Subgroups — #14521249

Order 1984.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#92481	C₂	1	2	yes	maximal, complemented
3	#11758451	C₂	1	1	yes	maximal, characteristic
4	#11758397	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×D₁₂₄	—	2	1	no	—
8	C₂×D₁₂₄	—	2	1	no	—
9	C₂×D₁₂₄	C₂²	1	2	yes	—
10	C₄⋊C₄×C₃₁	—	2	1	no	—
11	C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₄²×C₃₁	—	2	1	no	—
13	C₄²×C₃₁	C₂²	1	1	yes	characteristic
14	#92481	—	2	4	no	—
15	C₂×C₃₁⋊C₈	C₂²	1	2	yes	—
16	C₂²×D₃₁	—	2	2	no	—
17	Q₈×C₃₁	—	2	1	no	—
18	Q₈×C₃₁	D₄	1	2	yes	complemented
19	C₂×C₄×C₃₁	—	2	1	no	—
20	C₂×C₄×C₃₁	—	2	1	no	—
21	C₂×C₄×C₃₁	D₄	1	1	yes	characteristic
22	C₂×C₄×C₃₁	D₄	1	1	yes	characteristic
23	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
24	D₁₂₄	—	2	2	no	—
25	D₁₂₄	—	2	2	no	—
26	D₁₂₄	—	4	2	no	—
27	D₁₂₄	—	4	2	no	—
28	C₃₁⋊C₈	—	2	2	no	—
29	C₂²×C₃₁	C₂×D₄	1	1	yes	characteristic
30	C₂×D₃₁	—	4	2	no	—
31	C₂×D₃₁	—	4	2	no	—
32	C₂×D₃₁	—	4	2	no	—
33	C₄×C₃₁	—	2	2	no	—
34	C₄×C₃₁	—	2	1	no	—
35	C₄×C₃₁	—	4	1	no	—
36	C₄×C₃₁	SD₁₆	1	2	yes	complemented
37	C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
38	C₄×C₃₁	#41	1	1	yes	characteristic
39	#460	—	31	1	no	maximal, Sylow
40	C₂×C₃₁	#121	1	1	yes	characteristic
41	C₂×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
42	C₂×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
43	D₃₁	—	8	2	no	—
44	C₂×SD₁₆	—	31	2	no	—
45	#127	—	31	1	no	—
46	C₄×Q₈	D₃₁	1	1	yes	characteristic, complemented
47	C₄⋊C₈	—	31	1	no	—
48	#102	—	31	2	no	—
49	C₃₁	#460	1	1	yes	characteristic, complemented, Sylow
50	C₂×Q₈	C₂×D₃₁	1	1	yes	characteristic
51	C₂×D₄	—	31	2	no	—
52	C₂×D₄	—	62	1	no	—
53	C₂×D₄	—	62	1	no	—
54	SD₁₆	—	62	4	no	—
55	C₂×C₈	—	31	2	no	—
56	C₄⋊C₄	—	2	1	no	—
57	C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
58	C₄²	—	2	1	no	—
59	C₄²	C₂×D₃₁	1	1	yes	characteristic
60	C₂³	—	62	2	no	—
61	Q₈	—	2	1	no	—
62	Q₈	D₁₂₄	1	2	yes	complemented
63	D₄	—	62	2	no	—
64	D₄	—	62	2	no	—
65	D₄	—	124	2	no	—
66	D₄	—	124	2	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
70	C₂×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
72	C₈	—	62	2	no	—
73	C₂²	—	124	2	no	—
74	C₂²	—	124	2	no	—
75	C₂²	—	124	2	no	—
76	C₂²	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	—	2	1	no	—
79	C₄	—	4	1	no	—
80	C₄	#92481	1	2	yes	complemented
81	C₄	C₂×D₁₂₄	1	1	yes	characteristic
82	C₄	#92495	1	1	yes	characteristic
83	C₂	—	248	2	no	—
84	C₂	#11758507	1	1	yes	characteristic
85	C₂	C₂×#92481	1	1	yes	characteristic
86	C₂	#11758526	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂³
Frattini	C₂×C₄	C₂²×D₃₁
Fitting	C₄×Q₈×C₃₁	C₂
socle	C₂²×C₃₁	C₂×D₄
prime radical	#11758451	C₂
hypercenter	C₄×Q₈	D₃₁
hypocenter	C₃₁	#460
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. C₄×Q₈×C₃₁	C₂	semidirect
2. C₂×#92481	C₂	semidirect
4. #11758397	C₂	semidirect
5. C₄⋊(C₃₁⋊C₈)	C₂	semidirect
18. Q₈×C₃₁	D₄	semidirect
36. C₄×C₃₁	SD₁₆	semidirect
46. C₄×Q₈	D₃₁	semidirect
49. C₃₁	#460	semidirect
62. Q₈	D₁₂₄	semidirect
80. C₄	#92481	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×C₃₁	2
C₂×#92481	2
#11758451	2
#11758397	2
C₄⋊(C₃₁⋊C₈)	2
#460	31