Group #22559708 — subgroups

Subgroups — #22559708

Order 8576.

160 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16863794	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863686	C₂	1	1	yes	maximal, characteristic
3	#16863601	C₂	1	1	yes	maximal, characteristic
4	#16863149	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863144	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863141	C₂	1	1	yes	maximal, characteristic, complemented
7	#436×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#11761616	—	2	1	no	—
9	#14524043	C₂²	1	1	yes	characteristic
10	#14524043	C₂²	1	1	yes	characteristic
11	#14523975	—	2	1	no	—
12	#14523975	—	2	1	no	—
13	#14523974	—	2	1	no	—
14	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
16	#14523966	—	2	1	no	—
17	#14523964	—	2	1	no	—
18	C₂×M₁₆×C₆₇	—	2	1	no	—
19	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²⋊C₈×C₆₇	—	2	1	no	—
22	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₃₄	—	2	1	no	—
24	C₂×Dic₁₃₄	—	2	1	no	—
25	#11761616	—	4	2	no	—
26	C₂×C₆₇⋊C₈	—	2	1	no	—
27	C₂×C₆₇⋊C₈	—	2	1	no	—
28	C₄⋊Dic₆₇	—	2	1	no	—
29	C₄⋊Dic₆₇	—	2	1	no	—
30	C₄⋊Dic₆₇	—	2	1	no	—
31	C₄⋊Dic₆₇	—	2	1	no	—
32	C₂×D₄×C₆₇	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₆₇	—	2	1	no	—
34	M₁₆×C₆₇	—	4	2	no	—
35	C₂×C₈×C₆₇	—	2	1	no	—
36	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
37	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
39	C₂²⋊C₄×C₆₇	—	2	1	no	—
40	C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	C₂³×C₆₇	—	2	1	no	—
42	D₄×C₆₇	—	2	1	no	—
43	D₄×C₆₇	—	2	1	no	—
44	C₂×C₄×C₆₇	—	2	1	no	—
45	C₂×C₄×C₆₇	—	4	1	no	—
46	C₂×C₄×C₆₇	—	4	1	no	—
47	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
50	C₂×Dic₆₇	—	4	1	no	—
51	C₂×Dic₆₇	—	4	1	no	—
52	Dic₁₃₄	—	4	1	no	—
53	Dic₁₃₄	—	4	1	no	—
54	Dic₁₃₄	—	8	1	no	—
55	C₈×C₆₇	—	2	1	no	—
56	C₈×C₆₇	—	4	1	no	—
57	C₈×C₆₇	SD₁₆	1	2	yes	complemented
58	C₆₇⋊C₈	—	4	1	no	—
59	C₆₇⋊C₈	—	4	1	no	—
60	C₂²×C₆₇	—	4	1	no	—
61	C₂²×C₆₇	—	4	1	no	—
62	C₂²×C₆₇	—	4	1	no	—
63	C₂²×C₆₇	#121	1	1	yes	characteristic
64	C₄×C₆₇	—	2	1	no	—
65	C₄×C₆₇	—	4	1	no	—
66	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
67	C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₄×C₆₇	#137	1	1	yes	characteristic
69	C₄×C₆₇	#137	1	1	yes	characteristic
70	Dic₆₇	—	8	1	no	—
71	Dic₆₇	—	8	1	no	—
72	C₂×C₆₇	—	8	1	no	—
73	C₂×C₆₇	#461	1	1	yes	characteristic
74	C₂×C₆₇	#468	1	1	yes	characteristic
75	C₂×C₆₇	#471	1	1	yes	characteristic
76	#1691	—	67	1	no	maximal, Sylow
77	C₆₇	#1691	1	1	yes	characteristic, complemented, Sylow
78	#498	—	67	1	no	—
79	#476	—	67	1	no	—
80	#461	—	67	1	no	—
81	#436	D₆₇	1	1	yes	characteristic
82	#335	—	67	1	no	—
83	#333	—	67	1	no	—
84	#332	—	67	1	no	—
85	C₂×SD₁₆	—	134	1	no	—
86	C₂×M₁₆	—	2	1	no	—
87	C₄⋊Q₈	—	67	1	no	—
88	C₄⋊Q₈	—	67	1	no	—
89	C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
90	#106	—	134	1	no	—
91	#106	—	134	1	no	—
92	#106	—	134	1	no	—
93	C₄⋊C₈	—	67	1	no	—
94	C₄⋊C₈	—	67	1	no	—
95	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
96	#103	—	134	1	no	—
97	#102	—	134	1	no	—
98	C₂²⋊C₈	—	2	1	no	—
99	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
100	C₂×Q₈	—	134	1	no	—
101	C₂×Q₈	—	134	1	no	—
102	C₂×D₄	D₂₆₈	1	1	yes	characteristic
103	C₂²×C₄	—	2	1	no	—
104	SD₁₆	—	268	2	no	—
105	M₁₆	—	4	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	134	1	no	—
108	C₂×C₈	—	134	1	no	—
109	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
110	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
111	C₄⋊C₄	—	134	1	no	—
112	C₄⋊C₄	—	134	1	no	—
113	C₄⋊C₄	—	134	1	no	—
114	C₄⋊C₄	—	134	1	no	—
115	C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
116	C₂²⋊C₄	—	2	1	no	—
117	C₄²	C₂²×D₆₇	1	1	yes	characteristic
118	C₂³	—	2	1	no	—
119	Q₈	—	268	1	no	—
120	Q₈	—	268	1	no	—
121	Q₈	—	536	1	no	—
122	D₄	—	2	1	no	—
123	D₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	268	1	no	—
128	C₂×C₄	—	268	1	no	—
129	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
130	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
131	C₂×C₄	#11761628	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	268	1	no	—
135	C₈	—	268	1	no	—
136	C₈	#11761616	1	2	yes	complemented
137	C₂²	—	4	1	no	—
138	C₂²	—	4	1	no	—
139	C₂²	—	4	1	no	—
140	C₂²	#14524017	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	4	1	no	—
143	C₄	—	536	1	no	—
144	C₄	—	536	1	no	—
145	C₄	C₂×#11761616	1	1	yes	characteristic
146	C₄	#14524071	1	1	yes	characteristic
147	C₄	#14524073	1	1	yes	characteristic
148	C₄	#14524076	1	1	yes	characteristic
149	C₂	—	8	1	no	—
150	C₂	#16863601	1	1	yes	characteristic
151	C₂	#16863642	1	1	yes	characteristic
152	C₂	#16863659	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863794

C₂

yes

maximal, characteristic, complemented

#16863686

C₂

yes

maximal, characteristic

#16863601

C₂

yes

maximal, characteristic

#16863149

C₂

yes

maximal, characteristic, complemented

#16863144

C₂

yes

maximal, characteristic, complemented

#16863141

C₂

yes

maximal, characteristic, complemented

#436×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761616

—

#14524043

C₂²

yes

characteristic

#14524043

C₂²

yes

—

—

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

#14523966

—

#14523964

—

C₂×M₁₆×C₆₇

—

C₄×D₄×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

#11761616

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×D₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₄⋊C₄×C₆₇

D₄

yes

characteristic

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×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

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

SD₁₆

yes

complemented

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#121

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×SD₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

D₆₇

yes

—

—

—

C₂×SD₁₆

—

134

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×D₄

C₂×D₆₇

yes

characteristic

#106

—

134

—

#106

—

134

—

#106

—

134

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₆₇

yes

characteristic

#103

—

134

—

#102

—

134

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₆₇

yes

characteristic

100

C₂×Q₈

—

134

—

101

C₂×Q₈

—

134

—

102

C₂×D₄

D₂₆₈

yes

characteristic

103

C₂²×C₄

—

104

SD₁₆

—

268

—

105

M₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

134

—

108

C₂×C₈

—

134

—

109

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

110

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

111

C₄⋊C₄

—

134

—

112

C₄⋊C₄

—

134

—

113

C₄⋊C₄

—

134

—

114

C₄⋊C₄

—

134

—

115

C₄⋊C₄

D₂₆₈

yes

characteristic

116

C₂²⋊C₄

—

117

C₄²

C₂²×D₆₇

yes

characteristic

118

C₂³

—

119

Q₈

—

268

—

120

Q₈

—

268

—

121

Q₈

—

536

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

268

—

128

C₂×C₄

—

268

—

129

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

130

C₂×C₄

C₂×D₂₆₈

yes

characteristic

131

C₂×C₄

#11761628

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

268

—

135

C₈

—

268

—

136

C₈

#11761616

yes

complemented

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

#14524017

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

536

—

144

C₄

—

536

—

145

C₄

C₂×#11761616

yes

characteristic

146

C₄

#14524071

yes

characteristic

147

C₄

#14524073

yes

characteristic

148

C₄

#14524076

yes

characteristic

149

C₂

—

150

C₂

#16863601

yes

characteristic

151

C₂

#16863642

yes

characteristic

152

C₂

#16863659

yes

characteristic

	subgroup	G/H
center	C₂²	#14524017
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#436×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#436	D₆₇
hypocenter	C₆₇	#1691
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #436

subgroup

G/H

center

C₂²

#14524017

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

#436×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #436

N	H ≅ G/N	type
1. #16863794	C₂	semidirect
4. #16863149	C₂	semidirect
5. #16863144	C₂	semidirect
6. #16863141	C₂	semidirect
57. C₈×C₆₇	SD₁₆	semidirect
77. C₆₇	#1691	semidirect
136. C₈	#11761616	semidirect

H ≅ G/N

type

1. #16863794

C₂

semidirect

4. #16863149

C₂

semidirect

5. #16863144

C₂

semidirect

6. #16863141

C₂

semidirect

57. C₈×C₆₇

SD₁₆

semidirect

77. C₆₇

#1691

semidirect

136. C₈

#11761616

semidirect

Subgroups — #22559708

Conjugacy classes of subgroups

Complements