Subgroups — #15686500

Order 2944.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092834	C₂	1	8	yes	maximal, complemented
2	C₂×#10624994	C₂	1	6	yes	maximal, complemented
3	C₂×C₂₃×#131	C₂	1	1	yes	maximal, characteristic
4	C₂×#41×C₂₃	C₄	1	1	yes	characteristic
5	C₂₃×#131	C₄	1	4	yes	—
6	C₂₃×#131	C₂²	1	4	yes	—
7	C₂×M₁₆×C₂₃	C₄	1	3	yes	—
8	C₂²×C₈×C₂₃	C₂²	1	3	yes	—
9	#10624994	C₂²	1	4	yes	—
10	#10624994	C₂²	1	12	yes	complemented
11	C₂×C₂₃⋊C₁₆	C₂²	1	12	yes	complemented
12	#41×C₂₃	C₈	1	4	yes	—
13	#41×C₂₃	C₂×C₄	1	4	yes	—
14	C₂×Q₈×C₂₃	C₈	1	1	yes	characteristic
15	C₂×D₄×C₂₃	C₈	1	3	yes	—
16	C₂²×C₄×C₂₃	C₂×C₄	1	3	yes	—
17	M₁₆×C₂₃	C₂×C₄	1	12	yes	—
18	C₂×C₈×C₂₃	C₂×C₄	1	3	yes	—
19	C₂×C₈×C₂₃	C₂×C₄	1	6	yes	—
20	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
21	C₂×C₈×C₂₃	C₂³	1	6	yes	—
22	C₂₃⋊C₁₆	C₂³	1	8	yes	—
23	C₂³×C₂₃	C₂×C₈	1	3	yes	—
24	Q₈×C₂₃	C₂×C₈	1	4	yes	—
25	D₄×C₂₃	C₂×C₈	1	12	yes	—
26	C₂×C₄×C₂₃	C₂×C₈	1	3	yes	—
27	C₂×C₄×C₂₃	C₂×C₈	1	6	yes	—
28	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₂₃	C₂²×C₄	1	6	yes	—
30	C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
31	C₈×C₂₃	C₂²×C₄	1	6	yes	—
32	C₈×C₂₃	C₂⁴	1	1	yes	characteristic
33	#3407	—	23	1	no	maximal, Sylow
34	C₂²×C₂₃	—	4	3	no	—
35	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
36	C₂²×C₂₃	C₂²×C₈	1	6	yes	—
37	C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
38	C₄×C₂₃	C₂²×C₈	1	6	yes	—
39	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
40	C₂×#131	D₂₃	1	1	yes	characteristic
41	#504	—	23	8	no	—
42	C₂×M₃₂	—	23	6	no	—
43	C₂×C₂₃	—	2	1	no	—
44	C₂×C₂₃	—	2	6	no	—
45	C₂×C₂₃	C₂³×C₈	1	1	yes	characteristic
46	C₂×#41	Dic₂₃	1	1	yes	characteristic
47	#131	Dic₂₃	1	4	yes	—
48	#131	C₂×D₂₃	1	4	yes	—
49	C₂×M₁₆	Dic₂₃	1	3	yes	—
50	C₂²×C₈	C₂×D₂₃	1	3	yes	—
51	M₃₂	—	23	4	no	—
52	M₃₂	—	23	12	no	—
53	C₂×C₁₆	—	23	12	no	—
54	C₂₃	#3407	1	1	yes	characteristic, complemented, Sylow
55	#41	C₂₃⋊C₈	1	4	yes	—
56	#41	C₂×Dic₂₃	1	4	yes	—
57	C₂×Q₈	C₂₃⋊C₈	1	1	yes	characteristic
58	C₂×D₄	C₂₃⋊C₈	1	3	yes	—
59	C₂²×C₄	C₂×Dic₂₃	1	3	yes	—
60	M₁₆	C₂×Dic₂₃	1	12	yes	—
61	C₂×C₈	C₂×Dic₂₃	1	3	yes	—
62	C₂×C₈	C₂×Dic₂₃	1	6	yes	—
63	C₂×C₈	C₂²×D₂₃	1	6	yes	—
64	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
65	C₁₆	—	23	8	no	—
66	C₂³	C₂×C₂₃⋊C₈	1	3	yes	—
67	Q₈	C₂×C₂₃⋊C₈	1	4	yes	—
68	D₄	C₂×C₂₃⋊C₈	1	12	yes	—
69	C₂×C₄	C₂×C₂₃⋊C₈	1	3	yes	—
70	C₂×C₄	C₂×C₂₃⋊C₈	1	6	yes	—
71	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
72	C₂×C₄	C₂²×Dic₂₃	1	6	yes	—
73	C₈	C₂²×Dic₂₃	1	1	yes	characteristic
74	C₈	C₂²×Dic₂₃	1	6	yes	—
75	C₈	C₂³×D₂₃	1	1	yes	characteristic
76	C₂²	—	4	3	no	—
77	C₂²	C₂²×C₂₃⋊C₈	1	1	yes	characteristic
78	C₂²	C₂²×C₂₃⋊C₈	1	6	yes	—
79	C₄	C₂²×C₂₃⋊C₈	1	1	yes	characteristic
80	C₄	C₂²×C₂₃⋊C₈	1	6	yes	—
81	C₄	C₂³×Dic₂₃	1	1	yes	characteristic
82	C₂	—	2	1	no	—
83	C₂	—	2	6	no	—
84	C₂	C₂³×C₂₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂³×D₂₃
commutator, socle	C₂×C₂₃	C₂³×C₈
Fitting	C₂×C₂₃×#131	C₂
prime radical	C₂×#41×C₂₃	C₄
cosocle	C₈×C₂₃	C₂⁴
hypercenter	C₂×#131	D₂₃
hypocenter	C₂₃	#3407
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. #13092834	C₂	semidirect
2. C₂×#10624994	C₂	semidirect
10. #10624994	C₂²	semidirect
11. C₂×C₂₃⋊C₁₆	C₂²	semidirect
54. C₂₃	#3407	semidirect

Maximal subgroups

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

subgroup	index
#13092834	2
C₂×#10624994	2
C₂×C₂₃×#131	2
#3407	23