Subgroups — #35203949

Order 20288.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778779	C₂	1	3	yes	maximal
2	#23778772	C₂	1	3	yes	maximal
3	#23778771	C₂	1	3	yes	maximal
4	Q₈×Dic₃₁₇	C₂	1	1	yes	maximal, characteristic
5	Q₈×Dic₃₁₇	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₆₃₄	C₂	1	3	yes	maximal
7	C₄×Q₈×C₃₁₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₆₃₄	C₂²	1	3	yes	—
9	C₄⋊Dic₃₁₇	C₂²	1	3	yes	—
10	C₄⋊Dic₃₁₇	C₂²	1	3	yes	—
11	C₄⋊Dic₃₁₇	C₂²	1	3	yes	—
12	#19099553	C₂²	1	1	yes	characteristic
13	#19099553	C₂²	1	3	yes	—
14	#19099553	C₂²	1	6	yes	—
15	C₄×Dic₃₁₇	C₂²	1	3	yes	—
16	C₄×Dic₃₁₇	C₂²	1	3	yes	—
17	C₂×Q₈×C₃₁₇	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₃₁₇	C₂²	1	3	yes	—
19	C₄²×C₃₁₇	C₂²	1	3	yes	—
20	Dic₆₃₄	—	2	6	no	—
21	C₂×Dic₃₁₇	C₂³	1	1	yes	characteristic
22	C₂×Dic₃₁₇	C₂³	1	3	yes	—
23	C₂×Dic₃₁₇	C₂³	1	3	yes	—
24	C₂×Dic₃₁₇	C₂³	1	1	yes	characteristic
25	Q₈×C₃₁₇	Q₈	1	4	yes	complemented
26	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
27	C₂×C₄×C₃₁₇	C₂³	1	3	yes	—
28	C₂×C₄×C₃₁₇	C₂³	1	3	yes	—
29	C₂²×C₃₁₇	C₂⁴	1	1	yes	characteristic
30	C₄×C₃₁₇	—	2	3	no	—
31	C₄×C₃₁₇	—	2	1	no	—
32	C₄×C₃₁₇	C₂×Q₈	1	6	yes	—
33	Dic₃₁₇	—	2	3	no	—
34	Dic₃₁₇	—	2	1	no	—
35	Dic₃₁₇	—	2	3	no	—
36	Dic₃₁₇	#41	1	2	yes	—
37	C₂×C₃₁₇	C₂²×Q₈	1	1	yes	characteristic
38	C₂×C₃₁₇	C₂×#41	1	1	yes	characteristic
39	C₂×C₃₁₇	2₋¹⁺⁴	1	1	yes	characteristic
40	C₃₁₇	#557	1	1	yes	characteristic, complemented, Sylow
41	#557	—	317	1	no	maximal, Sylow
42	C₄⋊Q₈	—	317	3	no	—
43	#125	—	317	3	no	—
44	#125	—	317	3	no	—
45	C₄×Q₈	—	317	1	no	—
46	C₄×Q₈	—	317	1	no	—
47	C₄×Q₈	—	317	3	no	—
48	C₄×Q₈	D₃₁₇	1	1	yes	characteristic
49	C₂×Q₈	—	317	3	no	—
50	C₂×Q₈	C₂×D₃₁₇	1	1	yes	characteristic
51	C₄⋊C₄	—	317	1	no	—
52	C₄⋊C₄	—	317	3	no	—
53	C₄⋊C₄	—	317	3	no	—
54	C₄⋊C₄	—	317	3	no	—
55	C₄⋊C₄	—	317	3	no	—
56	C₄⋊C₄	—	317	6	no	—
57	C₄⋊C₄	C₂×D₃₁₇	1	3	yes	—
58	C₄²	—	317	3	no	—
59	C₄²	—	317	3	no	—
60	C₄²	C₂×D₃₁₇	1	3	yes	—
61	Q₈	—	634	6	no	—
62	Q₈	Dic₆₃₄	1	4	yes	complemented
63	C₂×C₄	—	317	1	no	—
64	C₂×C₄	—	317	1	no	—
65	C₂×C₄	—	317	3	no	—
66	C₂×C₄	—	317	3	no	—
67	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
68	C₂×C₄	C₂²×D₃₁₇	1	3	yes	—
69	C₂×C₄	C₂²×D₃₁₇	1	3	yes	—
70	C₂²	C₂³×D₃₁₇	1	1	yes	characteristic
71	C₄	—	2	3	no	—
72	C₄	—	2	1	no	—
73	C₄	—	317	2	no	—
74	C₄	—	634	3	no	—
75	C₄	—	634	1	no	—
76	C₄	—	634	3	no	—
77	C₄	C₂×Dic₆₃₄	1	6	yes	—
78	C₂	C₂²×Dic₆₃₄	1	1	yes	characteristic
79	C₂	#41×D₃₁₇	1	1	yes	characteristic
80	C₂	#23778828	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁₇
commutator, socle, prime radical, cosocle	C₂²×C₃₁₇	C₂⁴
Fitting	C₄×Q₈×C₃₁₇	C₂
hypercenter	C₄×Q₈	D₃₁₇
hypocenter	C₃₁₇	#557
LCS	G ▷ C₂²×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
25. Q₈×C₃₁₇	Q₈	semidirect
40. C₃₁₇	#557	semidirect
62. Q₈	Dic₆₃₄	semidirect

Maximal subgroups

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

subgroup	index
#23778779	2
#23778772	2
#23778771	2
Q₈×Dic₃₁₇	2
C₄×Dic₆₃₄	2
C₄×Q₈×C₃₁₇	2
#557	317