Group #22590439 — subgroups

Subgroups — #22590439

Order 9088.

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

normal subgroups only

This group has 164 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×#14556071	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₆×D₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986237	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986236	C₂	1	1	yes	maximal, characteristic
5	#17985826	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985801	C₂	1	1	yes	maximal, characteristic, complemented
7	#358×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₅₆₈	C₂²	1	1	yes	characteristic
9	C₂×C₈×D₇₁	C₂²	1	1	yes	characteristic
10	#14556104	—	2	1	no	—
11	#14556103	—	2	1	no	—
12	#14556071	—	2	2	no	—
13	C₂×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₁₆×D₇₁	—	2	2	no	—
15	#14556062	C₂²	1	1	yes	characteristic
16	#14556052	—	2	1	no	—
17	#14556048	—	2	1	no	—
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×D₂₈₄	—	2	1	no	—
22	C₂×D₂₈₄	—	4	1	no	—
23	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
24	D₅₆₈	—	2	1	no	—
25	D₅₆₈	—	2	1	no	—
26	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
27	C₈×D₇₁	—	2	1	no	—
28	C₈×D₇₁	—	2	1	no	—
29	C₄⋊Dic₇₁	—	2	1	no	—
30	Dic₇₁⋊C₄	—	4	1	no	—
31	(C₂×D₇₁)⋊C₄	—	4	1	no	—
32	(C₂×D₇₁)⋊C₄	—	4	1	no	—
33	C₇₁⋊C₁₆	D₄	1	2	yes	complemented
34	C₂×Q₈×C₇₁	—	2	1	no	—
35	Q₁₆×C₇₁	—	2	1	no	—
36	Q₁₆×C₇₁	—	2	1	no	—
37	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₇₁	—	2	1	no	—
39	C₁₆×C₇₁	—	2	1	no	—
40	C₂²×D₇₁	—	4	1	no	—
41	C₂²×D₇₁	D₈	1	1	yes	characteristic
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	D₂₈₄	—	4	1	no	—
50	D₂₈₄	—	4	1	no	—
51	D₂₈₄	—	8	2	no	—
52	C₄×D₇₁	—	2	1	no	—
53	C₄×D₇₁	—	2	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	C₂×D₈	1	1	yes	characteristic
58	C₂×D₇₁	—	2	1	no	—
59	C₂×D₇₁	—	2	1	no	—
60	C₂×D₇₁	—	8	1	no	—
61	C₂×D₇₁	—	8	1	no	—
62	C₂×D₇₁	—	8	1	no	—
63	C₂×D₇₁	SD₃₂	1	2	yes	complemented
64	C₄×C₇₁	—	8	1	no	—
65	C₄×C₇₁	—	8	1	no	—
66	C₄×C₇₁	#121	1	1	yes	characteristic
67	C₄×C₇₁	#135	1	1	yes	characteristic
68	Dic₇₁	—	2	1	no	—
69	Dic₇₁	—	8	1	no	—
70	C₂×C₇₁	#466	1	1	yes	characteristic
71	C₂×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
72	C₂×C₇₁	#508	1	1	yes	characteristic
73	D₇₁	—	2	2	no	—
74	D₇₁	—	16	1	no	—
75	#2217	—	71	1	no	maximal, Sylow
76	C₇₁	#2217	1	1	yes	characteristic, complemented, Sylow
77	C₂×SD₃₂	—	71	1	no	—
78	C₂²×C₁₆	—	71	1	no	—
79	#467	—	71	1	no	—
80	#466	—	71	1	no	—
81	#367	—	71	1	no	—
82	#358	D₇₁	1	1	yes	characteristic, complemented
83	#357	—	71	1	no	—
84	C₂×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
85	C₂×D₈	—	71	1	no	—
86	C₂²×C₈	—	71	1	no	—
87	C₂²⋊Q₈	—	142	1	no	—
88	#121	—	142	1	no	—
89	SD₃₂	—	142	2	no	—
90	C₂×C₁₆	—	71	1	no	—
91	C₂×C₁₆	—	142	2	no	—
92	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
93	#107	—	71	1	no	—
94	#107	C₂×D₇₁	1	1	yes	characteristic
95	#103	—	142	1	no	—
96	#102	—	142	1	no	—
97	C₂×Q₈	—	2	1	no	—
98	C₂×D₄	—	284	1	no	—
99	C₂×D₄	—	142	1	no	—
100	C₂²×C₄	—	71	1	no	—
101	Q₁₆	—	2	1	no	—
102	Q₁₆	—	2	1	no	—
103	D₈	—	142	1	no	—
104	D₈	—	142	1	no	—
105	C₂×C₈	—	71	1	no	—
106	C₂×C₈	—	142	1	no	—
107	C₂×C₈	—	142	1	no	—
108	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	284	1	no	—
111	C₄⋊C₄	—	142	1	no	—
112	C₂²⋊C₄	—	284	1	no	—
113	C₂²⋊C₄	—	284	1	no	—
114	C₁₆	—	2	1	no	—
115	C₁₆	—	71	2	no	—
116	C₂³	—	284	1	no	—
117	C₂³	—	71	1	no	—
118	Q₈	—	4	1	no	—
119	Q₈	—	4	1	no	—
120	D₄	—	284	1	no	—
121	D₄	—	284	1	no	—
122	D₄	—	568	2	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	284	1	no	—
126	C₂×C₄	—	71	1	no	—
127	C₂×C₄	—	142	1	no	—
128	C₂×C₄	—	142	1	no	—
129	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
130	C₈	—	142	1	no	—
131	C₈	D₄×D₇₁	1	1	yes	characteristic
132	C₈	#11765865	1	1	yes	characteristic
133	C₂²	—	568	1	no	—
134	C₂²	—	568	1	no	—
135	C₂²	—	568	1	no	—
136	C₂²	—	71	2	no	—
137	C₂²	—	142	1	no	—
138	C₂²	—	142	1	no	—
139	C₂²	D₈×D₇₁	1	1	yes	characteristic
140	C₄	—	8	1	no	—
141	C₄	—	8	1	no	—
142	C₄	—	568	1	no	—
143	C₄	—	142	1	no	—
144	C₄	#14556101	1	1	yes	characteristic
145	C₄	#14556149	1	1	yes	characteristic
146	C₂	—	1136	1	no	—
147	C₂	—	142	2	no	—
148	C₂	#17986234	1	1	yes	characteristic
149	C₂	SD₃₂×D₇₁	1	1	yes	characteristic
150	C₂	#17986432	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556071

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×D₇₁

C₂

yes

maximal, characteristic, complemented

#17986237

C₂

yes

maximal, characteristic, complemented

#17986236

C₂

yes

maximal, characteristic

#17985826

C₂

yes

maximal, characteristic, complemented

#17985801

C₂

yes

maximal, characteristic, complemented

#358×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×D₅₆₈

C₂²

yes

characteristic

C₂×C₈×D₇₁

C₂²

yes

—

—

—

C₂×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₁₆×D₇₁

—

#14556062

C₂²

yes

characteristic

#14556052

—

#14556048

—

C₂×Q₁₆×C₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₇₁

C₂²

yes

characteristic

#107×C₇₁

C₂²

yes

characteristic

C₂×D₂₈₄

—

C₂×D₂₈₄

—

C₂×C₄×D₇₁

D₄

yes

characteristic

D₅₆₈

—

D₅₆₈

—

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₈×D₇₁

—

C₈×D₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

(C₂×D₇₁)⋊C₄

—

(C₂×D₇₁)⋊C₄

—

C₇₁⋊C₁₆

D₄

yes

complemented

C₂×Q₈×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₁₆×C₇₁

—

C₂²×D₇₁

—

C₂²×D₇₁

D₈

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

D₈

yes

characteristic

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

C₄×D₇₁

—

C₄×D₇₁

—

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

—

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

SD₃₂

yes

complemented

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

#466

yes

characteristic

C₂×C₇₁

C₂×SD₃₂

yes

characteristic

C₂×C₇₁

#508

yes

characteristic

D₇₁

—

D₇₁

—

#2217

—

maximal, Sylow

C₇₁

#2217

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

C₂²×C₁₆

—

—

—

—

D₇₁

yes

characteristic, complemented

#357

—

C₂×Q₁₆

C₂×D₇₁

yes

characteristic

C₂×D₈

—

C₂²×C₈

—

C₂²⋊Q₈

—

142

—

#121

—

142

—

SD₃₂

—

142

—

C₂×C₁₆

—

C₂×C₁₆

—

142

—

C₂×C₁₆

C₂×D₇₁

yes

characteristic

#107

—

#107

C₂×D₇₁

yes

characteristic

#103

—

142

—

#102

—

142

—

C₂×Q₈

—

C₂×D₄

—

284

—

C₂×D₄

—

142

—

100

C₂²×C₄

—

101

Q₁₆

—

102

Q₁₆

—

103

D₈

—

142

—

104

D₈

—

142

—

105

C₂×C₈

—

106

C₂×C₈

—

142

—

107

C₂×C₈

—

142

—

108

C₂×C₈

C₂²×D₇₁

yes

characteristic

109

C₄⋊C₄

—

110

C₄⋊C₄

—

284

—

111

C₄⋊C₄

—

142

—

112

C₂²⋊C₄

—

284

—

113

C₂²⋊C₄

—

284

—

114

C₁₆

—

115

C₁₆

—

116

C₂³

—

284

—

117

C₂³

—

118

Q₈

—

119

Q₈

—

120

D₄

—

284

—

121

D₄

—

284

—

122

D₄

—

568

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

284

—

126

C₂×C₄

—

127

C₂×C₄

—

142

—

128

C₂×C₄

—

142

—

129

C₂×C₄

D₄×D₇₁

yes

characteristic

130

C₈

—

142

—

131

C₈

D₄×D₇₁

yes

characteristic

132

C₈

#11765865

yes

characteristic

133

C₂²

—

568

—

134

C₂²

—

568

—

135

C₂²

—

568

—

136

C₂²

—

137

C₂²

—

142

—

138

C₂²

—

142

—

139

C₂²

D₈×D₇₁

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

568

—

143

C₄

—

142

—

144

C₄

#14556101

yes

characteristic

145

C₄

#14556149

yes

characteristic

146

C₂

—

1136

—

147

C₂

—

142

—

148

C₂

#17986234

yes

characteristic

149

C₂

SD₃₂×D₇₁

yes

characteristic

150

C₂

#17986432

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	#2217
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
1. C₂×#14556071	C₂	semidirect
2. C₂×C₁₆×D₇₁	C₂	semidirect
3. #17986237	C₂	semidirect
5. #17985826	C₂	semidirect
6. #17985801	C₂	semidirect
7. #358×C₇₁	C₂	semidirect
33. C₇₁⋊C₁₆	D₄	semidirect
63. C₂×D₇₁	SD₃₂	semidirect
76. C₇₁	#2217	semidirect
82. #358	D₇₁	semidirect

H ≅ G/N

type

1. C₂×#14556071

C₂

semidirect

2. C₂×C₁₆×D₇₁

C₂

semidirect

3. #17986237

C₂

semidirect

5. #17985826

C₂

semidirect

6. #17985801

C₂

semidirect

7. #358×C₇₁

C₂

semidirect

33. C₇₁⋊C₁₆

D₄

semidirect

63. C₂×D₇₁

SD₃₂

semidirect

76. C₇₁

#2217

semidirect

82. #358

D₇₁

semidirect

Subgroups — #22590439

Conjugacy classes of subgroups

Complements