Subgroups — C₂₃×#19596 #19167222

Order 5888.

814 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3167	C₂	1	4	yes	maximal, complemented
2	C₂₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3147	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1708	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1707	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1706	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1705	C₂	1	2	yes	maximal, complemented
8	C₂×C₈²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#135	—	2	2	no	—
10	C₂₃×#533	C₂²	1	2	yes	—
11	C₂₃×#532	C₂²	1	2	yes	—
12	C₂₃×#500	C₂²	1	2	yes	—
13	C₂₃×#499	—	2	2	no	—
14	C₂₃×#498	C₂²	1	4	yes	—
15	C₂₃×#498	C₂²	1	2	yes	—
16	C₂₃×#495	—	2	2	no	—
17	C₂₃×#494	C₂²	1	1	yes	characteristic
18	C₂₃×#493	C₂²	1	1	yes	characteristic
19	C₂₃×#492	C₂²	1	4	yes	—
20	C₂₃×#492	C₂²	1	2	yes	—
21	C₂₃×#487	C₂²	1	4	yes	—
22	C₂₃×#486	C₂²	1	4	yes	—
23	C₂₃×#465	—	2	8	no	—
24	C₄×SD₁₆×C₂₃	—	2	8	no	—
25	C₂₃×#427	—	2	2	no	—
26	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₈²×C₂₃	C₂²	1	1	yes	characteristic
29	C₈²×C₂₃	C₂²	1	2	yes	—
30	C₈²×C₂₃	C₂²	1	1	yes	characteristic
31	C₂×#41×C₂₃	—	2	2	no	—
32	C₂₃×#135	—	4	8	no	—
33	C₂×Q₁₆×C₂₃	—	2	2	no	—
34	C₂×SD₁₆×C₂₃	—	2	4	no	—
35	C₂×SD₁₆×C₂₃	—	2	8	no	—
36	C₂×D₈×C₂₃	—	2	2	no	—
37	C₂²×C₈×C₂₃	D₄	1	4	yes	—
38	C₂²×C₈×C₂₃	D₄	1	2	yes	—
39	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
40	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
41	C₂₃×#127	C₂³	1	2	yes	—
42	C₂₃×#125	—	2	2	no	—
43	C₂₃×#124	—	2	2	no	—
44	C₂₃×C₂²⋊Q₈	—	2	4	no	—
45	C₂₃×#121	—	2	4	no	—
46	C₄×Q₈×C₂₃	—	2	4	no	—
47	C₄×D₄×C₂₃	—	2	4	no	—
48	C₂₃×#117	—	2	2	no	—
49	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
50	C₂₃×#107	—	2	4	no	—
51	C₂₃×#106	—	2	4	no	—
52	C₂₃×#106	—	2	8	no	—
53	C₂₃×#103	—	2	8	no	—
54	C₂₃×#102	—	2	8	no	—
55	C₄×C₈×C₂₃	D₄	1	4	yes	—
56	C₄×C₈×C₂₃	D₄	1	2	yes	—
57	C₄×C₈×C₂₃	C₂³	1	2	yes	—
58	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₂₃	C₂³	1	2	yes	—
60	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
61	#41×C₂₃	—	4	8	no	—
62	C₂×Q₈×C₂₃	—	2	4	no	—
63	C₂×Q₈×C₂₃	—	2	2	no	—
64	C₂×D₄×C₂₃	—	2	4	no	—
65	C₂×D₄×C₂₃	—	2	2	no	—
66	C₂×D₄×C₂₃	—	4	2	no	—
67	C₂²×C₄×C₂₃	—	4	2	no	—
68	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
70	Q₁₆×C₂₃	—	4	4	no	—
71	SD₁₆×C₂₃	—	4	8	no	—
72	SD₁₆×C₂₃	—	4	16	no	—
73	D₈×C₂₃	—	4	4	no	—
74	C₂×C₈×C₂₃	—	2	8	no	—
75	C₂×C₈×C₂₃	SD₁₆	1	8	yes	complemented
76	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
77	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
80	C₄⋊C₄×C₂₃	—	2	4	no	—
81	C₄⋊C₄×C₂₃	—	2	4	no	—
82	C₄⋊C₄×C₂₃	—	2	4	no	—
83	C₄⋊C₄×C₂₃	—	4	4	no	—
84	C₂²⋊C₄×C₂₃	—	4	2	no	—
85	C₂²⋊C₄×C₂₃	—	4	4	no	—
86	C₄²×C₂₃	—	4	2	no	—
87	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₂₃	C₂×D₄	1	2	yes	—
89	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
90	#19596	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₂₃	—	4	2	no	—
92	C₂³×C₂₃	#127	1	1	yes	characteristic
93	Q₈×C₂₃	—	4	4	no	—
94	Q₈×C₂₃	—	4	8	no	—
95	D₄×C₂₃	—	4	4	no	—
96	D₄×C₂₃	—	4	8	no	—
97	D₄×C₂₃	—	8	4	no	—
98	C₂×C₄×C₂₃	—	2	4	no	—
99	C₂×C₄×C₂₃	—	4	4	no	—
100	C₂×C₄×C₂₃	—	4	2	no	—
101	C₂×C₄×C₂₃	—	8	4	no	—
102	C₂×C₄×C₂₃	#127	1	2	yes	—
103	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
105	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
107	C₈×C₂₃	—	2	4	no	—
108	C₈×C₂₃	C₂×SD₁₆	1	4	yes	—
109	C₈×C₂₃	C₂×SD₁₆	1	4	yes	—
110	C₈×C₂₃	#135	1	8	yes	complemented
111	#3167	C₂×C₂₃	1	4	yes	complemented
112	#3160	C₂×C₂₃	1	1	yes	characteristic, complemented
113	#3147	C₂×C₂₃	1	1	yes	characteristic
114	#1708	C₂×C₂₃	1	2	yes	complemented
115	#1707	C₂×C₂₃	1	2	yes	complemented
116	#1706	C₂×C₂₃	1	2	yes	complemented
117	#1705	C₂×C₂₃	1	2	yes	complemented
118	C₂×C₈²	C₂×C₂₃	1	1	yes	characteristic, complemented
119	C₂²×C₂₃	—	2	2	no	—
120	C₂²×C₂₃	—	8	2	no	—
121	C₂²×C₂₃	—	8	4	no	—
122	C₂²×C₂₃	#492	1	2	yes	—
123	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
124	C₄×C₂₃	—	2	2	no	—
125	C₄×C₂₃	—	8	2	no	—
126	C₄×C₂₃	—	8	4	no	—
127	C₄×C₂₃	#492	1	2	yes	—
128	C₄×C₂₃	C₂²×SD₁₆	1	2	yes	—
129	C₄×C₂₃	C₂×#135	1	4	yes	—
130	C₂×#135	—	2	2	no	—
131	#533	C₂²×C₂₃	1	2	yes	—
132	#532	C₂²×C₂₃	1	2	yes	—
133	#500	C₂²×C₂₃	1	2	yes	—
134	#499	—	2	2	no	—
135	#498	C₂²×C₂₃	1	2	yes	—
136	#498	C₂²×C₂₃	1	4	yes	—
137	#495	—	2	2	no	—
138	#494	C₂²×C₂₃	1	1	yes	characteristic
139	#493	C₂²×C₂₃	1	1	yes	characteristic
140	#492	C₂²×C₂₃	1	2	yes	—
141	#492	C₂²×C₂₃	1	4	yes	—
142	#487	C₂²×C₂₃	1	4	yes	—
143	#486	C₂²×C₂₃	1	4	yes	—
144	#465	—	2	8	no	—
145	C₄×SD₁₆	—	2	8	no	—
146	#427	—	2	2	no	—
147	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
149	C₈²	C₂²×C₂₃	1	1	yes	characteristic
150	C₈²	C₂²×C₂₃	1	2	yes	—
151	C₈²	C₂²×C₂₃	1	1	yes	characteristic
152	C₂×C₂₃	—	2	2	no	—
153	C₂×C₂₃	—	16	2	no	—
154	C₂×C₂₃	C₂×#492	1	1	yes	characteristic
155	C₂×C₂₃	#3147	1	2	yes	—
156	C₂×#41	—	2	2	no	—
157	#135	—	4	8	no	—
158	C₂×Q₁₆	—	2	2	no	—
159	C₂×SD₁₆	—	2	4	no	—
160	C₂×SD₁₆	—	2	8	no	—
161	C₂×D₈	—	2	2	no	—
162	C₂²×C₈	D₄×C₂₃	1	4	yes	—
163	C₂²×C₈	D₄×C₂₃	1	2	yes	—
164	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
165	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
166	#127	C₂³×C₂₃	1	2	yes	—
167	#125	—	2	2	no	—
168	#124	—	2	2	no	—
169	C₂²⋊Q₈	—	2	4	no	—
170	#121	—	2	4	no	—
171	C₄×Q₈	—	2	4	no	—
172	C₄×D₄	—	2	4	no	—
173	#117	—	2	2	no	—
174	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
175	#107	—	2	4	no	—
176	#106	—	2	4	no	—
177	#106	—	2	8	no	—
178	#103	—	2	8	no	—
179	#102	—	2	8	no	—
180	C₄×C₈	D₄×C₂₃	1	4	yes	—
181	C₄×C₈	D₄×C₂₃	1	2	yes	—
182	C₄×C₈	C₂³×C₂₃	1	2	yes	—
183	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
184	C₄×C₈	C₂³×C₂₃	1	2	yes	—
185	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
186	C₂₃	#19596	1	1	yes	characteristic, direct product factor, Sylow
187	#41	—	4	8	no	—
188	C₂×Q₈	—	2	4	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	4	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
195	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
196	Q₁₆	—	4	4	no	—
197	SD₁₆	—	4	8	no	—
198	SD₁₆	—	4	16	no	—
199	D₈	—	4	4	no	—
200	C₂×C₈	—	2	8	no	—
201	C₂×C₈	SD₁₆×C₂₃	1	8	yes	complemented
202	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
203	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
204	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
205	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	4	4	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₂²⋊C₄	—	4	4	no	—
212	C₄²	—	4	2	no	—
213	C₄²	C₂×D₄×C₂₃	1	2	yes	—
214	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
215	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
216	C₂³	—	4	2	no	—
217	C₂³	C₂₃×#127	1	1	yes	characteristic
218	Q₈	—	4	4	no	—
219	Q₈	—	4	8	no	—
220	D₄	—	4	4	no	—
221	D₄	—	4	8	no	—
222	D₄	—	8	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	8	4	no	—
227	C₂×C₄	C₂₃×#127	1	2	yes	—
228	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
229	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
230	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
231	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
232	C₈	—	2	4	no	—
233	C₈	C₂×SD₁₆×C₂₃	1	4	yes	—
234	C₈	C₂×SD₁₆×C₂₃	1	4	yes	—
235	C₈	C₂₃×#135	1	8	yes	complemented
236	C₂²	—	2	2	no	—
237	C₂²	—	8	2	no	—
238	C₂²	—	8	4	no	—
239	C₂²	C₂₃×#492	1	2	yes	—
240	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	8	2	no	—
243	C₄	—	8	4	no	—
244	C₄	C₂₃×#492	1	2	yes	—
245	C₄	C₂²×SD₁₆×C₂₃	1	2	yes	—
246	C₄	C₂×C₂₃×#135	1	4	yes	—
247	C₂	—	2	2	no	—
248	C₂	—	16	2	no	—
249	C₂	C₂×C₂₃×#492	1	1	yes	characteristic
250	C₂	C₂₃×#3147	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×#127
prime radical	C₂₃×#3147	C₂
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×#3167	C₂	semidirect
2. C₂₃×#3160	C₂	semidirect
4. C₂₃×#1708	C₂	semidirect
5. C₂₃×#1707	C₂	semidirect
6. C₂₃×#1706	C₂	semidirect
7. C₂₃×#1705	C₂	semidirect
8. C₂×C₈²×C₂₃	C₂	semidirect
75. C₂×C₈×C₂₃	SD₁₆	semidirect
90. #19596	C₂₃	direct (N × H)
110. C₈×C₂₃	#135	semidirect
111. #3167	C₂×C₂₃	semidirect
112. #3160	C₂×C₂₃	semidirect
114. #1708	C₂×C₂₃	semidirect
115. #1707	C₂×C₂₃	semidirect
116. #1706	C₂×C₂₃	semidirect
117. #1705	C₂×C₂₃	semidirect
118. C₂×C₈²	C₂×C₂₃	semidirect
186. C₂₃	#19596	direct (N × H)
201. C₂×C₈	SD₁₆×C₂₃	semidirect
235. C₈	C₂₃×#135	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3167	2
C₂₃×#3160	2
C₂₃×#3147	2
C₂₃×#1708	2
C₂₃×#1707	2
C₂₃×#1706	2
C₂₃×#1705	2
C₂×C₈²×C₂₃	2
#19596	23

Subgroups — C23×#19596 #19167222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19596 #19167222