Group #30749482 — subgroups

Subgroups — #30749482

Order 17152.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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	#22559962	C₂	1	1	yes	maximal, characteristic
2	#22558955	C₂	1	2	yes	maximal
3	#22558054	C₂	1	1	yes	maximal, characteristic
4	#22558037	C₂	1	2	yes	maximal
5	C₆₇×#1281	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	#16863242	C₂²	1	1	yes	characteristic
9	#16863150	C₄	1	2	yes	—
10	#16863150	C₄	1	2	yes	—
11	#16863148	—	2	2	no	—
12	#16863146	—	2	2	no	—
13	#431×C₆₇	C₂²	1	2	yes	—
14	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
18	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
19	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
20	C₄×C₆₇⋊C₈	—	2	2	no	—
21	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
22	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
23	C₄×C₆₇⋊C₈	Q₈	1	2	yes	—
24	C₆₇⋊#95	—	2	2	no	—
25	C₂²×C₈×C₆₇	C₂×C₄	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	C₂²×Dic₆₇	—	2	1	no	—
32	C₂²×Dic₆₇	—	2	1	no	—
33	C₂×C₆₇⋊C₈	—	2	2	no	—
34	C₂×C₆₇⋊C₈	—	2	2	no	—
35	C₂×C₆₇⋊C₈	—	4	2	no	—
36	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
37	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
38	C₄×Dic₆₇	—	2	2	no	—
39	C₄×Dic₆₇	—	2	2	no	—
40	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	#41	1	2	yes	—
42	C₂×C₈×C₆₇	—	2	2	no	—
43	C₂×C₈×C₆₇	—	2	2	no	—
44	C₂×C₈×C₆₇	—	2	2	no	—
45	C₂×C₈×C₆₇	—	4	2	no	—
46	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₆₇	#41	1	2	yes	—
49	C₂³×C₆₇	#117	1	1	yes	characteristic
50	C₂×C₄×C₆₇	—	4	2	no	—
51	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
53	C₂×C₄×C₆₇	#124	1	2	yes	—
54	C₂×C₄×C₆₇	#125	1	2	yes	—
55	C₂×C₄×C₆₇	#131	1	2	yes	—
56	C₂×C₄×C₆₇	#131	1	2	yes	—
57	C₂×Dic₆₇	—	4	2	no	—
58	C₂×Dic₆₇	—	4	2	no	—
59	C₂×Dic₆₇	—	4	1	no	—
60	C₂×Dic₆₇	—	4	1	no	—
61	C₈×C₆₇	—	4	2	no	—
62	C₈×C₆₇	—	8	2	no	—
63	C₆₇⋊C₈	—	2	2	no	—
64	C₆₇⋊C₈	—	2	2	no	—
65	C₆₇⋊C₈	—	8	2	no	—
66	C₂²×C₆₇	—	2	2	no	—
67	C₂²×C₆₇	#382	1	1	yes	characteristic
68	C₂²×C₆₇	#433	1	1	yes	characteristic
69	C₂²×C₆₇	#433	1	1	yes	characteristic
70	C₄×C₆₇	—	2	2	no	—
71	C₄×C₆₇	—	2	2	no	—
72	C₄×C₆₇	#424	1	2	yes	—
73	C₄×C₆₇	#433	1	2	yes	—
74	Dic₆₇	—	8	1	no	—
75	Dic₆₇	—	8	1	no	—
76	#8782	—	67	1	no	maximal, Sylow
77	C₂×C₆₇	—	2	1	no	—
78	C₂×C₆₇	—	2	1	no	—
79	C₂×C₆₇	#1832	1	1	yes	characteristic
80	C₂×C₆₇	#1833	1	2	yes	—
81	#1746	—	67	1	no	—
82	#1564	—	67	2	no	—
83	#1281	—	67	1	no	—
84	#1281	D₆₇	1	1	yes	characteristic
85	#1274	—	67	2	no	—
86	C₆₇	#8782	1	1	yes	characteristic, complemented, Sylow
87	#431	—	67	2	no	—
88	#431	C₂×D₆₇	1	2	yes	—
89	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
90	C₂×C₄×C₈	—	67	1	no	—
91	#376	—	67	1	no	—
92	#336	—	67	2	no	—
93	#336	—	67	2	no	—
94	#335	—	134	2	no	—
95	#334	—	134	2	no	—
96	C₂²×C₈	—	67	2	no	—
97	C₂²×C₈	C₄×D₆₇	1	2	yes	—
98	C₂×C₄²	—	67	1	no	—
99	C₂×C₄²	—	67	1	no	—
100	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
101	C₄⋊C₈	—	2	2	no	—
102	C₄⋊C₈	—	134	2	no	—
103	C₂²⋊C₈	—	2	2	no	—
104	C₂²⋊C₈	—	134	2	no	—
105	#97	—	2	2	no	—
106	C₄×C₈	—	2	2	no	—
107	C₄×C₈	—	67	1	no	—
108	C₄×C₈	—	67	1	no	—
109	C₄×C₈	—	67	2	no	—
110	C₄×C₈	—	134	2	no	—
111	#95	—	134	2	no	—
112	C₂²×C₄	—	134	1	no	—
113	C₂²×C₄	—	134	1	no	—
114	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
115	C₂²×C₄	#11761628	1	2	yes	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	268	2	no	—
121	C₂×C₈	—	67	2	no	—
122	C₂×C₈	—	67	2	no	—
123	C₂×C₈	—	134	2	no	—
124	C₂×C₈	—	134	2	no	—
125	C₄²	—	134	2	no	—
126	C₄²	—	134	2	no	—
127	C₄²	D₄×D₆₇	1	1	yes	characteristic
128	C₄²	Q₈×D₆₇	1	1	yes	characteristic
129	C₄²	#11761627	1	2	yes	—
130	C₂³	#14524000	1	1	yes	characteristic
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	268	2	no	—
133	C₂×C₄	—	268	2	no	—
134	C₂×C₄	—	268	1	no	—
135	C₂×C₄	—	268	1	no	—
136	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
137	C₂×C₄	#14523999	1	1	yes	characteristic
138	C₂×C₄	#14524029	1	2	yes	—
139	C₂×C₄	#14524033	1	2	yes	—
140	C₂×C₄	#14524049	1	2	yes	—
141	C₂×C₄	#14524051	1	2	yes	—
142	C₈	—	4	2	no	—
143	C₈	—	8	2	no	—
144	C₈	—	536	2	no	—
145	C₈	—	134	2	no	—
146	C₈	—	134	2	no	—
147	C₂²	—	2	2	no	—
148	C₂²	#16863260	1	1	yes	characteristic
149	C₂²	#16863455	1	1	yes	characteristic
150	C₂²	#16863458	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	536	1	no	—
154	C₄	—	536	1	no	—
155	C₄	#16863421	1	2	yes	—
156	C₄	#16863454	1	2	yes	—
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	#22560327	1	1	yes	characteristic
160	C₂	#22560330	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22559962

C₂

yes

maximal, characteristic

#22558955

C₂

yes

maximal

#22558054

C₂

yes

maximal, characteristic

#22558037

C₂

yes

maximal

C₆₇×#1281

C₂

yes

maximal, characteristic

#16863449

C₂²

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

characteristic

#16863242

C₂²

yes

characteristic

#16863150

C₄

yes

—

#16863150

C₄

yes

—

#16863148

—

#16863146

—

#431×C₆₇

C₂²

yes

—

C₂×#97×C₆₇

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₆₇

C₂×C₄

yes

characteristic

C₂×C₄×Dic₆₇

C₂×C₄

yes

characteristic

C₄⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

D₄

yes

characteristic

C₄×C₆₇⋊C₈

D₄

yes

characteristic

C₄×C₆₇⋊C₈

Q₈

yes

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

#97×C₆₇

—

C₄×C₈×C₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

C₂²×C₄

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

C₂×Q₈

yes

characteristic

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

#117

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂×C₄×C₆₇

#125

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#424

yes

—

C₄×C₆₇

#433

yes

—

Dic₆₇

—

Dic₆₇

—

#8782

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#1832

yes

characteristic

C₂×C₆₇

#1833

yes

—

—

—

—

D₆₇

yes

characteristic

#1274

—

C₆₇

#8782

yes

characteristic, complemented, Sylow

#431

—

#431

C₂×D₆₇

yes

—

C₂×#97

C₂×D₆₇

yes

characteristic

C₂×C₄×C₈

—

—

—

—

—

134

—

#334

—

134

—

C₂²×C₈

—

C₂²×C₈

C₄×D₆₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

100

C₂×C₄²

C₂²×D₆₇

yes

characteristic

101

C₄⋊C₈

—

102

C₄⋊C₈

—

134

—

103

C₂²⋊C₈

—

104

C₂²⋊C₈

—

134

—

105

#97

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

134

—

111

#95

—

134

—

112

C₂²×C₄

—

134

—

113

C₂²×C₄

—

134

—

114

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

115

C₂²×C₄

#11761628

yes

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

268

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

134

—

124

C₂×C₈

—

134

—

125

C₄²

—

134

—

126

C₄²

—

134

—

127

C₄²

D₄×D₆₇

yes

characteristic

128

C₄²

Q₈×D₆₇

yes

characteristic

129

C₄²

#11761627

yes

—

130

C₂³

#14524000

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

268

—

133

C₂×C₄

—

268

—

134

C₂×C₄

—

268

—

135

C₂×C₄

—

268

—

136

C₂×C₄

C₄⋊C₄×D₆₇

yes

characteristic

137

C₂×C₄

#14523999

yes

characteristic

138

C₂×C₄

#14524029

yes

—

139

C₂×C₄

#14524033

yes

—

140

C₂×C₄

#14524049

yes

—

141

C₂×C₄

#14524051

yes

—

142

C₈

—

143

C₈

—

144

C₈

—

536

—

145

C₈

—

134

—

146

C₈

—

134

—

147

C₂²

—

148

C₂²

#16863260

yes

characteristic

149

C₂²

#16863455

yes

characteristic

150

C₂²

#16863458

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

536

—

154

C₄

—

536

—

155

C₄

#16863421

yes

—

156

C₄

#16863454

yes

—

157

C₂

—

158

C₂

—

159

C₂

#22560327

yes

characteristic

160

C₂

#22560330

yes

—

	subgroup	G/H
center	C₂×C₄	#14523999
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1281	C₂
socle	C₂²×C₆₇	#382
prime radical	C₂³×C₆₇	#117
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1281	D₆₇
hypocenter	C₆₇	#8782
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1281

subgroup

G/H

center

C₂×C₄

#14523999

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1281

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1281

N	H ≅ G/N	type
86. C₆₇	#8782	semidirect

H ≅ G/N

type

86. C₆₇

#8782

semidirect

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

subgroup	index
#22559962	2
#22558955	2
#22558054	2
#22558037	2
C₆₇×#1281	2
#8782	67

subgroup

index

C₆₇×#1281

Subgroups — #30749482

Conjugacy classes of subgroups

Complements

Maximal subgroups