Group #16845707 — subgroups

Subgroups — #16845707

Order 3968.

272 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 113 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	#14521073	C₂	1	2	yes	maximal
2	C₂×#11758424	C₂	1	2	yes	maximal
3	#14520818	C₂	1	2	yes	maximal
4	C₂×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
6	C₄×Q₈×C₃₁	—	2	2	no	—
7	C₄×Q₈×C₃₁	—	2	2	no	—
8	C₂×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
9	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₃₁	C₄	1	2	yes	—
11	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×#92474	C₂²	1	4	yes	—
13	#11758424	—	2	4	no	—
14	C₄⋊(C₃₁⋊C₈)	—	2	2	no	—
15	C₄⋊(C₃₁⋊C₈)	—	2	2	no	—
16	C₂×Q₈×C₃₁	—	2	2	no	—
17	C₂×Q₈×C₃₁	—	2	2	no	—
18	C₂×Q₈×C₃₁	D₄	1	2	yes	—
19	C₂×Q₈×C₃₁	Q₈	1	2	yes	—
20	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₃₁	—	2	4	no	—
26	C₄⋊C₄×C₃₁	—	2	2	no	—
27	C₄²×C₃₁	—	2	2	no	—
28	C₄²×C₃₁	—	2	2	no	—
29	C₄²×C₃₁	D₄	1	2	yes	—
30	C₄²×C₃₁	D₄	1	2	yes	—
31	#92474	—	2	4	no	—
32	#92474	—	2	4	no	—
33	C₂×C₃₁⋊C₈	—	2	4	no	—
34	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₃₁	—	2	4	no	—
36	Q₈×C₃₁	—	2	4	no	—
37	C₂×C₄×C₃₁	—	2	2	no	—
38	C₂×C₄×C₃₁	—	2	2	no	—
39	C₂×C₄×C₃₁	—	2	2	no	—
40	C₂×C₄×C₃₁	—	2	2	no	—
41	C₂×C₄×C₃₁	—	4	1	no	—
42	C₂×C₄×C₃₁	—	4	2	no	—
43	C₂×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
47	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₃₁	#41	1	2	yes	—
51	C₃₁⋊C₈	—	4	4	no	—
52	#1801	—	31	1	no	maximal, Sylow
53	C₂²×C₃₁	—	2	1	no	—
54	C₂²×C₃₁	—	2	1	no	—
55	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₃₁	#117	1	1	yes	characteristic
58	C₄×C₃₁	—	2	4	no	—
59	C₄×C₃₁	—	2	1	no	—
60	C₄×C₃₁	—	2	2	no	—
61	C₄×C₃₁	—	4	2	no	—
62	C₄×C₃₁	#101	1	2	yes	complemented
63	C₄×C₃₁	#121	1	2	yes	—
64	C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
65	C₂×C₄×Q₈	D₃₁	1	1	yes	characteristic
66	#423	—	31	2	no	—
67	C₂×#101	—	31	2	no	—
68	#343	—	31	2	no	—
69	C₂×C₃₁	—	2	1	no	—
70	C₂×C₃₁	—	2	1	no	—
71	C₂×C₃₁	#380	1	1	yes	characteristic
72	C₂×C₃₁	C₂×#101	1	1	yes	characteristic
73	C₂×C₃₁	#413	1	1	yes	characteristic
74	C₂²×Q₈	C₂×D₃₁	1	1	yes	characteristic
75	C₂×M₁₆	—	31	4	no	—
76	C₄×Q₈	—	2	2	no	—
77	C₄×Q₈	—	2	2	no	—
78	C₂×C₄⋊C₄	Dic₃₁	1	2	yes	—
79	C₂×C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
80	C₂×C₄²	Dic₃₁	1	2	yes	—
81	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
82	C₄⋊C₈	—	62	2	no	—
83	C₄⋊C₈	—	62	2	no	—
84	#101	—	62	4	no	—
85	C₃₁	#1801	1	1	yes	characteristic, complemented, Sylow
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	Dic₆₂	1	2	yes	—
89	C₂×Q₈	D₁₂₄	1	2	yes	—
90	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
91	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
92	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
93	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
94	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
95	M₁₆	—	62	4	no	—
96	M₁₆	—	62	4	no	—
97	C₂×C₈	—	62	4	no	—
98	C₄⋊C₄	—	2	4	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄²	—	2	2	no	—
101	C₄²	—	2	2	no	—
102	C₄²	C₂²⋊D₃₁	1	2	yes	—
103	C₄²	C₂²⋊D₃₁	1	2	yes	—
104	C₂³	C₂²×Dic₃₁	1	1	yes	characteristic
105	Q₈	—	2	4	no	—
106	Q₈	—	2	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	C₄⋊Dic₃₁	1	4	yes	—
114	C₂×C₄	C₂²⋊Dic₃₁	1	1	yes	characteristic
115	C₂×C₄	C₂²⋊Dic₃₁	1	2	yes	—
116	C₂×C₄	C₂²⋊Dic₃₁	1	1	yes	characteristic
117	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
118	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
119	C₂×C₄	#92495	1	2	yes	—
120	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
121	C₈	—	124	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	C₂×C₄⋊Dic₃₁	1	1	yes	characteristic
125	C₂²	#11758503	1	1	yes	characteristic
126	C₂²	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
127	C₄	—	2	4	no	—
128	C₄	—	2	1	no	—
129	C₄	—	2	2	no	—
130	C₄	—	4	2	no	—
131	C₄	#11758424	1	2	yes	complemented
132	C₄	#11758501	1	2	yes	—
133	C₄	#11758507	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#14520909	1	1	yes	characteristic
137	C₂	C₂×#11758424	1	1	yes	characteristic
138	C₂	#14521033	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521073

C₂

yes

maximal

C₂×#11758424

C₂

yes

maximal

#14520818

C₂

yes

maximal

C₂×C₄×Q₈×C₃₁

C₂

yes

maximal, characteristic

C₂²×Q₈×C₃₁

C₂²

yes

characteristic

C₄×Q₈×C₃₁

—

C₄×Q₈×C₃₁

—

C₂×C₄⋊C₄×C₃₁

C₄

yes

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

characteristic

C₂×C₄²×C₃₁

C₄

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×#92474

C₂²

yes

—

#11758424

—

C₄⋊(C₃₁⋊C₈)

—

C₄⋊(C₃₁⋊C₈)

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

D₄

yes

—

C₂×Q₈×C₃₁

Q₈

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

D₄

yes

—

#92474

—

#92474

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

C₂²×C₄

yes

characteristic

Q₈×C₃₁

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₃₁⋊C₈

—

#1801

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₁

#117

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#101

yes

complemented

C₄×C₃₁

#121

yes

—

C₄×C₃₁

C₂²⋊Q₈

yes

—

C₂×C₄×Q₈

D₃₁

yes

—

—

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂²×Q₈

C₂×D₃₁

yes

characteristic

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₃₁

yes

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

characteristic

C₂×C₄²

Dic₃₁

yes

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#101

—

C₃₁

#1801

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₆₂

yes

—

C₂×Q₈

D₁₂₄

yes

—

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂×Dic₃₁

yes

characteristic

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂×Dic₃₁

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂²⋊D₃₁

yes

—

103

C₄²

C₂²⋊D₃₁

yes

—

104

C₂³

C₂²×Dic₃₁

yes

characteristic

105

Q₈

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₄⋊Dic₃₁

yes

—

114

C₂×C₄

C₂²⋊Dic₃₁

yes

characteristic

115

C₂×C₄

C₂²⋊Dic₃₁

yes

—

116

C₂×C₄

C₂²⋊Dic₃₁

yes

characteristic

117

C₂×C₄

C₂×Dic₆₂

yes

characteristic

118

C₂×C₄

C₂×D₁₂₄

yes

characteristic

119

C₂×C₄

#92495

yes

—

120

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

121

C₈

—

124

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×C₄⋊Dic₃₁

yes

characteristic

125

C₂²

#11758503

yes

characteristic

126

C₂²

C₂×C₂²⋊Dic₃₁

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#11758424

yes

complemented

132

C₄

#11758501

yes

—

133

C₄

#11758507

yes

—

134

C₂

—

135

C₂

—

136

C₂

#14520909

yes

characteristic

137

C₂

C₂×#11758424

yes

characteristic

138

C₂

#14521033

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₃₁
commutator, prime radical	C₂³×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂×C₄×Q₈×C₃₁	C₂
socle	C₂²×C₃₁	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×C₄×Q₈	D₃₁
hypocenter	C₃₁	#1801
LCS	G ▷ C₂³×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₃₁

commutator, prime radical

C₂³×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂×C₄×Q₈×C₃₁

C₂

socle

C₂²×C₃₁

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₂×C₄×Q₈

D₃₁

hypocenter

C₃₁

#1801

LCS

G ▷ C₂³×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
62. C₄×C₃₁	#101	semidirect
85. C₃₁	#1801	semidirect
131. C₄	#11758424	semidirect

H ≅ G/N

type

62. C₄×C₃₁

#101

semidirect

85. C₃₁

#1801

semidirect

131. C₄

#11758424

semidirect

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

subgroup	index
#14521073	2
C₂×#11758424	2
#14520818	2
C₂×C₄×Q₈×C₃₁	2
#1801	31

subgroup

index

C₂×C₄×Q₈×C₃₁

Subgroups — #16845707

Conjugacy classes of subgroups

Complements

Maximal subgroups