Group #30749483 — subgroups

Subgroups — #30749483

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	#22559961	C₂	1	1	yes	maximal, characteristic
2	#22558955	C₂	1	2	yes	maximal
3	#22558053	C₂	1	2	yes	maximal
4	#22558038	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1274	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	2	yes	—
7	C₂×#14523954	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	#16863151	C₄	1	2	yes	—
10	#16863151	C₄	1	2	yes	—
11	#16863148	—	2	2	no	—
12	#16863146	—	2	2	no	—
13	#431×C₆₇	C₂²	1	2	yes	—
14	#376×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
18	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
19	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
20	#14523954	—	2	2	no	—
21	C₄×C₆₇⋊C₈	—	2	2	no	—
22	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
23	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
24	C₄×C₆₇⋊C₈	Q₈	1	2	yes	—
25	C₂×C₄²×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	C₄×C₈×C₆₇	—	2	2	no	—
30	#95×C₆₇	—	2	2	no	—
31	C₂×C₆₇⋊C₈	—	2	2	no	—
32	C₂×C₆₇⋊C₈	—	2	2	no	—
33	C₂×C₆₇⋊C₈	—	2	1	no	—
34	C₂×C₆₇⋊C₈	—	2	1	no	—
35	C₂×C₆₇⋊C₈	—	2	2	no	—
36	C₂×C₆₇⋊C₈	—	2	2	no	—
37	C₂×C₆₇⋊C₈	—	4	2	no	—
38	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
39	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₆₇	—	2	2	no	—
41	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	#41	1	2	yes	—
43	C₂×C₈×C₆₇	—	4	2	no	—
44	C₄²×C₆₇	—	2	2	no	—
45	C₄²×C₆₇	—	2	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	—	4	4	no	—
52	C₂×C₄×C₆₇	—	4	2	no	—
53	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
55	C₂×C₄×C₆₇	#124	1	2	yes	—
56	C₂×C₄×C₆₇	#125	1	2	yes	—
57	C₂×C₄×C₆₇	#131	1	2	yes	—
58	C₂×C₄×C₆₇	#131	1	2	yes	—
59	C₈×C₆₇	—	8	2	no	—
60	C₆₇⋊C₈	—	2	2	no	—
61	C₆₇⋊C₈	—	2	2	no	—
62	C₆₇⋊C₈	—	4	1	no	—
63	C₆₇⋊C₈	—	4	1	no	—
64	C₆₇⋊C₈	—	8	2	no	—
65	C₂²×C₆₇	—	2	2	no	—
66	C₂²×C₆₇	#382	1	1	yes	characteristic
67	C₂²×C₆₇	#433	1	2	yes	—
68	C₄×C₆₇	—	2	2	no	—
69	C₄×C₆₇	—	2	2	no	—
70	C₄×C₆₇	—	8	2	no	—
71	C₄×C₆₇	#424	1	2	yes	—
72	C₄×C₆₇	#433	1	1	yes	characteristic
73	C₄×C₆₇	#433	1	1	yes	characteristic
74	#8782	—	67	1	no	maximal, Sylow
75	C₂×C₆₇	—	2	2	no	—
76	C₂×C₆₇	#1832	1	1	yes	characteristic
77	C₂×C₆₇	#1833	1	2	yes	—
78	#1746	—	67	1	no	—
79	#1564	—	67	2	no	—
80	#1281	—	67	2	no	—
81	#1274	—	67	1	no	—
82	#1274	D₆₇	1	1	yes	characteristic
83	C₆₇	#8782	1	1	yes	characteristic, complemented, Sylow
84	#431	—	67	2	no	—
85	#431	C₂×D₆₇	1	2	yes	—
86	C₂×#97	—	67	1	no	—
87	C₂×C₄×C₈	—	67	1	no	—
88	#376	C₂×D₆₇	1	1	yes	characteristic
89	#336	—	67	2	no	—
90	#336	—	67	2	no	—
91	#335	—	134	2	no	—
92	#334	—	134	2	no	—
93	C₂²×C₈	—	67	2	no	—
94	C₂²×C₈	—	67	1	no	—
95	C₂²×C₈	—	67	1	no	—
96	C₂×C₄²	C₄×D₆₇	1	2	yes	—
97	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
98	C₄⋊C₈	—	2	2	no	—
99	C₄⋊C₈	—	134	2	no	—
100	C₂²⋊C₈	—	2	2	no	—
101	C₂²⋊C₈	—	134	2	no	—
102	#97	—	134	2	no	—
103	C₄×C₈	—	2	2	no	—
104	C₄×C₈	—	67	1	no	—
105	C₄×C₈	—	67	1	no	—
106	C₄×C₈	—	67	2	no	—
107	C₄×C₈	—	134	2	no	—
108	#95	—	2	2	no	—
109	C₂²×C₄	—	2	2	no	—
110	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
111	C₂²×C₄	#11761627	1	2	yes	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	268	2	no	—
114	C₂×C₈	—	67	2	no	—
115	C₂×C₈	—	67	2	no	—
116	C₂×C₈	—	134	1	no	—
117	C₂×C₈	—	134	1	no	—
118	C₂×C₈	—	134	2	no	—
119	C₂×C₈	—	134	2	no	—
120	C₂×C₈	—	134	2	no	—
121	C₂×C₈	—	134	2	no	—
122	C₄²	—	2	2	no	—
123	C₄²	—	2	2	no	—
124	C₄²	D₄×D₆₇	1	1	yes	characteristic
125	C₄²	Q₈×D₆₇	1	1	yes	characteristic
126	C₄²	#11761628	1	2	yes	—
127	C₂³	#14523999	1	1	yes	characteristic
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
132	C₂×C₄	#14524000	1	1	yes	characteristic
133	C₂×C₄	#14524029	1	2	yes	—
134	C₂×C₄	#14524033	1	2	yes	—
135	C₂×C₄	#14524049	1	2	yes	—
136	C₂×C₄	#14524051	1	2	yes	—
137	C₈	—	8	2	no	—
138	C₈	—	268	1	no	—
139	C₈	—	268	1	no	—
140	C₈	—	536	2	no	—
141	C₈	—	134	2	no	—
142	C₈	—	134	2	no	—
143	C₂²	—	2	2	no	—
144	C₂²	#16863260	1	1	yes	characteristic
145	C₂²	#16863454	1	2	yes	—
146	C₄	—	2	2	no	—
147	C₄	—	2	2	no	—
148	C₄	—	8	2	no	—
149	C₄	#16863421	1	2	yes	—
150	C₄	#16863455	1	1	yes	characteristic
151	C₄	#16863458	1	1	yes	characteristic
152	C₂	—	2	2	no	—
153	C₂	#22560327	1	1	yes	characteristic
154	C₂	#22560331	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22559961

C₂

yes

maximal, characteristic

#22558955

C₂

yes

maximal

#22558053

C₂

yes

maximal

#22558038

C₂

yes

maximal, characteristic

C₆₇×#1274

C₂

yes

maximal, characteristic

#16863449

C₂²

yes

—

C₂×#14523954

C₂²

yes

characteristic

C₂×C₄×C₆₇⋊C₈

C₂²

yes

characteristic

#16863151

C₄

yes

—

#16863151

C₄

yes

—

#16863148

—

#16863146

—

#431×C₆₇

C₂²

yes

—

#376×C₆₇

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₆₇⋊C₈

C₂×C₄

yes

—

C₄⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

#14523954

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

D₄

yes

characteristic

C₄×C₆₇⋊C₈

D₄

yes

characteristic

C₄×C₆₇⋊C₈

Q₈

yes

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

—

#95×C₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

—

C₄²×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×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

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#382

yes

characteristic

C₂²×C₆₇

#433

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#424

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₂×C₆₇

—

C₂×C₆₇

#1832

yes

characteristic

C₂×C₆₇

#1833

yes

—

—

—

—

—

D₆₇

yes

characteristic

C₆₇

#8782

yes

characteristic, complemented, Sylow

#431

—

#431

C₂×D₆₇

yes

—

C₂×#97

—

C₂×C₄×C₈

—

#376

C₂×D₆₇

yes

—

—

—

134

—

#334

—

134

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₄×D₆₇

yes

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

134

—

100

C₂²⋊C₈

—

101

C₂²⋊C₈

—

134

—

102

#97

—

134

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

134

—

108

#95

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

111

C₂²×C₄

#11761627

yes

—

112

C₂×C₈

—

113

C₂×C₈

—

268

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

134

—

117

C₂×C₈

—

134

—

118

C₂×C₈

—

134

—

119

C₂×C₈

—

134

—

120

C₂×C₈

—

134

—

121

C₂×C₈

—

134

—

122

C₄²

—

123

C₄²

—

124

C₄²

D₄×D₆₇

yes

characteristic

125

C₄²

Q₈×D₆₇

yes

characteristic

126

C₄²

#11761628

yes

—

127

C₂³

#14523999

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₄⋊C₄×D₆₇

yes

characteristic

132

C₂×C₄

#14524000

yes

characteristic

133

C₂×C₄

#14524029

yes

—

134

C₂×C₄

#14524033

yes

—

135

C₂×C₄

#14524049

yes

—

136

C₂×C₄

#14524051

yes

—

137

C₈

—

138

C₈

—

268

—

139

C₈

—

268

—

140

C₈

—

536

—

141

C₈

—

134

—

142

C₈

—

134

—

143

C₂²

—

144

C₂²

#16863260

yes

characteristic

145

C₂²

#16863454

yes

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#16863421

yes

—

150

C₄

#16863455

yes

characteristic

151

C₄

#16863458

yes

characteristic

152

C₂

—

153

C₂

#22560327

yes

characteristic

154

C₂

#22560331

yes

—

	subgroup	G/H
center	C₂×C₄	#14524000
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1274	C₂
socle	C₂²×C₆₇	#382
prime radical	C₂³×C₆₇	#117
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1274	D₆₇
hypocenter	C₆₇	#8782
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ #1274

subgroup

G/H

center

C₂×C₄

#14524000

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1274

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ #376 ◁ #1274

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

H ≅ G/N

type

83. C₆₇

#8782

semidirect

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

subgroup	index
#22559961	2
#22558955	2
#22558053	2
#22558038	2
C₆₇×#1274	2
#8782	67

subgroup

index

C₆₇×#1274

Subgroups — #30749483

Conjugacy classes of subgroups

Complements

Maximal subgroups