Subgroups — C₂²×#19099574 #35204050

Order 20288.

890 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₁₇
commutator, cosocle	C₂×C₃₁₇	C₂⁵
Frattini	C₂	C₂⁴×D₃₁₇
Fitting	C₂³×C₄×C₃₁₇	C₂
socle	C₂³×C₃₁₇	C₂³
hypercenter	C₂³×C₄	D₃₁₇
hypocenter	C₃₁₇	C₂²×#41
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#19099574	C₂	direct (N × H)
2. C₂²×Dic₆₃₄	C₂	semidirect
3. C₂²×D₁₂₆₈	C₂	semidirect
4. C₂²×C₂²⋊D₃₁₇	C₂	semidirect
5. C₂²×C₄×D₃₁₇	C₂	semidirect
6. C₂³×C₄×C₃₁₇	C₂	semidirect
8. #19099574	C₂²	direct (N × H)
9. C₂×Dic₆₃₄	C₂²	semidirect
10. C₂×D₁₂₆₈	C₂²	semidirect
11. C₂×C₂²⋊D₃₁₇	C₂²	semidirect
13. C₂×C₄×D₃₁₇	C₂²	semidirect
17. C₂²×C₄×C₃₁₇	C₂²	semidirect
20. Dic₆₃₄	C₂³	semidirect
21. D₁₂₆₈	C₂³	semidirect
22. C₂²⋊D₃₁₇	C₂³	semidirect
24. C₄×D₃₁₇	C₂³	semidirect
30. C₂×C₄×C₃₁₇	C₂³	semidirect
32. C₂²×C₃₁₇	#41	semidirect
41. C₂×C₃₁₇	C₂×#41	semidirect
44. C₃₁₇	C₂²×#41	semidirect
51. C₂³×C₄	D₃₁₇	semidirect
62. C₂²×C₄	C₂×D₃₁₇	semidirect
75. C₂×C₄	C₂²×D₃₁₇	semidirect
79. C₂²	#19099574	direct (N × H)
87. C₂	C₂×#19099574	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
C₂²×Dic₆₃₄	2
C₂²×D₁₂₆₈	2
C₂²×C₂²⋊D₃₁₇	2
C₂²×C₄×D₃₁₇	2
C₂³×C₄×C₃₁₇	2
C₂²×#41	317

Subgroups — C22×#19099574 #35204050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19099574 #35204050