Group #16856224 — subgroups

Subgroups — #16856224

Order 3968.

356 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 141 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	#14521605	C₂	1	2	yes	maximal
2	#14521453	C₂	1	2	yes	maximal, complemented
3	#14521451	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521448	C₂	1	2	yes	maximal, complemented
5	#14521446	C₂	1	2	yes	maximal, complemented
6	#14521086	C₂	1	2	yes	maximal
7	#14521084	C₂	1	2	yes	maximal
8	C₄×C₈×D₃₁	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#499	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#125	C₂²	1	2	yes	—
11	C₂×C₈×D₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₈×D₃₁	C₂²	1	1	yes	characteristic
13	#11758473	—	2	4	no	—
14	#11758469	C₂²	1	4	yes	—
15	#11758465	C₂²	1	4	yes	—
16	#11758464	—	2	4	no	—
17	C₄²×D₃₁	C₂²	1	1	yes	characteristic
18	#11758447	C₂²	1	2	yes	—
19	C₃₁×#107	C₂²	1	2	yes	—
20	C₃₁×#106	C₂²	1	2	yes	—
21	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	#11758406	C₂²	1	2	yes	—
23	#11758405	C₂²	1	2	yes	—
24	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
25	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
26	#11758396	C₂²	1	4	yes	—
27	#11758395	C₂²	1	4	yes	—
28	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₃₁	D₄	1	2	yes	—
30	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
31	C₂×Dic₆₂	—	2	4	no	—
32	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₃₁	—	2	4	no	—
35	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
36	C₄²×C₃₁	C₂³	1	1	yes	characteristic
37	(C₂×D₃₁)⋊C₄	—	2	4	no	—
38	C₄⋊Dic₃₁	C₂³	1	4	yes	—
39	Dic₃₁⋊C₄	—	2	4	no	—
40	Dic₃₁⋊C₄	—	2	4	no	—
41	C₄×Dic₃₁	—	2	4	no	—
42	C₄×Dic₃₁	D₄	1	2	yes	—
43	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
44	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
45	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
46	C₈×D₃₁	Q₈	1	4	yes	complemented
47	C₈×D₃₁	Q₈	1	4	yes	complemented
48	C₂²×D₃₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₃₁	—	2	4	no	—
50	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
52	C₂×Dic₃₁	—	2	4	no	—
53	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₃₁	C₂×D₄	1	2	yes	—
55	C₄×D₃₁	—	2	4	no	—
56	C₄×D₃₁	C₂×Q₈	1	2	yes	—
57	C₄×D₃₁	C₂×Q₈	1	2	yes	—
58	Dic₆₂	—	4	8	no	—
59	C₈×C₃₁	C₂×Q₈	1	2	yes	—
60	C₈×C₃₁	C₂×Q₈	1	2	yes	—
61	C₃₁⋊C₈	C₂×Q₈	1	2	yes	—
62	C₃₁⋊C₈	C₂×Q₈	1	2	yes	—
63	#3160	—	31	1	no	maximal, Sylow
64	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
65	C₂×D₃₁	—	2	2	no	—
66	C₂×D₃₁	C₄⋊Q₈	1	2	yes	—
67	C₄×C₃₁	—	2	2	no	—
68	C₄×C₃₁	—	4	4	no	—
69	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
70	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
71	Dic₃₁	—	4	4	no	—
72	Dic₃₁	C₄⋊Q₈	1	2	yes	—
73	Dic₃₁	#135	1	4	yes	—
74	#533	—	31	2	no	—
75	#500	—	31	2	no	—
76	#499	—	31	2	no	—
77	#499	—	31	1	no	—
78	#499	D₃₁	1	1	yes	characteristic, complemented
79	#498	—	31	2	no	—
80	#427	—	31	2	no	—
81	#427	—	31	2	no	—
82	C₂×C₄×C₈	—	31	1	no	—
83	C₂×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
84	C₂×C₃₁	C₂×#135	1	2	yes	—
85	D₃₁	—	2	2	no	—
86	C₂²×C₈	—	31	1	no	—
87	C₂²×C₈	—	31	1	no	—
88	C₄⋊Q₈	—	31	4	no	—
89	#125	—	31	2	no	—
90	#125	C₂×D₃₁	1	2	yes	—
91	C₂²⋊Q₈	—	62	4	no	—
92	C₄×Q₈	—	62	4	no	—
93	#117	—	31	4	no	—
94	C₂×C₄²	—	31	1	no	—
95	#107	—	31	2	no	—
96	#107	—	31	4	no	—
97	#107	C₂×D₃₁	1	2	yes	—
98	#106	—	31	2	no	—
99	#106	—	31	4	no	—
100	#106	C₂×D₃₁	1	2	yes	—
101	C₄×C₈	—	31	1	no	—
102	C₄×C₈	—	31	1	no	—
103	C₄×C₈	—	31	1	no	—
104	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
105	C₃₁	#3160	1	1	yes	characteristic, complemented, Sylow
106	C₂×Q₈	—	62	4	no	—
107	C₂²×C₄	—	31	1	no	—
108	C₂²×C₄	—	31	2	no	—
109	C₂×C₈	—	31	1	no	—
110	C₂×C₈	—	31	1	no	—
111	C₂×C₈	—	31	4	no	—
112	C₂×C₈	—	31	4	no	—
113	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
114	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	31	4	no	—
117	C₄⋊C₄	—	62	4	no	—
118	C₄⋊C₄	—	62	4	no	—
119	C₄⋊C₄	C₂²×D₃₁	1	4	yes	—
120	C₂²⋊C₄	—	62	4	no	—
121	C₄²	—	31	2	no	—
122	C₄²	—	31	1	no	—
123	C₄²	—	62	4	no	—
124	C₄²	C₂²×D₃₁	1	1	yes	characteristic
125	C₂³	—	31	1	no	—
126	Q₈	—	124	8	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	31	1	no	—
129	C₂×C₄	—	31	2	no	—
130	C₂×C₄	—	31	2	no	—
131	C₂×C₄	—	31	2	no	—
132	C₂×C₄	—	62	4	no	—
133	C₂×C₄	—	62	4	no	—
134	C₂×C₄	D₄×D₃₁	1	2	yes	—
135	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
136	C₈	—	31	2	no	—
137	C₈	—	31	2	no	—
138	C₈	Q₈×D₃₁	1	2	yes	—
139	C₈	Q₈×D₃₁	1	2	yes	—
140	C₂²	—	31	2	no	—
141	C₂²	—	62	2	no	—
142	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	4	4	no	—
145	C₄	—	31	4	no	—
146	C₄	—	31	2	no	—
147	C₄	—	124	4	no	—
148	C₄	C₂×Q₈×D₃₁	1	1	yes	characteristic
149	C₄	C₂×Q₈×D₃₁	1	1	yes	characteristic
150	C₂	—	62	2	no	—
151	C₂	C₄⋊Q₈×D₃₁	1	1	yes	characteristic
152	C₂	#135×D₃₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521605

C₂

yes

maximal

#14521453

C₂

yes

maximal, complemented

#14521451

C₂

yes

maximal, characteristic, complemented

#14521448

C₂

yes

maximal, complemented

#14521446

C₂

yes

maximal, complemented

#14521086

C₂

yes

maximal

#14521084

C₂

yes

maximal

C₄×C₈×D₃₁

C₂

yes

maximal, characteristic

C₃₁×#499

C₂

yes

maximal, characteristic, complemented

C₃₁×#125

C₂²

yes

—

C₂×C₈×D₃₁

C₂²

yes

characteristic

C₂×C₈×D₃₁

C₂²

yes

characteristic

#11758473

—

#11758469

C₂²

yes

—

#11758465

C₂²

yes

—

#11758464

—

C₄²×D₃₁

C₂²

yes

characteristic

#11758447

C₂²

yes

—

C₃₁×#107

C₂²

yes

—

C₃₁×#106

C₂²

yes

—

C₄×C₈×C₃₁

C₂²

yes

characteristic

#11758406

C₂²

yes

—

#11758405

C₂²

yes

—

C₈×Dic₃₁

C₂²

yes

characteristic

C₈×Dic₃₁

C₂²

yes

characteristic

#11758396

C₂²

yes

—

#11758395

C₂²

yes

—

C₄×C₃₁⋊C₈

C₂²

yes

characteristic

C₂×C₄×D₃₁

D₄

yes

—

C₂×C₄×D₃₁

C₂³

yes

characteristic

C₂×Dic₆₂

—

C₂×C₈×C₃₁

C₂³

yes

characteristic

C₂×C₈×C₃₁

C₂³

yes

characteristic

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

C₂³

yes

—

C₄²×C₃₁

C₂³

yes

characteristic

(C₂×D₃₁)⋊C₄

—

C₄⋊Dic₃₁

C₂³

yes

—

Dic₃₁⋊C₄

—

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

C₄×Dic₃₁

D₄

yes

—

C₄×Dic₃₁

C₂³

yes

characteristic

C₂×C₃₁⋊C₈

C₂³

yes

characteristic

C₂×C₃₁⋊C₈

C₂³

yes

characteristic

C₈×D₃₁

Q₈

yes

complemented

C₈×D₃₁

Q₈

yes

complemented

C₂²×D₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

C₂×D₄

yes

characteristic

C₂×Dic₃₁

C₂×D₄

yes

—

C₄×D₃₁

—

C₄×D₃₁

C₂×Q₈

yes

—

C₄×D₃₁

C₂×Q₈

yes

—

Dic₆₂

—

C₈×C₃₁

C₂×Q₈

yes

—

C₈×C₃₁

C₂×Q₈

yes

—

C₃₁⋊C₈

C₂×Q₈

yes

—

C₃₁⋊C₈

C₂×Q₈

yes

—

#3160

—

maximal, Sylow

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

C₄⋊Q₈

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂²×Q₈

yes

characteristic

C₄×C₃₁

C₂²×Q₈

yes

characteristic

Dic₃₁

—

Dic₃₁

C₄⋊Q₈

yes

—

Dic₃₁

#135

yes

—

—

—

—

—

D₃₁

yes

characteristic, complemented

#498

—

#427

—

#427

—

C₂×C₄×C₈

—

C₂×C₃₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₃₁

C₂×#135

yes

—

D₃₁

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

#125

—

#125

C₂×D₃₁

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

#107

—

#107

—

#107

C₂×D₃₁

yes

—

#106

—

#106

—

100

#106

C₂×D₃₁

yes

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

C₂×D₃₁

yes

characteristic

105

C₃₁

#3160

yes

characteristic, complemented, Sylow

106

C₂×Q₈

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₂²×D₃₁

yes

characteristic

114

C₂×C₈

C₂²×D₃₁

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

C₂²×D₃₁

yes

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₄²

C₂²×D₃₁

yes

characteristic

125

C₂³

—

126

Q₈

—

124

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

D₄×D₃₁

yes

—

135

C₂×C₄

C₂³×D₃₁

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

Q₈×D₃₁

yes

—

139

C₈

Q₈×D₃₁

yes

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂×D₄×D₃₁

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

124

—

148

C₄

C₂×Q₈×D₃₁

yes

characteristic

149

C₄

C₂×Q₈×D₃₁

yes

characteristic

150

C₂

—

151

C₂

C₄⋊Q₈×D₃₁

yes

characteristic

152

C₂

#135×D₃₁

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#499	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂²×D₃₁	C₂×D₄
hypercenter	#499	D₃₁
hypocenter	C₃₁	#3160
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

subgroup

G/H

center

C₂²

C₂×D₄×D₃₁

commutator, cosocle

C₂×C₄×C₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₃₁×#499

C₂

socle

C₂²×C₃₁

C₂²×D₄

prime radical

C₂²×D₃₁

C₂×D₄

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #499

N	H ≅ G/N	type
2. #14521453	C₂	semidirect
3. #14521451	C₂	semidirect
4. #14521448	C₂	semidirect
5. #14521446	C₂	semidirect
9. C₃₁×#499	C₂	semidirect
46. C₈×D₃₁	Q₈	semidirect
47. C₈×D₃₁	Q₈	semidirect
78. #499	D₃₁	semidirect
105. C₃₁	#3160	semidirect

H ≅ G/N

type

2. #14521453

C₂

semidirect

3. #14521451

C₂

semidirect

4. #14521448

C₂

semidirect

5. #14521446

C₂

semidirect

9. C₃₁×#499

C₂

semidirect

46. C₈×D₃₁

Q₈

semidirect

47. C₈×D₃₁

Q₈

semidirect

78. #499

D₃₁

semidirect

105. C₃₁

#3160

semidirect

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

subgroup	index
#14521605	2
#14521453	2
#14521451	2
#14521448	2
#14521446	2
#14521086	2
#14521084	2
C₄×C₈×D₃₁	2
C₃₁×#499	2
#3160	31

subgroup

index

C₄×C₈×D₃₁

C₃₁×#499

#3160

Subgroups — #16856224

Conjugacy classes of subgroups

Complements

Maximal subgroups