Finite Group Databasebeta

← #22597585

Subgroups — #22597585

Order 9088.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#11765865C211yesmaximal, characteristic, complemented
2C2×#14556161C211yesmaximal, characteristic, complemented
3C2×#14556156C211yesmaximal, characteristic, complemented
4#17986510C211yesmaximal, characteristic, complemented
5#17986163C214yesmaximal, complemented
6#17986146C214yesmaximal, complemented
7#17986009C211yesmaximal, characteristic, complemented
8#17986005C211yesmaximal, characteristic, complemented
9#407×C71C211yesmaximal, characteristic, complemented
10C2×#11765865—24no—
11C2×#11765865—24no—
12C2×#11765865—22no—
13C2×#11765865C2214yes—
14C2×#11765865C2211yescharacteristic
15C22×Dic142C2211yescharacteristic
16C22×D284C2211yescharacteristic
17C22×C22⋊D71—21no—
18C22×C4×D71—21no—
19#14556161—24no—
20#14556156—24no—
21C2×Dic284C2212yescomplemented
22C2×#11765854C2212yescomplemented
23C2×#11765854C2212yescomplemented
24C2×D568C2212yescomplemented
25#14556111—21no—
26#14556107C2212yescomplemented
27#14556102C2212yescomplemented
28#14556097—21no—
29C4×C22⋊D71—22no—
30#14556087C2211yescharacteristic
31#14556053C2214yescomplemented
32#14556050C2214yescomplemented
33C23×C4×C71C2211yescharacteristic
34C2×M16×C71C2211yescharacteristic
35C2×M16×C71C2211yescharacteristic
36C22⋊C8×C71C2214yescomplemented
37C23×D71—21no—
38#11765865—216no—
39#11765865—28no—
40#11765865—22no—
41#11765865—22no—
42#11765865—28no—
43C2×Dic142—22no—
44C2×Dic142—21no—
45C2×Dic142D414yescomplemented
46C2×Dic142C2311yescharacteristic
47C2×Dic142C2312yes—
48C2×D284—22no—
49C2×D284—21no—
50C2×D284D414yescomplemented
51C2×D284C2311yescharacteristic
52C2×D284C2312yes—
53C2×C22⋊D71—28no—
54C2×C22⋊D71—24no—
55C2×C22⋊D71—21no—
56C2×C22⋊D71—22no—
57C22×Dic71—21no—
58C2×C4×D71—28no—
59C2×C4×D71—24no—
60C2×C4×D71—21no—
61Dic284—24no—
62#11765854—24no—
63#11765854—24no—
64D568—24no—
65C4⋊Dic71C2312yes—
66Dic71⋊C4—22no—
67C22⋊Dic71—21no—
68C22⋊Dic71—22no—
69(C2×D71)⋊C4—22no—
70C4×Dic71—21no—
71C24×C71D411yescharacteristic
72C22×C4×C71—22no—
73C22×C4×C71—22no—
74C22×C4×C71D411yescharacteristic
75C22×C4×C71D412yes—
76C22×C4×C71C2312yes—
77C22×C4×C71C2311yescharacteristic
78M16×C71—22no—
79M16×C71—22no—
80C2×C8×C71C2312yes—
81C2×C8×C71C2312yes—
82C23×C71—22no—
83C23×C71—42no—
84C23×C71C2×D412yes—
85C23×C71C2×D411yescharacteristic
86C22×D71—24no—
87C22×D71—22no—
88C22×D71—21no—
89C22×D71—44no—
90C2×C4×C71—22no—
91C2×C4×C71—24no—
92C2×C4×C71—22no—
93C2×C4×C71—22no—
94C2×C4×C71C2×D412yes—
95C2×C4×C71C2×D412yes—
96C2×C4×C71C2×D412yes—
97C2×C4×C71C2×D411yescharacteristic
98C2×C4×C71C2411yescharacteristic
99C22⋊D71—216no—
100C22⋊D71—44no—
101C22⋊D71—42no—
102C2×Dic71—24no—
103C2×Dic71—22no—
104C2×Dic71—22no—
105C2×Dic71—21no—
106D284—24no—
107D284—22no—
108D284—21no—
109D284—21no—
110D284C2×D414yes—
111C4×D71—216no—
112C4×D71—42no—
113Dic142—24no—
114Dic142—22no—
115Dic142—21no—
116Dic142—21no—
117Dic142C2×D414yes—
118C8×C71—22no—
119C8×C71—22no—
120C22×C71—22no—
121C22×C71—22no—
122C22×C71—22no—
123C22×C71—22no—
124C22×C71—44no—
125C22×C71#12012yes—
126C22×C71C22×D411yescharacteristic
127C2×D71—24no—
128C2×D71—48no—
129C2×D71—44no—
130C2×D71—41no—
131C2×D71—42no—
132C4×C71—22no—
133C4×C71#12012yes—
134C4×C71C22×D411yescharacteristic
135C4×C71C22×D411yescharacteristic
136Dic71—24no—
137Dic71—41no—
138Dic71—42no—
139C2×C71—22no—
140C2×C71—42no—
141C2×C71C2×#12011yescharacteristic
142C2×C71#57512yes—
143D71—44no—
144D71—81no—
145#3005—711nomaximal, Sylow
146C71#300511yescharacteristic, complemented, Sylow
147C22×#41—711no—
148C2×#137—711no—
149C2×AGL(1,ℤ/8)—711no—
150#525—711no—
151#452—714no—
152#449—714no—
153#419—711no—
154#418—711no—
155#407D7111yescharacteristic, complemented
156C2×#41—711no—
157C2×#41—714no—
158C2×#41—1424no—
159C2×#41—1424no—
160C2×#41—1422no—
161C22×Q8—711no—
162C22×D4—711no—
163C22×D4—1421no—
164C23×C4—1421no—
165C23×C4C2×D7111yescharacteristic
166#137—1424no—
167AGL(1,ℤ/8)—1424no—
168C2×Q16—712no—
169C2×SD16—712no—
170C2×SD16—712no—
171C2×D8—712no—
172C2×M16C2×D7111yescharacteristic
173C2×M16C2×D7111yescharacteristic
174#123—1421no—
175C22⋊Q8—712no—
176#121—712no—
177#120—1421no—
178C4×D4—1422no—
179#117—711no—
180#103—714no—
181#102—714no—
182C22⋊C8C2×D7114yescomplemented
183C24—1421no—
184C24D28411yescharacteristic
185#41—1428no—
186#41—1428no—
187#41—14216no—
188#41—1422no—
189#41—1422no—
190C2×Q8—714no—
191C2×Q8—712no—
192C2×Q8—711no—
193C2×Q8—1422no—
194C2×Q8—1421no—
195C2×D4—714no—
196C2×D4—712no—
197C2×D4—711no—
198C2×D4—1422no—
199C2×D4—1421no—
200C2×D4—1428no—
201C2×D4—1424no—
202C2×D4—1422no—
203C2×D4—1421no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—1428no—
207C22×C4—1421no—
208C22×C4—1421no—
209C22×C4—1424no—
210C22×C4D28412yes—
211C22×C4D28411yescharacteristic
212C22×C4C22×D7112yes—
213C22×C4C22×D7111yescharacteristic
214Q16—1424no—
215SD16—1424no—
216SD16—1424no—
217D8—1424no—
218M16—22no—
219M16—22no—
220C2×C8C22×D7112yes—
221C2×C8C22×D7112yes—
222C4⋊C4—712no—
223C4⋊C4—1422no—
224C22⋊C4—1421no—
225C22⋊C4—1422no—
226C22⋊C4—1422no—
227C42—1421no—
228C23—22no—
229C23—42no—
230C23—2844no—
231C23—1421no—
232C23—1422no—
233C23—1424no—
234C23C2×D28412yes—
235C23C2×D28411yescharacteristic
236Q8—714no—
237Q8—1421no—
238Q8—1421no—
239Q8—1422no—
240Q8—1424no—
241D4—2844no—
242D4—2842no—
243D4—714no—
244D4—14216no—
245D4—1421no—
246D4—1421no—
247D4—1422no—
248D4—1424no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—2842no—
254C2×C4—14216no—
255C2×C4—1422no—
256C2×C4—1421no—
257C2×C4—1422no—
258C2×C4—1424no—
259C2×C4D4×D7112yes—
260C2×C4D4×D7112yes—
261C2×C4C2×D28412yes—
262C2×C4C2×D28411yescharacteristic
263C2×C4C23×D7111yescharacteristic
264C8—22no—
265C8—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—2848no—
272C22—2844no—
273C22—2841no—
274C22—2842no—
275C22—1424no—
276C22#1455609812yes—
277C22C22×D28411yescharacteristic
278C4—22no—
279C4—2841no—
280C4—2842no—
281C4—1424no—
282C4#1455609812yes—
283C4C2×D4×D7111yescharacteristic
284C4C2×D4×D7111yescharacteristic
285C2—22no—
286C2—42no—
287C2—2844no—
288C2—5681no—
289C2C2×#1455609811yescharacteristic
290C2#1798679912yes—
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#407×C71C2
socleC22×C71C22×D4
hypercenter#407D71
hypocenterC71#3005
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#11765865C2semidirect
2. C2×#14556161C2semidirect
3. C2×#14556156C2semidirect
4. #17986510C2semidirect
5. #17986163C2semidirect
6. #17986146C2semidirect
7. #17986009C2semidirect
8. #17986005C2semidirect
9. #407×C71C2semidirect
21. C2×Dic284C22semidirect
22. C2×#11765854C22semidirect
23. C2×#11765854C22semidirect
24. C2×D568C22semidirect
26. #14556107C22semidirect
27. #14556102C22semidirect
31. #14556053C22semidirect
32. #14556050C22semidirect
36. C22⋊C8×C71C22semidirect
45. C2×Dic142D4semidirect
50. C2×D284D4semidirect
146. C71#3005semidirect
155. #407D71semidirect
182. C22⋊C8C2×D71semidirect