Group #22561068 — subgroups

Subgroups — #22561068

Order 8576.

312 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂×#14523966	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
4	#16863301	C₂	1	2	yes	maximal
5	C₂×#107×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₃₄	C₂²	1	2	yes	—
7	C₂×Dic₂₆₈	—	2	2	no	—
8	C₂×Dic₂₆₈	—	2	2	no	—
9	C₂×Dic₂₆₈	C₄	1	4	yes	complemented
10	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
12	#14523966	—	2	4	no	—
13	C₈×Dic₆₇	—	2	2	no	—
14	C₆₇⋊#95	—	2	4	no	—
15	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
17	#107×C₆₇	—	2	2	no	—
18	C₂×Dic₁₃₄	—	2	2	no	—
19	C₂×Dic₁₃₄	—	2	2	no	—
20	C₂×Dic₁₃₄	—	2	2	no	—
21	C₂×Dic₁₃₄	—	2	2	no	—
22	C₂×Dic₁₃₄	—	2	2	no	—
23	C₂×Dic₁₃₄	C₂×C₄	1	4	yes	—
24	C₂²×Dic₆₇	—	2	2	no	—
25	C₂²×Dic₆₇	D₄	1	2	yes	—
26	Dic₂₆₈	—	2	8	no	—
27	C₂×C₆₇⋊C₈	—	2	1	no	—
28	C₂×C₆₇⋊C₈	—	2	1	no	—
29	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
30	C₄×Dic₆₇	—	2	1	no	—
31	C₄×Dic₆₇	—	2	1	no	—
32	C₂²×C₄×C₆₇	—	2	2	no	—
33	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
35	C₂×C₈×C₆₇	D₄	1	2	yes	—
36	C₂×C₈×C₆₇	D₄	1	2	yes	—
37	C₄⋊C₄×C₆₇	—	2	2	no	—
38	C₄⋊C₄×C₆₇	—	2	2	no	—
39	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₆₇	—	4	2	no	—
41	C₂×C₄×C₆₇	—	4	4	no	—
42	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
47	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
48	C₂×Dic₆₇	—	2	2	no	—
49	C₂×Dic₆₇	—	2	2	no	—
50	C₂×Dic₆₇	—	2	4	no	—
51	C₂×Dic₆₇	—	4	2	no	—
52	C₂×Dic₆₇	—	4	2	no	—
53	C₂×Dic₆₇	Q₁₆	1	4	yes	complemented
54	Dic₁₃₄	—	2	4	no	—
55	Dic₁₃₄	—	2	4	no	—
56	Dic₁₃₄	—	4	2	no	—
57	Dic₁₃₄	—	4	2	no	—
58	C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
59	C₆₇⋊C₈	—	2	2	no	—
60	C₂²×C₆₇	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₆₇	#121	1	1	yes	characteristic
62	C₂²×C₆₇	#127	1	1	yes	characteristic
63	C₂²×C₆₇	C₂×Q₁₆	1	2	yes	—
64	C₂²×C₆₇	#135	1	2	yes	—
65	C₄×C₆₇	—	8	2	no	—
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	#124	1	1	yes	characteristic
70	Dic₆₇	—	2	4	no	—
71	Dic₆₇	—	4	4	no	—
72	C₂×C₆₇	#390	1	1	yes	characteristic
73	C₂×C₆₇	C₄×Q₁₆	1	2	yes	—
74	C₂×C₆₇	#467	1	2	yes	—
75	C₂×C₆₇	#494	1	1	yes	characteristic
76	C₂×C₆₇	#495	1	1	yes	characteristic
77	#1969	—	67	1	no	maximal, Sylow
78	C₆₇	#1969	1	1	yes	characteristic, complemented, Sylow
79	C₂²×Q₁₆	—	67	1	no	—
80	C₂×#107	D₆₇	1	1	yes	characteristic
81	C₂×#103	—	67	2	no	—
82	C₂×C₄×C₈	—	67	1	no	—
83	#391	—	67	2	no	—
84	C₂²×Q₈	—	67	2	no	—
85	C₂×Q₁₆	—	67	4	no	—
86	C₂×Q₁₆	—	134	2	no	—
87	C₂×Q₁₆	—	134	2	no	—
88	C₂²×C₈	—	67	1	no	—
89	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
90	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
91	C₂×C₄²	—	67	1	no	—
92	#107	—	2	2	no	—
93	#103	—	134	4	no	—
94	C₄×C₈	—	134	2	no	—
95	#95	—	134	4	no	—
96	C₂×Q₈	—	67	4	no	—
97	C₂×Q₈	—	134	2	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	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	67	2	no	—
104	C₂²×C₄	—	134	2	no	—
105	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
106	Q₁₆	—	134	8	no	—
107	C₂×C₈	—	67	2	no	—
108	C₂×C₈	—	134	1	no	—
109	C₂×C₈	—	134	1	no	—
110	C₂×C₈	C₄×D₆₇	1	2	yes	—
111	C₂×C₈	D₂₆₈	1	2	yes	—
112	C₂×C₈	C₂²⋊D₆₇	1	2	yes	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄²	—	134	1	no	—
116	C₄²	—	134	1	no	—
117	C₂³	D₄×D₆₇	1	1	yes	characteristic
118	Q₈	—	268	2	no	—
119	Q₈	—	268	2	no	—
120	Q₈	—	134	4	no	—
121	Q₈	—	134	4	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	4	no	—
124	C₂×C₄	—	268	2	no	—
125	C₂×C₄	—	268	2	no	—
126	C₂×C₄	—	67	4	no	—
127	C₂×C₄	—	134	4	no	—
128	C₂×C₄	—	134	2	no	—
129	C₂×C₄	—	134	2	no	—
130	C₂×C₄	C₂×C₄×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	C₂×C₄	#11761629	1	1	yes	characteristic
136	C₈	—	134	2	no	—
137	C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
138	C₂²	#14524005	1	1	yes	characteristic
139	C₂²	#14524018	1	1	yes	characteristic
140	C₂²	#14524039	1	1	yes	characteristic
141	C₂²	Q₁₆×D₆₇	1	2	yes	—
142	C₂²	#14524062	1	2	yes	—
143	C₄	—	8	2	no	—
144	C₄	—	268	4	no	—
145	C₄	—	134	4	no	—
146	C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
147	C₄	#14524005	1	1	yes	characteristic
148	C₄	#14524018	1	1	yes	characteristic
149	C₄	#14524032	1	1	yes	characteristic
150	C₂	#16863297	1	1	yes	characteristic
151	C₂	#16863490	1	2	yes	—
152	C₂	#16863638	1	2	yes	—
153	C₂	#16863774	1	1	yes	characteristic
154	C₂	#16863779	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₆₈

C₂

yes

maximal, characteristic

C₂×#14523966

C₂

yes

maximal

C₂×C₈×Dic₆₇

C₂

yes

maximal, characteristic

#16863301

C₂

yes

maximal

C₂×#107×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

C₄

yes

complemented

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

#14523966

—

C₈×Dic₆₇

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

#107×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₂×C₄

yes

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

D₄

yes

—

Dic₂₆₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

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

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

Q₁₆

yes

complemented

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

C₂²⋊C₄

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₄×D₄

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

C₂×Q₁₆

yes

—

C₂²×C₆₇

#135

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₇

C₄×D₄

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

#390

yes

characteristic

C₂×C₆₇

C₄×Q₁₆

yes

—

C₂×C₆₇

#467

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×#107

D₆₇

yes

characteristic

C₂×#103

—

C₂×C₄×C₈

—

#391

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

134

—

C₂×Q₁₆

—

134

—

C₂²×C₈

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₄²

—

#107

—

#103

—

134

—

C₄×C₈

—

134

—

#95

—

134

—

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

100

C₂×Q₈

—

134

—

101

C₂×Q₈

—

134

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

134

—

105

C₂²×C₄

C₂²×D₆₇

yes

characteristic

106

Q₁₆

—

134

—

107

C₂×C₈

—

108

C₂×C₈

—

134

—

109

C₂×C₈

—

134

—

110

C₂×C₈

C₄×D₆₇

yes

—

111

C₂×C₈

D₂₆₈

yes

—

112

C₂×C₈

C₂²⋊D₆₇

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

—

134

—

116

C₄²

—

134

—

117

C₂³

D₄×D₆₇

yes

characteristic

118

Q₈

—

268

—

119

Q₈

—

268

—

120

Q₈

—

134

—

121

Q₈

—

134

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

268

—

125

C₂×C₄

—

268

—

126

C₂×C₄

—

127

C₂×C₄

—

134

—

128

C₂×C₄

—

134

—

129

C₂×C₄

—

134

—

130

C₂×C₄

C₂×C₄×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

C₂×C₄

#11761629

yes

characteristic

136

C₈

—

134

—

137

C₈

(C₂×D₆₇)⋊C₄

yes

—

138

C₂²

#14524005

yes

characteristic

139

C₂²

#14524018

yes

characteristic

140

C₂²

#14524039

yes

characteristic

141

C₂²

Q₁₆×D₆₇

yes

—

142

C₂²

#14524062

yes

—

143

C₄

—

144

C₄

—

268

—

145

C₄

—

134

—

146

C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

147

C₄

#14524005

yes

characteristic

148

C₄

#14524018

yes

characteristic

149

C₄

#14524032

yes

characteristic

150

C₂

#16863297

yes

characteristic

151

C₂

#16863490

yes

—

152

C₂

#16863638

yes

—

153

C₂

#16863774

yes

characteristic

154

C₂

#16863779

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×#107×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂×#107	D₆₇
hypocenter	C₆₇	#1969
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂×#107×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
9. C₂×Dic₂₆₈	C₄	semidirect
53. C₂×Dic₆₇	Q₁₆	semidirect
78. C₆₇	#1969	semidirect

H ≅ G/N

type

9. C₂×Dic₂₆₈

C₄

semidirect

53. C₂×Dic₆₇

Q₁₆

semidirect

78. C₆₇

#1969

semidirect

Subgroups — #22561068

Conjugacy classes of subgroups

Complements