Group #22559560 — subgroups

Subgroups — #22559560

Order 8576.

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

normal subgroups only

This group has 172 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	#16863770	C₂	1	1	yes	maximal, characteristic
2	#16863678	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863597	C₂	1	1	yes	maximal, characteristic
4	#16863520	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863146	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863139	C₂	1	1	yes	maximal, characteristic
7	#332×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11761617	—	2	1	no	—
9	C₂×#11761615	—	2	1	no	—
10	C₂×#11761615	—	2	1	no	—
11	#14524040	C₂²	1	1	yes	characteristic
12	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
13	#14523974	—	2	1	no	—
14	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
15	Dic₆₇⋊C₈	—	2	1	no	—
16	#14523966	—	2	1	no	—
17	#14523965	—	2	1	no	—
18	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₈×Dic₆₇	—	2	1	no	—
20	C₄⋊Q₈×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	#11761615	—	4	2	no	—
29	#11761615	—	4	2	no	—
30	C₂×C₆₇⋊C₈	—	2	1	no	—
31	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
32	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
33	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
34	Dic₆₇⋊C₄	—	2	1	no	—
35	C₄×Dic₆₇	—	2	1	no	—
36	C₂×Q₈×C₆₇	—	2	1	no	—
37	C₂×C₈×C₆₇	—	2	1	no	—
38	C₂×C₈×C₆₇	—	2	1	no	—
39	C₄⋊C₄×C₆₇	—	2	1	no	—
40	C₄⋊C₄×C₆₇	—	2	1	no	—
41	C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	C₂²×D₆₇	—	4	1	no	—
43	Q₈×C₆₇	—	4	1	no	—
44	Q₈×C₆₇	—	4	1	no	—
45	C₂×C₄×C₆₇	—	4	1	no	—
46	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
49	C₂×Dic₆₇	—	2	1	no	—
50	C₂×Dic₆₇	—	2	1	no	—
51	D₂₆₈	—	4	1	no	—
52	D₂₆₈	—	4	1	no	—
53	D₂₆₈	—	4	1	no	—
54	D₂₆₈	—	4	1	no	—
55	D₂₆₈	—	8	1	no	—
56	Dic₁₃₄	—	2	1	no	—
57	Dic₁₃₄	SD₁₆	1	2	yes	complemented
58	C₈×C₆₇	—	4	1	no	—
59	C₈×C₆₇	—	4	1	no	—
60	C₆₇⋊C₈	—	2	1	no	—
61	C₆₇⋊C₈	—	4	1	no	—
62	C₆₇⋊C₈	SD₁₆	1	2	yes	complemented
63	C₂²×C₆₇	#121	1	1	yes	characteristic
64	C₂×D₆₇	—	8	1	no	—
65	C₂×D₆₇	—	8	1	no	—
66	C₂×D₆₇	—	8	1	no	—
67	C₄×C₆₇	—	2	1	no	—
68	C₄×C₆₇	—	8	1	no	—
69	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
70	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
71	C₄×C₆₇	#135	1	1	yes	characteristic
72	C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
73	Dic₆₇	—	2	2	no	—
74	Dic₆₇	—	4	1	no	—
75	C₂×C₆₇	#460	1	1	yes	characteristic
76	C₂×C₆₇	#465	1	1	yes	characteristic
77	C₂×C₆₇	#471	1	1	yes	characteristic
78	D₆₇	—	16	1	no	—
79	#1666	—	67	1	no	maximal, Sylow
80	C₆₇	#1666	1	1	yes	characteristic, complemented, Sylow
81	#492	—	67	1	no	—
82	#475	—	67	1	no	—
83	#460	—	67	1	no	—
84	C₈×Q₈	—	67	1	no	—
85	#334	—	67	1	no	—
86	#332	D₆₇	1	1	yes	characteristic, complemented
87	#331	—	67	1	no	—
88	C₂×SD₁₆	—	134	1	no	—
89	C₂×SD₁₆	—	134	1	no	—
90	C₂×SD₁₆	—	134	1	no	—
91	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
92	#127	—	67	1	no	—
93	C₄×Q₈	—	67	1	no	—
94	#106	—	134	1	no	—
95	C₄⋊C₈	—	67	1	no	—
96	C₄⋊C₈	—	134	1	no	—
97	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
98	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
99	#103	—	134	1	no	—
100	#102	—	134	1	no	—
101	C₄×C₈	—	67	1	no	—
102	C₄×C₈	—	134	1	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	—	67	1	no	—
105	C₂×D₄	—	134	1	no	—
106	C₂×D₄	—	134	1	no	—
107	C₂×D₄	—	134	1	no	—
108	SD₁₆	—	268	2	no	—
109	SD₁₆	—	268	2	no	—
110	SD₁₆	—	268	2	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	67	1	no	—
114	C₂×C₈	—	67	1	no	—
115	C₂×C₈	—	134	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	67	1	no	—
119	C₄⋊C₄	—	134	1	no	—
120	C₄²	—	134	1	no	—
121	C₄²	C₂²×D₆₇	1	1	yes	characteristic
122	C₂³	—	268	1	no	—
123	Q₈	—	4	1	no	—
124	Q₈	—	4	1	no	—
125	Q₈	—	67	2	no	—
126	Q₈	—	134	1	no	—
127	D₄	—	268	1	no	—
128	D₄	—	268	1	no	—
129	D₄	—	268	1	no	—
130	D₄	—	268	1	no	—
131	D₄	—	536	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	134	1	no	—
134	C₂×C₄	—	134	1	no	—
135	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
136	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
137	C₂×C₄	#11761628	1	1	yes	characteristic
138	C₈	—	4	1	no	—
139	C₈	—	4	1	no	—
140	C₈	—	268	1	no	—
141	C₈	—	67	2	no	—
142	C₈	—	134	1	no	—
143	C₂²	—	536	1	no	—
144	C₂²	—	536	1	no	—
145	C₂²	—	536	1	no	—
146	C₂²	#14524016	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	8	1	no	—
149	C₄	—	268	1	no	—
150	C₄	—	134	2	no	—
151	C₄	SD₁₆×D₆₇	1	1	yes	characteristic
152	C₄	SD₁₆×D₆₇	1	1	yes	characteristic
153	C₄	#14524066	1	1	yes	characteristic
154	C₄	#14524067	1	1	yes	characteristic
155	C₂	—	1072	1	no	—
156	C₂	#16863595	1	1	yes	characteristic
157	C₂	#16863628	1	1	yes	characteristic
158	C₂	#16863662	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863770

C₂

yes

maximal, characteristic

#16863678

C₂

yes

maximal, characteristic, complemented

#16863597

C₂

yes

maximal, characteristic

#16863520

C₂

yes

maximal, characteristic, complemented

#16863146

C₂

yes

maximal, characteristic, complemented

#16863139

C₂

yes

maximal, characteristic

#332×C₆₇

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

characteristic

C₄×Dic₁₃₄

C₂²

yes

characteristic

#14523974

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

Dic₆₇⋊C₈

—

#14523966

—

#14523965

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₈×Dic₆₇

—

C₄⋊Q₈×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

—

#11761617

—

#11761615

—

#11761615

—

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×Q₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂²×D₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×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

SD₁₆

yes

complemented

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

SD₁₆

yes

complemented

C₂²×C₆₇

#121

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₄×C₆₇

C₂×SD₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

—

#1666

—

maximal, Sylow

C₆₇

#1666

yes

characteristic, complemented, Sylow

#492

—

#475

—

#460

—

C₈×Q₈

—

#334

—

#332

D₆₇

yes

characteristic, complemented

#331

—

C₂×SD₁₆

—

134

—

C₂×SD₁₆

—

134

—

C₂×SD₁₆

—

134

—

C₄⋊Q₈

C₂×D₆₇

yes

characteristic

#127

—

C₄×Q₈

—

#106

—

134

—

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₄⋊C₈

C₂×D₆₇

yes

characteristic

C₄⋊C₈

C₂×D₆₇

yes

characteristic

#103

—

134

—

100

#102

—

134

—

101

C₄×C₈

—

102

C₄×C₈

—

134

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×D₄

—

134

—

106

C₂×D₄

—

134

—

107

C₂×D₄

—

134

—

108

SD₁₆

—

268

—

109

SD₁₆

—

268

—

110

SD₁₆

—

268

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

134

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

134

—

120

C₄²

—

134

—

121

C₄²

C₂²×D₆₇

yes

characteristic

122

C₂³

—

268

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

126

Q₈

—

134

—

127

D₄

—

268

—

128

D₄

—

268

—

129

D₄

—

268

—

130

D₄

—

268

—

131

D₄

—

536

—

132

C₂×C₄

—

133

C₂×C₄

—

134

—

134

C₂×C₄

—

134

—

135

C₂×C₄

D₄×D₆₇

yes

characteristic

136

C₂×C₄

D₄×D₆₇

yes

characteristic

137

C₂×C₄

#11761628

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

268

—

141

C₈

—

142

C₈

—

134

—

143

C₂²

—

536

—

144

C₂²

—

536

—

145

C₂²

—

536

—

146

C₂²

#14524016

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

268

—

150

C₄

—

134

—

151

C₄

SD₁₆×D₆₇

yes

characteristic

152

C₄

SD₁₆×D₆₇

yes

characteristic

153

C₄

#14524066

yes

characteristic

154

C₄

#14524067

yes

characteristic

155

C₂

—

1072

—

156

C₂

#16863595

yes

characteristic

157

C₂

#16863628

yes

characteristic

158

C₂

#16863662

yes

characteristic

	subgroup	G/H
center	C₂²	#14524016
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#332×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#332	D₆₇
hypocenter	C₆₇	#1666
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#14524016

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

#332×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
2. #16863678	C₂	semidirect
4. #16863520	C₂	semidirect
5. #16863146	C₂	semidirect
7. #332×C₆₇	C₂	semidirect
57. Dic₁₃₄	SD₁₆	semidirect
62. C₆₇⋊C₈	SD₁₆	semidirect
80. C₆₇	#1666	semidirect
86. #332	D₆₇	semidirect

H ≅ G/N

type

2. #16863678

C₂

semidirect

4. #16863520

C₂

semidirect

5. #16863146

C₂

semidirect

7. #332×C₆₇

C₂

semidirect

57. Dic₁₃₄

SD₁₆

semidirect

62. C₆₇⋊C₈

SD₁₆

semidirect

80. C₆₇

#1666

semidirect

86. #332

D₆₇

semidirect

Subgroups — #22559560

Conjugacy classes of subgroups

Complements