Group #22587092 — subgroups

Subgroups — #22587092

Order 9088.

180 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 113 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	#17986052	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#17985752	C₂	1	2	yes	maximal, complemented
4	#17985750	C₂	1	2	yes	maximal, complemented
5	C₄×M₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₈	—	2	1	no	—
7	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
9	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
10	C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
11	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
12	C₄×C₇₁⋊C₈	—	2	1	no	—
13	C₂×M₁₆×C₇₁	C₄	1	1	yes	characteristic
14	C₂×M₁₆×C₇₁	C₄	1	1	yes	characteristic
15	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
16	#97×C₇₁	C₄	1	2	yes	—
17	C₄×C₈×C₇₁	C₂²	1	2	yes	—
18	C₂×C₇₁⋊C₈	—	2	1	no	—
19	C₂×C₇₁⋊C₈	—	2	1	no	—
20	C₂×C₇₁⋊C₈	—	2	2	no	—
21	C₂×C₇₁⋊C₈	—	4	2	no	—
22	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₇₁	Q₈	1	1	yes	characteristic
25	M₁₆×C₇₁	—	2	2	no	—
26	M₁₆×C₇₁	C₈	1	4	yes	complemented
27	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
29	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₇₁	D₄	1	1	yes	characteristic
31	C₄²×C₇₁	Q₈	1	1	yes	characteristic
32	C₄²×C₇₁	C₂³	1	1	yes	characteristic
33	C₂³×C₇₁	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₇₁	—	2	1	no	—
35	C₂×C₄×C₇₁	—	2	1	no	—
36	C₂×C₄×C₇₁	—	4	1	no	—
37	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₇₁	C₂×C₈	1	2	yes	—
41	C₂×C₄×C₇₁	M₁₆	1	2	yes	—
42	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₇₁	—	2	2	no	—
46	C₈×C₇₁	C₂×C₈	1	4	yes	complemented
47	C₇₁⋊C₈	—	4	1	no	—
48	C₇₁⋊C₈	—	4	2	no	—
49	C₇₁⋊C₈	—	4	1	no	—
50	C₂²×C₇₁	—	2	1	no	—
51	C₂²×C₇₁	—	2	1	no	—
52	C₂²×C₇₁	C₄⋊C₈	1	2	yes	—
53	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₇₁	—	2	1	no	—
55	C₄×C₇₁	—	2	1	no	—
56	C₄×C₇₁	—	2	1	no	—
57	C₄×C₇₁	C₄⋊C₈	1	2	yes	—
58	C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
59	C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
60	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₄×C₇₁	#137	1	1	yes	characteristic
62	C₂×C₇₁	—	2	2	no	—
63	C₂×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₇₁	#428	1	1	yes	characteristic
65	C₂×C₇₁	#430	1	1	yes	characteristic
66	#1565	—	71	1	no	maximal, Sylow
67	C₇₁	#1565	1	1	yes	characteristic, complemented, Sylow
68	#431	—	71	1	no	—
69	C₂×C₄⋊C₈	—	71	1	no	—
70	C₄×M₁₆	D₇₁	1	1	yes	characteristic
71	#335	—	71	2	no	—
72	#334	—	71	2	no	—
73	C₂×M₁₆	Dic₇₁	1	1	yes	characteristic
74	C₂×M₁₆	Dic₇₁	1	1	yes	characteristic
75	C₂²×C₈	—	142	1	no	—
76	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
77	C₄⋊C₈	—	71	1	no	—
78	C₄⋊C₈	—	71	2	no	—
79	C₄⋊C₈	—	71	1	no	—
80	C₄⋊C₈	—	142	1	no	—
81	C₂²⋊C₈	—	142	1	no	—
82	#97	Dic₇₁	1	2	yes	—
83	C₄×C₈	—	142	1	no	—
84	C₄×C₈	C₂×D₇₁	1	2	yes	—
85	C₂²×C₄	Dic₁₄₂	1	1	yes	characteristic
86	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
87	C₂²×C₄	C₂×Dic₇₁	1	1	yes	characteristic
88	M₁₆	—	2	2	no	—
89	M₁₆	C₇₁⋊C₈	1	4	yes	complemented
90	C₂×C₈	—	284	2	no	—
91	C₂×C₈	—	142	1	no	—
92	C₂×C₈	—	142	1	no	—
93	C₂×C₈	—	142	2	no	—
94	C₂×C₈	C₂×Dic₇₁	1	2	yes	—
95	C₂×C₈	C₂×Dic₇₁	1	2	yes	—
96	C₄²	Dic₁₄₂	1	1	yes	characteristic
97	C₄²	D₂₈₄	1	1	yes	characteristic
98	C₄²	C₂×Dic₇₁	1	1	yes	characteristic
99	C₄²	C₂²×D₇₁	1	1	yes	characteristic
100	C₂³	C₄⋊Dic₇₁	1	1	yes	characteristic
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
105	C₂×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
106	C₂×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
107	C₂×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
108	C₂×C₄	#11765849	1	2	yes	—
109	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
110	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
111	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	284	1	no	—
114	C₈	—	284	2	no	—
115	C₈	—	284	1	no	—
116	C₈	C₂×C₇₁⋊C₈	1	4	yes	complemented
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
120	C₂²	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
125	C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
126	C₄	C₂×#11765849	1	1	yes	characteristic
127	C₄	#14556156	1	1	yes	characteristic
128	C₄	#14556161	1	1	yes	characteristic
129	C₂	—	2	2	no	—
130	C₂	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
131	C₂	#17986041	1	1	yes	characteristic
132	C₂	#17986048	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986052

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic

#17985752

C₂

yes

maximal, complemented

#17985750

C₂

yes

maximal, complemented

C₄×M₁₆×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

C₂²⋊(C₇₁⋊C₈)

—

C₄×C₇₁⋊C₈

—

C₂×M₁₆×C₇₁

C₄

yes

characteristic

C₂×M₁₆×C₇₁

C₄

yes

characteristic

C₂×C₄²×C₇₁

C₂²

yes

characteristic

#97×C₇₁

C₄

yes

—

C₄×C₈×C₇₁

C₂²

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

Q₈

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

C₈

yes

complemented

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

Q₈

yes

characteristic

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×C₈

yes

—

C₂×C₄×C₇₁

M₁₆

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

C₂×C₈

yes

complemented

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₄⋊C₈

yes

—

C₂²×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₄⋊C₈

yes

—

C₄×C₇₁

C₂²×C₈

yes

characteristic

C₄×C₇₁

C₂×M₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₂×C₇₁

—

C₂×C₇₁

C₂×C₄⋊C₈

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#431

—

C₂×C₄⋊C₈

—

C₄×M₁₆

D₇₁

yes

characteristic

#335

—

#334

—

C₂×M₁₆

Dic₇₁

yes

characteristic

C₂×M₁₆

Dic₇₁

yes

characteristic

C₂²×C₈

—

142

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₂²⋊C₈

—

142

—

#97

Dic₇₁

yes

—

C₄×C₈

—

142

—

C₄×C₈

C₂×D₇₁

yes

—

C₂²×C₄

Dic₁₄₂

yes

characteristic

C₂²×C₄

D₂₈₄

yes

characteristic

C₂²×C₄

C₂×Dic₇₁

yes

characteristic

M₁₆

—

M₁₆

C₇₁⋊C₈

yes

complemented

C₂×C₈

—

284

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

C₂×Dic₇₁

yes

—

C₂×C₈

C₂×Dic₇₁

yes

—

C₄²

Dic₁₄₂

yes

characteristic

C₄²

D₂₈₄

yes

characteristic

C₄²

C₂×Dic₇₁

yes

characteristic

C₄²

C₂²×D₇₁

yes

characteristic

100

C₂³

C₄⋊Dic₇₁

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₄⋊Dic₇₁

yes

characteristic

105

C₂×C₄

C₄⋊Dic₇₁

yes

characteristic

106

C₂×C₄

C₄⋊Dic₇₁

yes

characteristic

107

C₂×C₄

C₂×C₇₁⋊C₈

yes

—

108

C₂×C₄

#11765849

yes

—

109

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

110

C₂×C₄

C₂×D₂₈₄

yes

characteristic

111

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

112

C₈

—

113

C₈

—

284

—

114

C₈

—

284

—

115

C₈

—

284

—

116

C₈

C₂×C₇₁⋊C₈

yes

complemented

117

C₂²

—

118

C₂²

—

119

C₂²

C₄⋊(C₇₁⋊C₈)

yes

—

120

C₂²

C₂×C₄⋊Dic₇₁

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

C₄⋊(C₇₁⋊C₈)

yes

—

125

C₄

C₂²×C₇₁⋊C₈

yes

characteristic

126

C₄

C₂×#11765849

yes

characteristic

127

C₄

#14556156

yes

characteristic

128

C₄

#14556161

yes

characteristic

129

C₂

—

130

C₂

C₂×C₄⋊(C₇₁⋊C₈)

yes

characteristic

131

C₂

#17986041

yes

characteristic

132

C₂

#17986048

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₈₄
commutator	C₄×C₇₁	C₂²×C₈
Frattini	C₄²	C₂²×D₇₁
Fitting	C₄×M₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₄⋊C₄
hypercenter	C₄×M₁₆	D₇₁
hypocenter	C₇₁	#1565
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₂₈₄

commutator

C₄×C₇₁

C₂²×C₈

Frattini

C₄²

C₂²×D₇₁

Fitting

C₄×M₁₆×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₄⋊C₄

hypercenter

C₄×M₁₆

D₇₁

hypocenter

C₇₁

#1565

LCS

G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
3. #17985752	C₂	semidirect
4. #17985750	C₂	semidirect
26. M₁₆×C₇₁	C₈	semidirect
46. C₈×C₇₁	C₂×C₈	semidirect
67. C₇₁	#1565	semidirect
89. M₁₆	C₇₁⋊C₈	semidirect
116. C₈	C₂×C₇₁⋊C₈	semidirect

H ≅ G/N

type

3. #17985752

C₂

semidirect

4. #17985750

C₂

semidirect

26. M₁₆×C₇₁

C₈

semidirect

46. C₈×C₇₁

C₂×C₈

semidirect

67. C₇₁

#1565

semidirect

89. M₁₆

C₇₁⋊C₈

semidirect

116. C₈

C₂×C₇₁⋊C₈

semidirect

Subgroups — #22587092

Conjugacy classes of subgroups

Complements