Group #22590401 — subgroups

Subgroups — #22590401

Order 9088.

158 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 47 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×Dic₅₆₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14556072	C₂	1	1	yes	maximal, characteristic
3	#17986382	C₂	1	1	yes	maximal, characteristic
4	#17986093	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985802	C₂	1	1	yes	maximal, characteristic, complemented
7	#357×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₈₄	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₈₄	C₂²	1	1	yes	characteristic
10	C₂×#11765853	—	2	1	no	—
11	#14556115	—	2	1	no	—
12	#14556093	—	2	1	no	—
13	Dic₅₆₈	—	2	2	no	—
14	#14556072	—	2	2	no	—
15	C₂×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
16	#14556051	—	2	1	no	—
17	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
18	C₂×D₈×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×Dic₁₄₂	—	2	1	no	—
23	Dic₂₈₄	—	2	1	no	—
24	Dic₂₈₄	—	2	1	no	—
25	Dic₂₈₄	—	2	1	no	—
26	Dic₂₈₄	D₄	1	2	yes	—
27	#11765853	—	4	2	no	—
28	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
29	Dic₇₁⋊C₄	—	4	1	no	—
30	C₂²⋊Dic₇₁	—	4	1	no	—
31	C₂²⋊Dic₇₁	—	4	1	no	—
32	C₄×Dic₇₁	—	4	1	no	—
33	C₄×Dic₇₁	D₄	1	1	yes	characteristic
34	C₇₁⋊C₁₆	—	2	1	no	—
35	C₂×D₄×C₇₁	—	2	1	no	—
36	D₈×C₇₁	—	2	1	no	—
37	D₈×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	C₂³×C₇₁	—	4	1	no	—
42	D₄×C₇₁	—	4	1	no	—
43	D₄×C₇₁	—	4	1	no	—
44	C₂×C₄×C₇₁	—	4	1	no	—
45	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
46	C₂×Dic₇₁	—	4	1	no	—
47	C₂×Dic₇₁	—	4	1	no	—
48	C₂×Dic₇₁	D₈	1	1	yes	characteristic
49	C₂×Dic₇₁	D₈	1	1	yes	characteristic
50	Dic₁₄₂	—	2	2	no	—
51	Dic₁₄₂	—	4	1	no	—
52	Dic₁₄₂	—	4	1	no	—
53	Dic₁₄₂	—	4	1	no	—
54	C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
55	C₈×C₇₁	#41	1	1	yes	characteristic
56	C₇₁⋊C₈	—	2	1	no	—
57	C₂²×C₇₁	—	8	1	no	—
58	C₂²×C₇₁	—	8	1	no	—
59	C₂²×C₇₁	—	8	1	no	—
60	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
61	C₄×C₇₁	—	8	1	no	—
62	C₄×C₇₁	#121	1	1	yes	characteristic
63	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
64	Dic₇₁	—	2	1	no	—
65	Dic₇₁	—	2	1	no	—
66	Dic₇₁	—	4	2	no	—
67	Dic₇₁	—	8	1	no	—
68	C₂×C₇₁	—	16	1	no	—
69	C₂×C₇₁	#459	1	1	yes	characteristic
70	C₂×C₇₁	#508	1	1	yes	characteristic
71	C₂×C₇₁	#510	1	1	yes	characteristic
72	#2211	—	71	1	no	maximal, Sylow
73	C₇₁	#2211	1	1	yes	characteristic, complemented, Sylow
74	C₂×Q₃₂	—	71	1	no	—
75	C₂×SD₃₂	—	71	1	no	—
76	#495	—	71	1	no	—
77	C₄×Q₁₆	—	71	1	no	—
78	C₄⋊C₁₆	—	71	1	no	—
79	#358	—	71	1	no	—
80	#357	D₇₁	1	1	yes	characteristic
81	C₂×Q₁₆	—	71	1	no	—
82	C₂×Q₁₆	—	71	1	no	—
83	C₂×SD₁₆	—	142	1	no	—
84	C₂×D₈	C₂×D₇₁	1	1	yes	characteristic
85	#124	—	142	1	no	—
86	C₄×Q₈	—	142	1	no	—
87	Q₃₂	—	142	2	no	—
88	SD₃₂	—	142	2	no	—
89	C₂×C₁₆	—	71	1	no	—
90	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
91	#107	C₂×D₇₁	1	1	yes	characteristic
92	#103	—	142	1	no	—
93	C₄×C₈	—	71	1	no	—
94	C₂×Q₈	—	142	1	no	—
95	C₂×Q₈	—	142	1	no	—
96	C₂×D₄	—	2	1	no	—
97	Q₁₆	—	71	2	no	—
98	Q₁₆	—	142	1	no	—
99	Q₁₆	—	142	1	no	—
100	Q₁₆	—	142	1	no	—
101	SD₁₆	—	284	2	no	—
102	D₈	—	2	1	no	—
103	D₈	—	2	1	no	—
104	C₂×C₈	—	71	1	no	—
105	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	284	1	no	—
108	C₂²⋊C₄	—	284	1	no	—
109	C₂²⋊C₄	—	284	1	no	—
110	C₄²	—	284	1	no	—
111	C₄²	—	71	1	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	—	142	1	no	—
114	C₂³	—	4	1	no	—
115	Q₈	—	284	1	no	—
116	Q₈	—	284	1	no	—
117	Q₈	—	284	1	no	—
118	Q₈	—	142	2	no	—
119	D₄	—	4	1	no	—
120	D₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	284	1	no	—
123	C₂×C₄	—	284	1	no	—
124	C₂×C₄	—	71	1	no	—
125	C₂×C₄	—	71	1	no	—
126	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
127	C₈	—	142	1	no	—
128	C₈	D₄×D₇₁	1	1	yes	characteristic
129	C₈	#11765865	1	1	yes	characteristic
130	C₂²	—	8	1	no	—
131	C₂²	—	8	1	no	—
132	C₂²	—	8	1	no	—
133	C₂²	D₈×D₇₁	1	1	yes	characteristic
134	C₄	—	8	1	no	—
135	C₄	—	284	2	no	—
136	C₄	—	568	1	no	—
137	C₄	—	142	1	no	—
138	C₄	—	142	1	no	—
139	C₄	#14556101	1	1	yes	characteristic
140	C₄	#14556155	1	1	yes	characteristic
141	C₂	—	16	1	no	—
142	C₂	#17986193	1	1	yes	characteristic
143	C₂	#17986428	1	1	yes	characteristic
144	C₂	#17986438	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₅₆₈

C₂

yes

maximal, characteristic, complemented

C₂×#14556072

C₂

yes

maximal, characteristic

#17986382

C₂

yes

maximal, characteristic

#17986093

C₂

yes

maximal, characteristic, complemented

Dic₇₁⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#17985802

C₂

yes

maximal, characteristic, complemented

#357×C₇₁

C₂

yes

maximal, characteristic

C₂×Dic₂₈₄

C₂²

yes

characteristic

C₂×Dic₂₈₄

C₂²

yes

—

—

—

Dic₅₆₈

—

#14556072

—

C₂×C₇₁⋊C₁₆

C₂²

yes

characteristic

#14556051

—

C₈×Dic₇₁

C₂²

yes

characteristic

C₂×D₈×C₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₇₁

C₂²

yes

characteristic

#107×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

D₄

yes

—

#11765853

—

C₂×C₇₁⋊C₈

D₄

yes

characteristic

Dic₇₁⋊C₄

—

C₂²⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

D₄

yes

characteristic

C₇₁⋊C₁₆

—

C₂×D₄×C₇₁

—

D₈×C₇₁

—

D₈×C₇₁

—

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₁₆×C₇₁

—

C₂³×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

D₈

yes

characteristic

C₂×Dic₇₁

D₈

yes

characteristic

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

C₂×D₄

yes

characteristic

C₈×C₇₁

#41

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×D₈

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×SD₃₂

—

#495

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

#358

—

#357

D₇₁

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

142

—

C₂×D₈

C₂×D₇₁

yes

characteristic

#124

—

142

—

C₄×Q₈

—

142

—

Q₃₂

—

142

—

SD₃₂

—

142

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₁

yes

characteristic

#107

C₂×D₇₁

yes

characteristic

#103

—

142

—

C₄×C₈

—

C₂×Q₈

—

142

—

C₂×Q₈

—

142

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

142

—

Q₁₆

—

142

—

100

Q₁₆

—

142

—

101

SD₁₆

—

284

—

102

D₈

—

103

D₈

—

104

C₂×C₈

—

105

C₂×C₈

C₂²×D₇₁

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

284

—

108

C₂²⋊C₄

—

284

—

109

C₂²⋊C₄

—

284

—

110

C₄²

—

284

—

111

C₄²

—

112

C₁₆

—

113

C₁₆

—

142

—

114

C₂³

—

115

Q₈

—

284

—

116

Q₈

—

284

—

117

Q₈

—

284

—

118

Q₈

—

142

—

119

D₄

—

120

D₄

—

121

C₂×C₄

—

122

C₂×C₄

—

284

—

123

C₂×C₄

—

284

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

D₄×D₇₁

yes

characteristic

127

C₈

—

142

—

128

C₈

D₄×D₇₁

yes

characteristic

129

C₈

#11765865

yes

characteristic

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

D₈×D₇₁

yes

characteristic

134

C₄

—

135

C₄

—

284

—

136

C₄

—

568

—

137

C₄

—

142

—

138

C₄

—

142

—

139

C₄

#14556101

yes

characteristic

140

C₄

#14556155

yes

characteristic

141

C₂

—

142

C₂

#17986193

yes

characteristic

143

C₂

#17986428

yes

characteristic

144

C₂

#17986438

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₇₁
commutator	C₂×C₈×C₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	#357×C₇₁	C₂
socle	C₂²×C₇₁	C₂×D₈
hypercenter	#357	D₇₁
hypocenter	C₇₁	#2211
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₇₁

commutator

C₂×C₈×C₇₁

C₂³

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

#357×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 ◁ #357

N	H ≅ G/N	type
1. C₂×Dic₅₆₈	C₂	semidirect
4. #17986093	C₂	semidirect
5. Dic₇₁⋊C₁₆	C₂	semidirect
6. #17985802	C₂	semidirect
73. C₇₁	#2211	semidirect

H ≅ G/N

type

1. C₂×Dic₅₆₈

C₂

semidirect

4. #17986093

C₂

semidirect

5. Dic₇₁⋊C₁₆

C₂

semidirect

6. #17985802

C₂

semidirect

73. C₇₁

#2211

semidirect

Subgroups — #22590401

Conjugacy classes of subgroups

Complements