Group #22596950 — subgroups

Subgroups — #22596950

Order 9088.

440 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 209 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	#17986765	C₂	1	2	yes	maximal
2	C₂×#14556136	C₂	1	1	yes	maximal, characteristic
3	#17986048	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986046	C₂	1	4	yes	maximal, complemented
5	C₂×#14556063	C₂	1	2	yes	maximal, complemented
6	#108×D₇₁	C₂	1	4	yes	maximal, complemented
7	#430×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11765865	C₂²	1	1	yes	characteristic
9	#14556136	—	2	4	no	—
10	#14556136	C₄	1	8	yes	—
11	#14556134	—	2	4	no	—
12	C₂×#11765849	C₂²	1	2	yes	—
13	C₂×#11765848	C₂²	1	2	yes	—
14	M₁₆×D₇₁	C₂²	1	8	yes	—
15	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₈×D₇₁	C₂²	1	2	yes	—
17	#14556064	C₂²	1	4	yes	—
18	#14556063	C₂²	1	4	yes	—
19	#14556063	C₂²	1	4	yes	—
20	C₂×M₁₆×C₇₁	C₂²	1	2	yes	—
21	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
22	#108×C₇₁	C₂²	1	4	yes	—
23	#11765865	C₂×C₄	1	4	yes	—
24	#11765865	D₄	1	4	yes	—
25	C₂×Dic₁₄₂	Q₈	1	1	yes	characteristic
26	C₂×D₂₈₄	Q₈	1	1	yes	characteristic
27	C₂×C₂²⋊D₇₁	Q₈	1	2	yes	—
28	C₂×C₄×D₇₁	C₂³	1	2	yes	—
29	#11765849	—	2	2	no	—
30	#11765849	C₂³	1	4	yes	—
31	#11765848	—	2	8	no	—
32	#11765848	C₂×C₄	1	8	yes	—
33	C₂×C₇₁⋊C₈	—	2	2	no	—
34	C₂×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
35	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
36	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
37	C₈×D₇₁	—	2	8	no	—
38	C₈×D₇₁	C₂×C₄	1	8	yes	—
39	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
40	M₁₆×C₇₁	—	2	2	no	—
41	M₁₆×C₇₁	C₂×C₄	1	4	yes	—
42	M₁₆×C₇₁	C₂³	1	4	yes	—
43	C₂×C₈×C₇₁	—	2	2	no	—
44	C₂×C₈×C₇₁	C₂³	1	2	yes	—
45	C₂³×C₇₁	C₂×Q₈	1	1	yes	characteristic
46	C₂²×D₇₁	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₇₁	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
51	C₂²⋊D₇₁	C₄⋊C₄	1	8	yes	—
52	C₂×Dic₇₁	C₂×Q₈	1	2	yes	—
53	D₂₈₄	C₄⋊C₄	1	4	yes	—
54	C₄×D₇₁	C₂²×C₄	1	4	yes	—
55	C₄×D₇₁	C₂×D₄	1	4	yes	—
56	Dic₁₄₂	C₄⋊C₄	1	4	yes	—
57	C₈×C₇₁	—	2	4	no	—
58	C₈×C₇₁	C₂²×C₄	1	4	yes	—
59	C₇₁⋊C₈	—	2	4	no	—
60	C₇₁⋊C₈	C₂²×C₄	1	2	yes	—
61	C₇₁⋊C₈	C₂²×C₄	1	2	yes	—
62	C₂²×C₇₁	—	4	1	no	—
63	C₂²×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
64	C₂²×C₇₁	C₂²×Q₈	1	1	yes	characteristic
65	C₂×D₇₁	—	4	2	no	—
66	C₂×D₇₁	C₂×C₄⋊C₄	1	4	yes	—
67	C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
68	C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
70	Dic₇₁	C₂×C₄⋊C₄	1	4	yes	—
71	C₂×C₇₁	—	2	1	no	—
72	C₂×C₇₁	—	2	2	no	—
73	C₂×C₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
74	D₇₁	—	2	4	no	—
75	#2916	—	71	1	no	maximal, Sylow
76	C₇₁	#2916	1	1	yes	characteristic, complemented, Sylow
77	#568	—	71	2	no	—
78	C₂×#131	—	71	1	no	—
79	#430	—	71	1	no	—
80	#430	—	71	4	no	—
81	#430	D₇₁	1	1	yes	characteristic, complemented
82	C₂×#108	—	71	2	no	—
83	C₂×#108	—	71	4	no	—
84	C₂×#41	—	71	1	no	—
85	#131	—	71	8	no	—
86	#131	—	142	4	no	—
87	#131	—	142	4	no	—
88	C₂×M₁₆	—	71	2	no	—
89	C₂×M₁₆	—	71	2	no	—
90	C₂×M₁₆	—	71	8	no	—
91	C₂×M₁₆	C₂×D₇₁	1	2	yes	—
92	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
93	C₂²×C₈	—	71	1	no	—
94	C₂²×C₈	—	71	2	no	—
95	#108	—	71	4	no	—
96	#108	—	71	4	no	—
97	#108	—	71	4	no	—
98	#108	C₂×D₇₁	1	4	yes	—
99	#41	—	71	4	no	—
100	#41	—	71	4	no	—
101	C₂×Q₈	—	71	1	no	—
102	C₂×D₄	—	71	2	no	—
103	C₂×D₄	—	71	1	no	—
104	C₂²×C₄	—	71	2	no	—
105	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
106	M₁₆	—	2	2	no	—
107	M₁₆	—	71	8	no	—
108	M₁₆	—	71	4	no	—
109	M₁₆	—	142	8	no	—
110	M₁₆	—	142	2	no	—
111	M₁₆	C₄×D₇₁	1	4	yes	—
112	M₁₆	C₂²×D₇₁	1	4	yes	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	71	4	no	—
115	C₂×C₈	—	71	1	no	—
116	C₂×C₈	—	71	1	no	—
117	C₂×C₈	—	71	8	no	—
118	C₂×C₈	—	142	8	no	—
119	C₂×C₈	—	142	2	no	—
120	C₂×C₈	C₂²×D₇₁	1	2	yes	—
121	C₂³	—	71	2	no	—
122	C₂³	Q₈×D₇₁	1	1	yes	characteristic
123	Q₈	—	71	4	no	—
124	D₄	—	71	8	no	—
125	D₄	—	71	4	no	—
126	C₂×C₄	—	71	2	no	—
127	C₂×C₄	—	71	4	no	—
128	C₂×C₄	—	71	4	no	—
129	C₂×C₄	C₂×C₄×D₇₁	1	2	yes	—
130	C₂×C₄	D₄×D₇₁	1	2	yes	—
131	C₂×C₄	Q₈×D₇₁	1	1	yes	characteristic
132	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
133	C₈	—	2	4	no	—
134	C₈	—	71	2	no	—
135	C₈	—	71	2	no	—
136	C₈	—	142	4	no	—
137	C₈	C₂×C₄×D₇₁	1	4	yes	—
138	C₂²	—	4	1	no	—
139	C₂²	—	284	2	no	—
140	C₂²	—	71	4	no	—
141	C₂²	C₄⋊C₄×D₇₁	1	2	yes	—
142	C₂²	C₂×Q₈×D₇₁	1	1	yes	characteristic
143	C₄	—	71	4	no	—
144	C₄	C₄⋊C₄×D₇₁	1	2	yes	—
145	C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
146	C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	2	no	—
149	C₂	—	142	4	no	—
150	C₂	C₂×C₄⋊C₄×D₇₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986765

C₂

yes

maximal

C₂×#14556136

C₂

yes

maximal, characteristic

#17986048

C₂

yes

maximal, characteristic, complemented

#17986046

C₂

yes

maximal, complemented

C₂×#14556063

C₂

yes

maximal, complemented

#108×D₇₁

C₂

yes

maximal, complemented

#430×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×#11765865

C₂²

yes

characteristic

#14556136

—

#14556136

C₄

yes

—

#14556134

—

C₂×#11765849

C₂²

yes

—

C₂×#11765848

C₂²

yes

—

M₁₆×D₇₁

C₂²

yes

—

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₇₁

C₂²

yes

—

#14556064

C₂²

yes

—

#14556063

C₂²

yes

—

#14556063

C₂²

yes

—

C₂×M₁₆×C₇₁

C₂²

yes

—

C₂×M₁₆×C₇₁

C₂²

yes

characteristic

#108×C₇₁

C₂²

yes

—

#11765865

C₂×C₄

yes

—

#11765865

D₄

yes

—

C₂×Dic₁₄₂

Q₈

yes

characteristic

C₂×D₂₈₄

Q₈

yes

characteristic

C₂×C₂²⋊D₇₁

Q₈

yes

—

C₂×C₄×D₇₁

C₂³

yes

—

#11765849

—

#11765849

C₂³

yes

—

#11765848

—

#11765848

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

C₂³

yes

characteristic

C₂×C₇₁⋊C₈

C₂³

yes

characteristic

C₈×D₇₁

—

C₈×D₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

C₂×C₄

yes

—

M₁₆×C₇₁

C₂³

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂³

yes

—

C₂³×C₇₁

C₂×Q₈

yes

characteristic

C₂²×D₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₂²⋊D₇₁

C₄⋊C₄

yes

—

C₂×Dic₇₁

C₂×Q₈

yes

—

D₂₈₄

C₄⋊C₄

yes

—

C₄×D₇₁

C₂²×C₄

yes

—

C₄×D₇₁

C₂×D₄

yes

—

Dic₁₄₂

C₄⋊C₄

yes

—

C₈×C₇₁

—

C₈×C₇₁

C₂²×C₄

yes

—

C₇₁⋊C₈

—

C₇₁⋊C₈

C₂²×C₄

yes

—

C₇₁⋊C₈

C₂²×C₄

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×C₄⋊C₄

yes

—

C₂²×C₇₁

C₂²×Q₈

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

C₂×C₄⋊C₄

yes

—

C₄×C₇₁

C₂×C₄⋊C₄

yes

—

C₄×C₇₁

C₂³×C₄

yes

characteristic

C₄×C₇₁

C₂²×D₄

yes

characteristic

Dic₇₁

C₂×C₄⋊C₄

yes

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

C₂²×C₄⋊C₄

yes

characteristic

D₇₁

—

#2916

—

maximal, Sylow

C₇₁

#2916

yes

characteristic, complemented, Sylow

—

—

—

—

D₇₁

yes

characteristic, complemented

C₂×#108

—

C₂×#108

—

C₂×#41

—

#131

—

#131

—

142

—

#131

—

142

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₁

yes

—

C₂×M₁₆

C₂×D₇₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

—

—

—

C₂×D₇₁

yes

—

#41

—

100

#41

—

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂²×D₇₁

yes

characteristic

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

142

—

110

M₁₆

—

142

—

111

M₁₆

C₄×D₇₁

yes

—

112

M₁₆

C₂²×D₇₁

yes

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

142

—

119

C₂×C₈

—

142

—

120

C₂×C₈

C₂²×D₇₁

yes

—

121

C₂³

—

122

C₂³

Q₈×D₇₁

yes

characteristic

123

Q₈

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×C₄×D₇₁

yes

—

130

C₂×C₄

D₄×D₇₁

yes

—

131

C₂×C₄

Q₈×D₇₁

yes

characteristic

132

C₂×C₄

C₂³×D₇₁

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

142

—

137

C₈

C₂×C₄×D₇₁

yes

—

138

C₂²

—

139

C₂²

—

284

—

140

C₂²

—

141

C₂²

C₄⋊C₄×D₇₁

yes

—

142

C₂²

C₂×Q₈×D₇₁

yes

characteristic

143

C₄

—

144

C₄

C₄⋊C₄×D₇₁

yes

—

145

C₄

C₂²×C₄×D₇₁

yes

characteristic

146

C₄

C₂×D₄×D₇₁

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

—

142

—

150

C₂

C₂×C₄⋊C₄×D₇₁

yes

characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₇₁
commutator	C₄×C₇₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#430×C₇₁	C₂
socle	C₂×C₇₁	C₂²×C₄⋊C₄
hypercenter	#430	D₇₁
hypocenter	C₇₁	#2916
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

subgroup

G/H

center

C₄

C₂×D₄×D₇₁

commutator

C₄×C₇₁

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₇₁

Fitting

#430×C₇₁

C₂

socle

C₂×C₇₁

C₂²×C₄⋊C₄

D₇₁

C₇₁

G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

N	H ≅ G/N	type
3. #17986048	C₂	semidirect
4. #17986046	C₂	semidirect
5. C₂×#14556063	C₂	semidirect
6. #108×D₇₁	C₂	semidirect
7. #430×C₇₁	C₂	semidirect
76. C₇₁	#2916	semidirect
81. #430	D₇₁	semidirect

H ≅ G/N

type

3. #17986048

C₂

semidirect

4. #17986046

C₂

semidirect

5. C₂×#14556063

C₂

semidirect

6. #108×D₇₁

C₂

semidirect

7. #430×C₇₁

C₂

semidirect

76. C₇₁

#2916

semidirect

81. #430

D₇₁

semidirect

Subgroups — #22596950

Conjugacy classes of subgroups

Complements