Subgroups — C₂×#23778827 #35204056

Order 20288.

794 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778827	C₂	1	16	yes	maximal, direct product factor
2	C₂×#19099574	C₂	1	3	yes	maximal, complemented
3	C₂×#19099573	C₂	1	6	yes	maximal, complemented
4	C₂²×Dic₆₃₄	C₂	1	3	yes	maximal, complemented
5	C₂×Q₈×D₃₁₇	C₂	1	2	yes	maximal, complemented
6	C₂×#41×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#19099574	C₂²	1	24	yes	complemented
8	#19099573	C₂²	1	48	yes	complemented
9	C₂×Dic₆₃₄	C₂²	1	6	yes	—
10	C₂×Dic₆₃₄	C₂²	1	3	yes	—
11	C₂×Dic₆₃₄	C₂²	1	24	yes	complemented
12	Q₈×D₃₁₇	C₂²	1	16	yes	complemented
13	C₂×D₁₂₆₈	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₃₁₇	C₂²	1	6	yes	—
15	C₂²×Dic₃₁₇	C₂²	1	6	yes	—
16	C₂×C₄×D₃₁₇	C₂²	1	6	yes	—
17	#41×C₃₁₇	C₂²	1	8	yes	complemented
18	C₂×Q₈×C₃₁₇	C₂²	1	1	yes	characteristic
19	C₂×D₄×C₃₁₇	C₂²	1	3	yes	—
20	C₂²×C₄×C₃₁₇	C₂²	1	3	yes	—
21	C₂²×D₃₁₇	C₂³	1	2	yes	—
22	Dic₆₃₄	C₂³	1	24	yes	—
23	Dic₆₃₄	C₂³	1	12	yes	—
24	D₁₂₆₈	C₂³	1	4	yes	—
25	C₂²⋊D₃₁₇	C₂³	1	24	yes	—
26	C₂×Dic₃₁₇	C₂³	1	6	yes	—
27	C₂×Dic₃₁₇	C₂³	1	24	yes	—
28	C₄×D₃₁₇	C₂³	1	24	yes	—
29	C₂³×C₃₁₇	C₂³	1	3	yes	—
30	Q₈×C₃₁₇	C₂³	1	4	yes	—
31	D₄×C₃₁₇	C₂³	1	12	yes	—
32	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₃₁₇	C₂³	1	3	yes	—
34	C₂×C₄×C₃₁₇	C₂³	1	12	yes	—
35	C₂²×C₃₁₇	—	2	6	no	—
36	C₂²×C₃₁₇	C₂⁴	1	6	yes	—
37	C₂²×C₃₁₇	C₂⁴	1	1	yes	characteristic
38	C₂×D₃₁₇	—	2	4	no	—
39	C₂×D₃₁₇	C₂⁴	1	4	yes	—
40	C₄×C₃₁₇	C₂⁴	1	2	yes	—
41	C₄×C₃₁₇	C₂⁴	1	6	yes	—
42	Dic₃₁₇	C₂⁴	1	12	yes	—
43	C₂×C₃₁₇	—	2	6	no	—
44	C₂×C₃₁₇	2₋¹⁺⁴	1	2	yes	complemented
45	C₂×C₃₁₇	C₂⁵	1	1	yes	characteristic
46	D₃₁₇	—	2	4	no	—
47	C₃₁₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
48	C₂×2₋¹⁺⁴	—	317	1	no	maximal, Sylow
49	2₋¹⁺⁴	—	317	16	no	—
50	C₂×#41	—	317	6	no	—
51	C₂×#41	—	317	3	no	—
52	C₂×#41	D₃₁₇	1	1	yes	characteristic, complemented
53	C₂²×Q₈	—	317	2	no	—
54	C₂²×Q₈	—	317	3	no	—
55	#41	—	317	48	no	—
56	#41	—	317	24	no	—
57	#41	C₂×D₃₁₇	1	8	yes	complemented
58	C₂×Q₈	—	317	16	no	—
59	C₂×Q₈	—	317	6	no	—
60	C₂×Q₈	—	317	3	no	—
61	C₂×Q₈	—	317	24	no	—
62	C₂×Q₈	C₂×D₃₁₇	1	1	yes	characteristic
63	C₂×D₄	—	317	6	no	—
64	C₂×D₄	—	317	1	no	—
65	C₂×D₄	C₂×D₃₁₇	1	3	yes	—
66	C₂²×C₄	—	317	6	no	—
67	C₂²×C₄	—	317	6	no	—
68	C₂²×C₄	C₂×D₃₁₇	1	3	yes	—
69	C₂³	—	317	2	no	—
70	C₂³	C₂²×D₃₁₇	1	3	yes	—
71	Q₈	—	317	24	no	—
72	Q₈	—	317	12	no	—
73	Q₈	C₂²×D₃₁₇	1	4	yes	—
74	D₄	—	317	24	no	—
75	D₄	—	317	4	no	—
76	D₄	C₂²×D₃₁₇	1	12	yes	—
77	C₂×C₄	—	317	24	no	—
78	C₂×C₄	—	317	6	no	—
79	C₂×C₄	—	317	24	no	—
80	C₂×C₄	C₂²×D₃₁₇	1	12	yes	—
81	C₂×C₄	C₂²×D₃₁₇	1	3	yes	—
82	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
83	C₂²	—	2	6	no	—
84	C₂²	—	317	4	no	—
85	C₂²	—	634	4	no	—
86	C₂²	C₂³×D₃₁₇	1	6	yes	—
87	C₂²	C₂³×D₃₁₇	1	1	yes	characteristic
88	C₄	—	317	12	no	—
89	C₄	C₂³×D₃₁₇	1	6	yes	—
90	C₄	C₂³×D₃₁₇	1	2	yes	—
91	C₂	—	2	6	no	—
92	C₂	—	634	4	no	—
93	C₂	#23778827	1	2	yes	direct product factor
94	C₂	C₂⁴×D₃₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×D₃₁₇
commutator, cosocle	C₂×C₃₁₇	C₂⁵
Frattini	C₂	C₂⁴×D₃₁₇
Fitting	C₂×#41×C₃₁₇	C₂
socle	C₂²×C₃₁₇	C₂⁴
hypercenter	C₂×#41	D₃₁₇
hypocenter	C₃₁₇	C₂×2₋¹⁺⁴
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂² ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #23778827	C₂	direct (N × H)
2. C₂×#19099574	C₂	semidirect
3. C₂×#19099573	C₂	semidirect
4. C₂²×Dic₆₃₄	C₂	semidirect
5. C₂×Q₈×D₃₁₇	C₂	semidirect
6. C₂×#41×C₃₁₇	C₂	semidirect
7. #19099574	C₂²	semidirect
8. #19099573	C₂²	semidirect
11. C₂×Dic₆₃₄	C₂²	semidirect
12. Q₈×D₃₁₇	C₂²	semidirect
17. #41×C₃₁₇	C₂²	semidirect
44. C₂×C₃₁₇	2₋¹⁺⁴	semidirect
47. C₃₁₇	C₂×2₋¹⁺⁴	semidirect
52. C₂×#41	D₃₁₇	semidirect
57. #41	C₂×D₃₁₇	semidirect
93. C₂	#23778827	direct (N × H)

Maximal subgroups

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

subgroup	index
#23778827	2
C₂×#19099574	2
C₂×#19099573	2
C₂²×Dic₆₃₄	2
C₂×Q₈×D₃₁₇	2
C₂×#41×C₃₁₇	2
C₂×2₋¹⁺⁴	317

Subgroups — C2×#23778827 #35204056

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23778827 #35204056