Subgroups — #13092426

Order 1472.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	#10625097	C₂	1	1	yes	maximal, characteristic
3	C₂×#67267	C₂	1	1	yes	maximal, characteristic
4	#10625017	C₂	1	4	yes	maximal, complemented
5	C₂×Q₈×C₂₃	—	2	2	no	—
6	C₂×Q₈×C₂₃	—	2	1	no	—
7	C₂×Q₈×C₂₃	C₄	1	4	yes	complemented
8	C₂×Q₈×C₂₃	C₂²	1	2	yes	—
9	C₂²×C₄×C₂₃	—	2	1	no	—
10	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²⋊Dic₂₃	—	2	1	no	—
12	C₄⋊Dic₂₃	C₂²	1	2	yes	—
13	C₄×Dic₂₃	—	2	1	no	—
14	#67267	—	2	2	no	—
15	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
16	C₂³×C₂₃	D₄	1	1	yes	characteristic
17	Q₈×C₂₃	—	2	4	no	—
18	Q₈×C₂₃	—	2	2	no	—
19	Q₈×C₂₃	C₂×C₄	1	4	yes	—
20	C₂×C₄×C₂₃	—	2	4	no	—
21	C₂×C₄×C₂₃	—	2	2	no	—
22	C₂×C₄×C₂₃	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
24	C₂×C₄×C₂₃	D₄	1	2	yes	—
25	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₂×Dic₂₃	—	2	2	no	—
27	C₂₃⋊C₈	—	2	2	no	—
28	C₂²×C₂₃	—	2	2	no	—
29	C₂²×C₂₃	C₂²⋊C₄	1	2	yes	—
30	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
31	C₄×C₂₃	—	2	4	no	—
32	C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
33	C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
34	C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
35	Dic₂₃	—	4	2	no	—
36	#419	—	23	1	no	maximal, Sylow
37	C₂×C₂₃	—	2	2	no	—
38	C₂×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₂₃	#137	1	2	yes	—
40	C₂²×Q₈	D₂₃	1	1	yes	characteristic
41	C₂×M₁₆	—	23	1	no	—
42	#117	—	23	1	no	—
43	#103	—	23	4	no	—
44	C₂₃	#419	1	1	yes	characteristic, complemented, Sylow
45	C₂×Q₈	—	2	2	no	—
46	C₂×Q₈	—	2	1	no	—
47	C₂×Q₈	Dic₂₃	1	4	yes	complemented
48	C₂×Q₈	C₂×D₂₃	1	2	yes	—
49	C₂²×C₄	—	2	1	no	—
50	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
51	M₁₆	—	46	2	no	—
52	C₂×C₈	—	23	2	no	—
53	C₄⋊C₄	—	23	2	no	—
54	C₂²⋊C₄	—	46	1	no	—
55	C₄²	—	46	1	no	—
56	C₂³	C₂²⋊D₂₃	1	1	yes	characteristic
57	Q₈	—	2	4	no	—
58	Q₈	—	2	2	no	—
59	Q₈	C₂×Dic₂₃	1	4	yes	—
60	C₂×C₄	—	2	4	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	46	2	no	—
63	C₂×C₄	C₂×Dic₂₃	1	2	yes	—
64	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
65	C₂×C₄	C₂²⋊D₂₃	1	2	yes	—
66	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
67	C₈	—	46	2	no	—
68	C₂²	—	2	2	no	—
69	C₂²	C₂²⋊Dic₂₃	1	2	yes	—
70	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
71	C₄	—	2	4	no	—
72	C₄	—	92	2	no	—
73	C₄	C₂²⋊Dic₂₃	1	2	yes	—
74	C₄	C₂²×Dic₂₃	1	1	yes	characteristic
75	C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
76	C₂	—	2	2	no	—
77	C₂	#10625113	1	2	yes	—
78	C₂	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₃
commutator	C₄×C₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₃
Fitting	C₂²×Q₈×C₂₃	C₂
socle	C₂²×C₂₃	C₂×D₄
prime radical	C₂³×C₂₃	D₄
cosocle	C₂×C₄×C₂₃	C₂³
hypercenter	C₂²×Q₈	D₂₃
hypocenter	C₂₃	#419
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
4. #10625017	C₂	semidirect
7. C₂×Q₈×C₂₃	C₄	semidirect
44. C₂₃	#419	semidirect
47. C₂×Q₈	Dic₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₈×C₂₃	2
#10625097	2
C₂×#67267	2
#10625017	2
#419	23