Subgroups — C₂×#23778769 #35203741

Order 20288.

330 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₂₆₈	C₂	1	1	yes	maximal, characteristic
2	#23778769	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19099551	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×Dic₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×Q₈×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₃₁₇	C₂²	1	2	yes	—
7	C₂×D₁₂₆₈	—	2	4	no	—
8	C₂×D₁₂₆₈	C₂²	1	4	yes	—
9	C₂²×Dic₃₁₇	C₂²	1	2	yes	—
10	#19099551	C₂²	1	16	yes	complemented
11	C₄×Dic₃₁₇	C₂²	1	4	yes	complemented
12	C₂×Q₈×C₃₁₇	—	2	4	no	—
13	C₂×Q₈×C₃₁₇	C₂²	1	4	yes	complemented
14	C₂²×C₄×C₃₁₇	C₂²	1	2	yes	—
15	C₂²×C₄×C₃₁₇	C₂²	1	1	yes	characteristic
16	C₂²×D₃₁₇	—	2	4	no	—
17	C₂²×D₃₁₇	—	2	8	no	—
18	C₂²×D₃₁₇	C₂³	1	4	yes	—
19	D₁₂₆₈	—	2	8	no	—
20	C₂×Dic₃₁₇	—	2	4	no	—
21	C₂×Dic₃₁₇	C₂³	1	4	yes	—
22	C₂³×C₃₁₇	C₂³	1	1	yes	characteristic
23	Q₈×C₃₁₇	—	2	8	no	—
24	C₂×C₄×C₃₁₇	—	2	4	no	—
25	C₂×C₄×C₃₁₇	D₄	1	4	yes	—
26	C₂×C₄×C₃₁₇	C₂³	1	4	yes	—
27	C₂×C₄×C₃₁₇	C₂³	1	2	yes	—
28	C₂²×C₃₁₇	C₂×D₄	1	2	yes	—
29	C₂²×C₃₁₇	#41	1	4	yes	—
30	C₂²×C₃₁₇	C₂⁴	1	1	yes	characteristic
31	C₂×D₃₁₇	—	2	4	no	—
32	C₂×D₃₁₇	—	2	8	no	—
33	C₂×D₃₁₇	—	4	8	no	—
34	C₄×C₃₁₇	—	2	4	no	—
35	C₄×C₃₁₇	C₂×D₄	1	4	yes	—
36	Dic₃₁₇	—	2	4	no	—
37	C₂×C₃₁₇	#124	1	4	yes	complemented
38	C₂×C₃₁₇	C₂²×D₄	1	1	yes	characteristic
39	C₂×C₃₁₇	C₂×#41	1	2	yes	—
40	D₃₁₇	—	4	4	no	—
41	C₃₁₇	C₂×#124	1	1	yes	characteristic, complemented, Sylow
42	C₂×#124	—	317	1	no	maximal, Sylow
43	C₂²×Q₈	D₃₁₇	1	1	yes	characteristic, complemented
44	C₂²×D₄	—	317	1	no	—
45	#124	—	317	8	no	—
46	C₂×C₂²⋊C₄	—	317	4	no	—
47	C₂×C₄²	—	317	1	no	—
48	C₂⁴	—	317	2	no	—
49	C₂×Q₈	—	2	4	no	—
50	C₂×Q₈	C₂×D₃₁₇	1	4	yes	complemented
51	C₂×D₄	—	317	4	no	—
52	C₂×D₄	—	634	4	no	—
53	C₂²×C₄	—	317	2	no	—
54	C₂²×C₄	C₂×D₃₁₇	1	2	yes	—
55	C₂²×C₄	C₂×D₃₁₇	1	1	yes	characteristic
56	C₂²⋊C₄	—	317	16	no	—
57	C₄²	—	317	4	no	—
58	C₂³	—	317	4	no	—
59	C₂³	—	634	4	no	—
60	C₂³	—	634	8	no	—
61	C₂³	C₂²×D₃₁₇	1	1	yes	characteristic
62	Q₈	—	2	8	no	—
63	D₄	—	634	8	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	317	4	no	—
66	C₂×C₄	—	634	4	no	—
67	C₂×C₄	C₂²⋊D₃₁₇	1	4	yes	—
68	C₂×C₄	C₂²×D₃₁₇	1	4	yes	—
69	C₂×C₄	C₂²×D₃₁₇	1	2	yes	—
70	C₂²	—	634	4	no	—
71	C₂²	—	634	8	no	—
72	C₂²	—	1268	8	no	—
73	C₂²	C₂×C₂²⋊D₃₁₇	1	2	yes	—
74	C₂²	#19099575	1	4	yes	—
75	C₂²	C₂³×D₃₁₇	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	634	4	no	—
78	C₄	C₂×C₂²⋊D₃₁₇	1	4	yes	—
79	C₂	—	1268	4	no	—
80	C₂	#23778769	1	4	yes	direct product factor
81	C₂	C₂²×C₂²⋊D₃₁₇	1	1	yes	characteristic
82	C₂	C₂×#19099575	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₃₁₇
commutator, cosocle	C₂²×C₃₁₇	C₂⁴
Frattini	C₂²	C₂³×D₃₁₇
Fitting	C₂²×Q₈×C₃₁₇	C₂
socle	C₂³×C₃₁₇	C₂³
prime radical	C₂²×D₁₂₆₈	C₂
hypercenter	C₂²×Q₈	D₃₁₇
hypocenter	C₃₁₇	C₂×#124
LCS	G ▷ C₂²×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
2. #23778769	C₂	direct (N × H)
3. C₂×#19099551	C₂	semidirect
4. C₂×C₄×Dic₃₁₇	C₂	semidirect
5. C₂²×Q₈×C₃₁₇	C₂	semidirect
10. #19099551	C₂²	semidirect
11. C₄×Dic₃₁₇	C₂²	semidirect
13. C₂×Q₈×C₃₁₇	C₂²	semidirect
37. C₂×C₃₁₇	#124	semidirect
41. C₃₁₇	C₂×#124	semidirect
43. C₂²×Q₈	D₃₁₇	semidirect
50. C₂×Q₈	C₂×D₃₁₇	semidirect
80. C₂	#23778769	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₁₂₆₈	2
#23778769	2
C₂×#19099551	2
C₂×C₄×Dic₃₁₇	2
C₂²×Q₈×C₃₁₇	2
C₂×#124	317

Subgroups — C2×#23778769 #35203741

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23778769 #35203741