Group #22561529 — subgroups

Subgroups — #22561529

Order 8576.

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

normal subgroups only

This group has 288 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	C₂×Dic₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523967	C₂	1	2	yes	maximal
3	#16863303	C₂	1	2	yes	maximal
4	#16863159	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
8	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
9	Dic₆₇⋊C₈	—	2	2	no	—
10	#14523967	—	2	4	no	—
11	C₆₇⋊#95	—	2	2	no	—
12	C₆₇⋊#95	—	2	2	no	—
13	C₂²×Q₈×C₆₇	C₂²	1	2	yes	—
14	C₂×Q₁₆×C₆₇	—	2	4	no	—
15	C₂×Q₁₆×C₆₇	—	2	2	no	—
16	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×Dic₆₇	—	2	2	no	—
18	C₂²×Dic₆₇	D₄	1	1	yes	characteristic
19	C₂²×Dic₆₇	D₄	1	1	yes	characteristic
20	C₂×C₆₇⋊C₈	—	2	2	no	—
21	C₂×C₆₇⋊C₈	—	2	1	no	—
22	C₄⋊Dic₆₇	—	2	2	no	—
23	C₄⋊Dic₆₇	—	2	2	no	—
24	C₄×Dic₆₇	—	2	1	no	—
25	C₄×Dic₆₇	—	2	1	no	—
26	C₂×Q₈×C₆₇	—	2	2	no	—
27	C₂×Q₈×C₆₇	—	2	2	no	—
28	C₂×Q₈×C₆₇	—	2	2	no	—
29	C₂×Q₈×C₆₇	—	2	2	no	—
30	C₂×Q₈×C₆₇	—	2	2	no	—
31	C₂×Q₈×C₆₇	D₄	1	4	yes	—
32	C₂²×C₄×C₆₇	—	2	2	no	—
33	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
34	Q₁₆×C₆₇	—	2	8	no	—
35	C₂×C₈×C₆₇	—	2	2	no	—
36	C₂×C₈×C₆₇	D₄	1	2	yes	—
37	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₆₇	—	2	4	no	—
39	Q₈×C₆₇	—	2	4	no	—
40	Q₈×C₆₇	—	4	2	no	—
41	Q₈×C₆₇	—	4	2	no	—
42	C₂×C₄×C₆₇	—	2	4	no	—
43	C₂×C₄×C₆₇	—	4	2	no	—
44	C₂×C₄×C₆₇	—	4	2	no	—
45	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
48	C₂×C₄×C₆₇	#41	1	2	yes	—
49	C₂×Dic₆₇	—	2	1	no	—
50	C₂×Dic₆₇	—	2	1	no	—
51	C₂×Dic₆₇	—	2	1	no	—
52	C₂×Dic₆₇	—	2	2	no	—
53	C₂×Dic₆₇	—	4	2	no	—
54	C₂×Dic₆₇	—	4	2	no	—
55	C₂×Dic₆₇	—	4	2	no	—
56	C₂×Dic₆₇	Q₁₆	1	2	yes	complemented
57	C₈×C₆₇	—	2	2	no	—
58	C₆₇⋊C₈	—	4	1	no	—
59	C₂²×C₆₇	#121	1	1	yes	characteristic
60	C₂²×C₆₇	#121	1	2	yes	—
61	C₂²×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
62	C₂²×C₆₇	#135	1	1	yes	characteristic
63	C₂²×C₆₇	#137	1	2	yes	—
64	C₄×C₆₇	—	4	4	no	—
65	C₄×C₆₇	#120	1	1	yes	characteristic
66	C₄×C₆₇	#123	1	2	yes	—
67	C₄×C₆₇	#124	1	1	yes	characteristic
68	Dic₆₇	—	2	2	no	—
69	Dic₆₇	—	4	1	no	—
70	Dic₆₇	—	8	2	no	—
71	C₂×C₆₇	#394	1	1	yes	characteristic
72	C₂×C₆₇	#462	1	2	yes	—
73	C₂×C₆₇	#464	1	2	yes	—
74	C₂×C₆₇	#467	1	1	yes	characteristic
75	C₂×C₆₇	#470	1	1	yes	characteristic
76	#2056	—	67	1	no	maximal, Sylow
77	C₆₇	#2056	1	1	yes	characteristic, complemented, Sylow
78	C₂²×Q₁₆	D₆₇	1	1	yes	characteristic
79	C₂×C₄⋊C₈	—	67	1	no	—
80	C₂×#103	—	67	2	no	—
81	#391	—	67	2	no	—
82	#340	—	67	1	no	—
83	C₂²×Q₈	C₂×D₆₇	1	2	yes	—
84	C₂×Q₁₆	—	2	4	no	—
85	C₂×Q₁₆	—	2	2	no	—
86	C₂²×C₈	—	67	1	no	—
87	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
88	C₂×C₄⋊C₄	—	67	2	no	—
89	C₂×C₄²	—	67	1	no	—
90	C₄⋊C₈	—	134	2	no	—
91	#103	—	134	4	no	—
92	#95	—	134	2	no	—
93	#95	—	134	2	no	—
94	C₂×Q₈	—	2	2	no	—
95	C₂×Q₈	—	2	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	C₂²⋊D₆₇	1	4	yes	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	—	67	1	no	—
102	C₂²×C₄	—	67	1	no	—
103	C₂²×C₄	—	134	2	no	—
104	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
105	Q₁₆	—	2	8	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	134	2	no	—
108	C₂×C₈	—	134	1	no	—
109	C₂×C₈	C₂²⋊D₆₇	1	2	yes	—
110	C₄⋊C₄	—	134	2	no	—
111	C₄⋊C₄	—	134	2	no	—
112	C₄²	—	134	1	no	—
113	C₄²	—	134	1	no	—
114	C₂³	D₄×D₆₇	1	1	yes	characteristic
115	Q₈	—	2	4	no	—
116	Q₈	—	2	4	no	—
117	Q₈	—	4	2	no	—
118	Q₈	—	4	2	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	268	2	no	—
123	C₂×C₄	—	268	2	no	—
124	C₂×C₄	—	268	2	no	—
125	C₂×C₄	—	67	2	no	—
126	C₂×C₄	—	134	1	no	—
127	C₂×C₄	—	134	2	no	—
128	C₂×C₄	—	134	1	no	—
129	C₂×C₄	—	134	1	no	—
130	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
131	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
132	C₂×C₄	#11761627	1	1	yes	characteristic
133	C₂×C₄	#11761627	1	2	yes	—
134	C₈	—	2	2	no	—
135	C₈	—	268	1	no	—
136	C₂²	#14524014	1	1	yes	characteristic
137	C₂²	#14524015	1	2	yes	—
138	C₂²	Q₁₆×D₆₇	1	1	yes	characteristic
139	C₂²	#14524066	1	1	yes	characteristic
140	C₂²	#14524075	1	2	yes	—
141	C₄	—	4	4	no	—
142	C₄	—	268	1	no	—
143	C₄	—	536	2	no	—
144	C₄	—	134	2	no	—
145	C₄	#14524012	1	1	yes	characteristic
146	C₄	#14524024	1	2	yes	—
147	C₄	#14524030	1	1	yes	characteristic
148	C₂	#16863310	1	1	yes	characteristic
149	C₂	#16863604	1	2	yes	—
150	C₂	#16863620	1	2	yes	—
151	C₂	#16863634	1	1	yes	characteristic
152	C₂	#16863651	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₆₇⋊C₈

C₂

yes

maximal, characteristic

C₂×#14523967

C₂

yes

maximal

#16863303

C₂

yes

maximal

#16863159

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

Dic₆₇⋊C₈

—

#14523967

—

C₆₇⋊#95

—

C₆₇⋊#95

—

C₂²×Q₈×C₆₇

C₂²

yes

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂²×Dic₆₇

—

C₂²×Dic₆₇

D₄

yes

characteristic

C₂²×Dic₆₇

D₄

yes

characteristic

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

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

—

C₂×Dic₆₇

—

C₂×Dic₆₇

Q₁₆

yes

complemented

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

#121

yes

characteristic

C₂²×C₆₇

#121

yes

—

C₂²×C₆₇

C₂×Q₁₆

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₄×C₆₇

—

C₄×C₆₇

#120

yes

characteristic

C₄×C₆₇

#123

yes

—

C₄×C₆₇

#124

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

#394

yes

characteristic

C₂×C₆₇

#462

yes

—

C₂×C₆₇

#464

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

D₆₇

yes

characteristic

C₂×C₄⋊C₈

—

C₂×#103

—

#391

—

#340

—

C₂²×Q₈

C₂×D₆₇

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

134

—

#103

—

134

—

#95

—

134

—

#95

—

134

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₆₇

yes

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

134

—

104

C₂²×C₄

C₂²×D₆₇

yes

characteristic

105

Q₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

134

—

108

C₂×C₈

—

134

—

109

C₂×C₈

C₂²⋊D₆₇

yes

—

110

C₄⋊C₄

—

134

—

111

C₄⋊C₄

—

134

—

112

C₄²

—

134

—

113

C₄²

—

134

—

114

C₂³

D₄×D₆₇

yes

characteristic

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

268

—

123

C₂×C₄

—

268

—

124

C₂×C₄

—

268

—

125

C₂×C₄

—

126

C₂×C₄

—

134

—

127

C₂×C₄

—

134

—

128

C₂×C₄

—

134

—

129

C₂×C₄

—

134

—

130

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

131

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

132

C₂×C₄

#11761627

yes

characteristic

133

C₂×C₄

#11761627

yes

—

134

C₈

—

135

C₈

—

268

—

136

C₂²

#14524014

yes

characteristic

137

C₂²

#14524015

yes

—

138

C₂²

Q₁₆×D₆₇

yes

characteristic

139

C₂²

#14524066

yes

characteristic

140

C₂²

#14524075

yes

—

141

C₄

—

142

C₄

—

268

—

143

C₄

—

536

—

144

C₄

—

134

—

145

C₄

#14524012

yes

characteristic

146

C₄

#14524024

yes

—

147

C₄

#14524030

yes

characteristic

148

C₂

#16863310

yes

characteristic

149

C₂

#16863604

yes

—

150

C₂

#16863620

yes

—

151

C₂

#16863634

yes

characteristic

152

C₂

#16863651

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₆₇
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂²×Q₁₆×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂²×Q₁₆	D₆₇
hypocenter	C₆₇	#2056
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂³

D₄×D₆₇

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂²×Q₁₆×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

hypercenter

C₂²×Q₁₆

D₆₇

hypocenter

C₆₇

#2056

LCS

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
56. C₂×Dic₆₇	Q₁₆	semidirect
77. C₆₇	#2056	semidirect

H ≅ G/N

type

56. C₂×Dic₆₇

Q₁₆

semidirect

77. C₆₇

#2056

semidirect

Subgroups — #22561529

Conjugacy classes of subgroups

Complements