Subgroups — C₂₃×#19632 #19167258

Order 5888.

750 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3230	C₂	1	2	yes	maximal
2	C₂₃×#3170	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3165	C₂	1	2	yes	maximal
4	C₂×C₂₃×#494	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1723	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1713	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1707	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₂₃	—	2	2	no	—
10	C₂₃×#533	C₂²	1	2	yes	—
11	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₂₃×#501	—	2	4	no	—
13	C₂₃×#500	C₂²	1	4	yes	—
14	C₂₃×#500	C₂²	1	1	yes	characteristic
15	C₂₃×#500	C₂²	1	4	yes	—
16	C₂₃×#498	C₂²	1	4	yes	—
17	C₂₃×#494	—	2	2	no	—
18	C₂₃×#494	C₂²	1	1	yes	characteristic
19	C₂₃×#494	C₂²	1	2	yes	—
20	C₂₃×#494	C₂²	1	2	yes	—
21	C₂₃×#487	C₂²	1	2	yes	—
22	C₂₃×#487	C₂²	1	4	yes	—
23	C₂₃×#484	—	2	4	no	—
24	C₂₃×#477	—	2	4	no	—
25	C₂₃×#467	—	2	2	no	—
26	C₄×Q₁₆×C₂₃	—	2	2	no	—
27	C₂₃×#431	C₂²	1	1	yes	characteristic
28	C₂₃×#428	—	2	4	no	—
29	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
31	C₂₃×#322	C₂²	1	2	yes	—
32	C₈²×C₂₃	C₂²	1	2	yes	—
33	C₂²×Q₈×C₂₃	—	2	2	no	—
34	C₂×Q₁₆×C₂₃	—	2	4	no	—
35	C₂×Q₁₆×C₂₃	—	2	4	no	—
36	C₂×Q₁₆×C₂₃	—	2	2	no	—
37	C₂×Q₁₆×C₂₃	—	4	8	no	—
38	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
39	C₂²×C₈×C₂₃	D₄	1	2	yes	—
40	C₂₃×C₄⋊Q₈	—	2	2	no	—
41	C₂₃×C₄⋊Q₈	—	2	2	no	—
42	C₂₃×C₄⋊Q₈	—	2	2	no	—
43	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
44	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
45	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
46	C₂₃×#125	—	2	2	no	—
47	C₂₃×C₂²⋊Q₈	—	2	2	no	—
48	C₂₃×C₂²⋊Q₈	—	2	2	no	—
49	C₄×Q₈×C₂₃	—	2	2	no	—
50	C₄×Q₈×C₂₃	—	2	2	no	—
51	C₂₃×#117	—	2	2	no	—
52	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
53	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
54	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
55	C₂₃×#107	—	2	8	no	—
56	C₂₃×#107	—	2	2	no	—
57	C₂₃×#107	—	2	8	no	—
58	C₂₃×#106	—	2	8	no	—
59	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
60	C₂₃×#103	—	2	4	no	—
61	C₂₃×#103	—	2	8	no	—
62	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
63	C₂₃×#97	D₄	1	2	yes	—
64	C₄×C₈×C₂₃	D₄	1	2	yes	—
65	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₂₃	C₂³	1	2	yes	—
67	C₄×C₈×C₂₃	C₂³	1	2	yes	—
68	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
69	C₂×Q₈×C₂₃	—	2	2	no	—
70	C₂×Q₈×C₂₃	—	2	2	no	—
71	C₂×Q₈×C₂₃	—	2	4	no	—
72	C₂×Q₈×C₂₃	—	4	8	no	—
73	C₂×Q₈×C₂₃	—	4	2	no	—
74	C₂²×C₄×C₂₃	—	4	2	no	—
75	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
78	Q₁₆×C₂₃	—	4	8	no	—
79	Q₁₆×C₂₃	—	4	4	no	—
80	Q₁₆×C₂₃	—	4	8	no	—
81	M₁₆×C₂₃	Q₁₆	1	8	yes	complemented
82	C₂×C₈×C₂₃	—	2	4	no	—
83	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
84	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
86	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
87	C₄⋊C₄×C₂₃	—	2	4	no	—
88	C₄⋊C₄×C₂₃	—	2	2	no	—
89	C₄⋊C₄×C₂₃	—	2	2	no	—
90	C₄⋊C₄×C₂₃	—	2	4	no	—
91	C₄⋊C₄×C₂₃	—	2	2	no	—
92	C₄⋊C₄×C₂₃	—	2	2	no	—
93	C₄⋊C₄×C₂₃	—	4	2	no	—
94	C₄⋊C₄×C₂₃	—	4	2	no	—
95	C₄⋊C₄×C₂₃	—	4	2	no	—
96	C₄⋊C₄×C₂₃	—	4	2	no	—
97	C₂²⋊C₄×C₂₃	—	4	2	no	—
98	C₄²×C₂₃	—	4	2	no	—
99	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
103	#19632	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂³×C₂₃	#127	1	1	yes	characteristic
105	Q₈×C₂₃	—	4	8	no	—
106	Q₈×C₂₃	—	4	4	no	—
107	Q₈×C₂₃	—	4	4	no	—
108	Q₈×C₂₃	—	8	4	no	—
109	C₂×C₄×C₂₃	—	2	2	no	—
110	C₂×C₄×C₂₃	—	2	2	no	—
111	C₂×C₄×C₂₃	—	4	2	no	—
112	C₂×C₄×C₂₃	—	4	2	no	—
113	C₂×C₄×C₂₃	—	4	4	no	—
114	C₂×C₄×C₂₃	—	8	4	no	—
115	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
116	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
117	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
118	C₂×C₄×C₂₃	C₂×Q₁₆	1	4	yes	—
119	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
122	C₈×C₂₃	—	2	2	no	—
123	C₈×C₂₃	—	2	4	no	—
124	C₈×C₂₃	C₂×Q₁₆	1	8	yes	complemented
125	C₈×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
126	#3230	C₂×C₂₃	1	2	yes	—
127	#3170	C₂×C₂₃	1	1	yes	characteristic
128	#3165	C₂×C₂₃	1	2	yes	—
129	C₂×#494	C₂×C₂₃	1	1	yes	characteristic
130	#1723	C₂×C₂₃	1	4	yes	complemented
131	#1713	C₂×C₂₃	1	2	yes	complemented
132	#1707	C₂×C₂₃	1	2	yes	complemented
133	C₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic
134	C₂²×C₂₃	—	2	1	no	—
135	C₂²×C₂₃	—	2	1	no	—
136	C₂²×C₂₃	#494	1	2	yes	—
137	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
138	C₄×C₂₃	—	2	1	no	—
139	C₄×C₂₃	—	2	1	no	—
140	C₄×C₂₃	—	8	4	no	—
141	C₄×C₂₃	—	8	2	no	—
142	C₄×C₂₃	—	8	2	no	—
143	C₄×C₂₃	#494	1	2	yes	—
144	C₄×C₂₃	C₂²×Q₁₆	1	2	yes	—
145	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
146	C₄×C₂₃	#578	1	2	yes	—
147	C₂²×Q₁₆	—	2	2	no	—
148	#533	C₂²×C₂₃	1	2	yes	—
149	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
150	#501	—	2	4	no	—
151	#500	C₂²×C₂₃	1	4	yes	—
152	#500	C₂²×C₂₃	1	1	yes	characteristic
153	#500	C₂²×C₂₃	1	4	yes	—
154	#498	C₂²×C₂₃	1	4	yes	—
155	#494	—	2	2	no	—
156	#494	C₂²×C₂₃	1	2	yes	—
157	#494	C₂²×C₂₃	1	2	yes	—
158	#494	C₂²×C₂₃	1	1	yes	characteristic
159	#487	C₂²×C₂₃	1	2	yes	—
160	#487	C₂²×C₂₃	1	4	yes	—
161	#484	—	2	4	no	—
162	#477	—	2	4	no	—
163	#467	—	2	2	no	—
164	C₄×Q₁₆	—	2	2	no	—
165	#431	C₂²×C₂₃	1	1	yes	characteristic
166	#428	—	2	4	no	—
167	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
168	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
169	#322	C₂²×C₂₃	1	2	yes	—
170	C₈²	C₂²×C₂₃	1	2	yes	—
171	C₂×C₂₃	—	2	2	no	—
172	C₂×C₂₃	C₂×#494	1	1	yes	characteristic
173	C₂×C₂₃	#3151	1	1	yes	characteristic
174	C₂×C₂₃	#3156	1	1	yes	characteristic
175	C₂²×Q₈	—	2	2	no	—
176	C₂×Q₁₆	—	2	4	no	—
177	C₂×Q₁₆	—	2	4	no	—
178	C₂×Q₁₆	—	2	2	no	—
179	C₂×Q₁₆	—	4	8	no	—
180	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
181	C₂²×C₈	D₄×C₂₃	1	2	yes	—
182	C₄⋊Q₈	—	2	2	no	—
183	C₄⋊Q₈	—	2	2	no	—
184	C₄⋊Q₈	—	2	2	no	—
185	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
186	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
187	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
188	#125	—	2	2	no	—
189	C₂²⋊Q₈	—	2	2	no	—
190	C₂²⋊Q₈	—	2	2	no	—
191	C₄×Q₈	—	2	2	no	—
192	C₄×Q₈	—	2	2	no	—
193	#117	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
197	#107	—	2	8	no	—
198	#107	—	2	2	no	—
199	#107	—	2	8	no	—
200	#106	—	2	8	no	—
201	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
202	#103	—	2	4	no	—
203	#103	—	2	8	no	—
204	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
205	#97	D₄×C₂₃	1	2	yes	—
206	C₄×C₈	D₄×C₂₃	1	2	yes	—
207	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
208	C₄×C₈	C₂³×C₂₃	1	2	yes	—
209	C₄×C₈	C₂³×C₂₃	1	2	yes	—
210	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
211	C₂₃	#19632	1	1	yes	characteristic, direct product factor, Sylow
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	4	no	—
215	C₂×Q₈	—	4	8	no	—
216	C₂×Q₈	—	4	2	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
221	Q₁₆	—	4	8	no	—
222	Q₁₆	—	4	8	no	—
223	Q₁₆	—	4	4	no	—
224	M₁₆	Q₁₆×C₂₃	1	8	yes	complemented
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
227	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
228	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
229	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₄²	—	4	2	no	—
242	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
243	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
244	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
245	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
246	C₂³	C₂₃×#127	1	1	yes	characteristic
247	Q₈	—	4	8	no	—
248	Q₈	—	4	4	no	—
249	Q₈	—	4	4	no	—
250	Q₈	—	8	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	8	4	no	—
257	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
258	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
259	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
260	C₂×C₄	C₂×Q₁₆×C₂₃	1	4	yes	—
261	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
262	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
263	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	2	4	no	—
266	C₈	C₂×Q₁₆×C₂₃	1	8	yes	complemented
267	C₈	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	C₂₃×#494	1	2	yes	—
271	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	8	4	no	—
275	C₄	—	8	2	no	—
276	C₄	—	8	2	no	—
277	C₄	C₂₃×#494	1	2	yes	—
278	C₄	C₂²×Q₁₆×C₂₃	1	2	yes	—
279	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
280	C₄	C₂₃×#578	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	C₂×C₂₃×#494	1	1	yes	characteristic
283	C₂	C₂₃×#3151	1	1	yes	characteristic
284	C₂	C₂₃×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂³×C₂₃	#127
LCS	G ▷ C₄² ▷ C₂² ▷ 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
5. C₂₃×#1723	C₂	semidirect
6. C₂₃×#1713	C₂	semidirect
7. C₂₃×#1707	C₂	semidirect
81. M₁₆×C₂₃	Q₁₆	semidirect
103. #19632	C₂₃	direct (N × H)
124. C₈×C₂₃	C₂×Q₁₆	semidirect
130. #1723	C₂×C₂₃	semidirect
131. #1713	C₂×C₂₃	semidirect
132. #1707	C₂×C₂₃	semidirect
211. C₂₃	#19632	direct (N × H)
224. M₁₆	Q₁₆×C₂₃	semidirect
266. C₈	C₂×Q₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3230	2
C₂₃×#3170	2
C₂₃×#3165	2
C₂×C₂₃×#494	2
C₂₃×#1723	2
C₂₃×#1713	2
C₂₃×#1707	2
C₈×M₁₆×C₂₃	2
#19632	23

Subgroups — C23×#19632 #19167258

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19632 #19167258