Subgroups — #35203424

Order 20288.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099563	C₂	1	2	yes	maximal, complemented
2	#23778777	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₆₃₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#23778702	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₈×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₆₃₄	C₂²	1	1	yes	characteristic
7	C₂×D₁₂₆₈	—	2	1	no	—
8	C₂×D₁₂₆₈	—	2	1	no	—
9	C₂×D₁₂₆₈	C₂²	1	2	yes	—
10	#19099563	—	2	4	no	—
11	C₄⋊Dic₃₁₇	C₂²	1	1	yes	characteristic
12	#19099553	—	2	1	no	—
13	C₄×Dic₃₁₇	—	2	1	no	—
14	C₂×C₈×C₃₁₇	C₂²	1	2	yes	—
15	C₄²×C₃₁₇	C₂²	1	1	yes	characteristic
16	C₂²×D₃₁₇	—	2	2	no	—
17	Dic₆₃₄	—	2	1	no	—
18	Dic₆₃₄	D₄	1	2	yes	complemented
19	D₁₂₆₈	—	2	2	no	—
20	D₁₂₆₈	—	2	2	no	—
21	D₁₂₆₈	—	4	2	no	—
22	D₁₂₆₈	—	4	2	no	—
23	C₂×Dic₃₁₇	—	2	1	no	—
24	C₂×Dic₃₁₇	—	2	1	no	—
25	C₂×C₄×C₃₁₇	D₄	1	1	yes	characteristic
26	C₂×C₄×C₃₁₇	D₄	1	1	yes	characteristic
27	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
28	C₈×C₃₁₇	—	2	2	no	—
29	C₂²×C₃₁₇	C₂×D₄	1	1	yes	characteristic
30	C₂×D₃₁₇	—	4	2	no	—
31	C₂×D₃₁₇	—	4	2	no	—
32	C₂×D₃₁₇	—	4	2	no	—
33	C₄×C₃₁₇	—	2	1	no	—
34	C₄×C₃₁₇	SD₁₆	1	2	yes	complemented
35	C₄×C₃₁₇	C₂×D₄	1	1	yes	characteristic
36	C₄×C₃₁₇	#41	1	1	yes	characteristic
37	Dic₃₁₇	—	2	2	no	—
38	Dic₃₁₇	—	4	1	no	—
39	C₂×C₃₁₇	#121	1	1	yes	characteristic
40	C₂×C₃₁₇	C₂×SD₁₆	1	1	yes	characteristic
41	C₂×C₃₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
42	D₃₁₇	—	8	2	no	—
43	C₃₁₇	#460	1	1	yes	characteristic, complemented, Sylow
44	#460	—	317	1	no	maximal, Sylow
45	C₂×SD₁₆	—	317	2	no	—
46	#127	—	317	1	no	—
47	C₄×Q₈	—	317	1	no	—
48	C₄⋊C₈	D₃₁₇	1	1	yes	characteristic, complemented
49	#102	—	317	2	no	—
50	C₂×Q₈	—	317	1	no	—
51	C₂×D₄	—	317	2	no	—
52	C₂×D₄	—	634	1	no	—
53	C₂×D₄	—	634	1	no	—
54	SD₁₆	—	634	4	no	—
55	C₂×C₈	C₂×D₃₁₇	1	2	yes	—
56	C₄⋊C₄	—	317	1	no	—
57	C₄⋊C₄	—	634	1	no	—
58	C₄²	—	634	1	no	—
59	C₄²	C₂×D₃₁₇	1	1	yes	characteristic
60	C₂³	—	634	2	no	—
61	Q₈	—	317	2	no	—
62	Q₈	—	634	1	no	—
63	D₄	—	634	2	no	—
64	D₄	—	634	2	no	—
65	D₄	—	1268	2	no	—
66	D₄	—	1268	2	no	—
67	C₂×C₄	—	634	1	no	—
68	C₂×C₄	—	634	1	no	—
69	C₂×C₄	D₁₂₆₈	1	1	yes	characteristic
70	C₂×C₄	D₁₂₆₈	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
72	C₈	—	2	2	no	—
73	C₂²	—	1268	2	no	—
74	C₂²	—	1268	2	no	—
75	C₂²	—	1268	2	no	—
76	C₂²	C₂×D₁₂₆₈	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	634	2	no	—
79	C₄	—	1268	1	no	—
80	C₄	#19099563	1	2	yes	complemented
81	C₄	D₄×D₃₁₇	1	1	yes	characteristic
82	C₄	#19099575	1	1	yes	characteristic
83	C₂	—	2536	2	no	—
84	C₂	#23778755	1	1	yes	characteristic
85	C₂	C₂×#19099563	1	1	yes	characteristic
86	C₂	#23778808	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₂₆₈
commutator, cosocle	C₂×C₄×C₃₁₇	C₂³
Frattini	C₂×C₄	C₂²×D₃₁₇
Fitting	C₄⋊C₈×C₃₁₇	C₂
socle	C₂²×C₃₁₇	C₂×D₄
prime radical	#23778777	C₂
hypercenter	C₄⋊C₈	D₃₁₇
hypocenter	C₃₁₇	#460
LCS	G ▷ C₂×C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#19099563	C₂	semidirect
3. C₄×Dic₆₃₄	C₂	semidirect
4. #23778702	C₂	semidirect
5. C₄⋊C₈×C₃₁₇	C₂	semidirect
18. Dic₆₃₄	D₄	semidirect
34. C₄×C₃₁₇	SD₁₆	semidirect
43. C₃₁₇	#460	semidirect
48. C₄⋊C₈	D₃₁₇	semidirect
80. C₄	#19099563	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19099563	2
#23778777	2
C₄×Dic₆₃₄	2
#23778702	2
C₄⋊C₈×C₃₁₇	2
#460	317