Group #22560744 — subgroups

Subgroups — #22560744

Order 8576.

314 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 85 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	#16863970	C₂	1	1	yes	maximal, characteristic
2	#16863874	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863412	C₂	1	2	yes	maximal
4	#16863153	C₂	1	2	yes	maximal, complemented
5	C₂×#101×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11761628	C₂²	1	1	yes	characteristic
7	#14524041	C₄	1	2	yes	complemented
8	#14524032	C₄	1	2	yes	complemented
9	#14524026	—	2	2	no	—
10	#14524019	—	2	2	no	—
11	Q₈×Dic₆₇	—	2	4	no	—
12	#14524002	—	2	2	no	—
13	#14524002	C₂²	1	1	yes	characteristic
14	#14524002	C₂²	1	2	yes	—
15	#14523969	—	2	4	no	—
16	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₆₇	C₂²	1	2	yes	—
18	#101×C₆₇	—	2	2	no	—
19	#11761628	—	2	2	no	—
20	#11761628	—	2	2	no	—
21	C₂×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
22	C₂×D₂₆₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₂²⋊D₆₇	—	2	1	no	—
24	C₂×C₄×D₆₇	—	2	1	no	—
25	C₄⋊Dic₆₇	—	2	4	no	—
26	C₄⋊Dic₆₇	—	2	2	no	—
27	C₄⋊Dic₆₇	—	2	1	no	—
28	Dic₆₇⋊C₄	—	2	4	no	—
29	C₂²⋊Dic₆₇	—	2	2	no	—
30	C₂²⋊Dic₆₇	—	2	1	no	—
31	(C₂×D₆₇)⋊C₄	—	2	4	no	—
32	C₄×Dic₆₇	—	2	4	no	—
33	C₄×Dic₆₇	C₂×C₄	1	2	yes	—
34	C₄×Dic₆₇	D₄	1	4	yes	—
35	C₂×Q₈×C₆₇	—	2	2	no	—
36	C₂×Q₈×C₆₇	—	2	2	no	—
37	C₂×Q₈×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×Q₈×C₆₇	D₄	1	2	yes	—
39	C₂²×C₄×C₆₇	D₄	1	2	yes	—
40	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
41	M₁₆×C₆₇	—	2	2	no	—
42	M₁₆×C₆₇	—	2	2	no	—
43	C₂×C₈×C₆₇	—	2	2	no	—
44	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×D₆₇	—	2	1	no	—
46	Q₈×C₆₇	—	2	4	no	—
47	Q₈×C₆₇	—	2	4	no	—
48	C₂×C₄×C₆₇	—	2	2	no	—
49	C₂×C₄×C₆₇	—	2	2	no	—
50	C₂×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
51	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₆₇	#41	1	2	yes	—
55	C₂²⋊D₆₇	—	4	1	no	—
56	C₂²⋊D₆₇	—	4	1	no	—
57	C₂×Dic₆₇	—	2	2	no	—
58	C₂×Dic₆₇	—	2	1	no	—
59	C₂×Dic₆₇	—	2	4	no	—
60	D₂₆₈	—	2	2	no	—
61	C₄×D₆₇	—	4	2	no	—
62	Dic₁₃₄	—	2	2	no	—
63	C₈×C₆₇	—	4	2	no	—
64	C₂²×C₆₇	—	4	1	no	—
65	C₂²×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₆₇	#120	1	1	yes	characteristic
67	C₂²×C₆₇	#123	1	1	yes	characteristic
68	C₂×D₆₇	—	4	2	no	—
69	C₂×D₆₇	—	4	1	no	—
70	C₄×C₆₇	—	2	4	no	—
71	C₄×C₆₇	C₄×D₄	1	2	yes	—
72	C₄×C₆₇	#121	1	2	yes	—
73	Dic₆₇	—	4	4	no	—
74	Dic₆₇	—	4	2	no	—
75	Dic₆₇	—	4	1	no	—
76	C₂×C₆₇	—	2	1	no	—
77	C₂×C₆₇	—	2	1	no	—
78	C₂×C₆₇	—	2	1	no	—
79	C₂×C₆₇	#386	1	1	yes	characteristic
80	D₆₇	—	8	1	no	—
81	#1912	—	67	1	no	maximal, Sylow
82	C₆₇	#1912	1	1	yes	characteristic, complemented, Sylow
83	#536	—	67	1	no	—
84	#519	—	67	1	no	—
85	#421	—	67	2	no	—
86	C₂×#101	D₆₇	1	1	yes	characteristic, complemented
87	H(3,ℤ/4)	—	67	2	no	—
88	C₂×#41	—	67	1	no	—
89	C₂²×Q₈	C₂×D₆₇	1	1	yes	characteristic
90	C₂×M₁₆	C₂×D₆₇	1	2	yes	—
91	C₄⋊Q₈	—	67	2	no	—
92	#124	—	67	2	no	—
93	#123	—	134	2	no	—
94	C₂²⋊Q₈	—	134	2	no	—
95	C₄×Q₈	—	134	4	no	—
96	#117	—	67	2	no	—
97	#117	—	67	1	no	—
98	#117	—	134	2	no	—
99	#104	—	134	4	no	—
100	#101	—	2	2	no	—
101	#41	—	134	2	no	—
102	#41	—	134	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	—	67	1	no	—
106	C₂×Q₈	C₄×D₆₇	1	2	yes	—
107	C₂×Q₈	C₂²⋊D₆₇	1	2	yes	—
108	C₂×D₄	—	67	1	no	—
109	C₂×D₄	—	134	1	no	—
110	C₂²×C₄	—	134	1	no	—
111	C₂²×C₄	D₂₆₈	1	2	yes	—
112	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
113	M₁₆	—	2	2	no	—
114	M₁₆	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₄⋊C₄	—	134	4	no	—
117	C₄⋊C₄	—	134	4	no	—
118	C₄⋊C₄	—	134	2	no	—
119	C₄⋊C₄	—	134	1	no	—
120	C₂²⋊C₄	—	134	2	no	—
121	C₂²⋊C₄	—	134	1	no	—
122	C₂²⋊C₄	—	134	4	no	—
123	C₄²	—	67	2	no	—
124	C₄²	—	67	4	no	—
125	C₄²	—	134	4	no	—
126	C₂³	—	134	1	no	—
127	C₂³	C₂×D₂₆₈	1	1	yes	characteristic
128	Q₈	—	2	4	no	—
129	Q₈	—	2	4	no	—
130	Q₈	—	134	2	no	—
131	D₄	—	268	1	no	—
132	D₄	—	268	1	no	—
133	D₄	—	134	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	268	2	no	—
137	C₂×C₄	—	134	2	no	—
138	C₂×C₄	—	134	1	no	—
139	C₂×C₄	—	134	4	no	—
140	C₂×C₄	(C₂×D₆₇)⋊C₄	1	4	yes	—
141	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
142	C₂×C₄	D₄×D₆₇	1	2	yes	—
143	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
144	C₂×C₄	#11761627	1	2	yes	—
145	C₈	—	4	2	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	268	2	no	—
148	C₂²	—	268	1	no	—
149	C₂²	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
150	C₂²	#14524013	1	1	yes	characteristic
151	C₂²	#14524027	1	1	yes	characteristic
152	C₄	—	2	4	no	—
153	C₄	—	268	4	no	—
154	C₄	—	268	2	no	—
155	C₄	—	268	1	no	—
156	C₄	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
157	C₄	#14524015	1	2	yes	—
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	—	2	1	no	—
161	C₂	—	536	1	no	—
162	C₂	#16863273	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863970

C₂

yes

maximal, characteristic

#16863874

C₂

yes

maximal, characteristic, complemented

#16863412

C₂

yes

maximal

#16863153

C₂

yes

maximal, complemented

C₂×#101×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761628

C₂²

yes

characteristic

#14524041

C₄

yes

complemented

#14524032

C₄

yes

complemented

#14524026

—

#14524019

—

Q₈×Dic₆₇

—

#14524002

—

#14524002

C₂²

yes

characteristic

#14524002

C₂²

yes

—

#14523969

—

C₂²×Q₈×C₆₇

C₂²

yes

characteristic

C₂×M₁₆×C₆₇

C₂²

yes

—

#101×C₆₇

—

#11761628

—

#11761628

—

C₂×Dic₁₃₄

C₂×C₄

yes

characteristic

C₂×D₂₆₈

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₆₇

—

C₂×C₄×D₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

(C₂×D₆₇)⋊C₄

—

C₄×Dic₆₇

—

C₄×Dic₆₇

C₂×C₄

yes

—

C₄×Dic₆₇

D₄

yes

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

C₂×C₄

yes

—

C₂×Q₈×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂²×D₆₇

—

Q₈×C₆₇

—

Q₈×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

—

C₂²⋊D₆₇

—

C₂²⋊D₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

C₄×D₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₂²⋊C₄

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₄×D₄

yes

—

C₄×C₆₇

#121

yes

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#386

yes

characteristic

D₆₇

—

#1912

—

maximal, Sylow

C₆₇

#1912

yes

characteristic, complemented, Sylow

—

—

—

D₆₇

yes

characteristic, complemented

H(3,ℤ/4)

—

C₂×#41

—

C₂²×Q₈

C₂×D₆₇

yes

characteristic

C₂×M₁₆

C₂×D₆₇

yes

—

C₄⋊Q₈

—

#124

—

#123

—

134

—

C₂²⋊Q₈

—

134

—

C₄×Q₈

—

134

—

#117

—

#117

—

#117

—

134

—

#104

—

134

—

100

#101

—

101

#41

—

134

—

102

#41

—

134

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

C₄×D₆₇

yes

—

107

C₂×Q₈

C₂²⋊D₆₇

yes

—

108

C₂×D₄

—

109

C₂×D₄

—

134

—

110

C₂²×C₄

—

134

—

111

C₂²×C₄

D₂₆₈

yes

—

112

C₂²×C₄

C₂²×D₆₇

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

C₂×C₈

—

116

C₄⋊C₄

—

134

—

117

C₄⋊C₄

—

134

—

118

C₄⋊C₄

—

134

—

119

C₄⋊C₄

—

134

—

120

C₂²⋊C₄

—

134

—

121

C₂²⋊C₄

—

134

—

122

C₂²⋊C₄

—

134

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

134

—

126

C₂³

—

134

—

127

C₂³

C₂×D₂₆₈

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

134

—

131

D₄

—

268

—

132

D₄

—

268

—

133

D₄

—

134

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

268

—

137

C₂×C₄

—

134

—

138

C₂×C₄

—

134

—

139

C₂×C₄

—

134

—

140

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

—

141

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

142

C₂×C₄

D₄×D₆₇

yes

—

143

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

144

C₂×C₄

#11761627

yes

—

145

C₈

—

146

C₂²

—

147

C₂²

—

268

—

148

C₂²

—

268

—

149

C₂²

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

150

C₂²

#14524013

yes

characteristic

151

C₂²

#14524027

yes

characteristic

152

C₄

—

153

C₄

—

268

—

154

C₄

—

268

—

155

C₄

—

268

—

156

C₄

(C₂²⋊D₆₇)⋊C₄

yes

—

157

C₄

#14524015

yes

—

158

C₂

—

159

C₂

—

160

C₂

—

161

C₂

—

536

—

162

C₂

#16863273

yes

characteristic

	subgroup	G/H
center	C₂	#16863273
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×#101×C₆₇	C₂
socle	C₂×C₆₇	#386
hypercenter	C₂×#101	D₆₇
hypocenter	C₆₇	#1912
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂

#16863273

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂×#101×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
2. #16863874	C₂	semidirect
4. #16863153	C₂	semidirect
5. C₂×#101×C₆₇	C₂	semidirect
7. #14524041	C₄	semidirect
8. #14524032	C₄	semidirect
82. C₆₇	#1912	semidirect
86. C₂×#101	D₆₇	semidirect

H ≅ G/N

type

2. #16863874

C₂

semidirect

4. #16863153

C₂

semidirect

5. C₂×#101×C₆₇

C₂

semidirect

7. #14524041

C₄

semidirect

8. #14524032

C₄

semidirect

82. C₆₇

#1912

semidirect

86. C₂×#101

D₆₇

semidirect

Subgroups — #22560744

Conjugacy classes of subgroups

Complements