Subgroups — C₂×#23778837 #35204192

Order 20288.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14579268	C₂	1	2	yes	maximal, complemented
2	#23778837	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₄×D₃₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579268	—	2	4	no	—
5	C₂×#14579268	C₂²	1	4	yes	complemented
6	C₂³×D₃₁₇	C₄	1	1	yes	characteristic
7	C₂²×Dic₃₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₃₁₇	—	2	4	no	—
9	C₂×C₄×D₃₁₇	C₄	1	2	yes	—
10	C₂×C₄×D₃₁₇	C₂²	1	2	yes	—
11	C₂²×C₄×C₃₁₇	C₄	1	1	yes	characteristic
12	#14579268	—	2	4	no	—
13	C₂²×D₃₁₇	—	2	2	no	—
14	C₂²×D₃₁₇	—	2	2	no	—
15	C₂²×D₃₁₇	C₈	1	4	yes	complemented
16	C₂²×D₃₁₇	C₂×C₄	1	2	yes	—
17	C₂×Dic₃₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₃₁₇	D₄	1	4	yes	—
19	C₂×Dic₃₁₇	C₂³	1	1	yes	characteristic
20	C₄×D₃₁₇	—	2	4	no	—
21	C₄×D₃₁₇	—	2	4	no	—
22	C₂³×C₃₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₃₁₇	—	2	2	no	—
24	C₂×C₄×C₃₁₇	C₂×C₄	1	2	yes	—
25	C₂²×C₃₁₇	C₂²⋊C₄	1	2	yes	—
26	C₂²×C₃₁₇	C₂×C₈	1	2	yes	—
27	C₂²×C₃₁₇	M₁₆	1	2	yes	—
28	C₂²×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×D₃₁₇	—	2	8	no	—
30	C₂×D₃₁₇	—	2	2	no	—
31	C₂×D₃₁₇	—	2	2	no	—
32	C₂×D₃₁₇	C₂×C₈	1	4	yes	complemented
33	C₄×C₃₁₇	—	2	2	no	—
34	Dic₃₁₇	C₂²⋊C₄	1	2	yes	—
35	Dic₃₁₇	C₂×D₄	1	2	yes	—
36	C₂×C₃₁₇	C₂²⋊C₈	1	4	yes	complemented
37	C₂×C₃₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₃₁₇	C₂²×C₈	1	1	yes	characteristic
39	C₂×C₃₁₇	C₂×M₁₆	1	1	yes	characteristic
40	D₃₁₇	—	2	4	no	—
41	C₃₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₂²⋊C₈	—	317	1	no	maximal, Sylow
43	C₂³×C₄	—	317	1	no	—
44	C₂²×C₈	—	317	2	no	—
45	C₂²⋊C₈	—	317	4	no	—
46	C₂⁴	—	317	1	no	—
47	C₂²×C₄	—	317	1	no	—
48	C₂²×C₄	—	317	2	no	—
49	C₂²×C₄	—	317	2	no	—
50	C₂²×C₄	—	634	4	no	—
51	C₂²×C₄	#11929639	1	1	yes	characteristic
52	C₂×C₈	—	317	4	no	—
53	C₂×C₈	—	634	4	no	—
54	C₂³	—	317	4	no	—
55	C₂³	—	317	2	no	—
56	C₂³	—	634	2	no	—
57	C₂³	—	634	2	no	—
58	C₂³	C₂×#11929639	1	1	yes	characteristic
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	317	4	no	—
61	C₂×C₄	—	317	1	no	—
62	C₂×C₄	—	317	1	no	—
63	C₂×C₄	—	634	4	no	—
64	C₂×C₄	—	634	4	no	—
65	C₂×C₄	C₂×#11929639	1	2	yes	—
66	C₈	—	634	4	no	—
67	C₂²	—	317	4	no	—
68	C₂²	—	634	8	no	—
69	C₂²	—	634	2	no	—
70	C₂²	—	634	2	no	—
71	C₂²	C₂²⋊#11929639	1	2	yes	—
72	C₂²	D₃₁₇⋊C₈	1	2	yes	—
73	C₂²	#19099584	1	2	yes	—
74	C₂²	C₂²×#11929639	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	317	2	no	—
77	C₄	—	317	2	no	—
78	C₂	—	634	4	no	—
79	C₂	#23778837	1	4	yes	direct product factor
80	C₂	C₂×C₂²⋊#11929639	1	1	yes	characteristic
81	C₂	C₂×D₃₁₇⋊C₈	1	1	yes	characteristic
82	C₂	C₂×#19099584	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11929639
commutator	C₂×C₃₁₇	C₂²×C₈
Frattini	C₂²	C₂²×#11929639
Fitting	C₂²×C₄×C₃₁₇	C₄
socle	C₂³×C₃₁₇	C₂×C₄
prime radical	C₂³×D₃₁₇	C₄
cosocle	C₂×Dic₃₁₇	C₂³
hypercenter	C₂²×C₄	#11929639
hypocenter	C₃₁₇	C₂×C₂²⋊C₈
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#14579268	C₂	semidirect
2. #23778837	C₂	direct (N × H)
5. C₂×#14579268	C₂²	semidirect
15. C₂²×D₃₁₇	C₈	semidirect
32. C₂×D₃₁₇	C₂×C₈	semidirect
36. C₂×C₃₁₇	C₂²⋊C₈	semidirect
41. C₃₁₇	C₂×C₂²⋊C₈	semidirect
79. C₂	#23778837	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14579268	2
#23778837	2
C₂²×C₄×D₃₁₇	2
C₂×C₂²⋊C₈	317

Subgroups — C2×#23778837 #35204192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23778837 #35204192