Subgroups — #14521779

Order 1984.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#11758521	C₂	1	4	yes	maximal
3	#11758465	C₂	1	4	yes	maximal
4	#11758464	C₂	1	4	yes	maximal
5	C₄×Dic₆₂	C₂	1	2	yes	maximal
6	C₂×Q₈×C₃₁	C₂²	1	2	yes	—
7	C₂×Dic₆₂	C₂²	1	4	yes	—
8	C₂×Dic₆₂	C₂²	1	2	yes	—
9	C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
10	C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₃₁	C₂²	1	2	yes	—
12	Dic₃₁⋊C₄	C₂²	1	4	yes	—
13	Dic₃₁⋊C₄	C₂²	1	8	yes	—
14	C₄×Dic₃₁	C₂²	1	4	yes	—
15	C₄×Dic₃₁	C₂²	1	4	yes	—
16	Q₈×C₃₁	—	4	2	no	—
17	C₂×C₄×C₃₁	C₂³	1	4	yes	—
18	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₃₁	C₂³	1	2	yes	—
20	C₂×Dic₃₁	C₂³	1	4	yes	—
21	C₂×Dic₃₁	C₂³	1	4	yes	—
22	Dic₆₂	—	4	4	no	—
23	Dic₆₂	Q₈	1	8	yes	complemented
24	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
25	C₄×C₃₁	—	2	4	no	—
26	C₄×C₃₁	—	2	1	no	—
27	C₄×C₃₁	C₂×Q₈	1	4	yes	—
28	Dic₃₁	—	2	4	no	—
29	Dic₃₁	C₂×Q₈	1	8	yes	—
30	Q₈²	—	31	1	no	maximal, Sylow
31	C₂×C₃₁	C₂²×Q₈	1	2	yes	—
32	C₂×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
33	C₄⋊Q₈	—	31	4	no	—
34	C₄⋊Q₈	—	31	4	no	—
35	C₄⋊Q₈	D₃₁	1	1	yes	characteristic
36	C₄×Q₈	—	31	4	no	—
37	C₄×Q₈	—	31	2	no	—
38	C₃₁	Q₈²	1	1	yes	characteristic, complemented, Sylow
39	C₂×Q₈	—	31	2	no	—
40	C₂×Q₈	—	31	4	no	—
41	C₂×Q₈	C₂×D₃₁	1	2	yes	—
42	C₄⋊C₄	—	31	8	no	—
43	C₄⋊C₄	—	31	4	no	—
44	C₄⋊C₄	—	31	2	no	—
45	C₄⋊C₄	C₂×D₃₁	1	4	yes	—
46	C₄²	—	31	4	no	—
47	C₄²	—	31	4	no	—
48	C₄²	C₂×D₃₁	1	1	yes	characteristic
49	Q₈	—	4	2	no	—
50	Q₈	—	31	8	no	—
51	Q₈	—	124	4	no	—
52	C₂×C₄	—	31	4	no	—
53	C₂×C₄	—	31	4	no	—
54	C₂×C₄	C₂²×D₃₁	1	4	yes	—
55	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
56	C₂×C₄	C₂²×D₃₁	1	2	yes	—
57	C₂²	C₂³×D₃₁	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	4	no	—
60	C₄	—	31	8	no	—
61	C₄	—	62	4	no	—
62	C₄	Q₈×D₃₁	1	4	yes	—
63	C₂	#11758561	1	1	yes	characteristic
64	C₂	C₂×Q₈×D₃₁	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁
commutator, socle, prime radical, cosocle	C₂²×C₃₁	C₂⁴
Fitting	C₃₁×C₄⋊Q₈	C₂
hypercenter	C₄⋊Q₈	D₃₁
hypocenter	C₃₁	Q₈²
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
23. Dic₆₂	Q₈	semidirect
38. C₃₁	Q₈²	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×C₄⋊Q₈	2
#11758521	2
#11758465	2
#11758464	2
C₄×Dic₆₂	2
Q₈²	31