Subgroups — #35203220

Order 20288.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099574	C₂	1	1	yes	maximal, characteristic
2	#23778739	C₂	1	1	yes	maximal, characteristic, complemented
3	#23778705	C₂	1	2	yes	maximal, complemented
4	#23778702	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#19099574	—	2	2	no	—
7	#19099574	C₄	1	4	yes	complemented
8	C₂×Dic₆₃₄	C₂²	1	1	yes	characteristic
9	C₂×D₁₂₆₈	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₃₁₇	—	2	1	no	—
11	C₂×C₄×D₃₁₇	—	2	1	no	—
12	C₄⋊Dic₃₁₇	C₂²	1	2	yes	—
13	C₂²⋊Dic₃₁₇	—	2	1	no	—
14	C₄×Dic₃₁₇	—	2	1	no	—
15	C₂²×C₄×C₃₁₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₃₁₇	—	2	2	no	—
17	C₂×C₈×C₃₁₇	C₂²	1	2	yes	—
18	C₂²×D₃₁₇	—	2	1	no	—
19	Dic₆₃₄	—	2	1	no	—
20	Dic₆₃₄	C₂×C₄	1	2	yes	—
21	D₁₂₆₈	—	2	1	no	—
22	D₁₂₆₈	C₂×C₄	1	2	yes	—
23	C₂²⋊D₃₁₇	—	2	4	no	—
24	C₂×Dic₃₁₇	—	2	1	no	—
25	C₂×Dic₃₁₇	—	2	2	no	—
26	C₄×D₃₁₇	—	2	4	no	—
27	C₂³×C₃₁₇	D₄	1	1	yes	characteristic
28	C₂×C₄×C₃₁₇	C₂×C₄	1	2	yes	—
29	C₂×C₄×C₃₁₇	D₄	1	2	yes	—
30	C₂×C₄×C₃₁₇	D₄	1	1	yes	characteristic
31	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
32	C₈×C₃₁₇	—	2	2	no	—
33	C₂²×C₃₁₇	—	2	2	no	—
34	C₂²×C₃₁₇	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₃₁₇	C₂×D₄	1	1	yes	characteristic
36	C₂×D₃₁₇	—	2	2	no	—
37	C₂×D₃₁₇	—	4	2	no	—
38	C₄×C₃₁₇	C₂²⋊C₄	1	2	yes	—
39	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
40	C₄×C₃₁₇	C₂×D₄	1	1	yes	characteristic
41	Dic₃₁₇	—	2	2	no	—
42	Dic₃₁₇	—	4	2	no	—
43	C₂×C₃₁₇	—	2	2	no	—
44	C₂×C₃₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₃₁₇	#135	1	2	yes	—
46	D₃₁₇	—	4	2	no	—
47	C₃₁₇	#416	1	1	yes	characteristic, complemented, Sylow
48	#416	—	317	1	no	maximal, Sylow
49	C₂×#41	—	317	1	no	—
50	C₂²×C₈	D₃₁₇	1	1	yes	characteristic, complemented
51	#117	—	317	1	no	—
52	#103	—	317	2	no	—
53	#102	—	317	2	no	—
54	#41	—	317	4	no	—
55	#41	—	634	2	no	—
56	C₂×Q₈	—	317	1	no	—
57	C₂×D₄	—	317	1	no	—
58	C₂×D₄	—	634	1	no	—
59	C₂²×C₄	—	634	1	no	—
60	C₂²×C₄	C₂×D₃₁₇	1	1	yes	characteristic
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	C₂×D₃₁₇	1	2	yes	—
63	C₄⋊C₄	—	317	2	no	—
64	C₂²⋊C₄	—	634	1	no	—
65	C₄²	—	634	1	no	—
66	C₂³	—	634	1	no	—
67	C₂³	D₁₂₆₈	1	1	yes	characteristic
68	Q₈	—	317	2	no	—
69	Q₈	—	634	1	no	—
70	D₄	—	317	2	no	—
71	D₄	—	634	4	no	—
72	D₄	—	634	1	no	—
73	C₂×C₄	—	634	1	no	—
74	C₂×C₄	—	634	4	no	—
75	C₂×C₄	—	634	2	no	—
76	C₂×C₄	C₄×D₃₁₇	1	2	yes	—
77	C₂×C₄	C₂²⋊D₃₁₇	1	2	yes	—
78	C₂×C₄	D₁₂₆₈	1	1	yes	characteristic
79	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
80	C₈	—	2	2	no	—
81	C₂²	—	2	2	no	—
82	C₂²	—	634	2	no	—
83	C₂²	—	1268	2	no	—
84	C₂²	#19099551	1	2	yes	—
85	C₂²	C₂×D₁₂₆₈	1	1	yes	characteristic
86	C₄	—	634	2	no	—
87	C₄	—	1268	2	no	—
88	C₄	#19099551	1	2	yes	—
89	C₄	C₂×C₄×D₃₁₇	1	1	yes	characteristic
90	C₄	C₂×C₂²⋊D₃₁₇	1	1	yes	characteristic
91	C₂	—	2	2	no	—
92	C₂	—	1268	2	no	—
93	C₂	C₂×#19099551	1	1	yes	characteristic
94	C₂	#23778802	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₁₂₆₈
commutator	C₄×C₃₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₁₇
Fitting	C₂²×C₈×C₃₁₇	C₂
socle	C₂²×C₃₁₇	C₂×D₄
prime radical	C₂×#19099574	C₂
cosocle	C₂×C₄×C₃₁₇	C₂³
hypercenter	C₂²×C₈	D₃₁₇
hypocenter	C₃₁₇	#416
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. #23778739	C₂	semidirect
3. #23778705	C₂	semidirect
4. #23778702	C₂	semidirect
5. C₂²×C₈×C₃₁₇	C₂	semidirect
7. #19099574	C₄	semidirect
47. C₃₁₇	#416	semidirect
50. C₂²×C₈	D₃₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19099574	2
#23778739	2
#23778705	2
#23778702	2
C₂²×C₈×C₃₁₇	2
#416	317