Subgroups — #35204080

Order 20288.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778835	C₂	1	1	yes	maximal, characteristic
2	#23778835	C₂	1	1	yes	maximal, characteristic
3	#23778771	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579268	—	2	1	no	—
5	C₂×#14579268	—	2	1	no	—
6	C₄⋊Dic₃₁₇	—	2	1	no	—
7	C₄⋊Dic₃₁₇	C₄	1	1	yes	characteristic
8	#19099553	—	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	—	2	2	no	—
13	C₂×Dic₃₁₇	—	2	1	no	—
14	C₂×Dic₃₁₇	D₄	1	1	yes	characteristic
15	C₂×Dic₃₁₇	D₄	1	1	yes	characteristic
16	C₂×C₄×C₃₁₇	—	2	1	no	—
17	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
19	C₄×C₃₁₇	—	2	1	no	—
20	C₄×C₃₁₇	—	4	1	no	—
21	Dic₃₁₇	—	2	1	no	—
22	Dic₃₁₇	—	2	1	no	—
23	Dic₃₁₇	—	4	1	no	—
24	C₂×C₃₁₇	#101	1	1	yes	characteristic
25	C₂×C₃₁₇	#104	1	1	yes	characteristic
26	C₂×C₃₁₇	#104	1	1	yes	characteristic
27	C₃₁₇	#330	1	1	yes	characteristic, complemented, Sylow
28	#330	—	317	1	no	maximal, Sylow
29	#125	—	317	1	no	—
30	#97	—	317	1	no	—
31	#97	—	317	1	no	—
32	C₂×C₈	—	634	1	no	—
33	C₂×C₈	—	634	1	no	—
34	C₄⋊C₄	—	317	1	no	—
35	C₄⋊C₄	—	634	1	no	—
36	C₄⋊C₄	—	634	1	no	—
37	C₄⋊C₄	#11929639	1	1	yes	characteristic
38	C₄²	—	317	1	no	—
39	C₂×C₄	—	2	1	no	—
40	C₂×C₄	—	317	1	no	—
41	C₂×C₄	—	317	1	no	—
42	C₂×C₄	—	634	1	no	—
43	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
44	C₈	—	634	2	no	—
45	C₈	—	634	2	no	—
46	C₂²	C₂²⋊#11929639	1	1	yes	characteristic
47	C₄	—	2	1	no	—
48	C₄	—	4	1	no	—
49	C₄	—	634	1	no	—
50	C₄	—	634	1	no	—
51	C₄	—	1268	1	no	—
52	C₂	#23778841	1	1	yes	characteristic
53	C₂	#23778844	1	1	yes	characteristic
54	C₂	#23778845	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#11929639
commutator	C₂×C₄×C₃₁₇	C₂×C₄
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	#330
LCS	G ▷ C₂×C₄×C₃₁₇ ▷ C₂²×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
27. C₃₁₇	#330	semidirect

Maximal subgroups

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

subgroup	index
#23778835	2
#23778771	2
#330	317