Subgroups — C₂₃×#3175 #15670794

Order 2944.

362 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#542	C₂	1	2	yes	maximal, complemented
2	C₂₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#489	C₂	1	2	yes	maximal, complemented
5	C₂₃×#468	C₂	1	2	yes	maximal, complemented
6	C₂₃×#467	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#466	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#440	C₂	1	2	yes	maximal, complemented
9	C₄×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
18	C₂₃×#126	—	2	2	no	—
19	C₂₃×#124	—	2	2	no	—
20	C₂₃×#124	C₂²	1	2	yes	—
21	C₂₃×#123	—	2	2	no	—
22	C₂₃×C₂²⋊Q₈	—	2	2	no	—
23	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
24	C₂₃×#121	C₂²	1	2	yes	—
25	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
26	C₄×D₄×C₂₃	C₂²	1	2	yes	—
27	C₂₃×#117	C₂²	1	1	yes	characteristic
28	C₂₃×#107	C₂²	1	1	yes	characteristic
29	C₂₃×#107	C₂²	1	1	yes	characteristic
30	C₂₃×#107	C₂²	1	2	yes	—
31	C₂₃×#103	C₂²	1	2	yes	—
32	C₂₃×#103	C₂²	1	2	yes	—
33	C₂₃×#102	C₂²	1	2	yes	—
34	C₂₃×#102	C₂²	1	2	yes	—
35	C₂₃×#97	C₂²	1	1	yes	characteristic
36	C₂₃×#97	C₂²	1	1	yes	characteristic
37	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
39	C₂×Q₈×C₂₃	—	2	2	no	—
40	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
41	C₂×D₄×C₂₃	—	2	2	no	—
42	C₂×D₄×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₃	—	2	2	no	—
44	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	Q₁₆×C₂₃	—	2	2	no	—
46	SD₁₆×C₂₃	—	2	4	no	—
47	D₈×C₂₃	—	2	2	no	—
48	M₁₆×C₂₃	—	2	2	no	—
49	C₂×C₈×C₂₃	—	2	2	no	—
50	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₂₃	—	2	2	no	—
55	C₄⋊C₄×C₂₃	—	2	2	no	—
56	C₄⋊C₄×C₂₃	—	2	2	no	—
57	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
58	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
59	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
60	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
61	C₂²⋊C₄×C₂₃	—	2	2	no	—
62	C₂²⋊C₄×C₂₃	—	2	2	no	—
63	C₂²⋊C₄×C₂₃	—	2	2	no	—
64	C₂²⋊C₄×C₂₃	—	2	2	no	—
65	C₂²⋊C₄×C₂₃	—	2	2	no	—
66	C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
67	C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
68	C₄²×C₂₃	—	2	2	no	—
69	C₄²×C₂₃	C₂³	1	1	yes	characteristic
70	C₄²×C₂₃	C₂³	1	1	yes	characteristic
71	C₂³×C₂₃	—	2	2	no	—
72	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
73	Q₈×C₂₃	—	2	2	no	—
74	Q₈×C₂₃	—	2	2	no	—
75	Q₈×C₂₃	—	8	2	no	—
76	D₄×C₂₃	—	2	2	no	—
77	D₄×C₂₃	—	2	2	no	—
78	D₄×C₂₃	—	8	2	no	—
79	C₂×C₄×C₂₃	—	2	1	no	—
80	C₂×C₄×C₂₃	—	2	1	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	8	2	no	—
85	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
91	C₈×C₂₃	—	2	1	no	—
92	C₈×C₂₃	—	2	1	no	—
93	C₈×C₂₃	#41	1	2	yes	—
94	C₈×C₂₃	#41	1	2	yes	—
95	#3175	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂²×C₂₃	—	2	1	no	—
97	C₂²×C₂₃	—	2	1	no	—
98	C₂²×C₂₃	—	2	1	no	—
99	C₂²×C₂₃	—	4	2	no	—
100	C₂²×C₂₃	—	4	2	no	—
101	C₂²×C₂₃	—	4	2	no	—
102	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
103	C₄×C₂₃	—	2	1	no	—
104	C₄×C₂₃	—	2	1	no	—
105	C₄×C₂₃	—	2	1	no	—
106	C₄×C₂₃	—	2	1	no	—
107	C₄×C₂₃	—	2	1	no	—
108	C₄×C₂₃	—	4	2	no	—
109	C₄×C₂₃	—	4	2	no	—
110	C₄×C₂₃	—	4	2	no	—
111	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
112	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
113	#542	C₂×C₂₃	1	2	yes	complemented
114	#500	C₂×C₂₃	1	1	yes	characteristic, complemented
115	#495	C₂×C₂₃	1	1	yes	characteristic, complemented
116	#489	C₂×C₂₃	1	2	yes	complemented
117	#468	C₂×C₂₃	1	2	yes	complemented
118	#467	C₂×C₂₃	1	1	yes	characteristic, complemented
119	#466	C₂×C₂₃	1	1	yes	characteristic
120	#440	C₂×C₂₃	1	2	yes	complemented
121	C₄×Q₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
122	C₄×D₈	C₂×C₂₃	1	1	yes	characteristic, complemented
123	#405	C₂×C₂₃	1	1	yes	characteristic, complemented
124	C₂×C₂₃	—	4	1	no	—
125	C₂×C₂₃	—	8	2	no	—
126	C₂×C₂₃	#532	1	1	yes	characteristic
127	C₂×C₂₃	#576	1	1	yes	characteristic
128	C₂×C₂₃	#578	1	1	yes	characteristic
129	C₂×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
130	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
131	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic
132	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
133	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
134	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
135	#126	—	2	2	no	—
136	#124	—	2	2	no	—
137	#124	C₂²×C₂₃	1	2	yes	—
138	#123	—	2	2	no	—
139	C₂²⋊Q₈	—	2	2	no	—
140	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
141	#121	C₂²×C₂₃	1	2	yes	—
142	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
143	C₄×D₄	C₂²×C₂₃	1	2	yes	—
144	#117	C₂²×C₂₃	1	1	yes	characteristic
145	#107	C₂²×C₂₃	1	1	yes	characteristic
146	#107	C₂²×C₂₃	1	1	yes	characteristic
147	#107	C₂²×C₂₃	1	2	yes	—
148	#103	C₂²×C₂₃	1	2	yes	—
149	#103	C₂²×C₂₃	1	2	yes	—
150	#102	C₂²×C₂₃	1	2	yes	—
151	#102	C₂²×C₂₃	1	2	yes	—
152	#97	C₂²×C₂₃	1	1	yes	characteristic
153	#97	C₂²×C₂₃	1	1	yes	characteristic
154	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
155	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
156	C₂₃	#3175	1	1	yes	characteristic, direct product factor, Sylow
157	C₂×Q₈	—	2	2	no	—
158	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	C₂³×C₂₃	1	2	yes	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
163	Q₁₆	—	2	2	no	—
164	SD₁₆	—	2	4	no	—
165	D₈	—	2	2	no	—
166	M₁₆	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
169	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
170	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
171	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
176	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
177	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
178	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
179	C₂²⋊C₄	—	2	2	no	—
180	C₂²⋊C₄	—	2	2	no	—
181	C₂²⋊C₄	—	2	2	no	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
185	C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
186	C₄²	—	2	2	no	—
187	C₄²	C₂³×C₂₃	1	1	yes	characteristic
188	C₄²	C₂³×C₂₃	1	1	yes	characteristic
189	C₂³	—	2	2	no	—
190	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
191	Q₈	—	2	2	no	—
192	Q₈	—	2	2	no	—
193	Q₈	—	8	2	no	—
194	D₄	—	2	2	no	—
195	D₄	—	2	2	no	—
196	D₄	—	8	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	8	2	no	—
203	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
204	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
205	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
206	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
207	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
208	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	#41×C₂₃	1	2	yes	—
212	C₈	#41×C₂₃	1	2	yes	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	2	no	—
227	C₄	—	4	2	no	—
228	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
229	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
230	C₂	—	4	1	no	—
231	C₂	—	8	2	no	—
232	C₂	C₂₃×#532	1	1	yes	characteristic
233	C₂	C₂₃×#576	1	1	yes	characteristic
234	C₂	C₂₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#466	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#542	C₂	semidirect
2. C₂₃×#500	C₂	semidirect
3. C₂₃×#495	C₂	semidirect
4. C₂₃×#489	C₂	semidirect
5. C₂₃×#468	C₂	semidirect
6. C₂₃×#467	C₂	semidirect
8. C₂₃×#440	C₂	semidirect
9. C₄×Q₁₆×C₂₃	C₂	semidirect
10. C₄×D₈×C₂₃	C₂	semidirect
11. C₂₃×#405	C₂	semidirect
95. #3175	C₂₃	direct (N × H)
113. #542	C₂×C₂₃	semidirect
114. #500	C₂×C₂₃	semidirect
115. #495	C₂×C₂₃	semidirect
116. #489	C₂×C₂₃	semidirect
117. #468	C₂×C₂₃	semidirect
118. #467	C₂×C₂₃	semidirect
120. #440	C₂×C₂₃	semidirect
121. C₄×Q₁₆	C₂×C₂₃	semidirect
122. C₄×D₈	C₂×C₂₃	semidirect
123. #405	C₂×C₂₃	semidirect
156. C₂₃	#3175	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×#542	2
C₂₃×#500	2
C₂₃×#495	2
C₂₃×#489	2
C₂₃×#468	2
C₂₃×#467	2
C₂₃×#466	2
C₂₃×#440	2
C₄×Q₁₆×C₂₃	2
C₄×D₈×C₂₃	2
C₂₃×#405	2
#3175	23

Subgroups — C23×#3175 #15670794

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3175 #15670794