Group #22560956 — subgroups

Subgroups — #22560956

Order 8576.

208 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 83 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×#14523978	C₂	1	1	yes	maximal, characteristic
2	#16863432	C₂	1	1	yes	maximal, characteristic
3	#16863424	C₂	1	1	yes	maximal, characteristic
4	#16863423	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#16863156	C₂	1	1	yes	maximal, characteristic
7	#339×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#11761612	C₂²	1	1	yes	characteristic
9	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
11	#14524002	C₂²	1	1	yes	characteristic
12	#14523978	—	2	2	no	—
13	#14523976	C₄	1	2	yes	complemented
14	#14523974	C₄	1	2	yes	complemented
15	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
16	Dic₆₇⋊C₈	—	2	2	no	—
17	C₄×C₆₇⋊C₈	—	2	2	no	—
18	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	#117×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	#11761612	—	2	1	no	—
22	#11761612	—	2	1	no	—
23	C₂×C₆₇⋊C₈	—	2	1	no	—
24	C₂×C₆₇⋊C₈	—	2	1	no	—
25	C₂×C₆₇⋊C₈	—	2	1	no	—
26	C₂×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
29	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
30	C₂×C₆₇⋊C₈	Q₈	1	2	yes	—
31	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
32	C₂²⋊Dic₆₇	—	2	1	no	—
33	C₄×Dic₆₇	—	2	1	no	—
34	C₂²×C₄×C₆₇	—	2	1	no	—
35	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	M₁₆×C₆₇	—	2	1	no	—
37	M₁₆×C₆₇	—	2	1	no	—
38	C₂×C₈×C₆₇	—	2	1	no	—
39	C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
40	C₂²⋊C₄×C₆₇	—	2	1	no	—
41	C₄²×C₆₇	—	2	1	no	—
42	C₄²×C₆₇	—	2	1	no	—
43	C₄²×C₆₇	Q₈	1	2	yes	—
44	C₂³×C₆₇	#41	1	1	yes	characteristic
45	C₂×C₄×C₆₇	—	2	4	no	—
46	C₂×C₄×C₆₇	—	2	2	no	—
47	C₂×C₄×C₆₇	—	2	2	no	—
48	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
54	C₂×Dic₆₇	—	2	2	no	—
55	C₈×C₆₇	—	4	1	no	—
56	C₆₇⋊C₈	—	2	2	no	—
57	C₆₇⋊C₈	—	4	1	no	—
58	C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
59	C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
60	C₂²×C₆₇	—	2	2	no	—
61	C₂²×C₆₇	C₄×D₄	1	1	yes	characteristic
62	C₂²×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₆₇	#125	1	1	yes	characteristic
64	C₄×C₆₇	—	2	4	no	—
65	C₄×C₆₇	—	4	2	no	—
66	C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
68	C₄×C₆₇	#121	1	1	yes	characteristic
69	C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
70	Dic₆₇	—	4	2	no	—
71	C₂×C₆₇	—	2	1	no	—
72	C₂×C₆₇	—	2	1	no	—
73	C₂×C₆₇	#389	1	1	yes	characteristic
74	C₂×C₆₇	#443	1	2	yes	—
75	#1945	—	67	1	no	maximal, Sylow
76	C₆₇	#1945	1	1	yes	characteristic, complemented, Sylow
77	C₂×#108	—	67	1	no	—
78	#427	—	67	1	no	—
79	#424	—	67	1	no	—
80	#424	—	67	1	no	—
81	C₂×C₄×C₈	—	67	1	no	—
82	#339	—	67	1	no	—
83	#339	D₆₇	1	1	yes	characteristic
84	C₂×M₁₆	—	67	1	no	—
85	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
86	C₂²×C₈	—	67	1	no	—
87	C₂²×C₈	—	67	1	no	—
88	#117	—	67	1	no	—
89	#117	C₂×D₆₇	1	1	yes	characteristic
90	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
91	#108	—	134	2	no	—
92	#107	—	67	2	no	—
93	#106	—	67	2	no	—
94	C₄⋊C₈	—	134	2	no	—
95	C₄⋊C₈	—	134	2	no	—
96	C₄×C₈	—	134	2	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
99	M₁₆	—	2	1	no	—
100	M₁₆	—	2	1	no	—
101	M₁₆	—	134	1	no	—
102	M₁₆	—	134	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	67	1	no	—
105	C₂×C₈	—	67	1	no	—
106	C₂×C₈	—	67	2	no	—
107	C₂×C₈	—	67	2	no	—
108	C₂×C₈	—	67	2	no	—
109	C₂×C₈	—	134	1	no	—
110	C₂×C₈	—	134	1	no	—
111	C₂×C₈	—	134	1	no	—
112	C₄⋊C₄	—	67	2	no	—
113	C₄⋊C₄	C₄×D₆₇	1	2	yes	—
114	C₂²⋊C₄	—	2	1	no	—
115	C₂²⋊C₄	—	134	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	134	1	no	—
119	C₄²	Dic₁₃₄	1	2	yes	—
120	C₂³	#11761627	1	1	yes	characteristic
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	134	2	no	—
125	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
126	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
127	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₆₇	1	1	yes	characteristic
129	C₂×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
130	C₂×C₄	#11761628	1	1	yes	characteristic
131	C₈	—	4	1	no	—
132	C₈	—	268	1	no	—
133	C₈	—	67	2	no	—
134	C₈	—	67	2	no	—
135	C₈	—	134	2	no	—
136	C₂²	—	2	2	no	—
137	C₂²	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
138	C₂²	#14524020	1	1	yes	characteristic
139	C₂²	#14524033	1	1	yes	characteristic
140	C₄	—	2	4	no	—
141	C₄	—	4	2	no	—
142	C₄	—	268	2	no	—
143	C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
144	C₄	C₄×Dic₁₃₄	1	1	yes	characteristic
145	C₄	#14524016	1	1	yes	characteristic
146	C₄	#14524042	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#16863293	1	1	yes	characteristic
150	C₂	#16863510	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523978

C₂

yes

maximal, characteristic

#16863432

C₂

yes

maximal, characteristic

#16863424

C₂

yes

maximal, characteristic

#16863423

C₂

yes

maximal, characteristic

C₂×C₄×C₆₇⋊C₈

C₂

yes

maximal, characteristic

#16863156

C₂

yes

maximal, characteristic

#339×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

#14524002

C₂²

yes

characteristic

#14523978

—

#14523976

C₄

yes

complemented

#14523974

C₄

yes

complemented

C₄⋊(C₆₇⋊C₈)

—

Dic₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

#117×C₆₇

C₂²

yes

characteristic

C₂×C₄²×C₆₇

C₂²

yes

characteristic

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₈

D₄

yes

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₂×C₆₇⋊C₈

Q₈

yes

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

Q₈

yes

—

C₂³×C₆₇

#41

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

C₄⋊C₄

yes

—

C₆₇⋊C₈

C₄⋊C₄

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄×D₄

yes

characteristic

C₂²×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₆₇

#125

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₇

C₄×Q₈

yes

characteristic

C₄×C₆₇

#121

yes

characteristic

C₄×C₆₇

C₄⋊Q₈

yes

characteristic

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#389

yes

characteristic

C₂×C₆₇

#443

yes

—

#1945

—

maximal, Sylow

C₆₇

#1945

yes

characteristic, complemented, Sylow

—

—

—

—

C₂×C₄×C₈

—

#339

—

#339

D₆₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#117

—

#117

C₂×D₆₇

yes

characteristic

C₂×C₄²

C₂×D₆₇

yes

characteristic

#108

—

134

—

#107

—

#106

—

C₄⋊C₈

—

134

—

C₄⋊C₈

—

134

—

C₄×C₈

—

134

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

M₁₆

—

100

M₁₆

—

101

M₁₆

—

134

—

102

M₁₆

—

134

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

134

—

110

C₂×C₈

—

134

—

111

C₂×C₈

—

134

—

112

C₄⋊C₄

—

113

C₄⋊C₄

C₄×D₆₇

yes

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

134

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

134

—

119

C₄²

Dic₁₃₄

yes

—

120

C₂³

#11761627

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

134

—

125

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

126

C₂×C₄

D₄×D₆₇

yes

characteristic

127

C₂×C₄

D₄×D₆₇

yes

characteristic

128

C₂×C₄

Q₈×D₆₇

yes

characteristic

129

C₂×C₄

C₂×Dic₁₃₄

yes

characteristic

130

C₂×C₄

#11761628

yes

characteristic

131

C₈

—

132

C₈

—

268

—

133

C₈

—

134

C₈

—

135

C₈

—

134

—

136

C₂²

—

137

C₂²

(C₂²⋊D₆₇)⋊C₄

yes

characteristic

138

C₂²

#14524020

yes

characteristic

139

C₂²

#14524033

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

268

—

143

C₄

C₄⋊C₄×D₆₇

yes

characteristic

144

C₄

C₄×Dic₁₃₄

yes

characteristic

145

C₄

#14524016

yes

characteristic

146

C₄

#14524042

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

#16863293

yes

characteristic

150

C₂

#16863510

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#339×C₆₇	C₂
socle	C₂²×C₆₇	C₄×D₄
hypercenter	#339	D₆₇
hypocenter	C₆₇	#1945
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

subgroup

G/H

center

C₂×C₄

D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#339×C₆₇

C₂

socle

C₂²×C₆₇

C₄×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

N	H ≅ G/N	type
13. #14523976	C₄	semidirect
14. #14523974	C₄	semidirect
76. C₆₇	#1945	semidirect

H ≅ G/N

type

13. #14523976

C₄

semidirect

14. #14523974

C₄

semidirect

76. C₆₇

#1945

semidirect

Subgroups — #22560956

Conjugacy classes of subgroups

Complements