Group #22560377 — subgroups

Subgroups — #22560377

Order 8576.

256 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 141 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	2	yes	maximal
2	C₂×#14523975	C₂	1	2	yes	maximal
3	#16863266	C₂	1	1	yes	maximal, characteristic
4	#16863262	C₂	1	1	yes	maximal, characteristic
5	C₂×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₆₇	—	2	2	no	—
7	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
10	#14523977	—	2	4	no	—
11	#14523975	—	2	4	no	—
12	C₆₇⋊#95	—	2	2	no	—
13	C₂²×C₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	#97×C₆₇	C₄	1	4	yes	complemented
16	C₂²×Dic₆₇	—	2	1	no	—
17	C₂²×Dic₆₇	—	2	1	no	—
18	C₂²×Dic₆₇	—	2	2	no	—
19	C₄⋊Dic₆₇	—	2	2	no	—
20	C₄⋊Dic₆₇	—	2	2	no	—
21	C₄⋊Dic₆₇	—	2	2	no	—
22	C₄⋊Dic₆₇	—	2	2	no	—
23	C₄⋊Dic₆₇	—	4	4	no	—
24	C₂²×C₄×C₆₇	D₄	1	2	yes	—
25	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₆₇	C₂×C₄	1	4	yes	—
27	C₂×C₈×C₆₇	D₄	1	4	yes	—
28	C₂×C₈×C₆₇	Q₈	1	4	yes	—
29	C₄²×C₆₇	C₂×C₄	1	2	yes	—
30	C₄²×C₆₇	Q₈	1	2	yes	—
31	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₆₇	—	2	2	no	—
33	C₂×C₄×C₆₇	—	2	2	no	—
34	C₂×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
35	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×Dic₆₇	—	4	2	no	—
40	C₂×Dic₆₇	—	4	1	no	—
41	C₂×Dic₆₇	—	4	1	no	—
42	C₂×Dic₆₇	—	4	4	no	—
43	C₂×Dic₆₇	—	4	2	no	—
44	C₂×Dic₆₇	—	4	2	no	—
45	C₈×C₆₇	C₄⋊C₄	1	8	yes	complemented
46	C₂²×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₆₇	#127	1	1	yes	characteristic
48	C₂²×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
49	C₂²×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
50	C₂²×C₆₇	#137	1	2	yes	—
51	C₄×C₆₇	—	2	4	no	—
52	C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
53	C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
54	Dic₆₇	—	8	2	no	—
55	Dic₆₇	—	8	1	no	—
56	Dic₆₇	—	8	1	no	—
57	C₂×C₆₇	#384	1	1	yes	characteristic
58	C₂×C₆₇	#428	1	2	yes	—
59	C₂×C₆₇	#496	1	1	yes	characteristic
60	C₂×C₆₇	#497	1	1	yes	characteristic
61	C₂×C₆₇	#501	1	2	yes	—
62	#1847	—	67	1	no	maximal, Sylow
63	C₆₇	#1847	1	1	yes	characteristic, complemented, Sylow
64	C₂×#107	—	67	2	no	—
65	C₂×#106	—	67	2	no	—
66	C₂×#97	D₆₇	1	1	yes	characteristic
67	#384	—	67	1	no	—
68	#382	—	67	1	no	—
69	C₂²×C₈	C₂×D₆₇	1	2	yes	—
70	C₂×C₄⋊C₄	—	67	1	no	—
71	C₂×C₄⋊C₄	—	67	1	no	—
72	C₂×C₄⋊C₄	—	67	2	no	—
73	C₂×C₄⋊C₄	—	134	2	no	—
74	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
75	#107	—	134	4	no	—
76	#106	—	134	4	no	—
77	#97	Dic₆₇	1	4	yes	complemented
78	#95	—	134	2	no	—
79	C₂²×C₄	—	134	1	no	—
80	C₂²×C₄	—	134	1	no	—
81	C₂²×C₄	—	134	2	no	—
82	C₂²×C₄	D₂₆₈	1	2	yes	—
83	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
84	C₂×C₈	Dic₁₃₄	1	4	yes	—
85	C₂×C₈	D₂₆₈	1	4	yes	—
86	C₂×C₈	C₂×Dic₆₇	1	4	yes	—
87	C₄⋊C₄	—	268	4	no	—
88	C₄⋊C₄	—	134	2	no	—
89	C₄⋊C₄	—	134	2	no	—
90	C₄⋊C₄	—	134	2	no	—
91	C₄⋊C₄	—	134	2	no	—
92	C₄²	Dic₁₃₄	1	2	yes	—
93	C₄²	C₂×Dic₆₇	1	2	yes	—
94	C₂³	C₂×D₂₆₈	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	268	2	no	—
98	C₂×C₄	—	268	1	no	—
99	C₂×C₄	—	268	1	no	—
100	C₂×C₄	—	268	4	no	—
101	C₂×C₄	—	268	2	no	—
102	C₂×C₄	—	268	2	no	—
103	C₂×C₄	C₄⋊Dic₆₇	1	4	yes	—
104	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
105	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
106	C₂×C₄	C₂×Dic₁₃₄	1	2	yes	—
107	C₂×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
108	C₈	C₄⋊Dic₆₇	1	8	yes	complemented
109	C₂²	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
110	C₂²	#14524040	1	1	yes	characteristic
111	C₂²	#14524043	1	1	yes	characteristic
112	C₂²	#14524071	1	2	yes	—
113	C₂²	#14524076	1	2	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	536	2	no	—
116	C₄	—	536	1	no	—
117	C₄	—	536	1	no	—
118	C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
119	C₄	#14524043	1	2	yes	—
120	C₂	#16863266	1	1	yes	characteristic
121	C₂	#16863438	1	2	yes	—
122	C₂	#16863785	1	1	yes	characteristic
123	C₂	#16863790	1	1	yes	characteristic
124	C₂	#16863805	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523977

C₂

yes

maximal

C₂×#14523975

C₂

yes

maximal

#16863266

C₂

yes

maximal, characteristic

#16863262

C₂

yes

maximal, characteristic

C₂×#97×C₆₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

—

#14523977

—

#14523975

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

#97×C₆₇

C₄

yes

complemented

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

Q₈

yes

—

C₄²×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

Q₈

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄⋊C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

C₄⋊C₄

yes

complemented

C₂²×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₆₇

#127

yes

characteristic

C₂²×C₆₇

C₄⋊Q₈

yes

characteristic

C₂²×C₆₇

AGL(1,ℤ/8)

yes

—

C₂²×C₆₇

#137

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₂×C₄⋊C₄

yes

—

C₄×C₆₇

C₄⋊Q₈

yes

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

#384

yes

characteristic

C₂×C₆₇

#428

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

#1847

—

maximal, Sylow

C₆₇

#1847

yes

characteristic, complemented, Sylow

C₂×#107

—

C₂×#106

—

C₂×#97

D₆₇

yes

characteristic

#384

—

#382

—

C₂²×C₈

C₂×D₆₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

#107

—

134

—

#106

—

134

—

#97

Dic₆₇

yes

complemented

#95

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

D₂₆₈

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂×C₈

Dic₁₃₄

yes

—

C₂×C₈

D₂₆₈

yes

—

C₂×C₈

C₂×Dic₆₇

yes

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄²

Dic₁₃₄

yes

—

C₄²

C₂×Dic₆₇

yes

—

C₂³

C₂×D₂₆₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

268

—

C₂×C₄

—

268

—

C₂×C₄

—

268

—

100

C₂×C₄

—

268

—

101

C₂×C₄

—

268

—

102

C₂×C₄

—

268

—

103

C₂×C₄

C₄⋊Dic₆₇

yes

—

104

C₂×C₄

C₂²×Dic₆₇

yes

characteristic

105

C₂×C₄

C₂×D₂₆₈

yes

—

106

C₂×C₄

C₂×Dic₁₃₄

yes

—

107

C₂×C₄

C₂×Dic₁₃₄

yes

characteristic

108

C₈

C₄⋊Dic₆₇

yes

complemented

109

C₂²

C₂×C₄⋊Dic₆₇

yes

characteristic

110

C₂²

#14524040

yes

characteristic

111

C₂²

#14524043

yes

characteristic

112

C₂²

#14524071

yes

—

113

C₂²

#14524076

yes

—

114

C₄

—

115

C₄

—

536

—

116

C₄

—

536

—

117

C₄

—

536

—

118

C₄

C₂×C₄⋊Dic₆₇

yes

—

119

C₄

#14524043

yes

—

120

C₂

#16863266

yes

characteristic

121

C₂

#16863438

yes

—

122

C₂

#16863785

yes

characteristic

123

C₂

#16863790

yes

characteristic

124

C₂

#16863805

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×#97×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂×#97	D₆₇
hypocenter	C₆₇	#1847
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₂₆₈

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂×#97×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

hypercenter

C₂×#97

D₆₇

hypocenter

C₆₇

#1847

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
15. #97×C₆₇	C₄	semidirect
45. C₈×C₆₇	C₄⋊C₄	semidirect
63. C₆₇	#1847	semidirect
77. #97	Dic₆₇	semidirect
108. C₈	C₄⋊Dic₆₇	semidirect

H ≅ G/N

type

15. #97×C₆₇

C₄

semidirect

45. C₈×C₆₇

C₄⋊C₄

semidirect

63. C₆₇

#1847

semidirect

77. #97

Dic₆₇

semidirect

108. C₈

C₄⋊Dic₆₇

semidirect

Subgroups — #22560377

Conjugacy classes of subgroups

Complements