Subgroups — C₂×#23778771 #35203744

Order 20288.

226 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778771	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊Dic₃₁₇	C₂	1	2	yes	maximal
3	C₂×C₄⋊Dic₃₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#19099553	C₂	1	2	yes	maximal
5	C₂×C₄×Dic₃₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊C₄×C₃₁₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₁₇	C₂²	1	1	yes	characteristic
8	C₂²×Dic₃₁₇	C₂²	1	1	yes	characteristic
9	C₂²×Dic₃₁₇	C₂²	1	2	yes	—
10	C₄⋊Dic₃₁₇	C₂²	1	8	yes	—
11	C₄⋊Dic₃₁₇	C₂²	1	4	yes	—
12	#19099553	C₂²	1	8	yes	—
13	C₄×Dic₃₁₇	C₂²	1	4	yes	—
14	C₂²×C₄×C₃₁₇	C₂²	1	2	yes	—
15	C₂²×C₄×C₃₁₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₃₁₇	C₂²	1	4	yes	—
17	C₂×Dic₃₁₇	—	2	2	no	—
18	C₂×Dic₃₁₇	—	2	2	no	—
19	C₂×Dic₃₁₇	—	2	4	no	—
20	C₂×Dic₃₁₇	C₂³	1	2	yes	—
21	C₂×Dic₃₁₇	C₂³	1	2	yes	—
22	C₂×Dic₃₁₇	C₂³	1	4	yes	—
23	C₂³×C₃₁₇	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₃₁₇	—	2	4	no	—
25	C₂×C₄×C₃₁₇	Q₈	1	4	yes	—
26	C₂×C₄×C₃₁₇	C₂³	1	4	yes	—
27	C₂×C₄×C₃₁₇	C₂³	1	2	yes	—
28	C₂²×C₃₁₇	C₂×Q₈	1	2	yes	—
29	C₂²×C₃₁₇	#41	1	2	yes	—
30	C₂²×C₃₁₇	#41	1	2	yes	—
31	C₂²×C₃₁₇	C₂⁴	1	1	yes	characteristic
32	C₄×C₃₁₇	—	2	4	no	—
33	C₄×C₃₁₇	C₂×Q₈	1	4	yes	—
34	Dic₃₁₇	—	2	2	no	—
35	Dic₃₁₇	—	2	2	no	—
36	Dic₃₁₇	—	2	4	no	—
37	C₂×C₃₁₇	#125	1	4	yes	complemented
38	C₂×C₃₁₇	C₂²×Q₈	1	1	yes	characteristic
39	C₂×C₃₁₇	C₂×#41	1	1	yes	characteristic
40	C₂×C₃₁₇	C₂×#41	1	1	yes	characteristic
41	C₃₁₇	C₂×#125	1	1	yes	characteristic, complemented, Sylow
42	C₂×#125	—	317	1	no	maximal, Sylow
43	#125	—	317	8	no	—
44	C₂×C₄⋊C₄	—	317	2	no	—
45	C₂×C₄⋊C₄	—	317	2	no	—
46	C₂×C₄⋊C₄	—	317	1	no	—
47	C₂×C₄⋊C₄	D₃₁₇	1	1	yes	characteristic
48	C₂×C₄²	—	317	1	no	—
49	C₂²×C₄	—	317	1	no	—
50	C₂²×C₄	—	317	1	no	—
51	C₂²×C₄	—	317	2	no	—
52	C₂²×C₄	C₂×D₃₁₇	1	2	yes	—
53	C₂²×C₄	C₂×D₃₁₇	1	1	yes	characteristic
54	C₄⋊C₄	—	317	8	no	—
55	C₄⋊C₄	—	317	8	no	—
56	C₄⋊C₄	—	317	4	no	—
57	C₄⋊C₄	C₂×D₃₁₇	1	4	yes	—
58	C₄²	—	317	4	no	—
59	C₂³	C₂²×D₃₁₇	1	1	yes	characteristic
60	C₂×C₄	—	2	4	no	—
61	C₂×C₄	—	317	2	no	—
62	C₂×C₄	—	317	2	no	—
63	C₂×C₄	—	317	4	no	—
64	C₂×C₄	—	634	2	no	—
65	C₂×C₄	—	634	2	no	—
66	C₂×C₄	—	634	4	no	—
67	C₂×C₄	Dic₆₃₄	1	4	yes	—
68	C₂×C₄	C₂²×D₃₁₇	1	4	yes	—
69	C₂×C₄	C₂²×D₃₁₇	1	2	yes	—
70	C₂²	C₂×Dic₆₃₄	1	2	yes	—
71	C₂²	#19099573	1	2	yes	—
72	C₂²	#19099575	1	2	yes	—
73	C₂²	C₂³×D₃₁₇	1	1	yes	characteristic
74	C₄	—	2	4	no	—
75	C₄	—	634	2	no	—
76	C₄	—	634	2	no	—
77	C₄	—	634	4	no	—
78	C₄	C₂×Dic₆₃₄	1	4	yes	—
79	C₂	#23778771	1	4	yes	direct product factor
80	C₂	C₂²×Dic₆₃₄	1	1	yes	characteristic
81	C₂	C₂×#19099573	1	1	yes	characteristic
82	C₂	C₂×#19099575	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₃₁₇
commutator, cosocle	C₂²×C₃₁₇	C₂⁴
Frattini	C₂²	C₂³×D₃₁₇
Fitting	C₂×C₄⋊C₄×C₃₁₇	C₂
socle, prime radical	C₂³×C₃₁₇	C₂³
hypercenter	C₂×C₄⋊C₄	D₃₁₇
hypocenter	C₃₁₇	C₂×#125
LCS	G ▷ C₂²×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. #23778771	C₂	direct (N × H)
37. C₂×C₃₁₇	#125	semidirect
41. C₃₁₇	C₂×#125	semidirect
79. C₂	#23778771	direct (N × H)

Maximal subgroups

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

subgroup	index
#23778771	2
C₂×C₄⋊Dic₃₁₇	2
C₂×#19099553	2
C₂×C₄×Dic₃₁₇	2
C₂×C₄⋊C₄×C₃₁₇	2
C₂×#125	317

Subgroups — C2×#23778771 #35203744

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23778771 #35203744