Subgroups — #35203431

Order 20288.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099564	C₂	1	1	yes	maximal, characteristic
2	#23778778	C₂	1	1	yes	maximal, characteristic
3	Q₈×Dic₃₁₇	C₂	1	1	yes	maximal, characteristic
4	#23778709	C₂	1	1	yes	maximal, characteristic
5	#23778705	C₂	1	1	yes	maximal, characteristic
6	#23778704	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₁₆×C₃₁₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₆₃₄	C₂²	1	1	yes	characteristic
9	#19099564	—	2	2	no	—
10	C₂×#14579261	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₃₁₇	—	2	1	no	—
12	C₄⋊Dic₃₁₇	C₂²	1	1	yes	characteristic
13	#19099553	—	2	1	no	—
14	#19099553	—	2	1	no	—
15	C₄×Dic₃₁₇	—	2	1	no	—
16	C₄×Dic₃₁₇	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₃₁₇	C₂²	1	1	yes	characteristic
18	C₂×Q₈×C₃₁₇	C₂²	1	1	yes	characteristic
19	Q₁₆×C₃₁₇	—	2	2	no	—
20	C₂×C₈×C₃₁₇	C₂²	1	1	yes	characteristic
21	Dic₆₃₄	—	2	1	no	—
22	Dic₆₃₄	—	2	1	no	—
23	C₂×Dic₃₁₇	—	2	1	no	—
24	C₂×Dic₃₁₇	—	2	1	no	—
25	C₂×Dic₃₁₇	D₄	1	1	yes	characteristic
26	C₂×Dic₃₁₇	D₄	1	1	yes	characteristic
27	#14579261	—	2	1	no	—
28	Q₈×C₃₁₇	—	2	1	no	—
29	Q₈×C₃₁₇	—	2	1	no	—
30	Q₈×C₃₁₇	—	2	1	no	—
31	Q₈×C₃₁₇	D₄	1	2	yes	—
32	C₂×C₄×C₃₁₇	—	2	1	no	—
33	C₂×C₄×C₃₁₇	—	2	1	no	—
34	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
35	C₈×C₃₁₇	—	2	1	no	—
36	C₂²×C₃₁₇	C₂×D₄	1	1	yes	characteristic
37	C₄×C₃₁₇	—	2	2	no	—
38	C₄×C₃₁₇	—	4	1	no	—
39	C₄×C₃₁₇	C₂×D₄	1	1	yes	characteristic
40	C₄×C₃₁₇	#41	1	1	yes	characteristic
41	Dic₃₁₇	—	2	1	no	—
42	Dic₃₁₇	—	4	1	no	—
43	Dic₃₁₇	—	4	1	no	—
44	Dic₃₁₇	Q₁₆	1	2	yes	complemented
45	C₂×C₃₁₇	#121	1	1	yes	characteristic
46	C₂×C₃₁₇	C₂×Q₁₆	1	1	yes	characteristic
47	C₂×C₃₁₇	#137	1	1	yes	characteristic
48	C₃₁₇	#462	1	1	yes	characteristic, complemented, Sylow
49	#462	—	317	1	no	maximal, Sylow
50	C₂×Q₁₆	—	317	1	no	—
51	C₂×Q₁₆	D₃₁₇	1	1	yes	characteristic
52	C₄⋊Q₈	—	317	1	no	—
53	C₄×Q₈	—	317	1	no	—
54	C₄⋊C₈	—	317	1	no	—
55	#103	—	317	1	no	—
56	#103	—	317	1	no	—
57	C₂×Q₈	—	317	1	no	—
58	C₂×Q₈	C₂×D₃₁₇	1	1	yes	characteristic
59	C₂×Q₈	C₂×D₃₁₇	1	1	yes	characteristic
60	Q₁₆	—	2	2	no	—
61	Q₁₆	—	634	2	no	—
62	C₂×C₈	—	317	1	no	—
63	C₂×C₈	C₂×D₃₁₇	1	1	yes	characteristic
64	C₄⋊C₄	—	317	1	no	—
65	C₄⋊C₄	—	634	1	no	—
66	C₄⋊C₄	—	634	1	no	—
67	C₄⋊C₄	—	634	1	no	—
68	C₄²	—	317	1	no	—
69	C₄²	—	634	1	no	—
70	Q₈	—	2	1	no	—
71	Q₈	—	2	1	no	—
72	Q₈	—	2	1	no	—
73	Q₈	—	634	1	no	—
74	Q₈	—	634	1	no	—
75	Q₈	C₂²⋊D₃₁₇	1	2	yes	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	317	1	no	—
79	C₂×C₄	—	317	1	no	—
80	C₂×C₄	—	634	1	no	—
81	C₂×C₄	—	634	1	no	—
82	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
83	C₈	—	2	1	no	—
84	C₈	—	634	1	no	—
85	C₂²	D₄×D₃₁₇	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	4	1	no	—
88	C₄	—	317	2	no	—
89	C₄	—	634	1	no	—
90	C₄	—	1268	1	no	—
91	C₄	—	1268	1	no	—
92	C₄	C₂×C₂²⋊D₃₁₇	1	1	yes	characteristic
93	C₄	#19099573	1	1	yes	characteristic
94	C₂	#23778752	1	1	yes	characteristic
95	C₂	Q₁₆×D₃₁₇	1	1	yes	characteristic
96	C₂	#23778812	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₃₁₇
commutator, cosocle	C₂×C₄×C₃₁₇	C₂³
Frattini	C₂×C₄	C₂²×D₃₁₇
Fitting	C₂×Q₁₆×C₃₁₇	C₂
socle, prime radical	C₂²×C₃₁₇	C₂×D₄
hypercenter	C₂×Q₁₆	D₃₁₇
hypocenter	C₃₁₇	#462
LCS	G ▷ C₂×C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
44. Dic₃₁₇	Q₁₆	semidirect
48. C₃₁₇	#462	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19099564	2
#23778778	2
Q₈×Dic₃₁₇	2
#23778709	2
#23778705	2
#23778704	2
C₂×Q₁₆×C₃₁₇	2
#462	317