Group #30743091 — subgroups

Subgroups — #30743091

Order 17152.

212 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 87 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	C₄×#14523979	C₂	1	1	yes	maximal, characteristic
2	#22558965	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2163	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523979	—	2	1	no	—
5	#16863420	—	2	1	no	—
6	C₄×#11761612	—	2	1	no	—
7	#16863161	C₄	1	2	yes	—
8	#16863147	—	2	1	no	—
9	#16863147	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₄⋊C₁₆×C₆₇	C₄	1	2	yes	—
12	C₂²⋊C₁₆×C₆₇	—	2	1	no	—
13	#346×C₆₇	—	2	1	no	—
14	C₂×#11761612	—	2	2	no	—
15	C₂×#11761612	—	4	1	no	—
16	#14523979	—	2	4	no	—
17	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
18	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
19	#14523954	—	4	1	no	—
20	C₄×C₆₇⋊C₈	—	4	1	no	—
21	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₆₇	—	2	2	no	—
25	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
28	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
29	#11761612	—	2	4	no	—
30	#11761612	—	4	2	no	—
31	#11761612	—	8	2	no	—
32	C₂×C₆₇⋊C₈	—	4	2	no	—
33	C₂×C₆₇⋊C₈	—	4	2	no	—
34	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₆₇	Q₁₆	1	1	yes	characteristic
37	C₂×C₈×C₆₇	—	2	2	no	—
38	C₂×C₈×C₆₇	—	4	1	no	—
39	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₆₇	D₈	1	1	yes	characteristic
46	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
47	C₁₆×C₆₇	—	4	2	no	—
48	C₂³×C₆₇	#103	1	1	yes	characteristic
49	C₂×C₄×C₆₇	—	2	1	no	—
50	C₂×C₄×C₆₇	—	2	1	no	—
51	C₂×C₄×C₆₇	—	2	2	no	—
52	C₂×C₄×C₆₇	—	4	1	no	—
53	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
54	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
55	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
56	C₂×C₄×C₆₇	#103	1	1	yes	characteristic
57	C₂×C₄×C₆₇	#106	1	1	yes	characteristic
58	C₂×C₄×C₆₇	#107	1	1	yes	characteristic
59	C₈×C₆₇	—	2	1	no	—
60	C₈×C₆₇	—	2	1	no	—
61	C₈×C₆₇	—	2	1	no	—
62	C₈×C₆₇	#104	1	2	yes	—
63	C₆₇⋊C₈	—	4	4	no	—
64	C₆₇⋊C₈	—	8	2	no	—
65	C₂²×C₆₇	—	2	1	no	—
66	C₂²×C₆₇	—	2	1	no	—
67	C₂²×C₆₇	—	2	1	no	—
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	#339	1	1	yes	characteristic
74	C₄×C₆₇	#344	1	1	yes	characteristic
75	C₄×C₆₇	#365	1	2	yes	—
76	#7300	—	67	1	no	maximal, Sylow
77	C₂×C₆₇	—	4	1	no	—
78	C₂×C₆₇	#1303	1	1	yes	characteristic
79	C₂×C₆₇	#1383	1	1	yes	characteristic
80	C₂×C₆₇	#1388	1	1	yes	characteristic
81	#2163	D₆₇	1	1	yes	characteristic
82	C₄×#108	—	67	1	no	—
83	#1566	—	67	1	no	—
84	C₆₇	#7300	1	1	yes	characteristic, complemented, Sylow
85	C₂×#108	—	134	1	no	—
86	#423	—	134	1	no	—
87	C₄×M₁₆	—	134	1	no	—
88	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
89	C₄⋊C₁₆	Dic₆₇	1	2	yes	—
90	C₂²⋊C₁₆	—	2	1	no	—
91	#346	—	2	1	no	—
92	#341	—	67	2	no	—
93	#334	—	67	2	no	—
94	#334	—	134	1	no	—
95	C₂×M₁₆	—	268	1	no	—
96	C₂×M₁₆	—	134	2	no	—
97	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
98	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
99	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
100	C₂×C₁₆	—	2	2	no	—
101	#108	—	134	4	no	—
102	C₄⋊C₈	—	268	1	no	—
103	C₄⋊C₈	—	134	2	no	—
104	#97	—	268	1	no	—
105	C₄×C₈	—	268	1	no	—
106	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
107	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
108	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
109	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
110	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
111	C₂²×C₄	#11761617	1	1	yes	characteristic
112	C₂²×C₄	Dic₂₆₈	1	1	yes	characteristic
113	M₁₆	—	268	2	no	—
114	M₁₆	—	536	2	no	—
115	M₁₆	—	134	4	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	268	2	no	—
119	C₂×C₈	—	268	2	no	—
120	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
121	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
122	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
123	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
124	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
125	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
126	C₄²	D₅₃₆	1	1	yes	characteristic
127	C₄²	#11761617	1	1	yes	characteristic
128	C₁₆	—	4	2	no	—
129	C₂³	#14523968	1	1	yes	characteristic
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
135	C₂×C₄	#14523965	1	1	yes	characteristic
136	C₂×C₄	#14523965	1	1	yes	characteristic
137	C₂×C₄	#14523968	1	1	yes	characteristic
138	C₂×C₄	#14523975	1	1	yes	characteristic
139	C₂×C₄	#14523977	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	268	4	no	—
144	C₈	—	536	2	no	—
145	C₈	#14523970	1	2	yes	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	#16863159	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	#16863155	1	1	yes	characteristic
155	C₄	#16863168	1	1	yes	characteristic
156	C₄	#16863219	1	2	yes	—
157	C₂	—	4	1	no	—
158	C₂	#22558111	1	1	yes	characteristic
159	C₂	#22558308	1	1	yes	characteristic
160	C₂	#22558321	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14523979

C₂

yes

maximal, characteristic

#22558965

C₂

yes

maximal, characteristic

C₆₇×#2163

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

—

#16863147

—

#16863147

C₄

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₁₆×C₆₇

C₄

yes

—

C₂²⋊C₁₆×C₆₇

—

#346×C₆₇

—

C₂×#11761612

—

C₂×#11761612

—

#14523979

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

#14523954

—

C₄×C₆₇⋊C₈

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₁₆×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

Q₈

yes

—

—

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₆₇

SD₁₆

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

C₄²×C₆₇

C₂²⋊C₄

yes

characteristic

C₄²×C₆₇

C₄⋊C₄

yes

characteristic

C₄²×C₆₇

D₈

yes

characteristic

C₄²×C₆₇

SD₁₆

yes

characteristic

C₁₆×C₆₇

—

C₂³×C₆₇

#103

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

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

yes

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#104

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#340

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

#7300

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

yes

characteristic

C₄×#108

—

#1566

—

C₆₇

#7300

yes

characteristic, complemented, Sylow

C₂×#108

—

134

—

#423

—

134

—

C₄×M₁₆

—

134

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

C₄⋊C₁₆

Dic₆₇

yes

—

C₂²⋊C₁₆

—

—

—

—

—

134

—

C₂×M₁₆

—

268

—

C₂×M₁₆

—

134

—

C₂²×C₈

C₄×D₆₇

yes

characteristic

C₂²×C₈

C₂²⋊D₆₇

yes

characteristic

C₂×C₄²

D₂₆₈

yes

characteristic

100

C₂×C₁₆

—

101

#108

—

134

—

102

C₄⋊C₈

—

268

—

103

C₄⋊C₈

—

134

—

104

#97

—

268

—

105

C₄×C₈

—

268

—

106

C₄×C₈

Dic₁₃₄

yes

characteristic

107

C₄×C₈

C₄×D₆₇

yes

characteristic

108

C₄×C₈

C₂×Dic₆₇

yes

characteristic

109

C₄×C₈

C₂²⋊D₆₇

yes

characteristic

110

C₂²×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

111

C₂²×C₄

#11761617

yes

characteristic

112

C₂²×C₄

Dic₂₆₈

yes

characteristic

113

M₁₆

—

268

—

114

M₁₆

—

536

—

115

M₁₆

—

134

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

268

—

119

C₂×C₈

—

268

—

120

C₂×C₈

C₄×Dic₆₇

yes

characteristic

121

C₂×C₈

C₂²⋊Dic₆₇

yes

characteristic

122

C₂×C₈

Dic₆₇⋊C₄

yes

characteristic

123

C₂×C₈

Dic₆₇⋊C₄

yes

characteristic

124

C₄²

(C₂×D₆₇)⋊C₄

yes

characteristic

125

C₄²

C₄⋊Dic₆₇

yes

characteristic

126

C₄²

D₅₃₆

yes

characteristic

127

C₄²

#11761617

yes

characteristic

128

C₁₆

—

129

C₂³

#14523968

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

C₆₇⋊#95

yes

characteristic

135

C₂×C₄

#14523965

yes

characteristic

136

C₂×C₄

#14523965

yes

characteristic

137

C₂×C₄

#14523968

yes

characteristic

138

C₂×C₄

#14523975

yes

characteristic

139

C₂×C₄

#14523977

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

268

—

144

C₈

—

536

—

145

C₈

#14523970

yes

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

#16863159

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#16863155

yes

characteristic

155

C₄

#16863168

yes

characteristic

156

C₄

#16863219

yes

—

157

C₂

—

158

C₂

#22558111

yes

characteristic

159

C₂

#22558308

yes

characteristic

160

C₂

#22558321

yes

characteristic

	subgroup	G/H
center	C₂²	#16863159
commutator	C₂×C₈×C₆₇	C₄²
Frattini	C₂×C₄×C₈	C₂×D₆₇
Fitting	C₆₇×#2163	C₂
socle	C₂²×C₆₇	#340
prime radical	C₂³×C₆₇	#103
cosocle	C₂×C₄×C₈×C₆₇	C₂²
hypercenter	#2163	D₆₇
hypocenter	C₆₇	#7300
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2163

subgroup

G/H

center

C₂²

#16863159

commutator

C₂×C₈×C₆₇

C₄²

Frattini

C₂×C₄×C₈

C₂×D₆₇

Fitting

C₆₇×#2163

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄×C₈×C₆₇

C₂²

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2163

N	H ≅ G/N	type
84. C₆₇	#7300	semidirect

H ≅ G/N

type

84. C₆₇

#7300

semidirect

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

subgroup	index
C₄×#14523979	2
#22558965	2
C₆₇×#2163	2
#7300	67

subgroup

index

C₄×#14523979

#22558965

C₆₇×#2163

#7300

Subgroups — #30743091

Conjugacy classes of subgroups

Complements

Maximal subgroups