Subgroups — #14521411

Order 1984.

100 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, complemented
2	#11758452	C₂	1	1	yes	maximal, characteristic
3	#11758398	C₂	1	2	yes	maximal, complemented
4	#11758397	C₂	1	2	yes	maximal
5	#11758391	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₃₁	—	2	1	no	—
7	C₂×D₁₂₄	C₂²	1	1	yes	characteristic
8	C₂×Dic₆₂	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×C₃₁	—	2	1	no	—
10	C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
11	C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	(C₂×D₃₁)⋊C₄	—	2	1	no	—
13	(C₂×D₃₁)⋊C₄	—	2	1	no	—
14	C₂×C₃₁⋊C₈	C₂²	1	2	yes	—
15	C₂²×D₃₁	—	2	1	no	—
16	Q₈×C₃₁	—	4	2	no	—
17	C₂×C₄×C₃₁	—	2	2	no	—
18	C₂×C₄×C₃₁	D₄	1	1	yes	characteristic
19	C₂×C₄×C₃₁	D₄	1	1	yes	characteristic
20	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
21	C₂×Dic₃₁	—	2	1	no	—
22	D₁₂₄	—	2	2	no	—
23	Dic₆₂	—	2	2	no	—
24	C₃₁⋊C₈	—	2	2	no	—
25	C₂²×C₃₁	C₂×D₄	1	1	yes	characteristic
26	C₂×D₃₁	—	4	1	no	—
27	C₂×D₃₁	—	4	1	no	—
28	C₂×D₃₁	—	4	1	no	—
29	C₄×C₃₁	—	2	1	no	—
30	C₄×C₃₁	—	2	1	no	—
31	C₄×C₃₁	—	4	2	no	—
32	C₄×C₃₁	#41	1	2	yes	—
33	Dic₃₁	—	4	1	no	—
34	#489	—	31	1	no	maximal, Sylow
35	C₂×C₃₁	#124	1	1	yes	characteristic
36	C₂×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
37	C₂×C₃₁	#137	1	1	yes	characteristic
38	D₃₁	—	8	1	no	—
39	C₄⋊Q₈	D₃₁	1	1	yes	characteristic, complemented
40	#124	—	31	1	no	—
41	#103	—	31	2	no	—
42	#102	—	31	2	no	—
43	#97	—	31	1	no	—
44	C₃₁	#489	1	1	yes	characteristic, complemented, Sylow
45	C₂×Q₈	—	2	1	no	—
46	C₂×Q₈	—	31	1	no	—
47	C₂×D₄	—	31	1	no	—
48	C₂×C₈	—	31	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	C₂×D₃₁	1	2	yes	—
51	C₂²⋊C₄	—	62	1	no	—
52	C₂²⋊C₄	—	62	1	no	—
53	C₄²	C₂×D₃₁	1	1	yes	characteristic
54	C₂³	—	62	1	no	—
55	Q₈	—	4	2	no	—
56	Q₈	—	62	2	no	—
57	D₄	—	62	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	62	1	no	—
60	C₂×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
61	C₂×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
63	C₈	—	62	2	no	—
64	C₂²	—	124	1	no	—
65	C₂²	—	124	1	no	—
66	C₂²	—	124	1	no	—
67	C₂²	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	2	1	no	—
70	C₄	—	4	2	no	—
71	C₄	—	124	1	no	—
72	C₄	#92499	1	2	yes	—
73	C₂	—	248	1	no	—
74	C₂	#11758509	1	1	yes	characteristic
75	C₂	#11758519	1	1	yes	characteristic
76	C₂	#11758524	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂³
Frattini	C₂×C₄	C₂²×D₃₁
Fitting	C₃₁×C₄⋊Q₈	C₂
socle	C₂²×C₃₁	C₂×D₄
prime radical	C₂×D₁₂₄	C₂²
hypercenter	C₄⋊Q₈	D₃₁
hypocenter	C₃₁	#489
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₃₁×C₄⋊Q₈	C₂	semidirect
3. #11758398	C₂	semidirect
5. #11758391	C₂	semidirect
39. C₄⋊Q₈	D₃₁	semidirect
44. C₃₁	#489	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
#11758452	2
#11758398	2
#11758397	2
#11758391	2
#489	31