Subgroups — #35203206

Order 20288.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099575	C₂	1	1	yes	maximal, characteristic
2	M₁₆×D₃₁₇	C₂	1	2	yes	maximal, complemented
3	#23778699	C₂	1	1	yes	maximal, characteristic, complemented
4	#23778696	C₂	1	2	yes	maximal, complemented
5	#101×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#19099575	—	2	2	no	—
7	#19099575	—	2	2	no	—
8	C₂×D₁₂₆₈	C₄	1	2	yes	—
9	C₂×D₁₂₆₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₃₁₇	C₄	1	2	yes	—
11	C₂×C₄×D₃₁₇	C₂²	1	1	yes	characteristic
12	#19099558	C₂²	1	2	yes	—
13	#19099557	—	2	2	no	—
14	C₈×D₃₁₇	—	2	2	no	—
15	C₂×Q₈×C₃₁₇	C₂²	1	1	yes	characteristic
16	M₁₆×C₃₁₇	C₂²	1	2	yes	—
17	C₂²×D₃₁₇	C₂×C₄	1	2	yes	—
18	C₂²×D₃₁₇	C₂×C₄	1	1	yes	characteristic
19	D₁₂₆₈	—	2	4	no	—
20	D₁₂₆₈	—	2	2	no	—
21	C₂×Dic₃₁₇	C₂×C₄	1	1	yes	characteristic
22	C₄×D₃₁₇	—	2	2	no	—
23	C₄×D₃₁₇	—	2	2	no	—
24	C₄×D₃₁₇	D₄	1	2	yes	—
25	C₄×D₃₁₇	D₄	1	2	yes	—
26	#14579261	—	2	2	no	—
27	Q₈×C₃₁₇	—	2	2	no	—
28	C₂×C₄×C₃₁₇	C₂×C₄	1	2	yes	—
29	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
30	C₈×C₃₁₇	—	2	2	no	—
31	C₂²×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×D₃₁₇	—	2	2	no	—
33	C₂×D₃₁₇	—	4	1	no	—
34	C₂×D₃₁₇	—	4	2	no	—
35	C₂×D₃₁₇	C₂²⋊C₄	1	2	yes	—
36	C₄×C₃₁₇	—	2	2	no	—
37	C₄×C₃₁₇	C₂×D₄	1	2	yes	—
38	Dic₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
39	Dic₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₃₁₇	—	2	1	no	—
41	C₂×C₃₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
42	D₃₁₇	—	2	2	no	—
43	D₃₁₇	—	4	2	no	—
44	C₃₁₇	#413	1	1	yes	characteristic, complemented, Sylow
45	#413	—	317	1	no	maximal, Sylow
46	C₂×#41	—	317	1	no	—
47	C₂×M₁₆	—	317	2	no	—
48	#101	—	317	1	no	—
49	#101	D₃₁₇	1	1	yes	characteristic, complemented
50	#100	—	317	2	no	—
51	#41	—	634	2	no	—
52	#41	—	634	2	no	—
53	C₂×Q₈	C₂×D₃₁₇	1	1	yes	characteristic
54	C₂×D₄	—	317	2	no	—
55	C₂×D₄	—	317	1	no	—
56	C₂²×C₄	—	317	1	no	—
57	C₂²×C₄	—	317	2	no	—
58	M₁₆	—	317	2	no	—
59	M₁₆	—	634	2	no	—
60	M₁₆	C₂×D₃₁₇	1	2	yes	—
61	C₂×C₈	—	634	2	no	—
62	C₂³	—	317	1	no	—
63	C₂³	—	317	2	no	—
64	Q₈	—	2	2	no	—
65	D₄	—	634	4	no	—
66	D₄	—	634	2	no	—
67	C₂×C₄	—	317	1	no	—
68	C₂×C₄	—	317	2	no	—
69	C₂×C₄	—	317	2	no	—
70	C₂×C₄	—	634	2	no	—
71	C₂×C₄	—	634	2	no	—
72	C₂×C₄	C₄×D₃₁₇	1	2	yes	—
73	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
74	C₈	—	2	2	no	—
75	C₈	—	634	2	no	—
76	C₂²	—	317	2	no	—
77	C₂²	—	634	2	no	—
78	C₂²	—	1268	1	no	—
79	C₂²	—	1268	2	no	—
80	C₂²	C₂×C₄×D₃₁₇	1	1	yes	characteristic
81	C₄	—	2	2	no	—
82	C₄	—	317	1	no	—
83	C₄	—	317	1	no	—
84	C₄	D₄×D₃₁₇	1	2	yes	—
85	C₂	—	2	1	no	—
86	C₂	—	634	2	no	—
87	C₂	—	1268	2	no	—
88	C₂	C₂²⋊C₄×D₃₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²⋊C₄×D₃₁₇
commutator	C₂²×C₃₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₁₇
Fitting	#101×C₃₁₇	C₂
socle	C₂×C₃₁₇	C₂×C₂²⋊C₄
prime radical	C₂×#19099575	C₂
cosocle	C₂×C₄×C₃₁₇	C₂³
hypercenter	#101	D₃₁₇
hypocenter	C₃₁₇	#413
LCS	G ▷ C₂²×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

Complements

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

N	H ≅ G/N	type
2. M₁₆×D₃₁₇	C₂	semidirect
3. #23778699	C₂	semidirect
4. #23778696	C₂	semidirect
5. #101×C₃₁₇	C₂	semidirect
44. C₃₁₇	#413	semidirect
49. #101	D₃₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19099575	2
M₁₆×D₃₁₇	2
#23778699	2
#23778696	2
#101×C₃₁₇	2
#413	317