Group #30765178 — subgroups

Subgroups — #30765178

Order 17152.

296 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 127 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	#22562008	C₂	1	2	yes	maximal
2	#22558151	C₂	1	4	yes	maximal
3	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₆₇⋊C₁₆)	—	2	2	no	—
5	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	—
6	C₄×C₆₇⋊C₁₆	—	2	2	no	—
7	#533×C₆₇	C₄	1	1	yes	characteristic
8	#446×C₆₇	—	2	2	no	—
9	C₈×Q₈×C₆₇	—	2	2	no	—
10	#424×C₆₇	C₂²	1	2	yes	—
11	#423×C₆₇	C₄	1	1	yes	characteristic
12	#405×C₆₇	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₆₇⋊C₁₆	—	2	4	no	—
15	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
18	C₄⋊Q₈×C₆₇	C₈	1	2	yes	—
19	#125×C₆₇	C₈	1	2	yes	—
20	C₂²⋊Q₈×C₆₇	—	2	2	no	—
21	C₄×Q₈×C₆₇	—	2	2	no	—
22	#117×C₆₇	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₄⋊C₈×C₆₇	—	2	4	no	—
25	C₄⋊C₈×C₆₇	—	2	2	no	—
26	#97×C₆₇	—	2	2	no	—
27	C₄×C₈×C₆₇	—	2	2	no	—
28	C₄×C₈×C₆₇	D₄	1	2	yes	—
29	C₄×C₈×C₆₇	D₄	1	2	yes	—
30	C₆₇⋊C₁₆	—	4	4	no	—
31	C₂×Q₈×C₆₇	—	2	2	no	—
32	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
33	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
34	M₁₆×C₆₇	—	4	2	no	—
35	C₂×C₈×C₆₇	—	2	2	no	—
36	C₂×C₈×C₆₇	—	2	2	no	—
37	C₂×C₈×C₆₇	—	4	1	no	—
38	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
41	C₄⋊C₄×C₆₇	—	2	4	no	—
42	C₄⋊C₄×C₆₇	—	2	2	no	—
43	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
44	C₂²⋊C₄×C₆₇	—	2	2	no	—
45	C₄²×C₆₇	—	2	2	no	—
46	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
47	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
48	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
49	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
50	Q₈×C₆₇	—	4	4	no	—
51	C₂×C₄×C₆₇	—	2	4	no	—
52	C₂×C₄×C₆₇	—	2	1	no	—
53	C₂×C₄×C₆₇	—	2	1	no	—
54	C₂×C₄×C₆₇	—	2	4	no	—
55	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
56	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
57	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
59	C₈×C₆₇	—	2	2	no	—
60	C₈×C₆₇	—	2	1	no	—
61	C₈×C₆₇	—	4	2	no	—
62	C₈×C₆₇	#101	1	2	yes	—
63	C₂²×C₆₇	—	2	2	no	—
64	C₂²×C₆₇	—	2	1	no	—
65	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
66	C₄×C₆₇	—	2	2	no	—
67	C₄×C₆₇	—	2	1	no	—
68	C₄×C₆₇	—	4	4	no	—
69	C₄×C₆₇	#350	1	4	yes	—
70	C₄×C₆₇	#410	1	1	yes	characteristic
71	C₄×C₆₇	C₂×#101	1	1	yes	characteristic
72	#11496	—	67	1	no	maximal, Sylow
73	C₂×C₆₇	—	4	1	no	—
74	C₂×C₆₇	#1461	1	1	yes	characteristic
75	C₂×C₆₇	C₂×#350	1	2	yes	—
76	#2960	D₆₇	1	1	yes	characteristic
77	#2162	—	67	2	no	—
78	#1323	—	67	4	no	—
79	C₆₇	#11496	1	1	yes	characteristic, complemented, Sylow
80	#533	Dic₆₇	1	1	yes	characteristic
81	#446	—	2	2	no	—
82	C₈×Q₈	—	2	2	no	—
83	#424	C₂×D₆₇	1	2	yes	—
84	#423	Dic₆₇	1	1	yes	characteristic
85	#405	Dic₆₇	1	2	yes	—
86	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
87	C₄⋊C₁₆	—	134	2	no	—
88	C₂²⋊C₁₆	—	67	4	no	—
89	C₄×C₁₆	—	134	2	no	—
90	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
91	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
92	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
93	C₄⋊Q₈	C₆₇⋊C₈	1	2	yes	—
94	#125	C₆₇⋊C₈	1	2	yes	—
95	C₂²⋊Q₈	—	2	2	no	—
96	C₄×Q₈	—	2	2	no	—
97	#117	C₂×Dic₆₇	1	2	yes	—
98	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
99	C₂×C₁₆	—	134	4	no	—
100	C₄⋊C₈	—	2	4	no	—
101	C₄⋊C₈	—	2	2	no	—
102	#97	—	2	2	no	—
103	C₄×C₈	—	2	2	no	—
104	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
105	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
106	C₂×Q₈	—	2	2	no	—
107	C₂²×C₄	#11761612	1	2	yes	—
108	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
109	M₁₆	—	4	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	4	1	no	—
113	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
114	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
115	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₄²	—	2	2	no	—
121	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
122	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
123	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
124	C₁₆	—	268	4	no	—
125	C₂³	C₂×#11761612	1	1	yes	characteristic
126	Q₈	—	4	4	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
132	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
133	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
134	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	2	1	no	—
137	C₈	—	4	2	no	—
138	C₈	#14523962	1	2	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
142	C₄	—	2	2	no	—
143	C₄	—	2	1	no	—
144	C₄	—	4	4	no	—
145	C₄	#16863180	1	4	yes	—
146	C₄	#16863370	1	1	yes	characteristic
147	C₄	C₂×#14523962	1	1	yes	characteristic
148	C₂	—	4	1	no	—
149	C₂	#22558533	1	1	yes	characteristic
150	C₂	C₂×#16863180	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22562008

C₂

yes

maximal

#22558151

C₂

yes

maximal

C₆₇×#2960

C₂

yes

maximal, characteristic

C₄⋊(C₆₇⋊C₁₆)

—

C₂²⋊(C₆₇⋊C₁₆)

C₂²

yes

—

C₄×C₆₇⋊C₁₆

—

#533×C₆₇

C₄

yes

characteristic

#446×C₆₇

—

C₈×Q₈×C₆₇

—

#424×C₆₇

C₂²

yes

—

#423×C₆₇

C₄

yes

characteristic

#405×C₆₇

C₄

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₆₇⋊C₁₆

—

C₂×M₁₆×C₆₇

C₂×C₄

yes

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊Q₈×C₆₇

C₈

yes

—

#125×C₆₇

C₈

yes

—

C₂²⋊Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

#117×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

#97×C₆₇

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₆₇⋊C₁₆

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

M₁₆

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂×C₈

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂²⋊C₄

yes

—

C₄²×C₆₇

C₂×C₈

yes

characteristic

C₄²×C₆₇

C₂×C₈

yes

characteristic

C₂³×C₆₇

C₂×M₁₆

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊C₈

yes

—

C₂×C₄×C₆₇

C₂²⋊C₈

yes

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₈

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#101

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#350

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₂×C₆₇

—

C₂×C₆₇

#1461

yes

characteristic

C₂×C₆₇

C₂×#350

yes

—

#2960

D₆₇

yes

characteristic

#2162

—

#1323

—

C₆₇

#11496

yes

characteristic, complemented, Sylow

#533

Dic₆₇

yes

characteristic

#446

—

C₈×Q₈

—

#424

C₂×D₆₇

yes

—

#423

Dic₆₇

yes

characteristic

#405

Dic₆₇

yes

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

C₄⋊C₁₆

—

134

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

134

—

C₂×M₁₆

C₂×Dic₆₇

yes

—

C₂²×C₈

C₂×Dic₆₇

yes

characteristic

C₂²×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊Q₈

C₆₇⋊C₈

yes

—

#125

C₆₇⋊C₈

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×Dic₆₇

yes

—

C₂×C₄²

C₂×Dic₆₇

yes

characteristic

C₂×C₁₆

—

134

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

#97

—

103

C₄×C₈

—

104

C₄×C₈

C₂²⋊D₆₇

yes

—

105

C₄×C₈

C₂²⋊D₆₇

yes

—

106

C₂×Q₈

—

107

C₂²×C₄

#11761612

yes

—

108

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₂²⋊Dic₆₇

yes

characteristic

114

C₂×C₈

C₂²⋊Dic₆₇

yes

characteristic

115

C₂×C₈

C₂×C₂²⋊D₆₇

yes

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

C₂×C₆₇⋊C₈

yes

—

119

C₂²⋊C₄

—

120

C₄²

—

121

C₄²

C₂²⋊Dic₆₇

yes

—

122

C₄²

C₂×C₆₇⋊C₈

yes

characteristic

123

C₄²

C₂×C₆₇⋊C₈

yes

characteristic

124

C₁₆

—

268

—

125

C₂³

C₂×#11761612

yes

characteristic

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₂²⋊(C₆₇⋊C₈)

yes

—

132

C₂×C₄

C₂²⋊(C₆₇⋊C₈)

yes

—

133

C₂×C₄

C₂×C₂²⋊Dic₆₇

yes

characteristic

134

C₂×C₄

C₂²×C₆₇⋊C₈

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

#14523962

yes

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂×C₂²⋊(C₆₇⋊C₈)

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

#16863180

yes

—

146

C₄

#16863370

yes

characteristic

147

C₄

C₂×#14523962

yes

characteristic

148

C₂

—

149

C₂

#22558533

yes

characteristic

150

C₂

C₂×#16863180

yes

—

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#2960	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₈
prime radical	C₂³×C₆₇	C₂×M₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#2960	D₆₇
hypocenter	C₆₇	#11496
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2960

subgroup

G/H

center

C₂×C₈

C₂²⋊Dic₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#2960

C₂

socle

C₂²×C₆₇

C₂×C₂²⋊C₈

prime radical

C₂³×C₆₇

C₂×M₁₆

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2960

N	H ≅ G/N	type
79. C₆₇	#11496	semidirect

H ≅ G/N

type

79. C₆₇

#11496

semidirect

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

subgroup	index
#22562008	2
#22558151	2
C₆₇×#2960	2
#11496	67

subgroup

index

#22562008

#22558151

C₆₇×#2960

#11496

Subgroups — #30765178

Conjugacy classes of subgroups

Complements

Maximal subgroups