Subgroups — C₂₃×#9588 #19157214

Order 5888.

304 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#1574	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#1543	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1492	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1484	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1452	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1299	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#532	C₂²	1	1	yes	characteristic
9	C₂₃×#431	C₂²	1	1	yes	characteristic
10	C₂₃×#431	C₂²	1	1	yes	characteristic
11	C₂₃×#431	C₂²	1	1	yes	characteristic
12	C₂₃×#423	C₂²	1	1	yes	characteristic
13	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#335	—	2	1	no	—
16	C₂₃×#334	—	2	1	no	—
17	C₂₃×#333	C₄	1	1	yes	characteristic
18	C₂₃×#333	C₄	1	1	yes	characteristic
19	C₂₃×#331	C₄	1	1	yes	characteristic
20	C₂₃×#331	C₄	1	1	yes	characteristic
21	C₂₃×D₄⋊C₈	—	2	2	no	—
22	C₂₃×D₄⋊C₈	—	2	1	no	—
23	C₂₃×D₄⋊C₈	—	2	1	no	—
24	C₂₃×#322	—	2	2	no	—
25	C₂×#41×C₂₃	—	2	1	no	—
26	C₂×M₁₆×C₂₃	—	2	1	no	—
27	C₂×M₁₆×C₂₃	—	2	1	no	—
28	C₂²×C₈×C₂₃	—	2	1	no	—
29	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
30	C₂₃×#127	C₂×C₄	1	1	yes	characteristic
31	C₂₃×#124	—	2	1	no	—
32	C₂₃×#121	—	2	1	no	—
33	C₂₃×#121	—	2	1	no	—
34	C₄×D₄×C₂₃	—	2	1	no	—
35	C₄×D₄×C₂₃	—	2	1	no	—
36	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
37	C₂₃×C₄⋊C₈	—	2	1	no	—
38	C₂₃×C₄⋊C₈	—	2	1	no	—
39	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
41	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
42	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
44	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
45	C₂₃×C₂²⋊C₈	—	2	1	no	—
46	C₂₃×C₂²⋊C₈	—	2	1	no	—
47	C₂₃×C₂²⋊C₈	—	2	1	no	—
48	C₂₃×#97	—	2	1	no	—
49	C₄×C₈×C₂₃	—	2	1	no	—
50	C₄×C₈×C₂₃	—	2	1	no	—
51	C₄×C₈×C₂₃	—	2	1	no	—
52	C₄×C₈×C₂₃	—	2	1	no	—
53	C₄×C₈×C₂₃	—	2	1	no	—
54	C₄×C₈×C₂₃	D₄	1	2	yes	—
55	#41×C₂₃	—	8	1	no	—
56	#41×C₂₃	—	8	1	no	—
57	C₂×Q₈×C₂₃	—	2	1	no	—
58	C₂×D₄×C₂₃	—	2	1	no	—
59	C₂×D₄×C₂₃	—	2	1	no	—
60	C₂×D₄×C₂₃	—	2	1	no	—
61	C₂×D₄×C₂₃	—	4	1	no	—
62	C₂²×C₄×C₂₃	—	4	1	no	—
63	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
66	M₁₆×C₂₃	—	2	2	no	—
67	M₁₆×C₂₃	—	4	1	no	—
68	M₁₆×C₂₃	—	8	1	no	—
69	C₂×C₈×C₂₃	—	2	1	no	—
70	C₂×C₈×C₂₃	—	2	1	no	—
71	C₂×C₈×C₂₃	—	2	1	no	—
72	C₂×C₈×C₂₃	—	2	1	no	—
73	C₂×C₈×C₂₃	—	2	1	no	—
74	C₂×C₈×C₂₃	—	2	1	no	—
75	C₂×C₈×C₂₃	—	2	2	no	—
76	C₂×C₈×C₂₃	—	4	1	no	—
77	C₂×C₈×C₂₃	—	4	1	no	—
78	C₂×C₈×C₂₃	—	4	1	no	—
79	C₂×C₈×C₂₃	—	4	1	no	—
80	C₄⋊C₄×C₂₃	—	2	1	no	—
81	C₄⋊C₄×C₂₃	—	2	1	no	—
82	C₂²⋊C₄×C₂₃	—	4	1	no	—
83	C₂²⋊C₄×C₂₃	—	4	1	no	—
84	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₂₃	#41	1	1	yes	characteristic
88	#9588	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₂₃	—	4	1	no	—
90	C₂³×C₂₃	#126	1	1	yes	characteristic
91	Q₈×C₂₃	—	8	1	no	—
92	D₄×C₂₃	—	4	1	no	—
93	D₄×C₂₃	—	4	1	no	—
94	D₄×C₂₃	—	4	1	no	—
95	D₄×C₂₃	—	4	1	no	—
96	D₄×C₂₃	—	8	1	no	—
97	D₄×C₂₃	—	8	2	no	—
98	C₂×C₄×C₂₃	—	2	1	no	—
99	C₂×C₄×C₂₃	—	2	1	no	—
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	2	1	no	—
102	C₂×C₄×C₂₃	—	2	1	no	—
103	C₂×C₄×C₂₃	—	2	1	no	—
104	C₂×C₄×C₂₃	—	4	1	no	—
105	C₂×C₄×C₂₃	—	8	1	no	—
106	C₂×C₄×C₂₃	—	8	1	no	—
107	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
108	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
111	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
112	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
113	C₈×C₂₃	—	4	2	no	—
114	C₈×C₂₃	—	4	2	no	—
115	C₈×C₂₃	—	4	1	no	—
116	C₈×C₂₃	—	4	1	no	—
117	C₈×C₂₃	—	4	1	no	—
118	C₈×C₂₃	—	4	1	no	—
119	C₈×C₂₃	—	4	1	no	—
120	C₈×C₂₃	—	4	1	no	—
121	C₈×C₂₃	—	8	1	no	—
122	#1574	C₂×C₂₃	1	1	yes	characteristic, complemented
123	#1543	C₂×C₂₃	1	1	yes	characteristic
124	#1492	C₂×C₂₃	1	1	yes	characteristic
125	#1484	C₂×C₂₃	1	1	yes	characteristic
126	#1452	C₂×C₂₃	1	1	yes	characteristic, complemented
127	#1299	C₂×C₂₃	1	1	yes	characteristic, complemented
128	#1278	C₂×C₂₃	1	1	yes	characteristic, complemented
129	C₂²×C₂₃	—	2	1	no	—
130	C₂²×C₂₃	—	2	1	no	—
131	C₂²×C₂₃	—	2	1	no	—
132	C₂²×C₂₃	—	8	1	no	—
133	C₂²×C₂₃	—	8	1	no	—
134	C₂²×C₂₃	—	8	1	no	—
135	C₂²×C₂₃	#388	1	1	yes	characteristic
136	C₄×C₂₃	—	2	1	no	—
137	C₄×C₂₃	—	2	1	no	—
138	C₄×C₂₃	—	2	1	no	—
139	C₄×C₂₃	—	8	1	no	—
140	C₄×C₂₃	#443	1	1	yes	characteristic
141	C₄×C₂₃	#444	1	1	yes	characteristic
142	C₄×C₂₃	#444	1	1	yes	characteristic
143	C₄×C₂₃	#444	1	1	yes	characteristic
144	C₄×C₂₃	#471	1	1	yes	characteristic
145	C₄×C₂₃	#472	1	1	yes	characteristic
146	#532	C₂²×C₂₃	1	1	yes	characteristic
147	#431	C₂²×C₂₃	1	1	yes	characteristic
148	#431	C₂²×C₂₃	1	1	yes	characteristic
149	#431	C₂²×C₂₃	1	1	yes	characteristic
150	#423	C₂²×C₂₃	1	1	yes	characteristic
151	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
153	#335	—	2	1	no	—
154	#334	—	2	1	no	—
155	#333	C₄×C₂₃	1	1	yes	characteristic
156	#333	C₄×C₂₃	1	1	yes	characteristic
157	#331	C₄×C₂₃	1	1	yes	characteristic
158	#331	C₄×C₂₃	1	1	yes	characteristic
159	D₄⋊C₈	—	2	1	no	—
160	D₄⋊C₈	—	2	1	no	—
161	D₄⋊C₈	—	2	2	no	—
162	#322	—	2	2	no	—
163	C₂×C₂₃	—	4	1	no	—
164	C₂×C₂₃	—	16	1	no	—
165	C₂×C₂₃	#1929	1	1	yes	characteristic
166	C₂×C₂₃	#1931	1	1	yes	characteristic
167	C₂×C₂₃	#1938	1	1	yes	characteristic
168	C₂×#41	—	2	1	no	—
169	C₂×M₁₆	—	2	1	no	—
170	C₂×M₁₆	—	2	1	no	—
171	C₂²×C₈	—	2	1	no	—
172	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
173	#127	C₂×C₄×C₂₃	1	1	yes	characteristic
174	#124	—	2	1	no	—
175	#121	—	2	1	no	—
176	#121	—	2	1	no	—
177	C₄×D₄	—	2	1	no	—
178	C₄×D₄	—	2	1	no	—
179	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
180	C₄⋊C₈	—	2	1	no	—
181	C₄⋊C₈	—	2	1	no	—
182	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
183	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
184	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
185	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
186	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
187	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
188	C₂²⋊C₈	—	2	1	no	—
189	C₂²⋊C₈	—	2	1	no	—
190	C₂²⋊C₈	—	2	1	no	—
191	#97	—	2	1	no	—
192	C₄×C₈	—	2	1	no	—
193	C₄×C₈	—	2	1	no	—
194	C₄×C₈	—	2	1	no	—
195	C₄×C₈	—	2	1	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	D₄×C₂₃	1	2	yes	—
198	C₂₃	#9588	1	1	yes	characteristic, direct product factor, Sylow
199	#41	—	8	1	no	—
200	#41	—	8	1	no	—
201	C₂×Q₈	—	2	1	no	—
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	—	2	1	no	—
204	C₂×D₄	—	2	1	no	—
205	C₂×D₄	—	4	1	no	—
206	C₂²×C₄	—	4	1	no	—
207	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
208	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
209	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
210	M₁₆	—	2	2	no	—
211	M₁₆	—	4	1	no	—
212	M₁₆	—	8	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₄⋊C₄	—	2	1	no	—
225	C₄⋊C₄	—	2	1	no	—
226	C₂²⋊C₄	—	4	1	no	—
227	C₂²⋊C₄	—	4	1	no	—
228	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
229	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
230	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
231	C₄²	#41×C₂₃	1	1	yes	characteristic
232	C₂³	—	4	1	no	—
233	C₂³	C₂₃×#126	1	1	yes	characteristic
234	Q₈	—	8	1	no	—
235	D₄	—	4	1	no	—
236	D₄	—	4	1	no	—
237	D₄	—	4	1	no	—
238	D₄	—	4	1	no	—
239	D₄	—	8	1	no	—
240	D₄	—	8	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	8	1	no	—
249	C₂×C₄	—	8	1	no	—
250	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
251	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
252	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
253	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
254	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
255	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
256	C₈	—	4	2	no	—
257	C₈	—	4	2	no	—
258	C₈	—	4	1	no	—
259	C₈	—	4	1	no	—
260	C₈	—	4	1	no	—
261	C₈	—	4	1	no	—
262	C₈	—	4	1	no	—
263	C₈	—	4	1	no	—
264	C₈	—	8	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	8	1	no	—
269	C₂²	—	8	1	no	—
270	C₂²	—	8	1	no	—
271	C₂²	C₂₃×#388	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	8	1	no	—
276	C₄	C₂₃×#443	1	1	yes	characteristic
277	C₄	C₂₃×#444	1	1	yes	characteristic
278	C₄	C₂₃×#444	1	1	yes	characteristic
279	C₄	C₂₃×#444	1	1	yes	characteristic
280	C₄	C₂₃×#471	1	1	yes	characteristic
281	C₄	C₂₃×#472	1	1	yes	characteristic
282	C₂	—	4	1	no	—
283	C₂	—	16	1	no	—
284	C₂	C₂₃×#1929	1	1	yes	characteristic
285	C₂	C₂₃×#1931	1	1	yes	characteristic
286	C₂	C₂₃×#1938	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#121
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#388
prime radical	C₂₃×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#1574	C₂	semidirect
5. C₂₃×#1452	C₂	semidirect
6. C₂₃×#1299	C₂	semidirect
7. C₂₃×#1278	C₂	semidirect
88. #9588	C₂₃	direct (N × H)
122. #1574	C₂×C₂₃	semidirect
126. #1452	C₂×C₂₃	semidirect
127. #1299	C₂×C₂₃	semidirect
128. #1278	C₂×C₂₃	semidirect
198. C₂₃	#9588	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×#1574	2
C₂₃×#1543	2
C₂₃×#1492	2
C₂₃×#1484	2
C₂₃×#1452	2
C₂₃×#1299	2
C₂₃×#1278	2
#9588	23

Subgroups — C23×#9588 #19157214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9588 #19157214