Group #22600771 — subgroups

Subgroups — #22600771

Order 9088.

340 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 125 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	#17986719	C₂	1	2	yes	maximal
2	#17986404	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986341	C₂	1	4	yes	maximal, complemented
4	#17986285	C₂	1	2	yes	maximal, complemented
5	#17986240	C₂	1	2	yes	maximal, complemented
6	C₄×Dic₂₈₄	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986037	C₂	1	2	yes	maximal, complemented
8	C₈×D₄×C₇₁	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₂₈₄	C₂²	1	1	yes	characteristic
10	#14556128	C₂²	1	2	yes	—
11	#14556121	—	2	4	no	—
12	#14556115	—	2	2	no	—
13	#14556107	C₂²	1	4	yes	—
14	C₄×Dic₁₄₂	C₂²	1	2	yes	—
15	#14556093	—	2	2	no	—
16	#14556087	C₂²	1	4	yes	—
17	#14556062	C₂²	1	2	yes	—
18	#14556062	C₂²	1	1	yes	characteristic
19	#14556062	C₂²	1	2	yes	—
20	#14556060	C₂²	1	4	yes	—
21	#14556053	C₂²	1	2	yes	—
22	#14556053	C₂²	1	4	yes	—
23	C₂²×C₈×C₇₁	C₂²	1	2	yes	—
24	C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
25	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
26	C₂²⋊C₈×C₇₁	C₂²	1	2	yes	—
27	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
28	C₂×Dic₁₄₂	—	2	2	no	—
29	C₂×Dic₁₄₂	C₂³	1	2	yes	—
30	Dic₂₈₄	—	2	2	no	—
31	C₄⋊Dic₇₁	—	2	2	no	—
32	C₄⋊Dic₇₁	C₂³	1	2	yes	—
33	C₄⋊Dic₇₁	C₂³	1	4	yes	—
34	Dic₇₁⋊C₄	—	2	4	no	—
35	Dic₇₁⋊C₄	—	2	2	no	—
36	C₂²⋊Dic₇₁	—	2	4	no	—
37	C₂²⋊Dic₇₁	—	2	4	no	—
38	C₄×Dic₇₁	—	2	4	no	—
39	C₄×Dic₇₁	—	2	2	no	—
40	C₂×D₄×C₇₁	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₁	C₂³	1	2	yes	—
42	C₂×C₈×C₇₁	—	2	4	no	—
43	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₇₁	C₂³	1	2	yes	—
45	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
47	C₂²⋊C₄×C₇₁	D₄	1	2	yes	—
48	C₄²×C₇₁	C₂³	1	1	yes	characteristic
49	C₂³×C₇₁	C₂×D₄	1	2	yes	—
50	D₄×C₇₁	—	2	2	no	—
51	C₂×C₄×C₇₁	—	2	2	no	—
52	C₂×C₄×C₇₁	—	2	2	no	—
53	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
57	C₂×Dic₇₁	—	2	2	no	—
58	C₂×Dic₇₁	—	2	2	no	—
59	C₂×Dic₇₁	—	2	4	no	—
60	Dic₁₄₂	—	2	2	no	—
61	Dic₁₄₂	—	2	2	no	—
62	Dic₁₄₂	—	4	4	no	—
63	C₈×C₇₁	—	2	2	no	—
64	C₈×C₇₁	—	2	1	no	—
65	C₈×C₇₁	#41	1	2	yes	—
66	C₂²×C₇₁	—	2	2	no	—
67	C₂²×C₇₁	—	2	2	no	—
68	C₂²×C₇₁	—	2	2	no	—
69	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
70	C₄×C₇₁	—	2	2	no	—
71	C₄×C₇₁	—	2	1	no	—
72	C₄×C₇₁	#135	1	2	yes	—
73	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
74	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
75	Dic₇₁	—	4	2	no	—
76	Dic₇₁	—	4	4	no	—
77	Dic₇₁	—	4	2	no	—
78	C₂×C₇₁	—	4	2	no	—
79	C₂×C₇₁	#547	1	1	yes	characteristic
80	C₂×C₇₁	C₂×#135	1	1	yes	characteristic
81	C₂×C₇₁	#578	1	1	yes	characteristic
82	#3342	—	71	1	no	maximal, Sylow
83	C₇₁	#3342	1	1	yes	characteristic, complemented, Sylow
84	#556	—	71	2	no	—
85	#500	—	71	1	no	—
86	#485	—	71	4	no	—
87	#475	—	71	2	no	—
88	#467	—	71	2	no	—
89	C₄×Q₁₆	—	71	1	no	—
90	C₈×D₄	D₇₁	1	1	yes	characteristic
91	#427	—	71	2	no	—
92	C₂×Q₁₆	—	71	1	no	—
93	C₂²×C₈	C₂×D₇₁	1	2	yes	—
94	C₄⋊Q₈	—	71	2	no	—
95	#126	—	142	4	no	—
96	#124	—	142	2	no	—
97	C₂²⋊Q₈	—	71	4	no	—
98	C₄×Q₈	—	71	2	no	—
99	C₄×Q₈	—	142	2	no	—
100	C₄×D₄	C₂×D₇₁	1	1	yes	characteristic
101	#117	—	71	4	no	—
102	#107	—	71	1	no	—
103	#107	—	71	2	no	—
104	#107	—	71	2	no	—
105	#106	—	71	4	no	—
106	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
107	#103	—	71	2	no	—
108	#103	—	71	4	no	—
109	C₂²⋊C₈	C₂×D₇₁	1	2	yes	—
110	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
111	C₂×Q₈	—	71	2	no	—
112	C₂×Q₈	—	142	2	no	—
113	C₂×D₄	D₂₈₄	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₇₁	1	2	yes	—
115	Q₁₆	—	142	2	no	—
116	C₂×C₈	—	2	4	no	—
117	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
118	C₂×C₈	C₂²×D₇₁	1	2	yes	—
119	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
120	C₄⋊C₄	—	71	2	no	—
121	C₄⋊C₄	—	71	4	no	—
122	C₄⋊C₄	—	142	2	no	—
123	C₄⋊C₄	—	142	4	no	—
124	C₄⋊C₄	—	142	2	no	—
125	C₄⋊C₄	D₂₈₄	1	1	yes	characteristic
126	C₂²⋊C₄	—	142	4	no	—
127	C₂²⋊C₄	—	142	4	no	—
128	C₂²⋊C₄	D₂₈₄	1	2	yes	—
129	C₄²	—	142	4	no	—
130	C₄²	—	142	2	no	—
131	C₄²	C₂²×D₇₁	1	1	yes	characteristic
132	C₂³	C₂×D₂₈₄	1	2	yes	—
133	Q₈	—	284	4	no	—
134	Q₈	—	142	2	no	—
135	Q₈	—	142	2	no	—
136	D₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	142	2	no	—
140	C₂×C₄	—	142	4	no	—
141	C₂×C₄	—	142	2	no	—
142	C₂×C₄	C₂×D₂₈₄	1	2	yes	—
143	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
144	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
145	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	2	1	no	—
148	C₈	#11765864	1	2	yes	—
149	C₂²	—	2	2	no	—
150	C₂²	—	2	2	no	—
151	C₂²	—	2	2	no	—
152	C₂²	C₂²×D₂₈₄	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	2	2	no	—
155	C₄	—	284	2	no	—
156	C₄	—	284	2	no	—
157	C₄	—	284	4	no	—
158	C₄	#14556150	1	2	yes	—
159	C₄	C₂×#11765864	1	1	yes	characteristic
160	C₄	#14556175	1	1	yes	characteristic
161	C₂	—	4	2	no	—
162	C₂	#17986653	1	1	yes	characteristic
163	C₂	C₂×#14556150	1	1	yes	characteristic
164	C₂	#17986816	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986719

C₂

yes

maximal

#17986404

C₂

yes

maximal, characteristic, complemented

#17986341

C₂

yes

maximal, complemented

#17986285

C₂

yes

maximal, complemented

#17986240

C₂

yes

maximal, complemented

C₄×Dic₂₈₄

C₂

yes

maximal, characteristic, complemented

#17986037

C₂

yes

maximal, complemented

C₈×D₄×C₇₁

C₂

yes

maximal, characteristic

C₂×Dic₂₈₄

C₂²

yes

characteristic

#14556128

C₂²

yes

—

#14556121

—

#14556115

—

#14556107

C₂²

yes

—

C₄×Dic₁₄₂

C₂²

yes

—

#14556093

—

#14556087

C₂²

yes

—

#14556062

C₂²

yes

—

#14556062

C₂²

yes

characteristic

#14556062

C₂²

yes

—

#14556060

C₂²

yes

—

#14556053

C₂²

yes

—

#14556053

C₂²

yes

—

C₂²×C₈×C₇₁

C₂²

yes

—

C₄×D₄×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₂²⋊C₈×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

C₂³

yes

—

Dic₂₈₄

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₂³

yes

—

C₄⋊Dic₇₁

C₂³

yes

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₂²⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₂×D₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂³

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

—

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

D₄

yes

characteristic

C₂²⋊C₄×C₇₁

D₄

yes

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

C₂×D₄

yes

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

#41

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂²×D₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#135

yes

—

C₄×C₇₁

C₂×#41

yes

characteristic

C₄×C₇₁

2₋¹⁺⁴

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

—

—

—

—

—

C₄×Q₁₆

—

C₈×D₄

D₇₁

yes

characteristic

#427

—

C₂×Q₁₆

—

C₂²×C₈

C₂×D₇₁

yes

—

C₄⋊Q₈

—

#126

—

142

—

#124

—

142

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

142

—

100

C₄×D₄

C₂×D₇₁

yes

characteristic

101

#117

—

102

#107

—

103

#107

—

104

#107

—

105

#106

—

106

C₄⋊C₈

C₂×D₇₁

yes

characteristic

107

#103

—

108

#103

—

109

C₂²⋊C₈

C₂×D₇₁

yes

—

110

C₄×C₈

C₂×D₇₁

yes

characteristic

111

C₂×Q₈

—

112

C₂×Q₈

—

142

—

113

C₂×D₄

D₂₈₄

yes

characteristic

114

C₂²×C₄

C₂²×D₇₁

yes

—

115

Q₁₆

—

142

—

116

C₂×C₈

—

117

C₂×C₈

C₂²×D₇₁

yes

characteristic

118

C₂×C₈

C₂²×D₇₁

yes

—

119

C₂×C₈

C₂²×D₇₁

yes

characteristic

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

142

—

123

C₄⋊C₄

—

142

—

124

C₄⋊C₄

—

142

—

125

C₄⋊C₄

D₂₈₄

yes

characteristic

126

C₂²⋊C₄

—

142

—

127

C₂²⋊C₄

—

142

—

128

C₂²⋊C₄

D₂₈₄

yes

—

129

C₄²

—

142

—

130

C₄²

—

142

—

131

C₄²

C₂²×D₇₁

yes

characteristic

132

C₂³

C₂×D₂₈₄

yes

—

133

Q₈

—

284

—

134

Q₈

—

142

—

135

Q₈

—

142

—

136

D₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

142

—

140

C₂×C₄

—

142

—

141

C₂×C₄

—

142

—

142

C₂×C₄

C₂×D₂₈₄

yes

—

143

C₂×C₄

C₂×D₂₈₄

yes

characteristic

144

C₂×C₄

C₂×D₂₈₄

yes

characteristic

145

C₂×C₄

C₂³×D₇₁

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

#11765864

yes

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂²×D₂₈₄

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

284

—

156

C₄

—

284

—

157

C₄

—

284

—

158

C₄

#14556150

yes

—

159

C₄

C₂×#11765864

yes

characteristic

160

C₄

#14556175

yes

characteristic

161

C₂

—

162

C₂

#17986653

yes

characteristic

163

C₂

C₂×#14556150

yes

characteristic

164

C₂

#17986816

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	C₈×D₄×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	C₈×D₄	D₇₁
hypocenter	C₇₁	#3342
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

subgroup

G/H

center

C₂²

C₂²×D₂₈₄

commutator

C₂×C₄×C₇₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₁

Fitting

C₈×D₄×C₇₁

C₂

socle

C₂²×C₇₁

C₂²×D₄

hypercenter

C₈×D₄

D₇₁

hypocenter

C₇₁

#3342

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

N	H ≅ G/N	type
2. #17986404	C₂	semidirect
3. #17986341	C₂	semidirect
4. #17986285	C₂	semidirect
5. #17986240	C₂	semidirect
6. C₄×Dic₂₈₄	C₂	semidirect
7. #17986037	C₂	semidirect
83. C₇₁	#3342	semidirect

H ≅ G/N

type

2. #17986404

C₂

semidirect

3. #17986341

C₂

semidirect

4. #17986285

C₂

semidirect

5. #17986240

C₂

semidirect

6. C₄×Dic₂₈₄

C₂

semidirect

7. #17986037

C₂

semidirect

83. C₇₁

#3342

semidirect

Subgroups — #22600771

Conjugacy classes of subgroups

Complements