Subgroups — #13092474

Order 1472.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊(C₂₃⋊C₈)	C₂	1	2	yes	maximal
3	C₄⋊(C₂₃⋊C₈)	C₂	1	2	yes	maximal, complemented
4	#10624985	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₂₃	C₄	1	1	yes	characteristic
6	C₂²×C₄×C₂₃	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₄²×C₂₃	C₄	1	1	yes	characteristic
9	C₄²×C₂₃	C₄	1	1	yes	characteristic
10	C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₂₃⋊C₈	C₂²	1	4	yes	—
13	C₂³×C₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₂₃	—	2	2	no	—
15	C₂×C₄×C₂₃	—	2	2	no	—
16	C₂×C₄×C₂₃	—	2	2	no	—
17	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	C₂₃⋊C₈	—	2	4	no	—
24	C₂²×C₂₃	—	2	1	no	—
25	C₂²×C₂₃	—	2	1	no	—
26	C₂²×C₂₃	M₁₆	1	2	yes	—
27	C₂²×C₂₃	C₂²×C₄	1	1	yes	characteristic
28	C₄×C₂₃	—	2	1	no	—
29	C₄×C₂₃	—	2	1	no	—
30	C₄×C₂₃	—	2	1	no	—
31	C₄×C₂₃	—	2	1	no	—
32	C₄×C₂₃	M₁₆	1	2	yes	—
33	C₄×C₂₃	#41	1	2	yes	—
34	#432	—	23	1	no	maximal, Sylow
35	C₂×C₂₃	—	2	2	no	—
36	C₂×C₂₃	#117	1	1	yes	characteristic
37	C₂×C₂₃	C₂×M₁₆	1	1	yes	characteristic
38	C₂×C₂₃	C₂×M₁₆	1	1	yes	characteristic
39	C₂×C₄²	D₂₃	1	1	yes	characteristic
40	C₄⋊C₈	—	23	2	no	—
41	C₂²⋊C₈	—	23	2	no	—
42	#97	—	23	2	no	—
43	C₂₃	#432	1	1	yes	characteristic, complemented, Sylow
44	C₂²×C₄	Dic₂₃	1	1	yes	characteristic
45	C₂²×C₄	Dic₂₃	1	1	yes	characteristic
46	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
47	C₂×C₈	—	23	4	no	—
48	C₄²	Dic₂₃	1	1	yes	characteristic
49	C₄²	Dic₂₃	1	1	yes	characteristic
50	C₄²	C₂×D₂₃	1	1	yes	characteristic
51	C₄²	C₂×D₂₃	1	1	yes	characteristic
52	C₂³	C₂×Dic₂₃	1	1	yes	characteristic
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	C₂×Dic₂₃	1	1	yes	characteristic
57	C₂×C₄	C₂×Dic₂₃	1	1	yes	characteristic
58	C₂×C₄	C₂×Dic₂₃	1	1	yes	characteristic
59	C₂×C₄	C₂×Dic₂₃	1	1	yes	characteristic
60	C₂×C₄	C₂×Dic₂₃	1	1	yes	characteristic
61	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
62	C₈	—	46	4	no	—
63	C₂²	—	2	1	no	—
64	C₂²	—	2	1	no	—
65	C₂²	#67267	1	2	yes	—
66	C₂²	C₂²×Dic₂₃	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	2	1	no	—
69	C₄	—	2	1	no	—
70	C₄	—	2	1	no	—
71	C₄	#67267	1	2	yes	—
72	C₄	#67288	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	C₂×#67267	1	1	yes	characteristic
75	C₂	C₂×#67267	1	1	yes	characteristic
76	C₂	#10625097	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₃
commutator, socle	C₂²×C₂₃	C₂²×C₄
Fitting	C₂×C₄²×C₂₃	C₂
prime radical	C₂³×C₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₃	C₂³
hypercenter	C₂×C₄²	D₂₃
hypocenter	C₂₃	#432
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₂₃⋊C₈)	C₂	semidirect
4. #10624985	C₂	semidirect
43. C₂₃	#432	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄²×C₂₃	2
C₂²⋊(C₂₃⋊C₈)	2
C₄⋊(C₂₃⋊C₈)	2
#10624985	2
#432	23