Subgroups — #35204188

Order 20288.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099584	C₂	1	1	yes	maximal, characteristic
2	C₂×D₃₁₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#23778835	C₂	1	2	yes	maximal, complemented
4	C₄×#14579268	C₂	1	2	yes	maximal, complemented
5	#23778738	C₂	1	1	yes	maximal, characteristic
6	#19099584	C₄	1	4	yes	complemented
7	D₃₁₇⋊C₈	C₄	1	4	yes	complemented
8	C₂×#14579268	C₂²	1	1	yes	characteristic
9	C₂×#14579268	C₂²	1	2	yes	—
10	C₂×#14579268	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₃₁₇	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₃₁₇	C₄	1	1	yes	characteristic
13	#19099551	C₄	1	2	yes	—
14	C₄×Dic₃₁₇	C₂²	1	2	yes	—
15	C₄⋊C₄×C₃₁₇	C₄	1	1	yes	characteristic
16	#14579268	C₂×C₄	1	2	yes	—
17	#14579268	C₂×C₄	1	2	yes	—
18	#14579268	C₂×C₄	1	4	yes	—
19	C₂²×D₃₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₃₁₇	C₂×C₄	1	2	yes	—
21	C₂×Dic₃₁₇	C₂³	1	1	yes	characteristic
22	C₄×D₃₁₇	C₂×C₄	1	2	yes	—
23	C₄×D₃₁₇	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₃₁₇	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₃₁₇	—	2	1	no	—
28	C₂×D₃₁₇	—	2	1	no	—
29	C₂×D₃₁₇	C₄²	1	2	yes	—
30	C₄×C₃₁₇	—	2	2	no	—
31	C₄×C₃₁₇	C₄²	1	2	yes	—
32	Dic₃₁₇	—	2	2	no	—
33	Dic₃₁₇	C₂²×C₄	1	1	yes	characteristic
34	Dic₃₁₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₃₁₇	C₂×C₄²	1	1	yes	characteristic
36	C₂×C₃₁₇	#131	1	1	yes	characteristic
37	C₂×C₃₁₇	#131	1	1	yes	characteristic
38	D₃₁₇	—	2	2	no	—
39	C₃₁₇	#405	1	1	yes	characteristic, complemented, Sylow
40	#405	—	317	1	no	maximal, Sylow
41	C₂×M₁₆	—	317	1	no	—
42	C₂²×C₈	—	317	1	no	—
43	#117	—	317	1	no	—
44	#97	—	317	2	no	—
45	C₄×C₈	—	317	2	no	—
46	C₂²×C₄	—	317	1	no	—
47	M₁₆	—	317	4	no	—
48	C₂×C₈	—	317	1	no	—
49	C₂×C₈	—	317	2	no	—
50	C₂×C₈	—	317	1	no	—
51	C₂×C₈	—	317	4	no	—
52	C₄⋊C₄	—	317	1	no	—
53	C₄⋊C₄	#11929639	1	1	yes	characteristic
54	C₂²⋊C₄	—	317	2	no	—
55	C₄²	—	317	2	no	—
56	C₂³	—	317	1	no	—
57	C₂×C₄	—	317	1	no	—
58	C₂×C₄	—	317	2	no	—
59	C₂×C₄	—	317	2	no	—
60	C₂×C₄	—	317	2	no	—
61	C₂×C₄	C₂×#11929639	1	2	yes	—
62	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
63	C₈	—	317	2	no	—
64	C₈	—	317	2	no	—
65	C₈	—	317	4	no	—
66	C₂²	—	317	2	no	—
67	C₂²	—	634	1	no	—
68	C₂²	—	634	1	no	—
69	C₂²	C₂²×#11929639	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	317	1	no	—
72	C₄	—	317	1	no	—
73	C₄	—	634	2	no	—
74	C₄	C₄×#11929639	1	2	yes	—
75	C₂	—	634	2	no	—
76	C₂	C₂×C₄×#11929639	1	1	yes	characteristic
77	C₂	#23778867	1	1	yes	characteristic
78	C₂	#23778868	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11929639
commutator	C₂×C₃₁₇	C₂×C₄²
Fitting	C₄⋊C₄×C₃₁₇	C₄
socle	C₂²×C₃₁₇	C₂²×C₄
prime radical	C₂²×D₃₁₇	C₂×C₄
cosocle	C₂×Dic₃₁₇	C₂³
hypercenter	C₄⋊C₄	#11929639
hypocenter	C₃₁₇	#405
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. #23778835	C₂	semidirect
4. C₄×#14579268	C₂	semidirect
6. #19099584	C₄	semidirect
7. D₃₁₇⋊C₈	C₄	semidirect
39. C₃₁₇	#405	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19099584	2
C₂×D₃₁₇⋊C₈	2
#23778835	2
C₄×#14579268	2
#23778738	2
#405	317