Subgroups — #34093882

Order 20032.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₆₂₆	C₂	1	1	yes	maximal, characteristic
2	C₄×#14578809	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×C₃₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₆₂₆	C₄	1	1	yes	characteristic
5	C₂×#14578809	—	2	1	no	—
6	C₄⋊Dic₃₁₃	C₄	1	1	yes	characteristic
7	#19097540	—	2	1	no	—
8	C₄×Dic₃₁₃	—	2	1	no	—
9	C₂×C₈×C₃₁₃	—	2	1	no	—
10	C₄²×C₃₁₃	C₂²	1	1	yes	characteristic
11	Dic₆₂₆	—	2	1	no	—
12	Dic₆₂₆	C₈	1	2	yes	complemented
13	C₂×Dic₃₁₃	—	2	1	no	—
14	C₂×Dic₃₁₃	—	2	1	no	—
15	#14578809	—	2	2	no	—
16	C₂×C₄×C₃₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₃₁₃	D₄	1	1	yes	characteristic
18	C₂×C₄×C₃₁₃	D₄	1	1	yes	characteristic
19	C₈×C₃₁₃	—	4	1	no	—
20	C₂²×C₃₁₃	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₃₁₃	—	2	1	no	—
22	C₄×C₃₁₃	C₂×C₈	1	1	yes	characteristic
23	C₄×C₃₁₃	M₁₆	1	1	yes	characteristic
24	C₄×C₃₁₃	SD₁₆	1	1	yes	characteristic
25	C₄×C₃₁₃	Q₁₆	1	1	yes	characteristic
26	Dic₃₁₃	—	2	2	no	—
27	Dic₃₁₃	—	4	1	no	—
28	C₂×C₃₁₃	C₂²⋊C₈	1	1	yes	characteristic
29	C₂×C₃₁₃	#103	1	1	yes	characteristic
30	C₂×C₃₁₃	#104	1	1	yes	characteristic
31	C₃₁₃	Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
32	Q₈⋊C₈	—	313	1	no	maximal, Sylow
33	C₄×Q₈	—	313	1	no	—
34	C₄⋊C₈	D₃₁₃	1	1	yes	characteristic
35	C₄×C₈	—	313	1	no	—
36	C₂×Q₈	—	313	1	no	—
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	626	1	no	—
39	C₄⋊C₄	—	313	1	no	—
40	C₄⋊C₄	—	626	1	no	—
41	C₄²	—	626	1	no	—
42	C₄²	C₂×D₃₁₃	1	1	yes	characteristic
43	Q₈	—	313	2	no	—
44	Q₈	—	626	1	no	—
45	C₂×C₄	—	626	1	no	—
46	C₂×C₄	—	626	1	no	—
47	C₂×C₄	C₄×D₃₁₃	1	1	yes	characteristic
48	C₂×C₄	C₂²⋊D₃₁₃	1	1	yes	characteristic
49	C₂×C₄	D₁₂₅₂	1	1	yes	characteristic
50	C₈	—	4	1	no	—
51	C₈	—	626	2	no	—
52	C₂²	#19097538	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	626	2	no	—
55	C₄	—	1252	1	no	—
56	C₄	C₈×D₃₁₃	1	1	yes	characteristic
57	C₄	#19097544	1	1	yes	characteristic
58	C₄	#19097549	1	1	yes	characteristic
59	C₄	#19097551	1	1	yes	characteristic
60	C₂	#23756409	1	1	yes	characteristic
61	C₂	#23756420	1	1	yes	characteristic
62	C₂	#23756423	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₃₁₃
commutator	C₄×C₃₁₃	C₂×C₈
Frattini	C₄²	C₂×D₃₁₃
Fitting	C₄⋊C₈×C₃₁₃	C₂
socle, prime radical	C₂²×C₃₁₃	C₂²⋊C₄
cosocle	C₄²×C₃₁₃	C₂²
hypercenter	C₄⋊C₈	D₃₁₃
hypocenter	C₃₁₃	Q₈⋊C₈
LCS	G ▷ C₄×C₃₁₃ ▷ C₂×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
12. Dic₆₂₆	C₈	semidirect
31. C₃₁₃	Q₈⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₆₂₆	2
C₄×#14578809	2
C₄⋊C₈×C₃₁₃	2
Q₈⋊C₈	313