Group #22589076 — subgroups

Subgroups — #22589076

Order 9088.

240 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 105 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×#14556062	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556038	C₂	1	1	yes	maximal, characteristic
3	#17985895	C₂	1	2	yes	maximal
4	#17985761	C₂	1	2	yes	maximal
5	C₂×#106×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₇₁	C₂²	1	2	yes	—
8	C₂×Dic₇₁⋊C₄	—	2	2	no	—
9	C₂×C₄×Dic₇₁	C₂²	1	1	yes	characteristic
10	#14556062	C₄	1	4	yes	complemented
11	#14556038	—	2	2	no	—
12	C₇₁⋊#95	—	2	2	no	—
13	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
15	#106×C₇₁	—	2	2	no	—
16	C₂²×Dic₇₁	—	2	2	no	—
17	C₂²×Dic₇₁	D₄	1	2	yes	—
18	C₂×C₇₁⋊C₈	—	2	1	no	—
19	C₂×C₇₁⋊C₈	—	2	1	no	—
20	C₂×C₇₁⋊C₈	Q₈	1	2	yes	—
21	C₄⋊Dic₇₁	—	2	2	no	—
22	C₄⋊Dic₇₁	C₂×C₄	1	4	yes	—
23	Dic₇₁⋊C₄	—	4	4	no	—
24	C₄×Dic₇₁	—	2	1	no	—
25	C₄×Dic₇₁	—	2	1	no	—
26	C₂²×C₄×C₇₁	—	2	2	no	—
27	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₇₁	D₄	1	2	yes	—
30	C₂×C₈×C₇₁	Q₈	1	2	yes	—
31	C₄⋊C₄×C₇₁	—	2	2	no	—
32	C₄⋊C₄×C₇₁	—	2	2	no	—
33	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₇₁	—	4	2	no	—
35	C₂×C₄×C₇₁	—	4	2	no	—
36	C₂×C₄×C₇₁	—	4	2	no	—
37	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
42	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
43	C₂×Dic₇₁	—	2	4	no	—
44	C₂×Dic₇₁	—	2	2	no	—
45	C₂×Dic₇₁	—	2	2	no	—
46	C₂×Dic₇₁	—	2	2	no	—
47	C₂×Dic₇₁	—	4	2	no	—
48	C₂×Dic₇₁	—	4	2	no	—
49	C₈×C₇₁	C₄⋊C₄	1	4	yes	complemented
50	C₇₁⋊C₈	—	2	2	no	—
51	C₂²×C₇₁	C₄×D₄	1	1	yes	characteristic
52	C₂²×C₇₁	#121	1	1	yes	characteristic
53	C₂²×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
54	C₂²×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
55	C₂²×C₇₁	#137	1	2	yes	—
56	C₄×C₇₁	—	8	2	no	—
57	C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
59	C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
60	C₄×C₇₁	#125	1	1	yes	characteristic
61	Dic₇₁	—	4	4	no	—
62	Dic₇₁	—	4	2	no	—
63	C₂×C₇₁	#389	1	1	yes	characteristic
64	C₂×C₇₁	#441	1	1	yes	characteristic
65	C₂×C₇₁	#442	1	1	yes	characteristic
66	C₂×C₇₁	#468	1	2	yes	—
67	C₂×C₇₁	#501	1	2	yes	—
68	#1943	—	71	1	no	maximal, Sylow
69	C₇₁	#1943	1	1	yes	characteristic, complemented, Sylow
70	C₂×#107	—	71	1	no	—
71	C₂×#106	D₇₁	1	1	yes	characteristic
72	C₂×#97	—	71	1	no	—
73	#389	—	71	2	no	—
74	#340	—	71	2	no	—
75	C₂²×C₈	—	71	1	no	—
76	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
77	C₂×C₄⋊C₄	—	71	2	no	—
78	C₂×C₄⋊C₄	—	142	2	no	—
79	C₂×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
80	C₂×C₄²	—	71	1	no	—
81	#107	—	71	4	no	—
82	#106	—	2	2	no	—
83	#97	—	142	2	no	—
84	#95	—	142	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	71	2	no	—
87	C₂²×C₄	—	142	2	no	—
88	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
89	C₂×C₈	—	71	2	no	—
90	C₂×C₈	—	142	1	no	—
91	C₂×C₈	—	142	1	no	—
92	C₂×C₈	Dic₁₄₂	1	2	yes	—
93	C₂×C₈	C₄×D₇₁	1	2	yes	—
94	C₂×C₈	C₂²⋊D₇₁	1	2	yes	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	284	4	no	—
98	C₄⋊C₄	—	71	4	no	—
99	C₄⋊C₄	—	142	2	no	—
100	C₄²	—	142	1	no	—
101	C₄²	—	142	1	no	—
102	C₂³	D₄×D₇₁	1	1	yes	characteristic
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	284	2	no	—
107	C₂×C₄	—	284	2	no	—
108	C₂×C₄	—	142	4	no	—
109	C₂×C₄	—	142	2	no	—
110	C₂×C₄	—	142	2	no	—
111	C₂×C₄	—	142	2	no	—
112	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
113	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
114	C₂×C₄	Q₈×D₇₁	1	1	yes	characteristic
115	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
116	C₂×C₄	#11765864	1	1	yes	characteristic
117	C₂×C₄	#11765866	1	1	yes	characteristic
118	C₈	—	142	2	no	—
119	C₈	Dic₇₁⋊C₄	1	4	yes	complemented
120	C₂²	#14556090	1	1	yes	characteristic
121	C₂²	#14556099	1	1	yes	characteristic
122	C₂²	#14556127	1	1	yes	characteristic
123	C₂²	#14556155	1	2	yes	—
124	C₂²	#14556157	1	2	yes	—
125	C₄	—	8	2	no	—
126	C₄	—	284	4	no	—
127	C₄	—	284	2	no	—
128	C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
129	C₄	#14556093	1	1	yes	characteristic
130	C₄	#14556104	1	1	yes	characteristic
131	C₄	#14556119	1	1	yes	characteristic
132	C₂	#17985895	1	1	yes	characteristic
133	C₂	#17986105	1	1	yes	characteristic
134	C₂	#17986110	1	1	yes	characteristic
135	C₂	#17986242	1	2	yes	—
136	C₂	#17986406	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556062

C₂

yes

maximal, characteristic

C₂×#14556038

C₂

yes

maximal, characteristic

#17985895

C₂

yes

maximal

#17985761

C₂

yes

maximal

C₂×#106×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₁

C₂²

yes

—

C₂×Dic₇₁⋊C₄

—

C₂×C₄×Dic₇₁

C₂²

yes

characteristic

#14556062

C₄

yes

complemented

#14556038

—

C₇₁⋊#95

—

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₁

C₂²

yes

—

#106×C₇₁

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

D₄

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

Q₈

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₂×C₄

yes

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

D₄

yes

—

C₂×C₈×C₇₁

Q₈

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₂³×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

C₄⋊C₄

yes

complemented

C₇₁⋊C₈

—

C₂²×C₇₁

C₄×D₄

yes

characteristic

C₂²×C₇₁

#121

yes

characteristic

C₂²×C₇₁

C₄⋊Q₈

yes

characteristic

C₂²×C₇₁

AGL(1,ℤ/8)

yes

—

C₂²×C₇₁

#137

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₁

C₄×Q₈

yes

characteristic

C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₄×C₇₁

#125

yes

characteristic

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×C₇₁

#501

yes

—

#1943

—

maximal, Sylow

C₇₁

#1943

yes

characteristic, complemented, Sylow

C₂×#107

—

C₂×#106

D₇₁

yes

characteristic

C₂×#97

—

#389

—

#340

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄⋊C₄

C₂×D₇₁

yes

—

C₂×C₄²

—

#107

—

#106

—

#97

—

142

—

#95

—

142

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

142

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

Dic₁₄₂

yes

—

C₂×C₈

C₄×D₇₁

yes

—

C₂×C₈

C₂²⋊D₇₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

C₄⋊C₄

—

142

—

100

C₄²

—

142

—

101

C₄²

—

142

—

102

C₂³

D₄×D₇₁

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

284

—

107

C₂×C₄

—

284

—

108

C₂×C₄

—

142

—

109

C₂×C₄

—

142

—

110

C₂×C₄

—

142

—

111

C₂×C₄

—

142

—

112

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

113

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

114

C₂×C₄

Q₈×D₇₁

yes

characteristic

115

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

116

C₂×C₄

#11765864

yes

characteristic

117

C₂×C₄

#11765866

yes

characteristic

118

C₈

—

142

—

119

C₈

Dic₇₁⋊C₄

yes

complemented

120

C₂²

#14556090

yes

characteristic

121

C₂²

#14556099

yes

characteristic

122

C₂²

#14556127

yes

characteristic

123

C₂²

#14556155

yes

—

124

C₂²

#14556157

yes

—

125

C₄

—

126

C₄

—

284

—

127

C₄

—

284

—

128

C₄

C₂×Dic₇₁⋊C₄

yes

characteristic

129

C₄

#14556093

yes

characteristic

130

C₄

#14556104

yes

characteristic

131

C₄

#14556119

yes

characteristic

132

C₂

#17985895

yes

characteristic

133

C₂

#17986105

yes

characteristic

134

C₂

#17986110

yes

characteristic

135

C₂

#17986242

yes

—

136

C₂

#17986406

yes

—

	subgroup	G/H
center	C₂³	D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	C₂×#106×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	C₂×#106	D₇₁
hypocenter	C₇₁	#1943
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

C₂×#106×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
10. #14556062	C₄	semidirect
49. C₈×C₇₁	C₄⋊C₄	semidirect
69. C₇₁	#1943	semidirect
119. C₈	Dic₇₁⋊C₄	semidirect

H ≅ G/N

type

10. #14556062

C₄

semidirect

49. C₈×C₇₁

C₄⋊C₄

semidirect

69. C₇₁

#1943

semidirect

119. C₈

Dic₇₁⋊C₄

semidirect

Subgroups — #22589076

Conjugacy classes of subgroups

Complements