Group #22589263 — subgroups

Subgroups — #22589263

Order 9088.

324 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 125 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×#14556128	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556051	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₄×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₄₂	C₂²	1	2	yes	—
6	C₂²×C₇₁⋊C₈	C₂²	1	2	yes	—
7	#14556128	—	2	2	no	—
8	#14556128	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₇₁	—	2	1	no	—
10	C₂×C₄⋊Dic₇₁	—	2	1	no	—
11	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
12	#14556051	—	2	8	no	—
13	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₇₁	—	2	1	no	—
15	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
16	#95×C₇₁	—	2	1	no	—
17	C₂×Dic₁₄₂	—	2	4	no	—
18	C₂×Dic₁₄₂	—	2	4	no	—
19	C₂×Dic₁₄₂	—	2	2	no	—
20	C₂×Dic₁₄₂	C₂×C₄	1	4	yes	—
21	C₂²×Dic₇₁	—	2	2	no	—
22	C₂×C₇₁⋊C₈	—	2	4	no	—
23	C₂×C₇₁⋊C₈	—	2	2	no	—
24	C₄⋊Dic₇₁	—	2	4	no	—
25	C₄⋊Dic₇₁	—	2	4	no	—
26	C₂²×C₄×C₇₁	—	2	2	no	—
27	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₇₁	—	2	2	no	—
31	C₄⋊C₄×C₇₁	D₄	1	4	yes	—
32	C₄²×C₇₁	—	4	2	no	—
33	C₄²×C₇₁	C₂×C₄	1	2	yes	—
34	C₄²×C₇₁	D₄	1	2	yes	—
35	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₇₁	—	2	2	no	—
37	C₂×C₄×C₇₁	—	2	4	no	—
38	C₂×C₄×C₇₁	—	4	2	no	—
39	C₂×C₄×C₇₁	—	4	2	no	—
40	C₂×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₇₁	SD₁₆	1	2	yes	—
43	C₂×C₄×C₇₁	Q₁₆	1	2	yes	—
44	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₇₁	#41	1	2	yes	—
48	C₂×Dic₇₁	—	2	4	no	—
49	C₂×Dic₇₁	—	4	4	no	—
50	Dic₁₄₂	—	2	8	no	—
51	Dic₁₄₂	—	4	2	no	—
52	Dic₁₄₂	—	4	2	no	—
53	C₇₁⋊C₈	—	4	2	no	—
54	C₂²×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₇₁	#121	1	2	yes	—
56	C₂²×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
57	C₂²×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×C₇₁	#137	1	2	yes	—
59	C₄×C₇₁	—	2	2	no	—
60	C₄×C₇₁	—	4	4	no	—
61	C₄×C₇₁	#103	1	4	yes	complemented
62	C₄×C₇₁	C₄×D₄	1	2	yes	—
63	C₄×C₇₁	#124	1	1	yes	characteristic
64	C₄×C₇₁	#127	1	1	yes	characteristic
65	Dic₇₁	—	4	4	no	—
66	C₂×C₇₁	#390	1	1	yes	characteristic
67	C₂×C₇₁	C₂×#103	1	1	yes	characteristic
68	C₂×C₇₁	#419	1	1	yes	characteristic
69	C₂×C₇₁	#461	1	2	yes	—
70	C₂×C₇₁	#462	1	2	yes	—
71	#1981	—	71	1	no	maximal, Sylow
72	C₇₁	#1981	1	1	yes	characteristic, complemented, Sylow
73	C₂×C₄⋊Q₈	—	71	1	no	—
74	C₂×C₄⋊C₈	—	71	1	no	—
75	C₂×#103	—	71	4	no	—
76	C₄×C₄⋊C₄	D₇₁	1	1	yes	characteristic
77	C₂²×Q₈	—	71	2	no	—
78	C₂²×C₈	—	71	2	no	—
79	C₄⋊Q₈	—	71	4	no	—
80	C₄⋊Q₈	—	142	2	no	—
81	C₂×C₄⋊C₄	—	142	1	no	—
82	C₂×C₄⋊C₄	—	142	1	no	—
83	C₂×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
84	C₂×C₄²	—	2	1	no	—
85	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
86	C₄⋊C₈	—	142	2	no	—
87	#103	—	142	8	no	—
88	#95	—	2	1	no	—
89	C₂×Q₈	—	71	4	no	—
90	C₂×Q₈	—	142	4	no	—
91	C₂×Q₈	—	142	4	no	—
92	C₂×Q₈	—	142	2	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	142	2	no	—
95	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
98	C₂×C₈	—	142	4	no	—
99	C₂×C₈	—	142	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	142	4	no	—
102	C₄⋊C₄	—	142	4	no	—
103	C₄⋊C₄	D₂₈₄	1	4	yes	—
104	C₄²	—	4	2	no	—
105	C₄²	C₄×D₇₁	1	2	yes	—
106	C₄²	D₂₈₄	1	2	yes	—
107	C₂³	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
108	Q₈	—	284	2	no	—
109	Q₈	—	284	2	no	—
110	Q₈	—	142	8	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	284	4	no	—
116	C₂×C₄	—	142	4	no	—
117	C₂×C₄	(C₂×D₇₁)⋊C₄	1	2	yes	—
118	C₂×C₄	(C₂×D₇₁)⋊C₄	1	2	yes	—
119	C₂×C₄	#11765853	1	2	yes	—
120	C₂×C₄	#11765855	1	2	yes	—
121	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
122	C₂×C₄	C₂×D₂₈₄	1	2	yes	—
123	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
124	C₂×C₄	#11765865	1	2	yes	—
125	C₈	—	284	2	no	—
126	C₂²	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
127	C₂²	#14556102	1	2	yes	—
128	C₂²	C₂×#11765853	1	1	yes	characteristic
129	C₂²	C₂×#11765855	1	1	yes	characteristic
130	C₂²	#14556158	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	—	4	4	no	—
133	C₄	—	284	4	no	—
134	C₄	#14556051	1	4	yes	complemented
135	C₄	C₄×D₂₈₄	1	2	yes	—
136	C₄	#14556116	1	1	yes	characteristic
137	C₄	#14556125	1	1	yes	characteristic
138	C₂	#17985901	1	1	yes	characteristic
139	C₂	C₂×#14556051	1	1	yes	characteristic
140	C₂	#17986007	1	1	yes	characteristic
141	C₂	#17986204	1	2	yes	—
142	C₂	#17986209	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556128

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic

C₂×#14556051

C₂

yes

maximal

C₄×C₄⋊C₄×C₇₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₂

C₂²

yes

—

C₂²×C₇₁⋊C₈

C₂²

yes

—

#14556128

—

#14556128

C₄

yes

complemented

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

C₄⋊(C₇₁⋊C₈)

—

#14556051

—

C₂×C₄⋊C₄×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

#95×C₇₁

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

C₂×C₄

yes

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

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

D₄

yes

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

D₄

yes

—

C₂³×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×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

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

#41

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₁

#121

yes

—

C₂²×C₇₁

C₂×SD₁₆

yes

characteristic

C₂²×C₇₁

C₂×Q₁₆

yes

characteristic

C₂²×C₇₁

#137

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#103

yes

complemented

C₄×C₇₁

C₄×D₄

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×C₇₁

#462

yes

—

#1981

—

maximal, Sylow

C₇₁

#1981

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₂×#103

—

C₄×C₄⋊C₄

D₇₁

yes

characteristic

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

142

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄⋊C₄

C₂×D₇₁

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

142

—

#103

—

142

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

142

—

C₂×Q₈

—

142

—

C₂×Q₈

—

142

—

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

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

142

—

102

C₄⋊C₄

—

142

—

103

C₄⋊C₄

D₂₈₄

yes

—

104

C₄²

—

105

C₄²

C₄×D₇₁

yes

—

106

C₄²

D₂₈₄

yes

—

107

C₂³

C₂×C₂²⋊D₇₁

yes

characteristic

108

Q₈

—

284

—

109

Q₈

—

284

—

110

Q₈

—

142

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

284

—

116

C₂×C₄

—

142

—

117

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

—

118

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

—

119

C₂×C₄

#11765853

yes

—

120

C₂×C₄

#11765855

yes

—

121

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

122

C₂×C₄

C₂×D₂₈₄

yes

—

123

C₂×C₄

C₂×D₂₈₄

yes

characteristic

124

C₂×C₄

#11765865

yes

—

125

C₈

—

284

—

126

C₂²

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

127

C₂²

#14556102

yes

—

128

C₂²

C₂×#11765853

yes

characteristic

129

C₂²

C₂×#11765855

yes

characteristic

130

C₂²

#14556158

yes

—

131

C₄

—

132

C₄

—

133

C₄

—

284

—

134

C₄

#14556051

yes

complemented

135

C₄

C₄×D₂₈₄

yes

—

136

C₄

#14556116

yes

characteristic

137

C₄

#14556125

yes

characteristic

138

C₂

#17985901

yes

characteristic

139

C₂

C₂×#14556051

yes

characteristic

140

C₂

#17986007

yes

characteristic

141

C₂

#17986204

yes

—

142

C₂

#17986209

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	C₄×C₄⋊C₄×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	C₄×C₄⋊C₄	D₇₁
hypocenter	C₇₁	#1981
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

C₄×C₄⋊C₄×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

hypercenter

C₄×C₄⋊C₄

D₇₁

hypocenter

C₇₁

#1981

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
8. #14556128	C₄	semidirect
61. C₄×C₇₁	#103	semidirect
72. C₇₁	#1981	semidirect
134. C₄	#14556051	semidirect

H ≅ G/N

type

8. #14556128

C₄

semidirect

61. C₄×C₇₁

#103

semidirect

72. C₇₁

#1981

semidirect

134. C₄

#14556051

semidirect

Subgroups — #22589263

Conjugacy classes of subgroups

Complements