Group #30746277 — subgroups

Subgroups — #30746277

Order 17152.

330 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 131 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	#22559896	C₂	1	1	yes	maximal, characteristic
2	#22558502	C₂	1	2	yes	maximal
3	#22558039	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	C₄	1	2	yes	—
8	C₂×#14523954	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₆₇⋊C₈	C₄	1	2	yes	—
10	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
11	#16863118	C₄	1	4	yes	complemented
12	#16863117	C₄	1	4	yes	complemented
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	4	no	—
21	#14523954	—	2	1	no	—
22	#14523954	—	2	1	no	—
23	C₄×C₆₇⋊C₈	—	2	4	no	—
24	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
25	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₆₇	—	2	2	no	—
30	C₂²⋊C₈×C₆₇	—	2	2	no	—
31	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
32	#95×C₆₇	—	2	2	no	—
33	C₂×C₆₇⋊C₈	—	2	2	no	—
34	C₂×C₆₇⋊C₈	—	2	2	no	—
35	C₂×C₆₇⋊C₈	—	2	2	no	—
36	C₂×C₆₇⋊C₈	—	2	2	no	—
37	C₂×C₆₇⋊C₈	—	2	4	no	—
38	C₂×C₆₇⋊C₈	C₄²	1	1	yes	characteristic
39	C₂×C₆₇⋊C₈	C₄²	1	2	yes	—
40	C₂×C₆₇⋊C₈	C₄²	1	1	yes	characteristic
41	C₂×C₆₇⋊C₈	M₁₆	1	4	yes	—
42	C₂²×C₄×C₆₇	—	2	2	no	—
43	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	#41	1	2	yes	—
45	C₂×C₈×C₆₇	—	2	4	no	—
46	C₄²×C₆₇	—	2	2	no	—
47	C₄²×C₆₇	M₁₆	1	4	yes	—
48	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
49	C₄²×C₆₇	#41	1	1	yes	characteristic
50	C₄²×C₆₇	#41	1	1	yes	characteristic
51	C₂³×C₆₇	#117	1	1	yes	characteristic
52	C₂×C₄×C₆₇	—	2	4	no	—
53	C₂×C₄×C₆₇	—	4	2	no	—
54	C₂×C₄×C₆₇	—	4	4	no	—
55	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
56	C₂×C₄×C₆₇	#117	1	2	yes	—
57	C₂×C₄×C₆₇	#117	1	2	yes	—
58	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
59	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
60	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
61	C₈×C₆₇	—	4	4	no	—
62	C₆₇⋊C₈	—	2	2	no	—
63	C₆₇⋊C₈	—	2	1	no	—
64	C₆₇⋊C₈	—	2	1	no	—
65	C₆₇⋊C₈	—	2	4	no	—
66	C₆₇⋊C₈	—	4	4	no	—
67	C₂²×C₆₇	—	2	2	no	—
68	C₂²×C₆₇	#376	1	1	yes	characteristic
69	C₂²×C₆₇	#432	1	2	yes	—
70	C₄×C₆₇	—	2	2	no	—
71	C₄×C₆₇	—	2	2	no	—
72	C₄×C₆₇	—	4	4	no	—
73	C₄×C₆₇	C₄×M₁₆	1	1	yes	characteristic
74	C₄×C₆₇	C₄×M₁₆	1	1	yes	characteristic
75	C₄×C₆₇	#432	1	2	yes	—
76	#8139	—	67	1	no	maximal, Sylow
77	C₂×C₆₇	—	2	2	no	—
78	C₂×C₆₇	#1745	1	1	yes	characteristic
79	C₂×C₆₇	#1747	1	2	yes	—
80	#1726	—	67	1	no	—
81	#1452	—	67	2	no	—
82	#1275	—	67	2	no	—
83	#1274	—	67	1	no	—
84	#1274	D₆₇	1	1	yes	characteristic
85	C₆₇	#8139	1	1	yes	characteristic, complemented, Sylow
86	#431	—	67	2	no	—
87	#431	C₂×D₆₇	1	2	yes	—
88	C₂×#97	—	67	2	no	—
89	C₂×#97	—	67	1	no	—
90	C₂×C₄×C₈	—	67	1	no	—
91	C₂×C₄×C₈	—	67	2	no	—
92	#376	C₂×D₆₇	1	1	yes	characteristic
93	#322	—	67	4	no	—
94	#322	—	67	4	no	—
95	C₂²×C₈	—	67	1	no	—
96	C₂²×C₈	—	67	1	no	—
97	C₂²×C₈	—	67	2	no	—
98	C₂×C₄²	C₄×D₆₇	1	2	yes	—
99	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
100	C₄⋊C₈	—	2	2	no	—
101	C₄⋊C₈	—	134	2	no	—
102	C₂²⋊C₈	—	2	2	no	—
103	C₂²⋊C₈	—	134	2	no	—
104	#97	—	134	4	no	—
105	#97	—	134	1	no	—
106	#97	—	134	1	no	—
107	C₄×C₈	—	67	2	no	—
108	C₄×C₈	—	67	2	no	—
109	C₄×C₈	—	67	4	no	—
110	C₄×C₈	—	134	4	no	—
111	C₄×C₈	C₄×D₆₇	1	4	yes	—
112	#95	—	2	2	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
115	C₂²×C₄	#11761627	1	2	yes	—
116	C₂×C₈	—	2	4	no	—
117	C₂×C₈	—	67	2	no	—
118	C₂×C₈	—	67	1	no	—
119	C₂×C₈	—	67	1	no	—
120	C₂×C₈	—	67	4	no	—
121	C₂×C₈	—	134	4	no	—
122	C₂×C₈	—	134	2	no	—
123	C₂×C₈	—	134	2	no	—
124	C₂×C₈	—	134	2	no	—
125	C₂×C₈	—	134	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	#11761611	1	4	yes	—
128	C₄²	C₂×C₄×D₆₇	1	2	yes	—
129	C₄²	#11761627	1	1	yes	characteristic
130	C₄²	#11761629	1	1	yes	characteristic
131	C₂³	#14523999	1	1	yes	characteristic
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
136	C₂×C₄	#14523999	1	2	yes	—
137	C₂×C₄	#14524001	1	2	yes	—
138	C₂×C₄	#14524001	1	1	yes	characteristic
139	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
140	C₂×C₄	C₂×#11761611	1	2	yes	—
141	C₈	—	4	4	no	—
142	C₈	—	268	4	no	—
143	C₈	—	134	2	no	—
144	C₈	—	134	1	no	—
145	C₈	—	134	1	no	—
146	C₈	—	134	4	no	—
147	C₂²	—	2	2	no	—
148	C₂²	#16863241	1	1	yes	characteristic
149	C₂²	#16863450	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	4	4	no	—
153	C₄	C₄×#11761611	1	1	yes	characteristic
154	C₄	#16863348	1	1	yes	characteristic
155	C₄	#16863452	1	2	yes	—
156	C₂	—	2	2	no	—
157	C₂	#22559956	1	1	yes	characteristic
158	C₂	#22559963	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22559896

C₂

yes

maximal, characteristic

#22558502

C₂

yes

maximal

#22558039

C₂

yes

maximal

#22558038

C₂

yes

maximal, characteristic

C₆₇×#1274

C₂

yes

maximal, characteristic

#16863449

C₂²

yes

—

C₂×#14523954

C₄

yes

—

C₂×#14523954

C₂²

yes

characteristic

C₂×C₄×C₆₇⋊C₈

C₄

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

characteristic

#16863118

C₄

yes

complemented

#16863117

C₄

yes

complemented

#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

—

#14523954

—

#14523954

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

C₂×C₄

yes

—

C₄×C₆₇⋊C₈

C₂×C₄

yes

—

C₄×C₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

#95×C₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₄²

yes

characteristic

C₂×C₆₇⋊C₈

C₄²

yes

—

C₂×C₆₇⋊C₈

C₄²

yes

characteristic

C₂×C₆₇⋊C₈

M₁₆

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

M₁₆

yes

—

C₄²×C₆₇

C₂²×C₄

yes

—

C₄²×C₆₇

yes

C₄²×C₆₇

yes

C₂³×C₆₇

yes

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

#117

yes

characteristic

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#376

yes

characteristic

C₂²×C₆₇

#432

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₄×M₁₆

yes

characteristic

C₄×C₆₇

C₄×M₁₆

yes

characteristic

C₄×C₆₇

#432

yes

—

#8139

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1745

yes

characteristic

C₂×C₆₇

#1747

yes

—

—

—

—

—

D₆₇

yes

characteristic

C₆₇

#8139

yes

characteristic, complemented, Sylow

#431

—

#431

C₂×D₆₇

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#376

C₂×D₆₇

yes

characteristic

#322

—

#322

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₄×D₆₇

yes

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

100

C₄⋊C₈

—

101

C₄⋊C₈

—

134

—

102

C₂²⋊C₈

—

103

C₂²⋊C₈

—

134

—

104

#97

—

134

—

105

#97

—

134

—

106

#97

—

134

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

134

—

111

C₄×C₈

C₄×D₆₇

yes

—

112

#95

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

115

C₂²×C₄

#11761627

yes

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

134

—

122

C₂×C₈

—

134

—

123

C₂×C₈

—

134

—

124

C₂×C₈

—

134

—

125

C₂×C₈

—

134

—

126

C₄²

—

127

C₄²

#11761611

yes

—

128

C₄²

C₂×C₄×D₆₇

yes

—

129

C₄²

#11761627

yes

characteristic

130

C₄²

#11761629

yes

characteristic

131

C₂³

#14523999

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₄²×D₆₇

yes

characteristic

136

C₂×C₄

#14523999

yes

—

137

C₂×C₄

#14524001

yes

—

138

C₂×C₄

#14524001

yes

characteristic

139

C₂×C₄

M₁₆×D₆₇

yes

—

140

C₂×C₄

C₂×#11761611

yes

—

141

C₈

—

142

C₈

—

268

—

143

C₈

—

134

—

144

C₈

—

134

—

145

C₈

—

134

—

146

C₈

—

134

—

147

C₂²

—

148

C₂²

#16863241

yes

characteristic

149

C₂²

#16863450

yes

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

C₄×#11761611

yes

characteristic

154

C₄

#16863348

yes

characteristic

155

C₄

#16863452

yes

—

156

C₂

—

157

C₂

#22559956

yes

characteristic

158

C₂

#22559963

yes

—

	subgroup	G/H
center	C₂×C₄	C₄²×D₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1274	C₂
socle	C₂²×C₆₇	#376
prime radical	C₂³×C₆₇	#117
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1274	D₆₇
hypocenter	C₆₇	#8139
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #1274

subgroup

G/H

center

C₂×C₄

C₄²×D₆₇

commutator

C₂×C₄×C₆₇

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1274

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₄ ◁ #1274

N	H ≅ G/N	type
11. #16863118	C₄	semidirect
12. #16863117	C₄	semidirect
85. C₆₇	#8139	semidirect

H ≅ G/N

type

11. #16863118

C₄

semidirect

12. #16863117

C₄

semidirect

85. C₆₇

#8139

semidirect

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

subgroup	index
#22559896	2
#22558502	2
#22558039	2
#22558038	2
C₆₇×#1274	2
#8139	67

subgroup

index

C₆₇×#1274

Subgroups — #30746277

Conjugacy classes of subgroups

Complements

Maximal subgroups