Group #30774365 — subgroups

Subgroups — #30774365

Order 17152.

334 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 135 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	#22569143	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863172	C₂	1	1	yes	maximal, characteristic
3	#22558299	C₂	1	4	yes	maximal
4	C₆₇×#2150	C₂	1	1	yes	maximal, characteristic
5	#16863953	C₄	1	1	yes	characteristic
6	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
7	#16863526	—	2	2	no	—
8	C₈×Dic₁₃₄	—	2	2	no	—
9	#16863422	C₄	1	2	yes	—
10	#16863420	C₄	1	1	yes	characteristic
11	#16863351	C₂²	1	2	yes	—
12	#16863172	—	2	2	no	—
13	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₆₇	—	2	2	no	—
16	C₂×#11761612	C₂×C₄	1	2	yes	—
17	#14524043	C₈	1	2	yes	—
18	#14524035	C₈	1	2	yes	—
19	#14524022	—	2	2	no	—
20	C₄×Dic₁₃₄	—	2	2	no	—
21	#14524002	C₂×C₄	1	2	yes	—
22	C₂×C₆₇⋊C₁₆	—	2	2	no	—
23	C₂×C₆₇⋊C₁₆	—	2	2	no	—
24	C₂×C₆₇⋊C₁₆	Q₈	1	4	yes	complemented
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	Dic₆₇⋊C₈	—	2	4	no	—
27	#14523953	—	2	2	no	—
28	C₈×Dic₆₇	—	2	2	no	—
29	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
32	M₃₂×C₆₇	—	2	4	no	—
33	C₂×C₁₆×C₆₇	—	2	2	no	—
34	C₄×C₈×C₆₇	D₄	1	2	yes	—
35	C₄×C₈×C₆₇	D₄	1	2	yes	—
36	C₂×Dic₁₃₄	—	2	2	no	—
37	#11761612	—	2	4	no	—
38	C₂×C₆₇⋊C₈	—	2	2	no	—
39	C₄⋊Dic₆₇	—	2	2	no	—
40	C₄⋊Dic₆₇	C₂×C₈	1	4	yes	—
41	Dic₆₇⋊C₄	—	2	4	no	—
42	C₂²⋊Dic₆₇	—	2	2	no	—
43	C₄×Dic₆₇	—	2	2	no	—
44	C₆₇⋊C₁₆	—	2	4	no	—
45	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
46	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₆₇	—	2	2	no	—
48	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
49	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₆₇	#41	1	2	yes	—
53	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
54	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
55	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
56	C₁₆×C₆₇	—	4	2	no	—
57	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
58	C₂×C₄×C₆₇	—	4	2	no	—
59	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
60	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
61	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
63	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₆₇	#131	1	2	yes	—
65	C₂×Dic₆₇	—	2	4	no	—
66	Dic₁₃₄	—	4	4	no	—
67	C₈×C₆₇	—	2	2	no	—
68	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
69	C₈×C₆₇	#124	1	1	yes	characteristic
70	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
71	C₆₇⋊C₈	—	4	2	no	—
72	C₂²×C₆₇	—	2	2	no	—
73	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
74	C₂²×C₆₇	#446	1	2	yes	—
75	C₄×C₆₇	—	2	2	no	—
76	C₄×C₆₇	—	2	2	no	—
77	C₄×C₆₇	#391	1	1	yes	characteristic
78	C₄×C₆₇	#408	1	1	yes	characteristic
79	C₄×C₆₇	C₈×Q₈	1	2	yes	—
80	Dic₆₇	—	4	4	no	—
81	#13267	—	67	1	no	maximal, Sylow
82	C₂×C₆₇	—	2	2	no	—
83	C₂×C₆₇	#1984	1	1	yes	characteristic
84	C₂×C₆₇	#2117	1	2	yes	—
85	#2960	—	67	1	no	—
86	#2150	D₆₇	1	1	yes	characteristic
87	C₂×#346	—	67	1	no	—
88	#1379	—	67	4	no	—
89	C₆₇	#13267	1	1	yes	characteristic, complemented, Sylow
90	#533	—	67	1	no	—
91	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
92	C₂²×C₁₆	—	67	2	no	—
93	#446	—	134	2	no	—
94	C₈×Q₈	—	134	2	no	—
95	#424	—	67	2	no	—
96	#423	—	67	1	no	—
97	#405	—	67	2	no	—
98	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
99	C₄⋊C₁₆	—	2	2	no	—
100	#346	—	134	2	no	—
101	C₂×M₁₆	—	67	2	no	—
102	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
104	C₄⋊Q₈	—	67	2	no	—
105	#125	—	67	2	no	—
106	C₂²⋊Q₈	—	134	2	no	—
107	C₄×Q₈	—	134	2	no	—
108	#117	—	67	2	no	—
109	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
110	M₃₂	—	2	4	no	—
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	—	67	4	no	—
113	C₂×C₁₆	—	134	2	no	—
114	C₂×C₁₆	—	134	2	no	—
115	C₄⋊C₈	—	134	4	no	—
116	C₄⋊C₈	—	134	2	no	—
117	#97	—	134	2	no	—
118	C₄×C₈	—	134	2	no	—
119	C₄×C₈	D₂₆₈	1	2	yes	—
120	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
121	C₂×Q₈	—	134	2	no	—
122	C₂²×C₄	#11761611	1	2	yes	—
123	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
124	M₁₆	—	134	4	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	134	2	no	—
127	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
128	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
129	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
130	C₂×C₈	Q₈×D₆₇	1	2	yes	—
131	C₂×C₈	#11761627	1	2	yes	—
132	C₄⋊C₄	—	67	4	no	—
133	C₄⋊C₄	—	134	4	no	—
134	C₄⋊C₄	—	134	2	no	—
135	C₂²⋊C₄	—	134	2	no	—
136	C₄²	—	134	2	no	—
137	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
138	C₄²	C₈×D₆₇	1	1	yes	characteristic
139	C₄²	C₈×D₆₇	1	1	yes	characteristic
140	C₁₆	—	4	2	no	—
141	C₁₆	—	134	4	no	—
142	C₂³	C₂×#11761611	1	1	yes	characteristic
143	Q₈	—	268	4	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	134	4	no	—
146	C₂×C₄	(C₂×D₆₇)⋊C₈	1	2	yes	—
147	C₂×C₄	(C₂×D₆₇)⋊C₈	1	2	yes	—
148	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
149	C₂×C₄	#14524008	1	2	yes	—
150	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
151	C₂×C₄	#14524051	1	2	yes	—
152	C₈	—	2	2	no	—
153	C₈	—	268	2	no	—
154	C₈	#14524023	1	2	yes	—
155	C₈	#14524030	1	1	yes	characteristic
156	C₈	#14524041	1	1	yes	characteristic
157	C₂²	—	2	2	no	—
158	C₂²	C₂×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
159	C₂²	#16863523	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	268	4	no	—
163	C₄	#16863301	1	1	yes	characteristic
164	C₄	#16863364	1	1	yes	characteristic
165	C₄	#16863520	1	2	yes	—
166	C₂	—	2	2	no	—
167	C₂	#22561144	1	1	yes	characteristic
168	C₂	#22561838	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22569143

C₂

yes

maximal, characteristic

C₂×#16863172

C₂

yes

maximal, characteristic

#22558299

C₂

yes

maximal

C₆₇×#2150

C₂

yes

maximal, characteristic

#16863953

C₄

yes

characteristic

C₂²×C₆₇⋊C₁₆

C₂²

yes

—

#16863526

—

C₈×Dic₁₃₄

—

#16863422

C₄

yes

—

#16863420

C₄

yes

characteristic

#16863351

C₂²

yes

—

#16863172

—

C₂×M₃₂×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₁₆×C₆₇

—

C₂×#11761612

C₂×C₄

yes

—

#14524043

C₈

yes

—

#14524035

C₈

yes

—

#14524022

—

C₄×Dic₁₃₄

—

#14524002

C₂×C₄

yes

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

Q₈

yes

complemented

C₄⋊(C₆₇⋊C₈)

—

Dic₆₇⋊C₈

—

#14523953

—

C₈×Dic₆₇

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₂×Dic₁₃₄

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂×C₈

yes

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

M₁₆

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

—

C₂×C₈×C₆₇

#41

yes

—

C₄²×C₆₇

C₂²⋊C₄

yes

—

C₄²×C₆₇

C₂×C₈

yes

characteristic

C₄²×C₆₇

C₂×C₈

yes

characteristic

C₁₆×C₆₇

—

C₂³×C₆₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊C₈

yes

—

C₂×C₄×C₆₇

C₂²⋊C₈

yes

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄×Q₈

yes

—

C₂×C₄×C₆₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₇

#131

yes

—

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

C₂²⋊Q₈

yes

—

C₈×C₆₇

#124

yes

characteristic

C₈×C₆₇

C₄⋊Q₈

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₆₇

#446

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

C₈×Q₈

yes

—

Dic₆₇

—

#13267

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1984

yes

characteristic

C₂×C₆₇

#2117

yes

—

#2960

—

#2150

D₆₇

yes

characteristic

C₂×#346

—

#1379

—

C₆₇

#13267

yes

characteristic, complemented, Sylow

#533

—

C₂×M₃₂

C₂×D₆₇

yes

—

C₂²×C₁₆

—

#446

—

134

—

C₈×Q₈

—

134

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

C₄⋊C₁₆

—

100

#346

—

134

—

101

C₂×M₁₆

—

102

C₂²×C₈

C₄×D₆₇

yes

characteristic

103

C₂²×C₈

C₂²×D₆₇

yes

characteristic

104

C₄⋊Q₈

—

105

#125

—

106

C₂²⋊Q₈

—

134

—

107

C₄×Q₈

—

134

—

108

#117

—

109

C₂×C₄²

C₄×D₆₇

yes

characteristic

110

M₃₂

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

113

C₂×C₁₆

—

134

—

114

C₂×C₁₆

—

134

—

115

C₄⋊C₈

—

134

—

116

C₄⋊C₈

—

134

—

117

#97

—

134

—

118

C₄×C₈

—

134

—

119

C₄×C₈

D₂₆₈

yes

—

120

C₄×C₈

C₂²⋊D₆₇

yes

—

121

C₂×Q₈

—

134

—

122

C₂²×C₄

#11761611

yes

—

123

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

124

M₁₆

—

134

—

125

C₂×C₈

—

126

C₂×C₈

—

134

—

127

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

128

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

129

C₂×C₈

C₂×D₂₆₈

yes

characteristic

130

C₂×C₈

Q₈×D₆₇

yes

—

131

C₂×C₈

#11761627

yes

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

—

134

C₄⋊C₄

—

134

—

135

C₂²⋊C₄

—

134

—

136

C₄²

—

134

—

137

C₄²

(C₂×D₆₇)⋊C₄

yes

—

138

C₄²

C₈×D₆₇

yes

characteristic

139

C₄²

C₈×D₆₇

yes

characteristic

140

C₁₆

—

141

C₁₆

—

134

—

142

C₂³

C₂×#11761611

yes

characteristic

143

Q₈

—

268

—

144

C₂×C₄

—

145

C₂×C₄

—

134

—

146

C₂×C₄

(C₂×D₆₇)⋊C₈

yes

—

147

C₂×C₄

(C₂×D₆₇)⋊C₈

yes

—

148

C₂×C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

149

C₂×C₄

#14524008

yes

—

150

C₂×C₄

C₂×C₈×D₆₇

yes

characteristic

151

C₂×C₄

#14524051

yes

—

152

C₈

—

153

C₈

—

268

—

154

C₈

#14524023

yes

—

155

C₈

#14524030

yes

characteristic

156

C₈

#14524041

yes

characteristic

157

C₂²

—

158

C₂²

C₂×(C₂×D₆₇)⋊C₈

yes

characteristic

159

C₂²

#16863523

yes

—

160

C₄

—

161

C₄

—

162

C₄

—

268

—

163

C₄

#16863301

yes

characteristic

164

C₄

#16863364

yes

characteristic

165

C₄

#16863520

yes

—

166

C₂

—

167

C₂

#22561144

yes

characteristic

168

C₂

#22561838

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#2150	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₈
prime radical	C₂³×C₆₇	C₂×M₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#2150	D₆₇
hypocenter	C₆₇	#13267
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#2150

C₂

socle

C₂²×C₆₇

C₂×C₂²⋊C₈

prime radical

C₂³×C₆₇

C₂×M₁₆

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
24. C₂×C₆₇⋊C₁₆	Q₈	semidirect
89. C₆₇	#13267	semidirect

H ≅ G/N

type

24. C₂×C₆₇⋊C₁₆

Q₈

semidirect

89. C₆₇

#13267

semidirect

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

subgroup	index
#22569143	2
C₂×#16863172	2
#22558299	2
C₆₇×#2150	2
#13267	67

subgroup

index

C₆₇×#2150

Subgroups — #30774365

Conjugacy classes of subgroups

Complements

Maximal subgroups