Group #22589302 — subgroups

Subgroups — #22589302

Order 9088.

300 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 145 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×C₄×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#17985869	C₂	1	1	yes	maximal, characteristic
3	#17985761	C₂	1	4	yes	maximal
4	C₂×C₄⋊Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₇₁	—	2	1	no	—
7	C₂×C₄⋊Dic₇₁	—	2	1	no	—
8	C₂×C₄⋊Dic₇₁	C₂²	1	2	yes	—
9	C₄×C₇₁⋊C₈	—	2	2	no	—
10	C₂²×Q₈×C₇₁	—	2	1	no	—
11	C₄⋊Q₈×C₇₁	—	2	2	no	—
12	C₄⋊Q₈×C₇₁	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
14	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
16	C₂²×Dic₇₁	—	2	2	no	—
17	C₂×C₇₁⋊C₈	—	2	2	no	—
18	C₂×C₇₁⋊C₈	—	2	2	no	—
19	C₂×C₇₁⋊C₈	Q₈	1	4	yes	complemented
20	C₄⋊Dic₇₁	—	2	4	no	—
21	C₄⋊Dic₇₁	—	4	2	no	—
22	C₄⋊Dic₇₁	—	4	2	no	—
23	C₂×Q₈×C₇₁	—	2	4	no	—
24	C₂×Q₈×C₇₁	—	4	2	no	—
25	C₂×Q₈×C₇₁	—	4	2	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	4	no	—
31	C₄⋊C₄×C₇₁	—	2	2	no	—
32	C₄⋊C₄×C₇₁	C₂×C₄	1	4	yes	—
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	Q₈×C₇₁	—	4	8	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	D₈	1	2	yes	—
43	C₂×C₄×C₇₁	SD₁₆	1	2	yes	—
44	C₂×C₄×C₇₁	SD₁₆	1	2	yes	—
45	C₂×C₄×C₇₁	Q₁₆	1	2	yes	—
46	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇₁	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₇₁	#41	1	2	yes	—
50	C₂×Dic₇₁	—	4	2	no	—
51	C₂×Dic₇₁	—	4	2	no	—
52	C₂×Dic₇₁	—	4	2	no	—
53	C₇₁⋊C₈	—	2	4	no	—
54	C₂²×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₇₁	#124	1	1	yes	characteristic
56	C₂²×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
57	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
58	C₂²×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
59	C₂²×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
60	C₂²×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
61	C₄×C₇₁	—	4	4	no	—
62	C₄×C₇₁	#102	1	4	yes	—
63	C₄×C₇₁	#103	1	4	yes	—
64	C₄×C₇₁	C₄×Q₈	1	2	yes	—
65	C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
66	Dic₇₁	—	8	2	no	—
67	C₂×C₇₁	#391	1	1	yes	characteristic
68	C₂×C₇₁	C₂×#102	1	1	yes	characteristic
69	C₂×C₇₁	C₂×#103	1	1	yes	characteristic
70	C₂×C₇₁	#486	1	1	yes	characteristic
71	C₂×C₇₁	#487	1	1	yes	characteristic
72	C₂×C₇₁	#498	1	1	yes	characteristic
73	C₂×C₇₁	#500	1	1	yes	characteristic
74	#1990	—	71	1	no	maximal, Sylow
75	C₇₁	#1990	1	1	yes	characteristic, complemented, Sylow
76	C₂×C₄⋊Q₈	D₇₁	1	1	yes	characteristic
77	C₂×C₄×C₈	—	71	1	no	—
78	#384	—	71	1	no	—
79	#340	—	71	4	no	—
80	C₂²×Q₈	—	2	1	no	—
81	C₂²×C₈	—	71	2	no	—
82	C₄⋊Q₈	—	2	2	no	—
83	C₄⋊Q₈	Dic₇₁	1	4	yes	complemented
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	71	2	no	—
86	C₂×C₄⋊C₄	—	142	1	no	—
87	C₂×C₄⋊C₄	—	142	1	no	—
88	C₂×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
89	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
90	C₄×C₈	—	142	2	no	—
91	C₂×Q₈	—	2	4	no	—
92	C₂×Q₈	—	4	2	no	—
93	C₂×Q₈	—	4	2	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	142	2	no	—
96	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
99	C₂×C₈	—	71	4	no	—
100	C₂×C₈	—	142	2	no	—
101	C₂×C₈	—	142	2	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	284	2	no	—
105	C₄⋊C₄	—	284	2	no	—
106	C₄⋊C₄	—	142	4	no	—
107	C₄⋊C₄	C₂×Dic₇₁	1	4	yes	—
108	C₄²	C₂×Dic₇₁	1	2	yes	—
109	C₄²	C₂²⋊D₇₁	1	2	yes	—
110	C₂³	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
111	Q₈	—	4	8	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	284	2	no	—
116	C₂×C₄	—	284	2	no	—
117	C₂×C₄	—	284	2	no	—
118	C₂×C₄	C₂²⋊Dic₇₁	1	2	yes	—
119	C₂×C₄	C₂²⋊Dic₇₁	1	2	yes	—
120	C₂×C₄	#11765850	1	2	yes	—
121	C₂×C₄	#11765852	1	2	yes	—
122	C₂×C₄	#11765853	1	2	yes	—
123	C₂×C₄	#11765855	1	2	yes	—
124	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
126	C₂×C₄	Q₈×D₇₁	1	2	yes	—
127	C₂×C₄	#11765866	1	2	yes	—
128	C₈	—	142	4	no	—
129	C₂²	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
130	C₂²	#14556117	1	1	yes	characteristic
131	C₂²	#14556126	1	1	yes	characteristic
132	C₂²	C₂×#11765850	1	1	yes	characteristic
133	C₂²	C₂×#11765852	1	1	yes	characteristic
134	C₂²	C₂×#11765853	1	1	yes	characteristic
135	C₂²	C₂×#11765855	1	1	yes	characteristic
136	C₄	—	4	4	no	—
137	C₄	—	568	2	no	—
138	C₄	#14556049	1	4	yes	—
139	C₄	#14556052	1	4	yes	—
140	C₄	Q₈×Dic₇₁	1	2	yes	—
141	C₄	#14556104	1	2	yes	—
142	C₂	#17985906	1	1	yes	characteristic
143	C₂	C₂×#14556049	1	1	yes	characteristic
144	C₂	C₂×#14556052	1	1	yes	characteristic
145	C₂	#17986347	1	1	yes	characteristic
146	C₂	#17986351	1	1	yes	characteristic
147	C₂	#17986395	1	1	yes	characteristic
148	C₂	#17986402	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₇₁⋊C₈

C₂

yes

maximal, characteristic

#17985869

C₂

yes

maximal, characteristic

#17985761

C₂

yes

maximal

C₂×C₄⋊Q₈×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

C₂²

yes

—

C₄×C₇₁⋊C₈

—

C₂²×Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

Q₈

yes

complemented

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

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

—

C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

D₄

yes

—

C₂³×C₇₁

C₂×D₄

yes

characteristic

Q₈×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

D₈

yes

—

C₂×C₄×C₇₁

SD₁₆

yes

—

C₂×C₄×C₇₁

SD₁₆

yes

—

C₂×C₄×C₇₁

Q₁₆

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

#41

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₁

#124

yes

characteristic

C₂²×C₇₁

C₄⋊Q₈

yes

characteristic

C₂²×C₇₁

C₂×D₈

yes

characteristic

C₂²×C₇₁

C₂×SD₁₆

yes

characteristic

C₂²×C₇₁

C₂×SD₁₆

yes

characteristic

C₂²×C₇₁

C₂×Q₁₆

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

#102

yes

—

C₄×C₇₁

#103

yes

—

C₄×C₇₁

C₄×Q₈

yes

—

C₄×C₇₁

C₂²⋊Q₈

yes

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₇₁

yes

characteristic

C₂×C₄×C₈

—

#384

—

#340

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₇₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄⋊C₄

C₂×D₇₁

yes

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄×C₈

—

142

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

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

—

100

C₂×C₈

—

142

—

101

C₂×C₈

—

142

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

284

—

105

C₄⋊C₄

—

284

—

106

C₄⋊C₄

—

142

—

107

C₄⋊C₄

C₂×Dic₇₁

yes

—

108

C₄²

C₂×Dic₇₁

yes

—

109

C₄²

C₂²⋊D₇₁

yes

—

110

C₂³

C₂×C₂²⋊D₇₁

yes

characteristic

111

Q₈

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

284

—

116

C₂×C₄

—

284

—

117

C₂×C₄

—

284

—

118

C₂×C₄

C₂²⋊Dic₇₁

yes

—

119

C₂×C₄

C₂²⋊Dic₇₁

yes

—

120

C₂×C₄

#11765850

yes

—

121

C₂×C₄

#11765852

yes

—

122

C₂×C₄

#11765853

yes

—

123

C₂×C₄

#11765855

yes

—

124

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

126

C₂×C₄

Q₈×D₇₁

yes

—

127

C₂×C₄

#11765866

yes

—

128

C₈

—

142

—

129

C₂²

C₂×C₂²⋊Dic₇₁

yes

characteristic

130

C₂²

#14556117

yes

characteristic

131

C₂²

#14556126

yes

characteristic

132

C₂²

C₂×#11765850

yes

characteristic

133

C₂²

C₂×#11765852

yes

characteristic

134

C₂²

C₂×#11765853

yes

characteristic

135

C₂²

C₂×#11765855

yes

characteristic

136

C₄

—

137

C₄

—

568

—

138

C₄

#14556049

yes

—

139

C₄

#14556052

yes

—

140

C₄

Q₈×Dic₇₁

yes

—

141

C₄

#14556104

yes

—

142

C₂

#17985906

yes

characteristic

143

C₂

C₂×#14556049

yes

characteristic

144

C₂

C₂×#14556052

yes

characteristic

145

C₂

#17986347

yes

characteristic

146

C₂

#17986351

yes

characteristic

147

C₂

#17986395

yes

characteristic

148

C₂

#17986402

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	C₂×C₄⋊Q₈×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₇₁
hypocenter	C₇₁	#1990
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

C₂×C₄⋊Q₈×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₇₁

hypocenter

C₇₁

#1990

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₇₁	C₄	semidirect
19. C₂×C₇₁⋊C₈	Q₈	semidirect
75. C₇₁	#1990	semidirect
83. C₄⋊Q₈	Dic₇₁	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₇₁

C₄

semidirect

19. C₂×C₇₁⋊C₈

Q₈

semidirect

75. C₇₁

#1990

semidirect

83. C₄⋊Q₈

Dic₇₁

semidirect

Subgroups — #22589302

Conjugacy classes of subgroups

Complements