Subgroups — #14520786

Order 1984.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃₁⋊#95	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊(C₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈×C₃₁	C₄	1	1	yes	characteristic, complemented
5	C₂²×C₄×C₃₁	C₂²	1	1	yes	characteristic
6	C₂²×Dic₃₁	—	2	1	no	—
7	C₄⋊C₄×C₃₁	—	2	1	no	—
8	C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic, complemented
9	C₂²⋊C₄×C₃₁	—	2	1	no	—
10	C₂×C₃₁⋊C₈	—	2	1	no	—
11	C₂³×C₃₁	D₄	1	1	yes	characteristic
12	Q₈×C₃₁	—	4	1	no	—
13	C₂×C₄×C₃₁	—	2	1	no	—
14	C₂×C₄×C₃₁	—	2	1	no	—
15	C₂×C₄×C₃₁	—	4	1	no	—
16	C₂×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₃₁	D₄	1	1	yes	characteristic
18	C₂×Dic₃₁	—	2	2	no	—
19	C₂×Dic₃₁	—	4	1	no	—
20	C₂×Dic₃₁	—	4	1	no	—
21	C₃₁⋊C₈	—	4	1	no	—
22	C₂²×C₃₁	—	2	1	no	—
23	C₂²×C₃₁	—	2	1	no	—
24	C₂²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₃₁	SD₁₆	1	1	yes	characteristic
26	C₂²×C₃₁	Q₁₆	1	1	yes	characteristic
27	C₄×C₃₁	—	2	1	no	—
28	C₄×C₃₁	—	2	1	no	—
29	C₄×C₃₁	—	4	1	no	—
30	C₄×C₃₁	—	4	1	no	—
31	Dic₃₁	—	4	2	no	—
32	#328	—	31	1	no	maximal, Sylow
33	C₂×C₃₁	—	2	1	no	—
34	C₂×C₃₁	—	2	1	no	—
35	C₂×C₃₁	#99	1	1	yes	characteristic
36	C₂×C₃₁	#103	1	1	yes	characteristic
37	C₂×C₃₁	#104	1	1	yes	characteristic
38	C₂²⋊Q₈	D₃₁	1	1	yes	characteristic
39	C₂²⋊C₈	—	31	1	no	—
40	#95	—	31	1	no	—
41	C₃₁	#328	1	1	yes	characteristic, complemented, Sylow
42	C₂×Q₈	Dic₃₁	1	1	yes	characteristic, complemented
43	C₂²×C₄	—	62	1	no	—
44	C₂²×C₄	C₂×D₃₁	1	1	yes	characteristic
45	C₂×C₈	—	62	1	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	Dic₃₁	1	1	yes	characteristic, complemented
48	C₂²⋊C₄	—	2	1	no	—
49	C₂³	C₂²⋊D₃₁	1	1	yes	characteristic
50	Q₈	—	4	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	—	62	2	no	—
55	C₂×C₄	—	124	1	no	—
56	C₂×C₄	—	124	1	no	—
57	C₂×C₄	C₂×Dic₃₁	1	1	yes	characteristic
58	C₂×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
59	C₈	—	124	1	no	—
60	C₂²	—	2	1	no	—
61	C₂²	—	2	1	no	—
62	C₂²	#92481	1	1	yes	characteristic
63	C₂²	#92482	1	1	yes	characteristic
64	C₂²	C₂²⋊Dic₃₁	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	—	2	1	no	—
67	C₄	—	4	1	no	—
68	C₄	—	4	1	no	—
69	C₄	—	124	2	no	—
70	C₂	—	2	1	no	—
71	C₂	—	2	1	no	—
72	C₂	#11758422	1	1	yes	characteristic
73	C₂	#11758423	1	1	yes	characteristic
74	C₂	#11758425	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊Dic₃₁
commutator	C₂×C₄×C₃₁	C₂×C₄
Frattini	C₂²×C₄	C₂×D₃₁
Fitting	C₃₁×C₂²⋊Q₈	C₂
socle	C₂²×C₃₁	C₂²⋊C₄
prime radical	C₂³×C₃₁	D₄
cosocle	C₂²×C₄×C₃₁	C₂²
hypercenter	C₂²⋊Q₈	D₃₁
hypocenter	C₃₁	#328
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
4. C₂×Q₈×C₃₁	C₄	semidirect
8. C₄⋊C₄×C₃₁	C₄	semidirect
41. C₃₁	#328	semidirect
42. C₂×Q₈	Dic₃₁	semidirect
47. C₄⋊C₄	Dic₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×C₂²⋊Q₈	2
C₃₁⋊#95	2
C₂²⋊(C₃₁⋊C₈)	2
#328	31