Group #30749534 — subgroups

Subgroups — #30749534

Order 17152.

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

normal subgroups only

This group has 270 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	#22560373	C₂	1	1	yes	maximal, characteristic
2	#22560347	C₂	1	1	yes	maximal, characteristic
3	#22558068	C₂	1	2	yes	maximal
4	#22558068	C₂	1	2	yes	maximal
5	C₆₇×#1452	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523977	—	2	1	no	—
7	C₂×#14523975	—	2	1	no	—
8	#16863420	C₂²	1	1	yes	characteristic
9	#16863420	C₂²	1	1	yes	characteristic
10	#16863262	C₂²	1	2	yes	—
11	#16863161	—	2	1	no	—
12	#16863161	—	2	1	no	—
13	#431×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#322×C₆₇	C₄	1	4	yes	complemented
16	C₂×#11761612	—	2	1	no	—
17	C₂×#11761612	—	2	1	no	—
18	C₂×C₄⋊Dic₆₇	—	2	2	no	—
19	#14523977	—	4	2	no	—
20	#14523975	—	4	2	no	—
21	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₆₇⋊#95	—	2	4	no	—
24	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₆₇	—	2	2	no	—
28	C₂²⋊C₈×C₆₇	—	2	2	no	—
29	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₆₇	Q₈	1	2	yes	—
32	C₂²×Dic₆₇	—	4	2	no	—
33	#11761612	—	4	1	no	—
34	#11761612	—	4	1	no	—
35	#11761612	—	4	1	no	—
36	#11761612	—	4	1	no	—
37	C₂×C₆₇⋊C₈	—	4	1	no	—
38	C₂×C₆₇⋊C₈	—	4	1	no	—
39	C₄⋊Dic₆₇	—	4	2	no	—
40	C₄⋊Dic₆₇	—	4	2	no	—
41	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	#41	1	2	yes	—
43	C₂×C₈×C₆₇	—	2	1	no	—
44	C₂×C₈×C₆₇	—	2	1	no	—
45	C₂×C₈×C₆₇	—	2	1	no	—
46	C₂×C₈×C₆₇	—	2	1	no	—
47	C₂×C₈×C₆₇	—	2	4	no	—
48	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
50	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₆₇	#41	1	2	yes	—
53	C₂³×C₆₇	#124	1	1	yes	characteristic
54	C₂×C₄×C₆₇	—	4	2	no	—
55	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₆₇	#117	1	2	yes	—
57	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
58	C₂×C₄×C₆₇	#125	1	2	yes	—
59	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
62	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
63	C₂×Dic₆₇	—	8	4	no	—
64	C₂×Dic₆₇	—	8	2	no	—
65	C₈×C₆₇	—	2	2	no	—
66	C₈×C₆₇	—	2	2	no	—
67	C₈×C₆₇	—	4	4	no	—
68	C₆₇⋊C₈	—	8	1	no	—
69	C₆₇⋊C₈	—	8	1	no	—
70	C₂²×C₆₇	—	2	2	no	—
71	C₂²×C₆₇	#382	1	1	yes	characteristic
72	C₂²×C₆₇	#490	1	1	yes	characteristic
73	C₂²×C₆₇	#491	1	1	yes	characteristic
74	C₄×C₆₇	—	2	2	no	—
75	C₄×C₆₇	—	2	2	no	—
76	C₄×C₆₇	#428	1	1	yes	characteristic
77	C₄×C₆₇	#428	1	1	yes	characteristic
78	C₄×C₆₇	#489	1	1	yes	characteristic
79	C₄×C₆₇	#489	1	1	yes	characteristic
80	Dic₆₇	—	16	2	no	—
81	#8795	—	67	1	no	maximal, Sylow
82	C₂×C₆₇	—	2	1	no	—
83	C₂×C₆₇	—	2	1	no	—
84	C₂×C₆₇	#1833	1	2	yes	—
85	C₂×C₆₇	#1836	1	1	yes	characteristic
86	#1846	—	67	1	no	—
87	#1837	—	67	1	no	—
88	#1452	D₆₇	1	1	yes	characteristic
89	#1286	—	67	2	no	—
90	#1286	—	67	2	no	—
91	C₆₇	#8795	1	1	yes	characteristic, complemented, Sylow
92	#431	C₂×D₆₇	1	2	yes	—
93	C₂×#107	—	134	1	no	—
94	C₂×#106	—	134	1	no	—
95	#423	—	67	1	no	—
96	#423	—	67	1	no	—
97	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
98	#382	—	67	2	no	—
99	#341	—	134	1	no	—
100	#341	—	134	1	no	—
101	#322	Dic₆₇	1	4	yes	complemented
102	C₂×M₁₆	—	134	1	no	—
103	C₂×M₁₆	—	134	1	no	—
104	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
105	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
106	C₂×C₄⋊C₄	—	134	2	no	—
107	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
108	#107	—	268	2	no	—
109	#106	—	268	2	no	—
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	134	2	no	—
112	C₄⋊C₈	—	134	2	no	—
113	C₂²⋊C₈	—	2	2	no	—
114	C₄×C₈	Dic₁₃₄	1	2	yes	—
115	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
116	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
117	#95	—	134	4	no	—
118	C₂²×C₄	—	268	2	no	—
119	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
120	C₂²×C₄	#11761628	1	2	yes	—
121	M₁₆	—	268	1	no	—
122	M₁₆	—	268	1	no	—
123	M₁₆	—	268	1	no	—
124	M₁₆	—	268	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	—	268	1	no	—
131	C₂×C₈	—	268	1	no	—
132	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
133	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
134	C₄⋊C₄	—	268	2	no	—
135	C₄⋊C₄	—	268	2	no	—
136	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
137	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
138	C₄²	#11761628	1	2	yes	—
139	C₂³	#14524031	1	1	yes	characteristic
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	536	4	no	—
142	C₂×C₄	—	536	2	no	—
143	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
144	C₂×C₄	#14524002	1	2	yes	—
145	C₂×C₄	#14524031	1	1	yes	characteristic
146	C₂×C₄	#14524035	1	2	yes	—
147	C₂×C₄	#14524071	1	1	yes	characteristic
148	C₂×C₄	#14524071	1	1	yes	characteristic
149	C₂×C₄	#14524076	1	1	yes	characteristic
150	C₂×C₄	#14524076	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₈	—	2	2	no	—
153	C₈	—	4	4	no	—
154	C₈	—	536	1	no	—
155	C₈	—	536	1	no	—
156	C₂²	—	2	2	no	—
157	C₂²	#16863262	1	1	yes	characteristic
158	C₂²	#16863761	1	1	yes	characteristic
159	C₂²	#16863765	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	1072	2	no	—
163	C₄	#16863438	1	1	yes	characteristic
164	C₄	#16863438	1	1	yes	characteristic
165	C₄	#16863756	1	1	yes	characteristic
166	C₄	#16863756	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#22560333	1	2	yes	—
170	C₂	#22560343	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560373

C₂

yes

maximal, characteristic

#22560347

C₂

yes

maximal, characteristic

#22558068

C₂

yes

maximal

#22558068

C₂

yes

maximal

C₆₇×#1452

C₂

yes

maximal, characteristic

C₂×#14523977

—

C₂×#14523975

—

#16863420

C₂²

yes

characteristic

#16863420

C₂²

yes

characteristic

#16863262

C₂²

yes

—

#16863161

—

#16863161

—

#431×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#322×C₆₇

C₄

yes

complemented

C₂×#11761612

—

C₂×#11761612

—

C₂×C₄⋊Dic₆₇

—

#14523977

—

#14523975

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

C₂²×Dic₆₇

—

—

—

—

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

#124

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

#124

yes

characteristic

C₂×C₄×C₆₇

#125

yes

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

#8795

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#1833

yes

—

C₂×C₆₇

yes

—

—

D₆₇

yes

characteristic

#1286

—

#1286

—

C₆₇

#8795

yes

characteristic, complemented, Sylow

#431

C₂×D₆₇

yes

—

C₂×#107

—

134

—

C₂×#106

—

134

—

#423

—

#423

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#382

—

#341

—

134

—

100

#341

—

134

—

101

#322

Dic₆₇

yes

complemented

102

C₂×M₁₆

—

134

—

103

C₂×M₁₆

—

134

—

104

C₂²×C₈

D₂₆₈

yes

characteristic

105

C₂²×C₈

D₂₆₈

yes

characteristic

106

C₂×C₄⋊C₄

—

134

—

107

C₂×C₄²

C₂²×D₆₇

yes

characteristic

108

#107

—

268

—

109

#106

—

268

—

110

C₄⋊C₈

—

111

C₄⋊C₈

—

134

—

112

C₄⋊C₈

—

134

—

113

C₂²⋊C₈

—

114

C₄×C₈

Dic₁₃₄

yes

—

115

C₄×C₈

C₂×Dic₆₇

yes

—

116

C₄×C₈

C₂×Dic₆₇

yes

—

117

#95

—

134

—

118

C₂²×C₄

—

268

—

119

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

120

C₂²×C₄

#11761628

yes

—

121

M₁₆

—

268

—

122

M₁₆

—

268

—

123

M₁₆

—

268

—

124

M₁₆

—

268

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

268

—

131

C₂×C₈

—

268

—

132

C₂×C₈

C₄⋊Dic₆₇

yes

—

133

C₂×C₈

C₄⋊Dic₆₇

yes

—

134

C₄⋊C₄

—

268

—

135

C₄⋊C₄

—

268

—

136

C₄²

C₂²×Dic₆₇

yes

characteristic

137

C₄²

C₂×Dic₁₃₄

yes

characteristic

138

C₄²

#11761628

yes

—

139

C₂³

#14524031

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

536

—

142

C₂×C₄

—

536

—

143

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

144

C₂×C₄

#14524002

yes

—

145

C₂×C₄

#14524031

yes

characteristic

146

C₂×C₄

#14524035

yes

—

147

C₂×C₄

#14524071

yes

characteristic

148

C₂×C₄

#14524071

yes

characteristic

149

C₂×C₄

#14524076

yes

characteristic

150

C₂×C₄

#14524076

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

536

—

155

C₈

—

536

—

156

C₂²

—

157

C₂²

#16863262

yes

characteristic

158

C₂²

#16863761

yes

characteristic

159

C₂²

#16863765

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

1072

—

163

C₄

#16863438

yes

characteristic

164

C₄

#16863438

yes

characteristic

165

C₄

#16863756

yes

characteristic

166

C₄

#16863756

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#22560333

yes

—

170

C₂

#22560343

yes

characteristic

	subgroup	G/H
center	C₂²	#16863262
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1452	C₂
socle	C₂²×C₆₇	#382
prime radical	C₂³×C₆₇	#124
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1452	D₆₇
hypocenter	C₆₇	#8795
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#16863262

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1452

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

N	H ≅ G/N	type
15. #322×C₆₇	C₄	semidirect
91. C₆₇	#8795	semidirect
101. #322	Dic₆₇	semidirect

H ≅ G/N

type

15. #322×C₆₇

C₄

semidirect

91. C₆₇

#8795

semidirect

101. #322

Dic₆₇

semidirect

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

subgroup	index
#22560373	2
#22560347	2
#22558068	2
C₆₇×#1452	2
#8795	67

subgroup

index

C₆₇×#1452

Subgroups — #30749534

Conjugacy classes of subgroups

Complements

Maximal subgroups