Subgroups — C₂₃×#9668 #19157294

Order 5888.

510 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1478	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1460	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₂₃	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
8	C₄×D₈×C₂₃	—	2	1	no	—
9	C₄×D₈×C₂₃	—	2	1	no	—
10	C₄×D₈×C₂₃	—	2	1	no	—
11	C₄×D₈×C₂₃	—	2	1	no	—
12	C₂₃×#431	C₂²	1	2	yes	—
13	C₂₃×#431	C₂²	1	1	yes	characteristic
14	C₂₃×#431	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#107	C₄	1	1	yes	characteristic
16	C₂×C₂₃×#102	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#335	—	2	1	no	—
19	C₂₃×#334	—	2	1	no	—
20	C₂₃×D₄⋊C₈	—	2	4	no	—
21	C₂₃×#323	—	2	4	no	—
22	C₂₃×#322	—	2	1	no	—
23	C₈²×C₂₃	—	2	1	no	—
24	C₂²×D₄×C₂₃	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₂₃	—	2	2	no	—
26	C₂×D₈×C₂₃	—	2	1	no	—
27	C₂×D₈×C₂₃	—	2	1	no	—
28	C₂×D₈×C₂₃	—	2	2	no	—
29	C₂×D₈×C₂₃	C₈	1	4	yes	complemented
30	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₄×D₄×C₂₃	—	2	2	no	—
33	C₄×D₄×C₂₃	—	2	2	no	—
34	C₄×D₄×C₂₃	—	2	2	no	—
35	C₄×D₄×C₂₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
37	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
38	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
39	C₂₃×#107	—	2	1	no	—
40	C₂₃×#107	—	2	1	no	—
41	C₂₃×C₄⋊C₈	—	2	2	no	—
42	C₂₃×C₄⋊C₈	—	2	1	no	—
43	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
44	C₂₃×#102	—	2	2	no	—
45	C₂₃×#102	—	2	2	no	—
46	C₂₃×C₂²⋊C₈	—	2	1	no	—
47	C₂₃×C₂²⋊C₈	—	2	2	no	—
48	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
49	C₄×C₈×C₂₃	—	2	1	no	—
50	C₄×C₈×C₂₃	—	2	1	no	—
51	C₄×C₈×C₂₃	—	2	2	no	—
52	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
54	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
55	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
56	C₂⁴×C₂₃	—	2	2	no	—
57	C₂×D₄×C₂₃	—	2	2	no	—
58	C₂×D₄×C₂₃	—	2	2	no	—
59	C₂×D₄×C₂₃	—	2	2	no	—
60	C₂×D₄×C₂₃	—	2	2	no	—
61	C₂×D₄×C₂₃	—	2	2	no	—
62	C₂×D₄×C₂₃	C₂×C₈	1	4	yes	—
63	C₂²×C₄×C₂₃	—	2	2	no	—
64	C₂²×C₄×C₂₃	—	4	2	no	—
65	C₂²×C₄×C₂₃	—	4	2	no	—
66	C₂²×C₄×C₂₃	—	8	4	no	—
67	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
70	D₈×C₂₃	—	2	4	no	—
71	D₈×C₂₃	—	2	4	no	—
72	C₂×C₈×C₂₃	—	2	1	no	—
73	C₂×C₈×C₂₃	—	2	1	no	—
74	C₂×C₈×C₂₃	—	2	2	no	—
75	C₂×C₈×C₂₃	—	2	2	no	—
76	C₂×C₈×C₂₃	—	4	2	no	—
77	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
82	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
83	C₄⋊C₄×C₂₃	—	2	2	no	—
84	C₄⋊C₄×C₂₃	—	2	2	no	—
85	C₂²⋊C₄×C₂₃	—	4	2	no	—
86	C₂²⋊C₄×C₂₃	—	4	2	no	—
87	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₂₃	#41	1	1	yes	characteristic
91	#9668	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₂₃	—	2	4	no	—
93	C₂³×C₂₃	—	4	2	no	—
94	C₂³×C₂₃	—	4	2	no	—
95	C₂³×C₂₃	—	8	2	no	—
96	C₂³×C₂₃	—	8	2	no	—
97	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
98	D₄×C₂₃	—	2	4	no	—
99	D₄×C₂₃	—	2	4	no	—
100	D₄×C₂₃	—	4	2	no	—
101	D₄×C₂₃	—	4	2	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	4	1	no	—
104	C₂×C₄×C₂₃	—	4	2	no	—
105	C₂×C₄×C₂₃	—	8	4	no	—
106	C₂×C₄×C₂₃	—	8	4	no	—
107	C₂×C₄×C₂₃	—	8	2	no	—
108	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	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	#121	1	1	yes	characteristic
112	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
113	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
116	C₂×C₄×C₂₃	#131	1	1	yes	characteristic
117	C₂×C₄×C₂₃	#131	1	1	yes	characteristic
118	C₈×C₂₃	—	2	1	no	—
119	C₈×C₂₃	—	2	1	no	—
120	C₈×C₂₃	—	4	2	no	—
121	C₈×C₂₃	—	4	2	no	—
122	C₈×C₂₃	—	8	2	no	—
123	C₈×C₂₃	C₂²⋊C₈	1	2	yes	—
124	C₈×C₂₃	C₂²⋊C₈	1	2	yes	—
125	C₂×C₄×D₈	C₂×C₂₃	1	1	yes	characteristic
126	#1564	C₂×C₂₃	1	1	yes	characteristic, complemented
127	#1478	C₂×C₂₃	1	2	yes	complemented
128	#1460	C₂×C₂₃	1	2	yes	complemented
129	#1450	C₂×C₂₃	1	1	yes	characteristic, complemented
130	C₂²×C₂₃	—	2	1	no	—
131	C₂²×C₂₃	—	2	1	no	—
132	C₂²×C₂₃	—	4	4	no	—
133	C₂²×C₂₃	—	8	4	no	—
134	C₂²×C₂₃	—	8	4	no	—
135	C₂²×C₂₃	—	8	2	no	—
136	C₂²×C₂₃	—	8	2	no	—
137	C₂²×C₂₃	#390	1	1	yes	characteristic
138	C₂²×C₂₃	C₈×D₄	1	1	yes	characteristic
139	C₂²×C₂₃	#436	1	1	yes	characteristic
140	C₄×C₂₃	—	2	1	no	—
141	C₄×C₂₃	—	2	1	no	—
142	C₄×C₂₃	—	2	1	no	—
143	C₄×C₂₃	—	8	2	no	—
144	C₄×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₄×C₂₃	#408	1	1	yes	characteristic
146	C₄×C₂₃	C₈×D₄	1	1	yes	characteristic
147	C₄×C₂₃	#436	1	1	yes	characteristic
148	C₄×C₂₃	#472	1	2	yes	—
149	C₂²×D₈	C₄×C₂₃	1	1	yes	characteristic
150	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
151	C₄×D₈	—	2	1	no	—
152	C₄×D₈	—	2	1	no	—
153	C₄×D₈	—	2	1	no	—
154	C₄×D₈	—	2	1	no	—
155	#431	C₂²×C₂₃	1	2	yes	—
156	#431	C₂²×C₂₃	1	1	yes	characteristic
157	#431	C₂²×C₂₃	1	1	yes	characteristic
158	C₂×#107	C₄×C₂₃	1	1	yes	characteristic
159	C₂×#102	C₄×C₂₃	1	2	yes	—
160	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
161	#335	—	2	1	no	—
162	#334	—	2	1	no	—
163	D₄⋊C₈	—	2	4	no	—
164	#323	—	2	4	no	—
165	#322	—	2	1	no	—
166	C₈²	—	2	1	no	—
167	C₂×C₂₃	—	2	1	no	—
168	C₂×C₂₃	—	2	1	no	—
169	C₂×C₂₃	—	8	4	no	—
170	C₂×C₂₃	#1954	1	1	yes	characteristic
171	C₂×C₂₃	#1976	1	1	yes	characteristic
172	C₂×C₂₃	#1978	1	1	yes	characteristic
173	C₂²×D₄	C₂×C₄×C₂₃	1	2	yes	—
174	C₂³×C₄	—	2	2	no	—
175	C₂×D₈	—	2	1	no	—
176	C₂×D₈	—	2	1	no	—
177	C₂×D₈	—	2	2	no	—
178	C₂×D₈	C₈×C₂₃	1	4	yes	complemented
179	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
180	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
181	C₄×D₄	—	2	2	no	—
182	C₄×D₄	—	2	2	no	—
183	C₄×D₄	—	2	2	no	—
184	C₄×D₄	—	2	2	no	—
185	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
186	C₂×C₂²⋊C₄	—	2	2	no	—
187	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
188	#107	—	2	1	no	—
189	#107	—	2	1	no	—
190	C₄⋊C₈	—	2	2	no	—
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
193	#102	—	2	2	no	—
194	#102	—	2	2	no	—
195	C₂²⋊C₈	—	2	1	no	—
196	C₂²⋊C₈	—	2	2	no	—
197	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
198	C₄×C₈	—	2	1	no	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	—	2	2	no	—
201	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
202	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
203	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
204	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
205	C₂₃	#9668	1	1	yes	characteristic, direct product factor, Sylow
206	C₂⁴	—	2	2	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	C₂×C₈×C₂₃	1	4	yes	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	—	8	4	no	—
217	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
219	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
220	D₈	—	2	4	no	—
221	D₈	—	2	4	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
228	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
229	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
230	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
231	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
232	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₂²⋊C₄	—	4	2	no	—
236	C₂²⋊C₄	—	4	2	no	—
237	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
238	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
239	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
240	C₄²	#41×C₂₃	1	1	yes	characteristic
241	C₂³	—	2	4	no	—
242	C₂³	—	4	2	no	—
243	C₂³	—	4	2	no	—
244	C₂³	—	8	2	no	—
245	C₂³	—	8	2	no	—
246	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
247	D₄	—	2	4	no	—
248	D₄	—	2	4	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	8	4	no	—
255	C₂×C₄	—	8	4	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
258	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
259	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
260	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
261	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
262	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
263	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
264	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
265	C₂×C₄	C₂₃×#131	1	1	yes	characteristic
266	C₂×C₄	C₂₃×#131	1	1	yes	characteristic
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	4	2	no	—
270	C₈	—	4	2	no	—
271	C₈	—	8	2	no	—
272	C₈	C₂₃×C₂²⋊C₈	1	2	yes	—
273	C₈	C₂₃×C₂²⋊C₈	1	2	yes	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	4	4	no	—
277	C₂²	—	8	4	no	—
278	C₂²	—	8	4	no	—
279	C₂²	—	8	2	no	—
280	C₂²	—	8	2	no	—
281	C₂²	C₂₃×#390	1	1	yes	characteristic
282	C₂²	C₈×D₄×C₂₃	1	1	yes	characteristic
283	C₂²	C₂₃×#436	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	8	2	no	—
288	C₄	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
289	C₄	C₂₃×#408	1	1	yes	characteristic
290	C₄	C₈×D₄×C₂₃	1	1	yes	characteristic
291	C₄	C₂₃×#436	1	1	yes	characteristic
292	C₄	C₂₃×#472	1	2	yes	—
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	8	4	no	—
296	C₂	C₂₃×#1954	1	1	yes	characteristic
297	C₂	C₂₃×#1976	1	1	yes	characteristic
298	C₂	C₂₃×#1978	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#121
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#390
prime radical	C₂²×D₈×C₂₃	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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
2. C₂₃×#1564	C₂	semidirect
3. C₂₃×#1478	C₂	semidirect
4. C₂₃×#1460	C₂	semidirect
5. C₂₃×#1450	C₂	semidirect
29. C₂×D₈×C₂₃	C₈	semidirect
91. #9668	C₂₃	direct (N × H)
126. #1564	C₂×C₂₃	semidirect
127. #1478	C₂×C₂₃	semidirect
128. #1460	C₂×C₂₃	semidirect
129. #1450	C₂×C₂₃	semidirect
178. C₂×D₈	C₈×C₂₃	semidirect
205. C₂₃	#9668	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₂₃	2
C₂₃×#1564	2
C₂₃×#1478	2
C₂₃×#1460	2
C₂₃×#1450	2
#9668	23

Subgroups — C23×#9668 #19157294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9668 #19157294