Group #30749546 — subgroups

Subgroups — #30749546

Order 17152.

220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 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	#22558972	C₂	1	1	yes	maximal, characteristic
2	#22558501	C₂	1	1	yes	maximal, characteristic
3	#22558072	C₂	1	2	yes	maximal
4	#22558070	C₂	1	2	yes	maximal
5	C₆₇×#1568	C₂	1	1	yes	maximal, characteristic
6	#16863453	C₂²	1	2	yes	—
7	#16863449	C₂²	1	1	yes	characteristic
8	#16863449	C₂²	1	1	yes	characteristic
9	#16863149	—	2	1	no	—
10	#16863147	—	2	1	no	—
11	#16863118	C₄	1	4	yes	—
12	#432×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#335×C₆₇	—	2	1	no	—
15	#334×C₆₇	—	2	1	no	—
16	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
17	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
18	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
19	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
20	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
21	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
22	#14523954	—	2	2	no	—
23	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₆₇	Q₈	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	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
32	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
33	C₂×C₆₇⋊C₈	—	2	2	no	—
34	C₂×C₆₇⋊C₈	—	2	2	no	—
35	C₂×C₆₇⋊C₈	—	4	2	no	—
36	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
37	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
38	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
39	C₂×C₈×C₆₇	—	4	2	no	—
40	C₂×C₈×C₆₇	—	4	2	no	—
41	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
43	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
44	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₄²×C₆₇	#41	1	1	yes	characteristic
46	C₄²×C₆₇	#41	1	1	yes	characteristic
47	C₂³×C₆₇	#125	1	1	yes	characteristic
48	C₂×C₄×C₆₇	—	2	2	no	—
49	C₂×C₄×C₆₇	—	4	1	no	—
50	C₂×C₄×C₆₇	—	4	1	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	#117	1	1	yes	characteristic
54	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
55	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
56	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
57	C₈×C₆₇	—	2	2	no	—
58	C₈×C₆₇	—	2	2	no	—
59	C₈×C₆₇	—	8	2	no	—
60	C₆₇⋊C₈	—	4	2	no	—
61	C₆₇⋊C₈	—	4	2	no	—
62	C₆₇⋊C₈	—	8	2	no	—
63	C₂²×C₆₇	—	2	1	no	—
64	C₂²×C₆₇	—	2	1	no	—
65	C₂²×C₆₇	—	2	1	no	—
66	C₂²×C₆₇	#382	1	1	yes	characteristic
67	C₄×C₆₇	—	2	1	no	—
68	C₄×C₆₇	—	2	1	no	—
69	C₄×C₆₇	—	2	1	no	—
70	C₄×C₆₇	—	2	1	no	—
71	C₄×C₆₇	—	2	1	no	—
72	C₄×C₆₇	#430	1	2	yes	—
73	#8799	—	67	1	no	maximal, Sylow
74	C₂×C₆₇	—	4	1	no	—
75	C₂×C₆₇	#1833	1	1	yes	characteristic
76	C₂×C₆₇	#1833	1	1	yes	characteristic
77	C₂×C₆₇	#1838	1	1	yes	characteristic
78	#1568	—	67	1	no	—
79	#1568	D₆₇	1	1	yes	characteristic
80	#1452	—	67	1	no	—
81	#1287	—	67	2	no	—
82	#1287	—	67	2	no	—
83	C₆₇	#8799	1	1	yes	characteristic, complemented, Sylow
84	#432	—	67	2	no	—
85	#432	C₂×D₆₇	1	2	yes	—
86	#431	—	67	1	no	—
87	#431	—	67	1	no	—
88	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
89	#335	—	2	1	no	—
90	#335	—	134	1	no	—
91	#334	—	2	1	no	—
92	#334	—	134	1	no	—
93	#322	—	67	4	no	—
94	C₂²×C₈	Dic₁₃₄	1	2	yes	—
95	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
96	C₄⋊C₈	—	2	2	no	—
97	C₄⋊C₈	—	134	2	no	—
98	C₄⋊C₈	—	134	1	no	—
99	C₄⋊C₈	—	134	1	no	—
100	C₂²⋊C₈	—	2	2	no	—
101	C₂²⋊C₈	—	134	2	no	—
102	C₂²⋊C₈	—	134	1	no	—
103	C₂²⋊C₈	—	134	1	no	—
104	#97	—	2	2	no	—
105	#97	—	134	2	no	—
106	C₄×C₈	—	67	2	no	—
107	C₄×C₈	—	67	2	no	—
108	C₄×C₈	C₄×D₆₇	1	2	yes	—
109	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
110	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
111	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
112	C₂²×C₄	#11761627	1	1	yes	characteristic
113	C₂²×C₄	#11761629	1	1	yes	characteristic
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	268	2	no	—
117	C₂×C₈	—	134	2	no	—
118	C₂×C₈	—	134	2	no	—
119	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
120	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
121	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
122	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
123	C₄²	#11761627	1	1	yes	characteristic
124	C₄²	#11761629	1	1	yes	characteristic
125	C₂³	#14524034	1	1	yes	characteristic
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
130	C₂×C₄	#14524001	1	1	yes	characteristic
131	C₂×C₄	#14524001	1	1	yes	characteristic
132	C₂×C₄	#14524030	1	1	yes	characteristic
133	C₂×C₄	#14524032	1	1	yes	characteristic
134	C₂×C₄	#14524034	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	2	2	no	—
137	C₈	—	8	2	no	—
138	C₈	—	268	2	no	—
139	C₈	—	268	2	no	—
140	C₈	—	536	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	#16863261	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	#16863444	1	2	yes	—
151	C₂	—	4	1	no	—
152	C₂	#22560332	1	1	yes	characteristic
153	C₂	#22560334	1	1	yes	characteristic
154	C₂	#22560349	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22558972

C₂

yes

maximal, characteristic

#22558501

C₂

yes

maximal, characteristic

#22558072

C₂

yes

maximal

#22558070

C₂

yes

maximal

C₆₇×#1568

C₂

yes

maximal, characteristic

#16863453

C₂²

yes

—

#16863449

C₂²

yes

characteristic

#16863449

C₂²

yes

—

—

C₄

yes

—

#432×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#335×C₆₇

—

#334×C₆₇

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

#14523954

—

C₄×C₆₇⋊C₈

C₂×C₄

yes

—

C₄×C₆₇⋊C₈

C₂×C₄

yes

—

C₂²×C₈×C₆₇

Q₈

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

#97×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

characteristic

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

C₂×Q₈

yes

C₂²×C₄×C₆₇

yes

C₂²×C₄×C₆₇

yes

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

C₄²×C₆₇

yes

C₄²×C₆₇

yes

C₂³×C₆₇

yes

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

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#382

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#430

yes

—

#8799

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

D₆₇

yes

—

—

—

C₆₇

#8799

yes

characteristic, complemented, Sylow

#432

—

#432

C₂×D₆₇

yes

—

#431

—

#431

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#335

—

#335

—

134

—

#334

—

#334

—

134

—

#322

—

C₂²×C₈

Dic₁₃₄

yes

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₄⋊C₈

—

134

—

C₄⋊C₈

—

134

—

100

C₂²⋊C₈

—

101

C₂²⋊C₈

—

134

—

102

C₂²⋊C₈

—

134

—

103

C₂²⋊C₈

—

134

—

104

#97

—

105

#97

—

134

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

C₄×D₆₇

yes

—

109

C₄×C₈

C₂²⋊D₆₇

yes

characteristic

110

C₄×C₈

C₂²⋊D₆₇

yes

characteristic

111

C₂²×C₄

C₂×Dic₁₃₄

yes

characteristic

112

C₂²×C₄

#11761627

yes

characteristic

113

C₂²×C₄

#11761629

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

268

—

117

C₂×C₈

—

134

—

118

C₂×C₈

—

134

—

119

C₂×C₈

Dic₆₇⋊C₄

yes

—

120

C₂×C₈

Dic₆₇⋊C₄

yes

—

121

C₄²

C₂×C₄×D₆₇

yes

characteristic

122

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

123

C₄²

#11761627

yes

characteristic

124

C₄²

#11761629

yes

characteristic

125

C₂³

#14524034

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

130

C₂×C₄

#14524001

yes

characteristic

131

C₂×C₄

#14524001

yes

characteristic

132

C₂×C₄

#14524030

yes

characteristic

133

C₂×C₄

#14524032

yes

characteristic

134

C₂×C₄

#14524034

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

268

—

139

C₈

—

268

—

140

C₈

—

536

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

#16863261

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#16863444

yes

—

151

C₂

—

152

C₂

#22560332

yes

characteristic

153

C₂

#22560334

yes

characteristic

154

C₂

#22560349

yes

characteristic

	subgroup	G/H
center	C₂²	#16863261
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1568	C₂
socle	C₂²×C₆₇	#382
prime radical	C₂³×C₆₇	#125
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1568	D₆₇
hypocenter	C₆₇	#8799
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1568

subgroup

G/H

center

C₂²

#16863261

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1568

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1568

N	H ≅ G/N	type
83. C₆₇	#8799	semidirect

H ≅ G/N

type

83. C₆₇

#8799

semidirect

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

subgroup	index
#22558972	2
#22558501	2
#22558072	2
#22558070	2
C₆₇×#1568	2
#8799	67

subgroup

index

C₆₇×#1568

Subgroups — #30749546

Conjugacy classes of subgroups

Complements

Maximal subgroups