Group #16844428 — subgroups

Subgroups — #16844428

Order 3968.

228 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 85 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	#14521106	C₂	1	1	yes	maximal, characteristic
2	#14521106	C₂	1	1	yes	maximal, characteristic
3	#14520795	C₂	1	4	yes	maximal, complemented
4	C₂×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₃₁	—	2	1	no	—
6	C₃₁×C₄⋊Q₈	—	2	1	no	—
7	C₃₁×C₄⋊Q₈	—	2	1	no	—
8	C₃₁×C₄⋊Q₈	C₄	1	2	yes	—
9	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
11	C₂×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
12	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²⋊(C₃₁⋊C₈)	—	2	1	no	—
14	C₂²⋊(C₃₁⋊C₈)	—	2	1	no	—
15	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
16	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
17	#11758391	—	2	1	no	—
18	#11758391	—	2	1	no	—
19	C₂×Q₈×C₃₁	—	2	2	no	—
20	C₂×Q₈×C₃₁	—	2	2	no	—
21	C₂×Q₈×C₃₁	—	4	4	no	—
22	C₂²×C₄×C₃₁	—	2	2	no	—
23	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
26	C₄⋊C₄×C₃₁	—	2	2	no	—
27	C₄⋊C₄×C₃₁	—	2	2	no	—
28	C₄⋊C₄×C₃₁	—	2	2	no	—
29	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
30	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₃₁	D₄	1	1	yes	characteristic
32	C₄²×C₃₁	D₄	1	1	yes	characteristic
33	C₄²×C₃₁	C₂³	1	1	yes	characteristic
34	C₂×C₃₁⋊C₈	—	2	2	no	—
35	C₂×C₃₁⋊C₈	—	2	2	no	—
36	C₂³×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
37	Q₈×C₃₁	—	4	4	no	—
38	Q₈×C₃₁	—	4	4	no	—
39	C₂×C₄×C₃₁	—	2	2	no	—
40	C₂×C₄×C₃₁	—	2	2	no	—
41	C₂×C₄×C₃₁	—	2	2	no	—
42	C₂×C₄×C₃₁	—	2	4	no	—
43	C₂×C₄×C₃₁	—	4	4	no	—
44	C₂×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
50	C₃₁⋊C₈	—	4	2	no	—
51	C₃₁⋊C₈	—	4	2	no	—
52	#1556	—	31	1	no	maximal, Sylow
53	C₂²×C₃₁	—	2	1	no	—
54	C₂²×C₃₁	—	2	1	no	—
55	C₂²×C₃₁	#101	1	2	yes	—
56	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₄×C₃₁	—	2	1	no	—
58	C₄×C₃₁	—	2	1	no	—
59	C₄×C₃₁	—	2	1	no	—
60	C₄×C₃₁	—	2	1	no	—
61	C₄×C₃₁	—	4	4	no	—
62	C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₃₁	#137	1	2	yes	—
64	C₂×C₄⋊Q₈	D₃₁	1	1	yes	characteristic
65	#432	—	31	1	no	—
66	#432	—	31	1	no	—
67	#332	—	31	4	no	—
68	C₂×C₃₁	—	2	2	no	—
69	C₂×C₃₁	C₂×#101	1	1	yes	characteristic
70	C₂×C₃₁	#418	1	1	yes	characteristic
71	C₂×C₃₁	#419	1	1	yes	characteristic
72	C₂²×Q₈	—	2	1	no	—
73	C₄⋊Q₈	—	2	1	no	—
74	C₄⋊Q₈	—	2	1	no	—
75	C₄⋊Q₈	Dic₃₁	1	2	yes	—
76	C₄⋊Q₈	C₂×D₃₁	1	2	yes	—
77	C₂×C₄⋊C₄	—	2	1	no	—
78	C₂×C₄⋊C₄	Dic₃₁	1	2	yes	—
79	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
80	C₄⋊C₈	—	31	2	no	—
81	C₄⋊C₈	—	31	2	no	—
82	C₂²⋊C₈	—	62	1	no	—
83	C₂²⋊C₈	—	62	1	no	—
84	#97	—	62	1	no	—
85	#97	—	62	1	no	—
86	C₃₁	#1556	1	1	yes	characteristic, complemented, Sylow
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	—	2	2	no	—
89	C₂×Q₈	—	4	4	no	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
92	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
93	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
94	C₂×C₈	—	62	2	no	—
95	C₂×C₈	—	62	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	—
100	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
101	C₄²	C₂²⋊D₃₁	1	1	yes	characteristic
102	C₄²	C₂²⋊D₃₁	1	1	yes	characteristic
103	C₄²	C₂²×D₃₁	1	1	yes	characteristic
104	C₂³	C₂²⋊Dic₃₁	1	1	yes	characteristic
105	Q₈	—	4	4	no	—
106	Q₈	—	4	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	C₂²⋊Dic₃₁	1	1	yes	characteristic
113	C₂×C₄	C₂²⋊Dic₃₁	1	1	yes	characteristic
114	C₂×C₄	C₂²⋊Dic₃₁	1	1	yes	characteristic
115	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
116	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
117	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
118	C₈	—	124	2	no	—
119	C₈	—	124	2	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	#11758424	1	2	yes	—
123	C₂²	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	4	4	no	—
129	C₄	#11758509	1	2	yes	—
130	C₄	#11758519	1	2	yes	—
131	C₂	—	2	2	no	—
132	C₂	C₂×#11758424	1	1	yes	characteristic
133	C₂	#14521054	1	1	yes	characteristic
134	C₂	#14521058	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521106

C₂

yes

maximal, characteristic

#14521106

C₂

yes

maximal, characteristic

#14520795

C₂

yes

maximal, complemented

C₂×C₃₁×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂²×Q₈×C₃₁

—

C₃₁×C₄⋊Q₈

—

C₃₁×C₄⋊Q₈

—

C₃₁×C₄⋊Q₈

C₄

yes

—

C₃₁×C₄⋊Q₈

C₂²

yes

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

C₄

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂²⋊(C₃₁⋊C₈)

—

C₂²⋊(C₃₁⋊C₈)

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

—

#11758391

—

#11758391

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

D₄

yes

characteristic

C₄²×C₃₁

D₄

yes

characteristic

C₄²×C₃₁

C₂³

yes

characteristic

C₂×C₃₁⋊C₈

—

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

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₃₁⋊C₈

—

C₃₁⋊C₈

—

#1556

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#101

yes

—

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

AGL(1,ℤ/8)

yes

—

C₄×C₃₁

#137

yes

—

C₂×C₄⋊Q₈

D₃₁

yes

—

—

—

C₂×C₃₁

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₃₁

yes

—

C₄⋊Q₈

C₂×D₃₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₃₁

yes

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₃₁

#1556

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₃₁

yes

characteristic

C₂²×C₄

C₂²⋊D₃₁

yes

characteristic

C₂²×C₄

C₂²⋊D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₃₁

yes

—

100

C₄²

C₂×Dic₃₁

yes

characteristic

101

C₄²

C₂²⋊D₃₁

yes

characteristic

102

C₄²

C₂²⋊D₃₁

yes

characteristic

103

C₄²

C₂²×D₃₁

yes

characteristic

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

C₂²⋊Dic₃₁

yes

characteristic

113

C₂×C₄

C₂²⋊Dic₃₁

yes

characteristic

114

C₂×C₄

C₂²⋊Dic₃₁

yes

characteristic

115

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

116

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

117

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

118

C₈

—

124

—

119

C₈

—

124

—

120

C₂²

—

121

C₂²

—

122

C₂²

#11758424

yes

—

123

C₂²

C₂×C₂²⋊Dic₃₁

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

#11758509

yes

—

130

C₄

#11758519

yes

—

131

C₂

—

132

C₂

C₂×#11758424

yes

characteristic

133

C₂

#14521054

yes

characteristic

134

C₂

#14521058

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₄²	C₂²×D₃₁
Fitting	C₂×C₃₁×C₄⋊Q₈	C₂
socle	C₂²×C₃₁	C₂×C₂²⋊C₄
prime radical	C₂³×C₃₁	C₂²⋊C₄
cosocle	C₄²×C₃₁	C₂³
hypercenter	C₂×C₄⋊Q₈	D₃₁
hypocenter	C₃₁	#1556
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₄²

C₂²×D₃₁

Fitting

C₂×C₃₁×C₄⋊Q₈

C₂

socle

C₂²×C₃₁

C₂×C₂²⋊C₄

prime radical

C₂³×C₃₁

C₂²⋊C₄

cosocle

C₄²×C₃₁

C₂³

hypercenter

C₂×C₄⋊Q₈

D₃₁

hypocenter

C₃₁

#1556

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
3. #14520795	C₂	semidirect
86. C₃₁	#1556	semidirect

H ≅ G/N

type

3. #14520795

C₂

semidirect

86. C₃₁

#1556

semidirect

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

subgroup	index
#14521106	2
#14520795	2
C₂×C₃₁×C₄⋊Q₈	2
#1556	31

subgroup

index

#14521106

#14520795

C₂×C₃₁×C₄⋊Q₈

#1556

Subgroups — #16844428

Conjugacy classes of subgroups

Complements

Maximal subgroups