Group #22560613 — subgroups

Subgroups — #22560613

Order 8576.

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

normal subgroups only

This group has 344 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, complemented
2	#16863401	C₂	1	2	yes	maximal
3	C₂×#14523962	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863156	C₂	1	2	yes	maximal, complemented
5	C₂×#124×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#11761612	C₂²	1	2	yes	—
7	Q₈×Dic₆₇	—	2	4	no	—
8	#14524002	—	2	2	no	—
9	#14524002	C₂²	1	2	yes	—
10	#14523964	—	2	4	no	—
11	#14523962	—	2	2	no	—
12	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
14	#124×C₆₇	—	2	2	no	—
15	#124×C₆₇	C₄	1	4	yes	complemented
16	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
17	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	#11761612	—	2	2	no	—
19	#11761612	—	2	2	no	—
20	C₂×C₆₇⋊C₈	—	2	2	no	—
21	C₄⋊Dic₆₇	—	2	4	no	—
22	C₄⋊Dic₆₇	D₄	1	4	yes	—
23	C₂²⋊Dic₆₇	—	2	2	no	—
24	C₄×Dic₆₇	—	2	4	no	—
25	C₄×Dic₆₇	—	2	2	no	—
26	C₂⁴×C₆₇	—	2	1	no	—
27	C₂×Q₈×C₆₇	—	2	2	no	—
28	C₂×Q₈×C₆₇	—	2	2	no	—
29	C₂×Q₈×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×Q₈×C₆₇	D₄	1	2	yes	—
31	C₂×D₄×C₆₇	—	2	2	no	—
32	C₂×D₄×C₆₇	—	2	1	no	—
33	C₂×D₄×C₆₇	—	2	2	no	—
34	C₂×D₄×C₆₇	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₆₇	—	2	1	no	—
36	C₂²×C₄×C₆₇	D₄	1	2	yes	—
37	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₂²⋊C₄×C₆₇	—	2	8	no	—
39	C₄²×C₆₇	—	2	1	no	—
40	C₄²×C₆₇	C₂×C₄	1	2	yes	—
41	C₂³×C₆₇	—	2	2	no	—
42	C₂³×C₆₇	—	4	4	no	—
43	C₂³×C₆₇	—	4	1	no	—
44	C₂³×C₆₇	—	4	1	no	—
45	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₆₇	—	2	4	no	—
47	Q₈×C₆₇	—	2	4	no	—
48	D₄×C₆₇	—	2	4	no	—
49	D₄×C₆₇	—	4	2	no	—
50	C₂×C₄×C₆₇	—	2	2	no	—
51	C₂×C₄×C₆₇	—	2	2	no	—
52	C₂×C₄×C₆₇	—	2	2	no	—
53	C₂×C₄×C₆₇	—	4	2	no	—
54	C₂×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
55	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₆₇	#41	1	2	yes	—
59	C₂×Dic₆₇	—	2	4	no	—
60	C₆₇⋊C₈	—	4	2	no	—
61	C₂²×C₆₇	—	2	2	no	—
62	C₂²×C₆₇	—	4	4	no	—
63	C₂²×C₆₇	—	4	2	no	—
64	C₂²×C₆₇	—	8	2	no	—
65	C₂²×C₆₇	—	8	2	no	—
66	C₂²×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₆₇	#120	1	1	yes	characteristic
68	C₂²×C₆₇	#123	1	1	yes	characteristic
69	C₄×C₆₇	—	2	4	no	—
70	C₄×C₆₇	—	4	2	no	—
71	C₄×C₆₇	C₄×D₄	1	2	yes	—
72	C₄×C₆₇	#121	1	2	yes	—
73	Dic₆₇	—	4	4	no	—
74	C₂×C₆₇	—	2	1	no	—
75	C₂×C₆₇	—	2	1	no	—
76	C₂×C₆₇	—	8	2	no	—
77	C₂×C₆₇	#386	1	1	yes	characteristic
78	C₂×C₆₇	#454	1	2	yes	—
79	#1888	—	67	1	no	maximal, Sylow
80	C₆₇	#1888	1	1	yes	characteristic, complemented, Sylow
81	C₂×#124	D₆₇	1	1	yes	characteristic
82	#519	—	67	1	no	—
83	#418	—	67	2	no	—
84	C₂×#101	—	67	1	no	—
85	#339	—	67	2	no	—
86	C₂²×Q₈	C₂×D₆₇	1	1	yes	characteristic
87	C₂²×D₄	C₂×D₆₇	1	1	yes	characteristic
88	C₂×M₁₆	—	67	2	no	—
89	#124	—	2	2	no	—
90	#124	Dic₆₇	1	4	yes	complemented
91	C₄×Q₈	—	134	4	no	—
92	#117	—	67	2	no	—
93	#117	—	134	2	no	—
94	C₂×C₂²⋊C₄	—	2	2	no	—
95	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
96	#102	—	134	4	no	—
97	#101	—	134	2	no	—
98	C₂⁴	—	2	1	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	C₂×Dic₆₇	1	2	yes	—
102	C₂×Q₈	C₂²⋊D₆₇	1	2	yes	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	—	2	2	no	—
106	C₂×D₄	C₂×Dic₆₇	1	2	yes	—
107	C₂²×C₄	—	2	1	no	—
108	C₂²×C₄	C₂²⋊D₆₇	1	2	yes	—
109	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
110	M₁₆	—	134	2	no	—
111	M₁₆	—	134	2	no	—
112	C₂×C₈	—	134	2	no	—
113	C₄⋊C₄	—	67	4	no	—
114	C₄⋊C₄	—	134	4	no	—
115	C₂²⋊C₄	—	2	8	no	—
116	C₂²⋊C₄	—	134	2	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	134	4	no	—
119	C₄²	—	134	2	no	—
120	C₄²	C₂×Dic₆₇	1	2	yes	—
121	C₂³	—	2	2	no	—
122	C₂³	—	4	4	no	—
123	C₂³	—	4	1	no	—
124	C₂³	—	4	1	no	—
125	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
126	Q₈	—	2	4	no	—
127	Q₈	—	2	4	no	—
128	D₄	—	2	4	no	—
129	D₄	—	4	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	134	4	no	—
135	C₂×C₄	C₂²⋊Dic₆₇	1	4	yes	—
136	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
137	C₂×C₄	D₄×D₆₇	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
139	C₂×C₄	#11761627	1	2	yes	—
140	C₈	—	268	2	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	4	4	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	8	2	no	—
145	C₂²	—	8	2	no	—
146	C₂²	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
147	C₂²	#14524011	1	1	yes	characteristic
148	C₂²	#14524024	1	1	yes	characteristic
149	C₄	—	2	4	no	—
150	C₄	—	4	2	no	—
151	C₄	—	268	4	no	—
152	C₄	D₄×Dic₆₇	1	2	yes	—
153	C₄	#14524015	1	2	yes	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	—	8	2	no	—
157	C₂	#16863276	1	1	yes	characteristic
158	C₂	#16863568	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16863874

C₂

yes

maximal, characteristic, complemented

#16863401

C₂

yes

maximal

C₂×#14523962

C₂

yes

maximal, characteristic, complemented

#16863156

C₂

yes

maximal, complemented

C₂×#124×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

C₂²

yes

—

Q₈×Dic₆₇

—

#14524002

—

#14524002

C₂²

yes

—

#14523964

—

#14523962

—

C₂²×Q₈×C₆₇

C₂²

yes

characteristic

C₂²×D₄×C₆₇

C₂²

yes

characteristic

#124×C₆₇

—

#124×C₆₇

C₄

yes

complemented

C₂×C₂²⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂⁴×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

C₂×C₄

yes

—

C₂×Q₈×C₆₇

D₄

yes

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×C₄

yes

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₂²⋊C₄

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₄×D₄

yes

—

C₄×C₆₇

#121

yes

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#386

yes

characteristic

C₂×C₆₇

#454

yes

—

#1888

—

maximal, Sylow

C₆₇

#1888

yes

characteristic, complemented, Sylow

C₂×#124

D₆₇

yes

—

—

—

—

C₂²×Q₈

C₂×D₆₇

yes

characteristic

C₂²×D₄

C₂×D₆₇

yes

characteristic

C₂×M₁₆

—

#124

—

#124

Dic₆₇

yes

complemented

C₄×Q₈

—

134

—

#117

—

#117

—

134

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

#102

—

134

—

#101

—

134

—

C₂⁴

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

C₂×Dic₆₇

yes

—

102

C₂×Q₈

C₂²⋊D₆₇

yes

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

C₂×Dic₆₇

yes

—

107

C₂²×C₄

—

108

C₂²×C₄

C₂²⋊D₆₇

yes

—

109

C₂²×C₄

C₂²×D₆₇

yes

characteristic

110

M₁₆

—

134

—

111

M₁₆

—

134

—

112

C₂×C₈

—

134

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

134

—

115

C₂²⋊C₄

—

116

C₂²⋊C₄

—

134

—

117

C₄²

—

118

C₄²

—

134

—

119

C₄²

—

134

—

120

C₄²

C₂×Dic₆₇

yes

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

126

Q₈

—

127

Q₈

—

128

D₄

—

129

D₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

134

—

135

C₂×C₄

C₂²⋊Dic₆₇

yes

—

136

C₂×C₄

C₂²×Dic₆₇

yes

characteristic

137

C₂×C₄

D₄×D₆₇

yes

—

138

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

139

C₂×C₄

#11761627

yes

—

140

C₈

—

268

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×C₂²⋊Dic₆₇

yes

characteristic

147

C₂²

#14524011

yes

characteristic

148

C₂²

#14524024

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

268

—

152

C₄

D₄×Dic₆₇

yes

—

153

C₄

#14524015

yes

—

154

C₂

—

155

C₂

—

156

C₂

—

157

C₂

#16863276

yes

characteristic

158

C₂

#16863568

yes

—

	subgroup	G/H
center	C₂²	#14524011
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×#124×C₆₇	C₂
socle	C₂²×C₆₇	#120
hypercenter	C₂×#124	D₆₇
hypocenter	C₆₇	#1888
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#124

subgroup

G/H

center

C₂²

#14524011

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂×#124×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#124

N	H ≅ G/N	type
1. #16863874	C₂	semidirect
3. C₂×#14523962	C₂	semidirect
4. #16863156	C₂	semidirect
15. #124×C₆₇	C₄	semidirect
80. C₆₇	#1888	semidirect
90. #124	Dic₆₇	semidirect

H ≅ G/N

type

1. #16863874

C₂

semidirect

3. C₂×#14523962

C₂

semidirect

4. #16863156

C₂

semidirect

15. #124×C₆₇

C₄

semidirect

80. C₆₇

#1888

semidirect

90. #124

Dic₆₇

semidirect

Subgroups — #22560613

Conjugacy classes of subgroups

Complements