Subgroups — #35204093

Order 20288.

110 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×C₄×#11929639	C₂	1	1	yes	maximal, characteristic
3	#23778738	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11929639	—	2	1	no	—
5	#19099584	—	2	1	no	—
6	#19099584	C₄	1	2	yes	complemented
7	C₂×#14579268	—	2	1	no	—
8	C₄×#11929639	—	2	1	no	—
9	C₄×#11929639	C₄	1	2	yes	complemented
10	C₂×C₄×D₃₁₇	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₃₁₇	C₄	1	1	yes	characteristic, complemented
12	#19099551	—	2	1	no	—
13	C₄×Dic₃₁₇	—	2	1	no	—
14	C₄⋊C₄×C₃₁₇	C₄	1	1	yes	characteristic, complemented
15	C₂×#11929639	—	2	2	no	—
16	C₂×#11929639	—	2	2	no	—
17	C₂×#11929639	—	2	2	no	—
18	#14579268	—	2	2	no	—
19	C₂²×D₃₁₇	D₄	1	1	yes	characteristic
20	C₂×Dic₃₁₇	—	2	1	no	—
21	C₂×Dic₃₁₇	D₄	1	1	yes	characteristic
22	C₄×D₃₁₇	C₂×C₄	1	1	yes	characteristic
23	C₄×D₃₁₇	C₂×C₄	1	1	yes	characteristic
24	C₄×D₃₁₇	D₄	1	1	yes	characteristic
25	C₄×D₃₁₇	Q₈	1	1	yes	characteristic
26	C₂×C₄×C₃₁₇	—	2	1	no	—
27	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×D₃₁₇	—	2	1	no	—
30	C₂×D₃₁₇	—	2	1	no	—
31	C₂×D₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×D₃₁₇	C₄⋊C₄	1	1	yes	characteristic
33	#11929639	—	2	4	no	—
34	C₄×C₃₁₇	—	4	1	no	—
35	C₄×C₃₁₇	C₄²	1	1	yes	characteristic
36	C₄×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
37	Dic₃₁₇	—	4	1	no	—
38	Dic₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
39	Dic₃₁₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₃₁₇	#95	1	1	yes	characteristic
41	C₂×C₃₁₇	#104	1	1	yes	characteristic
42	C₂×C₃₁₇	#104	1	1	yes	characteristic
43	D₃₁₇	—	2	1	no	—
44	D₃₁₇	—	2	1	no	—
45	C₃₁₇	#339	1	1	yes	characteristic, complemented, Sylow
46	#339	—	317	1	no	maximal, Sylow
47	C₂×M₁₆	—	317	1	no	—
48	#117	—	317	1	no	—
49	C₂×C₄²	—	317	1	no	—
50	C₂²×C₄	—	317	1	no	—
51	C₂²×C₄	—	634	1	no	—
52	M₁₆	—	317	2	no	—
53	M₁₆	—	634	1	no	—
54	C₂×C₈	—	634	1	no	—
55	C₄⋊C₄	—	317	1	no	—
56	C₄⋊C₄	#11929639	1	1	yes	characteristic, complemented
57	C₂²⋊C₄	—	634	1	no	—
58	C₄²	—	317	2	no	—
59	C₄²	—	634	1	no	—
60	C₄²	—	634	1	no	—
61	C₂³	—	317	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	317	1	no	—
64	C₂×C₄	—	317	1	no	—
65	C₂×C₄	—	317	1	no	—
66	C₂×C₄	—	317	1	no	—
67	C₂×C₄	—	317	1	no	—
68	C₂×C₄	—	634	2	no	—
69	C₂×C₄	—	634	2	no	—
70	C₂×C₄	—	634	2	no	—
71	C₂×C₄	—	634	1	no	—
72	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
73	C₈	—	634	2	no	—
74	C₂²	—	317	1	no	—
75	C₂²	—	317	1	no	—
76	C₂²	—	634	1	no	—
77	C₂²	—	634	1	no	—
78	C₂²	C₂²⋊#11929639	1	1	yes	characteristic
79	C₄	—	4	1	no	—
80	C₄	—	317	1	no	—
81	C₄	—	317	1	no	—
82	C₄	—	634	4	no	—
83	C₄	—	1268	1	no	—
84	C₄	C₄×#11929639	1	1	yes	characteristic
85	C₄	C₄⋊#11929639	1	1	yes	characteristic
86	C₂	—	634	1	no	—
87	C₂	—	634	1	no	—
88	C₂	(C₂×#11929639)⋊C₄	1	1	yes	characteristic
89	C₂	#23778844	1	1	yes	characteristic
90	C₂	#23778845	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#11929639
commutator	C₄×C₃₁₇	C₄²
Frattini	C₂×C₄	C₂×#11929639
Fitting	C₄⋊C₄×C₃₁₇	C₄
socle	C₂²×C₃₁₇	C₂²⋊C₄
prime radical	C₂²×D₃₁₇	D₄
cosocle	C₂×C₄×D₃₁₇	C₂²
hypercenter	C₄⋊C₄	#11929639
hypocenter	C₃₁₇	#339
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
6. #19099584	C₄	semidirect
9. C₄×#11929639	C₄	semidirect
11. C₄⋊Dic₃₁₇	C₄	semidirect
14. C₄⋊C₄×C₃₁₇	C₄	semidirect
45. C₃₁₇	#339	semidirect
56. C₄⋊C₄	#11929639	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×C₄×#11929639	2
#23778738	2
#339	317