Group #30749508 — subgroups

Subgroups — #30749508

Order 17152.

342 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 155 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	#22560361	C₂	1	1	yes	maximal, characteristic
2	#22560326	C₂	1	1	yes	maximal, characteristic
3	#22558044	C₂	1	4	yes	maximal
4	C₆₇×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
8	#16863151	—	2	2	no	—
9	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
10	C₄³×C₆₇	C₂²	1	1	yes	characteristic
11	#336×C₆₇	C₄	1	2	yes	—
12	#336×C₆₇	C₄	1	2	yes	—
13	C₂²×C₆₇⋊C₈	—	2	2	no	—
14	C₂²×C₆₇⋊C₈	—	2	2	no	—
15	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
16	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
17	C₄⋊(C₆₇⋊C₈)	—	4	4	no	—
18	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
19	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₆₇	D₄	1	2	yes	—
22	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
23	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
24	C₄×C₈×C₆₇	—	2	2	no	—
25	C₄×C₈×C₆₇	—	2	2	no	—
26	C₂×C₆₇⋊C₈	—	4	2	no	—
27	C₂×C₆₇⋊C₈	—	4	2	no	—
28	C₂×C₆₇⋊C₈	—	4	4	no	—
29	C₂×C₆₇⋊C₈	—	4	2	no	—
30	C₂×C₆₇⋊C₈	—	4	2	no	—
31	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
33	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈×C₆₇	—	2	4	no	—
37	C₂×C₈×C₆₇	—	2	4	no	—
38	C₂×C₈×C₆₇	—	2	4	no	—
39	C₄²×C₆₇	—	2	4	no	—
40	C₄²×C₆₇	—	2	2	no	—
41	C₄²×C₆₇	—	2	2	no	—
42	C₄²×C₆₇	—	2	2	no	—
43	C₄²×C₆₇	—	2	2	no	—
44	C₄²×C₆₇	#41	1	2	yes	—
45	C₄²×C₆₇	#41	1	2	yes	—
46	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₆₇	—	2	2	no	—
48	C₂×C₄×C₆₇	—	2	2	no	—
49	C₂×C₄×C₆₇	—	2	2	no	—
50	C₂×C₄×C₆₇	—	2	2	no	—
51	C₂×C₄×C₆₇	—	2	2	no	—
52	C₂×C₄×C₆₇	—	2	2	no	—
53	C₂×C₄×C₆₇	#108	1	4	yes	—
54	C₂×C₄×C₆₇	#117	1	2	yes	—
55	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
56	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
57	C₂×C₄×C₆₇	#125	1	2	yes	—
58	C₂×C₄×C₆₇	#131	1	2	yes	—
59	C₂×C₄×C₆₇	#131	1	2	yes	—
60	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
61	C₂×C₄×C₆₇	#137	1	2	yes	—
62	C₈×C₆₇	—	4	4	no	—
63	C₆₇⋊C₈	—	8	2	no	—
64	C₆₇⋊C₈	—	8	2	no	—
65	C₂²×C₆₇	#382	1	1	yes	characteristic
66	C₂²×C₆₇	#424	1	1	yes	characteristic
67	C₂²×C₆₇	#424	1	1	yes	characteristic
68	C₂²×C₆₇	#428	1	2	yes	—
69	C₂²×C₆₇	C₂×#108	1	2	yes	—
70	C₄×C₆₇	—	2	4	no	—
71	C₄×C₆₇	—	2	2	no	—
72	C₄×C₆₇	—	4	2	no	—
73	C₄×C₆₇	#433	1	4	yes	—
74	C₄×C₆₇	#489	1	2	yes	—
75	C₄×C₆₇	#490	1	1	yes	characteristic
76	C₄×C₆₇	#491	1	1	yes	characteristic
77	#8789	—	67	1	no	maximal, Sylow
78	C₂×C₆₇	#1573	1	2	yes	—
79	C₂×C₆₇	#1574	1	2	yes	—
80	C₂×C₆₇	#1832	1	1	yes	characteristic
81	C₂×C₆₇	#1836	1	1	yes	characteristic
82	C₂×C₆₇	#1837	1	1	yes	characteristic
83	#1842	—	67	1	no	—
84	#1831	—	67	1	no	—
85	#1744	D₆₇	1	1	yes	characteristic
86	#1277	—	67	4	no	—
87	C₆₇	#8789	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₄⋊C₈	—	67	2	no	—
89	C₂×C₄⋊C₈	—	67	2	no	—
90	C₂×C₄⋊C₈	—	134	2	no	—
91	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
92	C₄³	C₂×D₆₇	1	1	yes	characteristic
93	#336	—	134	2	no	—
94	#336	Dic₆₇	1	2	yes	—
95	#336	Dic₆₇	1	2	yes	—
96	C₂²×C₈	—	134	2	no	—
97	C₂²×C₈	—	134	2	no	—
98	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
99	C₂×C₄²	Dic₁₃₄	1	2	yes	—
100	C₂×C₄²	D₂₆₈	1	2	yes	—
101	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
102	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
103	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
104	C₄⋊C₈	—	268	4	no	—
105	C₄⋊C₈	—	134	4	no	—
106	C₄⋊C₈	—	134	4	no	—
107	C₄×C₈	—	2	2	no	—
108	C₄×C₈	—	2	2	no	—
109	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
110	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
111	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
112	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
113	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	2	4	no	—
116	C₂×C₈	—	2	4	no	—
117	C₂×C₈	—	268	2	no	—
118	C₂×C₈	—	268	2	no	—
119	C₂×C₈	—	268	4	no	—
120	C₂×C₈	—	268	2	no	—
121	C₂×C₈	—	268	2	no	—
122	C₄²	—	2	4	no	—
123	C₄²	—	2	2	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	#11761628	1	2	yes	—
128	C₄²	#11761628	1	2	yes	—
129	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	#14523979	1	4	yes	—
137	C₂×C₄	#14524002	1	2	yes	—
138	C₂×C₄	#14524031	1	1	yes	characteristic
139	C₂×C₄	#14524031	1	1	yes	characteristic
140	C₂×C₄	#14524035	1	2	yes	—
141	C₂×C₄	#14524050	1	2	yes	—
142	C₂×C₄	#14524050	1	2	yes	—
143	C₂×C₄	#14524071	1	2	yes	—
144	C₂×C₄	#14524076	1	2	yes	—
145	C₈	—	4	4	no	—
146	C₈	—	536	2	no	—
147	C₈	—	536	2	no	—
148	C₂²	#16863262	1	1	yes	characteristic
149	C₂²	#16863423	1	1	yes	characteristic
150	C₂²	#16863423	1	1	yes	characteristic
151	C₂²	#16863438	1	2	yes	—
152	C₂²	C₂×#14523979	1	2	yes	—
153	C₄	—	2	4	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	2	no	—
156	C₄	#16863457	1	4	yes	—
157	C₄	#16863756	1	2	yes	—
158	C₄	#16863761	1	1	yes	characteristic
159	C₄	#16863765	1	1	yes	characteristic
160	C₂	#22558994	1	2	yes	—
161	C₂	#22558998	1	2	yes	—
162	C₂	#22560329	1	1	yes	characteristic
163	C₂	#22560343	1	1	yes	characteristic
164	C₂	#22560347	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560361

C₂

yes

maximal, characteristic

#22560326

C₂

yes

maximal, characteristic

#22558044

C₂

yes

maximal

C₆₇×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₆₇⋊C₈)

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

#16863151

—

C₂×C₄×C₈×C₆₇

C₂²

yes

—

C₄³×C₆₇

C₂²

yes

characteristic

#336×C₆₇

C₄

yes

—

#336×C₆₇

C₄

yes

—

C₂²×C₆₇⋊C₈

—

C₂²×C₆₇⋊C₈

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

Q₈

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

#41

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#108

yes

—

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₂²×C₆₇

C₂×#108

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#433

yes

—

C₄×C₆₇

#489

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₂×C₆₇

yes

—

C₂×C₆₇

#1574

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

—

D₆₇

yes

characteristic

#1277

—

C₆₇

#8789

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

134

—

C₂×C₄×C₈

C₂×D₆₇

yes

—

C₄³

C₂×D₆₇

yes

characteristic

#336

—

134

—

#336

Dic₆₇

yes

—

#336

Dic₆₇

yes

—

C₂²×C₈

—

134

—

C₂²×C₈

—

134

—

C₂²×C₈

C₂×Dic₆₇

yes

—

C₂×C₄²

Dic₁₃₄

yes

—

100

C₂×C₄²

D₂₆₈

yes

—

101

C₂×C₄²

C₂×Dic₆₇

yes

characteristic

102

C₂×C₄²

C₂×Dic₆₇

yes

characteristic

103

C₂×C₄²

C₂²×D₆₇

yes

characteristic

104

C₄⋊C₈

—

268

—

105

C₄⋊C₈

—

134

—

106

C₄⋊C₈

—

134

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₂²×C₄

C₄⋊Dic₆₇

yes

—

110

C₂²×C₄

C₄⋊Dic₆₇

yes

—

111

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

112

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

113

C₂²×C₄

C₂×Dic₁₃₄

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

268

—

118

C₂×C₈

—

268

—

119

C₂×C₈

—

268

—

120

C₂×C₈

—

268

—

121

C₂×C₈

—

268

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

#11761628

yes

—

128

C₄²

#11761628

yes

—

129

C₂³

C₂×C₄⋊Dic₆₇

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#14523979

yes

—

137

C₂×C₄

#14524002

yes

—

138

C₂×C₄

#14524031

yes

characteristic

139

C₂×C₄

#14524031

yes

characteristic

140

C₂×C₄

#14524035

yes

—

141

C₂×C₄

#14524050

yes

—

142

C₂×C₄

#14524050

yes

—

143

C₂×C₄

#14524071

yes

—

144

C₂×C₄

#14524076

yes

—

145

C₈

—

146

C₈

—

536

—

147

C₈

—

536

—

148

C₂²

#16863262

yes

characteristic

149

C₂²

#16863423

yes

characteristic

150

C₂²

#16863423

yes

characteristic

151

C₂²

#16863438

yes

—

152

C₂²

C₂×#14523979

yes

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#16863457

yes

—

157

C₄

#16863756

yes

—

158

C₄

#16863761

yes

characteristic

159

C₄

#16863765

yes

characteristic

160

C₂

#22558994

yes

—

161

C₂

#22558998

yes

—

162

C₂

#22560329

yes

characteristic

163

C₂

#22560343

yes

characteristic

164

C₂

#22560347

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1744	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1744	D₆₇
hypocenter	C₆₇	#8789
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₂₆₈

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1744

C₂

socle, prime radical

C₂³×C₆₇

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
87. C₆₇	#8789	semidirect

H ≅ G/N

type

87. C₆₇

#8789

semidirect

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

subgroup	index
#22560361	2
#22560326	2
#22558044	2
C₆₇×#1744	2
#8789	67

subgroup

index

C₆₇×#1744

Subgroups — #30749508

Conjugacy classes of subgroups

Complements

Maximal subgroups