Subgroups — #35204129

Order 20288.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778839	C₂	1	2	yes	maximal
2	#23778769	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#11929639	—	2	2	no	—
4	C₂×D₁₂₆₈	C₂²	1	1	yes	characteristic
5	#19099551	—	2	2	no	—
6	C₄×Dic₃₁₇	C₄	1	1	yes	characteristic, complemented
7	C₂×Q₈×C₃₁₇	C₄	1	1	yes	characteristic, complemented
8	C₂×#11929639	—	4	2	no	—
9	C₂²×D₃₁₇	D₄	1	2	yes	—
10	D₁₂₆₈	—	4	1	no	—
11	C₂×Dic₃₁₇	—	2	1	no	—
12	Q₈×C₃₁₇	—	4	1	no	—
13	C₂×C₄×C₃₁₇	—	2	1	no	—
14	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
16	C₂×D₃₁₇	—	2	2	no	—
17	C₂×D₃₁₇	—	4	2	no	—
18	#11929639	—	8	2	no	—
19	C₄×C₃₁₇	—	2	1	no	—
20	C₄×C₃₁₇	—	4	1	no	—
21	Dic₃₁₇	—	4	1	no	—
22	C₂×C₃₁₇	—	2	1	no	—
23	C₂×C₃₁₇	#99	1	1	yes	characteristic
24	D₃₁₇	—	4	2	no	—
25	C₃₁₇	#354	1	1	yes	characteristic, complemented, Sylow
26	#354	—	317	1	no	maximal, Sylow
27	#124	—	317	1	no	—
28	#99	—	317	2	no	—
29	C₂×Q₈	#11929639	1	1	yes	characteristic, complemented
30	C₂×D₄	—	317	1	no	—
31	C₂²⋊C₄	—	634	2	no	—
32	C₂²⋊C₄	—	634	2	no	—
33	C₄²	—	317	1	no	—
34	C₂³	—	317	2	no	—
35	Q₈	—	4	1	no	—
36	D₄	—	1268	1	no	—
37	C₂×C₄	—	2	1	no	—
38	C₂×C₄	—	634	1	no	—
39	C₂×C₄	—	1268	2	no	—
40	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
41	C₂²	—	634	2	no	—
42	C₂²	—	1268	2	no	—
43	C₂²	C₂²⋊#11929639	1	1	yes	characteristic
44	C₄	—	2	1	no	—
45	C₄	—	4	1	no	—
46	C₄	—	2536	2	no	—
47	C₄	—	1268	1	no	—
48	C₂	—	2	1	no	—
49	C₂	—	1268	2	no	—
50	C₂	#23778838	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23778838
commutator	C₂×C₄×C₃₁₇	C₂×C₄
Frattini	C₂×C₄	C₂×#11929639
Fitting	C₂×Q₈×C₃₁₇	C₄
socle	C₂×C₃₁₇	#99
prime radical, cosocle	C₂×D₁₂₆₈	C₂²
hypercenter	C₂×Q₈	#11929639
hypocenter	C₃₁₇	#354
LCS	G ▷ C₂×C₄×C₃₁₇ ▷ C₂²×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
6. C₄×Dic₃₁₇	C₄	semidirect
7. C₂×Q₈×C₃₁₇	C₄	semidirect
25. C₃₁₇	#354	semidirect
29. C₂×Q₈	#11929639	semidirect

Maximal subgroups

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

subgroup	index
#23778839	2
#23778769	2
#354	317