Subgroups — C₂₃×#2931 #15670550

Order 2944.

500 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#131	C₂	1	2	yes	maximal, complemented
2	C₂₃×#532	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#436	C₂	1	4	yes	maximal, complemented
4	C₈×D₄×C₂₃	C₂	1	4	yes	maximal, complemented
5	C₂₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#408	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41×C₂₃	C₂²	1	2	yes	—
9	C₂₃×#131	—	2	8	no	—
10	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
11	C₂×M₁₆×C₂₃	C₂²	1	4	yes	—
12	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₂₃	C₂²	1	4	yes	—
14	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
16	C₂₃×#127	C₄	1	1	yes	characteristic
17	C₂₃×#124	C₄	1	2	yes	—
18	C₂₃×#121	C₄	1	4	yes	—
19	C₄×D₄×C₂₃	C₂²	1	4	yes	—
20	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
21	C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
22	C₂₃×C₂²⋊C₈	C₂²	1	8	yes	—
23	C₄×C₈×C₂₃	C₂²	1	2	yes	—
24	C₄×C₈×C₂₃	C₂²	1	2	yes	—
25	#41×C₂₃	—	2	4	no	—
26	#41×C₂₃	—	2	4	no	—
27	C₂×Q₈×C₂₃	C₂×C₄	1	2	yes	—
28	C₂×D₄×C₂₃	C₂×C₄	1	4	yes	—
29	C₂×D₄×C₂₃	C₂×C₄	1	4	yes	—
30	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
33	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
34	M₁₆×C₂₃	—	2	4	no	—
35	M₁₆×C₂₃	—	2	8	no	—
36	C₂×C₈×C₂₃	—	2	2	no	—
37	C₂×C₈×C₂₃	—	2	8	no	—
38	C₂×C₈×C₂₃	D₄	1	4	yes	complemented
39	C₂×C₈×C₂₃	C₂³	1	4	yes	—
40	C₂×C₈×C₂₃	C₂³	1	2	yes	—
41	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
42	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
44	C₂²⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
45	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₂₃	C₂³	1	2	yes	—
48	C₂³×C₂₃	C₂²×C₄	1	4	yes	—
49	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
50	Q₈×C₂₃	—	2	4	no	—
51	D₄×C₂₃	—	2	8	no	—
52	D₄×C₂₃	—	2	4	no	—
53	D₄×C₂₃	—	2	8	no	—
54	C₂×C₄×C₂₃	—	2	4	no	—
55	C₂×C₄×C₂₃	—	2	4	no	—
56	C₂×C₄×C₂₃	—	2	4	no	—
57	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
59	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₂₃	#41	1	2	yes	—
63	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
64	C₈×C₂₃	—	2	4	no	—
65	C₈×C₂₃	—	2	2	no	—
66	C₈×C₂₃	C₂×D₄	1	2	yes	—
67	C₈×C₂₃	C₂×D₄	1	2	yes	—
68	#2931	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂²×C₂₃	—	2	2	no	—
70	C₂²×C₂₃	—	2	4	no	—
71	C₂²×C₂₃	—	2	8	no	—
72	C₂²×C₂₃	C₄×D₄	1	2	yes	—
73	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
74	C₄×C₂₃	—	2	4	no	—
75	C₄×C₂₃	—	2	2	no	—
76	C₄×C₂₃	C₄×D₄	1	2	yes	—
77	C₄×C₂₃	#131	1	4	yes	complemented
78	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
79	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
80	C₂×#131	C₂×C₂₃	1	2	yes	complemented
81	#532	C₂×C₂₃	1	1	yes	characteristic
82	#436	C₂×C₂₃	1	4	yes	complemented
83	C₈×D₄	C₂×C₂₃	1	4	yes	complemented
84	#423	C₂×C₂₃	1	1	yes	characteristic, complemented
85	#408	C₂×C₂₃	1	2	yes	complemented
86	C₂×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
87	C₂×C₂₃	—	2	2	no	—
88	C₂×C₂₃	—	4	4	no	—
89	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
90	C₂×C₂₃	C₂×#131	1	2	yes	—
91	C₂×#41	C₂²×C₂₃	1	2	yes	—
92	#131	—	2	8	no	—
93	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
94	C₂×M₁₆	C₂²×C₂₃	1	4	yes	—
95	C₂²×C₈	C₂²×C₂₃	1	4	yes	—
96	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
97	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
98	C₄⋊Q₈	C₄×C₂₃	1	1	yes	characteristic
99	#127	C₄×C₂₃	1	1	yes	characteristic
100	#124	C₄×C₂₃	1	2	yes	—
101	#121	C₄×C₂₃	1	4	yes	—
102	C₄×D₄	C₂²×C₂₃	1	4	yes	—
103	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
104	C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
105	C₂²⋊C₈	C₂²×C₂₃	1	8	yes	—
106	C₄×C₈	C₂²×C₂₃	1	2	yes	—
107	C₄×C₈	C₂²×C₂₃	1	2	yes	—
108	C₂₃	#2931	1	1	yes	characteristic, direct product factor, Sylow
109	#41	—	2	4	no	—
110	#41	—	2	4	no	—
111	C₂×Q₈	C₂×C₄×C₂₃	1	2	yes	—
112	C₂×D₄	C₂×C₄×C₂₃	1	4	yes	—
113	C₂×D₄	C₂×C₄×C₂₃	1	4	yes	—
114	C₂×D₄	C₂×C₄×C₂₃	1	2	yes	—
115	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
116	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
117	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
118	M₁₆	—	2	4	no	—
119	M₁₆	—	2	8	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	2	8	no	—
122	C₂×C₈	D₄×C₂₃	1	4	yes	complemented
123	C₂×C₈	C₂³×C₂₃	1	2	yes	—
124	C₂×C₈	C₂³×C₂₃	1	4	yes	—
125	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
126	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
127	C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
128	C₂²⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
129	C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
130	C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
131	C₄²	C₂³×C₂₃	1	2	yes	—
132	C₂³	C₂²×C₄×C₂₃	1	4	yes	—
133	C₂³	C₂²×C₄×C₂₃	1	1	yes	characteristic
134	Q₈	—	2	4	no	—
135	D₄	—	2	8	no	—
136	D₄	—	2	4	no	—
137	D₄	—	2	8	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
142	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
143	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
144	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
145	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
146	C₂×C₄	#41×C₂₃	1	2	yes	—
147	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
148	C₈	—	2	4	no	—
149	C₈	—	2	2	no	—
150	C₈	C₂×D₄×C₂₃	1	2	yes	—
151	C₈	C₂×D₄×C₂₃	1	2	yes	—
152	C₂²	—	2	2	no	—
153	C₂²	—	2	8	no	—
154	C₂²	—	2	4	no	—
155	C₂²	C₄×D₄×C₂₃	1	2	yes	—
156	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	4	no	—
159	C₄	C₄×D₄×C₂₃	1	2	yes	—
160	C₄	C₂₃×#131	1	4	yes	complemented
161	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
162	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
163	C₂	—	2	2	no	—
164	C₂	—	4	4	no	—
165	C₂	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
166	C₂	C₂×C₂₃×#131	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂³
commutator	C₂²	C₂³×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂³×C₄
prime radical	C₂₃×#532	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#131	C₂	semidirect
3. C₂₃×#436	C₂	semidirect
4. C₈×D₄×C₂₃	C₂	semidirect
5. C₂₃×#423	C₂	semidirect
6. C₂₃×#408	C₂	semidirect
7. C₂×C₄×C₈×C₂₃	C₂	semidirect
38. C₂×C₈×C₂₃	D₄	semidirect
68. #2931	C₂₃	direct (N × H)
77. C₄×C₂₃	#131	semidirect
80. C₂×#131	C₂×C₂₃	semidirect
82. #436	C₂×C₂₃	semidirect
83. C₈×D₄	C₂×C₂₃	semidirect
84. #423	C₂×C₂₃	semidirect
85. #408	C₂×C₂₃	semidirect
86. C₂×C₄×C₈	C₂×C₂₃	semidirect
108. C₂₃	#2931	direct (N × H)
122. C₂×C₈	D₄×C₂₃	semidirect
160. C₄	C₂₃×#131	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#131	2
C₂₃×#532	2
C₂₃×#436	2
C₈×D₄×C₂₃	2
C₂₃×#423	2
C₂₃×#408	2
C₂×C₄×C₈×C₂₃	2
#2931	23

Subgroups — C23×#2931 #15670550

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2931 #15670550