Group #22598853 — subgroups

Subgroups — #22598853

Order 9088.

356 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 141 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	#17986555	C₂	1	2	yes	maximal
2	#17986403	C₂	1	2	yes	maximal, complemented
3	#17986401	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986398	C₂	1	2	yes	maximal, complemented
5	#17986396	C₂	1	2	yes	maximal, complemented
6	#17986036	C₂	1	2	yes	maximal
7	#17986034	C₂	1	2	yes	maximal
8	C₄×C₈×D₇₁	C₂	1	1	yes	maximal, characteristic
9	#499×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₈×D₇₁	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₇₁	C₂²	1	1	yes	characteristic
12	#14556127	C₂²	1	4	yes	—
13	#14556120	C₂²	1	2	yes	—
14	#14556106	—	2	4	no	—
15	#14556093	—	2	4	no	—
16	#14556086	C₂²	1	4	yes	—
17	C₄²×D₇₁	C₂²	1	1	yes	characteristic
18	#14556062	C₂²	1	2	yes	—
19	#14556061	C₂²	1	4	yes	—
20	#14556060	C₂²	1	2	yes	—
21	#14556059	C₂²	1	4	yes	—
22	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
23	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
24	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
25	#125×C₇₁	C₂²	1	2	yes	—
26	#107×C₇₁	C₂²	1	2	yes	—
27	#106×C₇₁	C₂²	1	2	yes	—
28	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
29	C₂×Dic₁₄₂	—	2	4	no	—
30	C₂×C₄×D₇₁	D₄	1	2	yes	—
31	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
32	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
33	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
34	C₈×D₇₁	Q₈	1	4	yes	complemented
35	C₈×D₇₁	Q₈	1	4	yes	complemented
36	C₄⋊Dic₇₁	C₂³	1	4	yes	—
37	Dic₇₁⋊C₄	—	2	4	no	—
38	Dic₇₁⋊C₄	—	2	4	no	—
39	(C₂×D₇₁)⋊C₄	—	2	4	no	—
40	C₄×Dic₇₁	—	2	4	no	—
41	C₄×Dic₇₁	D₄	1	2	yes	—
42	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₇₁	—	2	4	no	—
46	C₄⋊C₄×C₇₁	C₂³	1	4	yes	—
47	C₄²×C₇₁	C₂³	1	1	yes	characteristic
48	C₂²×D₇₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇₁	—	2	4	no	—
50	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
52	C₂×Dic₇₁	—	2	4	no	—
53	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₇₁	C₂×D₄	1	2	yes	—
55	C₄×D₇₁	—	2	4	no	—
56	C₄×D₇₁	C₂×Q₈	1	2	yes	—
57	C₄×D₇₁	C₂×Q₈	1	2	yes	—
58	Dic₁₄₂	—	4	8	no	—
59	C₈×C₇₁	C₂×Q₈	1	2	yes	—
60	C₈×C₇₁	C₂×Q₈	1	2	yes	—
61	C₇₁⋊C₈	C₂×Q₈	1	2	yes	—
62	C₇₁⋊C₈	C₂×Q₈	1	2	yes	—
63	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
64	C₂×D₇₁	—	2	2	no	—
65	C₂×D₇₁	C₄⋊Q₈	1	2	yes	—
66	C₄×C₇₁	—	2	2	no	—
67	C₄×C₇₁	—	4	4	no	—
68	C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
69	C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
70	Dic₇₁	—	4	4	no	—
71	Dic₇₁	C₄⋊Q₈	1	2	yes	—
72	Dic₇₁	#135	1	4	yes	—
73	C₂×C₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₇₁	C₂×#135	1	2	yes	—
75	D₇₁	—	2	2	no	—
76	#3160	—	71	1	no	maximal, Sylow
77	C₇₁	#3160	1	1	yes	characteristic, complemented, Sylow
78	#533	—	71	2	no	—
79	#500	—	71	2	no	—
80	#499	—	71	2	no	—
81	#499	—	71	1	no	—
82	#499	D₇₁	1	1	yes	characteristic, complemented
83	#498	—	71	2	no	—
84	#427	—	71	2	no	—
85	#427	—	71	2	no	—
86	C₂×C₄×C₈	—	71	1	no	—
87	C₂²×C₈	—	71	1	no	—
88	C₂²×C₈	—	71	1	no	—
89	C₄⋊Q₈	—	71	4	no	—
90	#125	—	71	2	no	—
91	#125	C₂×D₇₁	1	2	yes	—
92	C₂²⋊Q₈	—	142	4	no	—
93	C₄×Q₈	—	142	4	no	—
94	#117	—	71	4	no	—
95	C₂×C₄²	—	71	1	no	—
96	#107	—	71	2	no	—
97	#107	—	71	4	no	—
98	#107	C₂×D₇₁	1	2	yes	—
99	#106	—	71	2	no	—
100	#106	—	71	4	no	—
101	#106	C₂×D₇₁	1	2	yes	—
102	C₄×C₈	—	71	1	no	—
103	C₄×C₈	—	71	1	no	—
104	C₄×C₈	—	71	1	no	—
105	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
106	C₂×Q₈	—	142	4	no	—
107	C₂²×C₄	—	71	1	no	—
108	C₂²×C₄	—	71	2	no	—
109	C₂×C₈	—	71	1	no	—
110	C₂×C₈	—	71	1	no	—
111	C₂×C₈	—	71	4	no	—
112	C₂×C₈	—	71	4	no	—
113	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
114	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	71	4	no	—
117	C₄⋊C₄	—	142	4	no	—
118	C₄⋊C₄	—	142	4	no	—
119	C₄⋊C₄	C₂²×D₇₁	1	4	yes	—
120	C₂²⋊C₄	—	142	4	no	—
121	C₄²	—	71	2	no	—
122	C₄²	—	71	1	no	—
123	C₄²	—	142	4	no	—
124	C₄²	C₂²×D₇₁	1	1	yes	characteristic
125	C₂³	—	71	1	no	—
126	Q₈	—	284	8	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	71	1	no	—
129	C₂×C₄	—	71	2	no	—
130	C₂×C₄	—	71	2	no	—
131	C₂×C₄	—	71	2	no	—
132	C₂×C₄	—	142	4	no	—
133	C₂×C₄	—	142	4	no	—
134	C₂×C₄	D₄×D₇₁	1	2	yes	—
135	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
136	C₈	—	71	2	no	—
137	C₈	—	71	2	no	—
138	C₈	Q₈×D₇₁	1	2	yes	—
139	C₈	Q₈×D₇₁	1	2	yes	—
140	C₂²	—	71	2	no	—
141	C₂²	—	142	2	no	—
142	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	4	4	no	—
145	C₄	—	284	4	no	—
146	C₄	—	71	4	no	—
147	C₄	—	71	2	no	—
148	C₄	C₂×Q₈×D₇₁	1	1	yes	characteristic
149	C₄	C₂×Q₈×D₇₁	1	1	yes	characteristic
150	C₂	—	142	2	no	—
151	C₂	C₄⋊Q₈×D₇₁	1	1	yes	characteristic
152	C₂	#135×D₇₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17986555

C₂

yes

maximal

#17986403

C₂

yes

maximal, complemented

#17986401

C₂

yes

maximal, characteristic, complemented

#17986398

C₂

yes

maximal, complemented

#17986396

C₂

yes

maximal, complemented

#17986036

C₂

yes

maximal

#17986034

C₂

yes

maximal

C₄×C₈×D₇₁

C₂

yes

maximal, characteristic

#499×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₇₁

C₂²

yes

characteristic

C₂×C₈×D₇₁

C₂²

yes

characteristic

#14556127

C₂²

yes

—

#14556120

C₂²

yes

—

#14556106

—

#14556093

—

#14556086

C₂²

yes

—

C₄²×D₇₁

C₂²

yes

characteristic

#14556062

C₂²

yes

—

#14556061

C₂²

yes

—

#14556060

C₂²

yes

—

#14556059

C₂²

yes

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₈×Dic₇₁

C₂²

yes

characteristic

C₈×Dic₇₁

C₂²

yes

characteristic

#125×C₇₁

C₂²

yes

—

#107×C₇₁

C₂²

yes

—

#106×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×C₄×D₇₁

D₄

yes

—

C₂×C₄×D₇₁

C₂³

yes

characteristic

C₂×C₇₁⋊C₈

C₂³

yes

characteristic

C₂×C₇₁⋊C₈

C₂³

yes

characteristic

C₈×D₇₁

Q₈

yes

complemented

C₈×D₇₁

Q₈

yes

complemented

C₄⋊Dic₇₁

C₂³

yes

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

(C₂×D₇₁)⋊C₄

—

C₄×Dic₇₁

—

C₄×Dic₇₁

D₄

yes

—

C₄×Dic₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

C₂³

yes

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂²×D₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

C₂×D₄

yes

—

C₄×D₇₁

—

C₄×D₇₁

C₂×Q₈

yes

—

C₄×D₇₁

C₂×Q₈

yes

—

Dic₁₄₂

—

C₈×C₇₁

C₂×Q₈

yes

—

C₈×C₇₁

C₂×Q₈

yes

—

C₇₁⋊C₈

C₂×Q₈

yes

—

C₇₁⋊C₈

C₂×Q₈

yes

—

C₂²×C₇₁

C₂²×D₄

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

C₄⋊Q₈

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂²×Q₈

yes

characteristic

C₄×C₇₁

C₂²×Q₈

yes

characteristic

Dic₇₁

—

Dic₇₁

C₄⋊Q₈

yes

—

Dic₇₁

#135

yes

—

C₂×C₇₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₇₁

C₂×#135

yes

—

D₇₁

—

#3160

—

maximal, Sylow

C₇₁

#3160

yes

characteristic, complemented, Sylow

—

—

—

—

D₇₁

yes

characteristic, complemented

#498

—

#427

—

#427

—

C₂×C₄×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

#125

—

#125

C₂×D₇₁

yes

—

C₂²⋊Q₈

—

142

—

C₄×Q₈

—

142

—

#117

—

C₂×C₄²

—

#107

—

#107

—

#107

C₂×D₇₁

yes

—

#106

—

100

#106

—

101

#106

C₂×D₇₁

yes

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

C₂×D₇₁

yes

characteristic

106

C₂×Q₈

—

142

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₂²×D₇₁

yes

characteristic

114

C₂×C₈

C₂²×D₇₁

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

142

—

118

C₄⋊C₄

—

142

—

119

C₄⋊C₄

C₂²×D₇₁

yes

—

120

C₂²⋊C₄

—

142

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

142

—

124

C₄²

C₂²×D₇₁

yes

characteristic

125

C₂³

—

126

Q₈

—

284

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

142

—

133

C₂×C₄

—

142

—

134

C₂×C₄

D₄×D₇₁

yes

—

135

C₂×C₄

C₂³×D₇₁

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

Q₈×D₇₁

yes

—

139

C₈

Q₈×D₇₁

yes

—

140

C₂²

—

141

C₂²

—

142

—

142

C₂²

C₂×D₄×D₇₁

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

284

—

146

C₄

—

147

C₄

—

148

C₄

C₂×Q₈×D₇₁

yes

characteristic

149

C₄

C₂×Q₈×D₇₁

yes

characteristic

150

C₂

—

142

—

151

C₂

C₄⋊Q₈×D₇₁

yes

characteristic

152

C₂

#135×D₇₁

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#499×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#499	D₇₁
hypocenter	C₇₁	#3160
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

subgroup

G/H

center

C₂²

C₂×D₄×D₇₁

commutator

C₂×C₄×C₇₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₁

Fitting

#499×C₇₁

C₂

socle

C₂²×C₇₁

C₂²×D₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #499

N	H ≅ G/N	type
2. #17986403	C₂	semidirect
3. #17986401	C₂	semidirect
4. #17986398	C₂	semidirect
5. #17986396	C₂	semidirect
9. #499×C₇₁	C₂	semidirect
34. C₈×D₇₁	Q₈	semidirect
35. C₈×D₇₁	Q₈	semidirect
77. C₇₁	#3160	semidirect
82. #499	D₇₁	semidirect

H ≅ G/N

type

2. #17986403

C₂

semidirect

3. #17986401

C₂

semidirect

4. #17986398

C₂

semidirect

5. #17986396

C₂

semidirect

9. #499×C₇₁

C₂

semidirect

34. C₈×D₇₁

Q₈

semidirect

35. C₈×D₇₁

Q₈

semidirect

77. C₇₁

#3160

semidirect

82. #499

D₇₁

semidirect

Subgroups — #22598853

Conjugacy classes of subgroups

Complements