Group #16848460 — subgroups

Subgroups — #16848460

Order 3968.

460 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 237 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	#14520956	C₂	1	4	yes	maximal
2	#14520948	C₂	1	2	yes	maximal
3	#14520935	C₂	1	4	yes	maximal
4	#14520915	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₄×Dic₃₁	C₂	1	2	yes	maximal
6	#14520895	C₂	1	1	yes	maximal, characteristic
7	C₂×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₃₁	C₂²	1	2	yes	—
9	C₃₁×C₄⋊Q₈	C₄	1	8	yes	complemented
10	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
12	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₃₁	C₂²	1	4	yes	—
15	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₃₁	C₂²	1	4	yes	—
17	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
18	C₃₁⋊#95	C₂²	1	2	yes	—
19	C₃₁⋊#95	C₂²	1	2	yes	—
20	C₃₁⋊#95	C₂²	1	4	yes	—
21	C₃₁⋊#95	C₂²	1	2	yes	—
22	C₃₁⋊#95	C₂²	1	2	yes	—
23	C₃₁⋊#95	C₂²	1	4	yes	—
24	C₂×Q₈×C₃₁	—	4	4	no	—
25	C₂×Q₈×C₃₁	C₂×C₄	1	8	yes	—
26	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
29	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
30	C₂²×Dic₃₁	C₂³	1	2	yes	—
31	C₂²×Dic₃₁	C₂³	1	4	yes	—
32	C₂²×Dic₃₁	C₂³	1	2	yes	—
33	C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
34	C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
35	C₄²×C₃₁	C₂×C₄	1	4	yes	—
36	C₄⋊Dic₃₁	—	2	4	no	—
37	C₄⋊Dic₃₁	—	4	4	no	—
38	C₄×Dic₃₁	—	2	2	no	—
39	C₄×Dic₃₁	—	4	4	no	—
40	C₄×Dic₃₁	Q₈	1	4	yes	complemented
41	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
42	Q₈×C₃₁	—	4	8	no	—
43	C₂×C₄×C₃₁	—	2	4	no	—
44	C₂×C₄×C₃₁	—	2	4	no	—
45	C₂×C₄×C₃₁	—	2	2	no	—
46	C₂×C₄×C₃₁	—	2	2	no	—
47	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₃₁	C₂²×C₄	1	8	yes	—
50	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₃₁	#41	1	2	yes	—
53	C₂×Dic₃₁	—	2	2	no	—
54	C₂×Dic₃₁	—	2	2	no	—
55	C₂×Dic₃₁	—	2	2	no	—
56	C₂×Dic₃₁	—	2	4	no	—
57	C₂×Dic₃₁	—	2	4	no	—
58	C₂×Dic₃₁	—	2	2	no	—
59	C₂×Dic₃₁	—	2	2	no	—
60	C₂×Dic₃₁	—	2	4	no	—
61	C₂×Dic₃₁	C₂×Q₈	1	4	yes	—
62	#2331	—	31	1	no	maximal, Sylow
63	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
64	C₂²×C₃₁	C₂²×Q₈	1	1	yes	characteristic
65	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
66	C₂²×C₃₁	H(5,𝔽₂)	1	2	yes	—
67	C₂²×C₃₁	2₋¹⁺⁴	1	2	yes	—
68	C₄×C₃₁	—	2	8	no	—
69	C₄×C₃₁	—	2	2	no	—
70	C₄×C₃₁	—	2	2	no	—
71	C₄×C₃₁	C₄×Q₈	1	4	yes	—
72	Dic₃₁	—	2	4	no	—
73	Dic₃₁	—	4	2	no	—
74	Dic₃₁	—	4	4	no	—
75	C₂×C₄⋊Q₈	D₃₁	1	1	yes	characteristic
76	#391	—	31	4	no	—
77	#389	—	31	2	no	—
78	#387	—	31	4	no	—
79	#382	—	31	1	no	—
80	C₄×C₄⋊C₄	—	31	2	no	—
81	#376	—	31	1	no	—
82	C₂×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
83	C₂×C₃₁	#518	1	1	yes	characteristic
84	C₂×C₃₁	#519	1	1	yes	characteristic
85	C₂×C₃₁	#542	1	2	yes	—
86	C₂×C₃₁	#544	1	2	yes	—
87	C₂²×Q₈	C₂×D₃₁	1	2	yes	—
88	C₄⋊Q₈	Dic₃₁	1	8	yes	complemented
89	C₂×C₄⋊C₄	—	31	2	no	—
90	C₂×C₄⋊C₄	—	31	4	no	—
91	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
92	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
93	C₂×C₄²	—	31	1	no	—
94	C₂×C₄²	—	31	1	no	—
95	C₂×C₄²	—	31	4	no	—
96	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
97	#95	—	31	2	no	—
98	#95	—	31	2	no	—
99	#95	—	31	4	no	—
100	#95	—	31	2	no	—
101	#95	—	31	2	no	—
102	#95	—	31	4	no	—
103	C₃₁	#2331	1	1	yes	characteristic, complemented, Sylow
104	C₂×Q₈	—	4	4	no	—
105	C₂×Q₈	C₂×Dic₃₁	1	8	yes	—
106	C₂²×C₄	—	31	2	no	—
107	C₂²×C₄	—	31	4	no	—
108	C₂²×C₄	—	31	2	no	—
109	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
111	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
112	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
113	C₄⋊C₄	—	62	4	no	—
114	C₄⋊C₄	—	124	4	no	—
115	C₄⋊C₄	C₂×Dic₃₁	1	8	yes	—
116	C₄⋊C₄	C₂×Dic₃₁	1	8	yes	—
117	C₄²	—	31	4	no	—
118	C₄²	—	62	2	no	—
119	C₄²	—	124	4	no	—
120	C₄²	C₂×Dic₃₁	1	4	yes	—
121	C₂³	C₂³×D₃₁	1	1	yes	characteristic
122	Q₈	—	4	8	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	31	4	no	—
128	C₂×C₄	—	62	2	no	—
129	C₂×C₄	—	62	2	no	—
130	C₂×C₄	—	62	4	no	—
131	C₂×C₄	—	62	2	no	—
132	C₂×C₄	—	62	2	no	—
133	C₂×C₄	—	62	4	no	—
134	C₂×C₄	—	62	4	no	—
135	C₂×C₄	—	62	2	no	—
136	C₂×C₄	Q₈×D₃₁	1	2	yes	—
137	C₂×C₄	#92499	1	2	yes	—
138	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
139	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
140	C₂×C₄	C₂²×Dic₃₁	1	8	yes	—
141	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
142	C₂²	#11758561	1	2	yes	—
143	C₂²	C₂×Q₈×D₃₁	1	1	yes	characteristic
144	C₂²	C₂×#92499	1	1	yes	characteristic
145	C₂²	#11758564	1	2	yes	—
146	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	8	no	—
149	C₄	—	2	2	no	—
150	C₄	—	62	4	no	—
151	C₄	—	124	4	no	—
152	C₄	—	124	2	no	—
153	C₄	Q₈×Dic₃₁	1	4	yes	—
154	C₂	C₂×Q₈×Dic₃₁	1	1	yes	characteristic
155	C₂	#14521524	1	1	yes	characteristic
156	C₂	#14521527	1	1	yes	characteristic
157	C₂	#14521673	1	2	yes	—
158	C₂	#14521676	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14520956

C₂

yes

maximal

#14520948

C₂

yes

maximal

#14520935

C₂

yes

maximal

#14520915

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₃₁

C₂

yes

maximal

#14520895

C₂

yes

maximal, characteristic

C₂×C₃₁×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂²×Q₈×C₃₁

C₂²

yes

—

C₃₁×C₄⋊Q₈

C₄

yes

complemented

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

C₂×C₄×Dic₃₁

C₂²

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

C₃₁⋊#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

Q₈

yes

complemented

C₂³×C₃₁

C₂⁴

yes

characteristic

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₂×Q₈

yes

—

#2331

—

maximal, Sylow

C₂²×C₃₁

C₂³×C₄

yes

characteristic

C₂²×C₃₁

C₂²×Q₈

yes

characteristic

C₂²×C₃₁

C₂×#41

yes

characteristic

C₂²×C₃₁

H(5,𝔽₂)

yes

—

C₂²×C₃₁

2₋¹⁺⁴

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₄×Q₈

yes

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

C₂×C₄⋊Q₈

D₃₁

yes

—

—

—

—

C₄×C₄⋊C₄

—

#376

—

C₂×C₃₁

C₂×C₄×Q₈

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂×C₃₁

#544

yes

—

C₂²×Q₈

C₂×D₃₁

yes

—

C₄⋊Q₈

Dic₃₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₁

yes

—

—

—

100

#95

—

101

#95

—

102

#95

—

103

C₃₁

#2331

yes

characteristic, complemented, Sylow

104

C₂×Q₈

—

105

C₂×Q₈

C₂×Dic₃₁

yes

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂²×D₃₁

yes

characteristic

110

C₂²×C₄

C₂²×D₃₁

yes

characteristic

111

C₂²×C₄

C₂²×D₃₁

yes

—

112

C₂²×C₄

C₂²×D₃₁

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

124

—

115

C₄⋊C₄

C₂×Dic₃₁

yes

—

116

C₄⋊C₄

C₂×Dic₃₁

yes

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

124

—

120

C₄²

C₂×Dic₃₁

yes

—

121

C₂³

C₂³×D₃₁

yes

characteristic

122

Q₈

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

Q₈×D₃₁

yes

—

137

C₂×C₄

#92499

yes

—

138

C₂×C₄

C₂²×Dic₃₁

yes

—

139

C₂×C₄

C₂²×Dic₃₁

yes

—

140

C₂×C₄

C₂²×Dic₃₁

yes

—

141

C₂×C₄

C₂²×Dic₃₁

yes

—

142

C₂²

#11758561

yes

—

143

C₂²

C₂×Q₈×D₃₁

yes

characteristic

144

C₂²

C₂×#92499

yes

characteristic

145

C₂²

#11758564

yes

—

146

C₂²

C₂³×Dic₃₁

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

124

—

152

C₄

—

124

—

153

C₄

Q₈×Dic₃₁

yes

—

154

C₂

C₂×Q₈×Dic₃₁

yes

characteristic

155

C₂

#14521524

yes

characteristic

156

C₂

#14521527

yes

characteristic

157

C₂

#14521673

yes

—

158

C₂

#14521676

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₁
commutator	C₂²×C₃₁	C₂³×C₄
Fitting	C₂×C₃₁×C₄⋊Q₈	C₂
socle, prime radical, cosocle	C₂³×C₃₁	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₃₁
hypocenter	C₃₁	#2331
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃₁

commutator

C₂²×C₃₁

C₂³×C₄

Fitting

C₂×C₃₁×C₄⋊Q₈

C₂

socle, prime radical, cosocle

C₂³×C₃₁

C₂⁴

hypercenter

C₂×C₄⋊Q₈

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
9. C₃₁×C₄⋊Q₈	C₄	semidirect
40. C₄×Dic₃₁	Q₈	semidirect
88. C₄⋊Q₈	Dic₃₁	semidirect
103. C₃₁	#2331	semidirect

H ≅ G/N

type

9. C₃₁×C₄⋊Q₈

C₄

semidirect

40. C₄×Dic₃₁

Q₈

semidirect

88. C₄⋊Q₈

Dic₃₁

semidirect

103. C₃₁

#2331

semidirect

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

subgroup	index
#14520956	2
#14520948	2
#14520935	2
#14520915	2
C₄⋊C₄×Dic₃₁	2
#14520895	2
C₂×C₃₁×C₄⋊Q₈	2
#2331	31

subgroup

index

C₄⋊C₄×Dic₃₁

C₂×C₃₁×C₄⋊Q₈

Subgroups — #16848460

Conjugacy classes of subgroups

Complements

Maximal subgroups