Subgroups — D₈×#11929639 #35204259

Order 20288.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×#11929639	C₂	1	2	yes	maximal, complemented
2	#23778849	C₂	1	1	yes	maximal, characteristic, complemented
3	#23778842	C₂	1	2	yes	maximal, complemented
4	C₈×#11929639	C₂	1	1	yes	maximal, characteristic, complemented
5	D₈×D₃₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929639	—	2	2	no	—
7	D₃₁₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₄⋊#11929639	C₂²	1	2	yes	—
9	C₂²⋊#11929639	—	2	2	no	—
10	C₄×#11929639	C₂²	1	1	yes	characteristic
11	D₄×D₃₁₇	C₂²	1	2	yes	—
12	D₂₅₃₆	C₄	1	1	yes	characteristic, complemented
13	#19099559	C₄	1	2	yes	complemented
14	C₈×D₃₁₇	C₂²	1	1	yes	characteristic
15	D₈×C₃₁₇	C₄	1	1	yes	characteristic, complemented
16	C₂×#11929639	—	2	2	no	—
17	C₂×#11929639	—	4	4	no	—
18	C₂×#11929639	D₄	1	1	yes	characteristic
19	C₂×#11929639	D₄	1	1	yes	characteristic
20	#14579268	—	2	1	no	—
21	C₂²×D₃₁₇	—	2	2	no	—
22	D₁₂₆₈	C₂×C₄	1	2	yes	complemented
23	C₂²⋊D₃₁₇	—	2	2	no	—
24	C₄×D₃₁₇	C₂³	1	1	yes	characteristic
25	#14579261	C₂×C₄	1	1	yes	characteristic, complemented
26	D₄×C₃₁₇	C₂×C₄	1	2	yes	complemented
27	C₈×C₃₁₇	C₂×C₄	1	1	yes	characteristic, complemented
28	C₂²×C₃₁₇	—	2	2	no	—
29	C₂×D₃₁₇	—	2	2	no	—
30	C₂×D₃₁₇	—	4	2	no	—
31	C₂×D₃₁₇	—	4	2	no	—
32	C₂×D₃₁₇	C₂×D₄	1	1	yes	characteristic
33	#11929639	—	2	1	no	—
34	#11929639	—	4	2	no	—
35	#11929639	D₈	1	2	yes	direct product factor
36	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
37	Dic₃₁₇	#41	1	1	yes	characteristic
38	C₂×C₃₁₇	—	4	2	no	—
39	C₂×C₃₁₇	C₄×D₄	1	1	yes	characteristic
40	D₃₁₇	—	4	2	no	—
41	D₃₁₇	C₂×D₈	1	1	yes	characteristic
42	D₃₁₇	#135	1	1	yes	characteristic
43	C₃₁₇	C₄×D₈	1	1	yes	characteristic, complemented, Sylow
44	C₄×D₈	—	317	1	no	maximal, Sylow
45	C₂×D₈	—	317	1	no	—
46	C₄×D₄	—	317	2	no	—
47	#107	—	317	1	no	—
48	#102	—	317	2	no	—
49	C₄×C₈	—	317	1	no	—
50	C₂×D₄	—	317	2	no	—
51	C₂²×C₄	—	634	2	no	—
52	D₈	—	317	2	no	—
53	D₈	—	317	1	no	—
54	D₈	#11929639	1	1	yes	characteristic, direct product factor
55	C₂×C₈	—	317	1	no	—
56	C₂×C₈	—	317	1	no	—
57	C₄⋊C₄	—	317	2	no	—
58	C₂²⋊C₄	—	634	2	no	—
59	C₄²	—	317	1	no	—
60	C₂³	—	634	2	no	—
61	D₄	—	317	2	no	—
62	D₄	—	634	2	no	—
63	D₄	C₂×#11929639	1	2	yes	complemented
64	C₂×C₄	—	317	1	no	—
65	C₂×C₄	—	317	1	no	—
66	C₂×C₄	—	317	1	no	—
67	C₂×C₄	—	634	2	no	—
68	C₂×C₄	—	1268	4	no	—
69	C₈	—	317	1	no	—
70	C₈	—	634	1	no	—
71	C₈	C₂×#11929639	1	1	yes	characteristic, complemented
72	C₂²	—	2	2	no	—
73	C₂²	—	317	1	no	—
74	C₂²	—	634	2	no	—
75	C₂²	—	1268	2	no	—
76	C₂²	—	1268	2	no	—
77	C₄	—	317	2	no	—
78	C₄	—	317	1	no	—
79	C₄	—	634	1	no	—
80	C₄	—	1268	2	no	—
81	C₄	C₂²×#11929639	1	1	yes	characteristic
82	C₂	—	4	2	no	—
83	C₂	—	317	1	no	—
84	C₂	—	317	1	no	—
85	C₂	—	1268	2	no	—
86	C₂	D₄×#11929639	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#11929639
commutator	C₄×C₃₁₇	C₂²×C₄
Frattini	C₄	C₂²×#11929639
Fitting	D₈×C₃₁₇	C₄
socle	C₂×C₃₁₇	C₄×D₄
prime radical	D₈×D₃₁₇	C₂
cosocle	C₄×D₃₁₇	C₂³
hypercenter	D₈	#11929639
hypocenter	C₃₁₇	C₄×D₈
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. D₄×#11929639	C₂	semidirect
2. #23778849	C₂	semidirect
3. #23778842	C₂	semidirect
4. C₈×#11929639	C₂	semidirect
12. D₂₅₃₆	C₄	semidirect
13. #19099559	C₄	semidirect
15. D₈×C₃₁₇	C₄	semidirect
22. D₁₂₆₈	C₂×C₄	semidirect
25. #14579261	C₂×C₄	semidirect
26. D₄×C₃₁₇	C₂×C₄	semidirect
27. C₈×C₃₁₇	C₂×C₄	semidirect
35. #11929639	D₈	direct (N × H)
43. C₃₁₇	C₄×D₈	semidirect
54. D₈	#11929639	direct (N × H)
63. D₄	C₂×#11929639	semidirect
71. C₈	C₂×#11929639	semidirect

Maximal subgroups

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

subgroup	index
D₄×#11929639	2
#23778849	2
#23778842	2
C₈×#11929639	2
D₈×D₃₁₇	2
C₄×D₈	317

Subgroups — D8×#11929639 #35204259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₈×#11929639 #35204259