Subgroups — #35204234

Order 20288.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778859	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11929639	C₂	1	1	yes	maximal, characteristic
3	#23778849	C₂	1	2	yes	maximal, complemented
4	#23778848	C₂	1	2	yes	maximal, complemented
5	M₁₆×D₃₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929639	—	2	1	no	—
7	C₄⋊#11929639	—	2	1	no	—
8	C₄⋊#11929639	C₂²	1	1	yes	characteristic
9	C₄⋊#11929639	C₂²	1	2	yes	—
10	C₄⋊#11929639	C₂²	1	1	yes	characteristic
11	C₂²⋊#11929639	—	2	1	no	—
12	C₄×#11929639	—	2	1	no	—
13	C₂×C₄×D₃₁₇	C₂²	1	1	yes	characteristic
14	#19099558	C₄	1	1	yes	characteristic, complemented
15	#19099557	C₄	1	2	yes	complemented
16	C₈×D₃₁₇	C₂²	1	2	yes	—
17	M₁₆×C₃₁₇	C₄	1	1	yes	characteristic, complemented
18	C₂×#11929639	—	2	1	no	—
19	C₂×#11929639	—	2	1	no	—
20	C₂×#11929639	—	2	2	no	—
21	C₂×#11929639	—	4	2	no	—
22	C₂²×D₃₁₇	D₄	1	1	yes	characteristic
23	C₂×Dic₃₁₇	Q₈	1	1	yes	characteristic
24	C₄×D₃₁₇	C₂×C₄	1	1	yes	characteristic
25	C₄×D₃₁₇	D₄	1	1	yes	characteristic
26	C₄×D₃₁₇	Q₈	1	1	yes	characteristic
27	C₄×D₃₁₇	C₂³	1	1	yes	characteristic
28	#14579261	C₂×C₄	1	2	yes	complemented
29	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
30	C₈×C₃₁₇	C₂×C₄	1	2	yes	complemented
31	C₂²×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂×D₃₁₇	—	2	1	no	—
33	C₂×D₃₁₇	—	2	1	no	—
34	C₂×D₃₁₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×D₃₁₇	C₂×D₄	1	1	yes	characteristic
36	#11929639	—	4	1	no	—
37	#11929639	—	4	2	no	—
38	#11929639	—	4	1	no	—
39	C₄×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
40	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
41	Dic₃₁₇	C₄⋊C₄	1	1	yes	characteristic
42	Dic₃₁₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₃₁₇	—	2	1	no	—
44	C₂×C₃₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
45	D₃₁₇	—	2	1	no	—
46	D₃₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
47	D₃₁₇	#137	1	1	yes	characteristic
48	C₃₁₇	#428	1	1	yes	characteristic, complemented, Sylow
49	#428	—	317	1	no	maximal, Sylow
50	C₂×M₁₆	—	317	1	no	—
51	#117	—	317	1	no	—
52	C₂×C₄⋊C₄	—	317	1	no	—
53	#107	—	317	2	no	—
54	#106	—	317	2	no	—
55	C₂²×C₄	—	317	1	no	—
56	C₂²×C₄	—	634	1	no	—
57	M₁₆	—	317	1	no	—
58	M₁₆	—	317	2	no	—
59	M₁₆	#11929639	1	1	yes	characteristic, complemented
60	C₂×C₈	—	317	2	no	—
61	C₄⋊C₄	—	317	1	no	—
62	C₄⋊C₄	—	317	2	no	—
63	C₄⋊C₄	—	317	1	no	—
64	C₄⋊C₄	—	634	1	no	—
65	C₂²⋊C₄	—	634	1	no	—
66	C₄²	—	634	1	no	—
67	C₂³	—	317	1	no	—
68	C₂×C₄	—	317	1	no	—
69	C₂×C₄	—	317	1	no	—
70	C₂×C₄	—	317	1	no	—
71	C₂×C₄	—	317	1	no	—
72	C₂×C₄	—	317	1	no	—
73	C₂×C₄	—	634	1	no	—
74	C₂×C₄	—	634	1	no	—
75	C₂×C₄	—	634	2	no	—
76	C₂×C₄	—	1268	2	no	—
77	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
78	C₈	—	317	2	no	—
79	C₈	C₂×#11929639	1	2	yes	complemented
80	C₂²	—	317	1	no	—
81	C₂²	—	317	1	no	—
82	C₂²	—	634	1	no	—
83	C₂²	—	634	1	no	—
84	C₂²	C₄⋊#11929639	1	1	yes	characteristic
85	C₄	—	317	1	no	—
86	C₄	—	317	1	no	—
87	C₄	—	1268	2	no	—
88	C₄	—	1268	1	no	—
89	C₄	—	1268	1	no	—
90	C₄	C₄⋊#11929639	1	1	yes	characteristic
91	C₄	C₂²×#11929639	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	317	1	no	—
94	C₂	—	317	1	no	—
95	C₂	—	634	1	no	—
96	C₂	C₂×C₄⋊#11929639	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₄⋊#11929639
commutator	C₄×C₃₁₇	C₂²×C₄
Frattini	C₄	C₂²×#11929639
Fitting	M₁₆×C₃₁₇	C₄
socle	C₂×C₃₁₇	C₂×C₄⋊C₄
prime radical	C₂²×D₃₁₇	D₄
cosocle	C₄×D₃₁₇	C₂³
hypercenter	M₁₆	#11929639
hypocenter	C₃₁₇	#428
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
3. #23778849	C₂	semidirect
4. #23778848	C₂	semidirect
14. #19099558	C₄	semidirect
15. #19099557	C₄	semidirect
17. M₁₆×C₃₁₇	C₄	semidirect
28. #14579261	C₂×C₄	semidirect
30. C₈×C₃₁₇	C₂×C₄	semidirect
48. C₃₁₇	#428	semidirect
59. M₁₆	#11929639	semidirect
79. 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
#23778859	2
C₂×C₄⋊#11929639	2
#23778849	2
#23778848	2
M₁₆×D₃₁₇	2
#428	317