Subgroups — C₂×#23778802 #35204005

Order 20288.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099574	C₂	1	2	yes	maximal, complemented
2	#23778802	C₂	1	8	yes	maximal, direct product factor
3	C₂×Dic₁₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19099563	C₂	1	2	yes	maximal, complemented
5	C₂×D₂₅₃₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#19099574	—	2	4	no	—
8	#19099574	C₂²	1	8	yes	complemented
9	C₂×Dic₆₃₄	C₂²	1	2	yes	—
10	C₂×D₁₂₆₈	C₂²	1	2	yes	—
11	C₂×C₂²⋊D₃₁₇	—	2	2	no	—
12	C₂×C₄×D₃₁₇	—	2	2	no	—
13	Dic₁₂₆₈	C₂²	1	4	yes	complemented
14	#19099563	C₂²	1	8	yes	complemented
15	D₂₅₃₆	C₂²	1	4	yes	complemented
16	C₂²×C₄×C₃₁₇	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₃₁₇	C₂²	1	1	yes	characteristic
18	C₂×C₈×C₃₁₇	C₂²	1	1	yes	characteristic
19	C₂×C₈×C₃₁₇	C₂²	1	4	yes	complemented
20	C₂²×D₃₁₇	—	2	2	no	—
21	Dic₆₃₄	—	2	2	no	—
22	Dic₆₃₄	C₂³	1	4	yes	—
23	D₁₂₆₈	—	2	2	no	—
24	D₁₂₆₈	C₂³	1	4	yes	—
25	C₂²⋊D₃₁₇	—	2	8	no	—
26	C₂×Dic₃₁₇	—	2	2	no	—
27	C₄×D₃₁₇	—	2	8	no	—
28	C₂³×C₃₁₇	D₄	1	1	yes	characteristic
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	C₂×C₄×C₃₁₇	C₂³	1	2	yes	—
33	C₈×C₃₁₇	C₂³	1	2	yes	—
34	C₈×C₃₁₇	C₂³	1	2	yes	—
35	C₂²×C₃₁₇	—	2	2	no	—
36	C₂²×C₃₁₇	C₂×D₄	1	2	yes	—
37	C₂²×C₃₁₇	C₂×D₄	1	1	yes	characteristic
38	C₂×D₃₁₇	—	2	4	no	—
39	C₂×D₃₁₇	—	4	4	no	—
40	C₄×C₃₁₇	C₂×D₄	1	2	yes	—
41	C₄×C₃₁₇	C₂×D₄	1	1	yes	characteristic
42	C₄×C₃₁₇	C₂⁴	1	1	yes	characteristic
43	Dic₃₁₇	—	2	4	no	—
44	C₂×C₃₁₇	—	2	2	no	—
45	C₂×C₃₁₇	#135	1	2	yes	complemented
46	C₂×C₃₁₇	C₂²×D₄	1	1	yes	characteristic
47	D₃₁₇	—	4	4	no	—
48	C₃₁₇	C₂×#135	1	1	yes	characteristic, complemented, Sylow
49	C₂×#135	—	317	1	no	maximal, Sylow
50	C₂×#41	—	317	2	no	—
51	#135	—	317	8	no	—
52	C₂×Q₁₆	—	317	1	no	—
53	C₂×SD₁₆	—	317	2	no	—
54	C₂×D₈	—	317	1	no	—
55	C₂²×C₈	D₃₁₇	1	1	yes	characteristic, complemented
56	#41	—	317	8	no	—
57	#41	—	634	4	no	—
58	C₂×Q₈	—	317	2	no	—
59	C₂×D₄	—	317	2	no	—
60	C₂×D₄	—	634	2	no	—
61	C₂²×C₄	—	634	2	no	—
62	C₂²×C₄	C₂×D₃₁₇	1	1	yes	characteristic
63	Q₁₆	—	317	4	no	—
64	SD₁₆	—	317	8	no	—
65	D₈	—	317	4	no	—
66	C₂×C₈	C₂×D₃₁₇	1	1	yes	characteristic
67	C₂×C₈	C₂×D₃₁₇	1	1	yes	characteristic
68	C₂×C₈	C₂×D₃₁₇	1	4	yes	complemented
69	C₂³	—	634	2	no	—
70	C₂³	D₁₂₆₈	1	1	yes	characteristic
71	Q₈	—	317	4	no	—
72	Q₈	—	634	2	no	—
73	D₄	—	317	4	no	—
74	D₄	—	634	8	no	—
75	D₄	—	634	2	no	—
76	C₂×C₄	—	634	2	no	—
77	C₂×C₄	—	634	8	no	—
78	C₂×C₄	D₁₂₆₈	1	2	yes	—
79	C₂×C₄	D₁₂₆₈	1	1	yes	characteristic
80	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
81	C₂×C₄	C₂²×D₃₁₇	1	2	yes	—
82	C₈	C₂²×D₃₁₇	1	2	yes	—
83	C₈	C₂²×D₃₁₇	1	2	yes	—
84	C₂²	—	2	2	no	—
85	C₂²	—	634	4	no	—
86	C₂²	—	1268	4	no	—
87	C₂²	C₂×D₁₂₆₈	1	2	yes	—
88	C₂²	C₂×D₁₂₆₈	1	1	yes	characteristic
89	C₄	—	634	4	no	—
90	C₄	C₂×D₁₂₆₈	1	2	yes	—
91	C₄	C₂×D₁₂₆₈	1	1	yes	characteristic
92	C₄	C₂³×D₃₁₇	1	1	yes	characteristic
93	C₂	—	2	2	no	—
94	C₂	—	1268	4	no	—
95	C₂	#23778802	1	2	yes	direct product factor
96	C₂	C₂²×D₁₂₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₂₆₈
commutator, cosocle	C₄×C₃₁₇	C₂⁴
Frattini	C₄	C₂³×D₃₁₇
Fitting	C₂²×C₈×C₃₁₇	C₂
socle	C₂²×C₃₁₇	C₂×D₄
hypercenter	C₂²×C₈	D₃₁₇
hypocenter	C₃₁₇	C₂×#135
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
1. C₂×#19099574	C₂	semidirect
2. #23778802	C₂	direct (N × H)
3. C₂×Dic₁₂₆₈	C₂	semidirect
4. C₂×#19099563	C₂	semidirect
5. C₂×D₂₅₃₆	C₂	semidirect
6. C₂²×C₈×C₃₁₇	C₂	semidirect
8. #19099574	C₂²	semidirect
13. Dic₁₂₆₈	C₂²	semidirect
14. #19099563	C₂²	semidirect
15. D₂₅₃₆	C₂²	semidirect
19. C₂×C₈×C₃₁₇	C₂²	semidirect
45. C₂×C₃₁₇	#135	semidirect
48. C₃₁₇	C₂×#135	semidirect
55. C₂²×C₈	D₃₁₇	semidirect
68. C₂×C₈	C₂×D₃₁₇	semidirect
95. C₂	#23778802	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×#19099574	2
#23778802	2
C₂×Dic₁₂₆₈	2
C₂×#19099563	2
C₂×D₂₅₃₆	2
C₂²×C₈×C₃₁₇	2
C₂×#135	317

Subgroups — C2×#23778802 #35204005

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23778802 #35204005