Group #30748853 — subgroups

Subgroups — #30748853

Order 17152.

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

normal subgroups only

This group has 336 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	#22558965	C₂	1	2	yes	maximal
2	#22558947	C₂	1	2	yes	maximal
3	C₆₇⋊#1309	C₂	1	2	yes	maximal
4	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic
5	#16863420	C₂²	1	4	yes	—
6	#16863147	—	2	2	no	—
7	#16863147	—	2	2	no	—
8	#16863142	—	2	4	no	—
9	#533×C₆₇	C₂²	1	1	yes	characteristic
10	#446×C₆₇	—	2	2	no	—
11	C₈×Q₈×C₆₇	—	2	2	no	—
12	#424×C₆₇	C₄	1	2	yes	—
13	#423×C₆₇	C₂²	1	1	yes	characteristic
14	#405×C₆₇	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#11761612	—	2	4	no	—
17	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
18	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
19	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
23	#125×C₆₇	Q₈	1	2	yes	—
24	C₂²⋊Q₈×C₆₇	—	2	2	no	—
25	C₄×Q₈×C₆₇	—	2	2	no	—
26	#117×C₆₇	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₆₇	—	2	4	no	—
29	C₄⋊C₈×C₆₇	—	2	2	no	—
30	#97×C₆₇	—	2	2	no	—
31	C₄×C₈×C₆₇	—	2	2	no	—
32	C₄×C₈×C₆₇	D₄	1	2	yes	—
33	C₄×C₈×C₆₇	D₄	1	2	yes	—
34	#11761612	—	8	4	no	—
35	C₂×C₆₇⋊C₈	—	4	4	no	—
36	C₂×Q₈×C₆₇	—	2	2	no	—
37	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₇	#41	1	2	yes	—
39	M₁₆×C₆₇	—	4	2	no	—
40	C₂×C₈×C₆₇	—	2	2	no	—
41	C₂×C₈×C₆₇	—	2	2	no	—
42	C₂×C₈×C₆₇	—	4	1	no	—
43	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
44	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₄⋊C₄×C₆₇	—	2	2	no	—
47	C₄⋊C₄×C₆₇	—	2	4	no	—
48	C₄⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
49	C₂²⋊C₄×C₆₇	—	2	2	no	—
50	C₄²×C₆₇	—	2	2	no	—
51	C₄²×C₆₇	C₂×D₄	1	2	yes	—
52	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
54	C₂³×C₆₇	#117	1	1	yes	characteristic
55	Q₈×C₆₇	—	4	4	no	—
56	C₂×C₄×C₆₇	—	2	1	no	—
57	C₂×C₄×C₆₇	—	2	1	no	—
58	C₂×C₄×C₆₇	—	2	4	no	—
59	C₂×C₄×C₆₇	—	2	4	no	—
60	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₇	#121	1	2	yes	—
63	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
64	C₈×C₆₇	—	2	1	no	—
65	C₈×C₆₇	—	2	2	no	—
66	C₈×C₆₇	—	4	2	no	—
67	C₈×C₆₇	#101	1	2	yes	—
68	C₆₇⋊C₈	—	8	4	no	—
69	C₂²×C₆₇	—	2	1	no	—
70	C₂²×C₆₇	—	2	2	no	—
71	C₂²×C₆₇	#380	1	1	yes	characteristic
72	C₄×C₆₇	—	2	1	no	—
73	C₄×C₆₇	—	2	2	no	—
74	C₄×C₆₇	—	4	4	no	—
75	C₄×C₆₇	C₂×#101	1	1	yes	characteristic
76	C₄×C₆₇	#413	1	1	yes	characteristic
77	C₄×C₆₇	#471	1	4	yes	—
78	#8637	—	67	1	no	maximal, Sylow
79	C₂×C₆₇	—	4	1	no	—
80	C₂×C₆₇	#1801	1	1	yes	characteristic
81	C₂×C₆₇	#1815	1	2	yes	—
82	#2960	D₆₇	1	1	yes	characteristic
83	#1566	—	67	2	no	—
84	#1561	—	67	2	no	—
85	#1309	—	67	2	no	—
86	C₆₇	#8637	1	1	yes	characteristic, complemented, Sylow
87	#533	C₂×D₆₇	1	1	yes	characteristic
88	#446	—	2	2	no	—
89	C₈×Q₈	—	2	2	no	—
90	#424	Dic₆₇	1	2	yes	—
91	#423	—	67	4	no	—
92	#423	C₂×D₆₇	1	1	yes	characteristic
93	#405	Dic₆₇	1	2	yes	—
94	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
95	#334	—	134	2	no	—
96	#334	—	134	2	no	—
97	#332	—	134	4	no	—
98	C₂×M₁₆	—	134	4	no	—
99	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
100	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
101	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
102	C₄⋊Q₈	D₂₆₈	1	2	yes	—
103	#125	Dic₁₃₄	1	2	yes	—
104	C₂²⋊Q₈	—	2	2	no	—
105	C₄×Q₈	—	2	2	no	—
106	#117	C₂×Dic₆₇	1	2	yes	—
107	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
108	C₄⋊C₈	—	2	4	no	—
109	C₄⋊C₈	—	2	2	no	—
110	C₄⋊C₈	—	134	4	no	—
111	C₄⋊C₈	—	134	4	no	—
112	#97	—	2	2	no	—
113	C₄×C₈	—	2	2	no	—
114	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
115	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
116	C₂×Q₈	—	2	2	no	—
117	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
118	C₂²×C₄	#11761628	1	2	yes	—
119	M₁₆	—	4	2	no	—
120	M₁₆	—	536	4	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	4	1	no	—
124	C₂×C₈	—	268	4	no	—
125	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
126	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
127	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	4	no	—
130	C₄⋊C₄	C₄⋊Dic₆₇	1	4	yes	—
131	C₂²⋊C₄	—	2	2	no	—
132	C₄²	—	2	2	no	—
133	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
134	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
135	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
136	C₂³	#14524002	1	1	yes	characteristic
137	Q₈	—	4	4	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
143	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
144	C₂×C₄	#14524017	1	2	yes	—
145	C₂×C₄	#14524022	1	2	yes	—
146	C₈	—	2	1	no	—
147	C₈	—	2	2	no	—
148	C₈	—	4	2	no	—
149	C₈	—	536	4	no	—
150	C₈	#14523962	1	2	yes	—
151	C₂²	—	2	2	no	—
152	C₂²	—	2	1	no	—
153	C₂²	#16863256	1	1	yes	characteristic
154	C₄	—	2	2	no	—
155	C₄	—	2	1	no	—
156	C₄	—	4	4	no	—
157	C₄	C₂×#14523962	1	1	yes	characteristic
158	C₄	#16863380	1	1	yes	characteristic
159	C₄	#16863659	1	4	yes	—
160	C₂	—	4	1	no	—
161	C₂	#22560205	1	1	yes	characteristic
162	C₂	#22560268	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22558965

C₂

yes

maximal

#22558947

C₂

yes

maximal

C₆₇⋊#1309

C₂

yes

maximal

C₆₇×#2960

C₂

yes

maximal, characteristic

#16863420

C₂²

yes

—

#16863147

—

#16863147

—

#16863142

—

#533×C₆₇

C₂²

yes

characteristic

#446×C₆₇

—

C₈×Q₈×C₆₇

—

#424×C₆₇

C₄

yes

—

#423×C₆₇

C₂²

yes

characteristic

#405×C₆₇

C₄

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×#11761612

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₂×M₁₆×C₆₇

C₂×C₄

yes

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄⋊Q₈×C₆₇

D₄

yes

—

#125×C₆₇

Q₈

yes

—

C₂²⋊Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

#117×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

#97×C₆₇

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₄⋊C₄

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₂³×C₆₇

#117

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#101

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#380

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

#8637

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1801

yes

characteristic

C₂×C₆₇

#1815

yes

—

#2960

D₆₇

yes

—

—

—

C₆₇

#8637

yes

characteristic, complemented, Sylow

#533

C₂×D₆₇

yes

characteristic

#446

—

C₈×Q₈

—

#424

Dic₆₇

yes

—

#423

—

#423

C₂×D₆₇

yes

characteristic

#405

Dic₆₇

yes

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#334

—

134

—

#334

—

134

—

#332

—

134

—

C₂×M₁₆

—

134

—

C₂×M₁₆

C₂×Dic₆₇

yes

—

100

C₂²×C₈

C₂×Dic₆₇

yes

characteristic

101

C₂²×C₈

C₂×Dic₆₇

yes

characteristic

102

C₄⋊Q₈

D₂₆₈

yes

—

103

#125

Dic₁₃₄

yes

—

104

C₂²⋊Q₈

—

105

C₄×Q₈

—

106

#117

C₂×Dic₆₇

yes

—

107

C₂×C₄²

C₂²×D₆₇

yes

characteristic

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

134

—

111

C₄⋊C₈

—

134

—

112

#97

—

113

C₄×C₈

—

114

C₄×C₈

C₂²⋊D₆₇

yes

—

115

C₄×C₈

C₂²⋊D₆₇

yes

—

116

C₂×Q₈

—

117

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

118

C₂²×C₄

#11761628

yes

—

119

M₁₆

—

120

M₁₆

—

536

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

268

—

125

C₂×C₈

C₂²⋊Dic₆₇

yes

characteristic

126

C₂×C₈

C₂²⋊Dic₆₇

yes

—

127

C₂×C₈

C₂²⋊Dic₆₇

yes

characteristic

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

C₄⋊Dic₆₇

yes

—

131

C₂²⋊C₄

—

132

C₄²

—

133

C₄²

C₂×C₂²⋊D₆₇

yes

—

134

C₄²

C₂×D₂₆₈

yes

characteristic

135

C₄²

C₂×Dic₁₃₄

yes

characteristic

136

C₂³

#14524002

yes

characteristic

137

Q₈

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

C₂×C₂²⋊Dic₆₇

yes

characteristic

143

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

144

C₂×C₄

#14524017

yes

—

145

C₂×C₄

#14524022

yes

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

536

—

150

C₈

#14523962

yes

—

151

C₂²

—

152

C₂²

—

153

C₂²

#16863256

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

C₂×#14523962

yes

characteristic

158

C₄

#16863380

yes

characteristic

159

C₄

#16863659

yes

—

160

C₂

—

161

C₂

#22560205

yes

characteristic

162

C₂

#22560268

yes

—

	subgroup	G/H
center	C₂²	#16863256
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2960	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂³×C₆₇	#117
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2960	D₆₇
hypocenter	C₆₇	#8637
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

subgroup

G/H

center

C₂²

#16863256

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#2960

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

N	H ≅ G/N	type
86. C₆₇	#8637	semidirect

H ≅ G/N

type

86. C₆₇

#8637

semidirect

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

subgroup	index
#22558965	2
#22558947	2
C₆₇⋊#1309	2
C₆₇×#2960	2
#8637	67

subgroup

index

#22558965

#22558947

C₆₇⋊#1309

C₆₇×#2960

#8637

Subgroups — #30748853

Conjugacy classes of subgroups

Complements

Maximal subgroups