂₃	—	8	2	no	—
61	C₂×D₄×C₂₃	—	8	2	no	—
62	C₂×D₄×C₂₃	C₂×C₈	1	2	yes	—
63	C₂×D₄×C₂₃	C₂×C₈	1	2	yes	complemented
64	C₂×D₄×C₂₃	M₁₆	1	2	yes	—
65	C₂²×C₄×C₂₃	—	2	1	no	—
66	C₂²×C₄×C₂₃	—	2	2	no	—
67	C₂²×C₄×C₂₃	—	2	2	no	—
68	C₂²×C₄×C₂₃	—	2	1	no	—
69	C₂²×C₄×C₂₃	—	2	1	no	—
70	C₂²×C₄×C₂₃	—	4	2	no	—
71	C₂²×C₄×C₂₃	—	8	2	no	—
72	C₂²×C₄×C₂₃	—	8	2	no	—
73	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₂₃	—	2	4	no	—
78	C₂×C₈×C₂₃	—	4	2	no	—
79	C₂×C₈×C₂₃	—	8	4	no	—
80	C₄⋊C₄×C₂₃	—	2	2	no	—
81	C₄⋊C₄×C₂₃	—	2	2	no	—
82	C₂²⋊C₄×C₂₃	—	2	4	no	—
83	C₂²⋊C₄×C₂₃	—	4	1	no	—
84	C₂²⋊C₄×C₂₃	—	4	1	no	—
85	C₂²⋊C₄×C₂₃	—	8	2	no	—
86	C₂²⋊C₄×C₂₃	—	8	2	no	—
87	C₄²×C₂₃	—	2	1	no	—
88	C₄²×C₂₃	—	2	2	no	—
89	C₄²×C₂₃	C₂×C₈	1	2	yes	complemented
90	C₄²×C₂₃	C₂×D₄	1	2	yes	—
91	C₄²×C₂₃	#41	1	2	yes	—
92	#9101	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₂₃	—	2	2	no	—
94	C₂³×C₂₃	—	2	2	no	—
95	C₂³×C₂₃	—	2	2	no	—
96	C₂³×C₂₃	—	4	2	no	—
97	C₂³×C₂₃	—	4	2	no	—
98	C₂³×C₂₃	—	4	2	no	—
99	C₂³×C₂₃	—	4	2	no	—
100	C₂³×C₂₃	—	4	1	no	—
101	C₂³×C₂₃	—	4	1	no	—
102	C₂³×C₂₃	C₂²⋊C₈	1	4	yes	—
103	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	D₄×C₂₃	—	2	4	no	—
105	D₄×C₂₃	—	2	4	no	—
106	D₄×C₂₃	—	2	4	no	—
107	D₄×C₂₃	—	4	2	no	—
108	D₄×C₂₃	—	8	8	no	—
109	C₂×C₄×C₂₃	—	2	2	no	—
110	C₂×C₄×C₂₃	—	2	2	no	—
111	C₂×C₄×C₂₃	—	4	1	no	—
112	C₂×C₄×C₂₃	—	4	1	no	—
113	C₂×C₄×C₂₃	—	4	4	no	—
114	C₂×C₄×C₂₃	—	4	1	no	—
115	C₂×C₄×C₂₃	—	4	1	no	—
116	C₂×C₄×C₂₃	—	4	2	no	—
117	C₂×C₄×C₂₃	—	4	1	no	—
118	C₂×C₄×C₂₃	—	4	1	no	—
119	C₂×C₄×C₂₃	—	4	2	no	—
120	C₂×C₄×C₂₃	—	4	1	no	—
121	C₂×C₄×C₂₃	—	4	1	no	—
122	C₂×C₄×C₂₃	—	8	2	no	—
123	C₂×C₄×C₂₃	—	8	4	no	—
124	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
126	C₂×C₄×C₂₃	#121	1	2	yes	—
127	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
128	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
130	C₂×C₄×C₂₃	#131	1	2	yes	—
131	C₈×C₂₃	—	4	4	no	—
132	C₈×C₂₃	—	8	2	no	—
133	#2972	C₂×C₂₃	1	1	yes	characteristic, complemented
134	#2328	C₂×C₂₃	1	1	yes	characteristic
135	#1478	C₂×C₂₃	1	2	yes	complemented
136	#1460	C₂×C₂₃	1	1	yes	characteristic, complemented
137	#1274	C₂×C₂₃	1	2	yes	complemented
138	C₂²×C₂₃	—	2	1	no	—
139	C₂²×C₂₃	—	2	1	no	—
140	C₂²×C₂₃	—	2	4	no	—
141	C₂²×C₂₃	—	4	4	no	—
142	C₂²×C₂₃	—	4	4	no	—
143	C₂²×C₂₃	—	4	2	no	—
144	C₂²×C₂₃	—	4	2	no	—
145	C₂²×C₂₃	—	4	2	no	—
146	C₂²×C₂₃	—	8	2	no	—
147	C₂²×C₂₃	—	8	2	no	—
148	C₂²×C₂₃	—	8	1	no	—
149	C₂²×C₂₃	—	8	1	no	—
150	C₂²×C₂₃	#386	1	1	yes	characteristic
151	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
152	C₂²×C₂₃	#408	1	1	yes	characteristic
153	C₄×C₂₃	—	2	2	no	—
154	C₄×C₂₃	—	2	1	no	—
155	C₄×C₂₃	—	2	1	no	—
156	C₄×C₂₃	—	4	2	no	—
157	C₄×C₂₃	—	4	2	no	—
158	C₄×C₂₃	—	8	1	no	—
159	C₄×C₂₃	—	8	1	no	—
160	C₄×C₂₃	C₈×D₄	1	2	yes	—
161	C₄×C₂₃	#435	1	2	yes	—
162	C₂×#127	C₄×C₂₃	1	1	yes	characteristic
163	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
164	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
165	C₈×D₄	—	2	4	no	—
166	#431	C₂²×C₂₃	1	2	yes	—
167	#431	C₂²×C₂₃	1	2	yes	—
168	C₂×C₂²⋊C₈	—	2	2	no	—
169	#390	C₄×C₂₃	1	1	yes	characteristic
170	#390	C₄×C₂₃	1	1	yes	characteristic
171	#386	—	2	2	no	—
172	#386	—	2	2	no	—
173	#382	C₄×C₂₃	1	1	yes	characteristic
174	#376	C₂²×C₂₃	1	1	yes	characteristic
175	D₄⋊C₈	—	2	4	no	—
176	#323	—	2	2	no	—
177	C₂×C₂₃	—	2	1	no	—
178	C₂×C₂₃	—	2	1	no	—
179	C₂×C₂₃	—	4	4	no	—
180	C₂×C₂₃	—	8	2	no	—
181	C₂×C₂₃	#1869	1	1	yes	characteristic
182	C₂×C₂₃	#1910	1	1	yes	characteristic
183	C₂×C₂₃	#1911	1	1	yes	characteristic
184	C₂²×D₄	—	2	2	no	—
185	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
186	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
187	C₂³×C₄	—	2	1	no	—
188	C₂³×C₄	D₄×C₂₃	1	2	yes	—
189	C₂²×C₈	—	2	4	no	—
190	#127	—	2	2	no	—
191	#127	C₈×C₂₃	1	4	yes	complemented
192	C₄×D₄	—	2	2	no	—
193	C₄×D₄	—	2	2	no	—
194	C₄×D₄	—	2	2	no	—
195	C₄×D₄	—	2	2	no	—
196	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
197	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
198	C₂×C₂²⋊C₄	—	2	1	no	—
199	C₂×C₂²⋊C₄	—	2	2	no	—
200	C₂×C₂²⋊C₄	—	2	2	no	—
201	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
202	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
203	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
204	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
205	C₄⋊C₈	—	2	2	no	—
206	C₄⋊C₈	—	2	2	no	—
207	C₂²⋊C₈	—	2	4	no	—
208	C₂²⋊C₈	—	2	2	no	—
209	C₂²⋊C₈	—	2	2	no	—
210	C₄×C₈	—	2	2	no	—
211	C₄×C₈	D₄×C₂₃	1	4	yes	complemented
212	#95	—	2	2	no	—
213	#95	—	2	1	no	—
214	#95	—	2	1	no	—
215	C₂₃	#9101	1	1	yes	characteristic, direct product factor, Sylow
216	C₂⁴	—	2	1	no	—
217	C₂⁴	C₂²⋊C₄×C₂₃	1	2	yes	—
218	C₂×D₄	—	2	8	no	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	—	2	2	no	—
221	C₂×D₄	—	2	1	no	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	8	2	no	—
225	C₂×D₄	—	8	2	no	—
226	C₂×D₄	C₂×C₈×C₂₃	1	2	yes	—
227	C₂×D₄	C₂×C₈×C₂₃	1	2	yes	complemented
228	C₂×D₄	M₁₆×C₂₃	1	2	yes	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	8	2	no	—
236	C₂²×C₄	—	8	2	no	—
237	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
238	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	—	8	4	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₂²⋊C₄	—	2	4	no	—
247	C₂²⋊C₄	—	4	1	no	—
248	C₂²⋊C₄	—	4	1	no	—
249	C₂²⋊C₄	—	8	2	no	—
250	C₂²⋊C₄	—	8	2	no	—
251	C₄²	—	2	1	no	—
252	C₄²	—	2	2	no	—
253	C₄²	C₂×C₈×C₂₃	1	2	yes	complemented
254	C₄²	C₂×D₄×C₂₃	1	2	yes	—
255	C₄²	#41×C₂₃	1	2	yes	—
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	4	2	no	—
260	C₂³	—	4	2	no	—
261	C₂³	—	4	2	no	—
262	C₂³	—	4	2	no	—
263	C₂³	—	4	1	no	—
264	C₂³	—	4	1	no	—
265	C₂³	C₂₃×C₂²⋊C₈	1	4	yes	—
266	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
267	D₄	—	2	4	no	—
268	D₄	—	2	4	no	—
269	D₄	—	2	4	no	—
270	D₄	—	4	2	no	—
271	D₄	—	8	8	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	8	2	no	—
286	C₂×C₄	—	8	4	no	—
287	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
288	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
289	C₂×C₄	C₂₃×#121	1	2	yes	—
290	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
291	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
292	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
293	C₂×C₄	C₂₃×#131	1	2	yes	—
294	C₈	—	4	4	no	—
295	C₈	—	8	2	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	4	4	no	—
300	C₂²	—	4	4	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	8	1	no	—
307	C₂²	—	8	1	no	—
308	C₂²	C₂₃×#386	1	1	yes	characteristic
309	C₂²	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
310	C₂²	C₂₃×#408	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	4	2	no	—
315	C₄	—	4	2	no	—
316	C₄	—	8	1	no	—
317	C₄	—	8	1	no	—
318	C₄	C₈×D₄×C₂₃	1	2	yes	—
319	C₄	C₂₃×#435	1	2	yes	—
320	C₂	—	2	1	no	—
321	C₂	—	2	1	no	—
322	C₂	—	4	4	no	—
323	C₂	—	8	2	no	—
324	C₂	C₂₃×#1869	1	1	yes	characteristic
325	C₂	C₂₃×#1910	1	1	yes	characteristic
326	C₂	C₂₃×#1911	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#386
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂×C₂₃×#127	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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. C₂₃×#2972	C₂	semidirect
3. C₂₃×#1478	C₂	semidirect
4. C₂₃×#1460	C₂	semidirect
5. C₂₃×#1274	C₂	semidirect
28. C₂₃×#127	C₈	semidirect
48. C₄×C₈×C₂₃	D₄	semidirect
63. C₂×D₄×C₂₃	C₂×C₈	semidirect
89. C₄²×C₂₃	C₂×C₈	semidirect
92. #9101	C₂₃	direct (N × H)
133. #2972	C₂×C₂₃	semidirect
135. #1478	C₂×C₂₃	semidirect
136. #1460	C₂×C₂₃	semidirect
137. #1274	C₂×C₂₃	semidirect
191. #127	C₈×C₂₃	semidirect
211. C₄×C₈	D₄×C₂₃	semidirect
215. C₂₃	#9101	direct (N × H)
227. C₂×D₄	C₂×C₈×C₂₃	semidirect
253. C₄²	C₂×C₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#2972	2
C₂₃×#2328	2
C₂₃×#1478	2
C₂₃×#1460	2
C₂₃×#1274	2
#9101	23

Subgroups — C23×#9101 #19156727

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9101 #19156727