Group #30743092 — subgroups

Subgroups — #30743092

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	#22562012	C₂	1	1	yes	maximal, characteristic
2	#22558965	C₂	1	1	yes	maximal, characteristic
3	C₄×#108×C₆₇	C₂	1	1	yes	maximal, characteristic
4	#16863420	—	2	1	no	—
5	C₄⋊(C₆₇⋊C₁₆)	C₄	1	2	yes	—
6	C₂²⋊(C₆₇⋊C₁₆)	—	2	1	no	—
7	#16863172	—	2	1	no	—
8	#16863147	—	2	1	no	—
9	#16863147	C₄	1	2	yes	—
10	C₂×#108×C₆₇	—	2	1	no	—
11	C₄×M₁₆×C₆₇	—	2	1	no	—
12	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	#341×C₆₇	C₄	1	2	yes	—
14	C₂×#11761612	—	4	1	no	—
15	C₂×C₆₇⋊C₁₆	—	2	2	no	—
16	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
17	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
18	C₂×M₁₆×C₆₇	—	2	2	no	—
19	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
22	#108×C₆₇	—	2	4	no	—
23	#97×C₆₇	—	4	1	no	—
24	C₄×C₈×C₆₇	—	4	1	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	—	8	2	no	—
30	C₂×C₆₇⋊C₈	—	4	2	no	—
31	C₆₇⋊C₁₆	—	4	2	no	—
32	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
34	C₂²×C₄×C₆₇	Q₁₆	1	1	yes	characteristic
35	M₁₆×C₆₇	—	2	4	no	—
36	M₁₆×C₆₇	—	4	2	no	—
37	C₂×C₈×C₆₇	—	2	2	no	—
38	C₂×C₈×C₆₇	—	4	1	no	—
39	C₂×C₈×C₆₇	—	4	2	no	—
40	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₆₇	D₈	1	1	yes	characteristic
47	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
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	—	4	4	no	—
63	C₈×C₆₇	#104	1	2	yes	—
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	—	67	1	no	—
82	C₄×#108	D₆₇	1	1	yes	characteristic
83	#1566	—	67	1	no	—
84	C₆₇	#7300	1	1	yes	characteristic, complemented, Sylow
85	C₂×#108	—	2	1	no	—
86	#423	—	134	1	no	—
87	C₄×M₁₆	—	2	1	no	—
88	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
89	C₄⋊C₁₆	—	67	2	no	—
90	C₂²⋊C₁₆	—	134	1	no	—
91	#346	—	134	1	no	—
92	#341	Dic₆₇	1	2	yes	—
93	#334	—	67	2	no	—
94	#334	—	134	1	no	—
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	268	1	no	—
97	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
98	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
99	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
100	C₂×C₁₆	—	134	2	no	—
101	#108	—	2	4	no	—
102	C₄⋊C₈	—	268	1	no	—
103	C₄⋊C₈	—	134	2	no	—
104	#97	—	4	1	no	—
105	C₄×C₈	—	4	1	no	—
106	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
107	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
108	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
109	C₄×C₈	D₂₆₈	1	1	yes	characteristic
110	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
111	C₂²×C₄	#11761615	1	1	yes	characteristic
112	C₂²×C₄	#11761618	1	1	yes	characteristic
113	M₁₆	—	2	4	no	—
114	M₁₆	—	4	2	no	—
115	M₁₆	—	536	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	268	2	no	—
120	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
121	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
122	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
123	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
124	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
125	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
126	C₄²	#11761613	1	1	yes	characteristic
127	C₄²	#11761616	1	1	yes	characteristic
128	C₁₆	—	268	2	no	—
129	C₂³	#14523967	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	#14523963	1	1	yes	characteristic
136	C₂×C₄	#14523964	1	1	yes	characteristic
137	C₂×C₄	#14523966	1	1	yes	characteristic
138	C₂×C₄	#14523974	1	1	yes	characteristic
139	C₂×C₄	#14523976	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	4	4	no	—
144	C₈	—	536	2	no	—
145	C₈	#14523971	1	2	yes	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
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	#16863156	1	1	yes	characteristic
155	C₄	#16863166	1	1	yes	characteristic
156	C₄	#16863220	1	2	yes	—
157	C₂	—	4	1	no	—
158	C₂	#22558112	1	1	yes	characteristic
159	C₂	#22558309	1	1	yes	characteristic
160	C₂	#22558320	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562012

C₂

yes

maximal, characteristic

#22558965

C₂

yes

maximal, characteristic

C₄×#108×C₆₇

C₂

yes

maximal, characteristic

#16863420

—

C₄⋊(C₆₇⋊C₁₆)

C₄

yes

—

C₂²⋊(C₆₇⋊C₁₆)

—

#16863172

—

#16863147

—

#16863147

C₄

yes

—

C₂×#108×C₆₇

—

C₄×M₁₆×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#341×C₆₇

C₄

yes

—

C₂×#11761612

—

C₂×C₆₇⋊C₁₆

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

#108×C₆₇

—

#97×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

Q₈

yes

characteristic

#11761612

—

C₂×C₆₇⋊C₈

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₆₇

SD₁₆

yes

characteristic

C₂²×C₄×C₆₇

Q₁₆

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

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

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

—

C₈×C₆₇

#104

yes

—

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

#1566

—

C₆₇

#7300

yes

characteristic, complemented, Sylow

C₂×#108

—

#423

—

134

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

134

—

#346

—

134

—

#341

Dic₆₇

yes

—

#334

—

#334

—

134

—

C₂×M₁₆

—

C₂×M₁₆

—

268

—

C₂²×C₈

C₂×Dic₆₇

yes

characteristic

C₂²×C₈

C₂²⋊D₆₇

yes

characteristic

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

100

C₂×C₁₆

—

134

—

101

#108

—

102

C₄⋊C₈

—

268

—

103

C₄⋊C₈

—

134

—

104

#97

—

105

C₄×C₈

—

106

C₄×C₈

Dic₁₃₄

yes

characteristic

107

C₄×C₈

C₄×D₆₇

yes

characteristic

108

C₄×C₈

C₄×D₆₇

yes

characteristic

109

C₄×C₈

D₂₆₈

yes

characteristic

110

C₂²×C₄

C₂²⋊Dic₆₇

yes

characteristic

111

C₂²×C₄

#11761615

yes

characteristic

112

C₂²×C₄

#11761618

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

536

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

268

—

120

C₂×C₈

C₄×Dic₆₇

yes

characteristic

121

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

characteristic

122

C₂×C₈

Dic₆₇⋊C₄

yes

characteristic

123

C₂×C₈

C₄⋊Dic₆₇

yes

characteristic

124

C₄²

(C₂×D₆₇)⋊C₄

yes

characteristic

125

C₄²

Dic₆₇⋊C₄

yes

characteristic

126

C₄²

#11761613

yes

characteristic

127

C₄²

#11761616

yes

characteristic

128

C₁₆

—

268

—

129

C₂³

#14523967

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

C₆₇⋊#95

yes

characteristic

135

C₂×C₄

#14523963

yes

characteristic

136

C₂×C₄

#14523964

yes

characteristic

137

C₂×C₄

#14523966

yes

characteristic

138

C₂×C₄

#14523974

yes

characteristic

139

C₂×C₄

#14523976

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

536

—

145

C₈

#14523971

yes

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

#16863158

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#16863156

yes

characteristic

155

C₄

#16863166

yes

characteristic

156

C₄

#16863220

yes

—

157

C₂

—

158

C₂

#22558112

yes

characteristic

159

C₂

#22558309

yes

characteristic

160

C₂

#22558320

yes

characteristic

	subgroup	G/H
center	C₂²	#16863158
commutator	C₂×C₈×C₆₇	C₄²
Frattini	C₂×C₄×C₈	C₂×D₆₇
Fitting	C₄×#108×C₆₇	C₂
socle	C₂²×C₆₇	#340
prime radical	C₂³×C₆₇	#103
cosocle	C₂×C₄×C₈×C₆₇	C₂²
hypercenter	C₄×#108	D₆₇
hypocenter	C₆₇	#7300
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

subgroup

G/H

center

C₂²

#16863158

commutator

C₂×C₈×C₆₇

C₄²

Frattini

C₂×C₄×C₈

C₂×D₆₇

Fitting

C₄×#108×C₆₇

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 ◁ C₄×#108

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
#22562012	2
#22558965	2
C₄×#108×C₆₇	2
#7300	67

subgroup

index

#22562012

#22558965

C₄×#108×C₆₇

#7300

Subgroups — #30743092

Conjugacy classes of subgroups

Complements

Maximal subgroups