Group #16846440 — subgroups

Subgroups — #16846440

Order 3968.

248 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 117 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×#11758405	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₃₁	C₂	1	1	yes	maximal, characteristic
3	#14520948	C₂	1	2	yes	maximal
4	#14520811	C₂	1	2	yes	maximal
5	C₂×C₃₁×#106	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₃₁	—	2	2	no	—
9	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
10	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
11	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₃₁×#106	C₄	1	4	yes	complemented
13	C₃₁⋊#95	—	2	2	no	—
14	#11758405	—	2	2	no	—
15	C₈×Dic₃₁	—	2	2	no	—
16	C₂²×C₄×C₃₁	—	2	2	no	—
17	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
18	C₂²×Dic₃₁	—	2	2	no	—
19	C₂²×Dic₃₁	D₄	1	2	yes	—
20	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₃₁	D₄	1	2	yes	—
22	C₂×C₈×C₃₁	Q₈	1	2	yes	—
23	C₄⋊C₄×C₃₁	—	2	2	no	—
24	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
25	C₄⋊Dic₃₁	—	2	2	no	—
26	C₄⋊Dic₃₁	—	2	2	no	—
27	C₄⋊Dic₃₁	—	4	4	no	—
28	C₄×Dic₃₁	—	2	1	no	—
29	C₄×Dic₃₁	—	2	1	no	—
30	C₂×C₃₁⋊C₈	—	2	1	no	—
31	C₂×C₃₁⋊C₈	—	2	1	no	—
32	C₂×C₃₁⋊C₈	Q₈	1	2	yes	—
33	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₃₁	—	2	4	no	—
35	C₂×C₄×C₃₁	—	4	2	no	—
36	C₂×C₄×C₃₁	—	4	2	no	—
37	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
42	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
43	C₂×Dic₃₁	—	2	2	no	—
44	C₂×Dic₃₁	—	2	2	no	—
45	C₂×Dic₃₁	—	4	2	no	—
46	C₂×Dic₃₁	—	4	2	no	—
47	C₂×Dic₃₁	—	4	2	no	—
48	C₂×Dic₃₁	SD₁₆	1	4	yes	—
49	C₈×C₃₁	C₄⋊C₄	1	4	yes	complemented
50	C₃₁⋊C₈	—	2	2	no	—
51	#1941	—	31	1	no	maximal, Sylow
52	C₂²×C₃₁	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₃₁	#121	1	1	yes	characteristic
54	C₂²×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
55	C₂²×C₃₁	C₂×SD₁₆	1	2	yes	—
56	C₂²×C₃₁	#135	1	2	yes	—
57	C₄×C₃₁	—	4	4	no	—
58	C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
60	C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
61	C₄×C₃₁	#125	1	1	yes	characteristic
62	Dic₃₁	—	2	4	no	—
63	Dic₃₁	—	8	2	no	—
64	C₂×#106	—	31	1	no	—
65	C₂×#106	D₃₁	1	1	yes	characteristic
66	C₂×C₄×C₈	—	31	1	no	—
67	#389	—	31	2	no	—
68	#340	—	31	2	no	—
69	C₂×C₃₁	#389	1	1	yes	characteristic
70	C₂×C₃₁	C₄×SD₁₆	1	2	yes	—
71	C₂×C₃₁	#465	1	2	yes	—
72	C₂×C₃₁	#498	1	1	yes	characteristic
73	C₂×C₃₁	#499	1	1	yes	characteristic
74	C₂²×C₈	—	31	1	no	—
75	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
76	C₂×C₄⋊C₄	—	31	2	no	—
77	C₂×C₄⋊C₄	—	62	2	no	—
78	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
79	C₂×C₄²	—	31	1	no	—
80	#106	—	62	2	no	—
81	#106	Dic₃₁	1	4	yes	complemented
82	C₄×C₈	—	62	2	no	—
83	#95	—	62	2	no	—
84	C₃₁	#1941	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	31	2	no	—
87	C₂²×C₄	—	62	2	no	—
88	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
89	C₂×C₈	—	31	2	no	—
90	C₂×C₈	—	62	1	no	—
91	C₂×C₈	—	62	1	no	—
92	C₂×C₈	Dic₆₂	1	2	yes	—
93	C₂×C₈	D₁₂₄	1	2	yes	—
94	C₂×C₈	C₂×Dic₃₁	1	2	yes	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	62	2	no	—
97	C₄⋊C₄	—	62	2	no	—
98	C₄⋊C₄	—	124	4	no	—
99	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	—
100	C₄²	—	62	1	no	—
101	C₄²	—	62	1	no	—
102	C₂³	D₄×D₃₁	1	1	yes	characteristic
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	31	4	no	—
107	C₂×C₄	—	62	2	no	—
108	C₂×C₄	—	62	2	no	—
109	C₂×C₄	—	124	2	no	—
110	C₂×C₄	—	124	2	no	—
111	C₂×C₄	—	124	2	no	—
112	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
113	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
114	C₂×C₄	#92497	1	1	yes	characteristic
115	C₂×C₄	Q₈×D₃₁	1	1	yes	characteristic
116	C₂×C₄	#92499	1	1	yes	characteristic
117	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
118	C₈	—	62	2	no	—
119	C₈	C₄⋊Dic₃₁	1	4	yes	complemented
120	C₂²	#11758465	1	1	yes	characteristic
121	C₂²	#11758472	1	1	yes	characteristic
122	C₂²	SD₁₆×D₃₁	1	2	yes	—
123	C₂²	#11758493	1	2	yes	—
124	C₂²	D₄×Dic₃₁	1	1	yes	characteristic
125	C₄	—	4	4	no	—
126	C₄	—	62	4	no	—
127	C₄	—	248	2	no	—
128	C₄	#11758467	1	1	yes	characteristic
129	C₄	#11758474	1	1	yes	characteristic
130	C₄	C₂×C₄⋊Dic₃₁	1	1	yes	characteristic
131	C₄	Q₈×Dic₃₁	1	1	yes	characteristic
132	C₂	#14520948	1	1	yes	characteristic
133	C₂	SD₁₆×Dic₃₁	1	2	yes	—
134	C₂	#14521279	1	2	yes	—
135	C₂	#14521446	1	1	yes	characteristic
136	C₂	#14521449	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758405

C₂

yes

maximal, characteristic

C₂×C₈×Dic₃₁

C₂

yes

maximal, characteristic

#14520948

C₂

yes

maximal

#14520811

C₂

yes

maximal

C₂×C₃₁×#106

C₂

yes

maximal, characteristic

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

C₃₁×#106

C₄

yes

complemented

C₃₁⋊#95

—

#11758405

—

C₈×Dic₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂²×Dic₃₁

D₄

yes

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

D₄

yes

—

C₂×C₈×C₃₁

Q₈

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

Q₈

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

C₂×C₄×C₃₁

yes

C₂×C₄×C₃₁

yes

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

SD₁₆

yes

—

C₈×C₃₁

C₄⋊C₄

yes

complemented

C₃₁⋊C₈

—

#1941

—

maximal, Sylow

C₂²×C₃₁

C₄×D₄

yes

characteristic

C₂²×C₃₁

#121

yes

characteristic

C₂²×C₃₁

C₄⋊Q₈

yes

characteristic

C₂²×C₃₁

C₂×SD₁₆

yes

—

C₂²×C₃₁

#135

yes

—

C₄×C₃₁

—

C₄×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₃₁

C₄×Q₈

yes

characteristic

C₄×C₃₁

C₂²⋊Q₈

yes

characteristic

C₄×C₃₁

#125

yes

characteristic

Dic₃₁

—

Dic₃₁

—

C₂×#106

—

C₂×#106

D₃₁

yes

characteristic

C₂×C₄×C₈

—

#389

—

#340

—

C₂×C₃₁

#389

yes

characteristic

C₂×C₃₁

C₄×SD₁₆

yes

—

C₂×C₃₁

#465

yes

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂²×C₈

—

C₂²×C₈

C₂×D₃₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₄²

—

#106

—

#106

Dic₃₁

yes

complemented

C₄×C₈

—

#95

—

C₃₁

#1941

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₆₂

yes

—

C₂×C₈

D₁₂₄

yes

—

C₂×C₈

C₂×Dic₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

124

—

C₄⋊C₄

C₂×Dic₃₁

yes

—

100

C₄²

—

101

C₄²

—

102

C₂³

D₄×D₃₁

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

124

—

110

C₂×C₄

—

124

—

111

C₂×C₄

—

124

—

112

C₂×C₄

C₂×Dic₆₂

yes

characteristic

113

C₂×C₄

C₂×D₁₂₄

yes

characteristic

114

C₂×C₄

#92497

yes

characteristic

115

C₂×C₄

Q₈×D₃₁

yes

characteristic

116

C₂×C₄

#92499

yes

characteristic

117

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

118

C₈

—

119

C₈

C₄⋊Dic₃₁

yes

complemented

120

C₂²

#11758465

yes

characteristic

121

C₂²

#11758472

yes

characteristic

122

C₂²

SD₁₆×D₃₁

yes

—

123

C₂²

#11758493

yes

—

124

C₂²

D₄×Dic₃₁

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

248

—

128

C₄

#11758467

yes

characteristic

129

C₄

#11758474

yes

characteristic

130

C₄

C₂×C₄⋊Dic₃₁

yes

characteristic

131

C₄

Q₈×Dic₃₁

yes

characteristic

132

C₂

#14520948

yes

characteristic

133

C₂

SD₁₆×Dic₃₁

yes

—

134

C₂

#14521279

yes

—

135

C₂

#14521446

yes

characteristic

136

C₂

#14521449

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂×C₃₁×#106	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×#106	D₃₁
hypocenter	C₃₁	#1941
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂×C₃₁×#106

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

cosocle

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
12. C₃₁×#106	C₄	semidirect
49. C₈×C₃₁	C₄⋊C₄	semidirect
81. #106	Dic₃₁	semidirect
84. C₃₁	#1941	semidirect
119. C₈	C₄⋊Dic₃₁	semidirect

H ≅ G/N

type

12. C₃₁×#106

C₄

semidirect

49. C₈×C₃₁

C₄⋊C₄

semidirect

81. #106

Dic₃₁

semidirect

84. C₃₁

#1941

semidirect

119. C₈

C₄⋊Dic₃₁

semidirect

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

subgroup	index
C₂×#11758405	2
C₂×C₈×Dic₃₁	2
#14520948	2
#14520811	2
C₂×C₃₁×#106	2
#1941	31

subgroup

index

C₂×C₈×Dic₃₁

C₂×C₃₁×#106

Subgroups — #16846440

Conjugacy classes of subgroups

Complements

Maximal subgroups