Group #30775981 — subgroups

Subgroups — #30775981

Order 17152.

268 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 95 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	#22560366	C₂	1	1	yes	maximal, characteristic
2	#22558314	C₂	1	2	yes	maximal
3	#22558309	C₂	1	2	yes	maximal
4	#22558291	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1945	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	2	yes	—
7	C₂×#14523975	—	2	1	no	—
8	C₂×#14523975	C₂²	1	2	yes	—
9	#16863266	—	2	1	no	—
10	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
11	#427×C₆₇	C₂²	1	1	yes	characteristic
12	#424×C₆₇	—	2	1	no	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#339×C₆₇	—	2	1	no	—
15	C₂×C₄⋊Dic₆₇	—	2	2	no	—
16	C₂×C₄⋊Dic₆₇	—	4	2	no	—
17	#14523983	—	2	2	no	—
18	#14523983	—	2	2	no	—
19	C₂×C₆₇⋊C₁₆	—	2	2	no	—
20	#14523975	—	2	4	no	—
21	#14523975	—	4	2	no	—
22	C₂×M₁₆×C₆₇	—	2	1	no	—
23	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
25	#117×C₆₇	—	2	1	no	—
26	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
27	#108×C₆₇	—	2	2	no	—
28	#107×C₆₇	Q₈	1	2	yes	—
29	#106×C₆₇	—	2	1	no	—
30	C₄⋊C₈×C₆₇	—	4	2	no	—
31	C₄×C₈×C₆₇	—	2	2	no	—
32	C₂²×Dic₆₇	—	4	2	no	—
33	C₄⋊Dic₆₇	—	4	2	no	—
34	C₄⋊Dic₆₇	—	4	2	no	—
35	C₄⋊Dic₆₇	—	8	4	no	—
36	C₆₇⋊C₁₆	—	4	2	no	—
37	C₂²×C₄×C₆₇	—	2	1	no	—
38	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
39	M₁₆×C₆₇	—	4	1	no	—
40	M₁₆×C₆₇	—	4	1	no	—
41	C₂×C₈×C₆₇	—	2	1	no	—
42	C₂×C₈×C₆₇	—	2	1	no	—
43	C₂×C₈×C₆₇	—	4	1	no	—
44	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
45	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₆₇	#41	1	2	yes	—
47	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
48	C₂×C₈×C₆₇	#41	1	2	yes	—
49	C₄⋊C₄×C₆₇	—	2	2	no	—
50	C₂²⋊C₄×C₆₇	—	4	1	no	—
51	C₄²×C₆₇	—	2	1	no	—
52	C₄²×C₆₇	—	4	1	no	—
53	C₄²×C₆₇	Q₁₆	1	2	yes	—
54	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×C₄×C₆₇	—	2	4	no	—
56	C₂×C₄×C₆₇	—	2	2	no	—
57	C₂×C₄×C₆₇	—	4	2	no	—
58	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
60	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂×Dic₆₇	—	8	4	no	—
65	C₂×Dic₆₇	—	8	2	no	—
66	C₈×C₆₇	—	2	2	no	—
67	C₈×C₆₇	—	8	1	no	—
68	C₈×C₆₇	#125	1	2	yes	—
69	C₈×C₆₇	#126	1	2	yes	—
70	C₂²×C₆₇	—	2	2	no	—
71	C₂²×C₆₇	#476	1	1	yes	characteristic
72	C₂²×C₆₇	#484	1	1	yes	characteristic
73	C₂²×C₆₇	#490	1	1	yes	characteristic
74	C₄×C₆₇	—	2	4	no	—
75	C₄×C₆₇	—	8	2	no	—
76	C₄×C₆₇	#400	1	1	yes	characteristic
77	C₄×C₆₇	#477	1	1	yes	characteristic
78	C₄×C₆₇	#481	1	1	yes	characteristic
79	C₄×C₆₇	#487	1	1	yes	characteristic
80	Dic₆₇	—	16	2	no	—
81	#13543	—	67	1	no	maximal, Sylow
82	C₂×C₆₇	—	2	1	no	—
83	C₂×C₆₇	—	2	1	no	—
84	C₂×C₆₇	#2100	1	1	yes	characteristic
85	C₂×C₆₇	#2230	1	2	yes	—
86	#1945	D₆₇	1	1	yes	characteristic
87	#1844	—	67	1	no	—
88	#1386	—	67	2	no	—
89	#1383	—	67	2	no	—
90	#1376	—	67	1	no	—
91	C₆₇	#13543	1	1	yes	characteristic, complemented, Sylow
92	C₂×M₃₂	—	67	2	no	—
93	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
94	#427	C₂×D₆₇	1	1	yes	characteristic
95	C₂×#106	—	67	2	no	—
96	C₂×#106	—	134	1	no	—
97	#424	—	2	1	no	—
98	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
99	#384	—	134	1	no	—
100	#339	—	2	1	no	—
101	C₂×M₁₆	—	2	1	no	—
102	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
104	#117	—	2	1	no	—
105	C₂×C₄⋊C₄	—	268	2	no	—
106	C₂×C₄⋊C₄	—	134	2	no	—
107	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
108	M₃₂	—	134	2	no	—
109	M₃₂	—	134	2	no	—
110	C₂×C₁₆	—	134	2	no	—
111	#108	—	2	2	no	—
112	#107	Dic₁₃₄	1	2	yes	—
113	#106	—	2	1	no	—
114	#106	—	268	2	no	—
115	#106	—	134	4	no	—
116	C₄⋊C₈	—	4	2	no	—
117	C₄×C₈	—	2	2	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	268	2	no	—
120	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
121	M₁₆	—	4	1	no	—
122	M₁₆	—	4	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	4	1	no	—
126	C₂×C₈	#11761616	1	2	yes	—
127	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
128	C₂×C₈	#11761627	1	2	yes	—
129	C₂×C₈	#11761628	1	1	yes	characteristic
130	C₂×C₈	#11761629	1	2	yes	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	268	2	no	—
133	C₄⋊C₄	—	268	2	no	—
134	C₄⋊C₄	—	536	4	no	—
135	C₂²⋊C₄	—	4	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	4	1	no	—
138	C₄²	#11761618	1	2	yes	—
139	C₁₆	—	268	2	no	—
140	C₂³	#14524068	1	1	yes	characteristic
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	536	4	no	—
145	C₂×C₄	—	536	2	no	—
146	C₂×C₄	#14524022	1	1	yes	characteristic
147	C₂×C₄	#14524024	1	1	yes	characteristic
148	C₂×C₄	#14524032	1	1	yes	characteristic
149	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
150	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
151	C₂×C₄	#14524068	1	1	yes	characteristic
152	C₈	—	2	2	no	—
153	C₈	—	8	1	no	—
154	C₈	#14524034	1	2	yes	—
155	C₈	#14524038	1	2	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	#16863686	1	1	yes	characteristic
158	C₂²	#16863729	1	1	yes	characteristic
159	C₂²	#16863762	1	1	yes	characteristic
160	C₄	—	2	4	no	—
161	C₄	—	8	2	no	—
162	C₄	—	1072	2	no	—
163	C₄	#16863335	1	1	yes	characteristic
164	C₄	#16863691	1	1	yes	characteristic
165	C₄	#16863712	1	1	yes	characteristic
166	C₄	#16863748	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#22561772	1	1	yes	characteristic
170	C₂	#22562380	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22560366

C₂

yes

maximal, characteristic

#22558314

C₂

yes

maximal

#22558309

C₂

yes

maximal

#22558291

C₂

yes

maximal, characteristic

C₆₇×#1945

C₂

yes

maximal, characteristic

C₂×#14523983

C₂²

yes

—

C₂×#14523975

—

C₂×#14523975

C₂²

yes

—

#16863266

—

C₂×#108×C₆₇

C₂²

yes

characteristic

#427×C₆₇

C₂²

yes

characteristic

#424×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#339×C₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

#14523983

—

#14523983

—

C₂×C₆₇⋊C₁₆

—

#14523975

—

#14523975

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

#117×C₆₇

—

C₂×C₄²×C₆₇

D₄

yes

characteristic

#108×C₆₇

—

#107×C₆₇

Q₈

yes

—

#106×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₂×C₈×C₆₇

#41

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

Q₁₆

yes

—

C₂³×C₆₇

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#125

yes

—

C₈×C₆₇

#126

yes

—

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

—

#13543

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#2100

yes

characteristic

C₂×C₆₇

#2230

yes

—

#1945

D₆₇

yes

—

—

—

—

C₆₇

#13543

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×#108

C₂×D₆₇

yes

characteristic

#427

C₂×D₆₇

yes

characteristic

C₂×#106

—

C₂×#106

—

134

—

#424

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#384

—

134

—

100

#339

—

101

C₂×M₁₆

—

102

C₂²×C₈

C₂²⋊D₆₇

yes

characteristic

103

C₂²×C₈

C₂²×D₆₇

yes

characteristic

104

#117

—

105

C₂×C₄⋊C₄

—

268

—

106

C₂×C₄⋊C₄

—

134

—

107

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

108

M₃₂

—

134

—

109

M₃₂

—

134

—

110

C₂×C₁₆

—

134

—

111

#108

—

112

#107

Dic₁₃₄

yes

—

113

#106

—

114

#106

—

268

—

115

#106

—

134

—

116

C₄⋊C₈

—

117

C₄×C₈

—

118

C₂²×C₄

—

119

C₂²×C₄

—

268

—

120

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

121

M₁₆

—

122

M₁₆

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

#11761616

yes

—

127

C₂×C₈

C₂×Dic₁₃₄

yes

characteristic

128

C₂×C₈

#11761627

yes

—

129

C₂×C₈

#11761628

yes

characteristic

130

C₂×C₈

#11761629

yes

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

268

—

133

C₄⋊C₄

—

268

—

134

C₄⋊C₄

—

536

—

135

C₂²⋊C₄

—

136

C₄²

—

137

C₄²

—

138

C₄²

#11761618

yes

—

139

C₁₆

—

268

—

140

C₂³

#14524068

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

536

—

145

C₂×C₄

—

536

—

146

C₂×C₄

#14524022

yes

characteristic

147

C₂×C₄

#14524024

yes

characteristic

148

C₂×C₄

#14524032

yes

characteristic

149

C₂×C₄

C₂×#11761616

yes

characteristic

150

C₂×C₄

C₂×#11761618

yes

characteristic

151

C₂×C₄

#14524068

yes

characteristic

152

C₈

—

153

C₈

—

154

C₈

#14524034

yes

—

155

C₈

#14524038

yes

—

156

C₂²

—

157

C₂²

#16863686

yes

characteristic

158

C₂²

#16863729

yes

characteristic

159

C₂²

#16863762

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

1072

—

163

C₄

#16863335

yes

characteristic

164

C₄

#16863691

yes

characteristic

165

C₄

#16863712

yes

characteristic

166

C₄

#16863748

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#22561772

yes

characteristic

170

C₂

#22562380

yes

—

	subgroup	G/H
center	C₂²	#16863686
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1945	C₂
socle	C₂²×C₆₇	#476
prime radical	C₂³×C₆₇	AGL(1,ℤ/8)
hypercenter	#1945	D₆₇
hypocenter	C₆₇	#13543
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1945

subgroup

G/H

center

C₂²

#16863686

commutator, cosocle

C₂²×C₈×C₆₇

C₂³

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1945

C₂

C₂²×C₆₇

C₂³×C₆₇

D₆₇

C₆₇

G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1945

N	H ≅ G/N	type
91. C₆₇	#13543	semidirect

H ≅ G/N

type

91. C₆₇

#13543

semidirect

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

subgroup	index
#22560366	2
#22558314	2
#22558309	2
#22558291	2
C₆₇×#1945	2
#13543	67

subgroup

index

C₆₇×#1945

Subgroups — #30775981

Conjugacy classes of subgroups

Complements

Maximal subgroups