Group #16845079 — subgroups

Subgroups — #16845079

Order 3968.

156 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 57 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	#14521428	C₂	1	1	yes	maximal, characteristic
2	#14521334	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#14520801	C₂	1	1	yes	maximal, characteristic
5	#14520795	C₂	1	1	yes	maximal, characteristic
6	#14520794	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#475	C₂	1	1	yes	maximal, characteristic
8	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×#92482	—	2	1	no	—
11	C₂×#92482	—	2	1	no	—
12	#11758446	C₂²	1	1	yes	characteristic
13	C₃₁×#106	—	2	1	no	—
14	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₃₁×#103	—	2	1	no	—
16	#11758423	—	2	1	no	—
17	#11758396	—	2	1	no	—
18	C₄⋊(C₃₁⋊C₈)	—	2	1	no	—
19	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
20	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
21	C₄×C₃₁⋊C₈	—	2	1	no	—
22	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₃₁	—	2	1	no	—
24	C₂×Q₈×C₃₁	D₄	1	1	yes	characteristic
25	C₂×Dic₆₂	—	2	1	no	—
26	C₂×C₈×C₃₁	—	2	1	no	—
27	C₄⋊C₄×C₃₁	—	2	1	no	—
28	C₄⋊C₄×C₃₁	—	2	1	no	—
29	C₄⋊C₄×C₃₁	—	2	1	no	—
30	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
31	C₄²×C₃₁	—	2	1	no	—
32	C₄²×C₃₁	C₂³	1	1	yes	characteristic
33	#92482	—	4	2	no	—
34	#92482	—	4	2	no	—
35	C₄⋊Dic₃₁	—	2	1	no	—
36	C₄⋊Dic₃₁	—	2	1	no	—
37	C₂×C₃₁⋊C₈	—	2	1	no	—
38	C₂×C₃₁⋊C₈	—	2	1	no	—
39	C₂×C₃₁⋊C₈	D₄	1	1	yes	characteristic
40	C₂×C₃₁⋊C₈	D₄	1	1	yes	characteristic
41	Q₈×C₃₁	—	2	1	no	—
42	Q₈×C₃₁	—	4	1	no	—
43	Q₈×C₃₁	—	4	1	no	—
44	Q₈×C₃₁	Q₁₆	1	2	yes	complemented
45	C₂×C₄×C₃₁	—	2	1	no	—
46	C₂×C₄×C₃₁	—	2	1	no	—
47	C₂×C₄×C₃₁	—	4	1	no	—
48	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
51	C₂×Dic₃₁	—	4	1	no	—
52	Dic₆₂	—	4	1	no	—
53	Dic₆₂	—	4	1	no	—
54	C₈×C₃₁	—	4	1	no	—
55	C₃₁⋊C₈	—	2	1	no	—
56	C₃₁⋊C₈	—	4	1	no	—
57	C₃₁⋊C₈	—	4	1	no	—
58	C₃₁⋊C₈	SD₁₆	1	2	yes	—
59	#1670	—	31	1	no	maximal, Sylow
60	C₂²×C₃₁	#121	1	1	yes	characteristic
61	C₄×C₃₁	—	2	2	no	—
62	C₄×C₃₁	—	2	1	no	—
63	C₄×C₃₁	—	4	1	no	—
64	C₄×C₃₁	—	8	1	no	—
65	C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
66	C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
67	C₄×C₃₁	#135	1	1	yes	characteristic
68	C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
69	Dic₃₁	—	8	1	no	—
70	#494	—	31	1	no	—
71	#475	—	31	1	no	—
72	#475	D₃₁	1	1	yes	characteristic
73	C₈×Q₈	—	31	1	no	—
74	#335	—	31	1	no	—
75	#332	—	31	1	no	—
76	#332	—	31	1	no	—
77	C₂×C₃₁	#460	1	1	yes	characteristic
78	C₂×C₃₁	#467	1	1	yes	characteristic
79	C₂×C₃₁	#473	1	1	yes	characteristic
80	C₂×Q₁₆	—	62	1	no	—
81	C₂×Q₁₆	—	62	1	no	—
82	C₄⋊Q₈	—	31	1	no	—
83	C₄⋊Q₈	C₂×D₃₁	1	1	yes	characteristic
84	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic
85	#106	—	2	1	no	—
86	#106	—	62	1	no	—
87	C₄⋊C₈	—	31	1	no	—
88	C₄⋊C₈	—	31	1	no	—
89	C₄⋊C₈	—	62	1	no	—
90	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
91	#103	—	2	1	no	—
92	#103	—	62	1	no	—
93	C₄×C₈	—	31	1	no	—
94	C₄×C₈	—	62	1	no	—
95	C₃₁	#1670	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	62	1	no	—
98	C₂×Q₈	C₂²⋊D₃₁	1	1	yes	characteristic
99	Q₁₆	—	124	2	no	—
100	Q₁₆	—	124	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	31	1	no	—
103	C₂×C₈	—	31	1	no	—
104	C₂×C₈	—	62	1	no	—
105	C₂×C₈	—	62	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	62	1	no	—
110	C₄⋊C₄	—	62	1	no	—
111	C₄⋊C₄	C₂²⋊D₃₁	1	1	yes	characteristic
112	C₄²	—	2	1	no	—
113	C₄²	C₂²×D₃₁	1	1	yes	characteristic
114	Q₈	—	2	1	no	—
115	Q₈	—	4	1	no	—
116	Q₈	—	4	1	no	—
117	Q₈	—	124	1	no	—
118	Q₈	—	124	1	no	—
119	Q₈	#92482	1	2	yes	complemented
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	124	1	no	—
124	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
125	C₂×C₄	#92497	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
127	C₈	—	4	1	no	—
128	C₈	—	31	2	no	—
129	C₈	—	62	1	no	—
130	C₈	—	124	1	no	—
131	C₈	—	124	1	no	—
132	C₂²	#11758516	1	1	yes	characteristic
133	C₄	—	2	2	no	—
134	C₄	—	2	1	no	—
135	C₄	—	4	1	no	—
136	C₄	—	8	1	no	—
137	C₄	—	248	1	no	—
138	C₄	SD₁₆×D₃₁	1	1	yes	characteristic
139	C₄	#11758491	1	1	yes	characteristic
140	C₄	C₂×#92482	1	1	yes	characteristic
141	C₄	#11758527	1	1	yes	characteristic
142	C₂	#14521247	1	1	yes	characteristic
143	C₂	#14521288	1	1	yes	characteristic
144	C₂	#14521322	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521428

C₂

yes

maximal, characteristic

#14521334

C₂

yes

maximal, characteristic

Q₈×C₃₁⋊C₈

C₂

yes

maximal, characteristic

#14520801

C₂

yes

maximal, characteristic

#14520795

C₂

yes

maximal, characteristic

#14520794

C₂

yes

maximal, characteristic

C₃₁×#475

C₂

yes

maximal, characteristic

C₃₁×C₄⋊Q₈

C₂²

yes

characteristic

C₄×Q₈×C₃₁

C₂²

yes

—

—

C₂²

yes

characteristic

C₃₁×#106

—

C₃₁×C₄⋊C₈

C₂²

yes

characteristic

C₃₁×#103

—

#11758423

—

#11758396

—

C₄⋊(C₃₁⋊C₈)

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₃₁⋊C₈)

C₂²

yes

characteristic

C₄×C₃₁⋊C₈

—

C₄×C₃₁⋊C₈

C₂²

yes

characteristic

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

D₄

yes

characteristic

C₂×Dic₆₂

—

C₂×C₈×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

D₄

yes

characteristic

C₄²×C₃₁

—

C₄²×C₃₁

C₂³

yes

characteristic

#92482

—

#92482

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

D₄

yes

characteristic

C₂×C₃₁⋊C₈

D₄

yes

characteristic

Q₈×C₃₁

—

Q₈×C₃₁

—

Q₈×C₃₁

—

Q₈×C₃₁

Q₁₆

yes

complemented

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

characteristic

C₂×Dic₃₁

—

Dic₆₂

—

Dic₆₂

—

C₈×C₃₁

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

C₃₁⋊C₈

SD₁₆

yes

—

—

C₂²×C₃₁

yes

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×SD₁₆

yes

characteristic

C₄×C₃₁

C₂×Q₁₆

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

Dic₃₁

—

#494

—

#475

—

#475

D₃₁

yes

characteristic

C₈×Q₈

—

#335

—

#332

—

#332

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₁

yes

characteristic

C₄×Q₈

C₂×D₃₁

yes

characteristic

#106

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₃₁

yes

characteristic

#103

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₃₁

#1670

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₃₁

yes

characteristic

Q₁₆

—

124

—

100

Q₁₆

—

124

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂²⋊D₃₁

yes

characteristic

112

C₄²

—

113

C₄²

C₂²×D₃₁

yes

characteristic

114

Q₈

—

115

Q₈

—

116

Q₈

—

117

Q₈

—

124

—

118

Q₈

—

124

—

119

Q₈

#92482

yes

complemented

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

—

124

C₂×C₄

D₄×D₃₁

yes

characteristic

125

C₂×C₄

#92497

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

124

—

131

C₈

—

124

—

132

C₂²

#11758516

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

248

—

138

C₄

SD₁₆×D₃₁

yes

characteristic

139

C₄

#11758491

yes

characteristic

140

C₄

C₂×#92482

yes

characteristic

141

C₄

#11758527

yes

characteristic

142

C₂

#14521247

yes

characteristic

143

C₂

#14521288

yes

characteristic

144

C₂

#14521322

yes

characteristic

	subgroup	G/H
center	C₂²	#11758516
commutator, cosocle	C₄²×C₃₁	C₂³
Frattini	C₄²	C₂²×D₃₁
Fitting	C₃₁×#475	C₂
socle, prime radical	C₂²×C₃₁	#121
hypercenter	#475	D₃₁
hypocenter	C₃₁	#1670
LCS	G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

subgroup

G/H

center

C₂²

#11758516

commutator, cosocle

C₄²×C₃₁

C₂³

Frattini

C₄²

C₂²×D₃₁

Fitting

C₃₁×#475

C₂

C₂²×C₃₁

D₃₁

C₃₁

G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #475

N	H ≅ G/N	type
44. Q₈×C₃₁	Q₁₆	semidirect
95. C₃₁	#1670	semidirect
119. Q₈	#92482	semidirect

H ≅ G/N

type

44. Q₈×C₃₁

Q₁₆

semidirect

95. C₃₁

#1670

semidirect

119. Q₈

#92482

semidirect

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

subgroup	index
#14521428	2
#14521334	2
Q₈×C₃₁⋊C₈	2
#14520801	2
#14520795	2
#14520794	2
C₃₁×#475	2
#1670	31

subgroup

index

Q₈×C₃₁⋊C₈

C₃₁×#475

Subgroups — #16845079

Conjugacy classes of subgroups

Complements

Maximal subgroups