Group #22559568 — subgroups

Subgroups — #22559568

Order 8576.

188 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16863772	C₂	1	1	yes	maximal, characteristic
2	#16863597	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863593	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863520	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863148	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863139	C₂	1	1	yes	maximal, characteristic, complemented
7	#331×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11761617	—	2	1	no	—
9	C₂×#11761616	—	2	1	no	—
10	C₂×#11761613	—	2	1	no	—
11	C₂×#11761613	—	2	1	no	—
12	#14524040	C₂²	1	1	yes	characteristic
13	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
14	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
15	Dic₆₇⋊C₈	—	2	1	no	—
16	#14523965	—	2	1	no	—
17	#14523964	—	2	1	no	—
18	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₈×Dic₆₇	—	2	1	no	—
20	#127×C₆₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₃₄	D₄	1	1	yes	characteristic
24	C₂×D₂₆₈	—	2	1	no	—
25	C₂×D₂₆₈	—	2	1	no	—
26	C₂×D₂₆₈	—	2	1	no	—
27	#11761617	—	4	2	no	—
28	#11761616	—	4	2	no	—
29	#11761613	—	4	2	no	—
30	#11761613	—	4	2	no	—
31	C₂×C₆₇⋊C₈	—	2	1	no	—
32	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
33	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
34	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
35	Dic₆₇⋊C₄	—	2	1	no	—
36	C₄×Dic₆₇	—	2	1	no	—
37	C₂×D₄×C₆₇	—	2	1	no	—
38	C₂×D₄×C₆₇	—	2	1	no	—
39	C₂×D₄×C₆₇	—	2	1	no	—
40	C₂×C₈×C₆₇	—	2	1	no	—
41	C₂×C₈×C₆₇	—	2	1	no	—
42	C₄²×C₆₇	C₂³	1	1	yes	characteristic
43	C₂³×C₆₇	—	4	1	no	—
44	C₂²×D₆₇	—	4	1	no	—
45	D₄×C₆₇	—	4	1	no	—
46	D₄×C₆₇	—	4	1	no	—
47	D₄×C₆₇	—	4	1	no	—
48	D₄×C₆₇	—	4	1	no	—
49	D₄×C₆₇	—	8	1	no	—
50	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
53	C₂×Dic₆₇	—	2	1	no	—
54	C₂×Dic₆₇	—	2	1	no	—
55	D₂₆₈	—	4	1	no	—
56	D₂₆₈	—	4	1	no	—
57	D₂₆₈	—	4	1	no	—
58	D₂₆₈	—	4	1	no	—
59	D₂₆₈	—	8	1	no	—
60	Dic₁₃₄	—	2	1	no	—
61	Dic₁₃₄	D₈	1	2	yes	complemented
62	C₈×C₆₇	—	4	1	no	—
63	C₈×C₆₇	—	4	1	no	—
64	C₆₇⋊C₈	—	2	1	no	—
65	C₆₇⋊C₈	—	4	1	no	—
66	C₆₇⋊C₈	SD₁₆	1	2	yes	complemented
67	C₂²×C₆₇	—	8	1	no	—
68	C₂²×C₆₇	—	8	1	no	—
69	C₂²×C₆₇	—	8	1	no	—
70	C₂²×C₆₇	#121	1	1	yes	characteristic
71	C₂×D₆₇	—	8	1	no	—
72	C₂×D₆₇	—	8	1	no	—
73	C₂×D₆₇	—	8	1	no	—
74	C₄×C₆₇	—	2	1	no	—
75	C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
76	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
77	C₄×C₆₇	#135	1	1	yes	characteristic
78	C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
79	Dic₆₇	—	2	2	no	—
80	Dic₆₇	—	4	1	no	—
81	C₂×C₆₇	—	16	1	no	—
82	C₂×C₆₇	#460	1	1	yes	characteristic
83	C₂×C₆₇	#466	1	1	yes	characteristic
84	C₂×C₆₇	#472	1	1	yes	characteristic
85	D₆₇	—	16	1	no	—
86	#1668	—	67	1	no	maximal, Sylow
87	C₆₇	#1668	1	1	yes	characteristic, complemented, Sylow
88	#493	—	67	1	no	—
89	#460	—	67	1	no	—
90	#460	—	67	1	no	—
91	C₈×Q₈	—	67	1	no	—
92	#335	—	67	1	no	—
93	#331	—	67	1	no	—
94	#331	D₆₇	1	1	yes	characteristic, complemented
95	C₂×SD₁₆	—	134	1	no	—
96	C₂×SD₁₆	—	134	1	no	—
97	C₂×D₈	—	134	1	no	—
98	C₂×D₈	—	134	1	no	—
99	#127	—	67	1	no	—
100	#127	C₂×D₆₇	1	1	yes	characteristic
101	C₄×Q₈	—	67	1	no	—
102	C₄⋊C₈	—	67	1	no	—
103	C₄⋊C₈	—	134	1	no	—
104	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
105	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
106	#102	—	134	1	no	—
107	#102	—	134	1	no	—
108	C₄×C₈	—	67	1	no	—
109	C₄×C₈	—	134	1	no	—
110	C₂×Q₈	—	67	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	2	1	no	—
113	C₂×D₄	—	2	1	no	—
114	C₂×D₄	—	134	1	no	—
115	C₂×D₄	—	134	1	no	—
116	C₂×D₄	—	134	1	no	—
117	SD₁₆	—	268	2	no	—
118	SD₁₆	—	268	2	no	—
119	D₈	—	268	2	no	—
120	D₈	—	268	2	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	67	1	no	—
124	C₂×C₈	—	67	1	no	—
125	C₂×C₈	—	134	1	no	—
126	C₄⋊C₄	—	67	1	no	—
127	C₄⋊C₄	—	134	1	no	—
128	C₄²	—	134	1	no	—
129	C₄²	C₂²×D₆₇	1	1	yes	characteristic
130	C₂³	—	4	1	no	—
131	C₂³	—	268	1	no	—
132	Q₈	—	67	2	no	—
133	Q₈	—	134	1	no	—
134	D₄	—	4	1	no	—
135	D₄	—	4	1	no	—
136	D₄	—	4	1	no	—
137	D₄	—	4	1	no	—
138	D₄	—	8	1	no	—
139	D₄	—	268	1	no	—
140	D₄	—	268	1	no	—
141	D₄	—	268	1	no	—
142	D₄	—	268	1	no	—
143	D₄	—	536	1	no	—
144	C₂×C₄	—	134	1	no	—
145	C₂×C₄	—	134	1	no	—
146	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
147	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
148	C₂×C₄	#11761628	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	4	1	no	—
151	C₈	—	268	1	no	—
152	C₈	—	67	2	no	—
153	C₈	—	134	1	no	—
154	C₂²	—	8	1	no	—
155	C₂²	—	8	1	no	—
156	C₂²	—	8	1	no	—
157	C₂²	—	536	1	no	—
158	C₂²	—	536	1	no	—
159	C₂²	—	536	1	no	—
160	C₂²	#14524016	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	268	1	no	—
163	C₄	—	134	2	no	—
164	C₄	D₈×D₆₇	1	1	yes	characteristic
165	C₄	SD₁₆×D₆₇	1	1	yes	characteristic
166	C₄	#14524064	1	1	yes	characteristic
167	C₄	#14524067	1	1	yes	characteristic
168	C₂	—	16	1	no	—
169	C₂	—	1072	1	no	—
170	C₂	#16863595	1	1	yes	characteristic
171	C₂	#16863631	1	1	yes	characteristic
172	C₂	#16863665	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863772

C₂

yes

maximal, characteristic

#16863597

C₂

yes

maximal, characteristic, complemented

#16863593

C₂

yes

maximal, characteristic, complemented

#16863520

C₂

yes

maximal, characteristic, complemented

#16863148

C₂

yes

maximal, characteristic, complemented

#16863139

C₂

yes

maximal, characteristic, complemented

#331×C₆₇

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

characteristic

C₄×Dic₁₃₄

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

Dic₆₇⋊C₈

—

#14523965

—

#14523964

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₈×Dic₆₇

—

#127×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

D₄

yes

characteristic

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

—

—

—

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₄⋊Dic₆₇

D₄

yes

characteristic

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

—

C₂²×D₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

Dic₁₃₄

—

Dic₁₃₄

D₈

yes

complemented

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

SD₁₆

yes

complemented

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#121

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×D₈

yes

characteristic

C₄×C₆₇

C₂×SD₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

—

#1668

—

maximal, Sylow

C₆₇

#1668

yes

characteristic, complemented, Sylow

#493

—

#460

—

#460

—

C₈×Q₈

—

#335

—

#331

—

#331

D₆₇

yes

characteristic, complemented

C₂×SD₁₆

—

134

—

C₂×SD₁₆

—

134

—

C₂×D₈

—

134

—

C₂×D₈

—

134

—

#127

—

100

#127

C₂×D₆₇

yes

characteristic

101

C₄×Q₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

134

—

104

C₄⋊C₈

C₂×D₆₇

yes

characteristic

105

C₄⋊C₈

C₂×D₆₇

yes

characteristic

106

#102

—

134

—

107

#102

—

134

—

108

C₄×C₈

—

109

C₄×C₈

—

134

—

110

C₂×Q₈

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

—

134

—

115

C₂×D₄

—

134

—

116

C₂×D₄

—

134

—

117

SD₁₆

—

268

—

118

SD₁₆

—

268

—

119

D₈

—

268

—

120

D₈

—

268

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

134

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

134

—

128

C₄²

—

134

—

129

C₄²

C₂²×D₆₇

yes

characteristic

130

C₂³

—

131

C₂³

—

268

—

132

Q₈

—

133

Q₈

—

134

—

134

D₄

—

135

D₄

—

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

268

—

140

D₄

—

268

—

141

D₄

—

268

—

142

D₄

—

268

—

143

D₄

—

536

—

144

C₂×C₄

—

134

—

145

C₂×C₄

—

134

—

146

C₂×C₄

D₄×D₆₇

yes

characteristic

147

C₂×C₄

D₄×D₆₇

yes

characteristic

148

C₂×C₄

#11761628

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

268

—

152

C₈

—

153

C₈

—

134

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

536

—

158

C₂²

—

536

—

159

C₂²

—

536

—

160

C₂²

#14524016

yes

characteristic

161

C₄

—

162

C₄

—

268

—

163

C₄

—

134

—

164

C₄

D₈×D₆₇

yes

characteristic

165

C₄

SD₁₆×D₆₇

yes

characteristic

166

C₄

#14524064

yes

characteristic

167

C₄

#14524067

yes

characteristic

168

C₂

—

169

C₂

—

1072

—

170

C₂

#16863595

yes

characteristic

171

C₂

#16863631

yes

characteristic

172

C₂

#16863665

yes

characteristic

	subgroup	G/H
center	C₂²	#14524016
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#331×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#331	D₆₇
hypocenter	C₆₇	#1668
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

#14524016

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

#331×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
2. #16863597	C₂	semidirect
3. #16863593	C₂	semidirect
4. #16863520	C₂	semidirect
5. #16863148	C₂	semidirect
6. #16863139	C₂	semidirect
7. #331×C₆₇	C₂	semidirect
61. Dic₁₃₄	D₈	semidirect
66. C₆₇⋊C₈	SD₁₆	semidirect
87. C₆₇	#1668	semidirect
94. #331	D₆₇	semidirect

H ≅ G/N

type

2. #16863597

C₂

semidirect

3. #16863593

C₂

semidirect

4. #16863520

C₂

semidirect

5. #16863148

C₂

semidirect

6. #16863139

C₂

semidirect

7. #331×C₆₇

C₂

semidirect

61. Dic₁₃₄

D₈

semidirect

66. C₆₇⋊C₈

SD₁₆

semidirect

87. C₆₇

#1668

semidirect

94. #331

D₆₇

semidirect

Subgroups — #22559568

Conjugacy classes of subgroups

Complements