Finite Group Databasebeta

← #22599956

Subgroups — #22599956

Order 9088.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 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#17986670C211yesmaximal, characteristic, complemented
3#17986375C211yesmaximal, characteristic, complemented
4#17986250C212yesmaximal, complemented
5#17986196C211yesmaximal, characteristic, complemented
6#17986194C211yesmaximal, characteristic, complemented
7#17986142C212yesmaximal, complemented
8#17986133C212yesmaximal, complemented
9C4×#11765853C211yesmaximal, characteristic, complemented
10#17986077C211yesmaximal, characteristic, complemented
11D42×C71C211yesmaximal, characteristic, complemented
12C2×#11765865C2212yescomplemented
13#14556154—28no—
14C2×#11765853C2211yescharacteristic
15C2×#11765853C2212yescomplemented
16C2×#11765850C2212yes—
17C2×#11765850C2212yescomplemented
18C2×#11765850C2212yescomplemented
19C2×#11765849C2212yescomplemented
20#14556125C2211yescharacteristic, complemented
21#14556124—21no—
22#14556102C2212yescomplemented
23#14556101—22no—
24C4×Dic142C2211yescharacteristic, complemented
25#14556090—21no—
26#14556059C2211yescharacteristic, complemented
27C4⋊(C71⋊C8)C2211yescharacteristic, complemented
28#14556051C2211yescharacteristic, complemented
29#14556049C2211yescharacteristic
30#14556049C2212yescomplemented
31#14556048C2212yescomplemented
32C22⋊(C71⋊C8)C2212yescomplemented
33C4×C71⋊C8C2211yescharacteristic, complemented
34C22×D4×C71—21no—
35C22×D4×C71C2212yes—
36#127×C71C2211yescharacteristic
37#121×C71—21no—
38#121×C71C2212yes—
39#120×C71—22no—
40C4×D4×C71C2211yescharacteristic
41C4×D4×C71C2211yescharacteristic
42#11765865—24no—
43#11765865—42no—
44C2×Dic142C2311yescharacteristic
45C2×D284—21no—
46C2×D284—21no—
47C2×D284C2312yes—
48C2×C22⋊D71—22no—
49C2×C22⋊D71—22no—
50C2×C4×D71—22no—
51#11765853—24no—
52#11765853D414yescomplemented
53#11765850—24no—
54#11765850—24no—
55#11765850—42no—
56#11765849—24no—
57C2×C71⋊C8C2311yescharacteristic
58C2×C71⋊C8C2312yes—
59C2×C71⋊C8C2311yescharacteristic
60C4⋊Dic71C2311yescharacteristic
61Dic71⋊C4—21no—
62(C2×D71)⋊C4—22no—
63C4×Dic71—21no—
64C24×C71—22no—
65C2×D4×C71—24no—
66C2×D4×C71—22no—
67C2×D4×C71—22no—
68C2×D4×C71—21no—
69C2×D4×C71—21no—
70C2×D4×C71—42no—
71C2×D4×C71—44no—
72C2×D4×C71D411yescharacteristic
73C2×D4×C71C2312yes—
74C2×D4×C71C2311yescharacteristic
75C22×C4×C71—21no—
76C22×C4×C71C2312yes—
77C4⋊C4×C71D411yescharacteristic
78C4⋊C4×C71C2311yescharacteristic
79C22⋊C4×C71—21no—
80C22⋊C4×C71—22no—
81C22⋊C4×C71D412yes—
82C42×C71C2311yescharacteristic
83C23×C71—21no—
84C23×C71—22no—
85C23×C71—44no—
86C23×C71—44no—
87C23×C71—82no—
88C23×C71C2×D412yes—
89C22×D71—22no—
90D4×C71—22no—
91D4×C71—22no—
92D4×C71—21no—
93D4×C71—22no—
94D4×C71—44no—
95D4×C71—44no—
96D4×C71—42no—
97D4×C71—81no—
98D4×C71C2×D412yes—
99C2×C4×C71—22no—
100C2×C4×C71—22no—
101C2×C4×C71—21no—
102C2×C4×C71—42no—
103C2×C4×C71C2×D411yescharacteristic
104C2×C4×C71C2×D412yes—
105C2×C4×C71C2×D411yescharacteristic
106C2×C4×C71C2411yescharacteristic
107C22⋊D71—44no—
108C22⋊D71—82no—
109C2×Dic71—21no—
110C2×Dic71—21no—
111D284—22no—
112D284—22no—
113D284—42no—
114D284—81no—
115C4×D71—44no—
116Dic142—21no—
117Dic142C2×D412yescomplemented
118C71⋊C8—22no—
119C71⋊C8—21no—
120C71⋊C8C2×D412yescomplemented
121C22×C71—22no—
122C22×C71—22no—
123C22×C71—22no—
124C22×C71—22no—
125C22×C71—41no—
126C22×C71—41no—
127C22×C71—42no—
128C22×C71—42no—
129C22×C71—42no—
130C22×C71—88no—
131C22×C71C22×D411yescharacteristic
132C2×D71—42no—
133C2×D71—42no—
134C2×D71—42no—
135C4×C71—22no—
136C4×C71—21no—
137C4×C71—41no—
138C4×C71AGL(1,ℤ/8)12yescomplemented
139C4×C71C22×D411yescharacteristic
140C4×C71H(5,𝔽2)11yescharacteristic
141Dic71—22no—
142Dic71—41no—
143C2×C71—42no—
144C2×C71—42no—
145C2×C71—82no—
146C2×C71D4211yescharacteristic
147C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
148C2×C71#57611yescharacteristic
149D71—82no—
150#3274—711nomaximal, Sylow
151C71#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—712no—
153#550—711no—
154D42D7111yescharacteristic, complemented
155#493—711no—
156#469—712no—
157#460—711no—
158#459—711no—
159#449—712no—
160#447—712no—
161C4×SD16—711no—
162#436—711no—
163C2×#41—712no—
164C22×D4—21no—
165C22×D4C2×D7112yes—
166AGL(1,ℤ/8)—1428no—
167C2×SD16—711no—
168C2×SD16—712no—
169C2×D8—712no—
170C2×D8—712no—
171C2×D8—712no—
172C2×M16—712no—
173#127—711no—
174#127—1421no—
175#127C2×D7111yescharacteristic
176#121—21no—
177#121—712no—
178#121—1422no—
179#121C2×D7112yes—
180#120—22no—
181C4×Q8—711no—
182C4×D4—1421no—
183C4×D4C2×D7111yescharacteristic
184C4×D4C2×D7111yescharacteristic
185#106—711no—
186C4⋊C8—711no—
187#103—711no—
188#102—711no—
189#102—712no—
190#102—712no—
191C22⋊C8—712no—
192C4×C8—711no—
193C24—22no—
194#41—2842no—
195#41—1424no—
196C2×Q8—711no—
197C2×D4—24no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—42no—
203C2×D4—44no—
204C2×D4—712no—
205C2×D4—1422no—
206C2×D4—1422no—
207C2×D4—1421no—
208C2×D4—1421no—
209C2×D4C22⋊D7111yescharacteristic
210C2×D4C22×D7112yes—
211C2×D4C22×D7111yescharacteristic
212C22×C4—21no—
213C22×C4—1422no—
214C22×C4C22×D7112yes—
215SD16—714no—
216SD16—1424no—
217D8—2842no—
218D8—1424no—
219D8—1424no—
220M16—1424no—
221C2×C8—711no—
222C2×C8—711no—
223C2×C8—712no—
224C4⋊C4—711no—
225C4⋊C4—1421no—
226C4⋊C4C22⋊D7111yescharacteristic
227C4⋊C4C22×D7111yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—1422no—
231C22⋊C4C22⋊D7112yes—
232C42—1421no—
233C42C22×D7111yescharacteristic
234C23—21no—
235C23—22no—
236C23—44no—
237C23—44no—
238C23—82no—
239C23—1422no—
240C23C2×C22⋊D7112yes—
241Q8—712no—
242Q8—1421no—
243D4—22no—
244D4—21no—
245D4—22no—
246D4—22no—
247D4—44no—
248D4—44no—
249D4—42no—
250D4—81no—
251D4—2844no—
252D4—2842no—
253D4—5681no—
254D4—5682no—
255D4—1422no—
256D4—1422no—
257D4D4×D7112yes—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—2844no—
263C2×C4—1421no—
264C2×C4—1421no—
265C2×C4C2×C22⋊D7111yescharacteristic
266C2×C4C2×C22⋊D7112yes—
267C2×C4C2×C22⋊D7111yescharacteristic
268C2×C4C23×D7111yescharacteristic
269C8—712no—
270C8—1422no—
271C8—1421no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—88no—
282C22—2842no—
283C22—2842no—
284C22—2842no—
285C22C22×C22⋊D7111yescharacteristic
286C4—22no—
287C4—21no—
288C4—41no—
289C4—2841no—
290C4—1422no—
291C4#1455615412yescomplemented
292C4C2×D4×D7111yescharacteristic
293C4#1455617311yescharacteristic
294C2—42no—
295C2—42no—
296C2—82no—
297C2—5682no—
298C2D4×C22⋊D7111yescharacteristic
299C2C2×#1455615411yescharacteristic
300C2#1798680211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingD42×C71C2
socleC22×C71C22×D4
hypercenterD42D71
hypocenterC71#3274
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#14556154C2semidirect
2. #17986670C2semidirect
3. #17986375C2semidirect
4. #17986250C2semidirect
5. #17986196C2semidirect
6. #17986194C2semidirect
7. #17986142C2semidirect
8. #17986133C2semidirect
9. C4×#11765853C2semidirect
10. #17986077C2semidirect
11. D42×C71C2semidirect
12. C2×#11765865C22semidirect
15. C2×#11765853C22semidirect
17. C2×#11765850C22semidirect
18. C2×#11765850C22semidirect
19. C2×#11765849C22semidirect
20. #14556125C22semidirect
22. #14556102C22semidirect
24. C4×Dic142C22semidirect
26. #14556059C22semidirect
27. C4⋊(C71⋊C8)C22semidirect
28. #14556051C22semidirect
30. #14556049C22semidirect
31. #14556048C22semidirect
32. C22⋊(C71⋊C8)C22semidirect
33. C4×C71⋊C8C22semidirect
52. #11765853D4semidirect
117. Dic142C2×D4semidirect
120. C71⋊C8C2×D4semidirect
138. C4×C71AGL(1,ℤ/8)semidirect
151. C71#3274semidirect
154. D42D71semidirect
291. C4#14556154semidirect