Group #30742706 — subgroups

Subgroups — #30742706

Order 17152.

214 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 115 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	C₂×#16863172	C₂	1	1	yes	maximal, characteristic
2	#22558957	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1564	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₆₇⋊C₁₆	C₄	1	2	yes	—
5	#16863449	—	2	1	no	—
6	#16863172	—	2	1	no	—
7	#16863172	—	2	1	no	—
8	#16863149	—	2	1	no	—
9	#16863147	C₄	1	2	yes	—
10	#431×C₆₇	—	2	1	no	—
11	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	#335×C₆₇	—	2	1	no	—
13	#334×C₆₇	C₄	1	2	yes	—
14	C₂×C₆₇⋊C₁₆	—	2	2	no	—
15	C₂×C₆₇⋊C₁₆	—	2	2	no	—
16	C₂×C₆₇⋊C₁₆	C₈	1	4	yes	complemented
17	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
18	C₂²⋊(C₆₇⋊C₈)	—	4	1	no	—
19	C₄×C₆₇⋊C₈	—	4	1	no	—
20	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
23	C₄⋊C₈×C₆₇	—	2	2	no	—
24	C₂²⋊C₈×C₆₇	—	4	1	no	—
25	C₄×C₈×C₆₇	—	4	1	no	—
26	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
29	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
30	C₂×C₆₇⋊C₈	—	4	2	no	—
31	C₆₇⋊C₁₆	—	2	4	no	—
32	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₆₇	Q₁₆	1	1	yes	characteristic
35	C₂×C₈×C₆₇	—	2	2	no	—
36	C₂×C₈×C₆₇	—	4	2	no	—
37	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
42	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
43	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
46	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
47	C₂³×C₆₇	#107	1	1	yes	characteristic
48	C₂×C₄×C₆₇	—	4	1	no	—
49	C₂×C₄×C₆₇	—	4	1	no	—
50	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
51	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
52	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
53	C₂×C₄×C₆₇	#103	1	1	yes	characteristic
54	C₂×C₄×C₆₇	#103	1	1	yes	characteristic
55	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
56	C₂×C₄×C₆₇	#106	1	1	yes	characteristic
57	C₂×C₄×C₆₇	#108	1	2	yes	—
58	C₈×C₆₇	—	2	1	no	—
59	C₈×C₆₇	—	2	1	no	—
60	C₈×C₆₇	—	8	2	no	—
61	C₈×C₆₇	C₄×C₈	1	1	yes	characteristic
62	C₈×C₆₇	#97	1	1	yes	characteristic
63	C₈×C₆₇	C₂²⋊C₈	1	2	yes	—
64	C₆₇⋊C₈	—	8	2	no	—
65	C₂²×C₆₇	—	2	1	no	—
66	C₂²×C₆₇	—	2	1	no	—
67	C₂²×C₆₇	#335	1	2	yes	—
68	C₂²×C₆₇	#340	1	1	yes	characteristic
69	C₄×C₆₇	—	2	1	no	—
70	C₄×C₆₇	—	2	1	no	—
71	C₄×C₆₇	—	2	1	no	—
72	C₄×C₆₇	—	2	1	no	—
73	C₄×C₆₇	#334	1	2	yes	—
74	C₄×C₆₇	#336	1	1	yes	characteristic
75	C₄×C₆₇	#341	1	1	yes	characteristic
76	#7157	—	67	1	no	maximal, Sylow
77	C₂×C₆₇	—	2	2	no	—
78	C₂×C₆₇	#1277	1	1	yes	characteristic
79	C₂×C₆₇	#1384	1	1	yes	characteristic
80	C₂×C₆₇	#1385	1	1	yes	characteristic
81	C₂×#346	—	67	1	no	—
82	#1564	—	67	1	no	—
83	#1564	D₆₇	1	1	yes	characteristic
84	C₆₇	#7157	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₁₆	—	67	2	no	—
86	#431	—	2	1	no	—
87	#431	—	134	1	no	—
88	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
89	#346	—	134	1	no	—
90	#346	—	134	1	no	—
91	#335	—	2	1	no	—
92	#335	—	134	1	no	—
93	#334	—	67	2	no	—
94	#334	Dic₆₇	1	2	yes	—
95	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
96	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
97	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
98	C₂×C₁₆	—	67	4	no	—
99	C₂×C₁₆	—	134	2	no	—
100	C₂×C₁₆	—	134	2	no	—
101	C₄⋊C₈	—	2	2	no	—
102	C₄⋊C₈	—	134	2	no	—
103	C₂²⋊C₈	—	4	1	no	—
104	C₂²⋊C₈	—	268	1	no	—
105	C₄×C₈	—	4	1	no	—
106	C₄×C₈	—	268	1	no	—
107	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
108	C₄×C₈	D₂₆₈	1	1	yes	characteristic
109	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
110	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
111	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
112	C₂²×C₄	#11761613	1	1	yes	characteristic
113	C₂²×C₄	#11761618	1	1	yes	characteristic
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	268	2	no	—
117	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
118	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
119	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
120	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
121	C₂×C₈	C₈×D₆₇	1	2	yes	—
122	C₂×C₈	#11761611	1	2	yes	—
123	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
124	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
125	C₄²	#11761615	1	1	yes	characteristic
126	C₄²	#11761616	1	1	yes	characteristic
127	C₁₆	—	134	4	no	—
128	C₂³	#14523976	1	1	yes	characteristic
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
132	C₂×C₄	#14523963	1	1	yes	characteristic
133	C₂×C₄	#14523964	1	1	yes	characteristic
134	C₂×C₄	#14523966	1	1	yes	characteristic
135	C₂×C₄	#14523967	1	1	yes	characteristic
136	C₂×C₄	Dic₆₇⋊C₈	1	2	yes	—
137	C₂×C₄	#14523974	1	1	yes	characteristic
138	C₂×C₄	#14523978	1	2	yes	—
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	8	2	no	—
142	C₈	—	536	2	no	—
143	C₈	C₈×Dic₆₇	1	1	yes	characteristic
144	C₈	#14523953	1	1	yes	characteristic
145	C₈	(C₂×D₆₇)⋊C₈	1	2	yes	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	1	no	—
148	C₂²	#16863148	1	2	yes	—
149	C₂²	#16863158	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	#16863146	1	2	yes	—
155	C₄	#16863150	1	1	yes	characteristic
156	C₄	#16863160	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	#22558043	1	1	yes	characteristic
159	C₂	#22558310	1	1	yes	characteristic
160	C₂	#22558312	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863172

C₂

yes

maximal, characteristic

#22558957

C₂

yes

maximal, characteristic

C₆₇×#1564

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₁₆

C₄

yes

—

—

—

—

—

C₄

yes

—

#431×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#335×C₆₇

—

#334×C₆₇

C₄

yes

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

C₈

yes

complemented

C₄⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

Q₈

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

characteristic

C₂×C₆₇⋊C₈

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₆₇

D₈

yes

characteristic

C₂²×C₄×C₆₇

Q₁₆

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄²

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₈

yes

—

C₂×C₈×C₆₇

M₁₆

yes

—

C₄²×C₆₇

C₂²⋊C₄

yes

characteristic

C₄²×C₆₇

C₂²⋊C₄

yes

characteristic

C₄²×C₆₇

SD₁₆

yes

characteristic

C₄²×C₆₇

SD₁₆

yes

characteristic

C₂³×C₆₇

#107

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

C₄⋊C₈

yes

—

C₂×C₄×C₆₇

#106

yes

characteristic

C₂×C₄×C₆₇

#108

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₄×C₈

yes

characteristic

C₈×C₆₇

#97

yes

characteristic

C₈×C₆₇

C₂²⋊C₈

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#335

yes

—

C₂²×C₆₇

#340

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#334

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

—

D₆₇

yes

characteristic

C₆₇

#7157

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

#431

—

#431

—

134

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#346

—

134

—

#346

—

134

—

#335

—

#335

—

134

—

#334

—

#334

Dic₆₇

yes

—

C₂²×C₈

Dic₁₃₄

yes

characteristic

C₂²×C₈

C₄×D₆₇

yes

characteristic

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

134

—

100

C₂×C₁₆

—

134

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

134

—

103

C₂²⋊C₈

—

104

C₂²⋊C₈

—

268

—

105

C₄×C₈

—

106

C₄×C₈

—

268

—

107

C₄×C₈

C₄×D₆₇

yes

characteristic

108

C₄×C₈

D₂₆₈

yes

characteristic

109

C₄×C₈

C₂×Dic₆₇

yes

characteristic

110

C₄×C₈

C₂²⋊D₆₇

yes

characteristic

111

C₂²×C₄

Dic₆₇⋊C₄

yes

characteristic

112

C₂²×C₄

#11761613

yes

characteristic

113

C₂²×C₄

#11761618

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

268

—

117

C₂×C₈

C₄×Dic₆₇

yes

characteristic

118

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

characteristic

119

C₂×C₈

Dic₆₇⋊C₄

yes

characteristic

120

C₂×C₈

C₄⋊Dic₆₇

yes

characteristic

121

C₂×C₈

C₈×D₆₇

yes

—

122

C₂×C₈

#11761611

yes

—

123

C₄²

(C₂×D₆₇)⋊C₄

yes

characteristic

124

C₄²

C₂²⋊Dic₆₇

yes

characteristic

125

C₄²

#11761615

yes

characteristic

126

C₄²

#11761616

yes

characteristic

127

C₁₆

—

134

—

128

C₂³

#14523976

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₆₇⋊#95

yes

characteristic

132

C₂×C₄

#14523963

yes

characteristic

133

C₂×C₄

#14523964

yes

characteristic

134

C₂×C₄

#14523966

yes

characteristic

135

C₂×C₄

#14523967

yes

characteristic

136

C₂×C₄

Dic₆₇⋊C₈

yes

—

137

C₂×C₄

#14523974

yes

characteristic

138

C₂×C₄

#14523978

yes

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

536

—

143

C₈

C₈×Dic₆₇

yes

characteristic

144

C₈

#14523953

yes

characteristic

145

C₈

(C₂×D₆₇)⋊C₈

yes

—

146

C₂²

—

147

C₂²

—

148

C₂²

#16863148

yes

—

149

C₂²

#16863158

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#16863146

yes

—

155

C₄

#16863150

yes

characteristic

156

C₄

#16863160

yes

characteristic

157

C₂

—

158

C₂

#22558043

yes

characteristic

159

C₂

#22558310

yes

characteristic

160

C₂

#22558312

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14523974
commutator	C₈×C₆₇	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₆₇
Fitting	C₆₇×#1564	C₂
socle	C₂²×C₆₇	#340
prime radical	C₂³×C₆₇	#107
cosocle	C₂×C₄×C₈×C₆₇	C₂²
hypercenter	#1564	D₆₇
hypocenter	C₆₇	#7157
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

subgroup

G/H

center

C₂×C₄

#14523974

commutator

C₈×C₆₇

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₆₇

Fitting

C₆₇×#1564

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄×C₈×C₆₇

C₂²

D₆₇

C₆₇

G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

N	H ≅ G/N	type
16. C₂×C₆₇⋊C₁₆	C₈	semidirect
84. C₆₇	#7157	semidirect

H ≅ G/N

type

16. C₂×C₆₇⋊C₁₆

C₈

semidirect

84. C₆₇

#7157

semidirect

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

subgroup	index
C₂×#16863172	2
#22558957	2
C₆₇×#1564	2
#7157	67

subgroup

index

C₂×#16863172

#22558957

C₆₇×#1564

#7157

Subgroups — #30742706

Conjugacy classes of subgroups

Complements

Maximal subgroups