Subgroups — #14521336

Order 1984.

108 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	C₄×D₁₂₄	C₂	1	1	yes	maximal, characteristic
3	#11758397	C₂	1	2	yes	maximal
4	#11758396	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₃₁	—	2	1	no	—
7	C₂×D₁₂₄	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₃₁	—	2	1	no	—
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	C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
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	1	no	—
23	D₁₂₄	Q₈	1	2	yes	complemented
24	C₄×D₃₁	—	4	2	no	—
25	C₃₁⋊C₈	—	2	2	no	—
26	C₂²×C₃₁	C₂×D₄	1	1	yes	characteristic
27	C₂×D₃₁	—	2	2	no	—
28	C₂×D₃₁	—	4	1	no	—
29	C₂×D₃₁	—	4	1	no	—
30	C₄×C₃₁	—	2	1	no	—
31	C₄×C₃₁	—	4	2	no	—
32	C₄×C₃₁	SD₁₆	1	2	yes	—
33	C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₃₁	#41	1	1	yes	characteristic
35	Dic₃₁	—	4	1	no	—
36	#476	—	31	1	no	maximal, Sylow
37	C₂×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
38	C₂×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
39	C₂×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
40	D₃₁	—	4	2	no	—
41	C₄⋊Q₈	D₃₁	1	1	yes	characteristic, complemented
42	C₄×D₄	—	31	1	no	—
43	#106	—	31	2	no	—
44	C₄⋊C₈	—	31	1	no	—
45	#102	—	31	2	no	—
46	C₃₁	#476	1	1	yes	characteristic, complemented, Sylow
47	C₂×Q₈	—	2	1	no	—
48	C₂×D₄	—	31	1	no	—
49	C₂²×C₄	—	62	1	no	—
50	C₂×C₈	—	31	2	no	—
51	C₄⋊C₄	—	2	1	no	—
52	C₄⋊C₄	—	31	1	no	—
53	C₄⋊C₄	C₂×D₃₁	1	2	yes	—
54	C₂²⋊C₄	—	62	1	no	—
55	C₄²	C₂×D₃₁	1	1	yes	characteristic
56	C₂³	—	62	1	no	—
57	Q₈	—	4	2	no	—
58	D₄	—	31	2	no	—
59	D₄	—	62	1	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	62	1	no	—
62	C₂×C₄	—	124	2	no	—
63	C₂×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
64	C₂×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
66	C₈	—	62	2	no	—
67	C₂²	—	62	2	no	—
68	C₂²	—	124	1	no	—
69	C₂²	—	124	1	no	—
70	C₂²	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	4	2	no	—
73	C₄	—	124	1	no	—
74	C₄	#92481	1	2	yes	—
75	C₄	Q₈×D₃₁	1	1	yes	characteristic
76	C₄	#92499	1	1	yes	characteristic
77	C₂	—	124	2	no	—
78	C₂	#11758509	1	1	yes	characteristic
79	C₂	C₂×#92481	1	1	yes	characteristic
80	C₂	#11758523	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	#476
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
4. #11758396	C₂	semidirect
5. C₄⋊(C₃₁⋊C₈)	C₂	semidirect
23. D₁₂₄	Q₈	semidirect
41. C₄⋊Q₈	D₃₁	semidirect
46. C₃₁	#476	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
C₄×D₁₂₄	2
#11758397	2
#11758396	2
C₄⋊(C₃₁⋊C₈)	2
#476	31