Group #30749556 — subgroups

Subgroups — #30749556

Order 17152.

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

normal subgroups only

This group has 258 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	#22559958	C₂	1	1	yes	maximal, characteristic
2	#22558053	C₂	1	2	yes	maximal
3	#22558042	C₂	1	1	yes	maximal, characteristic
4	#22558039	C₂	1	2	yes	maximal
5	C₆₇×#1276	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	2	yes	—
7	C₂×#14523954	C₂²	1	1	yes	characteristic
8	C₂×#14523954	C₂²	1	1	yes	characteristic
9	#16863151	C₄	1	2	yes	—
10	#16863151	C₄	1	2	yes	—
11	#431×C₆₇	C₂²	1	2	yes	—
12	C₄³×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
16	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
17	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
18	#14523954	—	2	1	no	—
19	#14523954	—	2	2	no	—
20	#14523954	—	2	1	no	—
21	C₄×C₆₇⋊C₈	—	2	2	no	—
22	C₂×C₄²×C₆₇	—	2	2	no	—
23	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
25	C₄⋊C₈×C₆₇	—	2	2	no	—
26	C₂²⋊C₈×C₆₇	—	2	2	no	—
27	C₄×C₈×C₆₇	—	2	2	no	—
28	C₂×C₆₇⋊C₈	—	2	1	no	—
29	C₂×C₆₇⋊C₈	—	2	1	no	—
30	C₂×C₆₇⋊C₈	—	2	2	no	—
31	C₂×C₆₇⋊C₈	—	2	2	no	—
32	C₂×C₆₇⋊C₈	—	2	2	no	—
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	—	2	2	no	—
37	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₇	#41	1	2	yes	—
39	C₂×C₈×C₆₇	—	4	2	no	—
40	C₄²×C₆₇	—	2	2	no	—
41	C₄²×C₆₇	—	2	2	no	—
42	C₄²×C₆₇	—	4	4	no	—
43	C₄²×C₆₇	#41	1	1	yes	characteristic
44	C₄²×C₆₇	#41	1	2	yes	—
45	C₄²×C₆₇	#41	1	1	yes	characteristic
46	C₂³×C₆₇	#117	1	1	yes	characteristic
47	C₂×C₄×C₆₇	—	2	4	no	—
48	C₂×C₄×C₆₇	—	4	2	no	—
49	C₂×C₄×C₆₇	—	4	2	no	—
50	C₂×C₄×C₆₇	—	4	2	no	—
51	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
52	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
53	C₂×C₄×C₆₇	#126	1	2	yes	—
54	C₂×C₄×C₆₇	#126	1	2	yes	—
55	C₂×C₄×C₆₇	#131	1	2	yes	—
56	C₂×C₄×C₆₇	#131	1	2	yes	—
57	C₈×C₆₇	—	8	2	no	—
58	C₆₇⋊C₈	—	4	1	no	—
59	C₆₇⋊C₈	—	4	2	no	—
60	C₆₇⋊C₈	—	4	1	no	—
61	C₆₇⋊C₈	—	8	2	no	—
62	C₂²×C₆₇	—	2	2	no	—
63	C₂²×C₆₇	#383	1	1	yes	characteristic
64	C₂²×C₆₇	#433	1	2	yes	—
65	C₄×C₆₇	—	2	2	no	—
66	C₄×C₆₇	—	2	2	no	—
67	C₄×C₆₇	—	4	4	no	—
68	C₄×C₆₇	#433	1	1	yes	characteristic
69	C₄×C₆₇	#433	1	2	yes	—
70	C₄×C₆₇	#433	1	1	yes	characteristic
71	#8802	—	67	1	no	maximal, Sylow
72	C₂×C₆₇	—	2	2	no	—
73	C₂×C₆₇	#1840	1	1	yes	characteristic
74	C₂×C₆₇	#1841	1	2	yes	—
75	#1745	—	67	1	no	—
76	#1281	—	67	2	no	—
77	#1276	—	67	1	no	—
78	#1276	D₆₇	1	1	yes	characteristic
79	#1275	—	67	2	no	—
80	C₆₇	#8802	1	1	yes	characteristic, complemented, Sylow
81	#431	—	67	2	no	—
82	#431	C₂×D₆₇	1	2	yes	—
83	C₂×#97	—	67	1	no	—
84	C₂×#97	—	67	1	no	—
85	C₄³	C₂×D₆₇	1	1	yes	characteristic
86	#336	—	67	2	no	—
87	#336	—	67	2	no	—
88	C₂²×C₈	—	67	1	no	—
89	C₂²×C₈	—	67	1	no	—
90	C₂²×C₈	—	67	2	no	—
91	C₂×C₄²	—	2	2	no	—
92	C₂×C₄²	C₄×D₆₇	1	2	yes	—
93	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
94	C₄⋊C₈	—	2	2	no	—
95	C₄⋊C₈	—	134	2	no	—
96	C₂²⋊C₈	—	2	2	no	—
97	C₂²⋊C₈	—	134	2	no	—
98	#97	—	134	1	no	—
99	#97	—	134	2	no	—
100	#97	—	134	1	no	—
101	C₄×C₈	—	2	2	no	—
102	C₄×C₈	—	134	2	no	—
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
105	C₂²×C₄	#11761627	1	2	yes	—
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	—	268	2	no	—
108	C₂×C₈	—	134	1	no	—
109	C₂×C₈	—	134	1	no	—
110	C₂×C₈	—	134	2	no	—
111	C₂×C₈	—	134	2	no	—
112	C₂×C₈	—	134	2	no	—
113	C₂×C₈	—	134	2	no	—
114	C₂×C₈	—	134	2	no	—
115	C₄²	—	2	2	no	—
116	C₄²	—	2	2	no	—
117	C₄²	—	4	4	no	—
118	C₄²	#11761627	1	1	yes	characteristic
119	C₄²	#11761628	1	2	yes	—
120	C₄²	#11761629	1	1	yes	characteristic
121	C₂³	#14523999	1	1	yes	characteristic
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	#14524000	1	1	yes	characteristic
127	C₂×C₄	#14524001	1	1	yes	characteristic
128	C₂×C₄	#14524036	1	2	yes	—
129	C₂×C₄	#14524038	1	2	yes	—
130	C₂×C₄	#14524049	1	2	yes	—
131	C₂×C₄	#14524051	1	2	yes	—
132	C₈	—	8	2	no	—
133	C₈	—	268	1	no	—
134	C₈	—	268	2	no	—
135	C₈	—	268	1	no	—
136	C₈	—	536	2	no	—
137	C₂²	—	2	2	no	—
138	C₂²	#16863263	1	1	yes	characteristic
139	C₂²	#16863454	1	2	yes	—
140	C₄	—	2	2	no	—
141	C₄	—	2	2	no	—
142	C₄	—	4	4	no	—
143	C₄	#16863455	1	1	yes	characteristic
144	C₄	#16863456	1	2	yes	—
145	C₄	#16863458	1	1	yes	characteristic
146	C₂	—	2	2	no	—
147	C₂	#22560353	1	1	yes	characteristic
148	C₂	#22560355	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22559958

C₂

yes

maximal, characteristic

#22558053

C₂

yes

maximal

#22558042

C₂

yes

maximal, characteristic

#22558039

C₂

yes

maximal

C₆₇×#1276

C₂

yes

maximal, characteristic

#16863449

C₂²

yes

—

C₂×#14523954

C₂²

yes

characteristic

C₂×#14523954

C₂²

yes

characteristic

#16863151

C₄

yes

—

#16863151

C₄

yes

—

#431×C₆₇

C₂²

yes

—

C₄³×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

—

#14523954

—

#14523954

—

C₄×C₆₇⋊C₈

—

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

C₄×C₈×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

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

#41

yes

characteristic

C₄²×C₆₇

#41

yes

—

C₄²×C₆₇

yes

C₂³×C₆₇

yes

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂×C₄×C₆₇

#126

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#383

yes

characteristic

C₂²×C₆₇

#433

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#433

yes

characteristic

C₄×C₆₇

#433

yes

—

C₄×C₆₇

yes

—

C₂×C₆₇

—

C₂×C₆₇

#1840

yes

characteristic

C₂×C₆₇

#1841

yes

—

—

—

—

D₆₇

yes

characteristic

#1275

—

C₆₇

#8802

yes

characteristic, complemented, Sylow

#431

—

#431

C₂×D₆₇

yes

—

C₂×#97

—

C₂×#97

—

C₄³

C₂×D₆₇

yes

characteristic

#336

—

#336

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

C₄×D₆₇

yes

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₂²⋊C₈

—

C₂²⋊C₈

—

134

—

#97

—

134

—

#97

—

134

—

100

#97

—

134

—

101

C₄×C₈

—

102

C₄×C₈

—

134

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

105

C₂²×C₄

#11761627

yes

—

106

C₂×C₈

—

107

C₂×C₈

—

268

—

108

C₂×C₈

—

134

—

109

C₂×C₈

—

134

—

110

C₂×C₈

—

134

—

111

C₂×C₈

—

134

—

112

C₂×C₈

—

134

—

113

C₂×C₈

—

134

—

114

C₂×C₈

—

134

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

#11761627

yes

characteristic

119

C₄²

#11761628

yes

—

120

C₄²

#11761629

yes

characteristic

121

C₂³

#14523999

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

#14524000

yes

characteristic

127

C₂×C₄

#14524001

yes

characteristic

128

C₂×C₄

#14524036

yes

—

129

C₂×C₄

#14524038

yes

—

130

C₂×C₄

#14524049

yes

—

131

C₂×C₄

#14524051

yes

—

132

C₈

—

133

C₈

—

268

—

134

C₈

—

268

—

135

C₈

—

268

—

136

C₈

—

536

—

137

C₂²

—

138

C₂²

#16863263

yes

characteristic

139

C₂²

#16863454

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#16863455

yes

characteristic

144

C₄

#16863456

yes

—

145

C₄

#16863458

yes

characteristic

146

C₂

—

147

C₂

#22560353

yes

characteristic

148

C₂

#22560355

yes

—

	subgroup	G/H
center	C₂×C₄	#14524001
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1276	C₂
socle	C₂²×C₆₇	#383
prime radical	C₂³×C₆₇	#117
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1276	D₆₇
hypocenter	C₆₇	#8802
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1276

subgroup

G/H

center

C₂×C₄

#14524001

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1276

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1276

N	H ≅ G/N	type
80. C₆₇	#8802	semidirect

H ≅ G/N

type

80. C₆₇

#8802

semidirect

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

subgroup	index
#22559958	2
#22558053	2
#22558042	2
#22558039	2
C₆₇×#1276	2
#8802	67

subgroup

index

C₆₇×#1276

Subgroups — #30749556

Conjugacy classes of subgroups

Complements

Maximal subgroups