Group #30746261 — subgroups

Subgroups — #30746261

Order 17152.

446 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 227 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	#22559955	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₆₇⋊#1273	C₂	1	4	yes	maximal
4	C₆₇×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₆₇⋊C₈	C₄	1	2	yes	—
6	C₂×C₄×C₆₇⋊C₈	C₄	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
9	#16863151	C₄	1	4	yes	—
10	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
11	C₄³×C₆₇	C₂²	1	1	yes	characteristic
12	#336×C₆₇	C₄	1	4	yes	—
13	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
14	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
15	C₄×C₆₇⋊C₈	—	2	4	no	—
16	C₄×C₆₇⋊C₈	—	2	4	no	—
17	C₄×C₆₇⋊C₈	—	2	4	no	—
18	C₄×C₆₇⋊C₈	C₈	1	8	yes	complemented
19	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
20	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
25	C₄×C₈×C₆₇	—	2	4	no	—
26	C₂×C₆₇⋊C₈	—	2	4	no	—
27	C₂×C₆₇⋊C₈	—	2	4	no	—
28	C₂×C₆₇⋊C₈	—	2	4	no	—
29	C₂×C₆₇⋊C₈	—	2	8	no	—
30	C₂×C₆₇⋊C₈	C₂×C₈	1	8	yes	—
31	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
32	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
33	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₆₇	—	2	4	no	—
37	C₂×C₈×C₆₇	—	2	8	no	—
38	C₄²×C₆₇	—	2	4	no	—
39	C₄²×C₆₇	—	2	4	no	—
40	C₄²×C₆₇	C₂×C₈	1	4	yes	—
41	C₄²×C₆₇	M₁₆	1	4	yes	—
42	C₄²×C₆₇	#41	1	4	yes	—
43	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
44	C₂×C₄×C₆₇	—	2	2	no	—
45	C₂×C₄×C₆₇	—	2	2	no	—
46	C₂×C₄×C₆₇	—	2	2	no	—
47	C₂×C₄×C₆₇	—	2	2	no	—
48	C₂×C₄×C₆₇	C₄×C₈	1	4	yes	—
49	C₂×C₄×C₆₇	#97	1	4	yes	—
50	C₂×C₄×C₆₇	#117	1	2	yes	—
51	C₂×C₄×C₆₇	#117	1	2	yes	—
52	C₂×C₄×C₆₇	#117	1	2	yes	—
53	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
54	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
55	C₂×C₄×C₆₇	#131	1	4	yes	—
56	C₂×C₄×C₆₇	#131	1	4	yes	—
57	C₈×C₆₇	—	4	4	no	—
58	C₆₇⋊C₈	—	2	8	no	—
59	C₆₇⋊C₈	—	4	4	no	—
60	C₂²×C₆₇	#376	1	1	yes	characteristic
61	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
62	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
63	C₂²×C₆₇	#405	1	2	yes	—
64	C₂²×C₆₇	#405	1	2	yes	—
65	C₄×C₆₇	—	2	4	no	—
66	C₄×C₆₇	—	2	4	no	—
67	C₄×C₆₇	#431	1	4	yes	—
68	C₄×C₆₇	#433	1	4	yes	—
69	C₄×C₆₇	#433	1	4	yes	—
70	#8133	—	67	1	no	maximal, Sylow
71	C₂×C₆₇	#1450	1	2	yes	—
72	C₂×C₆₇	#1452	1	2	yes	—
73	C₂×C₆₇	#1744	1	1	yes	characteristic
74	C₂×C₆₇	#1746	1	1	yes	characteristic
75	C₂×C₆₇	#1746	1	1	yes	characteristic
76	#1744	—	67	1	no	—
77	#1744	D₆₇	1	1	yes	characteristic
78	C₄²×C₈	—	67	1	no	—
79	#1273	—	67	4	no	—
80	C₆₇	#8133	1	1	yes	characteristic, complemented, Sylow
81	C₂×C₄×C₈	—	67	2	no	—
82	C₂×C₄×C₈	—	67	2	no	—
83	C₂×C₄×C₈	—	67	2	no	—
84	C₂×C₄×C₈	—	67	2	no	—
85	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
86	C₄³	C₂×D₆₇	1	1	yes	characteristic
87	#336	—	67	4	no	—
88	#336	Dic₆₇	1	4	yes	—
89	C₂²×C₈	—	67	4	no	—
90	C₂²×C₈	—	67	4	no	—
91	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
92	C₂×C₄²	C₄×D₆₇	1	2	yes	—
93	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
94	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
95	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
96	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
97	C₄×C₈	—	2	4	no	—
98	C₄×C₈	—	67	8	no	—
99	C₄×C₈	—	134	4	no	—
100	C₄×C₈	—	134	4	no	—
101	C₄×C₈	—	134	4	no	—
102	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
103	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
104	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
105	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
106	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
107	C₂×C₈	—	2	4	no	—
108	C₂×C₈	—	2	8	no	—
109	C₂×C₈	—	67	8	no	—
110	C₂×C₈	—	134	4	no	—
111	C₂×C₈	—	134	8	no	—
112	C₂×C₈	—	134	4	no	—
113	C₂×C₈	—	134	4	no	—
114	C₄²	—	2	4	no	—
115	C₄²	—	2	4	no	—
116	C₄²	C₈×D₆₇	1	4	yes	—
117	C₄²	#11761611	1	4	yes	—
118	C₄²	#11761628	1	4	yes	—
119	C₂³	C₂×C₄×Dic₆₇	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	C₈×Dic₆₇	1	4	yes	—
125	C₂×C₄	#14523953	1	4	yes	—
126	C₂×C₄	#14524000	1	2	yes	—
127	C₂×C₄	#14524000	1	2	yes	—
128	C₂×C₄	#14524002	1	2	yes	—
129	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
130	C₂×C₄	C₂×#11761611	1	2	yes	—
131	C₂×C₄	#14524050	1	4	yes	—
132	C₂×C₄	#14524051	1	4	yes	—
133	C₈	—	4	4	no	—
134	C₈	—	268	4	no	—
135	C₈	—	134	8	no	—
136	C₂²	#16863242	1	1	yes	characteristic
137	C₂²	C₂×C₈×Dic₆₇	1	1	yes	characteristic
138	C₂²	C₂×#14523953	1	1	yes	characteristic
139	C₂²	#16863352	1	2	yes	—
140	C₂²	#16863354	1	2	yes	—
141	C₄	—	2	4	no	—
142	C₄	—	2	4	no	—
143	C₄	#16863447	1	4	yes	—
144	C₄	#16863457	1	4	yes	—
145	C₄	#16863458	1	4	yes	—
146	C₂	#22558493	1	2	yes	—
147	C₂	#22558502	1	2	yes	—
148	C₂	#22559954	1	1	yes	characteristic
149	C₂	#22559961	1	1	yes	characteristic
150	C₂	#22559962	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22559955

C₂

yes

maximal, characteristic

C₄²×C₆₇⋊C₈

C₂

yes

maximal, characteristic

C₆₇⋊#1273

C₂

yes

maximal

C₆₇×#1744

C₂

yes

maximal, characteristic

C₂×C₄×C₆₇⋊C₈

C₄

yes

—

C₂×C₄×C₆₇⋊C₈

C₄

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

#16863151

C₄

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

—

C₄³×C₆₇

C₂²

yes

characteristic

#336×C₆₇

C₄

yes

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

C₈

yes

complemented

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄×C₈×C₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×C₈

yes

—

C₂²×C₄×C₆₇

C₄²

yes

—

C₂²×C₄×C₆₇

C₄²

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×C₈

yes

—

C₄²×C₆₇

M₁₆

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄×C₈

yes

—

C₂×C₄×C₆₇

#97

yes

—

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

C₂²×C₈

yes

—

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

#376

yes

characteristic

C₂²×C₆₇

C₂×C₄×C₈

yes

characteristic

C₂²×C₆₇

C₂×#97

yes

characteristic

C₂²×C₆₇

#405

yes

—

C₂²×C₆₇

#405

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#431

yes

—

C₄×C₆₇

#433

yes

—

C₄×C₆₇

#433

yes

—

#8133

—

maximal, Sylow

C₂×C₆₇

#1450

yes

—

C₂×C₆₇

#1452

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

D₆₇

yes

characteristic

C₄²×C₈

—

#1273

—

C₆₇

#8133

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₆₇

yes

—

C₄³

C₂×D₆₇

yes

characteristic

#336

—

#336

Dic₆₇

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×Dic₆₇

yes

—

C₂×C₄²

C₄×D₆₇

yes

—

C₂×C₄²

C₄×D₆₇

yes

characteristic

C₂×C₄²

C₄×D₆₇

yes

characteristic

C₂×C₄²

C₂×Dic₆₇

yes

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

134

—

100

C₄×C₈

—

134

—

101

C₄×C₈

—

134

—

102

C₂²×C₄

C₄×Dic₆₇

yes

—

103

C₂²×C₄

C₄×Dic₆₇

yes

—

104

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

105

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

106

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

134

—

111

C₂×C₈

—

134

—

112

C₂×C₈

—

134

—

113

C₂×C₈

—

134

—

114

C₄²

—

115

C₄²

—

116

C₄²

C₈×D₆₇

yes

—

117

C₄²

#11761611

yes

—

118

C₄²

#11761628

yes

—

119

C₂³

C₂×C₄×Dic₆₇

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

C₈×Dic₆₇

yes

—

125

C₂×C₄

#14523953

yes

—

126

C₂×C₄

#14524000

yes

—

127

C₂×C₄

#14524000

yes

—

128

C₂×C₄

#14524002

yes

—

129

C₂×C₄

C₂×C₈×D₆₇

yes

—

130

C₂×C₄

C₂×#11761611

yes

—

131

C₂×C₄

#14524050

yes

—

132

C₂×C₄

#14524051

yes

—

133

C₈

—

134

C₈

—

268

—

135

C₈

—

134

—

136

C₂²

#16863242

yes

characteristic

137

C₂²

C₂×C₈×Dic₆₇

yes

characteristic

138

C₂²

C₂×#14523953

yes

characteristic

139

C₂²

#16863352

yes

—

140

C₂²

#16863354

yes

—

141

C₄

—

142

C₄

—

143

C₄

#16863447

yes

—

144

C₄

#16863457

yes

—

145

C₄

#16863458

yes

—

146

C₂

#22558493

yes

—

147

C₂

#22558502

yes

—

148

C₂

#22559954

yes

characteristic

149

C₂

#22559961

yes

characteristic

150

C₂

#22559962

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂×C₄×C₈
Fitting	C₆₇×#1744	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1744	D₆₇
hypocenter	C₆₇	#8133
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ #1744

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₆₇

commutator

C₂²×C₆₇

C₂×C₄×C₈

Fitting

C₆₇×#1744

C₂

socle, prime radical

C₂³×C₆₇

C₂×C₄²

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₄² ◁ #1744

N	H ≅ G/N	type
18. C₄×C₆₇⋊C₈	C₈	semidirect
80. C₆₇	#8133	semidirect

H ≅ G/N

type

18. C₄×C₆₇⋊C₈

C₈

semidirect

80. C₆₇

#8133

semidirect

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

subgroup	index
#22559955	2
C₄²×C₆₇⋊C₈	2
C₆₇⋊#1273	2
C₆₇×#1744	2
#8133	67

subgroup

index

#22559955

C₄²×C₆₇⋊C₈

C₆₇⋊#1273

C₆₇×#1744

#8133

Subgroups — #30746261

Conjugacy classes of subgroups

Complements

Maximal subgroups