Subgroups — #14521346

Order 1984.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₁₂₄	C₂	1	1	yes	maximal, characteristic
3	#11758397	C₂	1	1	yes	maximal, characteristic
4	#11758397	C₂	1	1	yes	maximal, characteristic
5	#11758396	C₂	1	1	yes	maximal, characteristic, complemented
6	#11758395	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊(C₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₁₂₄	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₃₁	—	2	1	no	—
10	C₄⋊C₄×C₃₁	—	2	1	no	—
11	C₄⋊C₄×C₃₁	—	2	1	no	—
12	C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₄²×C₃₁	C₂²	1	1	yes	characteristic
15	(C₂×D₃₁)⋊C₄	—	2	1	no	—
16	C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
19	C₂²×D₃₁	—	2	1	no	—
20	C₂×C₄×C₃₁	—	2	1	no	—
21	C₂×C₄×C₃₁	—	2	1	no	—
22	C₂×C₄×C₃₁	D₄	1	1	yes	characteristic
23	C₂×C₄×C₃₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
25	C₂×Dic₃₁	—	2	1	no	—
26	D₁₂₄	—	2	1	no	—
27	D₁₂₄	Q₈	1	2	yes	—
28	C₄×D₃₁	—	4	2	no	—
29	C₃₁⋊C₈	—	2	1	no	—
30	C₃₁⋊C₈	—	2	1	no	—
31	C₂²×C₃₁	C₂×D₄	1	1	yes	characteristic
32	C₂×D₃₁	—	2	2	no	—
33	C₂×D₃₁	—	4	1	no	—
34	C₂×D₃₁	—	4	1	no	—
35	C₄×C₃₁	—	2	1	no	—
36	C₄×C₃₁	—	2	1	no	—
37	C₄×C₃₁	—	4	1	no	—
38	C₄×C₃₁	—	4	1	no	—
39	C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
40	C₄×C₃₁	#41	1	1	yes	characteristic
41	Dic₃₁	—	4	1	no	—
42	#478	—	31	1	no	maximal, Sylow
43	C₂×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
44	C₂×C₃₁	#135	1	1	yes	characteristic
45	C₂×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
46	D₃₁	—	4	2	no	—
47	#125	D₃₁	1	1	yes	characteristic, complemented
48	C₄×D₄	—	31	1	no	—
49	#107	—	31	1	no	—
50	#106	—	31	1	no	—
51	C₄⋊C₈	—	31	1	no	—
52	#102	—	31	1	no	—
53	#102	—	31	1	no	—
54	C₃₁	#478	1	1	yes	characteristic, complemented, Sylow
55	C₂×D₄	—	31	1	no	—
56	C₂²×C₄	—	62	1	no	—
57	C₂×C₈	—	31	1	no	—
58	C₂×C₈	—	31	1	no	—
59	C₄⋊C₄	—	2	1	no	—
60	C₄⋊C₄	—	2	1	no	—
61	C₄⋊C₄	—	31	1	no	—
62	C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
63	C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
64	C₂²⋊C₄	—	62	1	no	—
65	C₄²	C₂×D₃₁	1	1	yes	characteristic
66	C₂³	—	62	1	no	—
67	D₄	—	31	2	no	—
68	D₄	—	62	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	62	1	no	—
72	C₂×C₄	—	124	2	no	—
73	C₂×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
74	C₂×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
75	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
76	C₈	—	62	1	no	—
77	C₈	—	62	1	no	—
78	C₂²	—	62	2	no	—
79	C₂²	—	124	1	no	—
80	C₂²	—	124	1	no	—
81	C₂²	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	2	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	124	1	no	—
87	C₄	Q₈×D₃₁	1	1	yes	characteristic
88	C₄	#92499	1	1	yes	characteristic
89	C₂	—	124	2	no	—
90	C₂	#11758523	1	1	yes	characteristic
91	C₂	#11758526	1	1	yes	characteristic
92	C₂	#11758527	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×#125	C₂
socle	C₂²×C₃₁	C₂×D₄
prime radical	C₂×D₁₂₄	C₂²
hypercenter	#125	D₃₁
hypocenter	C₃₁	#478
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
1. C₃₁×#125	C₂	semidirect
5. #11758396	C₂	semidirect
6. #11758395	C₂	semidirect
7. C₄⋊(C₃₁⋊C₈)	C₂	semidirect
47. #125	D₃₁	semidirect
54. C₃₁	#478	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#125	2
C₄×D₁₂₄	2
#11758397	2
#11758396	2
#11758395	2
C₄⋊(C₃₁⋊C₈)	2
#478	31