Subgroups — #15671680

Order 2944.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092470	C₂	1	2	yes	maximal
2	#13092138	C₂	1	2	yes	maximal, complemented
3	C₈×C₂₃⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×C₂₃	C₄	1	2	yes	—
6	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
7	C₄×C₈×C₂₃	C₄	1	2	yes	—
8	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²⋊(C₂₃⋊C₈)	C₄	1	4	yes	—
11	C₄⋊(C₂₃⋊C₈)	C₄	1	4	yes	—
12	C₄×C₂₃⋊C₈	C₂²	1	4	yes	—
13	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
14	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₂₃	C₈	1	8	yes	complemented
16	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
18	C₄²×C₂₃	C₂×C₄	1	2	yes	—
19	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₂₃	C₂³	1	1	yes	characteristic
21	C₂×C₂₃⋊C₈	C₂×C₄	1	8	yes	—
22	C₂³×C₂₃	C₄²	1	1	yes	characteristic
23	C₂×C₄×C₂₃	—	2	4	no	—
24	C₂×C₄×C₂₃	C₄²	1	2	yes	—
25	C₂×C₄×C₂₃	C₄²	1	1	yes	characteristic
26	C₂×C₄×C₂₃	C₂×C₈	1	4	yes	—
27	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
28	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
29	C₈×C₂₃	C₂×C₈	1	4	yes	—
30	C₈×C₂₃	C₂×C₈	1	4	yes	—
31	C₂₃⋊C₈	—	2	8	no	—
32	#1450	—	23	1	no	maximal, Sylow
33	C₂²×C₂₃	—	2	2	no	—
34	C₂²×C₂₃	C₄×C₈	1	2	yes	—
35	C₂²×C₂₃	C₂×C₄²	1	1	yes	characteristic
36	C₄×C₂₃	—	2	2	no	—
37	C₄×C₂₃	C₄×C₈	1	2	yes	—
38	C₄×C₂₃	C₂²×C₈	1	2	yes	—
39	C₄×C₂₃	#131	1	4	yes	—
40	#431	—	23	2	no	—
41	C₂×C₄×C₈	D₂₃	1	1	yes	characteristic
42	#322	—	23	2	no	—
43	C₈²	—	23	2	no	—
44	C₂×C₂₃	—	2	2	no	—
45	C₂×C₂₃	C₂×C₄×C₈	1	1	yes	characteristic
46	C₂×C₂₃	#405	1	2	yes	—
47	C₂²×C₈	Dic₂₃	1	2	yes	—
48	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
49	C₄⋊C₈	—	23	4	no	—
50	C₂²⋊C₈	—	23	4	no	—
51	C₄×C₈	—	23	4	no	—
52	C₄×C₈	Dic₂₃	1	2	yes	—
53	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
54	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
55	C₂₃	#1450	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₄	C₄×D₂₃	1	2	yes	—
57	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
58	C₂×C₈	—	23	8	no	—
59	C₂×C₈	C₂₃⋊C₈	1	8	yes	complemented
60	C₂×C₈	C₂×Dic₂₃	1	2	yes	—
61	C₂×C₈	C₂×Dic₂₃	1	2	yes	—
62	C₄²	C₄×D₂₃	1	2	yes	—
63	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
64	C₄²	C₂²×D₂₃	1	1	yes	characteristic
65	C₂³	C₄×Dic₂₃	1	1	yes	characteristic
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	C₂×C₂₃⋊C₈	1	4	yes	—
68	C₂×C₄	C₄×Dic₂₃	1	2	yes	—
69	C₂×C₄	C₄×Dic₂₃	1	1	yes	characteristic
70	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
71	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
72	C₈	—	46	8	no	—
73	C₈	C₂×C₂₃⋊C₈	1	4	yes	—
74	C₈	C₂×C₂₃⋊C₈	1	4	yes	—
75	C₂²	—	2	2	no	—
76	C₂²	C₄×C₂₃⋊C₈	1	2	yes	—
77	C₂²	C₂×C₄×Dic₂₃	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	C₄×C₂₃⋊C₈	1	2	yes	—
80	C₄	#10625072	1	4	yes	—
81	C₄	C₂²×C₂₃⋊C₈	1	2	yes	—
82	C₂	—	2	2	no	—
83	C₂	C₂×C₄×C₂₃⋊C₈	1	1	yes	characteristic
84	C₂	#13092372	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂×D₂₃
commutator	C₂×C₂₃	C₂×C₄×C₈
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂×C₄×C₈×C₂₃	C₂
socle	C₂²×C₂₃	C₂×C₄²
prime radical	C₂³×C₂₃	C₄²
cosocle	C₄²×C₂₃	C₂³
hypercenter	C₂×C₄×C₈	D₂₃
hypocenter	C₂₃	#1450
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13092138	C₂	semidirect
3. C₈×C₂₃⋊C₈	C₂	semidirect
15. C₂×C₈×C₂₃	C₈	semidirect
55. C₂₃	#1450	semidirect
59. C₂×C₈	C₂₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#13092470	2
#13092138	2
C₈×C₂₃⋊C₈	2
C₂×C₄×C₈×C₂₃	2
#1450	23