Subgroups — #35204238

Order 20288.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099584	C₂	1	1	yes	maximal, characteristic
2	D₃₁₇⋊M₁₆	C₂	1	1	yes	maximal, characteristic
3	#23778851	C₂	1	2	yes	maximal, complemented
4	#23778850	C₂	1	2	yes	maximal, complemented
5	M₁₆×D₃₁₇	C₂	1	1	yes	maximal, characteristic
6	#19099584	—	2	1	no	—
7	#19099584	C₂²	1	1	yes	characteristic
8	#19099584	C₂²	1	2	yes	—
9	#19099584	C₂²	1	1	yes	characteristic
10	#19099583	—	2	1	no	—
11	D₃₁₇⋊C₈	—	2	1	no	—
12	C₂×#14579268	—	2	1	no	—
13	C₂×C₄×D₃₁₇	C₂²	1	1	yes	characteristic
14	#19099558	C₄	1	1	yes	characteristic
15	#19099557	C₄	1	2	yes	—
16	C₈×D₃₁₇	C₂²	1	2	yes	—
17	M₁₆×C₃₁₇	C₄	1	1	yes	characteristic
18	#14579268	—	2	2	no	—
19	#14579268	—	2	1	no	—
20	#14579268	—	2	1	no	—
21	C₂²×D₃₁₇	Q₈	1	1	yes	characteristic
22	C₂×Dic₃₁₇	D₄	1	1	yes	characteristic
23	C₄×D₃₁₇	C₂×C₄	1	1	yes	characteristic
24	C₄×D₃₁₇	D₄	1	1	yes	characteristic
25	C₄×D₃₁₇	Q₈	1	1	yes	characteristic
26	C₄×D₃₁₇	C₂³	1	1	yes	characteristic
27	#14579261	C₂×C₄	1	2	yes	—
28	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
29	C₈×C₃₁₇	C₂×C₄	1	2	yes	—
30	C₂²×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂×D₃₁₇	—	4	1	no	—
32	C₂×D₃₁₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×D₃₁₇	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
35	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
36	Dic₃₁₇	C₄⋊C₄	1	1	yes	characteristic
37	Dic₃₁₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₃₁₇	—	2	1	no	—
39	C₂×C₃₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
40	D₃₁₇	—	2	1	no	—
41	D₃₁₇	—	2	1	no	—
42	C₃₁₇	#430	1	1	yes	characteristic, complemented, Sylow
43	#430	—	317	1	no	maximal, Sylow
44	C₂×M₁₆	—	317	1	no	—
45	C₂×M₁₆	—	317	1	no	—
46	C₂×M₁₆	—	317	1	no	—
47	#108	—	317	2	no	—
48	#108	—	317	2	no	—
49	C₂²×C₄	—	317	1	no	—
50	M₁₆	—	317	1	no	—
51	M₁₆	—	317	2	no	—
52	M₁₆	—	317	1	no	—
53	M₁₆	—	317	2	no	—
54	M₁₆	—	317	1	no	—
55	M₁₆	—	634	1	no	—
56	M₁₆	—	634	1	no	—
57	M₁₆	#11929639	1	1	yes	characteristic
58	C₂×C₈	—	317	2	no	—
59	C₂×C₈	—	634	1	no	—
60	C₂×C₈	—	634	1	no	—
61	C₂³	—	317	1	no	—
62	C₂×C₄	—	317	1	no	—
63	C₂×C₄	—	317	1	no	—
64	C₂×C₄	—	317	1	no	—
65	C₂×C₄	—	317	1	no	—
66	C₂×C₄	—	317	1	no	—
67	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
68	C₈	—	317	2	no	—
69	C₈	—	634	2	no	—
70	C₈	—	634	1	no	—
71	C₈	—	634	1	no	—
72	C₈	C₂×#11929639	1	2	yes	—
73	C₂²	—	317	1	no	—
74	C₂²	—	317	1	no	—
75	C₂²	—	1268	1	no	—
76	C₂²	C₄⋊#11929639	1	1	yes	characteristic
77	C₄	—	317	1	no	—
78	C₄	—	317	1	no	—
79	C₄	C₄⋊#11929639	1	1	yes	characteristic
80	C₄	C₂²×#11929639	1	1	yes	characteristic
81	C₂	—	2	1	no	—
82	C₂	—	634	1	no	—
83	C₂	—	634	1	no	—
84	C₂	C₂×C₄⋊#11929639	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₄⋊#11929639
commutator	C₄×C₃₁₇	C₂²×C₄
Frattini	C₄	C₂²×#11929639
Fitting	M₁₆×C₃₁₇	C₄
socle	C₂×C₃₁₇	C₂×C₄⋊C₄
prime radical	C₂²×D₃₁₇	Q₈
cosocle	C₄×D₃₁₇	C₂³
hypercenter	M₁₆	#11929639
hypocenter	C₃₁₇	#430
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
3. #23778851	C₂	semidirect
4. #23778850	C₂	semidirect
42. 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×#19099584	2
D₃₁₇⋊M₁₆	2
#23778851	2
#23778850	2
M₁₆×D₃₁₇	2
#430	317