Group #30749535 — subgroups

Subgroups — #30749535

Order 17152.

270 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 107 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	#22558501	C₂	1	1	yes	maximal, characteristic
3	#22558069	C₂	1	2	yes	maximal
4	#22558069	C₂	1	2	yes	maximal
5	C₆₇×#1837	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	2	yes	—
7	C₂×#14523977	—	2	1	no	—
8	C₂×#14523975	—	2	1	no	—
9	#16863262	C₂²	1	2	yes	—
10	#16863118	C₄	1	4	yes	complemented
11	#423×C₆₇	C₂²	1	1	yes	characteristic
12	#423×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#341×C₆₇	—	2	1	no	—
15	#341×C₆₇	—	2	1	no	—
16	C₂×C₄⋊Dic₆₇	—	2	2	no	—
17	#14523977	—	4	2	no	—
18	#14523975	—	4	2	no	—
19	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
20	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
21	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
22	C₆₇⋊#95	—	2	4	no	—
23	C₂×M₁₆×C₆₇	—	2	1	no	—
24	C₂×M₁₆×C₆₇	—	2	1	no	—
25	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₆₇	—	2	2	no	—
29	C₄⋊C₈×C₆₇	—	2	2	no	—
30	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₆₇	Q₈	1	2	yes	—
32	C₂²×Dic₆₇	—	4	2	no	—
33	C₂×C₆₇⋊C₈	—	2	4	no	—
34	C₄⋊Dic₆₇	—	4	2	no	—
35	C₄⋊Dic₆₇	—	4	2	no	—
36	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₆₇	#41	1	2	yes	—
38	M₁₆×C₆₇	—	4	1	no	—
39	M₁₆×C₆₇	—	4	1	no	—
40	M₁₆×C₆₇	—	4	1	no	—
41	M₁₆×C₆₇	—	4	1	no	—
42	C₂×C₈×C₆₇	—	2	1	no	—
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	—	4	1	no	—
47	C₂×C₈×C₆₇	—	4	1	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	—	8	1	no	—
68	C₈×C₆₇	—	8	1	no	—
69	C₆₇⋊C₈	—	4	4	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	D₆₇	1	1	yes	characteristic
88	#1452	—	67	1	no	—
89	#1286	—	67	2	no	—
90	#1286	—	67	2	no	—
91	C₆₇	#8795	1	1	yes	characteristic, complemented, Sylow
92	#431	—	67	2	no	—
93	C₂×#107	—	134	1	no	—
94	C₂×#106	—	134	1	no	—
95	#423	C₂×D₆₇	1	1	yes	characteristic
96	#423	C₂×D₆₇	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
98	#382	—	67	2	no	—
99	#341	—	2	1	no	—
100	#341	—	2	1	no	—
101	#322	—	67	4	no	—
102	C₂×M₁₆	—	2	1	no	—
103	C₂×M₁₆	—	2	1	no	—
104	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
105	C₂²×C₈	C₂²⋊D₆₇	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	—	2	2	no	—
112	C₄⋊C₈	—	134	2	no	—
113	C₂²⋊C₈	—	134	2	no	—
114	C₄×C₈	—	67	4	no	—
115	C₄×C₈	Dic₁₃₄	1	2	yes	—
116	C₄×C₈	C₄×D₆₇	1	2	yes	—
117	#95	—	134	4	no	—
118	C₂²×C₄	—	268	2	no	—
119	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
120	C₂²×C₄	#11761629	1	2	yes	—
121	M₁₆	—	4	1	no	—
122	M₁₆	—	4	1	no	—
123	M₁₆	—	4	1	no	—
124	M₁₆	—	4	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	—	4	1	no	—
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	—	134	4	no	—
132	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
133	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
134	C₄⋊C₄	—	268	2	no	—
135	C₄⋊C₄	—	268	2	no	—
136	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
137	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
138	C₄²	#11761627	1	2	yes	—
139	C₂³	#14524032	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×Dic₆₇⋊C₄	1	1	yes	characteristic
144	C₂×C₄	#14524001	1	2	yes	—
145	C₂×C₄	#14524030	1	1	yes	characteristic
146	C₂×C₄	#14524034	1	2	yes	—
147	C₂×C₄	#14524068	1	1	yes	characteristic
148	C₂×C₄	#14524068	1	1	yes	characteristic
149	C₂×C₄	#14524073	1	1	yes	characteristic
150	C₂×C₄	#14524073	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₈	—	2	2	no	—
153	C₈	—	8	1	no	—
154	C₈	—	8	1	no	—
155	C₈	—	268	4	no	—
156	C₂²	—	2	2	no	—
157	C₂²	#16863261	1	1	yes	characteristic
158	C₂²	#16863762	1	1	yes	characteristic
159	C₂²	#16863766	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	1072	2	no	—
163	C₄	#16863439	1	1	yes	characteristic
164	C₄	#16863439	1	1	yes	characteristic
165	C₄	#16863755	1	1	yes	characteristic
166	C₄	#16863755	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#22560334	1	2	yes	—
170	C₂	#22560342	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560373

C₂

yes

maximal, characteristic

#22558501

C₂

yes

maximal, characteristic

#22558069

C₂

yes

maximal

#22558069

C₂

yes

maximal

C₆₇×#1837

C₂

yes

maximal, characteristic

#16863449

C₂²

yes

—

C₂×#14523977

—

C₂×#14523975

—

#16863262

C₂²

yes

—

#16863118

C₄

yes

complemented

#423×C₆₇

C₂²

yes

characteristic

#423×C₆₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#341×C₆₇

—

#341×C₆₇

—

C₂×C₄⋊Dic₆₇

—

#14523977

—

#14523975

—

C₄⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

C₂×C₄

yes

—

C₆₇⋊#95

—

C₂×M₁₆×C₆₇

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

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

—

C₈×C₆₇

—

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

—

—

—

C₆₇

#8795

yes

characteristic, complemented, Sylow

#431

—

C₂×#107

—

134

—

C₂×#106

—

134

—

#423

C₂×D₆₇

yes

characteristic

#423

C₂×D₆₇

yes

characteristic

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#382

—

#341

—

100

#341

—

101

#322

—

102

C₂×M₁₆

—

103

C₂×M₁₆

—

104

C₂²×C₈

C₂²⋊D₆₇

yes

characteristic

105

C₂²×C₈

C₂²⋊D₆₇

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

—

112

C₄⋊C₈

—

134

—

113

C₂²⋊C₈

—

134

—

114

C₄×C₈

—

115

C₄×C₈

Dic₁₃₄

yes

—

116

C₄×C₈

C₄×D₆₇

yes

—

117

#95

—

134

—

118

C₂²×C₄

—

268

—

119

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

120

C₂²×C₄

#11761629

yes

—

121

M₁₆

—

122

M₁₆

—

123

M₁₆

—

124

M₁₆

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

134

—

132

C₂×C₈

Dic₆₇⋊C₄

yes

—

133

C₂×C₈

Dic₆₇⋊C₄

yes

—

134

C₄⋊C₄

—

268

—

135

C₄⋊C₄

—

268

—

136

C₄²

C₂×C₄×D₆₇

yes

characteristic

137

C₄²

C₂×Dic₁₃₄

yes

characteristic

138

C₄²

#11761627

yes

—

139

C₂³

#14524032

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

536

—

142

C₂×C₄

—

536

—

143

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

144

C₂×C₄

#14524001

yes

—

145

C₂×C₄

#14524030

yes

characteristic

146

C₂×C₄

#14524034

yes

—

147

C₂×C₄

#14524068

yes

characteristic

148

C₂×C₄

#14524068

yes

characteristic

149

C₂×C₄

#14524073

yes

characteristic

150

C₂×C₄

#14524073

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

—

268

—

156

C₂²

—

157

C₂²

#16863261

yes

characteristic

158

C₂²

#16863762

yes

characteristic

159

C₂²

#16863766

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

1072

—

163

C₄

#16863439

yes

characteristic

164

C₄

#16863439

yes

characteristic

165

C₄

#16863755

yes

characteristic

166

C₄

#16863755

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#22560334

yes

—

170

C₂

#22560342

yes

characteristic

	subgroup	G/H
center	C₂²	#16863261
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1837	C₂
socle	C₂²×C₆₇	#382
prime radical	C₂³×C₆₇	#124
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1837	D₆₇
hypocenter	C₆₇	#8795
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

subgroup

G/H

center

C₂²

#16863261

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1837

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

N	H ≅ G/N	type
10. #16863118	C₄	semidirect
91. C₆₇	#8795	semidirect

H ≅ G/N

type

10. #16863118

C₄

semidirect

91. C₆₇

#8795

semidirect

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

subgroup	index
#22560373	2
#22558501	2
#22558069	2
C₆₇×#1837	2
#8795	67

subgroup

index

C₆₇×#1837

Subgroups — #30749535

Conjugacy classes of subgroups

Complements

Maximal subgroups