Group #22590268 — subgroups

Subgroups — #22590268

Order 9088.

256 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₂²×D₅₆₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556071	C₂	1	2	yes	maximal, complemented
3	#17985799	C₂	1	2	yes	maximal, complemented
4	C₂²⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
5	#467×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₈₄	—	2	1	no	—
7	C₂×D₅₆₈	—	2	4	no	—
8	C₂×D₅₆₈	—	2	1	no	—
9	C₂×D₅₆₈	C₂²	1	2	yes	—
10	#14556071	—	2	4	no	—
11	C₂×C₇₁⋊C₁₆	C₂²	1	2	yes	complemented
12	C₂×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₂²⋊Q₈×C₇₁	—	2	1	no	—
15	#107×C₇₁	C₂²	1	1	yes	characteristic, complemented
16	#103×C₇₁	—	2	1	no	—
17	C₂³×D₇₁	—	4	1	no	—
18	C₂×D₂₈₄	—	2	4	no	—
19	C₂×D₂₈₄	—	2	2	no	—
20	C₂×D₂₈₄	—	4	2	no	—
21	C₂×D₂₈₄	—	4	1	no	—
22	D₅₆₈	—	2	4	no	—
23	D₅₆₈	—	2	2	no	—
24	D₅₆₈	D₄	1	4	yes	—
25	C₇₁⋊C₁₆	—	2	2	no	—
26	C₂×Q₈×C₇₁	—	2	1	no	—
27	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
28	Q₁₆×C₇₁	—	2	2	no	—
29	C₂×C₈×C₇₁	—	2	2	no	—
30	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
31	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₇₁	—	2	1	no	—
33	C₄⋊C₄×C₇₁	—	4	1	no	—
34	C₂²⋊C₄×C₇₁	—	4	1	no	—
35	C₂³×C₇₁	D₈	1	1	yes	characteristic
36	C₂²×D₇₁	—	4	4	no	—
37	C₂²×D₇₁	—	4	2	no	—
38	C₂²×D₇₁	—	8	2	no	—
39	C₂²×D₇₁	—	8	2	no	—
40	Q₈×C₇₁	—	4	1	no	—
41	Q₈×C₇₁	—	4	1	no	—
42	C₂×C₄×C₇₁	—	2	1	no	—
43	C₂×C₄×C₇₁	—	2	1	no	—
44	C₂×C₄×C₇₁	—	4	1	no	—
45	C₂×C₄×C₇₁	—	4	1	no	—
46	C₂×C₄×C₇₁	D₈	1	1	yes	characteristic
47	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
48	D₂₈₄	—	2	4	no	—
49	D₂₈₄	—	4	4	no	—
50	D₂₈₄	—	4	2	no	—
51	C₈×C₇₁	—	2	1	no	—
52	C₈×C₇₁	C₂×D₄	1	2	yes	—
53	C₂²×C₇₁	—	2	1	no	—
54	C₂²×C₇₁	—	2	1	no	—
55	C₂²×C₇₁	SD₃₂	1	2	yes	complemented
56	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
57	C₂×D₇₁	—	4	4	no	—
58	C₂×D₇₁	—	8	8	no	—
59	C₂×D₇₁	—	8	2	no	—
60	C₂×D₇₁	—	8	2	no	—
61	C₄×C₇₁	—	2	1	no	—
62	C₄×C₇₁	—	8	1	no	—
63	C₄×C₇₁	—	8	1	no	—
64	C₄×C₇₁	#120	1	1	yes	characteristic
65	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂×C₇₁	—	2	2	no	—
67	C₂×C₇₁	#447	1	1	yes	characteristic
68	C₂×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
69	C₂×C₇₁	#509	1	1	yes	characteristic
70	D₇₁	—	8	4	no	—
71	#2187	—	71	1	no	maximal, Sylow
72	C₇₁	#2187	1	1	yes	characteristic, complemented, Sylow
73	C₂²×D₈	—	71	1	no	—
74	C₂×SD₃₂	—	71	2	no	—
75	#467	D₇₁	1	1	yes	characteristic, complemented
76	#357	—	71	2	no	—
77	C₂²⋊C₁₆	—	71	1	no	—
78	C₂²×D₄	—	142	1	no	—
79	C₂×Q₁₆	C₂×D₇₁	1	1	yes	characteristic, complemented
80	C₂×D₈	—	71	2	no	—
81	C₂×D₈	—	142	4	no	—
82	C₂×D₈	—	142	1	no	—
83	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
84	C₂²⋊Q₈	—	2	1	no	—
85	SD₃₂	—	142	4	no	—
86	C₂×C₁₆	—	71	2	no	—
87	#107	C₂×D₇₁	1	1	yes	characteristic, complemented
88	#103	—	2	1	no	—
89	C₂⁴	—	284	1	no	—
90	C₂×Q₈	—	2	1	no	—
91	C₂×D₄	—	284	2	no	—
92	C₂×D₄	—	284	1	no	—
93	C₂×D₄	—	142	4	no	—
94	C₂×D₄	—	142	2	no	—
95	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
96	Q₁₆	—	2	2	no	—
97	D₈	—	71	4	no	—
98	D₈	—	142	4	no	—
99	D₈	—	142	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	4	1	no	—
105	C₂²⋊C₄	—	4	1	no	—
106	C₁₆	—	142	2	no	—
107	C₂³	—	284	4	no	—
108	C₂³	—	284	2	no	—
109	C₂³	—	568	2	no	—
110	C₂³	—	568	2	no	—
111	C₂³	#11765850	1	1	yes	characteristic
112	Q₈	—	4	1	no	—
113	Q₈	—	4	1	no	—
114	D₄	—	284	4	no	—
115	D₄	—	284	2	no	—
116	D₄	—	142	4	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	#11765850	1	1	yes	characteristic
122	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	D₄×D₇₁	1	2	yes	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	284	4	no	—
128	C₂²	—	568	8	no	—
129	C₂²	—	568	2	no	—
130	C₂²	—	568	2	no	—
131	C₂²	#14556071	1	2	yes	complemented
132	C₂²	C₂×#11765850	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	8	1	no	—
135	C₄	—	8	1	no	—
136	C₄	#14556096	1	1	yes	characteristic
137	C₄	#14556153	1	1	yes	characteristic
138	C₂	—	2	2	no	—
139	C₂	—	568	4	no	—
140	C₂	#17986132	1	1	yes	characteristic
141	C₂	C₂×#14556071	1	1	yes	characteristic
142	C₂	#17986434	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₅₆₈

C₂

yes

maximal, characteristic

C₂×#14556071

C₂

yes

maximal, complemented

#17985799

C₂

yes

maximal, complemented

C₂²⋊(C₇₁⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

#467×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₈₄

—

C₂×D₅₆₈

—

C₂×D₅₆₈

—

C₂×D₅₆₈

C₂²

yes

—

#14556071

—

C₂×C₇₁⋊C₁₆

C₂²

yes

complemented

C₂×Q₁₆×C₇₁

C₂²

yes

characteristic, complemented

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂²⋊Q₈×C₇₁

—

#107×C₇₁

C₂²

yes

characteristic, complemented

#103×C₇₁

—

C₂³×D₇₁

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

D₅₆₈

—

D₅₆₈

—

D₅₆₈

D₄

yes

—

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

D₄

yes

characteristic

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₂³×C₇₁

D₈

yes

characteristic

C₂²×D₇₁

—

C₂²×D₇₁

—

C₂²×D₇₁

—

C₂²×D₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

D₈

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

C₈×C₇₁

—

C₈×C₇₁

C₂×D₄

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

SD₃₂

yes

complemented

C₂²×C₇₁

C₂×D₈

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₂×C₇₁

—

C₂×C₇₁

#447

yes

characteristic

C₂×C₇₁

C₂×SD₃₂

yes

characteristic

C₂×C₇₁

#509

yes

characteristic

D₇₁

—

#2187

—

maximal, Sylow

C₇₁

#2187

yes

characteristic, complemented, Sylow

C₂²×D₈

—

C₂×SD₃₂

—

#467

D₇₁

yes

characteristic, complemented

#357

—

C₂²⋊C₁₆

—

C₂²×D₄

—

142

—

C₂×Q₁₆

C₂×D₇₁

yes

characteristic, complemented

C₂×D₈

—

C₂×D₈

—

142

—

C₂×D₈

—

142

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₂²⋊Q₈

—

SD₃₂

—

142

—

C₂×C₁₆

—

#107

C₂×D₇₁

yes

characteristic, complemented

#103

—

C₂⁴

—

284

—

C₂×Q₈

—

C₂×D₄

—

284

—

C₂×D₄

—

284

—

C₂×D₄

—

142

—

C₂×D₄

—

142

—

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

Q₁₆

—

D₈

—

D₈

—

142

—

D₈

—

142

—

100

C₂×C₈

—

101

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

102

C₂×C₈

C₂²×D₇₁

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₂²⋊C₄

—

106

C₁₆

—

142

—

107

C₂³

—

284

—

108

C₂³

—

284

—

109

C₂³

—

568

—

110

C₂³

—

568

—

111

C₂³

#11765850

yes

characteristic

112

Q₈

—

113

Q₈

—

114

D₄

—

284

—

115

D₄

—

284

—

116

D₄

—

142

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

#11765850

yes

characteristic

122

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

123

C₈

—

124

C₈

D₄×D₇₁

yes

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

284

—

128

C₂²

—

568

—

129

C₂²

—

568

—

130

C₂²

—

568

—

131

C₂²

#14556071

yes

complemented

132

C₂²

C₂×#11765850

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#14556096

yes

characteristic

137

C₄

#14556153

yes

characteristic

138

C₂

—

139

C₂

—

568

—

140

C₂

#17986132

yes

characteristic

141

C₂

C₂×#14556071

yes

characteristic

142

C₂

#17986434

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11765850
commutator	C₂×C₈×C₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	#467×C₇₁	C₂
socle	C₂²×C₇₁	C₂×D₈
hypercenter	#467	D₇₁
hypocenter	C₇₁	#2187
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #467

subgroup

G/H

center

C₂²

C₂×#11765850

commutator

C₂×C₈×C₇₁

C₂³

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

#467×C₇₁

C₂

socle

C₂²×C₇₁

C₂×D₈

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #467

N	H ≅ G/N	type
2. C₂×#14556071	C₂	semidirect
3. #17985799	C₂	semidirect
4. C₂²⋊(C₇₁⋊C₁₆)	C₂	semidirect
5. #467×C₇₁	C₂	semidirect
11. C₂×C₇₁⋊C₁₆	C₂²	semidirect
12. C₂×Q₁₆×C₇₁	C₂²	semidirect
15. #107×C₇₁	C₂²	semidirect
55. C₂²×C₇₁	SD₃₂	semidirect
72. C₇₁	#2187	semidirect
75. #467	D₇₁	semidirect
79. C₂×Q₁₆	C₂×D₇₁	semidirect
87. #107	C₂×D₇₁	semidirect
131. C₂²	#14556071	semidirect

H ≅ G/N

type

2. C₂×#14556071

C₂

semidirect

3. #17985799

C₂

semidirect

4. C₂²⋊(C₇₁⋊C₁₆)

C₂

semidirect

5. #467×C₇₁

C₂

semidirect

11. C₂×C₇₁⋊C₁₆

C₂²

semidirect

12. C₂×Q₁₆×C₇₁

C₂²

semidirect

15. #107×C₇₁

C₂²

semidirect

55. C₂²×C₇₁

SD₃₂

semidirect

72. C₇₁

#2187

semidirect

75. #467

D₇₁

semidirect

79. C₂×Q₁₆

C₂×D₇₁

semidirect

87. #107

C₂×D₇₁

semidirect

131. C₂²

#14556071

semidirect

Subgroups — #22590268

Conjugacy classes of subgroups

Complements