Group #30771209 — subgroups

Subgroups — #30771209

Order 17152.

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

normal subgroups only

This group has 308 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	#22561981	C₂	1	2	yes	maximal
2	M₃₂×Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	#22560090	C₂	1	2	yes	maximal
4	#22558313	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1384	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	1	yes	characteristic
7	C₂×#14523979	C₂²	1	2	yes	—
8	#16863352	—	2	2	no	—
9	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
10	#16863218	C₄	1	8	yes	complemented
11	#16863171	C₄	1	2	yes	complemented
12	C₁₆×Dic₆₇	C₄	1	2	yes	complemented
13	#16863155	—	2	2	no	—
14	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
15	#427×C₆₇	C₂²	1	2	yes	—
16	C₂×#11761612	—	2	2	no	—
17	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
18	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
19	#14523983	C₂×C₄	1	4	yes	—
20	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
21	#14523979	—	2	2	no	—
22	#14523979	C₂×C₄	1	4	yes	—
23	#14523954	—	4	2	no	—
24	C₄×C₆₇⋊C₈	—	4	2	no	—
25	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₆₇	Q₈	1	1	yes	characteristic
28	C₈×Dic₆₇	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
30	#117×C₆₇	—	2	2	no	—
31	M₃₂×C₆₇	—	2	2	no	—
32	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
33	#107×C₆₇	—	2	2	no	—
34	#106×C₆₇	C₂×C₄	1	4	yes	—
35	C₂²×Dic₆₇	—	2	1	no	—
36	#11761612	—	2	4	no	—
37	#11761612	—	4	2	no	—
38	C₂×C₆₇⋊C₈	—	4	2	no	—
39	C₂×C₆₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₆₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₆₇⋊C₈	SD₁₆	1	2	yes	—
42	C₂×C₆₇⋊C₈	SD₁₆	1	2	yes	—
43	C₄×Dic₆₇	—	2	1	no	—
44	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
45	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
46	C₆₇⋊C₁₆	C₄²	1	4	yes	—
47	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
50	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₆₇	#41	1	2	yes	—
52	C₄⋊C₄×C₆₇	—	2	4	no	—
53	C₂²⋊C₄×C₆₇	—	4	2	no	—
54	C₄²×C₆₇	—	4	2	no	—
55	C₁₆×C₆₇	—	2	2	no	—
56	C₂³×C₆₇	#135	1	1	yes	characteristic
57	C₂×C₄×C₆₇	—	4	4	no	—
58	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
60	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
61	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
62	C₂×Dic₆₇	—	2	1	no	—
63	C₂×Dic₆₇	—	2	1	no	—
64	C₂×Dic₆₇	—	4	2	no	—
65	C₈×C₆₇	C₂×C₄²	1	1	yes	characteristic
66	C₈×C₆₇	#117	1	1	yes	characteristic
67	C₈×C₆₇	C₄×Q₈	1	2	yes	—
68	C₆₇⋊C₈	—	4	4	no	—
69	C₆₇⋊C₈	#106	1	1	yes	characteristic
70	C₆₇⋊C₈	#106	1	2	yes	—
71	C₆₇⋊C₈	#106	1	1	yes	characteristic
72	C₂²×C₆₇	—	4	1	no	—
73	C₂²×C₆₇	#427	1	1	yes	characteristic
74	C₂²×C₆₇	C₄×SD₁₆	1	2	yes	—
75	C₄×C₆₇	—	8	4	no	—
76	C₄×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
77	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
78	C₄×C₆₇	C₄×SD₁₆	1	2	yes	—
79	Dic₆₇	—	2	2	no	—
80	Dic₆₇	—	4	1	no	—
81	#12668	—	67	1	no	maximal, Sylow
82	C₂×C₆₇	—	2	1	no	—
83	C₂×C₆₇	—	2	2	no	—
84	C₂×C₆₇	C₄×#106	1	1	yes	characteristic
85	#2155	—	67	2	no	—
86	C₄×M₃₂	—	67	1	no	—
87	#1772	—	67	2	no	—
88	#1385	—	67	1	no	—
89	#1384	D₆₇	1	1	yes	characteristic
90	C₆₇	#12668	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₃₂	—	67	1	no	—
92	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
93	C₂×#108	—	67	2	no	—
94	#427	C₂×D₆₇	1	2	yes	—
95	#405	—	134	2	no	—
96	C₂×C₄×C₈	—	67	1	no	—
97	#365	—	67	8	no	—
98	#346	—	67	2	no	—
99	C₄×C₁₆	—	67	2	no	—
100	#339	—	134	2	no	—
101	C₂×M₁₆	—	134	2	no	—
102	C₂²×C₈	—	67	1	no	—
103	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
104	#117	—	2	2	no	—
105	C₂×C₄²	—	67	1	no	—
106	M₃₂	—	2	2	no	—
107	M₃₂	—	67	4	no	—
108	C₂×C₁₆	—	67	2	no	—
109	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
110	#108	—	67	4	no	—
111	#108	—	134	2	no	—
112	#107	—	2	2	no	—
113	#106	C₄×D₆₇	1	4	yes	—
114	#97	—	268	2	no	—
115	C₄×C₈	—	268	2	no	—
116	C₄×C₈	—	67	1	no	—
117	C₄×C₈	—	67	1	no	—
118	C₄×C₈	—	67	1	no	—
119	C₄×C₈	—	67	1	no	—
120	C₂²×C₄	—	134	1	no	—
121	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
122	M₁₆	—	268	2	no	—
123	M₁₆	—	134	4	no	—
124	C₂×C₈	—	268	2	no	—
125	C₂×C₈	—	67	1	no	—
126	C₂×C₈	—	67	1	no	—
127	C₂×C₈	—	67	2	no	—
128	C₂×C₈	—	67	2	no	—
129	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
130	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
131	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
132	C₂×C₈	#11761627	1	2	yes	—
133	C₄⋊C₄	—	2	4	no	—
134	C₂²⋊C₄	—	4	2	no	—
135	C₄²	—	4	2	no	—
136	C₄²	—	67	1	no	—
137	C₄²	—	67	1	no	—
138	C₄²	—	134	1	no	—
139	C₁₆	—	2	2	no	—
140	C₁₆	—	67	4	no	—
141	C₂³	#14524062	1	1	yes	characteristic
142	C₂×C₄	—	4	4	no	—
143	C₂×C₄	—	268	2	no	—
144	C₂×C₄	—	134	1	no	—
145	C₂×C₄	—	134	1	no	—
146	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
147	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
148	C₂×C₄	SD₁₆×D₆₇	1	2	yes	—
149	C₂×C₄	#14524066	1	1	yes	characteristic
150	C₈	—	268	4	no	—
151	C₈	—	67	1	no	—
152	C₈	—	67	2	no	—
153	C₈	—	67	1	no	—
154	C₈	C₄²×D₆₇	1	1	yes	characteristic
155	C₈	#14523999	1	1	yes	characteristic
156	C₈	#14524008	1	2	yes	—
157	C₂²	—	4	1	no	—
158	C₂²	#16863433	1	1	yes	characteristic
159	C₂²	#16863487	1	2	yes	—
160	C₄	—	8	4	no	—
161	C₄	—	268	1	no	—
162	C₄	—	134	2	no	—
163	C₄	#16863247	1	1	yes	characteristic
164	C₄	#106×D₆₇	1	1	yes	characteristic
165	C₄	#16863481	1	2	yes	—
166	C₂	—	2	2	no	—
167	C₂	—	2	1	no	—
168	C₂	#22560097	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22561981

C₂

yes

maximal

M₃₂×Dic₆₇

C₂

yes

maximal, characteristic

#22560090

C₂

yes

maximal

#22558313

C₂

yes

maximal, characteristic

C₆₇×#1384

C₂

yes

maximal, characteristic

C₂×#14523983

C₂²

yes

characteristic

C₂×#14523979

C₂²

yes

—

#16863352

—

C₂×C₈×Dic₆₇

C₂²

yes

characteristic

#16863218

C₄

yes

complemented

#16863171

C₄

yes

complemented

C₁₆×Dic₆₇

C₄

yes

complemented

#16863155

—

C₂×M₃₂×C₆₇

C₂²

yes

characteristic

#427×C₆₇

C₂²

yes

—

C₂×#11761612

—

C₂²×C₆₇⋊C₈

D₄

yes

characteristic

C₂×C₄×Dic₆₇

D₄

yes

characteristic

#14523983

C₂×C₄

yes

—

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

—

#14523979

—

#14523979

C₂×C₄

yes

—

#14523954

—

C₄×C₆₇⋊C₈

—

C₈×Dic₆₇

C₂×C₄

yes

characteristic

C₈×Dic₆₇

C₂×C₄

yes

characteristic

C₈×Dic₆₇

Q₈

yes

characteristic

C₈×Dic₆₇

Q₈

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

#117×C₆₇

—

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

C₂×C₄

yes

—

#107×C₆₇

—

#106×C₆₇

C₂×C₄

yes

—

C₂²×Dic₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₆₇⋊C₈

SD₁₆

yes

—

C₂×C₆₇⋊C₈

SD₁₆

yes

—

C₄×Dic₆₇

—

C₄×Dic₆₇

C₄⋊C₄

yes

characteristic

C₄×Dic₆₇

C₄⋊C₄

yes

characteristic

C₆₇⋊C₁₆

C₄²

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂²×C₄

yes

—

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₁₆×C₆₇

—

C₂³×C₆₇

#135

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

C₂×C₄×C₆₇

#135

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

C₂×C₄²

yes

characteristic

C₈×C₆₇

#117

yes

characteristic

C₈×C₆₇

C₄×Q₈

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

#106

yes

characteristic

C₆₇⋊C₈

#106

yes

—

C₆₇⋊C₈

#106

yes

characteristic

C₂²×C₆₇

—

C₂²×C₆₇

#427

yes

characteristic

C₂²×C₆₇

C₄×SD₁₆

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₄×C₄⋊C₄

yes

characteristic

C₄×C₆₇

C₂×#106

yes

characteristic

C₄×C₆₇

C₄×SD₁₆

yes

—

Dic₆₇

—

Dic₆₇

—

#12668

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₄×#106

yes

characteristic

#2155

—

C₄×M₃₂

—

#1772

—

#1385

—

#1384

D₆₇

yes

characteristic

C₆₇

#12668

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₆₇

yes

characteristic

C₂×#108

—

#427

C₂×D₆₇

yes

—

#405

—

134

—

C₂×C₄×C₈

—

#365

—

#346

—

C₄×C₁₆

—

100

#339

—

134

—

101

C₂×M₁₆

—

134

—

102

C₂²×C₈

—

103

C₂²×C₈

C₂²×D₆₇

yes

characteristic

104

#117

—

105

C₂×C₄²

—

106

M₃₂

—

107

M₃₂

—

108

C₂×C₁₆

—

109

C₂×C₁₆

C₄×D₆₇

yes

—

110

#108

—

111

#108

—

134

—

112

#107

—

113

#106

C₄×D₆₇

yes

—

114

#97

—

268

—

115

C₄×C₈

—

268

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₂²×C₄

—

134

—

121

C₂²×C₄

D₄×D₆₇

yes

characteristic

122

M₁₆

—

268

—

123

M₁₆

—

134

—

124

C₂×C₈

—

268

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

130

C₂×C₈

C₂×C₄×D₆₇

yes

—

131

C₂×C₈

Q₈×D₆₇

yes

characteristic

132

C₂×C₈

#11761627

yes

—

133

C₄⋊C₄

—

134

C₂²⋊C₄

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

134

—

139

C₁₆

—

140

C₁₆

—

141

C₂³

#14524062

yes

characteristic

142

C₂×C₄

—

143

C₂×C₄

—

268

—

144

C₂×C₄

—

134

—

145

C₂×C₄

—

134

—

146

C₂×C₄

C₄⋊C₄×D₆₇

yes

characteristic

147

C₂×C₄

(C₂²⋊D₆₇)⋊C₄

yes

—

148

C₂×C₄

SD₁₆×D₆₇

yes

—

149

C₂×C₄

#14524066

yes

characteristic

150

C₈

—

268

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

C₄²×D₆₇

yes

characteristic

155

C₈

#14523999

yes

characteristic

156

C₈

#14524008

yes

—

157

C₂²

—

158

C₂²

#16863433

yes

characteristic

159

C₂²

#16863487

yes

—

160

C₄

—

161

C₄

—

268

—

162

C₄

—

134

—

163

C₄

#16863247

yes

characteristic

164

C₄

#106×D₆₇

yes

characteristic

165

C₄

#16863481

yes

—

166

C₂

—

167

C₂

—

168

C₂

#22560097

yes

characteristic

	subgroup	G/H
center	C₄	#106×D₆₇
commutator	C₈×C₆₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1384	C₂
socle	C₂×C₆₇	C₄×#106
prime radical	C₂³×C₆₇	#135
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1384	D₆₇
hypocenter	C₆₇	#12668
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1384

subgroup

G/H

center

C₄

#106×D₆₇

commutator

C₈×C₆₇

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1384

C₂

C₂×C₆₇

C₂³×C₆₇

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1384

N	H ≅ G/N	type
10. #16863218	C₄	semidirect
11. #16863171	C₄	semidirect
12. C₁₆×Dic₆₇	C₄	semidirect
90. C₆₇	#12668	semidirect

H ≅ G/N

type

10. #16863218

C₄

semidirect

11. #16863171

C₄

semidirect

12. C₁₆×Dic₆₇

C₄

semidirect

90. C₆₇

#12668

semidirect

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

subgroup	index
#22561981	2
M₃₂×Dic₆₇	2
#22560090	2
#22558313	2
C₆₇×#1384	2
#12668	67

subgroup

index

M₃₂×Dic₆₇

C₆₇×#1384

Subgroups — #30771209

Conjugacy classes of subgroups

Complements

Maximal subgroups