Subgroups — #35204302

Order 20288.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778868	C₂	1	2	yes	maximal, complemented
2	#23778867	C₂	1	6	yes	maximal, complemented
3	C₂×#19099584	C₂	1	3	yes	maximal, complemented
4	D₃₁₇⋊M₁₆	C₂	1	3	yes	maximal, complemented
5	#41×D₃₁₇	C₂	1	1	yes	maximal, characteristic
6	#19099584	C₂²	1	1	yes	characteristic
7	#19099584	C₂²	1	3	yes	—
8	#19099584	C₂²	1	6	yes	complemented
9	#19099583	C₂²	1	6	yes	complemented
10	D₃₁₇⋊C₈	C₂²	1	6	yes	complemented
11	C₂×#14579268	C₂²	1	6	yes	complemented
12	#19099575	C₂²	1	1	yes	characteristic
13	#19099574	C₄	1	3	yes	—
14	#19099573	C₂²	1	3	yes	—
15	Q₈×D₃₁₇	C₄	1	1	yes	characteristic
16	D₄×D₃₁₇	C₄	1	3	yes	—
17	C₂×C₄×D₃₁₇	C₂²	1	3	yes	—
18	#41×C₃₁₇	C₄	1	1	yes	characteristic
19	#14579268	C₂³	1	6	yes	—
20	#14579268	C₂³	1	2	yes	—
21	C₂²×D₃₁₇	C₂×C₄	1	3	yes	—
22	Dic₆₃₄	C₂×C₄	1	3	yes	—
23	D₁₂₆₈	C₂×C₄	1	3	yes	—
24	C₂²⋊D₃₁₇	C₂×C₄	1	6	yes	—
25	C₂×Dic₃₁₇	C₂³	1	3	yes	—
26	C₄×D₃₁₇	C₂×C₄	1	3	yes	—
27	C₄×D₃₁₇	C₂×C₄	1	3	yes	—
28	C₄×D₃₁₇	C₂³	1	1	yes	characteristic
29	C₄×D₃₁₇	C₂³	1	3	yes	—
30	Q₈×C₃₁₇	C₂×C₄	1	1	yes	characteristic
31	D₄×C₃₁₇	C₂×C₄	1	3	yes	—
32	C₂×C₄×C₃₁₇	C₂×C₄	1	3	yes	—
33	C₂²×C₃₁₇	C₂²×C₄	1	3	yes	—
34	C₂×D₃₁₇	—	4	3	no	—
35	C₂×D₃₁₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×D₃₁₇	C₂²×C₄	1	3	yes	—
37	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
38	C₄×C₃₁₇	C₂²×C₄	1	3	yes	—
39	Dic₃₁₇	C₂²×C₄	1	3	yes	—
40	Dic₃₁₇	C₂⁴	1	1	yes	characteristic
41	C₂×C₃₁₇	—	2	3	no	—
42	C₂×C₃₁₇	C₂³×C₄	1	1	yes	characteristic
43	D₃₁₇	—	2	1	no	—
44	D₃₁₇	—	2	3	no	—
45	C₃₁₇	#568	1	1	yes	characteristic, complemented, Sylow
46	#568	—	317	1	no	maximal, Sylow
47	C₂×#41	—	317	1	no	—
48	#131	—	317	6	no	—
49	#131	—	317	2	no	—
50	C₂×M₁₆	—	317	3	no	—
51	C₂×M₁₆	—	317	3	no	—
52	#41	—	317	1	no	—
53	#41	—	317	3	no	—
54	#41	—	317	3	no	—
55	#41	#11929639	1	1	yes	characteristic
56	C₂×Q₈	—	317	1	no	—
57	C₂×D₄	—	317	3	no	—
58	C₂²×C₄	—	317	3	no	—
59	M₁₆	—	317	1	no	—
60	M₁₆	—	317	3	no	—
61	M₁₆	—	317	6	no	—
62	M₁₆	—	317	6	no	—
63	C₂×C₈	—	317	6	no	—
64	C₂×C₈	—	317	6	no	—
65	C₂³	—	317	3	no	—
66	Q₈	—	317	3	no	—
67	Q₈	C₂×#11929639	1	1	yes	characteristic
68	D₄	—	317	6	no	—
69	D₄	—	317	3	no	—
70	D₄	C₂×#11929639	1	3	yes	—
71	C₂×C₄	—	317	3	no	—
72	C₂×C₄	—	317	3	no	—
73	C₂×C₄	—	317	1	no	—
74	C₂×C₄	—	317	3	no	—
75	C₂×C₄	—	317	3	no	—
76	C₂×C₄	C₂×#11929639	1	3	yes	—
77	C₈	—	317	6	no	—
78	C₈	—	317	2	no	—
79	C₂²	—	317	1	no	—
80	C₂²	—	317	3	no	—
81	C₂²	—	1268	3	no	—
82	C₂²	C₂²×#11929639	1	3	yes	—
83	C₄	—	317	1	no	—
84	C₄	—	317	3	no	—
85	C₄	C₂²×#11929639	1	1	yes	characteristic
86	C₄	C₂²×#11929639	1	3	yes	—
87	C₂	—	2	3	no	—
88	C₂	—	634	1	no	—
89	C₂	—	634	3	no	—
90	C₂	C₂³×#11929639	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×#11929639
commutator, socle	C₂×C₃₁₇	C₂³×C₄
Fitting	#41×C₃₁₇	C₄
prime radical	#41×D₃₁₇	C₂
cosocle	Dic₃₁₇	C₂⁴
hypercenter	#41	#11929639
hypocenter	C₃₁₇	#568
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #23778868	C₂	semidirect
2. #23778867	C₂	semidirect
3. C₂×#19099584	C₂	semidirect
4. D₃₁₇⋊M₁₆	C₂	semidirect
8. #19099584	C₂²	semidirect
9. #19099583	C₂²	semidirect
10. D₃₁₇⋊C₈	C₂²	semidirect
11. C₂×#14579268	C₂²	semidirect
45. C₃₁₇	#568	semidirect

Maximal subgroups

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

subgroup	index
#23778868	2
#23778867	2
C₂×#19099584	2
D₃₁₇⋊M₁₆	2
#41×D₃₁₇	2
#568	317