Group #22590218 — subgroups

Subgroups — #22590218

Order 9088.

146 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 53 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	#17986429	C₂	1	1	yes	maximal, characteristic
2	#17986116	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986115	C₂	1	1	yes	maximal, characteristic, complemented
4	#17985828	C₂	1	1	yes	maximal, characteristic, complemented
5	#17985807	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#359×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#14556150	C₂²	1	1	yes	characteristic
9	#14556136	—	2	1	no	—
10	#14556135	—	2	1	no	—
11	#14556074	C₄	1	1	yes	characteristic, complemented
12	#14556072	C₄	1	1	yes	characteristic, complemented
13	#14556071	C₄	1	1	yes	characteristic, complemented
14	#14556069	C₄	1	1	yes	characteristic, complemented
15	C₂×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
16	#14556064	C₂²	1	1	yes	characteristic
17	#14556055	—	2	1	no	—
18	#14556054	—	2	1	no	—
19	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
20	#135×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
22	#108×C₇₁	C₂²	1	1	yes	characteristic
23	#11765865	—	2	1	no	—
24	Dic₂₈₄	C₂×C₄	1	1	yes	characteristic
25	#11765854	—	2	1	no	—
26	D₅₆₈	C₂×C₄	1	1	yes	characteristic
27	#11765849	—	2	1	no	—
28	#11765849	—	4	1	no	—
29	#11765848	—	4	1	no	—
30	C₂×C₇₁⋊C₈	—	4	1	no	—
31	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
32	C₈×D₇₁	—	4	1	no	—
33	C₄×Dic₇₁	D₄	1	1	yes	characteristic
34	C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
35	C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
36	#41×C₇₁	—	2	1	no	—
37	Q₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
38	SD₁₆×C₇₁	—	2	1	no	—
39	D₈×C₇₁	C₂×C₄	1	1	yes	characteristic
40	M₁₆×C₇₁	—	2	1	no	—
41	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
42	C₁₆×C₇₁	—	2	1	no	—
43	Q₈×C₇₁	—	2	1	no	—
44	D₄×C₇₁	—	2	1	no	—
45	D₄×C₇₁	—	4	1	no	—
46	C₂×C₄×C₇₁	—	4	1	no	—
47	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₂²⋊D₇₁	—	4	1	no	—
49	C₂×Dic₇₁	—	2	1	no	—
50	D₂₈₄	—	2	1	no	—
51	C₄×D₇₁	—	4	1	no	—
52	Dic₁₄₂	—	2	1	no	—
53	C₈×C₇₁	—	4	1	no	—
54	C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
55	C₈×C₇₁	#41	1	1	yes	characteristic
56	C₇₁⋊C₈	—	4	1	no	—
57	C₇₁⋊C₈	D₈	1	1	yes	characteristic
58	C₇₁⋊C₈	D₈	1	1	yes	characteristic
59	C₂²×C₇₁	—	4	1	no	—
60	C₂²×C₇₁	#135	1	1	yes	characteristic
61	C₂×D₇₁	—	4	1	no	—
62	C₄×C₇₁	—	4	1	no	—
63	C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
64	C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
65	Dic₇₁	—	2	2	no	—
66	Dic₇₁	—	4	1	no	—
67	C₂×C₇₁	—	2	1	no	—
68	C₂×C₇₁	—	8	1	no	—
69	C₂×C₇₁	C₄×D₈	1	1	yes	characteristic
70	D₇₁	—	8	1	no	—
71	#2178	—	71	1	no	maximal, Sylow
72	C₇₁	#2178	1	1	yes	characteristic, complemented, Sylow
73	#508	—	71	1	no	—
74	#443	—	71	1	no	—
75	#443	—	71	1	no	—
76	#368	—	71	1	no	—
77	#359	—	71	1	no	—
78	#359	D₇₁	1	1	yes	characteristic, complemented
79	C₄×C₁₆	—	71	1	no	—
80	#135	—	71	1	no	—
81	#135	C₂×D₇₁	1	1	yes	characteristic
82	#131	—	142	1	no	—
83	#131	—	142	1	no	—
84	Q₃₂	—	71	1	no	—
85	SD₃₂	—	71	1	no	—
86	SD₃₂	—	71	1	no	—
87	D₁₆	—	71	1	no	—
88	C₂×C₁₆	—	71	1	no	—
89	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
90	#108	—	71	1	no	—
91	#108	C₂×D₇₁	1	1	yes	characteristic
92	#104	—	142	1	no	—
93	#104	—	142	1	no	—
94	C₄×C₈	—	71	1	no	—
95	#41	—	2	1	no	—
96	#41	—	142	1	no	—
97	Q₁₆	—	71	1	no	—
98	Q₁₆	C₄×D₇₁	1	1	yes	characteristic
99	SD₁₆	—	2	1	no	—
100	SD₁₆	—	142	1	no	—
101	D₈	—	71	1	no	—
102	D₈	C₄×D₇₁	1	1	yes	characteristic
103	M₁₆	—	2	1	no	—
104	M₁₆	—	284	1	no	—
105	M₁₆	—	284	1	no	—
106	M₁₆	—	142	1	no	—
107	C₂×C₈	—	284	1	no	—
108	C₂×C₈	—	284	1	no	—
109	C₂×C₈	—	71	1	no	—
110	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
111	C₄²	—	71	1	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	—	71	1	no	—
114	C₁₆	—	71	1	no	—
115	Q₈	—	2	1	no	—
116	Q₈	—	142	1	no	—
117	D₄	—	2	1	no	—
118	D₄	—	4	1	no	—
119	D₄	—	284	1	no	—
120	D₄	—	142	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	284	1	no	—
123	C₂×C₄	—	142	1	no	—
124	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
125	C₈	—	4	1	no	—
126	C₈	—	284	1	no	—
127	C₈	—	71	1	no	—
128	C₈	—	71	1	no	—
129	C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
130	C₈	#11765864	1	1	yes	characteristic
131	C₂²	—	4	1	no	—
132	C₂²	—	284	1	no	—
133	C₂²	#14556149	1	1	yes	characteristic
134	C₄	—	4	1	no	—
135	C₄	—	284	1	no	—
136	C₄	—	142	2	no	—
137	C₄	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
138	C₄	D₈×D₇₁	1	1	yes	characteristic
139	C₂	—	2	1	no	—
140	C₂	—	8	1	no	—
141	C₂	—	568	1	no	—
142	C₂	#17986079	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986429

C₂

yes

maximal, characteristic

#17986116

C₂

yes

maximal, characteristic, complemented

#17986115

C₂

yes

maximal, characteristic, complemented

#17985828

C₂

yes

maximal, characteristic, complemented

#17985807

C₂

yes

maximal, characteristic, complemented

C₁₆×Dic₇₁

C₂

yes

maximal, characteristic, complemented

#359×C₇₁

C₂

yes

maximal, characteristic, complemented

#14556150

C₂²

yes

—

—

C₄

yes

characteristic, complemented

#14556072

C₄

yes

characteristic, complemented

#14556071

C₄

yes

characteristic, complemented

#14556069

C₄

yes

characteristic, complemented

C₂×C₇₁⋊C₁₆

C₂²

yes

characteristic

#14556064

C₂²

yes

characteristic

#14556055

—

#14556054

—

C₈×Dic₇₁

C₂²

yes

characteristic

#135×C₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₇₁

C₂²

yes

characteristic

#108×C₇₁

C₂²

yes

characteristic

#11765865

—

Dic₂₈₄

C₂×C₄

yes

characteristic

#11765854

—

D₅₆₈

C₂×C₄

yes

—

—

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₈×D₇₁

—

C₄×Dic₇₁

D₄

yes

characteristic

C₇₁⋊C₁₆

C₂×C₄

yes

characteristic

C₇₁⋊C₁₆

C₂×C₄

yes

characteristic

#41×C₇₁

—

Q₁₆×C₇₁

C₂×C₄

yes

characteristic

SD₁₆×C₇₁

—

D₈×C₇₁

C₂×C₄

yes

characteristic

M₁₆×C₇₁

—

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₁₆×C₇₁

—

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²⋊D₇₁

—

C₂×Dic₇₁

—

D₂₈₄

—

C₄×D₇₁

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

C₂²×C₄

yes

characteristic

C₈×C₇₁

#41

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

D₈

yes

characteristic

C₇₁⋊C₈

D₈

yes

characteristic

C₂²×C₇₁

—

C₂²×C₇₁

#135

yes

characteristic

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₄×D₄

yes

characteristic

C₄×C₇₁

C₂×D₈

yes

characteristic

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

C₄×D₈

yes

characteristic

D₇₁

—

#2178

—

maximal, Sylow

C₇₁

#2178

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₇₁

yes

characteristic, complemented

C₄×C₁₆

—

#135

—

#135

C₂×D₇₁

yes

characteristic

#131

—

142

—

#131

—

142

—

Q₃₂

—

SD₃₂

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₁

yes

characteristic

#108

—

#108

C₂×D₇₁

yes

characteristic

#104

—

142

—

#104

—

142

—

C₄×C₈

—

#41

—

#41

—

142

—

Q₁₆

—

Q₁₆

C₄×D₇₁

yes

characteristic

SD₁₆

—

100

SD₁₆

—

142

—

101

D₈

—

102

D₈

C₄×D₇₁

yes

characteristic

103

M₁₆

—

104

M₁₆

—

284

—

105

M₁₆

—

284

—

106

M₁₆

—

142

—

107

C₂×C₈

—

284

—

108

C₂×C₈

—

284

—

109

C₂×C₈

—

110

C₂×C₈

C₂²×D₇₁

yes

characteristic

111

C₄²

—

112

C₁₆

—

113

C₁₆

—

114

C₁₆

—

115

Q₈

—

116

Q₈

—

142

—

117

D₄

—

118

D₄

—

119

D₄

—

284

—

120

D₄

—

142

—

121

C₂×C₄

—

122

C₂×C₄

—

284

—

123

C₂×C₄

—

142

—

124

C₂×C₄

D₄×D₇₁

yes

characteristic

125

C₈

—

126

C₈

—

284

—

127

C₈

—

128

C₈

—

129

C₈

C₂×C₄×D₇₁

yes

characteristic

130

C₈

#11765864

yes

characteristic

131

C₂²

—

132

C₂²

—

284

—

133

C₂²

#14556149

yes

characteristic

134

C₄

—

135

C₄

—

284

—

136

C₄

—

142

—

137

C₄

(C₂²⋊D₇₁)⋊C₄

yes

characteristic

138

C₄

D₈×D₇₁

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

—

568

—

142

C₂

#17986079

yes

characteristic

	subgroup	G/H
center	C₄	D₈×D₇₁
commutator	C₈×C₇₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	#359×C₇₁	C₂
socle	C₂×C₇₁	C₄×D₈
hypercenter	#359	D₇₁
hypocenter	C₇₁	#2178
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #359

subgroup

G/H

center

C₄

D₈×D₇₁

commutator

C₈×C₇₁

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

#359×C₇₁

C₂

socle

C₂×C₇₁

C₄×D₈

D₇₁

C₇₁

G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #359

N	H ≅ G/N	type
2. #17986116	C₂	semidirect
3. #17986115	C₂	semidirect
4. #17985828	C₂	semidirect
5. #17985807	C₂	semidirect
6. C₁₆×Dic₇₁	C₂	semidirect
7. #359×C₇₁	C₂	semidirect
11. #14556074	C₄	semidirect
12. #14556072	C₄	semidirect
13. #14556071	C₄	semidirect
14. #14556069	C₄	semidirect
72. C₇₁	#2178	semidirect
78. #359	D₇₁	semidirect

H ≅ G/N

type

2. #17986116

C₂

semidirect

3. #17986115

C₂

semidirect

4. #17985828

C₂

semidirect

5. #17985807

C₂

semidirect

6. C₁₆×Dic₇₁

C₂

semidirect

7. #359×C₇₁

C₂

semidirect

11. #14556074

C₄

semidirect

12. #14556072

C₄

semidirect

13. #14556071

C₄

semidirect

14. #14556069

C₄

semidirect

72. C₇₁

#2178

semidirect

78. #359

D₇₁

semidirect

Subgroups — #22590218

Conjugacy classes of subgroups

Complements