Subgroups — #35203283

Order 20288.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778737	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#14579261	C₂	1	1	yes	maximal, characteristic, complemented
3	#23778692	C₂	1	1	yes	maximal, characteristic
4	#23778692	C₂	1	1	yes	maximal, characteristic
5	#23778690	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×Dic₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊C₈×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₃₁₇	C₂²	1	1	yes	characteristic
9	C₂×#14579261	C₂²	1	1	yes	characteristic
10	C₂×#14579261	C₂²	1	1	yes	characteristic
11	#19099553	C₄	1	1	yes	characteristic
12	#19099553	C₄	1	1	yes	characteristic
13	#19099551	C₄	1	1	yes	characteristic
14	#19099551	C₄	1	1	yes	characteristic
15	C₄×Dic₃₁₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₃₁₇	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₃₁₇	C₂²	1	1	yes	characteristic
18	C₄²×C₃₁₇	C₂²	1	1	yes	characteristic
19	C₂²×D₃₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₃₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₃₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₃₁₇	C₂×C₄	1	1	yes	characteristic
23	C₄×D₃₁₇	—	2	1	no	—
24	C₄×D₃₁₇	—	2	1	no	—
25	#14579261	—	2	1	no	—
26	#14579261	—	2	1	no	—
27	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
30	C₈×C₃₁₇	—	2	1	no	—
31	C₈×C₃₁₇	—	2	1	no	—
32	C₂²×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×D₃₁₇	—	2	1	no	—
34	C₂×D₃₁₇	—	2	1	no	—
35	C₂×D₃₁₇	—	2	1	no	—
36	C₄×C₃₁₇	—	2	1	no	—
37	C₄×C₃₁₇	—	2	1	no	—
38	C₄×C₃₁₇	#41	1	1	yes	characteristic
39	C₄×C₃₁₇	#41	1	1	yes	characteristic
40	Dic₃₁₇	—	2	1	no	—
41	Dic₃₁₇	—	2	1	no	—
42	Dic₃₁₇	—	2	1	no	—
43	C₂×C₃₁₇	#117	1	1	yes	characteristic
44	C₂×C₃₁₇	#131	1	1	yes	characteristic
45	C₂×C₃₁₇	#131	1	1	yes	characteristic
46	D₃₁₇	—	4	1	no	—
47	C₃₁₇	#433	1	1	yes	characteristic, complemented, Sylow
48	#433	—	317	1	no	maximal, Sylow
49	#117	—	317	1	no	—
50	C₄⋊C₈	—	317	1	no	—
51	C₄⋊C₈	D₃₁₇	1	1	yes	characteristic, complemented
52	C₂²⋊C₈	—	317	1	no	—
53	C₂²⋊C₈	—	317	1	no	—
54	#97	—	317	1	no	—
55	C₄×C₈	—	317	1	no	—
56	C₂²×C₄	—	317	1	no	—
57	C₂×C₈	—	317	1	no	—
58	C₂×C₈	—	317	1	no	—
59	C₂×C₈	C₂×D₃₁₇	1	1	yes	characteristic
60	C₂×C₈	C₂×D₃₁₇	1	1	yes	characteristic
61	C₄⋊C₄	—	317	1	no	—
62	C₄⋊C₄	—	317	1	no	—
63	C₂²⋊C₄	—	317	1	no	—
64	C₂²⋊C₄	—	317	1	no	—
65	C₄²	—	317	1	no	—
66	C₄²	C₂×D₃₁₇	1	1	yes	characteristic
67	C₂³	—	317	1	no	—
68	C₂×C₄	—	317	1	no	—
69	C₂×C₄	—	317	1	no	—
70	C₂×C₄	—	317	1	no	—
71	C₂×C₄	—	634	1	no	—
72	C₂×C₄	—	634	1	no	—
73	C₂×C₄	C₄×D₃₁₇	1	1	yes	characteristic
74	C₂×C₄	C₄×D₃₁₇	1	1	yes	characteristic
75	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	—	2	1	no	—
78	C₈	—	634	1	no	—
79	C₈	—	634	1	no	—
80	C₂²	—	634	1	no	—
81	C₂²	—	634	1	no	—
82	C₂²	—	634	1	no	—
83	C₂²	C₂×C₄×D₃₁₇	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	2	1	no	—
86	C₄	—	634	1	no	—
87	C₄	—	634	1	no	—
88	C₄	—	634	1	no	—
89	C₄	#19099573	1	1	yes	characteristic
90	C₄	#19099575	1	1	yes	characteristic
91	C₂	—	1268	1	no	—
92	C₂	#23778738	1	1	yes	characteristic
93	C₂	#23778786	1	1	yes	characteristic
94	C₂	#23778788	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃₁₇
commutator, socle	C₂²×C₃₁₇	C₂²×C₄
Fitting	C₄⋊C₈×C₃₁₇	C₂
prime radical	C₂²×D₃₁₇	C₂×C₄
cosocle	C₂×C₄×C₃₁₇	C₂³
hypercenter	C₄⋊C₈	D₃₁₇
hypocenter	C₃₁₇	#433
LCS	G ▷ C₂²×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₄⋊#14579261	C₂	semidirect
5. #23778690	C₂	semidirect
6. C₈×Dic₃₁₇	C₂	semidirect
7. C₄⋊C₈×C₃₁₇	C₂	semidirect
47. C₃₁₇	#433	semidirect
51. C₄⋊C₈	D₃₁₇	semidirect

Maximal subgroups

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

subgroup	index
#23778737	2
C₄⋊#14579261	2
#23778692	2
#23778690	2
C₈×Dic₃₁₇	2
C₄⋊C₈×C₃₁₇	2
#433	317