Subgroups — #35203973

Order 20288.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778778	C₂	1	2	yes	maximal, complemented
2	#23778773	C₂	1	4	yes	maximal, complemented
3	#23778772	C₂	1	2	yes	maximal, complemented
4	#23778761	C₂	1	2	yes	maximal
5	#23778757	C₂	1	4	yes	maximal, complemented
6	#124×C₃₁₇	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₆₃₄	C₂²	1	2	yes	—
8	C₂²×Dic₃₁₇	C₂²	1	2	yes	—
9	C₄⋊Dic₃₁₇	C₂²	1	2	yes	—
10	C₄⋊Dic₃₁₇	C₂²	1	2	yes	—
11	#19099553	C₂²	1	4	yes	—
12	#19099553	C₂²	1	4	yes	—
13	#19099553	C₂²	1	4	yes	—
14	C₂²⋊Dic₃₁₇	C₂²	1	4	yes	—
15	C₂²⋊Dic₃₁₇	C₂²	1	2	yes	—
16	C₄×Dic₃₁₇	C₂²	1	2	yes	—
17	C₂×Q₈×C₃₁₇	C₂²	1	1	yes	characteristic
18	C₂×D₄×C₃₁₇	C₂²	1	1	yes	characteristic
19	C₂²⋊C₄×C₃₁₇	C₂²	1	4	yes	—
20	C₄²×C₃₁₇	C₂²	1	1	yes	characteristic
21	Dic₆₃₄	—	4	2	no	—
22	C₂×Dic₃₁₇	—	4	2	no	—
23	C₂×Dic₃₁₇	C₂³	1	4	yes	—
24	C₂×Dic₃₁₇	C₂³	1	4	yes	—
25	C₂³×C₃₁₇	C₂³	1	2	yes	—
26	Q₈×C₃₁₇	—	4	1	no	—
27	D₄×C₃₁₇	—	4	1	no	—
28	C₂×C₄×C₃₁₇	C₂³	1	2	yes	—
29	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₃₁₇	C₂³	1	2	yes	—
31	C₂²×C₃₁₇	—	2	2	no	—
32	C₂²×C₃₁₇	—	2	4	no	—
33	C₂²×C₃₁₇	C₂⁴	1	1	yes	characteristic
34	C₄×C₃₁₇	—	2	1	no	—
35	C₄×C₃₁₇	—	2	2	no	—
36	C₄×C₃₁₇	—	2	2	no	—
37	Dic₃₁₇	—	2	4	no	—
38	Dic₃₁₇	—	2	4	no	—
39	C₂×C₃₁₇	—	4	2	no	—
40	C₂×C₃₁₇	H(5,𝔽₂)	1	1	yes	characteristic
41	C₂×C₃₁₇	2₋¹⁺⁴	1	2	yes	—
42	C₃₁₇	#563	1	1	yes	characteristic, complemented, Sylow
43	#563	—	317	1	no	maximal, Sylow
44	C₄⋊Q₈	—	317	2	no	—
45	#126	—	317	4	no	—
46	#125	—	317	2	no	—
47	#124	D₃₁₇	1	1	yes	characteristic
48	#123	—	317	2	no	—
49	C₂²⋊Q₈	—	317	4	no	—
50	C₂×Q₈	—	317	2	no	—
51	C₂×Q₈	C₂×D₃₁₇	1	1	yes	characteristic
52	C₂×D₄	C₂×D₃₁₇	1	1	yes	characteristic
53	C₂²×C₄	—	317	2	no	—
54	C₄⋊C₄	—	317	2	no	—
55	C₄⋊C₄	—	317	4	no	—
56	C₄⋊C₄	—	317	4	no	—
57	C₄⋊C₄	—	317	4	no	—
58	C₄⋊C₄	—	317	2	no	—
59	C₂²⋊C₄	—	317	4	no	—
60	C₂²⋊C₄	—	317	2	no	—
61	C₂²⋊C₄	C₂×D₃₁₇	1	4	yes	—
62	C₄²	—	317	2	no	—
63	C₄²	C₂×D₃₁₇	1	1	yes	characteristic
64	C₂³	C₂²×D₃₁₇	1	2	yes	—
65	Q₈	—	4	1	no	—
66	Q₈	—	1268	2	no	—
67	D₄	—	4	1	no	—
68	C₂×C₄	—	317	4	no	—
69	C₂×C₄	—	317	4	no	—
70	C₂×C₄	—	1268	2	no	—
71	C₂×C₄	C₂²×D₃₁₇	1	2	yes	—
72	C₂×C₄	C₂²×D₃₁₇	1	2	yes	—
73	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
74	C₂²	—	2	2	no	—
75	C₂²	—	2	4	no	—
76	C₂²	C₂³×D₃₁₇	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	2	2	no	—
79	C₄	—	2	2	no	—
80	C₄	—	634	4	no	—
81	C₄	—	634	4	no	—
82	C₂	—	4	2	no	—
83	C₂	#23778825	1	1	yes	characteristic
84	C₂	#23778827	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁₇
commutator, socle, cosocle	C₂²×C₃₁₇	C₂⁴
Fitting	#124×C₃₁₇	C₂
prime radical	C₂×D₄×C₃₁₇	C₂²
hypercenter	#124	D₃₁₇
hypocenter	C₃₁₇	#563
LCS	G ▷ C₂²×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂² ◁ #124

Complements

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

N	H ≅ G/N	type
1. #23778778	C₂	semidirect
2. #23778773	C₂	semidirect
3. #23778772	C₂	semidirect
5. #23778757	C₂	semidirect
42. C₃₁₇	#563	semidirect

Maximal subgroups

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

subgroup	index
#23778778	2
#23778773	2
#23778772	2
#23778761	2
#23778757	2
#124×C₃₁₇	2
#563	317