Group #22559112 — subgroups

Subgroups — #22559112

Order 8576.

162 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 71 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	#16863527	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863474	C₂	1	1	yes	maximal, characteristic
3	#16863149	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#16863117	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#11761612	—	2	1	no	—
9	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
10	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
11	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
13	#14523954	—	2	1	no	—
14	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×SD₁₆×C₆₇	C₄	1	1	yes	characteristic
17	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
19	#106×C₆₇	C₄	1	1	yes	characteristic
20	#103×C₆₇	C₄	1	1	yes	characteristic
21	#102×C₆₇	C₄	1	1	yes	characteristic
22	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	#11761612	—	4	2	no	—
24	C₂×C₆₇⋊C₈	—	2	1	no	—
25	C₂×C₆₇⋊C₈	—	2	1	no	—
26	C₂×C₆₇⋊C₈	—	2	1	no	—
27	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
28	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
29	C₂×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₆₇	—	2	1	no	—
32	SD₁₆×C₆₇	—	2	2	no	—
33	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₄⋊C₄×C₆₇	—	2	1	no	—
36	C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂²⋊C₄×C₆₇	—	2	1	no	—
39	C₄²×C₆₇	—	2	1	no	—
40	C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	C₂³×C₆₇	—	2	1	no	—
42	Q₈×C₆₇	—	2	1	no	—
43	Q₈×C₆₇	—	2	1	no	—
44	D₄×C₆₇	—	2	1	no	—
45	D₄×C₆₇	—	2	1	no	—
46	C₂×C₄×C₆₇	—	2	1	no	—
47	C₂×C₄×C₆₇	—	2	1	no	—
48	C₂×C₄×C₆₇	—	2	1	no	—
49	C₂×C₄×C₆₇	—	4	1	no	—
50	C₂×C₄×C₆₇	—	4	1	no	—
51	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
54	C₈×C₆₇	—	2	1	no	—
55	C₈×C₆₇	M₁₆	1	2	yes	complemented
56	C₆₇⋊C₈	—	2	1	no	—
57	C₆₇⋊C₈	—	2	1	no	—
58	C₆₇⋊C₈	—	4	1	no	—
59	C₆₇⋊C₈	—	4	1	no	—
60	C₆₇⋊C₈	—	4	1	no	—
61	C₂²×C₆₇	—	4	1	no	—
62	C₂²×C₆₇	—	4	1	no	—
63	C₂²×C₆₇	—	4	1	no	—
64	C₂²×C₆₇	C₄×D₄	1	1	yes	characteristic
65	C₄×C₆₇	—	2	1	no	—
66	C₄×C₆₇	—	4	1	no	—
67	C₄×C₆₇	—	4	1	no	—
68	C₄×C₆₇	—	4	1	no	—
69	C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
70	C₄×C₆₇	#131	1	1	yes	characteristic
71	C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₄×C₆₇	#137	1	1	yes	characteristic
73	C₂×C₆₇	—	8	1	no	—
74	C₂×C₆₇	#436	1	1	yes	characteristic
75	C₂×C₆₇	#440	1	1	yes	characteristic
76	C₂×C₆₇	#444	1	1	yes	characteristic
77	#1592	—	67	1	no	maximal, Sylow
78	C₆₇	#1592	1	1	yes	characteristic, complemented, Sylow
79	#446	—	67	1	no	—
80	C₄×SD₁₆	D₆₇	1	1	yes	characteristic
81	#436	—	67	1	no	—
82	#335	—	67	1	no	—
83	Q₈⋊C₈	—	67	1	no	—
84	D₄⋊C₈	—	67	1	no	—
85	#322	—	67	1	no	—
86	C₂×SD₁₆	Dic₆₇	1	1	yes	characteristic
87	C₂×M₁₆	—	134	1	no	—
88	C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
89	C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
90	#106	Dic₆₇	1	1	yes	characteristic
91	C₄⋊C₈	—	67	1	no	—
92	C₄⋊C₈	—	67	1	no	—
93	C₄⋊C₈	—	134	1	no	—
94	#103	Dic₆₇	1	1	yes	characteristic
95	#102	Dic₆₇	1	1	yes	characteristic
96	C₂²⋊C₈	—	134	1	no	—
97	#97	—	134	1	no	—
98	C₄×C₈	—	67	1	no	—
99	C₄×C₈	—	67	1	no	—
100	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
101	C₂×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
102	C₂×D₄	C₂×Dic₆₇	1	1	yes	characteristic
103	C₂²×C₄	—	2	1	no	—
104	SD₁₆	—	2	2	no	—
105	M₁₆	—	268	2	no	—
106	C₂×C₈	—	67	1	no	—
107	C₂×C₈	—	67	1	no	—
108	C₂×C₈	—	134	1	no	—
109	C₂×C₈	—	134	1	no	—
110	C₂×C₈	—	134	1	no	—
111	C₂×C₈	C₂×Dic₆₇	1	1	yes	characteristic
112	C₂×C₈	C₂×Dic₆₇	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
115	C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
116	C₂²⋊C₄	—	2	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	C₂²×D₆₇	1	1	yes	characteristic
119	C₂³	—	2	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	—	2	1	no	—
122	D₄	—	2	1	no	—
123	D₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
130	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
131	C₂×C₄	#11761627	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	268	1	no	—
134	C₈	—	268	1	no	—
135	C₈	—	268	1	no	—
136	C₈	—	134	1	no	—
137	C₈	—	134	1	no	—
138	C₈	#11761612	1	2	yes	complemented
139	C₂²	—	4	1	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	4	1	no	—
142	C₂²	D₄×Dic₆₇	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	4	1	no	—
145	C₄	—	4	1	no	—
146	C₄	—	4	1	no	—
147	C₄	C₂×#11761612	1	1	yes	characteristic
148	C₄	#14524050	1	1	yes	characteristic
149	C₄	#14524070	1	1	yes	characteristic
150	C₄	#14524072	1	1	yes	characteristic
151	C₂	—	8	1	no	—
152	C₂	#16863474	1	1	yes	characteristic
153	C₂	#16863497	1	1	yes	characteristic
154	C₂	#16863518	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863527

C₂

yes

maximal, characteristic, complemented

#16863474

C₂

yes

maximal, characteristic

#16863149

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₆₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₆₇⋊C₈)

C₂

yes

maximal, characteristic

#16863117

C₂

yes

maximal, characteristic, complemented

C₄×SD₁₆×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

C₂²⋊(C₆₇⋊C₈)

—

#14523954

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₆₇

C₄

yes

characteristic

C₄×Q₈×C₆₇

C₂²

yes

characteristic

C₄×D₄×C₆₇

C₂²

yes

characteristic

#106×C₆₇

C₄

yes

characteristic

#103×C₆₇

C₄

yes

characteristic

#102×C₆₇

C₄

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

characteristic

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₂×Q₈×C₆₇

C₂×C₄

yes

characteristic

C₂×D₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

—

SD₁₆×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂×C₄

yes

characteristic

C₄⋊C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

M₁₆

yes

complemented

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×M₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#446

—

C₄×SD₁₆

D₆₇

yes

characteristic

#436

—

#335

—

Q₈⋊C₈

—

D₄⋊C₈

—

#322

—

C₂×SD₁₆

Dic₆₇

yes

characteristic

C₂×M₁₆

—

134

—

C₄×Q₈

C₂×D₆₇

yes

characteristic

C₄×D₄

C₂×D₆₇

yes

characteristic

#106

Dic₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

134

—

#103

Dic₆₇

yes

characteristic

#102

Dic₆₇

yes

characteristic

C₂²⋊C₈

—

134

—

#97

—

134

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

C₂×D₆₇

yes

characteristic

101

C₂×Q₈

C₂×Dic₆₇

yes

characteristic

102

C₂×D₄

C₂×Dic₆₇

yes

characteristic

103

C₂²×C₄

—

104

SD₁₆

—

105

M₁₆

—

268

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

134

—

109

C₂×C₈

—

134

—

110

C₂×C₈

—

134

—

111

C₂×C₈

C₂×Dic₆₇

yes

characteristic

112

C₂×C₈

C₂×Dic₆₇

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

C₂×Dic₆₇

yes

characteristic

115

C₄⋊C₄

C₂×Dic₆₇

yes

characteristic

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

C₂²×D₆₇

yes

characteristic

119

C₂³

—

120

Q₈

—

121

Q₈

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂²×Dic₆₇

yes

characteristic

130

C₂×C₄

D₄×D₆₇

yes

characteristic

131

C₂×C₄

#11761627

yes

characteristic

132

C₈

—

133

C₈

—

268

—

134

C₈

—

268

—

135

C₈

—

268

—

136

C₈

—

134

—

137

C₈

—

134

—

138

C₈

#11761612

yes

complemented

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

D₄×Dic₆₇

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

C₂×#11761612

yes

characteristic

148

C₄

#14524050

yes

characteristic

149

C₄

#14524070

yes

characteristic

150

C₄

#14524072

yes

characteristic

151

C₂

—

152

C₂

#16863474

yes

characteristic

153

C₂

#16863497

yes

characteristic

154

C₂

#16863518

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₄²	C₂²×D₆₇
Fitting	C₄×SD₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₄×D₄
hypercenter	C₄×SD₁₆	D₆₇
hypocenter	C₆₇	#1592
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

subgroup

G/H

center

C₂×C₄

D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₄²

C₂²×D₆₇

Fitting

C₄×SD₁₆×C₆₇

C₂

socle

C₂²×C₆₇

C₄×D₄

hypercenter

C₄×SD₁₆

D₆₇

hypocenter

C₆₇

#1592

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

N	H ≅ G/N	type
1. #16863527	C₂	semidirect
3. #16863149	C₂	semidirect
4. Q₈⋊(C₆₇⋊C₈)	C₂	semidirect
6. #16863117	C₂	semidirect
55. C₈×C₆₇	M₁₆	semidirect
78. C₆₇	#1592	semidirect
138. C₈	#11761612	semidirect

H ≅ G/N

type

1. #16863527

C₂

semidirect

3. #16863149

C₂

semidirect

4. Q₈⋊(C₆₇⋊C₈)

C₂

semidirect

6. #16863117

C₂

semidirect

55. C₈×C₆₇

M₁₆

semidirect

78. C₆₇

#1592

semidirect

138. C₈

#11761612

semidirect

Subgroups — #22559112

Conjugacy classes of subgroups

Complements