Group #16848897 — subgroups

Subgroups — #16848897

Order 3968.

408 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 213 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14520948	C₂	1	4	yes	maximal
2	#14520935	C₂	1	4	yes	maximal
3	#14520935	C₂	1	4	yes	maximal
4	#14520915	C₂	1	2	yes	maximal
5	C₂×C₃₁×#125	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#125	C₄	1	8	yes	complemented
7	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
8	C₂×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
9	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₃₁	C₂²	1	4	yes	—
13	C₂×C₄×Dic₃₁	C₂²	1	4	yes	—
14	C₃₁⋊#95	C₂²	1	4	yes	—
15	C₃₁⋊#95	C₂²	1	4	yes	—
16	C₃₁⋊#95	C₂²	1	4	yes	—
17	C₃₁⋊#95	C₂²	1	4	yes	—
18	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
19	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
20	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
21	C₂²×Dic₃₁	C₂³	1	4	yes	—
22	C₂²×Dic₃₁	C₂³	1	4	yes	—
23	C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
24	C₄⋊C₄×C₃₁	C₂×C₄	1	16	yes	—
25	C₄²×C₃₁	C₂×C₄	1	4	yes	—
26	C₄⋊Dic₃₁	—	4	2	no	—
27	C₄⋊Dic₃₁	—	4	4	no	—
28	C₄⋊Dic₃₁	—	4	2	no	—
29	C₄×Dic₃₁	—	4	4	no	—
30	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
31	C₂×C₄×C₃₁	—	2	4	no	—
32	C₂×C₄×C₃₁	—	2	2	no	—
33	C₂×C₄×C₃₁	—	2	1	no	—
34	C₂×C₄×C₃₁	—	2	1	no	—
35	C₂×C₄×C₃₁	—	2	4	no	—
36	C₂×C₄×C₃₁	—	2	2	no	—
37	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
39	C₂×C₄×C₃₁	C₂²×C₄	1	8	yes	—
40	C₂×Dic₃₁	—	2	4	no	—
41	C₂×Dic₃₁	—	2	4	no	—
42	C₂×Dic₃₁	—	2	4	no	—
43	C₂×Dic₃₁	—	2	4	no	—
44	C₂×Dic₃₁	—	2	4	no	—
45	C₂×Dic₃₁	—	2	4	no	—
46	#2382	—	31	1	no	maximal, Sylow
47	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
48	C₂²×C₃₁	H(5,𝔽₂)	1	2	yes	—
49	C₂²×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
50	C₂²×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
51	C₂²×C₃₁	2₋¹⁺⁴	1	2	yes	—
52	C₄×C₃₁	—	2	2	no	—
53	C₄×C₃₁	—	2	8	no	—
54	C₄×C₃₁	—	2	4	no	—
55	Dic₃₁	—	4	4	no	—
56	Dic₃₁	—	4	4	no	—
57	C₂×#125	D₃₁	1	1	yes	characteristic
58	#389	—	31	4	no	—
59	#387	—	31	4	no	—
60	#387	—	31	4	no	—
61	#382	—	31	2	no	—
62	C₂×C₃₁	#519	1	1	yes	characteristic
63	C₂×C₃₁	#520	1	2	yes	—
64	C₂×C₃₁	#562	1	1	yes	characteristic
65	C₂×C₃₁	#563	1	2	yes	—
66	C₂×C₃₁	#564	1	1	yes	characteristic
67	#125	Dic₃₁	1	8	yes	complemented
68	C₂×C₄⋊C₄	—	31	2	no	—
69	C₂×C₄⋊C₄	—	31	2	no	—
70	C₂×C₄⋊C₄	—	31	4	no	—
71	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
72	C₂×C₄⋊C₄	C₂×D₃₁	1	4	yes	—
73	C₂×C₄²	—	31	4	no	—
74	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
75	#95	—	31	4	no	—
76	#95	—	31	4	no	—
77	#95	—	31	4	no	—
78	#95	—	31	4	no	—
79	C₃₁	#2382	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₄	—	31	4	no	—
81	C₂²×C₄	—	31	4	no	—
82	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
83	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
84	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
85	C₄⋊C₄	—	124	4	no	—
86	C₄⋊C₄	—	124	2	no	—
87	C₄⋊C₄	—	124	2	no	—
88	C₄⋊C₄	C₂×Dic₃₁	1	8	yes	—
89	C₄⋊C₄	C₂×Dic₃₁	1	16	yes	—
90	C₄²	—	124	4	no	—
91	C₄²	C₂×Dic₃₁	1	4	yes	—
92	C₂³	C₂³×D₃₁	1	1	yes	characteristic
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	62	4	no	—
100	C₂×C₄	—	62	4	no	—
101	C₂×C₄	—	62	4	no	—
102	C₂×C₄	—	62	4	no	—
103	C₂×C₄	—	62	4	no	—
104	C₂×C₄	—	62	4	no	—
105	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
106	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
107	C₂×C₄	C₂²×Dic₃₁	1	8	yes	—
108	C₂²	#11758564	1	1	yes	characteristic
109	C₂²	#11758564	1	1	yes	characteristic
110	C₂²	#11758566	1	2	yes	—
111	C₂²	#11758567	1	2	yes	—
112	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	2	8	no	—
115	C₄	—	2	4	no	—
116	C₄	—	124	4	no	—
117	C₄	—	124	4	no	—
118	C₂	#14521527	1	1	yes	characteristic
119	C₂	#14521532	1	2	yes	—
120	C₂	#14521797	1	1	yes	characteristic
121	C₂	#14521803	1	2	yes	—
122	C₂	C₃₁⋊#564	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14520948

C₂

yes

maximal

#14520935

C₂

yes

maximal

#14520935

C₂

yes

maximal

#14520915

C₂

yes

maximal

C₂×C₃₁×#125

C₂

yes

maximal, characteristic

C₃₁×#125

C₄

yes

complemented

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄×Dic₃₁

—

C₂³×C₃₁

C₂⁴

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

#2382

—

maximal, Sylow

C₂²×C₃₁

C₂³×C₄

yes

characteristic

C₂²×C₃₁

H(5,𝔽₂)

yes

—

C₂²×C₃₁

2₋¹⁺⁴

yes

characteristic

C₂²×C₃₁

2₋¹⁺⁴

yes

characteristic

C₂²×C₃₁

2₋¹⁺⁴

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

Dic₃₁

—

Dic₃₁

—

C₂×#125

D₃₁

yes

—

—

—

—

C₂×C₃₁

#519

yes

characteristic

C₂×C₃₁

#520

yes

—

C₂×C₃₁

#562

yes

characteristic

C₂×C₃₁

#563

yes

—

C₂×C₃₁

#564

yes

characteristic

#125

Dic₃₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₁

yes

—

—

—

—

C₃₁

#2382

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

—

C₄⋊C₄

—

124

—

C₄⋊C₄

—

124

—

C₄⋊C₄

—

124

—

C₄⋊C₄

C₂×Dic₃₁

yes

—

C₄⋊C₄

C₂×Dic₃₁

yes

—

C₄²

—

124

—

C₄²

C₂×Dic₃₁

yes

—

C₂³

C₂³×D₃₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²×Dic₃₁

yes

—

106

C₂×C₄

C₂²×Dic₃₁

yes

—

107

C₂×C₄

C₂²×Dic₃₁

yes

—

108

C₂²

#11758564

yes

characteristic

109

C₂²

#11758564

yes

characteristic

110

C₂²

#11758566

yes

—

111

C₂²

#11758567

yes

—

112

C₂²

C₂³×Dic₃₁

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

124

—

117

C₄

—

124

—

118

C₂

#14521527

yes

characteristic

119

C₂

#14521532

yes

—

120

C₂

#14521797

yes

characteristic

121

C₂

#14521803

yes

—

122

C₂

C₃₁⋊#564

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₁
commutator	C₂²×C₃₁	C₂³×C₄
Fitting	C₂×C₃₁×#125	C₂
socle, prime radical, cosocle	C₂³×C₃₁	C₂⁴
hypercenter	C₂×#125	D₃₁
hypocenter	C₃₁	#2382
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃₁

commutator

C₂²×C₃₁

C₂³×C₄

Fitting

C₂×C₃₁×#125

C₂

socle, prime radical, cosocle

C₂³×C₃₁

C₂⁴

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
6. C₃₁×#125	C₄	semidirect
67. #125	Dic₃₁	semidirect
79. C₃₁	#2382	semidirect

H ≅ G/N

type

6. C₃₁×#125

C₄

semidirect

67. #125

Dic₃₁

semidirect

79. C₃₁

#2382

semidirect

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

subgroup	index
#14520948	2
#14520935	2
#14520915	2
C₂×C₃₁×#125	2
#2382	31

subgroup

index

C₂×C₃₁×#125

Subgroups — #16848897

Conjugacy classes of subgroups

Complements

Maximal subgroups