Group #22560109 — subgroups

Subgroups — #22560109

Order 8576.

272 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14523977	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523953	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×Dic₆₇	C₂	1	2	yes	maximal
4	#16863158	C₂	1	2	yes	maximal
5	C₂×#106×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
8	C₂×C₄×Dic₆₇	—	2	2	no	—
9	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
10	#14523977	C₄	1	4	yes	complemented
11	#14523953	C₄	1	4	yes	complemented
12	C₆₇⋊#95	—	2	2	no	—
13	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
15	#106×C₆₇	C₄	1	4	yes	complemented
16	C₂²×Dic₆₇	—	2	2	no	—
17	C₂²×Dic₆₇	D₄	1	2	yes	—
18	C₂×C₆₇⋊C₈	—	2	1	no	—
19	C₂×C₆₇⋊C₈	—	2	1	no	—
20	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
21	C₄⋊Dic₆₇	—	2	2	no	—
22	C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
23	C₄×Dic₆₇	—	4	4	no	—
24	C₄×Dic₆₇	C₂×C₄	1	2	yes	—
25	C₄×Dic₆₇	Q₈	1	2	yes	—
26	C₂²×C₄×C₆₇	—	2	2	no	—
27	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
31	C₄⋊C₄×C₆₇	—	2	2	no	—
32	C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
33	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₆₇	—	2	4	no	—
35	C₂×C₄×C₆₇	—	4	2	no	—
36	C₂×C₄×C₆₇	—	4	2	no	—
37	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
42	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
43	C₂×Dic₆₇	—	2	2	no	—
44	C₂×Dic₆₇	—	2	2	no	—
45	C₂×Dic₆₇	—	2	4	no	—
46	C₂×Dic₆₇	—	4	2	no	—
47	C₂×Dic₆₇	—	4	2	no	—
48	C₂×Dic₆₇	C₄⋊C₄	1	4	yes	—
49	C₈×C₆₇	C₄²	1	4	yes	complemented
50	C₆₇⋊C₈	—	2	2	no	—
51	C₂²×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₆₇	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₆₇	C₄×D₄	1	1	yes	characteristic
54	C₂²×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
55	C₂²×C₆₇	#137	1	2	yes	—
56	C₄×C₆₇	—	4	4	no	—
57	C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
58	C₄×C₆₇	#117	1	1	yes	characteristic
59	C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
60	C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
61	Dic₆₇	—	2	4	no	—
62	Dic₆₇	—	4	4	no	—
63	C₂×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₆₇	#428	1	2	yes	—
65	C₂×C₆₇	#440	1	2	yes	—
66	C₂×C₆₇	#441	1	1	yes	characteristic
67	C₂×C₆₇	#442	1	1	yes	characteristic
68	#1776	—	67	1	no	maximal, Sylow
69	C₆₇	#1776	1	1	yes	characteristic, complemented, Sylow
70	C₂×#107	—	67	1	no	—
71	C₂×#106	D₆₇	1	1	yes	characteristic
72	C₂×#97	—	67	1	no	—
73	C₄×C₄⋊C₄	—	67	2	no	—
74	#340	—	67	2	no	—
75	C₂²×C₈	—	67	1	no	—
76	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
77	C₂×C₄⋊C₄	—	67	2	no	—
78	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
79	C₂×C₄²	—	67	1	no	—
80	C₂×C₄²	—	134	2	no	—
81	#107	—	67	4	no	—
82	#106	Dic₆₇	1	4	yes	complemented
83	#97	—	67	4	no	—
84	#95	—	134	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	67	2	no	—
87	C₂²×C₄	—	134	2	no	—
88	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
89	C₂×C₈	—	67	2	no	—
90	C₂×C₈	—	134	1	no	—
91	C₂×C₈	—	134	1	no	—
92	C₂×C₈	C₄×D₆₇	1	2	yes	—
93	C₂×C₈	C₄×D₆₇	1	2	yes	—
94	C₂×C₈	C₂×Dic₆₇	1	2	yes	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	67	4	no	—
97	C₄⋊C₄	—	134	2	no	—
98	C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
99	C₄²	—	268	4	no	—
100	C₄²	—	67	2	no	—
101	C₄²	—	67	2	no	—
102	C₂³	D₄×D₆₇	1	1	yes	characteristic
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	268	2	no	—
107	C₂×C₄	—	268	2	no	—
108	C₂×C₄	—	67	4	no	—
109	C₂×C₄	—	134	4	no	—
110	C₂×C₄	—	134	2	no	—
111	C₂×C₄	—	134	2	no	—
112	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
113	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
114	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
115	C₂×C₄	Q₈×D₆₇	1	1	yes	characteristic
116	C₂×C₄	#11761627	1	1	yes	characteristic
117	C₂×C₄	#11761629	1	1	yes	characteristic
118	C₈	—	134	2	no	—
119	C₈	C₄×Dic₆₇	1	4	yes	complemented
120	C₂²	C₄⋊C₄×D₆₇	1	1	yes	characteristic
121	C₂²	#14524005	1	1	yes	characteristic
122	C₂²	D₄×Dic₆₇	1	1	yes	characteristic
123	C₂²	#14524070	1	2	yes	—
124	C₂²	#14524072	1	2	yes	—
125	C₄	—	4	4	no	—
126	C₄	—	268	4	no	—
127	C₄	—	134	4	no	—
128	C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
129	C₄	#14524001	1	1	yes	characteristic
130	C₄	#14524008	1	1	yes	characteristic
131	C₄	Q₈×Dic₆₇	1	1	yes	characteristic
132	C₂	C₄⋊C₄×Dic₆₇	1	1	yes	characteristic
133	C₂	#16863435	1	2	yes	—
134	C₂	#16863497	1	2	yes	—
135	C₂	#16863502	1	1	yes	characteristic
136	C₂	#16863507	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523977

C₂

yes

maximal, characteristic

C₂×#14523953

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₆₇

C₂

yes

maximal

#16863158

C₂

yes

maximal

C₂×#106×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₄×Dic₆₇

—

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

#14523977

C₄

yes

complemented

#14523953

C₄

yes

complemented

C₆₇⋊#95

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

#106×C₆₇

C₄

yes

complemented

C₂²×Dic₆₇

—

C₂²×Dic₆₇

D₄

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

C₄×Dic₆₇

—

C₄×Dic₆₇

C₂×C₄

yes

—

C₄×Dic₆₇

Q₈

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₄⋊C₄

yes

—

C₈×C₆₇

C₄²

yes

complemented

C₆₇⋊C₈

—

C₂²×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₆₇

C₄×D₄

yes

characteristic

C₂²×C₆₇

C₄×D₄

yes

characteristic

C₂²×C₆₇

AGL(1,ℤ/8)

yes

—

C₂²×C₆₇

#137

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₂×C₄²

yes

characteristic

C₄×C₆₇

#117

yes

characteristic

C₄×C₆₇

C₄×Q₈

yes

characteristic

C₄×C₆₇

C₄×Q₈

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₆₇

#428

yes

—

C₂×C₆₇

#440

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×#107

—

C₂×#106

D₆₇

yes

characteristic

C₂×#97

—

C₄×C₄⋊C₄

—

#340

—

C₂²×C₈

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

134

—

#107

—

#106

Dic₆₇

yes

complemented

#97

—

#95

—

134

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

134

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₂×C₈

C₄×D₆₇

yes

—

C₂×C₈

C₄×D₆₇

yes

—

C₂×C₈

C₂×Dic₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

134

—

C₄⋊C₄

C₂×Dic₆₇

yes

—

C₄²

—

268

—

100

C₄²

—

101

C₄²

—

102

C₂³

D₄×D₆₇

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

268

—

107

C₂×C₄

—

268

—

108

C₂×C₄

—

109

C₂×C₄

—

134

—

110

C₂×C₄

—

134

—

111

C₂×C₄

—

134

—

112

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

113

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

114

C₂×C₄

C₂²×Dic₆₇

yes

characteristic

115

C₂×C₄

Q₈×D₆₇

yes

characteristic

116

C₂×C₄

#11761627

yes

characteristic

117

C₂×C₄

#11761629

yes

characteristic

118

C₈

—

134

—

119

C₈

C₄×Dic₆₇

yes

complemented

120

C₂²

C₄⋊C₄×D₆₇

yes

characteristic

121

C₂²

#14524005

yes

characteristic

122

C₂²

D₄×Dic₆₇

yes

characteristic

123

C₂²

#14524070

yes

—

124

C₂²

#14524072

yes

—

125

C₄

—

126

C₄

—

268

—

127

C₄

—

134

—

128

C₄

C₂×C₄×Dic₆₇

yes

characteristic

129

C₄

#14524001

yes

characteristic

130

C₄

#14524008

yes

characteristic

131

C₄

Q₈×Dic₆₇

yes

characteristic

132

C₂

C₄⋊C₄×Dic₆₇

yes

characteristic

133

C₂

#16863435

yes

—

134

C₂

#16863497

yes

—

135

C₂

#16863502

yes

characteristic

136

C₂

#16863507

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₆₇
commutator	C₄×C₆₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×#106×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂×#106	D₆₇
hypocenter	C₆₇	#1776
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₆₇

commutator

C₄×C₆₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂×#106×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

D₆₇

C₆₇

G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
10. #14523977	C₄	semidirect
11. #14523953	C₄	semidirect
15. #106×C₆₇	C₄	semidirect
49. C₈×C₆₇	C₄²	semidirect
69. C₆₇	#1776	semidirect
82. #106	Dic₆₇	semidirect
119. C₈	C₄×Dic₆₇	semidirect

H ≅ G/N

type

10. #14523977

C₄

semidirect

11. #14523953

C₄

semidirect

15. #106×C₆₇

C₄

semidirect

49. C₈×C₆₇

C₄²

semidirect

69. C₆₇

#1776

semidirect

82. #106

Dic₆₇

semidirect

119. C₈

C₄×Dic₆₇

semidirect

Subgroups — #22560109

Conjugacy classes of subgroups

Complements