Group #22561122 — subgroups

Subgroups — #22561122

Order 8576.

276 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 89 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	C₂²×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic
2	#16863406	C₂	1	2	yes	maximal
3	C₂×#14523961	C₂	1	2	yes	maximal
4	#16863351	C₂	1	1	yes	maximal, characteristic
5	C₂×#108×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₃₄	C₂²	1	2	yes	—
7	C₂×Dic₂₆₈	—	2	1	no	—
8	C₂×Dic₂₆₈	—	2	1	no	—
9	C₂×Dic₂₆₈	—	2	2	no	—
10	C₂×Dic₂₆₈	C₄	1	4	yes	complemented
11	C₂×#11761612	C₂²	1	1	yes	characteristic
12	#14524002	C₂²	1	1	yes	characteristic
13	#14523968	—	2	4	no	—
14	#14523961	—	2	4	no	—
15	#14523953	—	2	1	no	—
16	C₈×Dic₆₇	—	2	1	no	—
17	C₂×M₁₆×C₆₇	C₂²	1	2	yes	—
18	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	#108×C₆₇	—	2	2	no	—
20	C₂×Dic₁₃₄	—	2	2	no	—
21	C₂×Dic₁₃₄	—	2	2	no	—
22	C₂×Dic₁₃₄	—	2	2	no	—
23	C₂×Dic₁₃₄	—	2	2	no	—
24	C₂×Dic₁₃₄	—	2	2	no	—
25	C₂×Dic₁₃₄	C₂×C₄	1	4	yes	—
26	C₂²×Dic₆₇	—	2	2	no	—
27	Dic₂₆₈	—	2	4	no	—
28	Dic₂₆₈	—	2	4	no	—
29	#11761612	—	2	1	no	—
30	#11761612	D₄	1	2	yes	—
31	C₂×C₆₇⋊C₈	—	2	1	no	—
32	C₄⋊Dic₆₇	D₄	1	2	yes	—
33	C₂²⋊Dic₆₇	—	2	1	no	—
34	C₄×Dic₆₇	—	2	1	no	—
35	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	M₁₆×C₆₇	—	2	2	no	—
37	M₁₆×C₆₇	—	2	2	no	—
38	C₂×C₈×C₆₇	—	2	2	no	—
39	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
40	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
41	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
42	C₂×C₈×C₆₇	D₄	1	2	yes	—
43	C₂³×C₆₇	#41	1	1	yes	characteristic
44	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
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	—	2	4	no	—
51	C₂×Dic₆₇	—	2	2	no	—
52	C₂×Dic₆₇	—	4	2	no	—
53	C₂×Dic₆₇	—	4	2	no	—
54	Dic₁₃₄	—	2	4	no	—
55	Dic₁₃₄	—	2	4	no	—
56	Dic₁₃₄	—	4	2	no	—
57	Dic₁₃₄	—	4	2	no	—
58	C₈×C₆₇	—	4	2	no	—
59	C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
60	C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
61	C₆₇⋊C₈	—	2	2	no	—
62	C₂²×C₆₇	—	2	2	no	—
63	C₂²×C₆₇	C₄×D₄	1	1	yes	characteristic
64	C₂²×C₆₇	#121	1	1	yes	characteristic
65	C₂²×C₆₇	#124	1	1	yes	characteristic
66	C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
68	C₄×C₆₇	#121	1	1	yes	characteristic
69	C₄×C₆₇	#127	1	1	yes	characteristic
70	Dic₆₇	—	4	4	no	—
71	Dic₆₇	—	4	2	no	—
72	C₂×C₆₇	—	2	1	no	—
73	C₂×C₆₇	—	2	1	no	—
74	C₂×C₆₇	#390	1	1	yes	characteristic
75	C₂×C₆₇	#473	1	2	yes	—
76	#1979	—	67	1	no	maximal, Sylow
77	C₆₇	#1979	1	1	yes	characteristic, complemented, Sylow
78	C₂²×Q₁₆	—	67	1	no	—
79	C₂×#108	D₆₇	1	1	yes	characteristic
80	#419	—	67	2	no	—
81	C₂×#101	—	67	2	no	—
82	#405	—	67	1	no	—
83	C₂²×Q₈	—	67	2	no	—
84	C₂×Q₁₆	—	67	4	no	—
85	C₂×Q₁₆	—	134	1	no	—
86	C₂×Q₁₆	—	134	1	no	—
87	C₂×Q₁₆	—	134	2	no	—
88	C₂×M₁₆	—	67	1	no	—
89	C₂×M₁₆	C₂×D₆₇	1	2	yes	—
90	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
91	#117	—	67	1	no	—
92	#108	—	2	2	no	—
93	#103	—	134	4	no	—
94	#101	—	134	4	no	—
95	#97	—	134	1	no	—
96	C₄×C₈	—	134	1	no	—
97	C₂×Q₈	—	67	4	no	—
98	C₂×Q₈	—	134	2	no	—
99	C₂×Q₈	—	134	2	no	—
100	C₂×Q₈	—	134	2	no	—
101	C₂×Q₈	—	134	2	no	—
102	C₂×Q₈	—	134	2	no	—
103	C₂²×C₄	—	134	2	no	—
104	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
105	Q₁₆	—	134	4	no	—
106	Q₁₆	—	134	4	no	—
107	M₁₆	—	2	2	no	—
108	M₁₆	—	2	2	no	—
109	M₁₆	—	67	2	no	—
110	M₁₆	—	134	1	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	134	1	no	—
113	C₂×C₈	C₄×D₆₇	1	2	yes	—
114	C₂×C₈	D₂₆₈	1	1	yes	characteristic
115	C₂×C₈	D₂₆₈	1	1	yes	characteristic
116	C₂×C₈	C₂²⋊D₆₇	1	2	yes	—
117	C₄⋊C₄	—	67	2	no	—
118	C₂²⋊C₄	—	134	1	no	—
119	C₄²	—	134	1	no	—
120	C₂³	#11761629	1	1	yes	characteristic
121	Q₈	—	268	2	no	—
122	Q₈	—	268	2	no	—
123	Q₈	—	134	4	no	—
124	Q₈	—	134	4	no	—
125	C₂×C₄	—	268	2	no	—
126	C₂×C₄	—	268	2	no	—
127	C₂×C₄	—	134	4	no	—
128	C₂×C₄	—	134	2	no	—
129	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
130	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
131	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
132	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
133	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
134	C₂×C₄	#11761629	1	1	yes	characteristic
135	C₈	—	4	2	no	—
136	C₈	—	134	2	no	—
137	C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
138	C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	#14524005	1	1	yes	characteristic
141	C₂²	#14524018	1	1	yes	characteristic
142	C₂²	#14524032	1	1	yes	characteristic
143	C₄	—	268	4	no	—
144	C₄	—	268	2	no	—
145	C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
146	C₄	#14524005	1	1	yes	characteristic
147	C₄	#14524018	1	1	yes	characteristic
148	C₄	#14524039	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	#16863297	1	1	yes	characteristic
152	C₂	#16863672	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₆₈

C₂

yes

maximal, characteristic

#16863406

C₂

yes

maximal

C₂×#14523961

C₂

yes

maximal

#16863351

C₂

yes

maximal, characteristic

C₂×#108×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

C₄

yes

complemented

C₂×#11761612

C₂²

yes

characteristic

#14524002

C₂²

yes

—

—

—

C₈×Dic₆₇

—

C₂×M₁₆×C₆₇

C₂²

yes

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

#108×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₂×C₄

yes

—

C₂²×Dic₆₇

—

Dic₂₆₈

—

Dic₂₆₈

—

#11761612

—

#11761612

D₄

yes

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

D₄

yes

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

—

C₂³×C₆₇

#41

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

C₂²⋊C₄

yes

—

C₈×C₆₇

C₂²⋊C₄

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄×D₄

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₇

C₄×D₄

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#390

yes

characteristic

C₂×C₆₇

#473

yes

—

#1979

—

maximal, Sylow

C₆₇

#1979

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×#108

D₆₇

yes

—

—

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

134

—

C₂×Q₁₆

—

134

—

C₂×Q₁₆

—

134

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

—

C₂²×C₈

C₂×D₆₇

yes

—

—

—

134

—

#101

—

134

—

#97

—

134

—

C₄×C₈

—

134

—

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

100

C₂×Q₈

—

134

—

101

C₂×Q₈

—

134

—

102

C₂×Q₈

—

134

—

103

C₂²×C₄

—

134

—

104

C₂²×C₄

C₂²×D₆₇

yes

characteristic

105

Q₁₆

—

134

—

106

Q₁₆

—

134

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

134

—

111

C₂×C₈

—

112

C₂×C₈

—

134

—

113

C₂×C₈

C₄×D₆₇

yes

—

114

C₂×C₈

D₂₆₈

yes

characteristic

115

C₂×C₈

D₂₆₈

yes

characteristic

116

C₂×C₈

C₂²⋊D₆₇

yes

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

134

—

119

C₄²

—

134

—

120

C₂³

#11761629

yes

characteristic

121

Q₈

—

268

—

122

Q₈

—

268

—

123

Q₈

—

134

—

124

Q₈

—

134

—

125

C₂×C₄

—

268

—

126

C₂×C₄

—

268

—

127

C₂×C₄

—

134

—

128

C₂×C₄

—

134

—

129

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

130

C₂×C₄

D₄×D₆₇

yes

characteristic

131

C₂×C₄

D₄×D₆₇

yes

characteristic

132

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

133

C₂×C₄

C₂×D₂₆₈

yes

characteristic

134

C₂×C₄

#11761629

yes

characteristic

135

C₈

—

136

C₈

—

134

—

137

C₈

(C₂×D₆₇)⋊C₄

yes

—

138

C₈

(C₂×D₆₇)⋊C₄

yes

—

139

C₂²

—

140

C₂²

#14524005

yes

characteristic

141

C₂²

#14524018

yes

characteristic

142

C₂²

#14524032

yes

characteristic

143

C₄

—

268

—

144

C₄

—

268

—

145

C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

146

C₄

#14524005

yes

characteristic

147

C₄

#14524018

yes

characteristic

148

C₄

#14524039

yes

characteristic

149

C₂

—

150

C₂

—

151

C₂

#16863297

yes

characteristic

152

C₂

#16863672

yes

—

	subgroup	G/H
center	C₂²	#14524018
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×#108×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	C₂×#108	D₆₇
hypocenter	C₆₇	#1979
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂²

#14524018

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂×#108×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
10. C₂×Dic₂₆₈	C₄	semidirect
77. C₆₇	#1979	semidirect

H ≅ G/N

type

10. C₂×Dic₂₆₈

C₄

semidirect

77. C₆₇

#1979

semidirect

Subgroups — #22561122

Conjugacy classes of subgroups

Complements