Subgroups — #35203271

Order 20288.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099558	C₂	1	1	yes	maximal, characteristic
2	C₂×#19099558	C₂	1	1	yes	maximal, characteristic
3	#23778715	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₃₁₇	C₂	1	1	yes	maximal, characteristic
5	#19099558	—	2	1	no	—
6	#19099558	C₄	1	2	yes	—
7	#19099558	C₄	1	2	yes	—
8	#19099558	C₂²	1	2	yes	—
9	C₂×#14579261	—	2	1	no	—
10	C₂×#14579261	C₂²	1	2	yes	—
11	C₂²×C₄×C₃₁₇	C₂²	1	1	yes	characteristic
12	M₁₆×C₃₁₇	—	2	1	no	—
13	M₁₆×C₃₁₇	C₂²	1	2	yes	—
14	C₂×C₈×C₃₁₇	—	2	1	no	—
15	#14579261	—	2	2	no	—
16	#14579261	C₂×C₄	1	4	yes	—
17	C₂³×C₃₁₇	Q₈	1	1	yes	characteristic
18	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₃₁₇	D₄	1	1	yes	characteristic
21	C₂×C₄×C₃₁₇	D₄	1	1	yes	characteristic
22	C₂×C₄×C₃₁₇	Q₈	1	1	yes	characteristic
23	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
24	C₈×C₃₁₇	—	2	2	no	—
25	C₂²×C₃₁₇	—	4	1	no	—
26	C₂²×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
28	C₂²×C₃₁₇	C₂×Q₈	1	1	yes	characteristic
29	C₄×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
30	C₄×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
31	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₃₁₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₃₁₇	—	2	1	no	—
34	C₂×C₃₁₇	—	2	1	no	—
35	C₂×C₃₁₇	—	2	1	no	—
36	C₂×C₃₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
37	C₃₁₇	#430	1	1	yes	characteristic, complemented, Sylow
38	#430	—	317	1	no	maximal, Sylow
39	C₂×M₁₆	—	317	1	no	—
40	C₂×M₁₆	—	317	1	no	—
41	C₂×M₁₆	D₃₁₇	1	1	yes	characteristic
42	#108	—	317	4	no	—
43	C₂²×C₄	C₂×D₃₁₇	1	1	yes	characteristic
44	M₁₆	—	2	1	no	—
45	M₁₆	—	317	2	no	—
46	M₁₆	—	317	2	no	—
47	M₁₆	—	317	2	no	—
48	M₁₆	—	634	1	no	—
49	M₁₆	C₂×D₃₁₇	1	2	yes	—
50	C₂×C₈	—	2	1	no	—
51	C₂×C₈	—	317	2	no	—
52	C₂×C₈	—	634	1	no	—
53	C₂³	Dic₆₃₄	1	1	yes	characteristic
54	C₂×C₄	C₄×D₃₁₇	1	1	yes	characteristic
55	C₂×C₄	C₄×D₃₁₇	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₃₁₇	1	1	yes	characteristic
57	C₂×C₄	C₂²⋊D₃₁₇	1	1	yes	characteristic
58	C₂×C₄	Dic₆₃₄	1	1	yes	characteristic
59	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
60	C₈	—	2	2	no	—
61	C₈	—	317	4	no	—
62	C₈	—	634	2	no	—
63	C₂²	—	4	1	no	—
64	C₂²	#19099553	1	1	yes	characteristic
65	C₂²	#19099553	1	1	yes	characteristic
66	C₂²	C₂×Dic₆₃₄	1	1	yes	characteristic
67	C₄	#19099553	1	1	yes	characteristic
68	C₄	#19099553	1	1	yes	characteristic
69	C₄	C₂×C₄×D₃₁₇	1	1	yes	characteristic
70	C₄	C₂×C₂²⋊D₃₁₇	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	—	2	1	no	—
73	C₂	—	2	1	no	—
74	C₂	C₂×#19099553	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₂²⋊D₃₁₇
commutator	C₄×C₃₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₁₇
Fitting	C₂×M₁₆×C₃₁₇	C₂
socle	C₂×C₃₁₇	C₂×C₄⋊C₄
prime radical	C₂³×C₃₁₇	Q₈
cosocle	C₂×C₄×C₃₁₇	C₂³
hypercenter	C₂×M₁₆	D₃₁₇
hypocenter	C₃₁₇	#430
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #23778715	C₂	semidirect
37. C₃₁₇	#430	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19099558	2
#23778715	2
C₂×M₁₆×C₃₁₇	2
#430	317