Subgroups — #35203233

Order 20288.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₆₃₄	C₂	1	1	yes	maximal, characteristic
2	#23778739	C₂	1	1	yes	maximal, characteristic
3	#23778705	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₃₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₆₃₄	—	2	2	no	—
6	C₂×Dic₆₃₄	—	2	1	no	—
7	C₂×Dic₆₃₄	C₄	1	4	yes	complemented
8	C₂×Dic₆₃₄	C₂²	1	2	yes	—
9	C₂²×Dic₃₁₇	—	2	1	no	—
10	C₄⋊Dic₃₁₇	C₂²	1	2	yes	—
11	C₂²⋊Dic₃₁₇	—	2	1	no	—
12	C₄×Dic₃₁₇	—	2	1	no	—
13	C₂²×C₄×C₃₁₇	C₂²	1	1	yes	characteristic
14	M₁₆×C₃₁₇	—	2	2	no	—
15	C₂×C₈×C₃₁₇	C₂²	1	2	yes	—
16	Dic₆₃₄	—	2	4	no	—
17	Dic₆₃₄	—	2	2	no	—
18	Dic₆₃₄	C₂×C₄	1	4	yes	—
19	C₂×Dic₃₁₇	—	2	4	no	—
20	C₂×Dic₃₁₇	—	2	2	no	—
21	C₂×Dic₃₁₇	—	2	2	no	—
22	C₂³×C₃₁₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₃₁₇	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₃₁₇	D₄	1	2	yes	—
25	C₂×C₄×C₃₁₇	D₄	1	1	yes	characteristic
26	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
27	C₈×C₃₁₇	—	2	2	no	—
28	C₂²×C₃₁₇	—	2	2	no	—
29	C₂²×C₃₁₇	C₂²⋊C₄	1	2	yes	—
30	C₂²×C₃₁₇	C₂×D₄	1	1	yes	characteristic
31	C₄×C₃₁₇	C₂²⋊C₄	1	2	yes	—
32	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
33	C₄×C₃₁₇	C₂×D₄	1	1	yes	characteristic
34	Dic₃₁₇	—	2	4	no	—
35	Dic₃₁₇	—	4	2	no	—
36	C₂×C₃₁₇	—	2	2	no	—
37	C₂×C₃₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₃₁₇	#137	1	2	yes	—
39	C₃₁₇	#419	1	1	yes	characteristic, complemented, Sylow
40	#419	—	317	1	no	maximal, Sylow
41	C₂²×Q₈	—	317	1	no	—
42	C₂×M₁₆	D₃₁₇	1	1	yes	characteristic
43	#117	—	317	1	no	—
44	#103	—	317	4	no	—
45	C₂×Q₈	—	317	4	no	—
46	C₂×Q₈	—	317	2	no	—
47	C₂×Q₈	—	634	2	no	—
48	C₂×Q₈	—	634	1	no	—
49	C₂²×C₄	—	634	1	no	—
50	C₂²×C₄	C₂×D₃₁₇	1	1	yes	characteristic
51	M₁₆	—	2	2	no	—
52	C₂×C₈	C₂×D₃₁₇	1	2	yes	—
53	C₄⋊C₄	—	317	2	no	—
54	C₂²⋊C₄	—	634	1	no	—
55	C₄²	—	634	1	no	—
56	C₂³	D₁₂₆₈	1	1	yes	characteristic
57	Q₈	—	317	4	no	—
58	Q₈	—	634	4	no	—
59	Q₈	—	634	2	no	—
60	C₂×C₄	—	634	4	no	—
61	C₂×C₄	—	634	2	no	—
62	C₂×C₄	—	634	2	no	—
63	C₂×C₄	C₄×D₃₁₇	1	2	yes	—
64	C₂×C₄	C₂²⋊D₃₁₇	1	2	yes	—
65	C₂×C₄	D₁₂₆₈	1	1	yes	characteristic
66	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
67	C₈	—	2	2	no	—
68	C₂²	—	2	2	no	—
69	C₂²	#19099551	1	2	yes	—
70	C₂²	C₂×D₁₂₆₈	1	1	yes	characteristic
71	C₄	—	634	4	no	—
72	C₄	—	1268	2	no	—
73	C₄	#19099551	1	2	yes	—
74	C₄	C₂×C₄×D₃₁₇	1	1	yes	characteristic
75	C₄	C₂×C₂²⋊D₃₁₇	1	1	yes	characteristic
76	C₂	—	2	2	no	—
77	C₂	C₂×#19099551	1	1	yes	characteristic
78	C₂	#23778813	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₂₆₈
commutator	C₄×C₃₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₁₇
Fitting	C₂×M₁₆×C₃₁₇	C₂
socle	C₂²×C₃₁₇	C₂×D₄
prime radical	C₂³×C₃₁₇	D₄
cosocle	C₂×C₄×C₃₁₇	C₂³
hypercenter	C₂×M₁₆	D₃₁₇
hypocenter	C₃₁₇	#419
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #23778705	C₂	semidirect
7. C₂×Dic₆₃₄	C₄	semidirect
39. C₃₁₇	#419	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₆₃₄	2
#23778739	2
#23778705	2
C₂×M₁₆×C₃₁₇	2
#419	317