Group #30760643 — subgroups

Subgroups — #30760643

Order 17152.

294 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 85 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	#22561029	C₂	1	1	yes	maximal, characteristic
2	#22560350	C₂	1	1	yes	maximal, characteristic
3	#22558880	C₂	1	2	yes	maximal
4	#22558068	C₂	1	2	yes	maximal
5	C₆₇×#1281	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524043	C₂²	1	1	yes	characteristic
7	#16863420	C₂²	1	2	yes	—
8	C₂×#14523968	—	2	2	no	—
9	#16863262	C₂²	1	1	yes	characteristic
10	#16863161	—	2	2	no	—
11	#16863141	—	2	4	no	—
12	#431×C₆₇	C₂²	1	2	yes	—
13	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₃₄	—	2	1	no	—
15	C₂×#11761612	—	2	2	no	—
16	#14524043	—	2	2	no	—
17	#14524043	—	2	1	no	—
18	#14524043	D₄	1	2	yes	—
19	C₂×C₄⋊Dic₆₇	—	2	1	no	—
20	C₂×C₄⋊Dic₆₇	—	2	2	no	—
21	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	#14523968	—	4	4	no	—
24	C₆₇⋊#95	—	2	2	no	—
25	C₂²×C₈×C₆₇	D₄	1	2	yes	—
26	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₆₇	—	2	2	no	—
28	C₂²⋊C₈×C₆₇	—	2	2	no	—
29	#97×C₆₇	—	2	2	no	—
30	C₄×C₈×C₆₇	—	2	2	no	—
31	C₂×Dic₁₃₄	—	2	2	no	—
32	C₂×Dic₁₃₄	—	4	2	no	—
33	C₂×Dic₁₃₄	—	4	2	no	—
34	C₂×Dic₁₃₄	—	4	1	no	—
35	C₂²×Dic₆₇	—	4	1	no	—
36	C₂²×Dic₆₇	—	4	1	no	—
37	#11761612	—	4	2	no	—
38	#11761612	—	4	2	no	—
39	C₂×C₆₇⋊C₈	—	4	2	no	—
40	C₄⋊Dic₆₇	—	2	4	no	—
41	C₄⋊Dic₆₇	—	4	2	no	—
42	C₄⋊Dic₆₇	—	4	2	no	—
43	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	#41	1	2	yes	—
45	C₂×C₈×C₆₇	—	2	2	no	—
46	C₂×C₈×C₆₇	—	2	2	no	—
47	C₂×C₈×C₆₇	—	2	2	no	—
48	C₂×C₈×C₆₇	—	4	2	no	—
49	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₆₇	#41	1	2	yes	—
51	C₄²×C₆₇	#41	1	1	yes	characteristic
52	C₂³×C₆₇	#124	1	1	yes	characteristic
53	C₂×C₄×C₆₇	—	4	2	no	—
54	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
55	C₂×C₄×C₆₇	#123	1	2	yes	—
56	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
57	C₂×C₄×C₆₇	#126	1	2	yes	—
58	C₂×C₄×C₆₇	#135	1	2	yes	—
59	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
60	C₂×Dic₆₇	—	4	2	no	—
61	C₂×Dic₆₇	—	8	1	no	—
62	C₂×Dic₆₇	—	8	1	no	—
63	C₂×Dic₆₇	—	8	2	no	—
64	C₂×Dic₆₇	—	8	1	no	—
65	Dic₁₃₄	—	4	4	no	—
66	Dic₁₃₄	—	8	2	no	—
67	C₈×C₆₇	—	4	2	no	—
68	C₈×C₆₇	—	8	2	no	—
69	C₆₇⋊C₈	—	8	2	no	—
70	C₂²×C₆₇	—	2	2	no	—
71	C₂²×C₆₇	#397	1	1	yes	characteristic
72	C₂²×C₆₇	#488	1	1	yes	characteristic
73	C₂²×C₆₇	#490	1	1	yes	characteristic
74	C₄×C₆₇	—	2	2	no	—
75	C₄×C₆₇	—	2	2	no	—
76	C₄×C₆₇	#463	1	2	yes	—
77	C₄×C₆₇	#482	1	2	yes	—
78	Dic₆₇	—	8	2	no	—
79	Dic₆₇	—	16	1	no	—
80	#10635	—	67	1	no	maximal, Sylow
81	C₂×C₆₇	—	2	1	no	—
82	C₂×C₆₇	—	2	1	no	—
83	C₂×C₆₇	#2073	1	1	yes	characteristic
84	C₂×C₆₇	#2082	1	2	yes	—
85	#1960	—	67	1	no	—
86	#1838	—	67	1	no	—
87	#1546	—	67	2	no	—
88	#1286	—	67	2	no	—
89	#1281	D₆₇	1	1	yes	characteristic
90	C₆₇	#10635	1	1	yes	characteristic, complemented, Sylow
91	C₂×C₄⋊Q₈	—	67	1	no	—
92	#431	C₂×D₆₇	1	2	yes	—
93	#423	—	67	2	no	—
94	C₂×#103	—	134	2	no	—
95	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
96	#382	—	67	1	no	—
97	#341	—	134	2	no	—
98	#332	—	134	4	no	—
99	C₂²×Q₈	—	134	1	no	—
100	C₂×M₁₆	—	134	2	no	—
101	C₂²×C₈	D₂₆₈	1	2	yes	—
102	C₄⋊Q₈	—	67	2	no	—
103	C₄⋊Q₈	—	134	2	no	—
104	C₄⋊Q₈	—	134	1	no	—
105	C₂×C₄⋊C₄	—	134	1	no	—
106	C₂×C₄⋊C₄	—	134	2	no	—
107	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
108	C₄⋊C₈	—	2	2	no	—
109	C₄⋊C₈	—	134	2	no	—
110	C₄⋊C₈	—	134	2	no	—
111	#103	—	268	4	no	—
112	C₂²⋊C₈	—	2	2	no	—
113	#97	—	2	2	no	—
114	C₄×C₈	—	2	2	no	—
115	#95	—	134	2	no	—
116	C₂×Q₈	—	268	1	no	—
117	C₂×Q₈	—	268	2	no	—
118	C₂×Q₈	—	268	2	no	—
119	C₂×Q₈	—	134	2	no	—
120	C₂²×C₄	—	268	1	no	—
121	C₂²×C₄	—	268	1	no	—
122	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
123	C₂²×C₄	#11761628	1	2	yes	—
124	M₁₆	—	268	2	no	—
125	M₁₆	—	268	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	268	2	no	—
131	C₄⋊C₄	—	268	2	no	—
132	C₄⋊C₄	—	268	2	no	—
133	C₄⋊C₄	—	134	4	no	—
134	C₄²	D₄×D₆₇	1	1	yes	characteristic
135	C₄²	#11761627	1	2	yes	—
136	C₄²	#11761629	1	1	yes	characteristic
137	C₂³	#14524031	1	1	yes	characteristic
138	Q₈	—	268	4	no	—
139	Q₈	—	536	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	268	2	no	—
142	C₂×C₄	—	536	1	no	—
143	C₂×C₄	—	536	1	no	—
144	C₂×C₄	—	536	2	no	—
145	C₂×C₄	—	536	1	no	—
146	C₂×C₄	#14524018	1	1	yes	characteristic
147	C₂×C₄	#14524025	1	2	yes	—
148	C₂×C₄	#14524027	1	1	yes	characteristic
149	C₂×C₄	#14524038	1	2	yes	—
150	C₂×C₄	#14524065	1	2	yes	—
151	C₂×C₄	#14524071	1	2	yes	—
152	C₈	—	4	2	no	—
153	C₈	—	8	2	no	—
154	C₈	—	536	2	no	—
155	C₂²	—	2	2	no	—
156	C₂²	#16863323	1	1	yes	characteristic
157	C₂²	#16863751	1	1	yes	characteristic
158	C₂²	#16863761	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	536	2	no	—
162	C₄	—	1072	1	no	—
163	C₄	#16863612	1	2	yes	—
164	C₄	#16863721	1	2	yes	—
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#22561624	1	1	yes	characteristic
168	C₂	#22561675	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22561029

C₂

yes

maximal, characteristic

#22560350

C₂

yes

maximal, characteristic

#22558880

C₂

yes

maximal

#22558068

C₂

yes

maximal

C₆₇×#1281

C₂

yes

maximal, characteristic

C₂×#14524043

C₂²

yes

characteristic

#16863420

C₂²

yes

—

C₂×#14523968

—

#16863262

C₂²

yes

characteristic

#16863161

—

#16863141

—

#431×C₆₇

C₂²

yes

—

C₂×#97×C₆₇

C₂²

yes

characteristic

C₂²×Dic₁₃₄

—

—

—

—

D₄

yes

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

#14523968

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

#97×C₆₇

—

C₄×C₈×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

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

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

#41

yes

—

C₄²×C₆₇

yes

C₂³×C₆₇

yes

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#121

yes

characteristic

C₂×C₄×C₆₇

#123

yes

—

C₂×C₄×C₆₇

#123

yes

characteristic

C₂×C₄×C₆₇

#126

yes

—

C₂×C₄×C₆₇

#135

yes

—

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#463

yes

—

C₄×C₆₇

#482

yes

—

Dic₆₇

—

Dic₆₇

—

#10635

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#2073

yes

characteristic

C₂×C₆₇

#2082

yes

—

—

—

—

—

D₆₇

yes

characteristic

C₆₇

#10635

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#431

C₂×D₆₇

yes

—

#423

—

C₂×#103

—

134

—

C₂×#97

C₂×D₆₇

yes

characteristic

#382

—

#341

—

134

—

#332

—

134

—

C₂²×Q₈

—

134

—

100

C₂×M₁₆

—

134

—

101

C₂²×C₈

D₂₆₈

yes

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

—

134

—

104

C₄⋊Q₈

—

134

—

105

C₂×C₄⋊C₄

—

134

—

106

C₂×C₄⋊C₄

—

134

—

107

C₂×C₄²

C₂²×D₆₇

yes

characteristic

108

C₄⋊C₈

—

109

C₄⋊C₈

—

134

—

110

C₄⋊C₈

—

134

—

111

#103

—

268

—

112

C₂²⋊C₈

—

113

#97

—

114

C₄×C₈

—

115

#95

—

134

—

116

C₂×Q₈

—

268

—

117

C₂×Q₈

—

268

—

118

C₂×Q₈

—

268

—

119

C₂×Q₈

—

134

—

120

C₂²×C₄

—

268

—

121

C₂²×C₄

—

268

—

122

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

123

C₂²×C₄

#11761628

yes

—

124

M₁₆

—

268

—

125

M₁₆

—

268

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

268

—

131

C₄⋊C₄

—

268

—

132

C₄⋊C₄

—

268

—

133

C₄⋊C₄

—

134

—

134

C₄²

D₄×D₆₇

yes

characteristic

135

C₄²

#11761627

yes

—

136

C₄²

#11761629

yes

characteristic

137

C₂³

#14524031

yes

characteristic

138

Q₈

—

268

—

139

Q₈

—

536

—

140

C₂×C₄

—

141

C₂×C₄

—

268

—

142

C₂×C₄

—

536

—

143

C₂×C₄

—

536

—

144

C₂×C₄

—

536

—

145

C₂×C₄

—

536

—

146

C₂×C₄

#14524018

yes

characteristic

147

C₂×C₄

#14524025

yes

—

148

C₂×C₄

#14524027

yes

characteristic

149

C₂×C₄

#14524038

yes

—

150

C₂×C₄

#14524065

yes

—

151

C₂×C₄

#14524071

yes

—

152

C₈

—

153

C₈

—

154

C₈

—

536

—

155

C₂²

—

156

C₂²

#16863323

yes

characteristic

157

C₂²

#16863751

yes

characteristic

158

C₂²

#16863761

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

536

—

162

C₄

—

1072

—

163

C₄

#16863612

yes

—

164

C₄

#16863721

yes

—

165

C₂

—

166

C₂

—

167

C₂

#22561624

yes

characteristic

168

C₂

#22561675

yes

—

	subgroup	G/H
center	C₂²	#16863323
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1281	C₂
socle	C₂²×C₆₇	#397
prime radical	C₂³×C₆₇	#124
hypercenter	#1281	D₆₇
hypocenter	C₆₇	#10635
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1281

subgroup

G/H

center

C₂²

#16863323

commutator, cosocle

C₂×C₄²×C₆₇

C₂³

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1281

C₂

C₂²×C₆₇

C₂³×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1281

N	H ≅ G/N	type
90. C₆₇	#10635	semidirect

H ≅ G/N

type

90. C₆₇

#10635

semidirect

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

subgroup	index
#22561029	2
#22560350	2
#22558880	2
#22558068	2
C₆₇×#1281	2
#10635	67

subgroup

index

C₆₇×#1281

Subgroups — #30760643

Conjugacy classes of subgroups

Complements

Maximal subgroups