Subgroups — #13093147

Order 1472.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#10625115	C₂	1	4	yes	maximal
3	#10625059	C₂	1	4	yes	maximal
4	#10625058	C₂	1	4	yes	maximal
5	C₄×Dic₄₆	C₂	1	2	yes	maximal
6	C₂×Q₈×C₂₃	C₂²	1	2	yes	—
7	C₂×Dic₄₆	C₂²	1	4	yes	—
8	C₂×Dic₄₆	C₂²	1	2	yes	—
9	C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
10	C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₃	C₂²	1	2	yes	—
12	Dic₂₃⋊C₄	C₂²	1	4	yes	—
13	Dic₂₃⋊C₄	C₂²	1	8	yes	—
14	C₄×Dic₂₃	C₂²	1	4	yes	—
15	C₄×Dic₂₃	C₂²	1	4	yes	—
16	Q₈×C₂₃	—	4	2	no	—
17	C₂×C₄×C₂₃	C₂³	1	4	yes	—
18	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₂₃	C₂³	1	2	yes	—
20	C₂×Dic₂₃	C₂³	1	4	yes	—
21	C₂×Dic₂₃	C₂³	1	4	yes	—
22	Dic₄₆	—	4	4	no	—
23	Dic₄₆	Q₈	1	8	yes	complemented
24	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
25	C₄×C₂₃	—	2	4	no	—
26	C₄×C₂₃	—	2	1	no	—
27	C₄×C₂₃	C₂×Q₈	1	4	yes	—
28	Dic₂₃	—	2	4	no	—
29	Dic₂₃	C₂×Q₈	1	8	yes	—
30	Q₈²	—	23	1	no	maximal, Sylow
31	C₂×C₂₃	C₂²×Q₈	1	2	yes	—
32	C₂×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
33	C₄⋊Q₈	—	23	4	no	—
34	C₄⋊Q₈	—	23	4	no	—
35	C₄⋊Q₈	D₂₃	1	1	yes	characteristic
36	C₄×Q₈	—	23	4	no	—
37	C₄×Q₈	—	23	2	no	—
38	C₂₃	Q₈²	1	1	yes	characteristic, complemented, Sylow
39	C₂×Q₈	—	23	2	no	—
40	C₂×Q₈	—	23	4	no	—
41	C₂×Q₈	C₂×D₂₃	1	2	yes	—
42	C₄⋊C₄	—	23	8	no	—
43	C₄⋊C₄	—	23	4	no	—
44	C₄⋊C₄	—	23	2	no	—
45	C₄⋊C₄	C₂×D₂₃	1	4	yes	—
46	C₄²	—	23	4	no	—
47	C₄²	—	23	4	no	—
48	C₄²	C₂×D₂₃	1	1	yes	characteristic
49	Q₈	—	4	2	no	—
50	Q₈	—	23	8	no	—
51	Q₈	—	92	4	no	—
52	C₂×C₄	—	23	4	no	—
53	C₂×C₄	—	23	4	no	—
54	C₂×C₄	C₂²×D₂₃	1	4	yes	—
55	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
56	C₂×C₄	C₂²×D₂₃	1	2	yes	—
57	C₂²	C₂³×D₂₃	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	4	no	—
60	C₄	—	23	8	no	—
61	C₄	—	46	4	no	—
62	C₄	Q₈×D₂₃	1	4	yes	—
63	C₂	#10625155	1	1	yes	characteristic
64	C₂	C₂×Q₈×D₂₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, prime radical, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂₃×C₄⋊Q₈	C₂
hypercenter	C₄⋊Q₈	D₂₃
hypocenter	C₂₃	Q₈²
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
23. Dic₄₆	Q₈	semidirect
38. C₂₃	Q₈²	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×C₄⋊Q₈	2
#10625115	2
#10625059	2
#10625058	2
C₄×Dic₄₆	2
Q₈²	23