Group #30746992 — subgroups

Subgroups — #30746992

Order 17152.

390 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 151 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	C₄²×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#22558955	C₂	1	2	yes	maximal
3	#22558502	C₂	1	2	yes	maximal
4	#22558042	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1276	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	C₄	1	2	yes	—
8	C₂×C₄×C₆₇⋊C₈	C₄	1	2	yes	—
9	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
11	#16863148	C₄	1	4	yes	complemented
12	#16863146	C₄	1	4	yes	complemented
13	#16863118	—	2	2	no	—
14	#16863117	—	2	2	no	—
15	#431×C₆₇	C₂²	1	2	yes	—
16	C₄³×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
19	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
21	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
22	C₄×C₆₇⋊C₈	—	2	4	no	—
23	C₄×C₆₇⋊C₈	—	2	4	no	—
24	C₄×C₆₇⋊C₈	—	2	2	no	—
25	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
27	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
28	C₄×C₆₇⋊C₈	Q₈	1	2	yes	—
29	C₂×C₄²×C₆₇	—	2	2	no	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
33	C₂²⋊C₈×C₆₇	—	2	2	no	—
34	C₄×C₈×C₆₇	—	2	2	no	—
35	C₂×C₆₇⋊C₈	—	2	2	no	—
36	C₂×C₆₇⋊C₈	—	2	2	no	—
37	C₂×C₆₇⋊C₈	—	2	2	no	—
38	C₂×C₆₇⋊C₈	—	2	4	no	—
39	C₂×C₆₇⋊C₈	C₄²	1	2	yes	—
40	C₂×C₆₇⋊C₈	C₄²	1	1	yes	characteristic
41	C₂×C₆₇⋊C₈	C₄²	1	1	yes	characteristic
42	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
43	C₂×C₆₇⋊C₈	M₁₆	1	4	yes	—
44	C₂²×C₄×C₆₇	—	2	2	no	—
45	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	#41	1	2	yes	—
47	C₂×C₈×C₆₇	—	2	4	no	—
48	C₄²×C₆₇	—	2	8	no	—
49	C₄²×C₆₇	—	2	2	no	—
50	C₄²×C₆₇	M₁₆	1	4	yes	—
51	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
52	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
54	C₂³×C₆₇	#117	1	1	yes	characteristic
55	C₂×C₄×C₆₇	—	2	8	no	—
56	C₂×C₄×C₆₇	—	2	4	no	—
57	C₂×C₄×C₆₇	—	2	4	no	—
58	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
59	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
61	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
62	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
63	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
64	C₈×C₆₇	—	4	4	no	—
65	C₆₇⋊C₈	—	2	2	no	—
66	C₆₇⋊C₈	—	2	4	no	—
67	C₆₇⋊C₈	—	4	4	no	—
68	C₆₇⋊C₈	#97	1	2	yes	—
69	C₆₇⋊C₈	#97	1	2	yes	—
70	C₂²×C₆₇	—	2	2	no	—
71	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₆₇	#432	1	2	yes	—
73	C₄×C₆₇	—	2	8	no	—
74	C₄×C₆₇	—	2	2	no	—
75	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
76	C₄×C₆₇	C₄×M₁₆	1	1	yes	characteristic
77	C₄×C₆₇	#423	1	2	yes	—
78	C₄×C₆₇	#443	1	4	yes	—
79	#8267	—	67	1	no	maximal, Sylow
80	C₂×C₆₇	—	2	2	no	—
81	C₂×C₆₇	#1767	1	1	yes	characteristic
82	C₂×C₆₇	#1772	1	2	yes	—
83	C₄²×C₈	—	67	1	no	—
84	#1564	—	67	2	no	—
85	#1452	—	67	2	no	—
86	#1276	—	67	1	no	—
87	#1276	D₆₇	1	1	yes	characteristic
88	C₆₇	#8267	1	1	yes	characteristic, complemented, Sylow
89	#431	—	67	2	no	—
90	#431	C₂×D₆₇	1	2	yes	—
91	C₂×C₄×C₈	—	67	1	no	—
92	C₂×C₄×C₈	—	67	1	no	—
93	C₂×C₄×C₈	—	67	2	no	—
94	C₂×C₄×C₈	—	67	2	no	—
95	C₄³	C₂×D₆₇	1	1	yes	characteristic
96	#335	—	67	4	no	—
97	#334	—	67	4	no	—
98	#322	—	134	2	no	—
99	#322	—	134	2	no	—
100	C₂²×C₈	—	67	1	no	—
101	C₂²×C₈	—	67	1	no	—
102	C₂²×C₈	—	67	2	no	—
103	C₂×C₄²	—	2	2	no	—
104	C₂×C₄²	C₄×D₆₇	1	2	yes	—
105	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
106	C₄⋊C₈	—	67	4	no	—
107	C₄⋊C₈	C₄×D₆₇	1	4	yes	—
108	C₂²⋊C₈	—	2	2	no	—
109	C₂²⋊C₈	—	134	2	no	—
110	C₄×C₈	—	2	2	no	—
111	C₄×C₈	—	67	2	no	—
112	C₄×C₈	—	67	2	no	—
113	C₄×C₈	—	67	2	no	—
114	C₄×C₈	—	67	2	no	—
115	C₄×C₈	—	134	4	no	—
116	C₄×C₈	—	134	4	no	—
117	C₄×C₈	—	134	2	no	—
118	C₂²×C₄	—	2	2	no	—
119	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
120	C₂²×C₄	#11761627	1	2	yes	—
121	C₂×C₈	—	2	4	no	—
122	C₂×C₈	—	67	1	no	—
123	C₂×C₈	—	67	1	no	—
124	C₂×C₈	—	67	4	no	—
125	C₂×C₈	—	67	2	no	—
126	C₂×C₈	—	67	4	no	—
127	C₂×C₈	—	134	4	no	—
128	C₂×C₈	—	134	2	no	—
129	C₂×C₈	—	134	2	no	—
130	C₂×C₈	—	134	2	no	—
131	C₄²	—	2	8	no	—
132	C₄²	—	2	2	no	—
133	C₄²	#11761611	1	4	yes	—
134	C₄²	C₂×C₄×D₆₇	1	2	yes	—
135	C₄²	D₄×D₆₇	1	1	yes	characteristic
136	C₄²	Q₈×D₆₇	1	1	yes	characteristic
137	C₂³	#14523999	1	1	yes	characteristic
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
142	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
143	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
144	C₂×C₄	#14524008	1	2	yes	—
145	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
146	C₂×C₄	C₂×#11761611	1	2	yes	—
147	C₈	—	4	4	no	—
148	C₈	—	268	4	no	—
149	C₈	—	67	2	no	—
150	C₈	—	67	2	no	—
151	C₈	—	134	2	no	—
152	C₈	—	134	4	no	—
153	C₂²	—	2	2	no	—
154	C₂²	#16863247	1	1	yes	characteristic
155	C₂²	#16863450	1	2	yes	—
156	C₄	—	2	8	no	—
157	C₄	—	2	2	no	—
158	C₄	#97×D₆₇	1	1	yes	characteristic
159	C₄	C₄×#11761611	1	1	yes	characteristic
160	C₄	#16863418	1	2	yes	—
161	C₄	#16863510	1	4	yes	—
162	C₂	—	2	2	no	—
163	C₂	#22560061	1	1	yes	characteristic
164	C₂	#22560087	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄²×C₆₇⋊C₈

C₂

yes

maximal, characteristic

#22558955

C₂

yes

maximal

#22558502

C₂

yes

maximal

#22558042

C₂

yes

maximal, characteristic

C₆₇×#1276

C₂

yes

maximal, characteristic

#16863449

C₂²

yes

—

C₂×C₄×C₆₇⋊C₈

C₄

yes

—

C₂×C₄×C₆₇⋊C₈

C₄

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₆₇⋊C₈

C₂²

yes

characteristic

#16863148

C₄

yes

complemented

#16863146

C₄

yes

complemented

#16863118

—

#16863117

—

#431×C₆₇

C₂²

yes

—

C₄³×C₆₇

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₆₇⋊C₈

C₂×C₄

yes

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂×C₄

yes

—

C₂²⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

C₂×C₄

yes

—

C₄×C₆₇⋊C₈

C₂×C₄

yes

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₄×C₆₇⋊C₈

Q₈

yes

—

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

C₂×C₄

yes

—

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₄²

yes

—

C₂×C₆₇⋊C₈

C₄²

yes

characteristic

C₂×C₆₇⋊C₈

C₄²

yes

characteristic

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

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

—

C₄²×C₆₇

M₁₆

yes

—

C₄²×C₆₇

C₂²×C₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₂³×C₆₇

#117

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

C₄×Q₈

yes

—

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

#97

yes

—

C₆₇⋊C₈

#97

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄×C₄⋊C₄

yes

characteristic

C₂²×C₆₇

#432

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#97

yes

characteristic

C₄×C₆₇

C₄×M₁₆

yes

characteristic

C₄×C₆₇

#423

yes

—

C₄×C₆₇

#443

yes

—

#8267

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1767

yes

characteristic

C₂×C₆₇

#1772

yes

—

C₄²×C₈

—

—

—

—

D₆₇

yes

characteristic

C₆₇

#8267

yes

characteristic, complemented, Sylow

#431

—

#431

C₂×D₆₇

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

C₂×D₆₇

yes

—

—

—

134

—

#322

—

134

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₂×C₄²

—

104

C₂×C₄²

C₄×D₆₇

yes

—

105

C₂×C₄²

C₂²×D₆₇

yes

characteristic

106

C₄⋊C₈

—

107

C₄⋊C₈

C₄×D₆₇

yes

—

108

C₂²⋊C₈

—

109

C₂²⋊C₈

—

134

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

134

—

116

C₄×C₈

—

134

—

117

C₄×C₈

—

134

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

120

C₂²×C₄

#11761627

yes

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

134

—

128

C₂×C₈

—

134

—

129

C₂×C₈

—

134

—

130

C₂×C₈

—

134

—

131

C₄²

—

132

C₄²

—

133

C₄²

#11761611

yes

—

134

C₄²

C₂×C₄×D₆₇

yes

—

135

C₄²

D₄×D₆₇

yes

characteristic

136

C₄²

Q₈×D₆₇

yes

characteristic

137

C₂³

#14523999

yes

characteristic

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

C₄²×D₆₇

yes

characteristic

142

C₂×C₄

C₄⋊C₄×D₆₇

yes

characteristic

143

C₂×C₄

(C₂²⋊D₆₇)⋊C₄

yes

—

144

C₂×C₄

#14524008

yes

—

145

C₂×C₄

M₁₆×D₆₇

yes

—

146

C₂×C₄

C₂×#11761611

yes

—

147

C₈

—

148

C₈

—

268

—

149

C₈

—

150

C₈

—

151

C₈

—

134

—

152

C₈

—

134

—

153

C₂²

—

154

C₂²

#16863247

yes

characteristic

155

C₂²

#16863450

yes

—

156

C₄

—

157

C₄

—

158

C₄

#97×D₆₇

yes

characteristic

159

C₄

C₄×#11761611

yes

characteristic

160

C₄

#16863418

yes

—

161

C₄

#16863510

yes

—

162

C₂

—

163

C₂

#22560061

yes

characteristic

164

C₂

#22560087

yes

—

	subgroup	G/H
center	C₄²	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1276	C₂
socle	C₂²×C₆₇	C₄×C₄⋊C₄
prime radical	C₂³×C₆₇	#117
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1276	D₆₇
hypocenter	C₆₇	#8267
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1276

subgroup

G/H

center

C₄²

D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1276

C₂

socle

C₂²×C₆₇

C₄×C₄⋊C₄

prime radical

C₂³×C₆₇

#117

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1276

N	H ≅ G/N	type
11. #16863148	C₄	semidirect
12. #16863146	C₄	semidirect
88. C₆₇	#8267	semidirect

H ≅ G/N

type

11. #16863148

C₄

semidirect

12. #16863146

C₄

semidirect

88. C₆₇

#8267

semidirect

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

subgroup	index
C₄²×C₆₇⋊C₈	2
#22558955	2
#22558502	2
#22558042	2
C₆₇×#1276	2
#8267	67

subgroup

index

C₄²×C₆₇⋊C₈

C₆₇×#1276

Subgroups — #30746992

Conjugacy classes of subgroups

Complements

Maximal subgroups