Group #22560972 — subgroups

Subgroups — #22560972

Order 8576.

256 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 125 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×#14523974	C₂	1	2	yes	maximal
2	C₂×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#16863266	C₂	1	1	yes	maximal, characteristic
4	#16863158	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₆₇	—	2	1	no	—
8	C₂×C₄⋊Dic₆₇	—	2	1	no	—
9	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
10	#14523974	—	2	4	no	—
11	C₄⋊(C₆₇⋊C₈)	C₄	1	4	yes	complemented
12	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₆₇	—	2	1	no	—
15	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	#95×C₆₇	—	2	1	no	—
17	C₂²×Dic₆₇	—	2	2	no	—
18	C₂×C₆₇⋊C₈	—	2	2	no	—
19	C₂×C₆₇⋊C₈	—	2	2	no	—
20	C₂×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
21	C₄⋊Dic₆₇	—	2	2	no	—
22	C₄⋊Dic₆₇	—	2	2	no	—
23	C₄⋊Dic₆₇	—	4	2	no	—
24	C₄⋊Dic₆₇	—	4	2	no	—
25	C₂²×C₄×C₆₇	—	2	1	no	—
26	C₂²×C₄×C₆₇	—	2	1	no	—
27	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₆₇	—	2	1	no	—
31	C₄⋊C₄×C₆₇	—	2	1	no	—
32	C₄⋊C₄×C₆₇	D₄	1	2	yes	—
33	C₄⋊C₄×C₆₇	Q₈	1	2	yes	—
34	C₄²×C₆₇	—	4	2	no	—
35	C₄²×C₆₇	C₂×C₄	1	2	yes	—
36	C₄²×C₆₇	Q₈	1	2	yes	—
37	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₆₇	—	2	1	no	—
39	C₂×C₄×C₆₇	—	2	1	no	—
40	C₂×C₄×C₆₇	—	2	2	no	—
41	C₂×C₄×C₆₇	—	2	2	no	—
42	C₂×C₄×C₆₇	—	4	1	no	—
43	C₂×C₄×C₆₇	—	4	1	no	—
44	C₂×C₄×C₆₇	—	4	1	no	—
45	C₂×C₄×C₆₇	—	4	1	no	—
46	C₂×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
48	C₂×C₄×C₆₇	SD₁₆	1	2	yes	—
49	C₂×C₄×C₆₇	SD₁₆	1	2	yes	—
50	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
55	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
56	C₂×Dic₆₇	—	4	2	no	—
57	C₂×Dic₆₇	—	4	2	no	—
58	C₂×Dic₆₇	—	4	2	no	—
59	C₆₇⋊C₈	—	2	4	no	—
60	C₂²×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₆₇	#121	1	1	yes	characteristic
62	C₂²×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₂²×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
65	C₂²×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂²×C₆₇	#137	1	1	yes	characteristic
67	C₄×C₆₇	—	2	2	no	—
68	C₄×C₆₇	—	4	2	no	—
69	C₄×C₆₇	—	4	2	no	—
70	C₄×C₆₇	#106	1	4	yes	complemented
71	C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
72	C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
73	C₄×C₆₇	#125	1	1	yes	characteristic
74	C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
75	Dic₆₇	—	8	2	no	—
76	C₂×C₆₇	#389	1	1	yes	characteristic
77	C₂×C₆₇	C₂×#106	1	1	yes	characteristic
78	C₂×C₆₇	#428	1	1	yes	characteristic
79	C₂×C₆₇	#460	1	1	yes	characteristic
80	C₂×C₆₇	#461	1	1	yes	characteristic
81	C₂×C₆₇	#475	1	1	yes	characteristic
82	C₂×C₆₇	#476	1	1	yes	characteristic
83	#1948	—	67	1	no	maximal, Sylow
84	C₆₇	#1948	1	1	yes	characteristic, complemented, Sylow
85	C₂×#106	—	67	2	no	—
86	C₂×C₄⋊C₈	—	67	1	no	—
87	#384	—	67	1	no	—
88	C₄×C₄⋊C₄	D₆₇	1	1	yes	characteristic
89	#340	—	67	2	no	—
90	C₂²×C₈	—	67	2	no	—
91	C₂×C₄⋊C₄	—	67	2	no	—
92	C₂×C₄⋊C₄	—	134	1	no	—
93	C₂×C₄⋊C₄	—	134	1	no	—
94	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
95	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
96	C₂×C₄²	—	2	1	no	—
97	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
98	#106	—	134	4	no	—
99	C₄⋊C₈	—	67	4	no	—
100	#95	—	2	1	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	—	134	2	no	—
104	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
107	C₂×C₈	—	67	4	no	—
108	C₂×C₈	—	134	2	no	—
109	C₂×C₈	—	134	2	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	268	2	no	—
113	C₄⋊C₄	—	268	2	no	—
114	C₄⋊C₄	—	134	2	no	—
115	C₄⋊C₄	—	134	2	no	—
116	C₄⋊C₄	Dic₁₃₄	1	2	yes	—
117	C₄⋊C₄	D₂₆₈	1	2	yes	—
118	C₄²	—	4	2	no	—
119	C₄²	Dic₁₃₄	1	2	yes	—
120	C₄²	C₄×D₆₇	1	2	yes	—
121	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	268	2	no	—
131	C₂×C₄	—	268	2	no	—
132	C₂×C₄	—	268	2	no	—
133	C₂×C₄	Dic₆₇⋊C₄	1	2	yes	—
134	C₂×C₄	Dic₆₇⋊C₄	1	2	yes	—
135	C₂×C₄	#11761615	1	2	yes	—
136	C₂×C₄	#11761616	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
138	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
139	C₂×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
140	C₂×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
141	C₂×C₄	#11761628	1	1	yes	characteristic
142	C₂×C₄	#11761628	1	1	yes	characteristic
143	C₈	—	134	4	no	—
144	C₂²	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
145	C₂²	#14524017	1	1	yes	characteristic
146	C₂²	#14524022	1	1	yes	characteristic
147	C₂²	C₂×#11761615	1	1	yes	characteristic
148	C₂²	C₂×#11761616	1	1	yes	characteristic
149	C₂²	#14524068	1	1	yes	characteristic
150	C₂²	#14524073	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	2	no	—
154	C₄	—	536	2	no	—
155	C₄	#14523974	1	4	yes	complemented
156	C₄	C₄×D₂₆₈	1	1	yes	characteristic
157	C₄	C₄×Dic₁₃₄	1	1	yes	characteristic
158	C₄	#14524035	1	1	yes	characteristic
159	C₄	#14524043	1	1	yes	characteristic
160	C₂	#16863294	1	1	yes	characteristic
161	C₂	C₂×#14523974	1	1	yes	characteristic
162	C₂	#16863439	1	1	yes	characteristic
163	C₂	#16863596	1	1	yes	characteristic
164	C₂	#16863601	1	1	yes	characteristic
165	C₂	#16863681	1	1	yes	characteristic
166	C₂	#16863686	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523974

C₂

yes

maximal

C₂×C₄⋊(C₆₇⋊C₈)

C₂

yes

maximal, characteristic

#16863266

C₂

yes

maximal, characteristic

#16863158

C₂

yes

maximal

C₄×C₄⋊C₄×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

#14523974

—

C₄⋊(C₆₇⋊C₈)

C₄

yes

complemented

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

#95×C₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

D₄

yes

—

C₄⋊C₄×C₆₇

Q₈

yes

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

Q₈

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄⋊C₄

yes

—

C₂×C₄×C₆₇

C₄⋊C₄

yes

—

C₂×C₄×C₆₇

SD₁₆

yes

—

C₂×C₄×C₆₇

SD₁₆

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₆₇

#121

yes

characteristic

C₂²×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₆₇

C₂×SD₁₆

yes

characteristic

C₂²×C₆₇

C₂×SD₁₆

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#106

yes

complemented

C₄×C₆₇

C₄×D₄

yes

characteristic

C₄×C₆₇

C₄×Q₈

yes

characteristic

C₄×C₆₇

#125

yes

characteristic

C₄×C₆₇

C₄⋊Q₈

yes

characteristic

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×#106

—

C₂×C₄⋊C₈

—

#384

—

C₄×C₄⋊C₄

D₆₇

yes

characteristic

#340

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

#106

—

134

—

C₄⋊C₈

—

100

#95

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

134

—

104

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

105

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

106

C₂²×C₄

C₂²×D₆₇

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

134

—

109

C₂×C₈

—

134

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

268

—

113

C₄⋊C₄

—

268

—

114

C₄⋊C₄

—

134

—

115

C₄⋊C₄

—

134

—

116

C₄⋊C₄

Dic₁₃₄

yes

—

117

C₄⋊C₄

D₂₆₈

yes

—

118

C₄²

—

119

C₄²

Dic₁₃₄

yes

—

120

C₄²

C₄×D₆₇

yes

—

121

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

268

—

131

C₂×C₄

—

268

—

132

C₂×C₄

—

268

—

133

C₂×C₄

Dic₆₇⋊C₄

yes

—

134

C₂×C₄

Dic₆₇⋊C₄

yes

—

135

C₂×C₄

#11761615

yes

—

136

C₂×C₄

#11761616

yes

—

137

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

138

C₂×C₄

C₂×D₂₆₈

yes

characteristic

139

C₂×C₄

C₂×Dic₁₃₄

yes

characteristic

140

C₂×C₄

C₂×Dic₁₃₄

yes

characteristic

141

C₂×C₄

#11761628

yes

characteristic

142

C₂×C₄

#11761628

yes

characteristic

143

C₈

—

134

—

144

C₂²

C₂×Dic₆₇⋊C₄

yes

characteristic

145

C₂²

#14524017

yes

characteristic

146

C₂²

#14524022

yes

characteristic

147

C₂²

C₂×#11761615

yes

characteristic

148

C₂²

C₂×#11761616

yes

characteristic

149

C₂²

#14524068

yes

characteristic

150

C₂²

#14524073

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

536

—

155

C₄

#14523974

yes

complemented

156

C₄

C₄×D₂₆₈

yes

characteristic

157

C₄

C₄×Dic₁₃₄

yes

characteristic

158

C₄

#14524035

yes

characteristic

159

C₄

#14524043

yes

characteristic

160

C₂

#16863294

yes

characteristic

161

C₂

C₂×#14523974

yes

characteristic

162

C₂

#16863439

yes

characteristic

163

C₂

#16863596

yes

characteristic

164

C₂

#16863601

yes

characteristic

165

C₂

#16863681

yes

characteristic

166

C₂

#16863686

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₄×C₄⋊C₄×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₄×C₄⋊C₄	D₆₇
hypocenter	C₆₇	#1948
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₄×C₄⋊C₄×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

hypercenter

C₄×C₄⋊C₄

D₆₇

hypocenter

C₆₇

#1948

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
11. C₄⋊(C₆₇⋊C₈)	C₄	semidirect
70. C₄×C₆₇	#106	semidirect
84. C₆₇	#1948	semidirect
155. C₄	#14523974	semidirect

H ≅ G/N

type

11. C₄⋊(C₆₇⋊C₈)

C₄

semidirect

70. C₄×C₆₇

#106

semidirect

84. C₆₇

#1948

semidirect

155. C₄

#14523974

semidirect

Subgroups — #22560972

Conjugacy classes of subgroups

Complements