Group #16844934 — subgroups

Subgroups — #16844934

Order 3968.

200 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 59 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	Q₈×Dic₆₂	C₂	1	1	yes	maximal, characteristic
2	#14521400	C₂	1	1	yes	maximal, characteristic
3	#14521335	C₂	1	1	yes	maximal, characteristic
4	#14521259	C₂	1	1	yes	maximal, characteristic
5	#14520797	C₂	1	1	yes	maximal, characteristic
6	#14520781	C₂	1	1	yes	maximal, characteristic
7	C₃₁×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
9	Q₈×Dic₃₁	—	2	1	no	—
10	C₂×#92482	—	2	1	no	—
11	#11758465	—	2	2	no	—
12	#11758465	—	2	1	no	—
13	#11758446	—	2	1	no	—
14	#11758446	C₂²	1	1	yes	characteristic
15	#11758446	C₂²	1	1	yes	characteristic
16	C₄×Dic₆₂	—	2	1	no	—
17	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
18	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	#11758405	—	2	1	no	—
21	#11758404	—	2	2	no	—
22	#11758404	—	2	1	no	—
23	#11758398	—	2	1	no	—
24	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₃₁	D₄	1	1	yes	characteristic
26	C₂×Dic₆₂	—	2	1	no	—
27	C₂×Dic₆₂	—	2	2	no	—
28	C₂×Dic₆₂	D₄	1	1	yes	characteristic
29	C₂×Dic₆₂	D₄	1	2	yes	—
30	C₂×C₈×C₃₁	—	2	1	no	—
31	C₂×C₈×C₃₁	—	2	1	no	—
32	C₄⋊C₄×C₃₁	—	2	1	no	—
33	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
34	C₄²×C₃₁	—	2	1	no	—
35	C₄²×C₃₁	C₂³	1	1	yes	characteristic
36	#92482	—	4	2	no	—
37	C₄⋊Dic₃₁	—	2	2	no	—
38	C₄⋊Dic₃₁	—	2	1	no	—
39	C₄⋊Dic₃₁	—	2	1	no	—
40	C₄⋊Dic₃₁	—	2	1	no	—
41	C₄⋊Dic₃₁	—	2	1	no	—
42	C₄⋊Dic₃₁	D₄	1	1	yes	characteristic
43	Dic₃₁⋊C₄	—	2	2	no	—
44	Dic₃₁⋊C₄	—	2	1	no	—
45	C₄×Dic₃₁	—	2	2	no	—
46	C₄×Dic₃₁	—	2	1	no	—
47	C₂×C₃₁⋊C₈	—	2	1	no	—
48	Q₈×C₃₁	—	2	1	no	—
49	Q₈×C₃₁	SD₁₆	1	2	yes	—
50	C₂×C₄×C₃₁	—	2	1	no	—
51	C₂×C₄×C₃₁	—	2	1	no	—
52	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₃₁	—	2	1	no	—
56	C₂×Dic₃₁	—	2	2	no	—
57	C₂×Dic₃₁	—	2	1	no	—
58	C₂×Dic₃₁	—	4	1	no	—
59	Dic₆₂	—	2	1	no	—
60	Dic₆₂	—	4	2	no	—
61	Dic₆₂	—	4	1	no	—
62	Dic₆₂	—	4	1	no	—
63	Dic₆₂	—	8	2	no	—
64	Dic₆₂	Q₁₆	1	2	yes	complemented
65	C₈×C₃₁	—	4	1	no	—
66	C₈×C₃₁	—	4	1	no	—
67	C₃₁⋊C₈	—	4	1	no	—
68	#1645	—	31	1	no	maximal, Sylow
69	C₂²×C₃₁	#120	1	1	yes	characteristic
70	C₄×C₃₁	—	2	2	no	—
71	C₄×C₃₁	—	2	1	no	—
72	C₄×C₃₁	—	4	1	no	—
73	C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
74	C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
75	C₄×C₃₁	#137	1	1	yes	characteristic
76	C₄×C₃₁	#137	1	1	yes	characteristic
77	Dic₃₁	—	2	2	no	—
78	Dic₃₁	—	4	2	no	—
79	Dic₃₁	—	4	1	no	—
80	Dic₃₁	—	8	1	no	—
81	Q₈²	—	31	1	no	—
82	#487	—	31	1	no	—
83	#475	—	31	1	no	—
84	#462	—	31	1	no	—
85	#333	—	31	1	no	—
86	Q₈⋊C₈	—	31	1	no	—
87	Q₈⋊C₈	D₃₁	1	1	yes	characteristic
88	C₂×C₃₁	#448	1	1	yes	characteristic
89	C₂×C₃₁	#451	1	1	yes	characteristic
90	C₂×C₃₁	#455	1	1	yes	characteristic
91	C₂×Q₁₆	—	62	1	no	—
92	C₄⋊Q₈	—	31	1	no	—
93	C₄⋊Q₈	—	31	1	no	—
94	C₄⋊Q₈	—	62	1	no	—
95	C₄⋊Q₈	—	62	2	no	—
96	C₄⋊Q₈	—	62	1	no	—
97	C₄×Q₈	—	31	1	no	—
98	C₄×Q₈	—	62	1	no	—
99	C₄×Q₈	—	62	1	no	—
100	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic
101	#106	—	62	1	no	—
102	C₄⋊C₈	—	31	1	no	—
103	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
104	#103	—	62	1	no	—
105	#103	—	62	2	no	—
106	#103	—	62	1	no	—
107	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
108	C₃₁	#1645	1	1	yes	characteristic, complemented, Sylow
109	C₂×Q₈	—	31	1	no	—
110	C₂×Q₈	—	31	2	no	—
111	C₂×Q₈	—	62	2	no	—
112	C₂×Q₈	—	62	1	no	—
113	C₂×Q₈	D₁₂₄	1	1	yes	characteristic
114	Q₁₆	—	124	2	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	62	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	31	1	no	—
120	C₄⋊C₄	—	62	2	no	—
121	C₄⋊C₄	—	62	2	no	—
122	C₄⋊C₄	—	62	1	no	—
123	C₄⋊C₄	—	62	1	no	—
124	C₄⋊C₄	—	62	1	no	—
125	C₄⋊C₄	—	62	1	no	—
126	C₄⋊C₄	—	62	1	no	—
127	C₄⋊C₄	D₁₂₄	1	1	yes	characteristic
128	C₄²	—	2	1	no	—
129	C₄²	—	62	2	no	—
130	C₄²	—	62	1	no	—
131	C₄²	C₂²×D₃₁	1	1	yes	characteristic
132	Q₈	—	2	1	no	—
133	Q₈	—	31	2	no	—
134	Q₈	—	62	1	no	—
135	Q₈	—	124	2	no	—
136	Q₈	—	124	1	no	—
137	Q₈	—	124	1	no	—
138	Q₈	—	248	2	no	—
139	Q₈	#92470	1	2	yes	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	62	1	no	—
143	C₂×C₄	—	62	1	no	—
144	C₂×C₄	—	62	2	no	—
145	C₂×C₄	—	124	1	no	—
146	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
147	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
148	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	4	1	no	—
151	C₈	—	124	1	no	—
152	C₂²	#11758459	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	1	no	—
155	C₄	—	4	1	no	—
156	C₄	—	62	2	no	—
157	C₄	—	124	2	no	—
158	C₄	—	124	1	no	—
159	C₄	—	248	1	no	—
160	C₄	C₂×#92470	1	1	yes	characteristic
161	C₄	#11758486	1	1	yes	characteristic
162	C₄	#11758492	1	1	yes	characteristic
163	C₄	Q₁₆×D₃₁	1	1	yes	characteristic
164	C₂	#14521190	1	1	yes	characteristic
165	C₂	#14521206	1	1	yes	characteristic
166	C₂	#14521228	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₆₂

C₂

yes

maximal, characteristic

#14521400

C₂

yes

maximal, characteristic

#14521335

C₂

yes

maximal, characteristic

#14521259

C₂

yes

maximal, characteristic

#14520797

C₂

yes

maximal, characteristic

#14520781

C₂

yes

maximal, characteristic

C₃₁×Q₈⋊C₈

C₂

yes

maximal, characteristic

C₄×Q₈×C₃₁

C₂²

yes

characteristic

Q₈×Dic₃₁

—

—

—

—

—

C₂²

yes

characteristic

#11758446

C₂²

yes

characteristic

C₄×Dic₆₂

—

C₄×Dic₆₂

C₂²

yes

characteristic

C₃₁×C₄⋊C₈

C₂²

yes

characteristic

C₄×C₈×C₃₁

C₂²

yes

—

—

—

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

characteristic

C₂×Q₈×C₃₁

D₄

yes

characteristic

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

D₄

yes

characteristic

C₂×Dic₆₂

D₄

yes

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

D₄

yes

characteristic

C₄²×C₃₁

—

C₄²×C₃₁

C₂³

yes

characteristic

#92482

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

D₄

yes

characteristic

Dic₃₁⋊C₄

—

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

C₄×Dic₃₁

—

C₂×C₃₁⋊C₈

—

Q₈×C₃₁

—

Q₈×C₃₁

SD₁₆

yes

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

Q₁₆

yes

complemented

C₈×C₃₁

—

C₈×C₃₁

—

C₃₁⋊C₈

—

—

C₂²×C₃₁

yes

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×SD₁₆

yes

characteristic

C₄×C₃₁

C₂×Q₁₆

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Q₈²

—

—

—

—

—

Q₈⋊C₈

—

Q₈⋊C₈

D₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

100

C₄×Q₈

C₂×D₃₁

yes

characteristic

101

#106

—

102

C₄⋊C₈

—

103

C₄⋊C₈

C₂×D₃₁

yes

characteristic

104

#103

—

105

#103

—

106

#103

—

107

C₄×C₈

C₂×D₃₁

yes

characteristic

108

C₃₁

#1645

yes

characteristic, complemented, Sylow

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

D₁₂₄

yes

characteristic

114

Q₁₆

—

124

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

D₁₂₄

yes

characteristic

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

C₂²×D₃₁

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

124

—

136

Q₈

—

124

—

137

Q₈

—

124

—

138

Q₈

—

248

—

139

Q₈

#92470

yes

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

124

—

146

C₂×C₄

C₂×D₁₂₄

yes

characteristic

147

C₂×C₄

D₄×D₃₁

yes

characteristic

148

C₂×C₄

D₄×D₃₁

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

124

—

152

C₂²

#11758459

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

124

—

158

C₄

—

124

—

159

C₄

—

248

—

160

C₄

C₂×#92470

yes

characteristic

161

C₄

#11758486

yes

characteristic

162

C₄

#11758492

yes

characteristic

163

C₄

Q₁₆×D₃₁

yes

characteristic

164

C₂

#14521190

yes

characteristic

165

C₂

#14521206

yes

characteristic

166

C₂

#14521228

yes

characteristic

	subgroup	G/H
center	C₂²	#11758459
commutator, cosocle	C₄²×C₃₁	C₂³
Frattini	C₄²	C₂²×D₃₁
Fitting	C₃₁×Q₈⋊C₈	C₂
socle, prime radical	C₂²×C₃₁	#120
hypercenter	Q₈⋊C₈	D₃₁
hypocenter	C₃₁	#1645
LCS	G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

subgroup

G/H

center

C₂²

#11758459

commutator, cosocle

C₄²×C₃₁

C₂³

Frattini

C₄²

C₂²×D₃₁

Fitting

C₃₁×Q₈⋊C₈

C₂

socle, prime radical

C₂²×C₃₁

#120

hypercenter

Q₈⋊C₈

D₃₁

hypocenter

C₃₁

#1645

LCS

G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
64. Dic₆₂	Q₁₆	semidirect
108. C₃₁	#1645	semidirect

H ≅ G/N

type

64. Dic₆₂

Q₁₆

semidirect

108. C₃₁

#1645

semidirect

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

subgroup	index
Q₈×Dic₆₂	2
#14521400	2
#14521335	2
#14521259	2
#14520797	2
#14520781	2
C₃₁×Q₈⋊C₈	2
#1645	31

subgroup

index

Q₈×Dic₆₂

C₃₁×Q₈⋊C₈

Subgroups — #16844934

Conjugacy classes of subgroups

Complements

Maximal subgroups