Subgroups — #11718739

Order 832.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#88123	C₂	1	1	yes	maximal, characteristic
3	#88047	C₂	1	4	yes	maximal
4	C₄×#1055	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈×C₁₃	—	2	1	no	—
6	C₂×Dic₂₆	—	2	1	no	—
7	C₁₃×C₄⋊C₄	—	2	1	no	—
8	C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
9	C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₃	—	2	1	no	—
11	C₄⋊Dic₁₃	C₂²	1	2	yes	—
12	C₂×#1055	C₂²	1	2	yes	—
13	Q₈×C₁₃	—	4	2	no	—
14	C₂×C₄×C₁₃	—	2	2	no	—
15	C₂×C₄×C₁₃	D₄	1	1	yes	characteristic
16	C₂×C₄×C₁₃	D₄	1	1	yes	characteristic
17	C₂×C₄×C₁₃	C₂³	1	1	yes	characteristic
18	C₂×Dic₁₃	—	2	2	no	—
19	Dic₂₆	—	4	2	no	—
20	#1055	Q₈	1	4	yes	complemented
21	#500	—	13	1	no	maximal, Sylow
22	C₂²×C₁₃	C₂×D₄	1	1	yes	characteristic
23	C₄×C₁₃	—	4	2	no	—
24	C₄×C₁₃	D₈	1	2	yes	—
25	C₄×C₁₃	Q₁₆	1	2	yes	—
26	C₄×C₁₃	C₂×Q₈	1	2	yes	—
27	Dic₁₃	—	4	2	no	—
28	C₄⋊Q₈	—	13	1	no	—
29	C₄⋊Q₈	D₁₃	1	1	yes	characteristic
30	#107	—	13	4	no	—
31	C₄×C₈	—	13	1	no	—
32	C₂×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
33	C₂×C₁₃	C₂×D₈	1	1	yes	characteristic
34	C₂×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
35	C₂×Q₈	—	2	1	no	—
36	C₂×Q₈	—	26	1	no	—
37	C₂×C₈	—	13	2	no	—
38	C₄⋊C₄	—	2	1	no	—
39	C₄⋊C₄	—	13	2	no	—
40	C₄⋊C₄	—	26	1	no	—
41	C₄⋊C₄	C₂×D₁₃	1	2	yes	—
42	C₄²	C₂×D₁₃	1	1	yes	characteristic
43	C₁₃	#500	1	1	yes	characteristic, complemented, Sylow
44	Q₈	—	4	2	no	—
45	Q₈	—	52	2	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	26	2	no	—
48	C₂×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₁₃	1	1	yes	characteristic
51	C₈	—	13	4	no	—
52	C₂²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	—	52	2	no	—
55	C₄	#6104	1	2	yes	—
56	C₄	#6107	1	2	yes	—
57	C₄	Q₈×D₁₃	1	2	yes	—
58	C₂	C₂×#6104	1	1	yes	characteristic
59	C₂	C₂×#6107	1	1	yes	characteristic
60	C₂	#88198	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₃
Fitting	C₁₃×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₁₃	C₂×D₄
hypercenter	C₄⋊Q₈	D₁₃
hypocenter	C₁₃	#500
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
20. #1055	Q₈	semidirect
43. C₁₃	#500	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×C₄⋊Q₈	2
#88123	2
#88047	2
C₄×#1055	2
#500	13