Group #16848891 — subgroups

Subgroups — #16848891

Order 3968.

424 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 173 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	C₂×#11758446	C₂	1	1	yes	maximal, characteristic
2	#14520953	C₂	1	4	yes	maximal
3	#14520934	C₂	1	4	yes	maximal
4	#14520932	C₂	1	4	yes	maximal
5	#14520917	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#383	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₆₂	C₂²	1	2	yes	—
8	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
11	C₂×Dic₃₁⋊C₄	C₂²	1	4	yes	—
12	C₂×C₄×Dic₃₁	C₂²	1	4	yes	—
13	#11758446	C₄	1	8	yes	complemented
14	C₃₁×#95	C₂²	1	2	yes	—
15	C₃₁×#95	C₂²	1	2	yes	—
16	C₃₁×#95	C₂²	1	2	yes	—
17	C₃₁⋊#95	C₂²	1	2	yes	—
18	C₃₁⋊#95	C₂²	1	2	yes	—
19	C₃₁⋊#95	C₂²	1	2	yes	—
20	C₃₁⋊#95	C₂²	1	8	yes	—
21	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
22	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
25	C₂²×Dic₃₁	C₂³	1	4	yes	—
26	C₂²×Dic₃₁	C₂³	1	4	yes	—
27	C₂×Dic₆₂	—	4	2	no	—
28	C₂×Dic₆₂	—	4	2	no	—
29	C₂×Dic₆₂	C₂×C₄	1	8	yes	—
30	C₄²×C₃₁	C₂×C₄	1	4	yes	—
31	C₄⋊Dic₃₁	C₂×C₄	1	8	yes	—
32	C₄⋊Dic₃₁	C₂×C₄	1	8	yes	—
33	Dic₃₁⋊C₄	—	4	4	no	—
34	C₄×Dic₃₁	—	4	4	no	—
35	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
36	C₂×C₄×C₃₁	—	2	2	no	—
37	C₂×C₄×C₃₁	—	2	2	no	—
38	C₂×C₄×C₃₁	—	2	2	no	—
39	C₂×C₄×C₃₁	—	2	4	no	—
40	C₂×C₄×C₃₁	—	2	4	no	—
41	C₂×C₄×C₃₁	—	2	4	no	—
42	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
43	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
45	C₂×Dic₃₁	—	2	4	no	—
46	C₂×Dic₃₁	—	2	4	no	—
47	C₂×Dic₃₁	—	2	4	no	—
48	C₂×Dic₃₁	—	2	4	no	—
49	C₂×Dic₃₁	—	2	4	no	—
50	C₂×Dic₃₁	C₂²×C₄	1	8	yes	—
51	Dic₆₂	—	4	8	no	—
52	#2381	—	31	1	no	maximal, Sylow
53	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
54	C₂²×C₃₁	H(5,𝔽₂)	1	2	yes	—
55	C₂²×C₃₁	2₋¹⁺⁴	1	2	yes	—
56	C₂²×C₃₁	2₋¹⁺⁴	1	2	yes	—
57	C₄×C₃₁	—	2	2	no	—
58	C₄×C₃₁	—	2	2	no	—
59	C₄×C₃₁	—	2	2	no	—
60	C₄×C₃₁	—	4	4	no	—
61	Dic₃₁	—	2	8	no	—
62	Dic₃₁	—	4	4	no	—
63	C₂×C₄⋊Q₈	—	31	1	no	—
64	#391	—	31	4	no	—
65	#387	—	31	4	no	—
66	#387	—	31	4	no	—
67	#383	—	31	1	no	—
68	#383	D₃₁	1	1	yes	characteristic
69	C₂×C₃₁	#518	1	1	yes	characteristic
70	C₂×C₃₁	#519	1	1	yes	characteristic
71	C₂×C₃₁	#519	1	1	yes	characteristic
72	C₂×C₃₁	#563	1	4	yes	—
73	C₂²×Q₈	—	31	2	no	—
74	C₄⋊Q₈	—	31	8	no	—
75	C₂×C₄⋊C₄	—	31	2	no	—
76	C₂×C₄⋊C₄	—	31	2	no	—
77	C₂×C₄⋊C₄	—	31	4	no	—
78	C₂×C₄²	—	31	4	no	—
79	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
80	#95	—	31	2	no	—
81	#95	—	31	2	no	—
82	#95	—	31	2	no	—
83	#95	—	31	8	no	—
84	#95	C₂×D₃₁	1	2	yes	—
85	#95	C₂×D₃₁	1	2	yes	—
86	#95	C₂×D₃₁	1	2	yes	—
87	C₃₁	#2381	1	1	yes	characteristic, complemented, Sylow
88	C₂×Q₈	—	31	8	no	—
89	C₂×Q₈	—	124	2	no	—
90	C₂×Q₈	—	124	2	no	—
91	C₂²×C₄	—	31	4	no	—
92	C₂²×C₄	—	31	4	no	—
93	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
94	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
97	C₄⋊C₄	—	31	8	no	—
98	C₄⋊C₄	—	31	8	no	—
99	C₄⋊C₄	—	124	4	no	—
100	C₄²	—	124	4	no	—
101	C₄²	C₄×D₃₁	1	4	yes	—
102	C₂³	C₂³×D₃₁	1	1	yes	characteristic
103	Q₈	—	124	8	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	31	8	no	—
111	C₂×C₄	—	62	4	no	—
112	C₂×C₄	—	62	4	no	—
113	C₂×C₄	—	62	4	no	—
114	C₂×C₄	—	62	4	no	—
115	C₂×C₄	—	62	4	no	—
116	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
118	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
119	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
120	C₂²	#11758566	1	2	yes	—
121	C₂²	#11758567	1	2	yes	—
122	C₂²	#11758567	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	—	4	4	no	—
127	C₄	—	62	8	no	—
128	C₄	—	124	4	no	—
129	C₂	#14521523	1	1	yes	characteristic
130	C₂	#14521526	1	1	yes	characteristic
131	C₂	#14521526	1	1	yes	characteristic
132	C₂	#14521801	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758446

C₂

yes

maximal, characteristic

#14520953

C₂

yes

maximal

#14520934

C₂

yes

maximal

#14520932

C₂

yes

maximal

#14520917

C₂

yes

maximal, characteristic

C₃₁×#383

C₂

yes

maximal, characteristic

C₂²×Dic₆₂

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×Dic₃₁⋊C₄

C₂²

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

—

#11758446

C₄

yes

complemented

C₃₁×#95

C₂²

yes

—

C₃₁×#95

C₂²

yes

—

C₃₁×#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

C₂×C₄

yes

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

Dic₃₁⋊C₄

—

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

C₂²×C₄

yes

—

Dic₆₂

—

#2381

—

maximal, Sylow

C₂²×C₃₁

C₂³×C₄

yes

characteristic

C₂²×C₃₁

H(5,𝔽₂)

yes

—

C₂²×C₃₁

2₋¹⁺⁴

yes

—

C₂²×C₃₁

2₋¹⁺⁴

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

Dic₃₁

—

Dic₃₁

—

C₂×C₄⋊Q₈

—

—

—

—

—

D₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₁

yes

—

—

—

—

C₂×D₃₁

yes

—

#95

C₂×D₃₁

yes

—

#95

C₂×D₃₁

yes

—

C₃₁

#2381

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

124

—

C₂×Q₈

—

124

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

124

—

100

C₄²

—

124

—

101

C₄²

C₄×D₃₁

yes

—

102

C₂³

C₂³×D₃₁

yes

characteristic

103

Q₈

—

124

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₂×C₄×D₃₁

yes

—

117

C₂×C₄

C₂×C₄×D₃₁

yes

—

118

C₂×C₄

C₂×C₄×D₃₁

yes

—

119

C₂²

C₂²×C₄×D₃₁

yes

characteristic

120

C₂²

#11758566

yes

—

121

C₂²

#11758567

yes

—

122

C₂²

#11758567

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

124

—

129

C₂

#14521523

yes

characteristic

130

C₂

#14521526

yes

characteristic

131

C₂

#14521526

yes

characteristic

132

C₂

#14521801

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₁
commutator	C₂²×C₃₁	C₂³×C₄
Fitting	C₃₁×#383	C₂
socle, prime radical, cosocle	C₂³×C₃₁	C₂⁴
hypercenter	#383	D₃₁
hypocenter	C₃₁	#2381
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ #383

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃₁

commutator

C₂²×C₃₁

C₂³×C₄

Fitting

C₃₁×#383

C₂

socle, prime radical, cosocle

C₂³×C₃₁

C₂⁴

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂³ ◁ #383

N	H ≅ G/N	type
13. #11758446	C₄	semidirect
87. C₃₁	#2381	semidirect

H ≅ G/N

type

13. #11758446

C₄

semidirect

87. C₃₁

#2381

semidirect

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

subgroup	index
C₂×#11758446	2
#14520953	2
#14520934	2
#14520932	2
#14520917	2
C₃₁×#383	2
#2381	31

subgroup

index

C₃₁×#383

Subgroups — #16848891

Conjugacy classes of subgroups

Complements

Maximal subgroups