Subgroups — #35202959

Order 20288.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778759	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊#14579261	C₂	1	1	yes	maximal, characteristic
3	#95×C₃₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₆₃₄	C₄	1	1	yes	characteristic, complemented
5	C₂×#14579261	—	2	1	no	—
6	C₄⋊Dic₃₁₇	C₄	1	1	yes	characteristic, complemented
7	#19099553	—	2	1	no	—
8	C₂²⋊Dic₃₁₇	—	2	1	no	—
9	C₂²×C₄×C₃₁₇	—	2	1	no	—
10	C₂²×C₄×C₃₁₇	C₂²	1	1	yes	characteristic
11	Dic₆₃₄	—	4	1	no	—
12	C₂×Dic₃₁₇	—	2	1	no	—
13	C₂×Dic₃₁₇	—	2	1	no	—
14	#14579261	—	4	1	no	—
15	C₂³×C₃₁₇	D₄	1	1	yes	characteristic
16	C₂×C₄×C₃₁₇	—	2	2	no	—
17	C₂×C₄×C₃₁₇	—	4	1	no	—
18	C₂×C₄×C₃₁₇	—	4	1	no	—
19	C₂×C₄×C₃₁₇	—	4	1	no	—
20	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₃₁₇	D₄	1	1	yes	characteristic
22	C₂²×C₃₁₇	—	2	1	no	—
23	C₂²×C₃₁₇	—	2	1	no	—
24	C₂²×C₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₃₁₇	SD₁₆	1	1	yes	characteristic
26	C₂²×C₃₁₇	Q₁₆	1	1	yes	characteristic
27	C₄×C₃₁₇	—	2	1	no	—
28	C₄×C₃₁₇	—	2	1	no	—
29	C₄×C₃₁₇	—	4	2	no	—
30	Dic₃₁₇	—	4	1	no	—
31	Dic₃₁₇	—	4	1	no	—
32	C₂×C₃₁₇	—	2	1	no	—
33	C₂×C₃₁₇	—	2	1	no	—
34	C₂×C₃₁₇	#99	1	1	yes	characteristic
35	C₂×C₃₁₇	#103	1	1	yes	characteristic
36	C₂×C₃₁₇	#104	1	1	yes	characteristic
37	C₃₁₇	#328	1	1	yes	characteristic, complemented, Sylow
38	#328	—	317	1	no	maximal, Sylow
39	C₂²⋊Q₈	—	317	1	no	—
40	C₂²⋊C₈	—	317	1	no	—
41	#95	D₃₁₇	1	1	yes	characteristic
42	C₂×Q₈	—	317	1	no	—
43	C₂²×C₄	—	2	1	no	—
44	C₂²×C₄	C₂×D₃₁₇	1	1	yes	characteristic
45	C₂×C₈	—	634	1	no	—
46	C₄⋊C₄	—	317	1	no	—
47	C₄⋊C₄	—	634	1	no	—
48	C₂²⋊C₄	—	634	1	no	—
49	C₂³	C₂²⋊D₃₁₇	1	1	yes	characteristic
50	Q₈	—	1268	1	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	4	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	—	4	1	no	—
55	C₂×C₄	—	634	1	no	—
56	C₂×C₄	—	634	1	no	—
57	C₂×C₄	C₄×D₃₁₇	1	1	yes	characteristic
58	C₂×C₄	D₁₂₆₈	1	1	yes	characteristic
59	C₈	—	1268	1	no	—
60	C₂²	—	2	1	no	—
61	C₂²	—	2	1	no	—
62	C₂²	#19099551	1	1	yes	characteristic
63	C₂²	#19099562	1	1	yes	characteristic
64	C₂²	#19099564	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	—	2	1	no	—
67	C₄	—	4	2	no	—
68	C₄	—	1268	1	no	—
69	C₄	—	1268	1	no	—
70	C₂	—	2	1	no	—
71	C₂	—	2	1	no	—
72	C₂	#23778694	1	1	yes	characteristic
73	C₂	#23778703	1	1	yes	characteristic
74	C₂	#23778708	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19099551
commutator	C₂×C₄×C₃₁₇	C₂×C₄
Frattini	C₂²×C₄	C₂×D₃₁₇
Fitting	#95×C₃₁₇	C₂
socle	C₂²×C₃₁₇	C₂²⋊C₄
prime radical	C₂³×C₃₁₇	D₄
cosocle	C₂²×C₄×C₃₁₇	C₂²
hypercenter	#95	D₃₁₇
hypocenter	C₃₁₇	#328
LCS	G ▷ C₂×C₄×C₃₁₇ ▷ C₂²×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #95

Complements

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

N	H ≅ G/N	type
4. C₂×Dic₆₃₄	C₄	semidirect
6. C₄⋊Dic₃₁₇	C₄	semidirect
37. C₃₁₇	#328	semidirect

Maximal subgroups

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

subgroup	index
#23778759	2
C₂²⋊#14579261	2
#95×C₃₁₇	2
#328	317