Group #22565198 — subgroups

Subgroups — #22565198

Order 8576.

540 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 189 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	C₂×#14524041	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
3	#16863327	C₂	1	4	yes	maximal
4	#16863303	C₂	1	2	yes	maximal
5	#16863292	C₂	1	4	yes	maximal
6	C₄×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₃₄	C₂²	1	1	yes	characteristic
9	#14524041	—	2	8	no	—
10	C₄×Dic₁₃₄	—	2	4	no	—
11	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
13	C₂×Dic₆₇⋊C₄	C₂²	1	8	yes	—
14	C₂×Dic₆₇⋊C₄	C₂²	1	2	yes	—
15	C₂×C₄×Dic₆₇	C₂²	1	2	yes	—
16	C₂×C₄×Dic₆₇	C₂²	1	2	yes	—
17	C₆₇⋊#95	C₂²	1	2	yes	—
18	C₆₇⋊#95	C₂²	1	8	yes	—
19	C₂²×Q₈×C₆₇	C₂²	1	2	yes	—
20	C₄⋊Q₈×C₆₇	—	2	4	no	—
21	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×Dic₁₃₄	—	2	2	no	—
25	C₂×Dic₁₃₄	—	2	2	no	—
26	C₂×Dic₁₃₄	D₄	1	4	yes	—
27	C₂²×Dic₆₇	C₂³	1	2	yes	—
28	C₂²×Dic₆₇	C₂³	1	2	yes	—
29	C₂²×Dic₆₇	C₂³	1	4	yes	—
30	C₄⋊Dic₆₇	—	2	2	no	—
31	C₄⋊Dic₆₇	Q₈	1	8	yes	complemented
32	Dic₆₇⋊C₄	—	2	4	no	—
33	Dic₆₇⋊C₄	—	2	16	no	—
34	C₄×Dic₆₇	—	2	4	no	—
35	C₄×Dic₆₇	—	2	4	no	—
36	C₂×Q₈×C₆₇	—	2	4	no	—
37	C₂×Q₈×C₆₇	—	2	4	no	—
38	C₂×Q₈×C₆₇	—	2	4	no	—
39	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
40	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₆₇	—	2	4	no	—
44	C₄⋊C₄×C₆₇	—	2	4	no	—
45	C₄²×C₆₇	D₄	1	4	yes	—
46	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
47	Q₈×C₆₇	—	4	8	no	—
48	C₂×C₄×C₆₇	—	2	2	no	—
49	C₂×C₄×C₆₇	—	2	4	no	—
50	C₂×C₄×C₆₇	—	2	4	no	—
51	C₂×C₄×C₆₇	—	2	4	no	—
52	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₆₇	C₂×Q₈	1	4	yes	—
57	C₂×C₄×C₆₇	#41	1	2	yes	—
58	C₂×Dic₆₇	—	2	4	no	—
59	C₂×Dic₆₇	—	2	2	no	—
60	C₂×Dic₆₇	—	2	2	no	—
61	C₂×Dic₆₇	—	2	4	no	—
62	C₂×Dic₆₇	—	2	2	no	—
63	C₂×Dic₆₇	—	2	4	no	—
64	C₂×Dic₆₇	C₂×D₄	1	4	yes	—
65	C₂×Dic₆₇	C₂×Q₈	1	8	yes	—
66	Dic₁₃₄	—	2	8	no	—
67	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
68	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
69	C₂²×C₆₇	C₂²×Q₈	1	2	yes	—
70	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
71	C₂²×C₆₇	2₋¹⁺⁴	1	2	yes	—
72	C₄×C₆₇	—	2	2	no	—
73	C₄×C₆₇	—	4	4	no	—
74	C₄×C₆₇	#121	1	4	yes	—
75	C₄×C₆₇	C₄⋊Q₈	1	4	yes	complemented
76	Dic₆₇	—	2	4	no	—
77	Dic₆₇	—	2	8	no	—
78	Dic₆₇	—	4	2	no	—
79	C₂×C₆₇	C₂×#121	1	1	yes	characteristic
80	C₂×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₆₇	#536	1	1	yes	characteristic
82	C₂×C₆₇	D₄×Q₈	1	2	yes	—
83	C₂×C₆₇	#557	1	2	yes	—
84	#2546	—	67	1	no	maximal, Sylow
85	C₆₇	#2546	1	1	yes	characteristic, complemented, Sylow
86	C₂×C₄⋊Q₈	—	67	2	no	—
87	C₂×C₄⋊Q₈	D₆₇	1	1	yes	characteristic
88	C₂×C₄×Q₈	—	67	1	no	—
89	#398	—	67	4	no	—
90	#391	—	67	2	no	—
91	#389	—	67	4	no	—
92	C₄×C₄⋊C₄	—	67	1	no	—
93	C₂²×Q₈	—	67	1	no	—
94	C₂²×Q₈	C₂×D₆₇	1	2	yes	—
95	C₄⋊Q₈	—	2	4	no	—
96	C₄⋊Q₈	—	134	8	no	—
97	C₄×Q₈	—	134	4	no	—
98	C₂×C₄⋊C₄	—	67	2	no	—
99	C₂×C₄⋊C₄	—	67	2	no	—
100	C₂×C₄⋊C₄	—	67	1	no	—
101	C₂×C₄⋊C₄	—	67	8	no	—
102	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
103	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
104	C₂×C₄²	—	67	2	no	—
105	C₂×C₄²	—	67	2	no	—
106	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
107	#95	—	67	2	no	—
108	#95	—	67	8	no	—
109	C₂×Q₈	—	2	4	no	—
110	C₂×Q₈	—	2	4	no	—
111	C₂×Q₈	—	2	4	no	—
112	C₂×Q₈	—	67	4	no	—
113	C₂×Q₈	—	134	2	no	—
114	C₂×Q₈	—	134	2	no	—
115	C₂²×C₄	—	67	4	no	—
116	C₂²×C₄	—	67	2	no	—
117	C₂²×C₄	—	67	2	no	—
118	C₂²×C₄	C₂²×D₆₇	1	4	yes	—
119	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
120	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
121	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
122	C₄⋊C₄	—	2	4	no	—
123	C₄⋊C₄	—	2	4	no	—
124	C₄⋊C₄	—	67	8	no	—
125	C₄⋊C₄	—	134	16	no	—
126	C₄⋊C₄	—	134	4	no	—
127	C₄⋊C₄	—	134	2	no	—
128	C₄²	—	134	4	no	—
129	C₄²	—	134	4	no	—
130	C₄²	C₂²⋊D₆₇	1	4	yes	—
131	C₂³	C₂³×D₆₇	1	1	yes	characteristic
132	Q₈	—	4	8	no	—
133	Q₈	—	134	8	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	67	4	no	—
139	C₂×C₄	—	67	8	no	—
140	C₂×C₄	—	134	2	no	—
141	C₂×C₄	—	134	4	no	—
142	C₂×C₄	—	134	4	no	—
143	C₂×C₄	—	134	2	no	—
144	C₂×C₄	—	134	2	no	—
145	C₂×C₄	—	134	4	no	—
146	C₂×C₄	D₄×D₆₇	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
148	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
149	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
150	C₂×C₄	Q₈×D₆₇	1	4	yes	—
151	C₂×C₄	#11761627	1	2	yes	—
152	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
153	C₂²	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
154	C₂²	C₂×Q₈×D₆₇	1	2	yes	—
155	C₂²	C₂×#11761627	1	1	yes	characteristic
156	C₂²	#14524091	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	4	4	no	—
159	C₄	—	268	2	no	—
160	C₄	—	134	8	no	—
161	C₄	—	134	4	no	—
162	C₄	#14524014	1	4	yes	—
163	C₄	#14524041	1	4	yes	complemented
164	C₂	C₂×#14524014	1	1	yes	characteristic
165	C₂	C₂×#14524041	1	1	yes	characteristic
166	C₂	#16863970	1	1	yes	characteristic
167	C₂	#16864061	1	2	yes	—
168	C₂	#16864121	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524041

C₂

yes

maximal

C₂×C₄×Dic₁₃₄

C₂

yes

maximal, characteristic

#16863327

C₂

yes

maximal

#16863303

C₂

yes

maximal

#16863292

C₂

yes

maximal

C₄×C₄⋊Dic₆₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

characteristic

#14524041

—

C₄×Dic₁₃₄

—

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×Dic₆₇⋊C₄

C₂²

yes

—

C₂×Dic₆₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₂²×Q₈×C₆₇

C₂²

yes

—

C₄⋊Q₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

D₄

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

Q₈

yes

complemented

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

D₄

yes

—

C₂³×C₆₇

C₂⁴

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×D₄

yes

—

C₂×Dic₆₇

C₂×Q₈

yes

—

Dic₁₃₄

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂²×C₆₇

C₂²×Q₈

yes

—

C₂²×C₆₇

C₂×#41

yes

characteristic

C₂²×C₆₇

2₋¹⁺⁴

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#121

yes

—

C₄×C₆₇

C₄⋊Q₈

yes

complemented

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

C₂×#121

yes

characteristic

C₂×C₆₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₆₇

#536

yes

characteristic

C₂×C₆₇

D₄×Q₈

yes

—

C₂×C₆₇

#557

yes

—

#2546

—

maximal, Sylow

C₆₇

#2546

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₆₇

yes

characteristic

C₂×C₄×Q₈

—

#398

—

#391

—

#389

—

C₄×C₄⋊C₄

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₆₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

134

—

C₄×Q₈

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

103

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

104

C₂×C₄²

—

105

C₂×C₄²

—

106

C₂×C₄²

C₂×D₆₇

yes

characteristic

107

#95

—

108

#95

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

134

—

114

C₂×Q₈

—

134

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

C₂²×D₆₇

yes

—

119

C₂²×C₄

C₂²×D₆₇

yes

characteristic

120

C₂²×C₄

C₂²×D₆₇

yes

characteristic

121

C₂²×C₄

C₂²×D₆₇

yes

characteristic

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

134

—

126

C₄⋊C₄

—

134

—

127

C₄⋊C₄

—

134

—

128

C₄²

—

134

—

129

C₄²

—

134

—

130

C₄²

C₂²⋊D₆₇

yes

—

131

C₂³

C₂³×D₆₇

yes

characteristic

132

Q₈

—

133

Q₈

—

134

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

134

—

141

C₂×C₄

—

134

—

142

C₂×C₄

—

134

—

143

C₂×C₄

—

134

—

144

C₂×C₄

—

134

—

145

C₂×C₄

—

134

—

146

C₂×C₄

D₄×D₆₇

yes

—

147

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

148

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

149

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

150

C₂×C₄

Q₈×D₆₇

yes

—

151

C₂×C₄

#11761627

yes

—

152

C₂²

C₂×D₄×D₆₇

yes

characteristic

153

C₂²

C₂²×C₂²⋊D₆₇

yes

characteristic

154

C₂²

C₂×Q₈×D₆₇

yes

—

155

C₂²

C₂×#11761627

yes

characteristic

156

C₂²

#14524091

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

268

—

160

C₄

—

134

—

161

C₄

—

134

—

162

C₄

#14524014

yes

—

163

C₄

#14524041

yes

complemented

164

C₂

C₂×#14524014

yes

characteristic

165

C₂

C₂×#14524041

yes

characteristic

166

C₂

#16863970

yes

characteristic

167

C₂

#16864061

yes

—

168

C₂

#16864121

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₆₇	C₂
hypercenter	C₂×C₄⋊Q₈	D₆₇
hypocenter	C₆₇	#2546
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₇

commutator, socle

C₂³×C₆₇

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₆₇

C₂

hypercenter

C₂×C₄⋊Q₈

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
31. C₄⋊Dic₆₇	Q₈	semidirect
75. C₄×C₆₇	C₄⋊Q₈	semidirect
85. C₆₇	#2546	semidirect
163. C₄	#14524041	semidirect

H ≅ G/N

type

31. C₄⋊Dic₆₇

Q₈

semidirect

75. C₄×C₆₇

C₄⋊Q₈

semidirect

85. C₆₇

#2546

semidirect

163. C₄

#14524041

semidirect

Subgroups — #22565198

Conjugacy classes of subgroups

Complements