Subgroups — C₄×#19099584 #35204185

Order 20288.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099584	C₂	1	2	yes	maximal
2	#23778835	C₂	1	2	yes	maximal, complemented
3	C₄×#14579268	C₂	1	2	yes	maximal, complemented
4	C₄²×D₃₁₇	C₂	1	1	yes	maximal, characteristic
5	#19099584	C₄	1	8	yes	direct product factor
6	C₂×#14579268	C₂²	1	4	yes	—
7	C₂×C₄×D₃₁₇	C₄	1	1	yes	characteristic
8	C₂×C₄×D₃₁₇	C₄	1	1	yes	characteristic
9	C₂×C₄×D₃₁₇	C₂²	1	1	yes	characteristic
10	C₄×Dic₃₁₇	C₄	1	1	yes	characteristic
11	C₄×Dic₃₁₇	C₂²	1	1	yes	characteristic
12	C₄×Dic₃₁₇	C₂²	1	1	yes	characteristic
13	C₄²×C₃₁₇	C₄	1	1	yes	characteristic
14	#14579268	C₂×C₄	1	8	yes	complemented
15	C₂²×D₃₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂×Dic₃₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₃₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₃₁₇	C₂³	1	1	yes	characteristic
19	C₄×D₃₁₇	—	2	2	no	—
20	C₄×D₃₁₇	—	2	2	no	—
21	C₄×D₃₁₇	C₂×C₄	1	4	yes	—
22	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
26	C₂×D₃₁₇	—	2	1	no	—
27	C₂×D₃₁₇	—	2	1	no	—
28	C₂×D₃₁₇	C₄²	1	2	yes	—
29	C₄×C₃₁₇	—	2	1	no	—
30	C₄×C₃₁₇	C₄²	1	2	yes	—
31	C₄×C₃₁₇	M₁₆	1	2	yes	complemented
32	Dic₃₁₇	—	2	1	no	—
33	Dic₃₁₇	M₁₆	1	2	yes	complemented
34	Dic₃₁₇	C₂²×C₄	1	2	yes	—
35	C₂×C₃₁₇	C₂×C₄²	1	1	yes	characteristic
36	C₂×C₃₁₇	C₂×M₁₆	1	1	yes	characteristic
37	C₂×C₃₁₇	C₂×M₁₆	1	1	yes	characteristic
38	D₃₁₇	—	2	2	no	—
39	C₃₁₇	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
40	C₄×M₁₆	—	317	1	no	maximal, Sylow
41	C₂×M₁₆	—	317	2	no	—
42	C₂×C₄²	—	317	1	no	—
43	#97	—	317	2	no	—
44	C₄×C₈	—	317	2	no	—
45	C₂²×C₄	—	317	1	no	—
46	C₂²×C₄	—	317	1	no	—
47	C₂²×C₄	—	317	1	no	—
48	M₁₆	—	317	8	no	—
49	C₂×C₈	—	317	4	no	—
50	C₄²	—	317	1	no	—
51	C₄²	—	317	1	no	—
52	C₄²	—	317	1	no	—
53	C₄²	#11929639	1	1	yes	characteristic
54	C₂³	—	317	1	no	—
55	C₂×C₄	—	317	1	no	—
56	C₂×C₄	—	317	1	no	—
57	C₂×C₄	—	317	4	no	—
58	C₂×C₄	—	317	1	no	—
59	C₂×C₄	—	634	2	no	—
60	C₂×C₄	—	634	2	no	—
61	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
62	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
63	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
64	C₈	—	317	8	no	—
65	C₂²	—	317	2	no	—
66	C₂²	—	634	1	no	—
67	C₂²	—	634	1	no	—
68	C₂²	C₂²×#11929639	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	317	2	no	—
71	C₄	—	317	2	no	—
72	C₄	—	634	1	no	—
73	C₄	C₄×#11929639	1	2	yes	—
74	C₄	#19099584	1	2	yes	direct product factor
75	C₂	—	634	2	no	—
76	C₂	C₂×C₄×#11929639	1	1	yes	characteristic
77	C₂	D₃₁₇⋊M₁₆	1	1	yes	characteristic
78	C₂	C₂×#19099584	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11929639
commutator	C₂×C₃₁₇	C₂×C₄²
Frattini	C₂²	C₂²×#11929639
Fitting	C₄²×C₃₁₇	C₄
socle	C₂²×C₃₁₇	C₂²×C₄
prime radical	C₂²×D₃₁₇	C₂×C₄
cosocle	C₂×Dic₃₁₇	C₂³
hypercenter	C₄²	#11929639
hypocenter	C₃₁₇	C₄×M₁₆
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. #23778835	C₂	semidirect
3. C₄×#14579268	C₂	semidirect
5. #19099584	C₄	direct (N × H)
14. #14579268	C₂×C₄	semidirect
31. C₄×C₃₁₇	M₁₆	semidirect
33. Dic₃₁₇	M₁₆	semidirect
39. C₃₁₇	C₄×M₁₆	semidirect
74. C₄	#19099584	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19099584	2
#23778835	2
C₄×#14579268	2
C₄²×D₃₁₇	2
C₄×M₁₆	317

Subgroups — C4×#19099584 #35204185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19099584 #35204185