Group #22587816 — subgroups

Subgroups — #22587816

Order 9088.

198 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 55 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	#17986191	C₂	1	2	yes	maximal, complemented
2	#17986078	C₂	1	1	yes	maximal, characteristic, complemented
3	#17985750	C₂	1	1	yes	maximal, characteristic, complemented
4	#17985743	C₂	1	2	yes	maximal, complemented
5	#493×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11765850	—	2	2	no	—
7	C₂×#11765848	—	2	1	no	—
8	C₄×D₂₈₄	C₂²	1	1	yes	characteristic
9	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
10	C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
11	#14556049	—	2	2	no	—
12	(C₂×D₇₁)⋊C₈	—	2	1	no	—
13	C₂×D₈×C₇₁	—	2	1	no	—
14	C₂×D₈×C₇₁	—	2	1	no	—
15	#127×C₇₁	C₂²	1	2	yes	—
16	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×D₂₈₄	D₄	1	1	yes	characteristic
18	C₂×C₄×D₇₁	—	2	1	no	—
19	#11765850	—	4	4	no	—
20	#11765848	—	4	2	no	—
21	C₂×C₇₁⋊C₈	—	2	1	no	—
22	C₂×C₇₁⋊C₈	—	2	2	no	—
23	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
24	(C₂×D₇₁)⋊C₄	—	2	1	no	—
25	C₂×D₄×C₇₁	—	2	2	no	—
26	C₂×D₄×C₇₁	—	2	2	no	—
27	C₂×D₄×C₇₁	—	2	2	no	—
28	D₈×C₇₁	—	4	2	no	—
29	D₈×C₇₁	—	4	2	no	—
30	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
31	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
32	C₄²×C₇₁	C₂³	1	1	yes	characteristic
33	C₂³×C₇₁	—	4	2	no	—
34	C₂²×D₇₁	—	2	1	no	—
35	D₄×C₇₁	—	4	2	no	—
36	D₄×C₇₁	—	4	2	no	—
37	D₄×C₇₁	—	4	2	no	—
38	D₄×C₇₁	—	4	2	no	—
39	D₄×C₇₁	—	8	2	no	—
40	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
43	C₂×Dic₇₁	—	2	1	no	—
44	D₂₈₄	—	2	2	no	—
45	C₄×D₇₁	—	4	1	no	—
46	C₄×D₇₁	—	4	1	no	—
47	C₈×C₇₁	—	2	1	no	—
48	C₈×C₇₁	D₈	1	2	yes	complemented
49	C₇₁⋊C₈	—	4	1	no	—
50	C₇₁⋊C₈	—	4	2	no	—
51	C₂²×C₇₁	—	8	2	no	—
52	C₂²×C₇₁	—	8	2	no	—
53	C₂²×C₇₁	—	8	2	no	—
54	C₂²×C₇₁	#121	1	1	yes	characteristic
55	C₂×D₇₁	—	4	1	no	—
56	C₂×D₇₁	—	4	1	no	—
57	C₂×D₇₁	—	4	1	no	—
58	C₄×C₇₁	—	2	1	no	—
59	C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
60	C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
61	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₇₁	—	4	1	no	—
63	C₂×C₇₁	—	16	2	no	—
64	C₂×C₇₁	#459	1	1	yes	characteristic
65	C₂×C₇₁	#469	1	1	yes	characteristic
66	C₂×C₇₁	#472	1	1	yes	characteristic
67	D₇₁	—	8	1	no	—
68	#1687	—	71	1	no	maximal, Sylow
69	C₇₁	#1687	1	1	yes	characteristic, complemented, Sylow
70	#493	D₇₁	1	1	yes	characteristic, complemented
71	#459	—	71	2	no	—
72	#436	—	71	1	no	—
73	#334	—	71	1	no	—
74	#331	—	71	2	no	—
75	C₂×D₈	—	2	1	no	—
76	C₂×D₈	—	2	1	no	—
77	C₂×D₈	—	142	2	no	—
78	C₂×M₁₆	—	142	1	no	—
79	#127	C₂×D₇₁	1	2	yes	—
80	C₄×D₄	—	71	1	no	—
81	C₄⋊C₈	—	71	1	no	—
82	C₄⋊C₈	—	71	2	no	—
83	#102	—	142	2	no	—
84	C₂²⋊C₈	—	142	1	no	—
85	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
86	C₂×D₄	—	2	2	no	—
87	C₂×D₄	—	2	2	no	—
88	C₂×D₄	—	2	2	no	—
89	C₂×D₄	—	71	1	no	—
90	C₂²×C₄	—	142	1	no	—
91	D₈	—	4	2	no	—
92	D₈	—	4	2	no	—
93	D₈	—	284	4	no	—
94	M₁₆	—	284	2	no	—
95	C₂×C₈	—	142	2	no	—
96	C₂×C₈	—	142	1	no	—
97	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
98	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
99	C₄⋊C₄	—	71	1	no	—
100	C₂²⋊C₄	—	142	1	no	—
101	C₄²	C₂²×D₇₁	1	1	yes	characteristic
102	C₂³	—	4	2	no	—
103	C₂³	—	142	1	no	—
104	D₄	—	4	2	no	—
105	D₄	—	4	2	no	—
106	D₄	—	4	2	no	—
107	D₄	—	4	2	no	—
108	D₄	—	8	2	no	—
109	D₄	—	142	2	no	—
110	C₂×C₄	—	284	1	no	—
111	C₂×C₄	—	284	1	no	—
112	C₂×C₄	—	142	1	no	—
113	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
114	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
115	C₂×C₄	#11765864	1	1	yes	characteristic
116	C₈	—	2	1	no	—
117	C₈	—	284	1	no	—
118	C₈	—	284	2	no	—
119	C₈	#11765850	1	2	yes	complemented
120	C₂²	—	8	2	no	—
121	C₂²	—	8	2	no	—
122	C₂²	—	8	2	no	—
123	C₂²	—	284	1	no	—
124	C₂²	—	284	1	no	—
125	C₂²	—	284	1	no	—
126	C₂²	#14556099	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	284	1	no	—
129	C₄	C₂×#11765850	1	1	yes	characteristic
130	C₄	#14556152	1	2	yes	—
131	C₄	#14556154	1	1	yes	characteristic
132	C₂	—	16	2	no	—
133	C₂	—	568	1	no	—
134	C₂	#17986191	1	1	yes	characteristic
135	C₂	#17986249	1	1	yes	characteristic
136	C₂	#17986266	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986191

C₂

yes

maximal, complemented

#17986078

C₂

yes

maximal, characteristic, complemented

#17985750

C₂

yes

maximal, characteristic, complemented

#17985743

C₂

yes

maximal, complemented

#493×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×#11765850

—

C₂×#11765848

—

C₄×D₂₈₄

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

#14556049

—

(C₂×D₇₁)⋊C₈

—

C₂×D₈×C₇₁

—

C₂×D₈×C₇₁

—

#127×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×D₂₈₄

D₄

yes

characteristic

C₂×C₄×D₇₁

—

#11765850

—

#11765848

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

D₄

yes

characteristic

(C₂×D₇₁)⋊C₄

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

D₈×C₇₁

—

D₈×C₇₁

—

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

—

C₂²×D₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

#41

yes

characteristic

C₂×Dic₇₁

—

D₂₈₄

—

C₄×D₇₁

—

C₄×D₇₁

—

C₈×C₇₁

—

C₈×C₇₁

D₈

yes

complemented

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#121

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×D₈

yes

characteristic

C₄×C₇₁

AGL(1,ℤ/8)

yes

—

C₄×C₇₁

AGL(1,ℤ/8)

yes

characteristic

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

#1687

—

maximal, Sylow

C₇₁

#1687

yes

characteristic, complemented, Sylow

#493

D₇₁

yes

characteristic, complemented

—

—

—

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

142

—

C₂×M₁₆

—

142

—

#127

C₂×D₇₁

yes

—

C₄×D₄

—

C₄⋊C₈

—

C₄⋊C₈

—

#102

—

142

—

C₂²⋊C₈

—

142

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

142

—

D₈

—

D₈

—

D₈

—

284

—

M₁₆

—

284

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₄⋊C₄

—

100

C₂²⋊C₄

—

142

—

101

C₄²

C₂²×D₇₁

yes

characteristic

102

C₂³

—

103

C₂³

—

142

—

104

D₄

—

105

D₄

—

106

D₄

—

107

D₄

—

108

D₄

—

109

D₄

—

142

—

110

C₂×C₄

—

284

—

111

C₂×C₄

—

284

—

112

C₂×C₄

—

142

—

113

C₂×C₄

D₄×D₇₁

yes

characteristic

114

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

115

C₂×C₄

#11765864

yes

characteristic

116

C₈

—

117

C₈

—

284

—

118

C₈

—

284

—

119

C₈

#11765850

yes

complemented

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

284

—

124

C₂²

—

284

—

125

C₂²

—

284

—

126

C₂²

#14556099

yes

characteristic

127

C₄

—

128

C₄

—

284

—

129

C₄

C₂×#11765850

yes

characteristic

130

C₄

#14556152

yes

—

131

C₄

#14556154

yes

characteristic

132

C₂

—

133

C₂

—

568

—

134

C₂

#17986191

yes

characteristic

135

C₂

#17986249

yes

characteristic

136

C₂

#17986266

yes

characteristic

	subgroup	G/H
center	C₂²	#14556099
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#493×C₇₁	C₂
socle	C₂²×C₇₁	#121
hypercenter	#493	D₇₁
hypocenter	C₇₁	#1687
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#14556099

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#493×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #17986191	C₂	semidirect
2. #17986078	C₂	semidirect
3. #17985750	C₂	semidirect
4. #17985743	C₂	semidirect
5. #493×C₇₁	C₂	semidirect
48. C₈×C₇₁	D₈	semidirect
69. C₇₁	#1687	semidirect
70. #493	D₇₁	semidirect
119. C₈	#11765850	semidirect

H ≅ G/N

type

1. #17986191

C₂

semidirect

2. #17986078

C₂

semidirect

3. #17985750

C₂

semidirect

4. #17985743

C₂

semidirect

5. #493×C₇₁

C₂

semidirect

48. C₈×C₇₁

D₈

semidirect

69. C₇₁

#1687

semidirect

70. #493

D₇₁

semidirect

119. C₈

#11765850

semidirect

Subgroups — #22587816

Conjugacy classes of subgroups

Complements