Group #30775982 — subgroups

Subgroups — #30775982

Order 17152.

268 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 93 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	#22560957	C₂	1	1	yes	maximal, characteristic
2	#22560366	C₂	1	1	yes	maximal, characteristic
3	#22558315	C₂	1	2	yes	maximal
4	#22558308	C₂	1	2	yes	maximal
5	C₆₇×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523979	C₂²	1	1	yes	characteristic
7	#16863434	C₂²	1	1	yes	characteristic
8	C₂×#14523975	—	2	1	no	—
9	C₂×#14523975	C₂²	1	2	yes	—
10	#16863422	—	2	1	no	—
11	#16863266	—	2	1	no	—
12	#16863156	—	2	1	no	—
13	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#11761612	—	2	1	no	—
16	#14524002	—	2	1	no	—
17	C₂×C₄⋊Dic₆₇	—	2	2	no	—
18	C₂×C₄⋊Dic₆₇	—	4	2	no	—
19	#14523979	—	2	2	no	—
20	#14523977	Q₈	1	2	yes	—
21	#14523975	—	2	1	no	—
22	#14523975	—	2	4	no	—
23	#14523975	—	4	2	no	—
24	Dic₆₇⋊C₈	—	4	2	no	—
25	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
28	M₃₂×C₆₇	—	2	2	no	—
29	M₃₂×C₆₇	—	2	2	no	—
30	C₂×C₁₆×C₆₇	—	2	2	no	—
31	C₄×C₈×C₆₇	—	2	2	no	—
32	C₂²×Dic₆₇	—	4	2	no	—
33	#11761612	—	4	1	no	—
34	#11761612	—	4	1	no	—
35	C₂×C₆₇⋊C₈	—	4	1	no	—
36	C₄⋊Dic₆₇	—	2	2	no	—
37	C₄⋊Dic₆₇	—	4	2	no	—
38	C₄⋊Dic₆₇	—	4	2	no	—
39	C₄⋊Dic₆₇	—	8	4	no	—
40	C₂²⋊Dic₆₇	—	4	1	no	—
41	C₄×Dic₆₇	—	4	1	no	—
42	C₂²×C₄×C₆₇	—	2	1	no	—
43	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₆₇	—	2	1	no	—
45	C₂×C₈×C₆₇	—	2	1	no	—
46	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
47	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₆₇	#41	1	2	yes	—
49	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
50	C₂×C₈×C₆₇	#41	1	2	yes	—
51	C₄²×C₆₇	—	2	1	no	—
52	C₄²×C₆₇	Q₁₆	1	2	yes	—
53	C₁₆×C₆₇	—	4	2	no	—
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	C₂²⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
59	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
60	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
62	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₂×Dic₆₇	—	4	2	no	—
64	C₂×Dic₆₇	—	8	4	no	—
65	C₂×Dic₆₇	—	8	2	no	—
66	C₈×C₆₇	—	2	2	no	—
67	C₈×C₆₇	#125	1	2	yes	—
68	C₈×C₆₇	#126	1	2	yes	—
69	C₆₇⋊C₈	—	8	1	no	—
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	#400	1	1	yes	characteristic
76	C₄×C₆₇	#477	1	1	yes	characteristic
77	C₄×C₆₇	#481	1	1	yes	characteristic
78	C₄×C₆₇	#487	1	1	yes	characteristic
79	Dic₆₇	—	8	2	no	—
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	—	67	1	no	—
87	#1844	—	67	1	no	—
88	#1386	—	67	2	no	—
89	#1383	—	67	2	no	—
90	#1376	D₆₇	1	1	yes	characteristic
91	C₆₇	#13543	1	1	yes	characteristic, complemented, Sylow
92	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
93	C₂×#108	—	67	1	no	—
94	#427	—	67	1	no	—
95	C₂×#106	—	67	2	no	—
96	C₂×#106	—	134	1	no	—
97	#424	—	134	1	no	—
98	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
99	#384	—	134	1	no	—
100	#339	—	134	1	no	—
101	C₂×M₁₆	—	134	1	no	—
102	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
104	#117	—	134	1	no	—
105	C₂×C₄⋊C₄	—	268	2	no	—
106	C₂×C₄⋊C₄	—	134	2	no	—
107	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
108	M₃₂	—	2	2	no	—
109	M₃₂	—	2	2	no	—
110	C₂×C₁₆	—	2	2	no	—
111	#108	—	134	2	no	—
112	#107	—	67	2	no	—
113	#106	—	268	2	no	—
114	#106	—	134	4	no	—
115	#106	—	134	1	no	—
116	C₄⋊C₈	—	268	2	no	—
117	C₄×C₈	—	2	2	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	268	2	no	—
120	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
121	M₁₆	—	268	1	no	—
122	M₁₆	—	268	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	268	1	no	—
126	C₂×C₈	#11761617	1	2	yes	—
127	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
128	C₂×C₈	#11761627	1	2	yes	—
129	C₂×C₈	#11761627	1	1	yes	characteristic
130	C₂×C₈	#11761628	1	2	yes	—
131	C₄⋊C₄	—	268	2	no	—
132	C₄⋊C₄	—	268	2	no	—
133	C₄⋊C₄	—	536	4	no	—
134	C₄⋊C₄	—	134	2	no	—
135	C₂²⋊C₄	—	268	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	268	1	no	—
138	C₄²	Dic₂₆₈	1	2	yes	—
139	C₁₆	—	4	2	no	—
140	C₂³	#14524071	1	1	yes	characteristic
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	268	2	no	—
144	C₂×C₄	—	536	4	no	—
145	C₂×C₄	—	536	2	no	—
146	C₂×C₄	#14524023	1	1	yes	characteristic
147	C₂×C₄	#14524027	1	1	yes	characteristic
148	C₂×C₄	#14524031	1	1	yes	characteristic
149	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
150	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
151	C₂×C₄	#14524071	1	1	yes	characteristic
152	C₈	—	2	2	no	—
153	C₈	—	536	1	no	—
154	C₈	#14524033	1	2	yes	—
155	C₈	#14524036	1	2	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	#16863687	1	1	yes	characteristic
158	C₂²	#16863732	1	1	yes	characteristic
159	C₂²	#16863761	1	1	yes	characteristic
160	C₄	—	2	4	no	—
161	C₄	—	536	2	no	—
162	C₄	—	1072	2	no	—
163	C₄	#16863336	1	1	yes	characteristic
164	C₄	#16863692	1	1	yes	characteristic
165	C₄	#16863715	1	1	yes	characteristic
166	C₄	#16863747	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#22561773	1	1	yes	characteristic
170	C₂	#22562381	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22560957

C₂

yes

maximal, characteristic

#22560366

C₂

yes

maximal, characteristic

#22558315

C₂

yes

maximal

#22558308

C₂

yes

maximal

C₆₇×#1376

C₂

yes

maximal, characteristic

C₂×#14523979

C₂²

yes

characteristic

#16863434

C₂²

yes

characteristic

C₂×#14523975

—

C₂×#14523975

C₂²

yes

—

#16863422

—

#16863266

—

#16863156

—

C₂×M₃₂×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×#11761612

—

#14524002

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

#14523979

—

#14523977

Q₈

yes

—

#14523975

—

#14523975

—

#14523975

—

Dic₆₇⋊C₈

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

M₃₂×C₆₇

—

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

—

C₄×C₈×C₆₇

—

C₂²×Dic₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

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

—

C₄²×C₆₇

—

C₄²×C₆₇

Q₁₆

yes

—

C₁₆×C₆₇

—

C₂³×C₆₇

AGL(1,ℤ/8)

yes

characteristic

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

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#125

yes

—

C₈×C₆₇

#126

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

#13543

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#2100

yes

characteristic

C₂×C₆₇

#2230

yes

—

—

—

—

—

D₆₇

yes

characteristic

C₆₇

#13543

yes

characteristic, complemented, Sylow

C₂×M₃₂

C₂×D₆₇

yes

—

—

—

—

—

134

—

#424

—

134

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#384

—

134

—

100

#339

—

134

—

101

C₂×M₁₆

—

134

—

102

C₂²×C₈

D₂₆₈

yes

characteristic

103

C₂²×C₈

C₂²×D₆₇

yes

characteristic

104

#117

—

134

—

105

C₂×C₄⋊C₄

—

268

—

106

C₂×C₄⋊C₄

—

134

—

107

C₂×C₄²

D₂₆₈

yes

characteristic

108

M₃₂

—

109

M₃₂

—

110

C₂×C₁₆

—

111

#108

—

134

—

112

#107

—

113

#106

—

268

—

114

#106

—

134

—

115

#106

—

134

—

116

C₄⋊C₈

—

268

—

117

C₄×C₈

—

118

C₂²×C₄

—

119

C₂²×C₄

—

268

—

120

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

121

M₁₆

—

268

—

122

M₁₆

—

268

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

268

—

126

C₂×C₈

#11761617

yes

—

127

C₂×C₈

Q₈×D₆₇

yes

characteristic

128

C₂×C₈

#11761627

yes

—

129

C₂×C₈

#11761627

yes

characteristic

130

C₂×C₈

#11761628

yes

—

131

C₄⋊C₄

—

268

—

132

C₄⋊C₄

—

268

—

133

C₄⋊C₄

—

536

—

134

C₄⋊C₄

—

134

—

135

C₂²⋊C₄

—

268

—

136

C₄²

—

137

C₄²

—

268

—

138

C₄²

Dic₂₆₈

yes

—

139

C₁₆

—

140

C₂³

#14524071

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

268

—

144

C₂×C₄

—

536

—

145

C₂×C₄

—

536

—

146

C₂×C₄

#14524023

yes

characteristic

147

C₂×C₄

#14524027

yes

characteristic

148

C₂×C₄

#14524031

yes

characteristic

149

C₂×C₄

C₂×#11761617

yes

characteristic

150

C₂×C₄

C₂×Dic₂₆₈

yes

characteristic

151

C₂×C₄

#14524071

yes

characteristic

152

C₈

—

153

C₈

—

536

—

154

C₈

#14524033

yes

—

155

C₈

#14524036

yes

—

156

C₂²

—

157

C₂²

#16863687

yes

characteristic

158

C₂²

#16863732

yes

characteristic

159

C₂²

#16863761

yes

characteristic

160

C₄

—

161

C₄

—

536

—

162

C₄

—

1072

—

163

C₄

#16863336

yes

characteristic

164

C₄

#16863692

yes

characteristic

165

C₄

#16863715

yes

characteristic

166

C₄

#16863747

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#22561773

yes

characteristic

170

C₂

#22562381

yes

—

	subgroup	G/H
center	C₂²	#16863687
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1376	C₂
socle	C₂²×C₆₇	#476
prime radical	C₂³×C₆₇	AGL(1,ℤ/8)
hypercenter	#1376	D₆₇
hypocenter	C₆₇	#13543
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

subgroup

G/H

center

C₂²

#16863687

commutator, cosocle

C₂²×C₈×C₆₇

C₂³

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1376

C₂

C₂²×C₆₇

C₂³×C₆₇

D₆₇

C₆₇

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

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
#22560957	2
#22560366	2
#22558315	2
#22558308	2
C₆₇×#1376	2
#13543	67

subgroup

index

C₆₇×#1376

Subgroups — #30775982

Conjugacy classes of subgroups

Complements

Maximal subgroups