Subgroups — #35203381

Order 20288.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₆₃₄	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₂₆₈	C₂	1	2	yes	maximal, complemented
3	#23778759	C₂	1	1	yes	maximal, characteristic
4	#23778705	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×C₃₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₃₄	—	2	4	no	—
7	C₂×Dic₆₃₄	—	2	1	no	—
8	C₂×Dic₆₃₄	C₂²	1	1	yes	characteristic
9	C₂×Dic₆₃₄	C₂²	1	2	yes	—
10	C₂²×Dic₃₁₇	—	2	1	no	—
11	Dic₁₂₆₈	—	2	4	no	—
12	C₄⋊Dic₃₁₇	C₂²	1	1	yes	characteristic
13	#19099553	—	2	1	no	—
14	C₂²⋊Dic₃₁₇	—	2	1	no	—
15	C₂²×C₄×C₃₁₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₃₁₇	C₂²	1	2	yes	—
17	Dic₆₃₄	—	2	4	no	—
18	Dic₆₃₄	—	2	2	no	—
19	Dic₆₃₄	—	2	2	no	—
20	Dic₆₃₄	D₄	1	4	yes	—
21	C₂×Dic₃₁₇	—	2	4	no	—
22	C₂×Dic₃₁₇	—	2	2	no	—
23	C₂×Dic₃₁₇	—	2	1	no	—
24	C₂×Dic₃₁₇	—	2	1	no	—
25	C₂³×C₃₁₇	D₄	1	1	yes	characteristic
26	C₂×C₄×C₃₁₇	—	2	2	no	—
27	C₂×C₄×C₃₁₇	D₄	1	1	yes	characteristic
28	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
29	C₈×C₃₁₇	—	2	2	no	—
30	C₂²×C₃₁₇	—	2	1	no	—
31	C₂²×C₃₁₇	—	2	1	no	—
32	C₂²×C₃₁₇	Q₁₆	1	2	yes	complemented
33	C₂²×C₃₁₇	C₂×D₄	1	1	yes	characteristic
34	C₄×C₃₁₇	—	2	1	no	—
35	C₄×C₃₁₇	C₂×D₄	1	2	yes	—
36	Dic₃₁₇	—	2	4	no	—
37	Dic₃₁₇	—	4	1	no	—
38	Dic₃₁₇	—	4	1	no	—
39	C₂×C₃₁₇	—	2	2	no	—
40	C₂×C₃₁₇	#120	1	1	yes	characteristic
41	C₂×C₃₁₇	C₂×Q₁₆	1	1	yes	characteristic
42	C₂×C₃₁₇	#137	1	1	yes	characteristic
43	C₃₁₇	#451	1	1	yes	characteristic, complemented, Sylow
44	#451	—	317	1	no	maximal, Sylow
45	C₂²×Q₈	—	317	1	no	—
46	C₂×Q₁₆	—	317	2	no	—
47	C₂²⋊Q₈	—	317	1	no	—
48	#103	—	317	2	no	—
49	C₂²⋊C₈	D₃₁₇	1	1	yes	characteristic
50	C₂×Q₈	—	317	2	no	—
51	C₂×Q₈	—	317	1	no	—
52	C₂×Q₈	—	634	4	no	—
53	C₂×Q₈	—	634	1	no	—
54	C₂²×C₄	—	634	1	no	—
55	C₂²×C₄	C₂×D₃₁₇	1	1	yes	characteristic
56	Q₁₆	—	634	4	no	—
57	C₂×C₈	C₂×D₃₁₇	1	2	yes	—
58	C₄⋊C₄	—	317	1	no	—
59	C₄⋊C₄	—	634	1	no	—
60	C₂²⋊C₄	—	634	1	no	—
61	C₂³	D₁₂₆₈	1	1	yes	characteristic
62	Q₈	—	317	4	no	—
63	Q₈	—	634	4	no	—
64	Q₈	—	634	2	no	—
65	Q₈	—	634	2	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	634	4	no	—
68	C₂×C₄	—	634	2	no	—
69	C₂×C₄	—	634	1	no	—
70	C₂×C₄	—	634	1	no	—
71	C₂×C₄	D₁₂₆₈	1	1	yes	characteristic
72	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
73	C₈	—	2	2	no	—
74	C₂²	—	2	1	no	—
75	C₂²	—	2	1	no	—
76	C₂²	Dic₁₂₆₈	1	2	yes	complemented
77	C₂²	C₂×D₁₂₆₈	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	634	4	no	—
80	C₄	—	1268	1	no	—
81	C₄	—	1268	1	no	—
82	C₄	D₄×D₃₁₇	1	2	yes	—
83	C₂	—	2	2	no	—
84	C₂	#23778750	1	1	yes	characteristic
85	C₂	C₂×Dic₁₂₆₈	1	1	yes	characteristic
86	C₂	#23778813	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₂₆₈
commutator, cosocle	C₂×C₄×C₃₁₇	C₂³
Frattini	C₂×C₄	C₂²×D₃₁₇
Fitting	C₂²⋊C₈×C₃₁₇	C₂
socle	C₂²×C₃₁₇	C₂×D₄
prime radical	C₂³×C₃₁₇	D₄
hypercenter	C₂²⋊C₈	D₃₁₇
hypocenter	C₃₁₇	#451
LCS	G ▷ C₂×C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₁₂₆₈	C₂	semidirect
4. #23778705	C₂	semidirect
32. C₂²×C₃₁₇	Q₁₆	semidirect
43. C₃₁₇	#451	semidirect
76. C₂²	Dic₁₂₆₈	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₆₃₄	2
C₂×Dic₁₂₆₈	2
#23778759	2
#23778705	2
C₂²⋊C₈×C₃₁₇	2
#451	317