Finite Group Databasebeta

← #22597567

Subgroups — #22597567

Order 9088.

994 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 173 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14556154C212yesmaximal, complemented
2#17986510C211yesmaximal, characteristic, complemented
3#17986150C214yesmaximal, complemented
4#17986133C214yesmaximal, complemented
5#17986004C212yesmaximal, complemented
6#17985963C211yesmaximal, characteristic, complemented
7C23×D4×C71C211yesmaximal, characteristic, complemented
8C2×#11765865C2211yescharacteristic, complemented
9#14556154—28no—
10C2×#11765853C2214yescomplemented
11C2×#11765850C2214yescomplemented
12C2×#11765849C2212yescomplemented
13#14556111—21no—
14#14556107C2212yescomplemented
15#14556102C2212yescomplemented
16#14556097—21no—
17C4×C22⋊D71—22no—
18#14556087C2211yescharacteristic, complemented
19#14556049C2218yescomplemented
20C22⋊(C71⋊C8)C2214yescomplemented
21C25×C71—21no—
22C22×D4×C71—28no—
23C22×D4×C71—22no—
24C22×D4×C71—21no—
25C22×D4×C71C2214yes—
26C22×D4×C71C2212yes—
27C23×C4×C71C2211yescharacteristic
28#11765865—24no—
29C2×Dic142C2311yescharacteristic, complemented
30C2×D284C2311yescharacteristic, complemented
31C2×C22⋊D71—21no—
32C2×C22⋊D71—22no—
33C2×C4×D71—21no—
34#11765853—28no—
35#11765850—28no—
36#11765849—24no—
37C2×C71⋊C8C2314yescomplemented
38C4⋊Dic71C2312yescomplemented
39Dic71⋊C4—22no—
40C22⋊Dic71—21no—
41C22⋊Dic71—22no—
42(C2×D71)⋊C4—22no—
43C4×Dic71—21no—
44C24×C71—28no—
45C24×C71—24no—
46C24×C71—22no—
47C24×C71—48no—
48C24×C71D411yescharacteristic
49C2×D4×C71—28no—
50C2×D4×C71—216no—
51C2×D4×C71—216no—
52C2×D4×C71—24no—
53C2×D4×C71—22no—
54C2×D4×C71—24no—
55C2×D4×C71D418yescomplemented
56C2×D4×C71C2314yes—
57C22×C4×C71—24no—
58C22×C4×C71D412yes—
59C22×C4×C71D411yescharacteristic
60C22×C4×C71C2312yes—
61C22×C4×C71C2311yescharacteristic
62C23×C71—22no—
63C23×C71—24no—
64C23×C71—216no—
65C23×C71—28no—
66C23×C71—24no—
67C23×C71—432no—
68C23×C71—48no—
69C23×C71—48no—
70C23×C71—48no—
71C23×C71C2×D411yescharacteristic
72C23×C71C2×D412yes—
73C22×D71—21no—
74D4×C71—216no—
75D4×C71—28no—
76D4×C71—24no—
77D4×C71C2×D418yes—
78C2×C4×C71—22no—
79C2×C4×C71—24no—
80C2×C4×C71—24no—
81C2×C4×C71C2×D412yes—
82C2×C4×C71C2×D411yescharacteristic
83C2×C4×C71C2×D414yes—
84C2×C4×C71C2411yescharacteristic
85C22⋊D71—42no—
86C22⋊D71—44no—
87C2×Dic71—22no—
88C2×Dic71—21no—
89D284—22no—
90C4×D71—42no—
91Dic142—22no—
92C71⋊C8—24no—
93C22×C71—28no—
94C22×C71—22no—
95C22×C71—22no—
96C22×C71—22no—
97C22×C71—28no—
98C22×C71—416no—
99C22×C71—432no—
100C22×C71—48no—
101C22×C71#12012yes—
102C22×C71AGL(1,ℤ/8)14yescomplemented
103C22×C71C22×D411yescharacteristic
104C2×D71—41no—
105C2×D71—42no—
106C4×C71—22no—
107C4×C71#12012yes—
108C4×C71C22×D412yes—
109Dic71—41no—
110Dic71—42no—
111C2×C71—22no—
112C2×C71—24no—
113C2×C71—48no—
114C2×C71C2×#12011yescharacteristic
115C2×C71C2×AGL(1,ℤ/8)12yes—
116D71—81no—
117#3003—711nomaximal, Sylow
118C71#300311yescharacteristic, complemented, Sylow
119C23×D4D7111yescharacteristic, complemented
120C2×AGL(1,ℤ/8)—712no—
121#525—711no—
122#450—714no—
123#447—714no—
124#418—712no—
125#407—711no—
126C25—21no—
127C2×#41—711no—
128C22×D4—28no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4C2×D7114yes—
132C22×D4C2×D7112yes—
133C23×C4C2×D7111yescharacteristic
134AGL(1,ℤ/8)—1428no—
135C2×SD16—714no—
136C2×D8—714no—
137C2×M16—712no—
138#123—1421no—
139C22⋊Q8—712no—
140#121—712no—
141#120—1421no—
142C4×D4—1422no—
143#117—711no—
144#102—718no—
145C22⋊C8—714no—
146C24—28no—
147C24—24no—
148C24—22no—
149C24—48no—
150C24C22⋊D7111yescharacteristic
151#41—1424no—
152C2×Q8—711no—
153C2×D4—28no—
154C2×D4—216no—
155C2×D4—216no—
156C2×D4—24no—
157C2×D4—24no—
158C2×D4—22no—
159C2×D4—711no—
160C2×D4—1422no—
161C2×D4—1421no—
162C2×D4C22⋊D7118yescomplemented
163C2×D4C22×D7114yes—
164C22×C4—24no—
165C22×C4—1421no—
166C22×C4C22⋊D7112yes—
167C22×C4C22⋊D7111yescharacteristic
168C22×C4C22×D7112yes—
169C22×C4C22×D7111yescharacteristic
170SD16—1428no—
171D8—1428no—
172M16—1424no—
173C2×C8—714no—
174C4⋊C4—712no—
175C4⋊C4—1422no—
176C22⋊C4—1421no—
177C22⋊C4—1422no—
178C22⋊C4—1422no—
179C42—1421no—
180C23—22no—
181C23—24no—
182C23—216no—
183C23—24no—
184C23—28no—
185C23—432no—
186C23—48no—
187C23—48no—
188C23—48no—
189C23—1421no—
190C23C2×C22⋊D7111yescharacteristic
191C23C2×C22⋊D7112yes—
192Q8—1422no—
193D4—216no—
194D4—24no—
195D4—28no—
196D4—2842no—
197D4—2844no—
198D4—1422no—
199D4C2×C22⋊D7118yes—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—2842no—
204C2×C4—1422no—
205C2×C4—1421no—
206C2×C4C2×C22⋊D7112yes—
207C2×C4C2×C22⋊D7114yes—
208C2×C4C2×C22⋊D7111yescharacteristic
209C2×C4C23×D7111yescharacteristic
210C8—1424no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—28no—
216C22—416no—
217C22—432no—
218C22—48no—
219C22—2841no—
220C22—2842no—
221C22#1455609712yes—
222C22#1455615414yescomplemented
223C22C22×C22⋊D7111yescharacteristic
224C4—22no—
225C4—2841no—
226C4—2842no—
227C4#1455609712yes—
228C4C22×C22⋊D7112yes—
229C2—22no—
230C2—24no—
231C2—48no—
232C2—5681no—
233C2C2×#1455609711yescharacteristic
234C2C2×#1455615412yes—
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC23×D4×C71C2
socleC22×C71C22×D4
hypercenterC23×D4D71
hypocenterC71#3003
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#14556154C2semidirect
2. #17986510C2semidirect
3. #17986150C2semidirect
4. #17986133C2semidirect
5. #17986004C2semidirect
6. #17985963C2semidirect
7. C23×D4×C71C2semidirect
8. C2×#11765865C22semidirect
10. C2×#11765853C22semidirect
11. C2×#11765850C22semidirect
12. C2×#11765849C22semidirect
14. #14556107C22semidirect
15. #14556102C22semidirect
18. #14556087C22semidirect
19. #14556049C22semidirect
20. C22⋊(C71⋊C8)C22semidirect
29. C2×Dic142C23semidirect
30. C2×D284C23semidirect
37. C2×C71⋊C8C23semidirect
38. C4⋊Dic71C23semidirect
55. C2×D4×C71D4semidirect
102. C22×C71AGL(1,ℤ/8)semidirect
118. C71#3003semidirect
119. C23×D4D71semidirect
162. C2×D4C22⋊D71semidirect
222. C22#14556154semidirect