Group #22558049 — subgroups

Subgroups — #22558049

Order 8576.

208 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 109 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×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₆₇⋊C₈	—	2	1	no	—
5	C₂×C₄⋊Dic₆₇	C₄	1	2	yes	—
6	C₂×C₄×Dic₆₇	—	2	1	no	—
7	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
8	C₄⋊(C₆₇⋊C₈)	C₄	1	2	yes	complemented
9	C₆₇⋊#95	—	2	1	no	—
10	C₂²×C₈×C₆₇	—	2	1	no	—
11	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₆₇	—	2	1	no	—
13	C₄×C₈×C₆₇	C₄	1	2	yes	complemented
14	C₂²×Dic₆₇	—	2	2	no	—
15	C₂×C₆₇⋊C₈	—	2	2	no	—
16	C₂×C₆₇⋊C₈	—	4	1	no	—
17	C₂×C₆₇⋊C₈	—	4	1	no	—
18	C₄⋊Dic₆₇	—	2	2	no	—
19	C₄⋊Dic₆₇	C₈	1	4	yes	complemented
20	C₄×Dic₆₇	—	4	1	no	—
21	C₄×Dic₆₇	—	4	1	no	—
22	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
25	C₂×C₈×C₆₇	—	2	4	no	—
26	C₂×C₈×C₆₇	—	2	2	no	—
27	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₆₇	D₄	1	1	yes	characteristic
30	C₄²×C₆₇	Q₈	1	1	yes	characteristic
31	C₂³×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₆₇	—	2	2	no	—
33	C₂×C₄×C₆₇	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₆₇	C₂×C₈	1	2	yes	—
40	C₂×C₄×C₆₇	M₁₆	1	2	yes	—
41	C₂×C₄×C₆₇	D₈	1	1	yes	characteristic
42	C₂×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
43	C₂×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
44	C₂×C₄×C₆₇	Q₁₆	1	1	yes	characteristic
45	C₂×Dic₆₇	—	2	4	no	—
46	C₂×Dic₆₇	—	4	2	no	—
47	C₂×Dic₆₇	—	4	2	no	—
48	C₈×C₆₇	—	2	4	no	—
49	C₆₇⋊C₈	—	4	2	no	—
50	C₂²×C₆₇	#95	1	1	yes	characteristic
51	C₂²×C₆₇	C₂²⋊C₈	1	2	yes	—
52	C₂²×C₆₇	#102	1	1	yes	characteristic
53	C₂²×C₆₇	#103	1	1	yes	characteristic
54	C₂²×C₆₇	#104	1	2	yes	—
55	C₄×C₆₇	—	2	1	no	—
56	C₄×C₆₇	—	2	1	no	—
57	C₄×C₆₇	C₄×C₈	1	1	yes	characteristic
58	C₄×C₆₇	#97	1	1	yes	characteristic
59	C₄×C₆₇	#102	1	1	yes	characteristic
60	C₄×C₆₇	#103	1	1	yes	characteristic
61	C₄×C₆₇	C₄⋊C₈	1	2	yes	—
62	C₄×C₆₇	#106	1	1	yes	characteristic
63	C₄×C₆₇	#107	1	1	yes	characteristic
64	Dic₆₇	—	4	4	no	—
65	C₂×C₆₇	D₄⋊C₈	1	2	yes	—
66	C₂×C₆₇	Q₈⋊C₈	1	2	yes	—
67	C₂×C₆₇	#336	1	1	yes	characteristic
68	C₂×C₆₇	#339	1	1	yes	characteristic
69	C₂×C₆₇	#340	1	1	yes	characteristic
70	#1279	—	67	1	no	maximal, Sylow
71	C₆₇	#1279	1	1	yes	characteristic, complemented, Sylow
72	C₂×C₄⋊C₈	—	67	1	no	—
73	C₂×C₄×C₈	D₆₇	1	1	yes	characteristic
74	C₄×C₄⋊C₄	—	67	1	no	—
75	C₂²×C₈	—	2	1	no	—
76	C₂²×C₈	—	134	1	no	—
77	C₂×C₄⋊C₄	—	67	2	no	—
78	C₂×C₄²	—	134	1	no	—
79	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
80	C₄⋊C₈	—	67	2	no	—
81	C₄⋊C₈	—	134	1	no	—
82	C₄×C₈	—	2	1	no	—
83	C₄×C₈	Dic₆₇	1	2	yes	complemented
84	#95	—	134	1	no	—
85	C₂²×C₄	—	134	2	no	—
86	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
87	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
88	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
89	C₂×C₈	—	2	4	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	268	1	no	—
92	C₂×C₈	—	268	1	no	—
93	C₂×C₈	—	134	2	no	—
94	C₄⋊C₄	—	67	4	no	—
95	C₄⋊C₄	—	134	2	no	—
96	C₄²	—	268	1	no	—
97	C₄²	—	268	1	no	—
98	C₄²	Dic₁₃₄	1	1	yes	characteristic
99	C₄²	C₄×D₆₇	1	1	yes	characteristic
100	C₄²	C₂×Dic₆₇	1	1	yes	characteristic
101	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
102	C₂³	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	268	2	no	—
105	C₂×C₄	—	268	2	no	—
106	C₂×C₄	—	134	4	no	—
107	C₂×C₄	C₄×Dic₆₇	1	1	yes	characteristic
108	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
109	C₂×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
110	C₂×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
111	C₂×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
112	C₂×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
113	C₂×C₄	C₈×D₆₇	1	2	yes	—
114	C₂×C₄	#11761611	1	2	yes	—
115	C₂×C₄	D₅₃₆	1	1	yes	characteristic
116	C₂×C₄	#11761617	1	1	yes	characteristic
117	C₂×C₄	#11761617	1	1	yes	characteristic
118	C₂×C₄	Dic₂₆₈	1	1	yes	characteristic
119	C₈	—	2	4	no	—
120	C₈	—	268	2	no	—
121	C₂²	C₆₇⋊#95	1	1	yes	characteristic
122	C₂²	(C₂×D₆₇)⋊C₈	1	2	yes	—
123	C₂²	#14523965	1	1	yes	characteristic
124	C₂²	#14523968	1	1	yes	characteristic
125	C₂²	#14523971	1	2	yes	—
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	268	4	no	—
129	C₄	C₈×Dic₆₇	1	1	yes	characteristic
130	C₄	#14523953	1	1	yes	characteristic
131	C₄	#14523965	1	1	yes	characteristic
132	C₄	#14523968	1	1	yes	characteristic
133	C₄	Dic₆₇⋊C₈	1	2	yes	—
134	C₄	#14523975	1	1	yes	characteristic
135	C₄	#14523977	1	1	yes	characteristic
136	C₂	#16863125	1	2	yes	—
137	C₂	#16863128	1	2	yes	—
138	C₂	#16863150	1	1	yes	characteristic
139	C₂	#16863156	1	1	yes	characteristic
140	C₂	#16863159	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₆₇⋊C₈)

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₆₇

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

—

C₂×C₄⋊Dic₆₇

C₄

yes

—

C₂×C₄×Dic₆₇

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

C₄

yes

complemented

C₆₇⋊#95

—

C₂²×C₈×C₆₇

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

C₄

yes

complemented

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₈

yes

complemented

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

Q₈

yes

characteristic

C₂³×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄²

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×C₈

yes

—

C₂×C₄×C₆₇

M₁₆

yes

—

C₂×C₄×C₆₇

D₈

yes

characteristic

C₂×C₄×C₆₇

SD₁₆

yes

characteristic

C₂×C₄×C₆₇

SD₁₆

yes

characteristic

C₂×C₄×C₆₇

Q₁₆

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

#95

yes

characteristic

C₂²×C₆₇

C₂²⋊C₈

yes

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₄×C₈

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

C₄⋊C₈

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

C₂×C₆₇

D₄⋊C₈

yes

—

C₂×C₆₇

Q₈⋊C₈

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄×C₈

D₆₇

yes

characteristic

C₄×C₄⋊C₄

—

C₂²×C₈

—

C₂²×C₈

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

134

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₄×C₈

—

C₄×C₈

Dic₆₇

yes

complemented

#95

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

C₄×D₆₇

yes

characteristic

C₂²×C₄

D₂₆₈

yes

characteristic

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

268

—

C₂×C₈

—

268

—

C₂×C₈

—

134

—

C₄⋊C₄

—

C₄⋊C₄

—

134

—

C₄²

—

268

—

C₄²

—

268

—

C₄²

Dic₁₃₄

yes

characteristic

C₄²

C₄×D₆₇

yes

characteristic

100

C₄²

C₂×Dic₆₇

yes

characteristic

101

C₄²

C₂²⋊D₆₇

yes

characteristic

102

C₂³

(C₂×D₆₇)⋊C₄

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

268

—

105

C₂×C₄

—

268

—

106

C₂×C₄

—

134

—

107

C₂×C₄

C₄×Dic₆₇

yes

characteristic

108

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

109

C₂×C₄

C₂²⋊Dic₆₇

yes

characteristic

110

C₂×C₄

Dic₆₇⋊C₄

yes

characteristic

111

C₂×C₄

Dic₆₇⋊C₄

yes

characteristic

112

C₂×C₄

C₄⋊Dic₆₇

yes

characteristic

113

C₂×C₄

C₈×D₆₇

yes

—

114

C₂×C₄

#11761611

yes

—

115

C₂×C₄

D₅₃₆

yes

characteristic

116

C₂×C₄

#11761617

yes

characteristic

117

C₂×C₄

#11761617

yes

characteristic

118

C₂×C₄

Dic₂₆₈

yes

characteristic

119

C₈

—

120

C₈

—

268

—

121

C₂²

C₆₇⋊#95

yes

characteristic

122

C₂²

(C₂×D₆₇)⋊C₈

yes

—

123

C₂²

#14523965

yes

characteristic

124

C₂²

#14523968

yes

characteristic

125

C₂²

#14523971

yes

—

126

C₄

—

127

C₄

—

128

C₄

—

268

—

129

C₄

C₈×Dic₆₇

yes

characteristic

130

C₄

#14523953

yes

characteristic

131

C₄

#14523965

yes

characteristic

132

C₄

#14523968

yes

characteristic

133

C₄

Dic₆₇⋊C₈

yes

—

134

C₄

#14523975

yes

characteristic

135

C₄

#14523977

yes

characteristic

136

C₂

#16863125

yes

—

137

C₂

#16863128

yes

—

138

C₂

#16863150

yes

characteristic

139

C₂

#16863156

yes

characteristic

140

C₂

#16863159

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	D₂₆₈
commutator	C₄×C₆₇	C₄×C₈
Frattini	C₂×C₄²	C₂×D₆₇
Fitting	C₂×C₄×C₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂²⋊C₄
hypercenter	C₂×C₄×C₈	D₆₇
hypocenter	C₆₇	#1279
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₄

D₂₆₈

commutator

C₄×C₆₇

C₄×C₈

Frattini

C₂×C₄²

C₂×D₆₇

Fitting

C₂×C₄×C₈×C₆₇

C₂

socle

C₂³×C₆₇

C₂²⋊C₄

hypercenter

C₂×C₄×C₈

D₆₇

hypocenter

C₆₇

#1279

LCS

G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
8. C₄⋊(C₆₇⋊C₈)	C₄	semidirect
13. C₄×C₈×C₆₇	C₄	semidirect
19. C₄⋊Dic₆₇	C₈	semidirect
71. C₆₇	#1279	semidirect
83. C₄×C₈	Dic₆₇	semidirect

H ≅ G/N

type

8. C₄⋊(C₆₇⋊C₈)

C₄

semidirect

13. C₄×C₈×C₆₇

C₄

semidirect

19. C₄⋊Dic₆₇

C₈

semidirect

71. C₆₇

#1279

semidirect

83. C₄×C₈

Dic₆₇

semidirect

Subgroups — #22558049

Conjugacy classes of subgroups

Complements