Group #30764866 — subgroups

Subgroups — #30764866

Order 17152.

290 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 147 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	#22562006	C₂	1	2	yes	maximal
2	C₈²×D₆₇	C₂	1	1	yes	maximal, characteristic
3	#22558127	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558125	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₈×D₆₇	C₄	1	1	yes	characteristic
7	C₄×C₈×D₆₇	C₄	1	1	yes	characteristic
8	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
9	Dic₆₇⋊C₁₆	C₄	1	4	yes	—
10	(C₂×D₆₇)⋊C₁₆	C₄	1	4	yes	—
11	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
12	C₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
13	C₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
14	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
16	C₈²×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₆₇	C₈	1	2	yes	—
18	C₂×C₈×D₆₇	C₈	1	2	yes	—
19	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
20	C₄²×D₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
22	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
24	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₈×Dic₆₇	C₈	1	2	yes	—
26	C₈×Dic₆₇	C₈	1	2	yes	—
27	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
28	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
32	C₂×C₄×D₆₇	C₄²	1	1	yes	characteristic
33	C₂×C₄×D₆₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×D₆₇	C₂×C₈	1	1	yes	characteristic
35	C₂×C₆₇⋊C₈	C₄²	1	2	yes	—
36	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
37	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
38	C₈×D₆₇	—	2	4	no	—
39	C₈×D₆₇	—	2	4	no	—
40	C₈×D₆₇	—	2	4	no	—
41	C₄×Dic₆₇	C₄²	1	1	yes	characteristic
42	C₄×Dic₆₇	C₂×C₈	1	1	yes	characteristic
43	C₄×Dic₆₇	C₂×C₈	1	1	yes	characteristic
44	C₆₇⋊C₁₆	—	2	4	no	—
45	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
48	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
49	C₁₆×C₆₇	—	2	4	no	—
50	C₂²×D₆₇	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
53	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
54	C₂×Dic₆₇	C₄×C₈	1	1	yes	characteristic
55	C₂×Dic₆₇	C₄×C₈	1	1	yes	characteristic
56	C₂×Dic₆₇	C₄×C₈	1	1	yes	characteristic
57	C₄×D₆₇	—	2	2	no	—
58	C₄×D₆₇	—	2	2	no	—
59	C₄×D₆₇	M₃₂	1	4	yes	—
60	C₈×C₆₇	—	2	2	no	—
61	C₈×C₆₇	M₃₂	1	4	yes	—
62	C₈×C₆₇	#131	1	4	yes	—
63	C₆₇⋊C₈	—	2	2	no	—
64	C₆₇⋊C₈	—	2	2	no	—
65	C₆₇⋊C₈	—	2	2	no	—
66	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
67	C₂×D₆₇	—	2	1	no	—
68	C₂×D₆₇	—	2	1	no	—
69	C₂×D₆₇	#346	1	2	yes	—
70	C₄×C₆₇	#405	1	2	yes	—
71	C₄×C₆₇	C₂×M₃₂	1	2	yes	—
72	C₄×C₆₇	C₂×M₃₂	1	2	yes	—
73	Dic₆₇	—	2	1	no	—
74	Dic₆₇	—	2	1	no	—
75	Dic₆₇	#346	1	2	yes	—
76	#11411	—	67	1	no	maximal, Sylow
77	C₂×C₆₇	#1450	1	1	yes	characteristic
78	C₂×C₆₇	C₂×#346	1	1	yes	characteristic
79	C₂×C₆₇	C₄×M₃₂	1	1	yes	characteristic
80	D₆₇	—	2	2	no	—
81	#2162	—	67	2	no	—
82	C₂×C₈²	—	67	1	no	—
83	#1312	—	67	1	no	—
84	#1312	D₆₇	1	1	yes	characteristic, complemented
85	#1311	—	67	2	no	—
86	C₆₇	#11411	1	1	yes	characteristic, complemented, Sylow
87	C₂×C₄×C₈	—	67	1	no	—
88	C₂×C₄×C₈	—	67	1	no	—
89	C₂×C₄×C₈	—	67	1	no	—
90	C₄⋊C₁₆	—	67	4	no	—
91	C₂²⋊C₁₆	—	67	4	no	—
92	C₄×C₁₆	—	67	2	no	—
93	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
94	C₈²	—	67	1	no	—
95	C₈²	—	67	1	no	—
96	C₈²	—	67	1	no	—
97	C₈²	C₂×D₆₇	1	1	yes	characteristic
98	C₂²×C₈	—	67	2	no	—
99	C₂²×C₈	—	67	2	no	—
100	C₂²×C₈	—	67	2	no	—
101	C₂×C₄²	—	67	1	no	—
102	C₂×C₁₆	—	67	4	no	—
103	C₂×C₁₆	C₄×D₆₇	1	4	yes	—
104	C₄×C₈	—	67	2	no	—
105	C₄×C₈	—	67	2	no	—
106	C₄×C₈	—	67	2	no	—
107	C₄×C₈	—	67	1	no	—
108	C₄×C₈	—	67	1	no	—
109	C₄×C₈	—	67	1	no	—
110	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
111	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
112	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
113	C₂²×C₄	—	67	1	no	—
114	C₂²×C₄	—	67	1	no	—
115	C₂²×C₄	—	67	1	no	—
116	C₂×C₈	—	67	2	no	—
117	C₂×C₈	—	67	2	no	—
118	C₂×C₈	—	67	2	no	—
119	C₂×C₈	—	134	4	no	—
120	C₂×C₈	—	134	4	no	—
121	C₂×C₈	—	134	4	no	—
122	C₂×C₈	C₈×D₆₇	1	2	yes	—
123	C₂×C₈	C₈×D₆₇	1	2	yes	—
124	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
125	C₄²	—	67	1	no	—
126	C₄²	—	67	1	no	—
127	C₄²	—	67	1	no	—
128	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
129	C₁₆	—	2	4	no	—
130	C₁₆	—	134	4	no	—
131	C₂³	—	67	1	no	—
132	C₂×C₄	—	67	1	no	—
133	C₂×C₄	—	67	4	no	—
134	C₂×C₄	—	67	1	no	—
135	C₂×C₄	—	67	1	no	—
136	C₂×C₄	—	134	2	no	—
137	C₂×C₄	—	134	2	no	—
138	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
139	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
140	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	134	2	no	—
143	C₈	—	134	2	no	—
144	C₈	—	134	2	no	—
145	C₈	#14523982	1	4	yes	—
146	C₈	#14524051	1	4	yes	—
147	C₂²	—	67	2	no	—
148	C₂²	—	134	1	no	—
149	C₂²	—	134	1	no	—
150	C₂²	C₄×C₈×D₆₇	1	1	yes	characteristic
151	C₄	—	67	2	no	—
152	C₄	—	134	1	no	—
153	C₄	—	134	1	no	—
154	C₄	#16863353	1	2	yes	—
155	C₄	M₃₂×D₆₇	1	2	yes	—
156	C₄	C₂×#14523982	1	2	yes	—
157	C₂	—	134	2	no	—
158	C₂	#22558494	1	1	yes	characteristic
159	C₂	#346×D₆₇	1	1	yes	characteristic
160	C₂	C₄×#14523982	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562006

C₂

yes

maximal

C₈²×D₆₇

C₂

yes

maximal, characteristic

#22558127

C₂

yes

maximal, characteristic, complemented

#22558125

C₂

yes

maximal, complemented

C₆₇×#1312

C₂

yes

maximal, characteristic, complemented

C₄×C₈×D₆₇

C₄

yes

characteristic

C₄×C₈×D₆₇

C₄

yes

characteristic

C₄×C₈×D₆₇

C₂²

yes

characteristic

Dic₆₇⋊C₁₆

C₄

yes

—

(C₂×D₆₇)⋊C₁₆

C₄

yes

—

C₄×C₆₇⋊C₁₆

C₂²

yes

—

C₈×C₆₇⋊C₈

C₄

yes

characteristic

C₈×C₆₇⋊C₈

C₄

yes

characteristic

C₈×C₆₇⋊C₈

C₂²

yes

characteristic

C₄×C₁₆×C₆₇

C₂²

yes

—

C₈²×C₆₇

C₂²

yes

characteristic

C₂×C₈×D₆₇

C₈

yes

—

C₂×C₈×D₆₇

C₈

yes

—

C₂×C₈×D₆₇

C₂×C₄

yes

—

C₄²×D₆₇

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

—

C₄×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₄×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₄×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₈×Dic₆₇

C₈

yes

—

C₈×Dic₆₇

C₈

yes

—

C₈×Dic₆₇

C₂×C₄

yes

—

C₂×C₁₆×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄×D₆₇

C₄²

yes

characteristic

C₂×C₄×D₆₇

C₂×C₈

yes

characteristic

C₂×C₄×D₆₇

C₂×C₈

yes

characteristic

C₂×C₆₇⋊C₈

C₄²

yes

—

C₂×C₆₇⋊C₈

C₂×C₈

yes

—

C₂×C₆₇⋊C₈

C₂×C₈

yes

—

C₈×D₆₇

—

C₈×D₆₇

—

C₈×D₆₇

—

C₄×Dic₆₇

C₄²

yes

characteristic

C₄×Dic₆₇

C₂×C₈

yes

characteristic

C₄×Dic₆₇

C₂×C₈

yes

characteristic

C₆₇⋊C₁₆

—

C₂×C₈×C₆₇

C₂×C₈

yes

—

C₂×C₈×C₆₇

C₂×C₈

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₁₆×C₆₇

—

C₂²×D₆₇

C₄×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₈

yes

characteristic

C₂×Dic₆₇

C₄×C₈

yes

characteristic

C₂×Dic₆₇

C₄×C₈

yes

characteristic

C₂×Dic₆₇

C₄×C₈

yes

characteristic

C₄×D₆₇

—

C₄×D₆₇

—

C₄×D₆₇

M₃₂

yes

—

C₈×C₆₇

—

C₈×C₆₇

M₃₂

yes

—

C₈×C₆₇

#131

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₂×C₄×C₈

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

#346

yes

—

C₄×C₆₇

#405

yes

—

C₄×C₆₇

C₂×M₃₂

yes

—

C₄×C₆₇

C₂×M₃₂

yes

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

#346

yes

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

C₄×M₃₂

yes

characteristic

D₆₇

—

#2162

—

C₂×C₈²

—

#1312

—

#1312

D₆₇

yes

characteristic, complemented

#1311

—

C₆₇

#11411

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₇

yes

—

C₈²

—

C₈²

—

C₈²

—

C₈²

C₂×D₆₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

100

C₂²×C₈

—

101

C₂×C₄²

—

102

C₂×C₁₆

—

103

C₂×C₁₆

C₄×D₆₇

yes

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

C₄×D₆₇

yes

characteristic

111

C₄×C₈

C₄×D₆₇

yes

characteristic

112

C₄×C₈

C₂²×D₆₇

yes

characteristic

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

134

—

120

C₂×C₈

—

134

—

121

C₂×C₈

—

134

—

122

C₂×C₈

C₈×D₆₇

yes

—

123

C₂×C₈

C₈×D₆₇

yes

—

124

C₂×C₈

C₂×C₄×D₆₇

yes

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

C₂×C₄×D₆₇

yes

characteristic

129

C₁₆

—

130

C₁₆

—

134

—

131

C₂³

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

134

—

137

C₂×C₄

—

134

—

138

C₂×C₄

C₄²×D₆₇

yes

characteristic

139

C₂×C₄

C₂×C₈×D₆₇

yes

characteristic

140

C₂×C₄

C₂×C₈×D₆₇

yes

characteristic

141

C₈

—

142

C₈

—

134

—

143

C₈

—

134

—

144

C₈

—

134

—

145

C₈

#14523982

yes

—

146

C₈

#14524051

yes

—

147

C₂²

—

148

C₂²

—

134

—

149

C₂²

—

134

—

150

C₂²

C₄×C₈×D₆₇

yes

characteristic

151

C₄

—

152

C₄

—

134

—

153

C₄

—

134

—

154

C₄

#16863353

yes

—

155

C₄

M₃₂×D₆₇

yes

—

156

C₄

C₂×#14523982

yes

—

157

C₂

—

134

—

158

C₂

#22558494

yes

characteristic

159

C₂

#346×D₆₇

yes

characteristic

160

C₂

C₄×#14523982

yes

characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₆₇
commutator, socle	C₂²×C₆₇	C₂×C₄×C₈
Fitting	C₆₇×#1312	C₂
prime radical	C₂²×D₆₇	C₄×C₈
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#1312	D₆₇
hypocenter	C₆₇	#11411
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄×C₈ ◁ #1312

subgroup

G/H

center, Frattini

C₄×C₈

C₂²×D₆₇

commutator, socle

C₂²×C₆₇

C₂×C₄×C₈

Fitting

C₆₇×#1312

C₂

prime radical

C₂²×D₆₇

C₄×C₈

cosocle

C₄×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₄×C₈ ◁ #1312

N	H ≅ G/N	type
3. #22558127	C₂	semidirect
4. #22558125	C₂	semidirect
5. C₆₇×#1312	C₂	semidirect
84. #1312	D₆₇	semidirect
86. C₆₇	#11411	semidirect

H ≅ G/N

type

3. #22558127

C₂

semidirect

4. #22558125

C₂

semidirect

5. C₆₇×#1312

C₂

semidirect

84. #1312

D₆₇

semidirect

86. C₆₇

#11411

semidirect

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

subgroup	index
#22562006	2
C₈²×D₆₇	2
#22558127	2
#22558125	2
C₆₇×#1312	2
#11411	67

subgroup

index

C₈²×D₆₇

C₆₇×#1312

Subgroups — #30764866

Conjugacy classes of subgroups

Complements

Maximal subgroups