Group #30869540 — subgroups

Subgroups — #30869540

Order 17152.

594 conjugacy classes of subgroups; only the 243 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 243 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	#22569150	C₂	1	2	yes	maximal
2	#22561187	C₂	1	2	yes	maximal
3	#22560810	C₂	1	8	yes	maximal
4	#22560350	C₂	1	1	yes	maximal, characteristic
5	#22559957	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#3162	C₂	1	1	yes	maximal, characteristic
7	#16863527	—	2	8	no	—
8	#16863423	C₂²	1	4	yes	—
9	#16863420	C₂²	1	2	yes	—
10	#16863352	C₂²	1	4	yes	—
11	C₂×#14523954	C₂²	1	1	yes	characteristic
12	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
13	#16863161	C₂²	1	4	yes	—
14	#16863155	C₂²	1	8	yes	—
15	#16863150	C₂²	1	4	yes	—
16	#533×C₆₇	C₂²	1	2	yes	—
17	#501×C₆₇	C₄	1	8	yes	complemented
18	#427×C₆₇	C₂²	1	4	yes	—
19	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#11761612	C₂³	1	4	yes	—
21	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
22	C₂×C₄×Dic₆₇	—	2	2	no	—
23	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
24	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
26	#14523954	—	2	4	no	—
27	#14523954	D₄	1	4	yes	—
28	C₄×C₆₇⋊C₈	—	2	4	no	—
29	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
30	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
31	#125×C₆₇	C₂×C₄	1	4	yes	—
32	C₂²⋊Q₈×C₆₇	—	2	4	no	—
33	C₄×Q₈×C₆₇	—	2	4	no	—
34	#117×C₆₇	C₂³	1	4	yes	—
35	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
36	#107×C₆₇	C₂×C₄	1	8	yes	—
37	#106×C₆₇	C₂×C₄	1	8	yes	—
38	#97×C₆₇	C₂×C₄	1	4	yes	—
39	C₂²×Dic₆₇	—	2	2	no	—
40	#11761612	—	2	8	no	—
41	C₂×C₆₇⋊C₈	—	2	4	no	—
42	C₂×C₆₇⋊C₈	—	2	4	no	—
43	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
44	C₄×Dic₆₇	—	2	4	no	—
45	C₄×Dic₆₇	—	4	4	no	—
46	C₂×Q₈×C₆₇	—	2	4	no	—
47	C₂²×C₄×C₆₇	#41	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
49	C₂×C₈×C₆₇	—	2	4	no	—
50	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
51	C₄⋊C₄×C₆₇	—	2	8	no	—
52	C₄⋊C₄×C₆₇	—	2	4	no	—
53	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
54	C₂²⋊C₄×C₆₇	—	2	4	no	—
55	C₄²×C₆₇	—	2	4	no	—
56	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
57	C₄²×C₆₇	C₂×D₄	1	2	yes	—
58	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
59	Q₈×C₆₇	—	8	4	no	—
60	C₂×C₄×C₆₇	—	2	8	no	—
61	C₂×C₄×C₆₇	—	4	2	no	—
62	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
63	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
65	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
66	C₂×Dic₆₇	—	2	4	no	—
67	C₂×Dic₆₇	—	4	2	no	—
68	C₂×Dic₆₇	—	4	2	no	—
69	C₈×C₆₇	—	2	4	no	—
70	C₆₇⋊C₈	—	2	4	no	—
71	C₆₇⋊C₈	—	4	4	no	—
72	C₂²×C₆₇	—	2	2	no	—
73	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
74	C₂²×C₆₇	#541	1	2	yes	—
75	C₄×C₆₇	—	2	4	no	—
76	C₄×C₆₇	—	4	8	no	—
77	C₄×C₆₇	#519	1	2	yes	—
78	C₄×C₆₇	#536	1	2	yes	—
79	Dic₆₇	—	4	4	no	—
80	#22259	—	67	1	no	maximal, Sylow
81	C₂×C₆₇	—	2	2	no	—
82	C₂×C₆₇	#2327	1	1	yes	characteristic
83	C₂×C₆₇	#2956	1	2	yes	—
84	#3162	D₆₇	1	1	yes	characteristic
85	#2961	—	67	2	no	—
86	#1994	—	67	2	no	—
87	#1922	—	67	8	no	—
88	#1838	—	67	1	no	—
89	#1745	—	67	1	no	—
90	C₆₇	#22259	1	1	yes	characteristic, complemented, Sylow
91	#533	C₂×D₆₇	1	2	yes	—
92	#501	Dic₆₇	1	8	yes	complemented
93	#446	—	134	8	no	—
94	#427	C₂×D₆₇	1	4	yes	—
95	#424	—	67	4	no	—
96	#423	—	67	2	no	—
97	#405	—	67	4	no	—
98	C₂×#97	—	67	1	no	—
99	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
100	C₄³	—	67	1	no	—
101	#341	—	67	4	no	—
102	#339	—	67	8	no	—
103	#336	—	67	4	no	—
104	C₂×M₁₆	—	67	4	no	—
105	C₂²×C₈	—	67	2	no	—
106	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
107	C₄⋊Q₈	C₂×Dic₆₇	1	4	yes	—
108	#125	C₂×Dic₆₇	1	4	yes	—
109	C₂²⋊Q₈	—	2	4	no	—
110	C₄×Q₈	—	2	4	no	—
111	#117	C₂²×D₆₇	1	4	yes	—
112	C₂×C₄²	—	67	2	no	—
113	C₂×C₄²	—	134	2	no	—
114	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
115	#107	C₂×Dic₆₇	1	8	yes	—
116	#106	C₂×Dic₆₇	1	8	yes	—
117	C₄⋊C₈	—	134	8	no	—
118	C₄⋊C₈	—	134	4	no	—
119	#97	—	67	4	no	—
120	#97	—	134	4	no	—
121	#97	C₂×Dic₆₇	1	4	yes	—
122	C₄×C₈	—	134	4	no	—
123	C₂×Q₈	—	2	4	no	—
124	C₂²×C₄	—	134	2	no	—
125	C₂²×C₄	#11761627	1	2	yes	—
126	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
127	M₁₆	—	134	8	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	67	4	no	—
130	C₂×C₈	—	134	4	no	—
131	C₂×C₈	—	134	4	no	—
132	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
133	C₄⋊C₄	—	2	8	no	—
134	C₄⋊C₄	—	2	4	no	—
135	C₄⋊C₄	C₂²×Dic₆₇	1	8	yes	—
136	C₂²⋊C₄	—	2	4	no	—
137	C₄²	—	2	4	no	—
138	C₄²	—	268	4	no	—
139	C₄²	—	134	4	no	—
140	C₄²	C₂²×Dic₆₇	1	2	yes	—
141	C₄²	D₄×D₆₇	1	2	yes	—
142	C₂³	C₂×#11761627	1	1	yes	characteristic
143	Q₈	—	8	4	no	—
144	C₂×C₄	—	2	8	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	268	2	no	—
147	C₂×C₄	—	268	2	no	—
148	C₂×C₄	—	134	4	no	—
149	C₂×C₄	D₄×Dic₆₇	1	4	yes	—
150	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
151	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
152	C₂×C₄	#14524091	1	4	yes	—
153	C₈	—	2	4	no	—
154	C₈	—	268	4	no	—
155	C₈	—	134	4	no	—
156	C₂²	—	2	2	no	—
157	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
158	C₂²	#16864007	1	2	yes	—
159	C₄	—	2	4	no	—
160	C₄	—	4	8	no	—
161	C₄	—	268	4	no	—
162	C₄	#16863874	1	2	yes	—
163	C₄	#16863971	1	2	yes	—
164	C₂	—	2	2	no	—
165	C₂	#22562929	1	1	yes	characteristic
166	C₂	#22569119	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22569150

C₂

yes

maximal

#22561187

C₂

yes

maximal

#22560810

C₂

yes

maximal

#22560350

C₂

yes

maximal, characteristic

#22559957

C₂

yes

maximal, characteristic

C₆₇×#3162

C₂

yes

maximal, characteristic

#16863527

—

#16863423

C₂²

yes

—

#16863420

C₂²

yes

—

#16863352

C₂²

yes

—

C₂×#14523954

C₂²

yes

characteristic

C₄²×Dic₆₇

C₂²

yes

characteristic

#16863161

C₂²

yes

—

#16863155

C₂²

yes

—

#16863150

C₂²

yes

—

#533×C₆₇

C₂²

yes

—

#501×C₆₇

C₄

yes

complemented

#427×C₆₇

C₂²

yes

—

C₂×#97×C₆₇

C₂²

yes

characteristic

C₂×#11761612

C₂³

yes

—

C₂²×C₆₇⋊C₈

C₂³

yes

—

C₂×C₄×Dic₆₇

—

C₂×C₄×Dic₆₇

C₂³

yes

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

#14523954

—

#14523954

D₄

yes

—

C₄×C₆₇⋊C₈

—

C₂²×C₈×C₆₇

C₂³

yes

—

C₄⋊Q₈×C₆₇

C₂×C₄

yes

—

#125×C₆₇

C₂×C₄

yes

—

C₂²⋊Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

#117×C₆₇

C₂³

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#107×C₆₇

C₂×C₄

yes

—

#106×C₆₇

C₂×C₄

yes

—

#97×C₆₇

C₂×C₄

yes

—

C₂²×Dic₆₇

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×D₄

yes

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

#41

yes

—

C₂²×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²×C₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂²×C₄

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂²×C₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

—

C₂³×C₆₇

C₂×#41

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

C₂³×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×D₄

yes

characteristic

C₂×C₄×C₆₇

2₋¹⁺⁴

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₄×D₄

yes

characteristic

C₂²×C₆₇

#541

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#519

yes

—

C₄×C₆₇

#536

yes

—

Dic₆₇

—

#22259

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#2327

yes

characteristic

C₂×C₆₇

#2956

yes

—

#3162

D₆₇

yes

—

—

—

—

—

C₆₇

#22259

yes

characteristic, complemented, Sylow

#533

C₂×D₆₇

yes

—

#501

Dic₆₇

yes

complemented

#446

—

134

—

#427

C₂×D₆₇

yes

—

#424

—

#423

—

#405

—

C₂×#97

—

C₂×#97

C₂×D₆₇

yes

characteristic

100

C₄³

—

101

#341

—

102

#339

—

103

#336

—

104

C₂×M₁₆

—

105

C₂²×C₈

—

106

C₂²×C₈

C₂²×D₆₇

yes

—

107

C₄⋊Q₈

C₂×Dic₆₇

yes

—

108

#125

C₂×Dic₆₇

yes

—

109

C₂²⋊Q₈

—

110

C₄×Q₈

—

111

#117

C₂²×D₆₇

yes

—

112

C₂×C₄²

—

113

C₂×C₄²

—

134

—

114

C₂×C₄²

C₂²×D₆₇

yes

characteristic

115

#107

C₂×Dic₆₇

yes

—

116

#106

C₂×Dic₆₇

yes

—

117

C₄⋊C₈

—

134

—

118

C₄⋊C₈

—

134

—

119

#97

—

120

#97

—

134

—

121

#97

C₂×Dic₆₇

yes

—

122

C₄×C₈

—

134

—

123

C₂×Q₈

—

124

C₂²×C₄

—

134

—

125

C₂²×C₄

#11761627

yes

—

126

C₂²×C₄

C₂³×D₆₇

yes

characteristic

127

M₁₆

—

134

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

134

—

131

C₂×C₈

—

134

—

132

C₂×C₈

C₂²×Dic₆₇

yes

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

C₂²×Dic₆₇

yes

—

136

C₂²⋊C₄

—

137

C₄²

—

138

C₄²

—

268

—

139

C₄²

—

134

—

140

C₄²

C₂²×Dic₆₇

yes

—

141

C₄²

D₄×D₆₇

yes

—

142

C₂³

C₂×#11761627

yes

characteristic

143

Q₈

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

268

—

147

C₂×C₄

—

268

—

148

C₂×C₄

—

134

—

149

C₂×C₄

D₄×Dic₆₇

yes

—

150

C₂×C₄

C₂³×Dic₆₇

yes

characteristic

151

C₂×C₄

C₂×D₄×D₆₇

yes

characteristic

152

C₂×C₄

#14524091

yes

—

153

C₈

—

154

C₈

—

268

—

155

C₈

—

134

—

156

C₂²

—

157

C₂²

C₂×D₄×Dic₆₇

yes

characteristic

158

C₂²

#16864007

yes

—

159

C₄

—

160

C₄

—

161

C₄

—

268

—

162

C₄

#16863874

yes

—

163

C₄

#16863971

yes

—

164

C₂

—

165

C₂

#22562929

yes

characteristic

166

C₂

#22569119

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#3162	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#3162	D₆₇
hypocenter	C₆₇	#22259
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3162

subgroup

G/H

center

C₂×C₄

C₂×D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₆₇

Fitting

C₆₇×#3162

C₂

socle

C₂²×C₆₇

C₂×C₄×D₄

prime radical

C₂³×C₆₇

C₂×#41

cosocle

C₂²×C₄×C₆₇

C₂⁴

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3162

N	H ≅ G/N	type
17. #501×C₆₇	C₄	semidirect
90. C₆₇	#22259	semidirect
92. #501	Dic₆₇	semidirect

H ≅ G/N

type

17. #501×C₆₇

C₄

semidirect

90. C₆₇

#22259

semidirect

92. #501

Dic₆₇

semidirect

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

subgroup	index
#22569150	2
#22561187	2
#22560810	2
#22560350	2
#22559957	2
C₆₇×#3162	2
#22259	67

subgroup

index

C₆₇×#3162

Subgroups — #30869540

Conjugacy classes of subgroups

Complements

Maximal subgroups