Subgroups — M₁₆×#11929639 #35204183

Order 20288.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃₁₇⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11929639	C₂	1	1	yes	maximal, characteristic
3	#23778834	C₂	1	2	yes	maximal, complemented
4	C₈×#11929639	C₂	1	2	yes	maximal, complemented
5	M₁₆×D₃₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929639	C₄	1	2	yes	—
7	#19099584	C₄	1	2	yes	complemented
8	#19099583	C₄	1	2	yes	complemented
9	D₃₁₇⋊C₈	C₂²	1	2	yes	—
10	C₄×#11929639	C₄	1	2	yes	—
11	C₄×#11929639	C₂²	1	1	yes	characteristic
12	C₄×#11929639	C₂²	1	1	yes	characteristic
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	4	no	—
19	C₂×#11929639	C₂×C₄	1	2	yes	—
20	C₂×#11929639	C₂×C₄	1	2	yes	—
21	#14579268	C₂×C₄	1	4	yes	complemented
22	C₂²×D₃₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×Dic₃₁₇	C₂×C₄	1	1	yes	characteristic
24	C₄×D₃₁₇	C₂×C₄	1	1	yes	characteristic
25	C₄×D₃₁₇	C₂×C₄	1	1	yes	characteristic
26	C₄×D₃₁₇	C₂×C₄	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	C₄²	1	1	yes	characteristic
32	C₂×D₃₁₇	—	2	2	no	—
33	C₂×D₃₁₇	C₄²	1	1	yes	characteristic
34	C₂×D₃₁₇	C₂²×C₄	1	1	yes	characteristic
35	#11929639	—	2	2	no	—
36	#11929639	M₁₆	1	4	yes	direct product factor
37	C₄×C₃₁₇	C₄²	1	1	yes	characteristic
38	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
39	Dic₃₁₇	C₄²	1	1	yes	characteristic
40	Dic₃₁₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₃₁₇	—	2	1	no	—
42	C₂×C₃₁₇	C₂×C₄²	1	1	yes	characteristic
43	D₃₁₇	—	2	1	no	—
44	D₃₁₇	C₂×M₁₆	1	2	yes	—
45	C₃₁₇	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
46	C₄×M₁₆	—	317	1	no	maximal, Sylow
47	C₂×M₁₆	—	317	1	no	—
48	C₂×M₁₆	—	317	1	no	—
49	C₂×C₄²	—	317	1	no	—
50	#97	—	317	2	no	—
51	C₄×C₈	—	317	2	no	—
52	C₂²×C₄	—	317	1	no	—
53	C₂²×C₄	—	317	2	no	—
54	M₁₆	—	317	2	no	—
55	M₁₆	—	317	2	no	—
56	M₁₆	—	317	1	no	—
57	M₁₆	—	317	2	no	—
58	M₁₆	#11929639	1	1	yes	characteristic, direct product factor
59	C₂×C₈	—	317	2	no	—
60	C₂×C₈	—	317	2	no	—
61	C₄²	—	317	1	no	—
62	C₄²	—	317	1	no	—
63	C₄²	—	317	2	no	—
64	C₂³	—	317	1	no	—
65	C₂×C₄	—	317	2	no	—
66	C₂×C₄	—	317	1	no	—
67	C₂×C₄	—	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	2	no	—
72	C₂×C₄	—	634	4	no	—
73	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
74	C₈	—	317	4	no	—
75	C₈	—	317	2	no	—
76	C₈	C₂×#11929639	1	2	yes	complemented
77	C₂²	—	317	1	no	—
78	C₂²	—	317	1	no	—
79	C₂²	—	634	2	no	—
80	C₂²	C₄×#11929639	1	1	yes	characteristic
81	C₄	—	317	4	no	—
82	C₄	—	317	1	no	—
83	C₄	—	317	1	no	—
84	C₄	—	634	2	no	—
85	C₄	C₄×#11929639	1	1	yes	characteristic
86	C₄	C₂²×#11929639	1	1	yes	characteristic
87	C₂	—	2	1	no	—
88	C₂	—	317	2	no	—
89	C₂	—	634	1	no	—
90	C₂	C₂×C₄×#11929639	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×#11929639
commutator, socle	C₂×C₃₁₇	C₂×C₄²
Fitting	M₁₆×C₃₁₇	C₄
prime radical	C₂²×D₃₁₇	C₂×C₄
cosocle	C₄×D₃₁₇	C₂³
hypercenter	M₁₆	#11929639
hypocenter	C₃₁₇	C₄×M₁₆
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
3. #23778834	C₂	semidirect
4. C₈×#11929639	C₂	semidirect
7. #19099584	C₄	semidirect
8. #19099583	C₄	semidirect
14. #19099558	C₄	semidirect
15. #19099557	C₄	semidirect
17. M₁₆×C₃₁₇	C₄	semidirect
21. #14579268	C₂×C₄	semidirect
28. #14579261	C₂×C₄	semidirect
30. C₈×C₃₁₇	C₂×C₄	semidirect
36. #11929639	M₁₆	direct (N × H)
45. C₃₁₇	C₄×M₁₆	semidirect
58. M₁₆	#11929639	direct (N × H)
76. 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
D₃₁₇⋊M₁₆	2
C₂×C₄×#11929639	2
#23778834	2
C₈×#11929639	2
M₁₆×D₃₁₇	2
C₄×M₁₆	317

Subgroups — M16×#11929639 #35204183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×#11929639 #35204183