Group #22560426 — subgroups

Subgroups — #22560426

Order 8576.

272 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 97 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	#16863874	C₂	1	1	yes	maximal, characteristic
2	#16863438	C₂	1	2	yes	maximal
3	#16863303	C₂	1	1	yes	maximal, characteristic
4	#16863157	C₂	1	2	yes	maximal
5	C₂×#101×C₆₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₆₇	—	2	2	no	—
7	Q₈×Dic₆₇	—	2	2	no	—
8	#14524002	—	2	2	no	—
9	#14524002	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
12	#14523977	—	2	2	no	—
13	#14523975	—	2	2	no	—
14	C₆₇⋊#95	—	2	2	no	—
15	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₆₇	C₂²	1	2	yes	—
17	#101×C₆₇	C₄	1	4	yes	complemented
18	C₂²×Dic₆₇	—	2	1	no	—
19	C₂²×Dic₆₇	—	2	1	no	—
20	C₄⋊Dic₆₇	—	2	2	no	—
21	C₄⋊Dic₆₇	—	2	2	no	—
22	C₄⋊Dic₆₇	—	2	2	no	—
23	C₄⋊Dic₆₇	D₄	1	2	yes	—
24	C₄⋊Dic₆₇	D₄	1	2	yes	—
25	C₂²⋊Dic₆₇	—	2	2	no	—
26	C₄×Dic₆₇	—	2	4	no	—
27	C₄×Dic₆₇	—	2	2	no	—
28	C₄×Dic₆₇	—	2	2	no	—
29	C₂×Q₈×C₆₇	—	2	2	no	—
30	C₂×Q₈×C₆₇	—	2	2	no	—
31	C₂×Q₈×C₆₇	C₂×C₄	1	2	yes	—
32	C₂×Q₈×C₆₇	Q₈	1	2	yes	—
33	C₂²×C₄×C₆₇	D₄	1	2	yes	—
34	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
35	M₁₆×C₆₇	—	2	2	no	—
36	M₁₆×C₆₇	C₂×C₄	1	4	yes	—
37	C₂×C₈×C₆₇	—	2	2	no	—
38	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₆₇	—	2	4	no	—
40	Q₈×C₆₇	—	2	4	no	—
41	C₂×C₄×C₆₇	—	2	2	no	—
42	C₂×C₄×C₆₇	—	2	2	no	—
43	C₂×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
44	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₆₇	#41	1	2	yes	—
48	C₂×Dic₆₇	—	2	2	no	—
49	C₂×Dic₆₇	—	2	2	no	—
50	C₂×Dic₆₇	—	2	2	no	—
51	C₂×Dic₆₇	—	4	1	no	—
52	C₂×Dic₆₇	—	4	1	no	—
53	C₂×Dic₆₇	—	4	2	no	—
54	C₂×Dic₆₇	—	4	1	no	—
55	C₈×C₆₇	—	2	4	no	—
56	C₂²×C₆₇	—	2	1	no	—
57	C₂²×C₆₇	—	2	1	no	—
58	C₂²×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₆₇	#120	1	1	yes	characteristic
60	C₂²×C₆₇	#123	1	1	yes	characteristic
61	C₄×C₆₇	—	2	4	no	—
62	C₄×C₆₇	C₄×D₄	1	2	yes	—
63	C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
64	Dic₆₇	—	4	2	no	—
65	Dic₆₇	—	4	2	no	—
66	Dic₆₇	—	4	2	no	—
67	Dic₆₇	—	8	1	no	—
68	C₂×C₆₇	—	2	1	no	—
69	C₂×C₆₇	—	2	1	no	—
70	C₂×C₆₇	#385	1	1	yes	characteristic
71	C₂×C₆₇	#454	1	1	yes	characteristic
72	C₂×C₆₇	#456	1	1	yes	characteristic
73	#1857	—	67	1	no	maximal, Sylow
74	C₆₇	#1857	1	1	yes	characteristic, complemented, Sylow
75	#519	—	67	1	no	—
76	#428	—	67	2	no	—
77	C₂×#101	D₆₇	1	1	yes	characteristic
78	#391	—	67	1	no	—
79	#339	—	67	2	no	—
80	C₂²×Q₈	C₂×D₆₇	1	1	yes	characteristic
81	C₂×M₁₆	C₂×D₆₇	1	2	yes	—
82	C₄×Q₈	—	134	2	no	—
83	C₄×Q₈	—	134	2	no	—
84	#117	—	67	2	no	—
85	#117	—	134	2	no	—
86	C₂×C₄⋊C₄	—	67	1	no	—
87	C₂×C₄²	—	67	1	no	—
88	#107	—	134	2	no	—
89	#106	—	134	2	no	—
90	#101	Dic₆₇	1	4	yes	complemented
91	#95	—	134	2	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	Dic₁₃₄	1	2	yes	—
95	C₂×Q₈	C₂×Dic₆₇	1	2	yes	—
96	C₂²×C₄	—	134	1	no	—
97	C₂²×C₄	—	134	1	no	—
98	C₂²×C₄	D₂₆₈	1	2	yes	—
99	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
100	M₁₆	—	2	2	no	—
101	M₁₆	C₂×Dic₆₇	1	4	yes	—
102	C₂×C₈	—	2	2	no	—
103	C₄⋊C₄	—	67	2	no	—
104	C₄⋊C₄	—	67	2	no	—
105	C₄⋊C₄	—	134	2	no	—
106	C₄⋊C₄	—	134	2	no	—
107	C₄⋊C₄	—	134	2	no	—
108	C₂²⋊C₄	—	134	2	no	—
109	C₄²	—	134	4	no	—
110	C₄²	—	134	2	no	—
111	C₄²	—	134	2	no	—
112	C₂³	C₂×D₂₆₈	1	1	yes	characteristic
113	Q₈	—	2	4	no	—
114	Q₈	—	2	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	268	1	no	—
118	C₂×C₄	—	268	1	no	—
119	C₂×C₄	—	268	2	no	—
120	C₂×C₄	—	268	1	no	—
121	C₂×C₄	—	134	2	no	—
122	C₂×C₄	—	134	2	no	—
123	C₂×C₄	—	134	2	no	—
124	C₂×C₄	C₄⋊Dic₆₇	1	4	yes	—
125	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
126	C₂×C₄	D₄×D₆₇	1	2	yes	—
127	C₂×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
128	C₂×C₄	#11761627	1	2	yes	—
129	C₈	—	2	4	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
133	C₂²	#14524013	1	1	yes	characteristic
134	C₂²	#14524027	1	1	yes	characteristic
135	C₄	—	2	4	no	—
136	C₄	—	268	2	no	—
137	C₄	—	268	2	no	—
138	C₄	—	268	2	no	—
139	C₄	—	536	1	no	—
140	C₄	D₄×Dic₆₇	1	2	yes	—
141	C₄	#14524020	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#16863271	1	1	yes	characteristic
145	C₂	#16863571	1	1	yes	characteristic
146	C₂	#16863581	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863874

C₂

yes

maximal, characteristic

#16863438

C₂

yes

maximal

#16863303

C₂

yes

maximal, characteristic

#16863157

C₂

yes

maximal

C₂×#101×C₆₇

C₂

yes

maximal, characteristic

Q₈×Dic₆₇

—

Q₈×Dic₆₇

—

#14524002

—

#14524002

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

#14523977

—

#14523975

—

C₆₇⋊#95

—

C₂²×Q₈×C₆₇

C₂²

yes

characteristic

C₂×M₁₆×C₆₇

C₂²

yes

—

#101×C₆₇

C₄

yes

complemented

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

—

C₄⋊Dic₆₇

D₄

yes

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

C₂×C₄

yes

—

C₂×Q₈×C₆₇

Q₈

yes

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄⋊C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₄⋊C₄

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

C₄×D₄

yes

—

C₄×C₆₇

C₂²⋊Q₈

yes

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#519

—

#428

—

C₂×#101

D₆₇

yes

characteristic

#391

—

#339

—

C₂²×Q₈

C₂×D₆₇

yes

characteristic

C₂×M₁₆

C₂×D₆₇

yes

—

C₄×Q₈

—

134

—

C₄×Q₈

—

134

—

#117

—

#117

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

#107

—

134

—

#106

—

134

—

#101

Dic₆₇

yes

complemented

#95

—

134

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₁₃₄

yes

—

C₂×Q₈

C₂×Dic₆₇

yes

—

C₂²×C₄

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

D₂₆₈

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

100

M₁₆

—

101

M₁₆

C₂×Dic₆₇

yes

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

134

—

106

C₄⋊C₄

—

134

—

107

C₄⋊C₄

—

134

—

108

C₂²⋊C₄

—

134

—

109

C₄²

—

134

—

110

C₄²

—

134

—

111

C₄²

—

134

—

112

C₂³

C₂×D₂₆₈

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

268

—

118

C₂×C₄

—

268

—

119

C₂×C₄

—

268

—

120

C₂×C₄

—

268

—

121

C₂×C₄

—

134

—

122

C₂×C₄

—

134

—

123

C₂×C₄

—

134

—

124

C₂×C₄

C₄⋊Dic₆₇

yes

—

125

C₂×C₄

C₂²×Dic₆₇

yes

characteristic

126

C₂×C₄

D₄×D₆₇

yes

—

127

C₂×C₄

C₂×Dic₁₃₄

yes

characteristic

128

C₂×C₄

#11761627

yes

—

129

C₈

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×C₄⋊Dic₆₇

yes

characteristic

133

C₂²

#14524013

yes

characteristic

134

C₂²

#14524027

yes

characteristic

135

C₄

—

136

C₄

—

268

—

137

C₄

—

268

—

138

C₄

—

268

—

139

C₄

—

536

—

140

C₄

D₄×Dic₆₇

yes

—

141

C₄

#14524020

yes

—

142

C₂

—

143

C₂

—

144

C₂

#16863271

yes

characteristic

145

C₂

#16863571

yes

characteristic

146

C₂

#16863581

yes

characteristic

	subgroup	G/H
center	C₂²	#14524013
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×#101×C₆₇	C₂
socle	C₂²×C₆₇	#120
hypercenter	C₂×#101	D₆₇
hypocenter	C₆₇	#1857
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂²

#14524013

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂×#101×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
17. #101×C₆₇	C₄	semidirect
74. C₆₇	#1857	semidirect
90. #101	Dic₆₇	semidirect

H ≅ G/N

type

17. #101×C₆₇

C₄

semidirect

74. C₆₇

#1857

semidirect

90. #101

Dic₆₇

semidirect

Subgroups — #22560426

Conjugacy classes of subgroups

Complements