Subgroups — #13093048

Order 1472.

206 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	#10625097	C₂	1	2	yes	maximal
3	#10625095	C₂	1	4	yes	maximal
4	#10625061	C₂	1	4	yes	maximal, complemented
5	#10625059	C₂	1	4	yes	maximal, complemented
6	C₂²×C₄×C₂₃	C₂²	1	2	yes	—
7	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×Dic₄₆	C₂²	1	4	yes	—
9	C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
10	C₂²⋊Dic₂₃	C₂²	1	4	yes	—
11	C₄⋊Dic₂₃	C₂²	1	2	yes	—
12	C₄⋊Dic₂₃	C₂²	1	4	yes	—
13	C₄⋊Dic₂₃	C₂²	1	2	yes	—
14	Dic₂₃⋊C₄	C₂²	1	8	yes	—
15	C₄×Dic₂₃	C₂²	1	4	yes	—
16	C₂³×C₂₃	C₂³	1	1	yes	characteristic
17	C₂×C₄×C₂₃	—	4	2	no	—
18	C₂×C₄×C₂₃	Q₈	1	4	yes	complemented
19	C₂×C₄×C₂₃	C₂³	1	4	yes	—
20	C₂×C₄×C₂₃	C₂³	1	2	yes	—
21	C₂×Dic₂₃	C₂³	1	4	yes	—
22	C₂×Dic₂₃	C₂³	1	4	yes	—
23	Dic₄₆	—	4	4	no	—
24	C₂²×C₂₃	—	2	1	no	—
25	C₂²×C₂₃	—	2	1	no	—
26	C₂²×C₂₃	C₂×Q₈	1	2	yes	—
27	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
28	C₄×C₂₃	—	2	4	no	—
29	C₄×C₂₃	C₂×Q₈	1	4	yes	—
30	Dic₂₃	—	2	4	no	—
31	Dic₂₃	—	2	4	no	—
32	#544	—	23	1	no	maximal, Sylow
33	C₂×C₂₃	—	2	2	no	—
34	C₂×C₂₃	C₂²×Q₈	1	1	yes	characteristic
35	C₂×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
36	C₂×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
37	C₄⋊Q₈	—	23	4	no	—
38	#125	—	23	4	no	—
39	C₂²⋊Q₈	—	23	4	no	—
40	#117	—	23	2	no	—
41	C₂×C₄⋊C₄	D₂₃	1	1	yes	characteristic
42	C₂₃	#544	1	1	yes	characteristic, complemented, Sylow
43	C₂×Q₈	—	23	4	no	—
44	C₂²×C₄	C₂×D₂₃	1	2	yes	—
45	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
46	C₄⋊C₄	—	23	2	no	—
47	C₄⋊C₄	—	23	2	no	—
48	C₄⋊C₄	—	23	4	no	—
49	C₄⋊C₄	—	23	8	no	—
50	C₄⋊C₄	C₂×D₂₃	1	4	yes	—
51	C₂²⋊C₄	—	23	4	no	—
52	C₄²	—	23	4	no	—
53	C₂³	C₂²×D₂₃	1	1	yes	characteristic
54	Q₈	—	92	4	no	—
55	C₂×C₄	—	4	2	no	—
56	C₂×C₄	—	23	4	no	—
57	C₂×C₄	—	23	4	no	—
58	C₂×C₄	Dic₄₆	1	4	yes	complemented
59	C₂×C₄	C₂²×D₂₃	1	4	yes	—
60	C₂×C₄	C₂²×D₂₃	1	2	yes	—
61	C₂²	—	2	1	no	—
62	C₂²	—	2	1	no	—
63	C₂²	C₂×Dic₄₆	1	2	yes	—
64	C₂²	C₂³×D₂₃	1	1	yes	characteristic
65	C₄	—	2	4	no	—
66	C₄	—	46	4	no	—
67	C₄	—	46	4	no	—
68	C₄	C₂×Dic₄₆	1	4	yes	—
69	C₂	—	2	2	no	—
70	C₂	C₂²×Dic₄₆	1	1	yes	characteristic
71	C₂	#10625155	1	1	yes	characteristic
72	C₂	#10625158	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂×C₄⋊C₄×C₂₃	C₂
prime radical	C₂³×C₂₃	C₂³
hypercenter	C₂×C₄⋊C₄	D₂₃
hypocenter	C₂₃	#544
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. #10625061	C₂	semidirect
5. #10625059	C₂	semidirect
18. C₂×C₄×C₂₃	Q₈	semidirect
42. C₂₃	#544	semidirect
58. C₂×C₄	Dic₄₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₄×C₂₃	2
#10625097	2
#10625095	2
#10625061	2
#10625059	2
#544	23