Group #22590384 — subgroups

Subgroups — #22590384

Order 9088.

166 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 49 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×#14556073	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556071	C₂	1	1	yes	maximal, characteristic
3	#17986377	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986081	C₂	1	1	yes	maximal, characteristic
5	Dic₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985802	C₂	1	1	yes	maximal, characteristic, complemented
7	#358×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₈₄	C₂²	1	1	yes	characteristic
9	C₂×#11765855	—	2	1	no	—
10	C₂×D₅₆₈	C₂²	1	1	yes	characteristic
11	#14556126	—	2	1	no	—
12	#14556090	—	2	1	no	—
13	#14556073	—	2	2	no	—
14	#14556071	—	2	2	no	—
15	C₂×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
16	#14556048	—	2	1	no	—
17	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
20	#107×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₄₂	—	2	1	no	—
22	C₂×D₂₈₄	—	2	1	no	—
23	C₂×C₄×D₇₁	—	4	1	no	—
24	Dic₂₈₄	—	2	1	no	—
25	Dic₂₈₄	—	2	1	no	—
26	#11765855	—	4	2	no	—
27	D₅₆₈	—	2	1	no	—
28	D₅₆₈	D₄	1	2	yes	—
29	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
30	Dic₇₁⋊C₄	—	4	1	no	—
31	Dic₇₁⋊C₄	—	4	1	no	—
32	(C₂×D₇₁)⋊C₄	—	4	1	no	—
33	C₄×Dic₇₁	D₄	1	1	yes	characteristic
34	C₇₁⋊C₁₆	—	2	1	no	—
35	C₂×Q₈×C₇₁	—	2	1	no	—
36	Q₁₆×C₇₁	—	2	1	no	—
37	Q₁₆×C₇₁	—	2	1	no	—
38	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₇₁	—	2	1	no	—
40	C₁₆×C₇₁	—	2	1	no	—
41	C₂²×D₇₁	—	4	1	no	—
42	Q₈×C₇₁	—	4	1	no	—
43	Q₈×C₇₁	—	4	1	no	—
44	C₂×C₄×C₇₁	—	4	1	no	—
45	C₂×C₄×C₇₁	—	4	1	no	—
46	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₂×Dic₇₁	—	4	1	no	—
48	C₂×Dic₇₁	D₈	1	1	yes	characteristic
49	C₂×Dic₇₁	D₈	1	1	yes	characteristic
50	D₂₈₄	—	2	2	no	—
51	D₂₈₄	—	4	1	no	—
52	C₄×D₇₁	—	8	2	no	—
53	Dic₁₄₂	—	4	1	no	—
54	Dic₁₄₂	—	4	1	no	—
55	C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
56	C₈×C₇₁	#41	1	1	yes	characteristic
57	C₇₁⋊C₈	—	2	1	no	—
58	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
59	C₂×D₇₁	—	4	2	no	—
60	C₂×D₇₁	—	8	1	no	—
61	C₂×D₇₁	—	8	1	no	—
62	C₄×C₇₁	—	8	1	no	—
63	C₄×C₇₁	—	8	1	no	—
64	C₄×C₇₁	#121	1	1	yes	characteristic
65	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
66	Dic₇₁	—	2	1	no	—
67	Dic₇₁	—	8	1	no	—
68	Dic₇₁	SD₃₂	1	2	yes	complemented
69	C₂×C₇₁	#459	1	1	yes	characteristic
70	C₂×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
71	C₂×C₇₁	#510	1	1	yes	characteristic
72	D₇₁	—	8	2	no	—
73	#2208	—	71	1	no	maximal, Sylow
74	C₇₁	#2208	1	1	yes	characteristic, complemented, Sylow
75	C₂×SD₃₂	—	71	1	no	—
76	C₂×SD₃₂	—	71	1	no	—
77	#494	—	71	1	no	—
78	C₄×D₈	—	71	1	no	—
79	C₄⋊C₁₆	—	71	1	no	—
80	#358	—	71	1	no	—
81	#358	D₇₁	1	1	yes	characteristic, complemented
82	C₂×Q₁₆	—	71	1	no	—
83	C₂×Q₁₆	—	142	1	no	—
84	C₂×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
85	C₂×D₈	—	71	1	no	—
86	C₄⋊Q₈	—	142	1	no	—
87	C₄×D₄	—	142	1	no	—
88	SD₃₂	—	142	2	no	—
89	SD₃₂	—	142	2	no	—
90	C₂×C₁₆	—	71	1	no	—
91	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
92	#107	C₂×D₇₁	1	1	yes	characteristic
93	#102	—	142	1	no	—
94	C₄×C₈	—	71	1	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	142	1	no	—
97	C₂×D₄	—	142	1	no	—
98	C₂²×C₄	—	284	1	no	—
99	Q₁₆	—	2	1	no	—
100	Q₁₆	—	2	1	no	—
101	Q₁₆	—	284	2	no	—
102	Q₁₆	—	142	1	no	—
103	Q₁₆	—	142	1	no	—
104	D₈	—	71	2	no	—
105	D₈	—	142	1	no	—
106	C₂×C₈	—	71	1	no	—
107	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	284	1	no	—
110	C₄⋊C₄	—	284	1	no	—
111	C₂²⋊C₄	—	284	1	no	—
112	C₄²	—	71	1	no	—
113	C₁₆	—	2	1	no	—
114	C₁₆	—	142	1	no	—
115	C₂³	—	284	1	no	—
116	Q₈	—	4	1	no	—
117	Q₈	—	4	1	no	—
118	Q₈	—	284	1	no	—
119	Q₈	—	284	1	no	—
120	D₄	—	284	1	no	—
121	D₄	—	142	2	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	284	1	no	—
125	C₂×C₄	—	568	2	no	—
126	C₂×C₄	—	71	1	no	—
127	C₂×C₄	—	71	1	no	—
128	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
129	C₈	—	142	1	no	—
130	C₈	D₄×D₇₁	1	1	yes	characteristic
131	C₈	#11765865	1	1	yes	characteristic
132	C₂²	—	284	2	no	—
133	C₂²	—	568	1	no	—
134	C₂²	—	568	1	no	—
135	C₂²	D₈×D₇₁	1	1	yes	characteristic
136	C₄	—	8	1	no	—
137	C₄	—	8	1	no	—
138	C₄	—	568	1	no	—
139	C₄	—	71	2	no	—
140	C₄	—	142	1	no	—
141	C₄	#14556101	1	1	yes	characteristic
142	C₄	#14556155	1	1	yes	characteristic
143	C₂	—	568	2	no	—
144	C₂	#17986193	1	1	yes	characteristic
145	C₂	SD₃₂×D₇₁	1	1	yes	characteristic
146	C₂	#17986441	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556073

C₂

yes

maximal, characteristic

C₂×#14556071

C₂

yes

maximal, characteristic

#17986377

C₂

yes

maximal, characteristic, complemented

#17986081

C₂

yes

maximal, characteristic

Dic₇₁⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#17985802

C₂

yes

maximal, characteristic, complemented

#358×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₈₄

C₂²

yes

characteristic

C₂×#11765855

—

C₂×D₅₆₈

C₂²

yes

—

—

—

—

C₂×C₇₁⋊C₁₆

C₂²

yes

characteristic

#14556048

—

C₈×Dic₇₁

C₂²

yes

characteristic

C₂×Q₁₆×C₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₇₁

C₂²

yes

characteristic

#107×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×D₂₈₄

—

C₂×C₄×D₇₁

—

Dic₂₈₄

—

Dic₂₈₄

—

#11765855

—

D₅₆₈

—

D₅₆₈

D₄

yes

—

C₂×C₇₁⋊C₈

D₄

yes

characteristic

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

(C₂×D₇₁)⋊C₄

—

C₄×Dic₇₁

D₄

yes

characteristic

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₁₆×C₇₁

—

C₂²×D₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

D₈

yes

characteristic

C₂×Dic₇₁

D₈

yes

characteristic

D₂₈₄

—

D₂₈₄

—

C₄×D₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

C₂×D₄

yes

characteristic

C₈×C₇₁

#41

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×D₈

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

Dic₇₁

SD₃₂

yes

complemented

C₂×C₇₁

#459

yes

characteristic

C₂×C₇₁

C₂×SD₃₂

yes

characteristic

C₂×C₇₁

#510

yes

characteristic

D₇₁

—

#2208

—

maximal, Sylow

C₇₁

#2208

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

C₂×SD₃₂

—

#494

—

C₄×D₈

—

C₄⋊C₁₆

—

#358

—

#358

D₇₁

yes

characteristic, complemented

C₂×Q₁₆

—

C₂×Q₁₆

—

142

—

C₂×Q₁₆

C₂×D₇₁

yes

characteristic

C₂×D₈

—

C₄⋊Q₈

—

142

—

C₄×D₄

—

142

—

SD₃₂

—

142

—

SD₃₂

—

142

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₁

yes

characteristic

#107

C₂×D₇₁

yes

characteristic

#102

—

142

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

142

—

C₂×D₄

—

142

—

C₂²×C₄

—

284

—

Q₁₆

—

100

Q₁₆

—

101

Q₁₆

—

284

—

102

Q₁₆

—

142

—

103

Q₁₆

—

142

—

104

D₈

—

105

D₈

—

142

—

106

C₂×C₈

—

107

C₂×C₈

C₂²×D₇₁

yes

characteristic

108

C₄⋊C₄

—

109

C₄⋊C₄

—

284

—

110

C₄⋊C₄

—

284

—

111

C₂²⋊C₄

—

284

—

112

C₄²

—

113

C₁₆

—

114

C₁₆

—

142

—

115

C₂³

—

284

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

284

—

119

Q₈

—

284

—

120

D₄

—

284

—

121

D₄

—

142

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

284

—

125

C₂×C₄

—

568

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

D₄×D₇₁

yes

characteristic

129

C₈

—

142

—

130

C₈

D₄×D₇₁

yes

characteristic

131

C₈

#11765865

yes

characteristic

132

C₂²

—

284

—

133

C₂²

—

568

—

134

C₂²

—

568

—

135

C₂²

D₈×D₇₁

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

568

—

139

C₄

—

140

C₄

—

142

—

141

C₄

#14556101

yes

characteristic

142

C₄

#14556155

yes

characteristic

143

C₂

—

568

—

144

C₂

#17986193

yes

characteristic

145

C₂

SD₃₂×D₇₁

yes

characteristic

146

C₂

#17986441

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₇₁
commutator	C₂×C₈×C₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	#358×C₇₁	C₂
socle	C₂²×C₇₁	C₂×D₈
hypercenter	#358	D₇₁
hypocenter	C₇₁	#2208
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₇₁

commutator

C₂×C₈×C₇₁

C₂³

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

#358×C₇₁

C₂

socle

C₂²×C₇₁

C₂×D₈

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
3. #17986377	C₂	semidirect
5. Dic₇₁⋊C₁₆	C₂	semidirect
6. #17985802	C₂	semidirect
7. #358×C₇₁	C₂	semidirect
68. Dic₇₁	SD₃₂	semidirect
74. C₇₁	#2208	semidirect
81. #358	D₇₁	semidirect

H ≅ G/N

type

3. #17986377

C₂

semidirect

5. Dic₇₁⋊C₁₆

C₂

semidirect

6. #17985802

C₂

semidirect

7. #358×C₇₁

C₂

semidirect

68. Dic₇₁

SD₃₂

semidirect

74. C₇₁

#2208

semidirect

81. #358

D₇₁

semidirect

Subgroups — #22590384

Conjugacy classes of subgroups

Complements