Group #22589289 — subgroups

Subgroups — #22589289

Order 9088.

292 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 129 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⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#17985761	C₂	1	4	yes	maximal
4	C₂×C₄⋊Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₇₁	C₂²	1	2	yes	—
7	C₂×C₄×Dic₇₁	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
9	C₇₁⋊#95	—	2	1	no	—
10	C₂²×Q₈×C₇₁	—	2	1	no	—
11	C₄⋊Q₈×C₇₁	—	2	2	no	—
12	C₄⋊Q₈×C₇₁	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
14	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
16	C₂²×Dic₇₁	—	2	2	no	—
17	C₂×C₇₁⋊C₈	—	2	2	no	—
18	C₂×C₇₁⋊C₈	—	2	4	no	—
19	C₄⋊Dic₇₁	—	2	2	no	—
20	C₄⋊Dic₇₁	Q₈	1	4	yes	complemented
21	C₄×Dic₇₁	—	4	2	no	—
22	C₂×Q₈×C₇₁	—	2	4	no	—
23	C₂×Q₈×C₇₁	—	4	4	no	—
24	C₂²×C₄×C₇₁	—	2	2	no	—
25	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₇₁	—	2	4	no	—
29	C₄⋊C₄×C₇₁	—	2	2	no	—
30	C₄⋊C₄×C₇₁	C₂×C₄	1	4	yes	—
31	C₄²×C₇₁	C₂×C₄	1	2	yes	—
32	C₄²×C₇₁	D₄	1	2	yes	—
33	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
34	Q₈×C₇₁	—	4	8	no	—
35	C₂×C₄×C₇₁	—	2	2	no	—
36	C₂×C₄×C₇₁	—	2	4	no	—
37	C₂×C₄×C₇₁	—	4	4	no	—
38	C₂×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₇₁	SD₁₆	1	2	yes	—
41	C₂×C₄×C₇₁	Q₁₆	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	2	yes	—
45	C₂×C₄×C₇₁	#41	1	2	yes	—
46	C₂×Dic₇₁	—	2	4	no	—
47	C₂×Dic₇₁	—	4	2	no	—
48	C₂×Dic₇₁	—	4	2	no	—
49	C₇₁⋊C₈	—	4	2	no	—
50	C₂²×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₇₁	C₂²⋊Q₈	1	2	yes	—
52	C₂²×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
53	C₂²×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
54	C₂²×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₇₁	—	2	2	no	—
56	C₄×C₇₁	—	4	4	no	—
57	C₄×C₇₁	#103	1	4	yes	—
58	C₄×C₇₁	C₄×Q₈	1	2	yes	—
59	C₄×C₇₁	#124	1	1	yes	characteristic
60	C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
61	Dic₇₁	—	4	4	no	—
62	C₂×C₇₁	#391	1	1	yes	characteristic
63	C₂×C₇₁	C₂×#103	1	1	yes	characteristic
64	C₂×C₇₁	#418	1	1	yes	characteristic
65	C₂×C₇₁	#476	1	2	yes	—
66	C₂×C₇₁	#477	1	2	yes	—
67	#1987	—	71	1	no	maximal, Sylow
68	C₇₁	#1987	1	1	yes	characteristic, complemented, Sylow
69	C₂×C₄⋊Q₈	D₇₁	1	1	yes	characteristic
70	C₂×C₄⋊C₈	—	71	1	no	—
71	C₄×C₄⋊C₄	—	71	1	no	—
72	#340	—	71	4	no	—
73	C₂²×Q₈	—	2	1	no	—
74	C₂²×C₈	—	71	2	no	—
75	C₄⋊Q₈	—	2	2	no	—
76	C₄⋊Q₈	Dic₇₁	1	4	yes	complemented
77	C₂×C₄⋊C₄	—	2	1	no	—
78	C₂×C₄⋊C₄	—	71	2	no	—
79	C₂×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
80	C₂×C₄²	—	142	1	no	—
81	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
82	C₄⋊C₈	—	142	2	no	—
83	#95	—	142	1	no	—
84	C₂×Q₈	—	2	4	no	—
85	C₂×Q₈	—	4	4	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	142	2	no	—
88	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
89	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
91	C₂×C₈	—	142	2	no	—
92	C₂×C₈	—	142	4	no	—
93	C₄⋊C₄	—	2	4	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	71	4	no	—
96	C₄⋊C₄	—	142	2	no	—
97	C₄⋊C₄	C₂×Dic₇₁	1	4	yes	—
98	C₄²	—	284	2	no	—
99	C₄²	C₂×Dic₇₁	1	2	yes	—
100	C₄²	C₂²⋊D₇₁	1	2	yes	—
101	C₂³	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
102	Q₈	—	4	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	4	4	no	—
106	C₂×C₄	—	284	2	no	—
107	C₂×C₄	—	284	2	no	—
108	C₂×C₄	—	142	4	no	—
109	C₂×C₄	C₂²⋊Dic₇₁	1	2	yes	—
110	C₂×C₄	C₂²⋊Dic₇₁	1	2	yes	—
111	C₂×C₄	#11765852	1	2	yes	—
112	C₂×C₄	#11765855	1	2	yes	—
113	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
114	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
115	C₂×C₄	Q₈×D₇₁	1	2	yes	—
116	C₂×C₄	#11765866	1	2	yes	—
117	C₈	—	284	2	no	—
118	C₂²	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
119	C₂²	#14556104	1	2	yes	—
120	C₂²	C₂×#11765852	1	1	yes	characteristic
121	C₂²	C₂×#11765855	1	1	yes	characteristic
122	C₂²	#14556154	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	284	4	no	—
126	C₄	#14556052	1	4	yes	—
127	C₄	Q₈×Dic₇₁	1	2	yes	—
128	C₄	#14556117	1	1	yes	characteristic
129	C₄	#14556126	1	1	yes	characteristic
130	C₂	#17985906	1	1	yes	characteristic
131	C₂	C₂×#14556052	1	1	yes	characteristic
132	C₂	#17986004	1	1	yes	characteristic
133	C₂	#17986286	1	2	yes	—
134	C₂	#17986291	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₇₁

C₂

yes

maximal, characteristic

#17985761

C₂

yes

maximal

C₂×C₄⋊Q₈×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₇₁

C₂²

yes

—

C₂×C₄×Dic₇₁

—

C₄⋊(C₇₁⋊C₈)

—

C₇₁⋊#95

—

C₂²×Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

Q₈

yes

complemented

C₄×Dic₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

D₄

yes

—

C₂³×C₇₁

C₂×D₄

yes

characteristic

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

—

C₂×C₄×C₇₁

SD₁₆

yes

—

C₂×C₄×C₇₁

Q₁₆

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

#41

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₁

C₂²⋊Q₈

yes

—

C₂²×C₇₁

C₂×SD₁₆

yes

characteristic

C₂²×C₇₁

C₂×Q₁₆

yes

characteristic

C₂²×C₇₁

AGL(1,ℤ/8)

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#103

yes

—

C₄×C₇₁

C₄×Q₈

yes

—

C₄×C₇₁

#124

yes

characteristic

C₄×C₇₁

C₄⋊Q₈

yes

characteristic

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×C₇₁

#477

yes

—

#1987

—

maximal, Sylow

C₇₁

#1987

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₇₁

yes

characteristic

C₂×C₄⋊C₈

—

C₄×C₄⋊C₄

—

#340

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₇₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₁

yes

—

C₂×C₄²

—

142

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

142

—

#95

—

142

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

142

—

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

142

—

C₄⋊C₄

C₂×Dic₇₁

yes

—

C₄²

—

284

—

C₄²

C₂×Dic₇₁

yes

—

100

C₄²

C₂²⋊D₇₁

yes

—

101

C₂³

C₂×C₂²⋊D₇₁

yes

characteristic

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

284

—

107

C₂×C₄

—

284

—

108

C₂×C₄

—

142

—

109

C₂×C₄

C₂²⋊Dic₇₁

yes

—

110

C₂×C₄

C₂²⋊Dic₇₁

yes

—

111

C₂×C₄

#11765852

yes

—

112

C₂×C₄

#11765855

yes

—

113

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

114

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

115

C₂×C₄

Q₈×D₇₁

yes

—

116

C₂×C₄

#11765866

yes

—

117

C₈

—

284

—

118

C₂²

C₂×C₂²⋊Dic₇₁

yes

characteristic

119

C₂²

#14556104

yes

—

120

C₂²

C₂×#11765852

yes

characteristic

121

C₂²

C₂×#11765855

yes

characteristic

122

C₂²

#14556154

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

284

—

126

C₄

#14556052

yes

—

127

C₄

Q₈×Dic₇₁

yes

—

128

C₄

#14556117

yes

characteristic

129

C₄

#14556126

yes

characteristic

130

C₂

#17985906

yes

characteristic

131

C₂

C₂×#14556052

yes

characteristic

132

C₂

#17986004

yes

characteristic

133

C₂

#17986286

yes

—

134

C₂

#17986291

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	C₂×C₄⋊Q₈×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₇₁
hypocenter	C₇₁	#1987
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

C₂×C₄⋊Q₈×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₇₁

hypocenter

C₇₁

#1987

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₇₁	C₄	semidirect
20. C₄⋊Dic₇₁	Q₈	semidirect
68. C₇₁	#1987	semidirect
76. C₄⋊Q₈	Dic₇₁	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₇₁

C₄

semidirect

20. C₄⋊Dic₇₁

Q₈

semidirect

68. C₇₁

#1987

semidirect

76. C₄⋊Q₈

Dic₇₁

semidirect

Subgroups — #22589289

Conjugacy classes of subgroups

Complements