Subgroups — #13093143

Order 1472.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic
3	Q₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic
4	#10625061	C₂	1	3	yes	maximal
5	#10625059	C₂	1	3	yes	maximal
6	#10625041	C₂	1	3	yes	maximal
7	C₄×Dic₄₆	C₂	1	3	yes	maximal
8	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×Dic₄₆	C₂²	1	3	yes	—
10	C₄⋊C₄×C₂₃	C₂²	1	3	yes	—
11	C₄²×C₂₃	C₂²	1	3	yes	—
12	C₄⋊Dic₂₃	C₂²	1	3	yes	—
13	C₄⋊Dic₂₃	C₂²	1	3	yes	—
14	C₄⋊Dic₂₃	C₂²	1	3	yes	—
15	Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
16	Dic₂₃⋊C₄	C₂²	1	3	yes	—
17	Dic₂₃⋊C₄	C₂²	1	6	yes	—
18	C₄×Dic₂₃	C₂²	1	3	yes	—
19	C₄×Dic₂₃	C₂²	1	3	yes	—
20	Q₈×C₂₃	Q₈	1	4	yes	complemented
21	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
22	C₂×C₄×C₂₃	C₂³	1	3	yes	—
23	C₂×C₄×C₂₃	C₂³	1	3	yes	—
24	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
25	C₂×Dic₂₃	C₂³	1	3	yes	—
26	C₂×Dic₂₃	C₂³	1	3	yes	—
27	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
28	Dic₄₆	—	2	6	no	—
29	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
30	C₄×C₂₃	—	2	3	no	—
31	C₄×C₂₃	—	2	1	no	—
32	C₄×C₂₃	C₂×Q₈	1	6	yes	—
33	Dic₂₃	—	2	3	no	—
34	Dic₂₃	—	2	1	no	—
35	Dic₂₃	—	2	3	no	—
36	Dic₂₃	#41	1	2	yes	—
37	#557	—	23	1	no	maximal, Sylow
38	C₂×C₂₃	C₂²×Q₈	1	1	yes	characteristic
39	C₂×C₂₃	C₂×#41	1	1	yes	characteristic
40	C₂×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
41	C₄⋊Q₈	—	23	3	no	—
42	#125	—	23	3	no	—
43	#125	—	23	3	no	—
44	C₄×Q₈	—	23	1	no	—
45	C₄×Q₈	—	23	1	no	—
46	C₄×Q₈	—	23	3	no	—
47	C₄×Q₈	D₂₃	1	1	yes	characteristic
48	C₂₃	#557	1	1	yes	characteristic, complemented, Sylow
49	C₂×Q₈	—	23	3	no	—
50	C₂×Q₈	C₂×D₂₃	1	1	yes	characteristic
51	C₄⋊C₄	—	23	1	no	—
52	C₄⋊C₄	—	23	3	no	—
53	C₄⋊C₄	—	23	3	no	—
54	C₄⋊C₄	—	23	3	no	—
55	C₄⋊C₄	—	23	3	no	—
56	C₄⋊C₄	—	23	6	no	—
57	C₄⋊C₄	C₂×D₂₃	1	3	yes	—
58	C₄²	—	23	3	no	—
59	C₄²	—	23	3	no	—
60	C₄²	C₂×D₂₃	1	3	yes	—
61	Q₈	—	46	6	no	—
62	Q₈	Dic₄₆	1	4	yes	complemented
63	C₂×C₄	—	23	1	no	—
64	C₂×C₄	—	23	1	no	—
65	C₂×C₄	—	23	3	no	—
66	C₂×C₄	—	23	3	no	—
67	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
68	C₂×C₄	C₂²×D₂₃	1	3	yes	—
69	C₂×C₄	C₂²×D₂₃	1	3	yes	—
70	C₂²	C₂³×D₂₃	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	2	3	no	—
73	C₄	—	23	2	no	—
74	C₄	—	46	1	no	—
75	C₄	—	46	3	no	—
76	C₄	—	46	3	no	—
77	C₄	C₂×Dic₄₆	1	6	yes	—
78	C₂	C₂²×Dic₄₆	1	1	yes	characteristic
79	C₂	#10625158	1	1	yes	characteristic
80	C₂	#41×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, prime radical, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₄×Q₈×C₂₃	C₂
hypercenter	C₄×Q₈	D₂₃
hypocenter	C₂₃	#557
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
20. Q₈×C₂₃	Q₈	semidirect
48. C₂₃	#557	semidirect
62. Q₈	Dic₄₆	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×C₂₃	2
Q₈×Dic₂₃	2
#10625061	2
#10625059	2
#10625041	2
C₄×Dic₄₆	2
#557	23