Subgroups — #35204073

Order 20288.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#14579268	C₂	1	1	yes	maximal, characteristic
2	C₄×#14579268	C₂	1	1	yes	maximal, characteristic
3	#23778745	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579268	—	2	1	no	—
5	C₂×#14579268	—	2	1	no	—
6	C₂×Dic₆₃₄	C₄	1	1	yes	characteristic
7	#19099553	—	2	1	no	—
8	C₄×Dic₃₁₇	—	2	1	no	—
9	C₄×Dic₃₁₇	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₃₁₇	C₄	1	1	yes	characteristic
11	#14579268	—	2	2	no	—
12	#14579268	—	4	1	no	—
13	Dic₆₃₄	—	2	1	no	—
14	Dic₆₃₄	C₈	1	2	yes	complemented
15	C₂×Dic₃₁₇	—	2	1	no	—
16	C₂×Dic₃₁₇	D₄	1	1	yes	characteristic
17	C₂×Dic₃₁₇	D₄	1	1	yes	characteristic
18	C₂×C₄×C₃₁₇	—	2	1	no	—
19	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₃₁₇	—	4	1	no	—
22	C₄×C₃₁₇	C₂×C₈	1	1	yes	characteristic
23	C₄×C₃₁₇	M₁₆	1	1	yes	characteristic
24	Dic₃₁₇	—	2	2	no	—
25	Dic₃₁₇	—	2	1	no	—
26	Dic₃₁₇	SD₁₆	1	1	yes	characteristic
27	Dic₃₁₇	Q₁₆	1	1	yes	characteristic
28	C₂×C₃₁₇	C₂²⋊C₈	1	1	yes	characteristic
29	C₂×C₃₁₇	#103	1	1	yes	characteristic
30	C₂×C₃₁₇	#104	1	1	yes	characteristic
31	C₃₁₇	Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
32	Q₈⋊C₈	—	317	1	no	maximal, Sylow
33	C₄×Q₈	—	317	1	no	—
34	C₄⋊C₈	—	317	1	no	—
35	C₄×C₈	—	317	1	no	—
36	C₂×Q₈	—	317	1	no	—
37	C₂×C₈	—	634	1	no	—
38	C₂×C₈	—	634	1	no	—
39	C₄⋊C₄	—	634	1	no	—
40	C₄⋊C₄	#11929639	1	1	yes	characteristic
41	C₄²	—	317	1	no	—
42	C₄²	—	634	1	no	—
43	Q₈	—	317	2	no	—
44	Q₈	—	634	1	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	317	1	no	—
47	C₂×C₄	—	317	1	no	—
48	C₂×C₄	—	634	1	no	—
49	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
50	C₈	—	634	2	no	—
51	C₈	—	1268	1	no	—
52	C₂²	C₂²⋊#11929639	1	1	yes	characteristic
53	C₄	—	4	1	no	—
54	C₄	—	317	1	no	—
55	C₄	—	317	1	no	—
56	C₄	—	634	2	no	—
57	C₄	—	634	1	no	—
58	C₄	D₃₁₇⋊C₈	1	1	yes	characteristic
59	C₄	#19099584	1	1	yes	characteristic
60	C₂	#23778837	1	1	yes	characteristic
61	C₂	#23778843	1	1	yes	characteristic
62	C₂	#23778844	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#11929639
commutator	C₄×C₃₁₇	C₂×C₈
Frattini	C₂×C₄	C₂×#11929639
Fitting	C₄⋊C₄×C₃₁₇	C₄
socle, prime radical	C₂²×C₃₁₇	C₂²⋊C₄
cosocle	C₄×Dic₃₁₇	C₂²
hypercenter	C₄⋊C₄	#11929639
hypocenter	C₃₁₇	Q₈⋊C₈
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
14. Dic₆₃₄	C₈	semidirect
31. C₃₁₇	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⋊#14579268	2
C₄×#14579268	2
#23778745	2
Q₈⋊C₈	317