Group #22560310 — subgroups

Subgroups — #22560310

Order 8576.

262 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 81 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	#41×Dic₆₇	C₂	1	1	yes	maximal, characteristic
2	#16863380	C₂	1	1	yes	maximal, characteristic
3	#16863155	C₂	1	2	yes	maximal, complemented
4	#16863152	C₂	1	2	yes	maximal, complemented
5	#410×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#11761612	C₂²	1	2	yes	—
7	Q₈×Dic₆₇	—	2	1	no	—
8	D₄×Dic₆₇	—	2	1	no	—
9	D₄×Dic₆₇	—	2	2	no	—
10	#14524002	C₄	1	2	yes	complemented
11	#14524002	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₆₇	C₄	1	2	yes	complemented
13	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
14	#14523962	—	2	1	no	—
15	#14523960	—	2	1	no	—
16	C₂×#41×C₆₇	C₂²	1	1	yes	characteristic
17	#117×C₆₇	C₂²	1	2	yes	—
18	#99×C₆₇	—	2	2	no	—
19	C₂²×Dic₆₇	—	2	2	no	—
20	C₂²×Dic₆₇	—	2	1	no	—
21	#11761612	—	2	2	no	—
22	#11761612	—	2	2	no	—
23	C₂×C₆₇⋊C₈	—	2	2	no	—
24	C₄⋊Dic₆₇	—	2	2	no	—
25	C₄⋊Dic₆₇	—	2	1	no	—
26	C₂²⋊Dic₆₇	—	2	2	no	—
27	C₂²⋊Dic₆₇	—	2	1	no	—
28	C₄×Dic₆₇	—	2	2	no	—
29	C₄×Dic₆₇	—	2	1	no	—
30	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₆₇	C₂×C₄	1	2	yes	—
32	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
33	#41×C₆₇	—	2	2	no	—
34	#41×C₆₇	—	2	2	no	—
35	C₂×Q₈×C₆₇	D₄	1	1	yes	characteristic
36	C₂×D₄×C₆₇	D₄	1	1	yes	characteristic
37	C₂×D₄×C₆₇	D₄	1	2	yes	—
38	C₂²×C₄×C₆₇	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₆₇	—	2	2	no	—
41	C₂²⋊C₄×C₆₇	—	2	2	no	—
42	C₄²×C₆₇	—	2	2	no	—
43	C₂³×C₆₇	C₂²⋊C₄	1	2	yes	—
44	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
45	Q₈×C₆₇	—	2	2	no	—
46	D₄×C₆₇	—	2	4	no	—
47	D₄×C₆₇	—	2	2	no	—
48	C₂×C₄×C₆₇	—	2	2	no	—
49	C₂×C₄×C₆₇	—	2	2	no	—
50	C₂×C₄×C₆₇	—	4	2	no	—
51	C₂×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₆₇	#41	1	2	yes	—
55	C₂×C₄×C₆₇	#41	1	2	yes	—
56	C₂×Dic₆₇	—	2	1	no	—
57	C₂×Dic₆₇	—	2	2	no	—
58	C₂×Dic₆₇	—	2	1	no	—
59	C₂×Dic₆₇	—	4	2	no	—
60	C₂×Dic₆₇	—	4	4	no	—
61	C₆₇⋊C₈	—	4	2	no	—
62	C₂²×C₆₇	—	2	2	no	—
63	C₂²×C₆₇	—	4	1	no	—
64	C₂²×C₆₇	—	4	2	no	—
65	C₂²×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₆₇	#123	1	2	yes	—
67	C₄×C₆₇	—	2	2	no	—
68	C₄×C₆₇	—	8	2	no	—
69	C₄×C₆₇	#117	1	1	yes	characteristic
70	C₄×C₆₇	#117	1	1	yes	characteristic
71	C₄×C₆₇	#123	1	2	yes	—
72	Dic₆₇	—	2	2	no	—
73	Dic₆₇	—	4	2	no	—
74	Dic₆₇	—	4	1	no	—
75	C₂×C₆₇	—	2	1	no	—
76	C₂×C₆₇	—	2	2	no	—
77	C₂×C₆₇	—	4	2	no	—
78	C₂×C₆₇	#381	1	1	yes	characteristic
79	#1827	—	67	1	no	maximal, Sylow
80	C₆₇	#1827	1	1	yes	characteristic, complemented, Sylow
81	C₄×#41	—	67	1	no	—
82	#413	—	67	1	no	—
83	#410	D₆₇	1	1	yes	characteristic
84	#339	—	67	2	no	—
85	H(3,ℤ/4)	—	67	2	no	—
86	C₂×#41	C₂×D₆₇	1	1	yes	characteristic
87	C₂×M₁₆	—	67	2	no	—
88	C₄×Q₈	—	134	1	no	—
89	C₄×D₄	—	134	1	no	—
90	C₄×D₄	—	134	2	no	—
91	#117	—	67	2	no	—
92	#117	—	67	1	no	—
93	#117	C₂×D₆₇	1	2	yes	—
94	C₂×C₄²	—	67	2	no	—
95	C₂×C₄²	—	67	1	no	—
96	#101	—	134	1	no	—
97	#100	—	134	1	no	—
98	#99	—	2	2	no	—
99	#41	—	2	2	no	—
100	#41	—	2	2	no	—
101	C₂×Q₈	D₂₆₈	1	1	yes	characteristic
102	C₂×D₄	D₂₆₈	1	1	yes	characteristic
103	C₂×D₄	C₂²⋊D₆₇	1	2	yes	—
104	C₂²×C₄	—	134	2	no	—
105	C₂²×C₄	—	134	1	no	—
106	C₂²×C₄	C₄×D₆₇	1	2	yes	—
107	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
108	M₁₆	—	134	2	no	—
109	M₁₆	—	134	2	no	—
110	C₂×C₈	—	134	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	134	2	no	—
113	C₄⋊C₄	—	134	1	no	—
114	C₂²⋊C₄	—	2	2	no	—
115	C₂²⋊C₄	—	134	2	no	—
116	C₂²⋊C₄	—	134	1	no	—
117	C₄²	—	2	2	no	—
118	C₄²	—	67	1	no	—
119	C₄²	—	67	2	no	—
120	C₄²	—	67	1	no	—
121	C₄²	—	134	2	no	—
122	C₄²	—	134	1	no	—
123	C₂³	(C₂×D₆₇)⋊C₄	1	2	yes	—
124	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	D₄	—	2	4	no	—
127	D₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	268	2	no	—
132	C₂×C₄	—	268	4	no	—
133	C₂×C₄	—	134	2	no	—
134	C₂×C₄	—	134	1	no	—
135	C₂×C₄	—	134	1	no	—
136	C₂×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
138	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
139	C₂×C₄	#11761627	1	2	yes	—
140	C₂×C₄	#11761627	1	2	yes	—
141	C₈	—	268	2	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	4	1	no	—
144	C₂²	—	4	2	no	—
145	C₂²	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
146	C₂²	#14524024	1	2	yes	—
147	C₄	—	2	2	no	—
148	C₄	—	8	2	no	—
149	C₄	—	268	1	no	—
150	C₄	—	268	2	no	—
151	C₄	—	134	2	no	—
152	C₄	#14523999	1	1	yes	characteristic
153	C₄	#14523999	1	1	yes	characteristic
154	C₄	#14524027	1	2	yes	—
155	C₂	—	2	2	no	—
156	C₂	—	2	1	no	—
157	C₂	—	4	2	no	—
158	C₂	#16863257	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#41×Dic₆₇

C₂

yes

maximal, characteristic

#16863380

C₂

yes

maximal, characteristic

#16863155

C₂

yes

maximal, complemented

#16863152

C₂

yes

maximal, complemented

#410×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

C₂²

yes

—

Q₈×Dic₆₇

—

D₄×Dic₆₇

—

D₄×Dic₆₇

—

#14524002

C₄

yes

complemented

#14524002

C₂²

yes

characteristic

C₂×C₄×Dic₆₇

C₄

yes

complemented

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

#14523962

—

#14523960

—

C₂×#41×C₆₇

C₂²

yes

characteristic

#117×C₆₇

C₂²

yes

—

#99×C₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

C₂×C₄

yes

characteristic

C₄×Dic₆₇

C₂×C₄

yes

—

C₄×Dic₆₇

C₂×C₄

yes

characteristic

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

D₄

yes

characteristic

C₂×D₄×C₆₇

D₄

yes

characteristic

C₂×D₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

C₂²⋊C₄

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×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

characteristic

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₇

#123

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₄×#41

—

#413

—

#410

D₆₇

yes

characteristic

#339

—

H(3,ℤ/4)

—

C₂×#41

C₂×D₆₇

yes

characteristic

C₂×M₁₆

—

C₄×Q₈

—

134

—

C₄×D₄

—

134

—

C₄×D₄

—

134

—

#117

—

#117

—

#117

C₂×D₆₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

#101

—

134

—

#100

—

134

—

#99

—

#41

—

100

#41

—

101

C₂×Q₈

D₂₆₈

yes

characteristic

102

C₂×D₄

D₂₆₈

yes

characteristic

103

C₂×D₄

C₂²⋊D₆₇

yes

—

104

C₂²×C₄

—

134

—

105

C₂²×C₄

—

134

—

106

C₂²×C₄

C₄×D₆₇

yes

—

107

C₂²×C₄

C₂²×D₆₇

yes

characteristic

108

M₁₆

—

134

—

109

M₁₆

—

134

—

110

C₂×C₈

—

134

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

134

—

113

C₄⋊C₄

—

134

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

134

—

116

C₂²⋊C₄

—

134

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

134

—

122

C₄²

—

134

—

123

C₂³

(C₂×D₆₇)⋊C₄

yes

—

124

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

268

—

132

C₂×C₄

—

268

—

133

C₂×C₄

—

134

—

134

C₂×C₄

—

134

—

135

C₂×C₄

—

134

—

136

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

—

137

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

138

C₂×C₄

C₂×D₂₆₈

yes

characteristic

139

C₂×C₄

#11761627

yes

—

140

C₂×C₄

#11761627

yes

—

141

C₈

—

268

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

146

C₂²

#14524024

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

268

—

150

C₄

—

268

—

151

C₄

—

134

—

152

C₄

#14523999

yes

characteristic

153

C₄

#14523999

yes

characteristic

154

C₄

#14524027

yes

—

155

C₂

—

156

C₂

—

157

C₂

—

158

C₂

#16863257

yes

characteristic

	subgroup	G/H
center	C₄	#14523999
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#410×C₆₇	C₂
socle	C₂×C₆₇	#381
hypercenter	#410	D₆₇
hypocenter	C₆₇	#1827
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

subgroup

G/H

center

C₄

#14523999

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#410×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

N	H ≅ G/N	type
3. #16863155	C₂	semidirect
4. #16863152	C₂	semidirect
10. #14524002	C₄	semidirect
12. C₂×C₄×Dic₆₇	C₄	semidirect
80. C₆₇	#1827	semidirect

H ≅ G/N

type

3. #16863155

C₂

semidirect

4. #16863152

C₂

semidirect

10. #14524002

C₄

semidirect

12. C₂×C₄×Dic₆₇

C₄

semidirect

80. C₆₇

#1827

semidirect

Subgroups — #22560310

Conjugacy classes of subgroups

Complements