Subgroups — #16858814

Order 3968.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521466	C₂	1	8	yes	maximal, complemented
2	C₂×#11758400	C₂	1	6	yes	maximal, complemented
3	C₂×C₃₁×#131	C₂	1	1	yes	maximal, characteristic
4	C₂×#41×C₃₁	C₄	1	1	yes	characteristic
5	C₃₁×#131	C₄	1	4	yes	—
6	C₃₁×#131	C₂²	1	4	yes	—
7	C₂×M₁₆×C₃₁	C₄	1	3	yes	—
8	C₂²×C₈×C₃₁	C₂²	1	3	yes	—
9	#11758400	C₂²	1	4	yes	—
10	#11758400	C₂²	1	12	yes	complemented
11	C₂×C₃₁⋊C₁₆	C₂²	1	12	yes	complemented
12	#41×C₃₁	C₈	1	4	yes	—
13	#41×C₃₁	C₂×C₄	1	4	yes	—
14	C₂×Q₈×C₃₁	C₈	1	1	yes	characteristic
15	C₂×D₄×C₃₁	C₈	1	3	yes	—
16	C₂²×C₄×C₃₁	C₂×C₄	1	3	yes	—
17	M₁₆×C₃₁	C₂×C₄	1	12	yes	—
18	C₂×C₈×C₃₁	C₂×C₄	1	3	yes	—
19	C₂×C₈×C₃₁	C₂×C₄	1	6	yes	—
20	C₂×C₈×C₃₁	C₂³	1	6	yes	—
21	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
22	C₃₁⋊C₁₆	C₂³	1	8	yes	—
23	C₂³×C₃₁	C₂×C₈	1	3	yes	—
24	Q₈×C₃₁	C₂×C₈	1	4	yes	—
25	D₄×C₃₁	C₂×C₈	1	12	yes	—
26	C₂×C₄×C₃₁	C₂×C₈	1	3	yes	—
27	C₂×C₄×C₃₁	C₂×C₈	1	6	yes	—
28	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₃₁	C₂²×C₄	1	6	yes	—
30	C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
31	C₈×C₃₁	C₂²×C₄	1	6	yes	—
32	C₈×C₃₁	C₂⁴	1	1	yes	characteristic
33	#3407	—	31	1	no	maximal, Sylow
34	C₂²×C₃₁	—	4	3	no	—
35	C₂²×C₃₁	C₂²×C₈	1	1	yes	characteristic
36	C₂²×C₃₁	C₂²×C₈	1	6	yes	—
37	C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
38	C₄×C₃₁	C₂²×C₈	1	6	yes	—
39	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
40	C₂×#131	D₃₁	1	1	yes	characteristic
41	#504	—	31	8	no	—
42	C₂×M₃₂	—	31	6	no	—
43	C₂×C₃₁	—	2	1	no	—
44	C₂×C₃₁	—	2	6	no	—
45	C₂×C₃₁	C₂³×C₈	1	1	yes	characteristic
46	C₂×#41	Dic₃₁	1	1	yes	characteristic
47	#131	Dic₃₁	1	4	yes	—
48	#131	C₂×D₃₁	1	4	yes	—
49	C₂×M₁₆	Dic₃₁	1	3	yes	—
50	C₂²×C₈	C₂×D₃₁	1	3	yes	—
51	M₃₂	—	31	4	no	—
52	M₃₂	—	31	12	no	—
53	C₂×C₁₆	—	31	12	no	—
54	C₃₁	#3407	1	1	yes	characteristic, complemented, Sylow
55	#41	C₃₁⋊C₈	1	4	yes	—
56	#41	C₂×Dic₃₁	1	4	yes	—
57	C₂×Q₈	C₃₁⋊C₈	1	1	yes	characteristic
58	C₂×D₄	C₃₁⋊C₈	1	3	yes	—
59	C₂²×C₄	C₂×Dic₃₁	1	3	yes	—
60	M₁₆	C₂×Dic₃₁	1	12	yes	—
61	C₂×C₈	C₂×Dic₃₁	1	3	yes	—
62	C₂×C₈	C₂×Dic₃₁	1	6	yes	—
63	C₂×C₈	C₂²×D₃₁	1	6	yes	—
64	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
65	C₁₆	—	31	8	no	—
66	C₂³	C₂×C₃₁⋊C₈	1	3	yes	—
67	Q₈	C₂×C₃₁⋊C₈	1	4	yes	—
68	D₄	C₂×C₃₁⋊C₈	1	12	yes	—
69	C₂×C₄	C₂×C₃₁⋊C₈	1	3	yes	—
70	C₂×C₄	C₂×C₃₁⋊C₈	1	6	yes	—
71	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
72	C₂×C₄	C₂²×Dic₃₁	1	6	yes	—
73	C₈	C₂²×Dic₃₁	1	1	yes	characteristic
74	C₈	C₂²×Dic₃₁	1	6	yes	—
75	C₈	C₂³×D₃₁	1	1	yes	characteristic
76	C₂²	—	4	3	no	—
77	C₂²	C₂²×C₃₁⋊C₈	1	1	yes	characteristic
78	C₂²	C₂²×C₃₁⋊C₈	1	6	yes	—
79	C₄	C₂²×C₃₁⋊C₈	1	1	yes	characteristic
80	C₄	C₂²×C₃₁⋊C₈	1	6	yes	—
81	C₄	C₂³×Dic₃₁	1	1	yes	characteristic
82	C₂	—	2	1	no	—
83	C₂	—	2	6	no	—
84	C₂	C₂³×C₃₁⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂³×D₃₁
commutator, socle	C₂×C₃₁	C₂³×C₈
Fitting	C₂×C₃₁×#131	C₂
prime radical	C₂×#41×C₃₁	C₄
cosocle	C₈×C₃₁	C₂⁴
hypercenter	C₂×#131	D₃₁
hypocenter	C₃₁	#3407
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. #14521466	C₂	semidirect
2. C₂×#11758400	C₂	semidirect
10. #11758400	C₂²	semidirect
11. C₂×C₃₁⋊C₁₆	C₂²	semidirect
54. C₃₁	#3407	semidirect

Maximal subgroups

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

subgroup	index
#14521466	2
C₂×#11758400	2
C₂×C₃₁×#131	2
#3407	31