Group #31011333 — subgroups

Subgroups — #31011333

Order 17152.

482 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 219 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	#22570723	C₂	1	2	yes	maximal
2	#22562495	C₂	1	8	yes	maximal, complemented
3	#22561993	C₂	1	4	yes	maximal
4	C₂×#346×C₆₇	C₂	1	1	yes	maximal, characteristic
5	#16863953	—	2	2	no	—
6	#16863801	C₂²	1	4	yes	—
7	#16863798	—	2	2	no	—
8	#16863798	C₂²	1	4	yes	—
9	#16863794	—	2	2	no	—
10	#16863434	—	2	2	no	—
11	#16863434	C₂²	1	4	yes	—
12	#16863224	C₂²	1	8	yes	—
13	#16863222	C₂²	1	8	yes	—
14	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	#346×C₆₇	C₂²	1	4	yes	—
17	#14524043	—	2	4	no	—
18	#14524035	—	2	4	no	—
19	#14524022	—	4	4	no	—
20	C₄×Dic₁₃₄	—	4	4	no	—
21	#14524002	—	2	4	no	—
22	#14523977	—	2	4	no	—
23	#14523977	C₂³	1	8	yes	—
24	#14523975	—	2	4	no	—
25	#14523975	—	2	4	no	—
26	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
29	C₂×C₁₆×C₆₇	Q₈	1	8	yes	complemented
30	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
31	C₄×C₈×C₆₇	D₄	1	2	yes	—
32	C₄×C₈×C₆₇	C₂³	1	2	yes	—
33	C₂×Dic₁₃₄	—	4	4	no	—
34	C₄⋊Dic₆₇	—	2	8	no	—
35	C₄⋊Dic₆₇	—	4	4	no	—
36	Dic₆₇⋊C₄	—	4	8	no	—
37	C₂²⋊Dic₆₇	—	4	4	no	—
38	C₄×Dic₆₇	—	4	4	no	—
39	C₂²×C₄×C₆₇	D₈	1	2	yes	—
40	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
42	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
44	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
45	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
46	C₄²×C₆₇	D₈	1	2	yes	—
47	C₄²×C₆₇	C₂×D₄	1	2	yes	—
48	C₁₆×C₆₇	C₂×Q₈	1	8	yes	—
49	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₆₇	—	4	2	no	—
51	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
52	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
53	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
55	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
56	C₂×Dic₆₇	—	4	8	no	—
57	Dic₁₃₄	—	8	8	no	—
58	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
59	C₈×C₆₇	C₂×Q₁₆	1	4	yes	—
60	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
61	C₂²×C₆₇	—	2	2	no	—
62	C₂²×C₆₇	#500	1	2	yes	—
63	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
64	C₄×C₆₇	—	2	4	no	—
65	C₄×C₆₇	#500	1	2	yes	—
66	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
67	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
68	Dic₆₇	—	8	8	no	—
69	#33728	—	67	1	no	maximal, Sylow
70	C₂×C₆₇	—	2	2	no	—
71	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
72	C₂×C₆₇	#3414	1	2	yes	—
73	#3160	—	67	2	no	—
74	#2255	—	67	8	no	—
75	#2158	—	67	4	no	—
76	C₂×#346	D₆₇	1	1	yes	characteristic
77	C₆₇	#33728	1	1	yes	characteristic, complemented, Sylow
78	#533	—	134	2	no	—
79	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
80	#500	—	67	4	no	—
81	#499	—	67	4	no	—
82	#499	—	134	2	no	—
83	#498	—	134	2	no	—
84	#427	—	67	4	no	—
85	#427	—	134	2	no	—
86	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
87	#367	—	67	8	no	—
88	#366	—	67	8	no	—
89	#346	C₂×D₆₇	1	4	yes	—
90	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
91	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
92	C₄⋊Q₈	—	134	4	no	—
93	#125	—	134	4	no	—
94	C₂²⋊Q₈	—	268	4	no	—
95	C₄×Q₈	—	268	4	no	—
96	#117	—	134	4	no	—
97	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
98	C₂×C₁₆	Dic₁₃₄	1	8	yes	complemented
99	C₂×C₁₆	C₂²×D₆₇	1	4	yes	—
100	#107	—	67	8	no	—
101	#107	—	134	4	no	—
102	#106	—	134	4	no	—
103	#106	—	134	4	no	—
104	C₄×C₈	D₂₆₈	1	2	yes	—
105	C₄×C₈	C₂²×D₆₇	1	2	yes	—
106	C₂×Q₈	—	268	4	no	—
107	C₂²×C₄	D₅₃₆	1	2	yes	—
108	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
109	C₂×C₈	Dic₂₆₈	1	4	yes	—
110	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
111	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
112	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
113	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
114	C₄⋊C₄	—	268	8	no	—
115	C₄⋊C₄	—	268	4	no	—
116	C₄⋊C₄	—	134	8	no	—
117	C₂²⋊C₄	—	268	4	no	—
118	C₄²	—	268	4	no	—
119	C₄²	D₅₃₆	1	2	yes	—
120	C₄²	C₂×D₂₆₈	1	2	yes	—
121	C₁₆	C₂×Dic₁₃₄	1	8	yes	—
122	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
123	Q₈	—	536	8	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	268	8	no	—
126	C₂×C₄	#14524043	1	2	yes	—
127	C₂×C₄	C₂×D₅₃₆	1	4	yes	—
128	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
129	C₂×C₄	C₂×Dic₂₆₈	1	2	yes	—
130	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
131	C₈	#14524043	1	2	yes	—
132	C₈	C₂×Dic₂₆₈	1	4	yes	—
133	C₈	C₂²×Dic₁₃₄	1	2	yes	—
134	C₂²	—	2	2	no	—
135	C₂²	#16863801	1	2	yes	—
136	C₂²	C₂²×D₅₃₆	1	1	yes	characteristic
137	C₄	—	2	4	no	—
138	C₄	—	536	8	no	—
139	C₄	#16863801	1	2	yes	—
140	C₄	C₂×#14524043	1	1	yes	characteristic
141	C₄	C₂²×Dic₂₆₈	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×#16863801	1	1	yes	characteristic
144	C₂	#22573345	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22570723

C₂

yes

maximal

#22562495

C₂

yes

maximal, complemented

#22561993

C₂

yes

maximal

C₂×#346×C₆₇

C₂

yes

maximal, characteristic

#16863953

—

#16863801

C₂²

yes

—

#16863798

—

#16863798

C₂²

yes

—

#16863794

—

#16863434

—

#16863434

C₂²

yes

—

#16863224

C₂²

yes

—

#16863222

C₂²

yes

—

C₂²×C₁₆×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#346×C₆₇

C₂²

yes

—

#14524043

—

#14524035

—

#14524022

—

C₄×Dic₁₃₄

—

#14524002

—

#14523977

—

#14523977

C₂³

yes

—

#14523975

—

#14523975

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₁₆×C₆₇

Q₈

yes

complemented

C₂×C₁₆×C₆₇

C₂³

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

C₂³

yes

—

C₂×Dic₁₃₄

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

D₈

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

Q₁₆

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×Q₈

yes

—

C₂×C₈×C₆₇

C₂⁴

yes

characteristic

C₄²×C₆₇

D₈

yes

—

C₄²×C₆₇

C₂×D₄

yes

—

C₁₆×C₆₇

C₂×Q₈

yes

—

C₂³×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄⋊Q₈

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₁₆

yes

—

C₂×C₄×C₆₇

C₂²×D₄

yes

characteristic

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

C₄⋊Q₈

yes

—

C₈×C₆₇

C₂×Q₁₆

yes

—

C₈×C₆₇

C₂²×Q₈

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

#500

yes

—

C₂²×C₆₇

C₂²×D₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

#500

yes

—

C₄×C₆₇

C₂×C₄⋊Q₈

yes

characteristic

C₄×C₆₇

C₂²×Q₁₆

yes

characteristic

Dic₆₇

—

#33728

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

C₂×#500

yes

characteristic

C₂×C₆₇

#3414

yes

—

—

—

—

D₆₇

yes

characteristic

C₆₇

#33728

yes

characteristic, complemented, Sylow

#533

—

134

—

C₂²×C₁₆

C₂×D₆₇

yes

—

#500

—

#499

—

#499

—

134

—

#498

—

134

—

#427

—

#427

—

134

—

C₂×C₄×C₈

C₂×D₆₇

yes

—

—

C₂×D₆₇

yes

—

C₂²×C₈

D₂₆₈

yes

characteristic

C₂²×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊Q₈

—

134

—

#125

—

134

—

C₂²⋊Q₈

—

268

—

C₄×Q₈

—

268

—

#117

—

134

—

C₂×C₄²

D₂₆₈

yes

characteristic

C₂×C₁₆

Dic₁₃₄

yes

complemented

C₂×C₁₆

C₂²×D₆₇

yes

—

100

#107

—

101

#107

—

134

—

102

#106

—

134

—

103

#106

—

134

—

104

C₄×C₈

D₂₆₈

yes

—

105

C₄×C₈

C₂²×D₆₇

yes

—

106

C₂×Q₈

—

268

—

107

C₂²×C₄

D₅₃₆

yes

—

108

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

109

C₂×C₈

Dic₂₆₈

yes

—

110

C₂×C₈

C₂×D₂₆₈

yes

characteristic

111

C₂×C₈

C₂×D₂₆₈

yes

—

112

C₂×C₈

C₂×Dic₁₃₄

yes

—

113

C₂×C₈

C₂³×D₆₇

yes

characteristic

114

C₄⋊C₄

—

268

—

115

C₄⋊C₄

—

268

—

116

C₄⋊C₄

—

134

—

117

C₂²⋊C₄

—

268

—

118

C₄²

—

268

—

119

C₄²

D₅₃₆

yes

—

120

C₄²

C₂×D₂₆₈

yes

—

121

C₁₆

C₂×Dic₁₃₄

yes

—

122

C₂³

C₂×D₅₃₆

yes

characteristic

123

Q₈

—

536

—

124

C₂×C₄

—

125

C₂×C₄

—

268

—

126

C₂×C₄

#14524043

yes

—

127

C₂×C₄

C₂×D₅₃₆

yes

—

128

C₂×C₄

C₂×D₅₃₆

yes

characteristic

129

C₂×C₄

C₂×Dic₂₆₈

yes

—

130

C₂×C₄

C₂²×D₂₆₈

yes

characteristic

131

C₈

#14524043

yes

—

132

C₈

C₂×Dic₂₆₈

yes

—

133

C₈

C₂²×Dic₁₃₄

yes

—

134

C₂²

—

135

C₂²

#16863801

yes

—

136

C₂²

C₂²×D₅₃₆

yes

characteristic

137

C₄

—

138

C₄

—

536

—

139

C₄

#16863801

yes

—

140

C₄

C₂×#14524043

yes

characteristic

141

C₄

C₂²×Dic₂₆₈

yes

characteristic

142

C₂

—

143

C₂

C₂×#16863801

yes

characteristic

144

C₂

#22573345

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₃₆
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#346×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂³×C₆₇	C₂×D₈
hypercenter	C₂×#346	D₆₇
hypocenter	C₆₇	#33728
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₅₃₆

commutator, cosocle

C₂×C₈×C₆₇

C₂⁴

Frattini

C₂×C₈

C₂³×D₆₇

Fitting

C₂×#346×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₈

prime radical

C₂³×C₆₇

C₂×D₈

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
2. #22562495	C₂	semidirect
29. C₂×C₁₆×C₆₇	Q₈	semidirect
77. C₆₇	#33728	semidirect
98. C₂×C₁₆	Dic₁₃₄	semidirect

H ≅ G/N

type

2. #22562495

C₂

semidirect

29. C₂×C₁₆×C₆₇

Q₈

semidirect

77. C₆₇

#33728

semidirect

98. C₂×C₁₆

Dic₁₃₄

semidirect

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

subgroup	index
#22570723	2
#22562495	2
#22561993	2
C₂×#346×C₆₇	2
#33728	67

subgroup

index

C₂×#346×C₆₇

Subgroups — #31011333

Conjugacy classes of subgroups

Complements

Maximal subgroups