Finite Group Databasebeta

← #22599095

Subgroups — #22599095

Order 9088.

468 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 121 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×#14556104C211yesmaximal, characteristic, complemented
2#121×D71C211yesmaximal, characteristic
3#17986328C212yesmaximal, complemented
4#17986318C211yesmaximal, characteristic, complemented
5#17986315C211yesmaximal, characteristic, complemented
6#17986231C212yesmaximal, complemented
7#17986225C212yesmaximal, complemented
8#17986148C212yesmaximal, complemented
9#17986138C211yesmaximal, characteristic, complemented
10C22⋊C8×D71C211yesmaximal, characteristic, complemented
11#448×C71C211yesmaximal, characteristic, complemented
12C2×D4×D71—21no—
13C2×D4×D71C2211yescharacteristic
14C22×C4×D71C2211yescharacteristic
15C2×#11765852C2212yescomplemented
16C2×C8×D71C2212yescomplemented
17#14556104—21no—
18#14556104—21no—
19#14556104C2212yes—
20#14556104C2212yescomplemented
21#14556103C2211yescharacteristic
22#14556102C2211yescharacteristic
23#14556101—21no—
24#14556100—21no—
25#14556099C2211yescharacteristic
26C2×C4⋊Dic71C2211yescharacteristic, complemented
27C2×Dic71⋊C4—21no—
28C4⋊C4×D71C2211yescharacteristic
29C2×(C2×D71)⋊C4—21no—
30C22⋊C4×D71—21no—
31#14556060C2212yescomplemented
32#14556059C2212yescomplemented
33#14556052C2212yescomplemented
34#14556050C2212yescomplemented
35#14556049C2212yescomplemented
36C22⋊(C71⋊C8)C2211yescharacteristic, complemented
37(C2×D71)⋊C8C2212yes—
38C22×Q8×C71C2211yescharacteristic, complemented
39C2×SD16×C71C2212yescomplemented
40#121×C71C2211yescharacteristic
41#103×C71C2212yescomplemented
42C22⋊C8×C71C2211yescharacteristic, complemented
43C23×D71—21no—
44C23×D71D411yescharacteristic
45C2×D284—21no—
46C2×D284C2311yescharacteristic
47C2×C22⋊D71—21no—
48C2×C22⋊D71—21no—
49C2×C22⋊D71—21no—
50D4×D71—42no—
51D4×D71—44no—
52C22×Dic71—21no—
53C22×Dic71D411yescharacteristic
54C2×C4×D71—21no—
55C2×C4×D71—42no—
56C2×C4×D71D411yescharacteristic
57C2×C4×D71D411yescharacteristic
58C2×C4×D71C2311yescharacteristic
59C2×C4×D71C2311yescharacteristic
60#11765852—42no—
61C2×C71⋊C8C2312yes—
62C8×D71—42no—
63C4⋊Dic71—21no—
64C4⋊Dic71C2312yes—
65C4⋊Dic71C2311yescharacteristic
66Dic71⋊C4—21no—
67Dic71⋊C4—22no—
68Dic71⋊C4—22no—
69C22⋊Dic71—21no—
70(C2×D71)⋊C4—22no—
71(C2×D71)⋊C4—22no—
72(C2×D71)⋊C4—21no—
73(C2×D71)⋊C4—21no—
74C2×Q8×C71—21no—
75C2×Q8×C71—42no—
76C2×Q8×C71C2312yes—
77C2×D4×C71—21no—
78C2×D4×C71C2311yescharacteristic
79C22×C4×C71—21no—
80C22×C4×C71C2311yescharacteristic
81SD16×C71—42no—
82C2×C8×C71C2312yes—
83C4⋊C4×C71C2311yescharacteristic
84C22⋊C4×C71—21no—
85C23×C71—21no—
86C23×C71C2×D411yescharacteristic
87C22×D71—21no—
88C22×D71—41no—
89C22×D71—41no—
90C22×D71—42no—
91C22×D71—42no—
92C22×D71—42no—
93C22×D71SD1614yescomplemented
94C22×D71C2×D411yescharacteristic
95C22×D71C2×D411yescharacteristic
96Q8×C71—22no—
97Q8×C71—22no—
98Q8×C71—42no—
99D4×C71—22no—
100D4×C71—42no—
101C2×C4×C71—22no—
102C2×C4×C71—21no—
103C2×C4×C71—22no—
104C2×C4×C71—42no—
105C2×C4×C71C2×D411yescharacteristic
106C2×C4×C71C2411yescharacteristic
107C22⋊D71—42no—
108C22⋊D71—42no—
109C22⋊D71—42no—
110C2×Dic71—22no—
111C2×Dic71—21no—
112C2×Dic71—41no—
113C2×Dic71—42no—
114C2×Dic71C2×D411yescharacteristic
115C2×Dic71C2×D411yescharacteristic
116D284—22no—
117D284—42no—
118C4×D71—22no—
119C4×D71—22no—
120C4×D71—41no—
121C4×D71—41no—
122C4×D71—42no—
123C8×C71—22no—
124C71⋊C8—22no—
125C22×C71—21no—
126C22×C71—21no—
127C22×C71—41no—
128C22×C71—41no—
129C22×C71—41no—
130C22×C71C2×SD1612yes—
131C22×C71C22×D411yescharacteristic
132C2×D71—21no—
133C2×D71—21no—
134C2×D71—21no—
135C2×D71—21no—
136C2×D71—44no—
137C2×D71—41no—
138C2×D71—41no—
139C2×D71—41no—
140C2×D71—84no—
141C2×D71C2×SD1612yes—
142C2×D71C2×SD1612yes—
143C4×C71—21no—
144C4×C71—42no—
145C4×C71—41no—
146C4×C71H(5,𝔽2)12yes—
147Dic71—21no—
148Dic71—21no—
149Dic71—41no—
150Dic71—42no—
151C2×C71—22no—
152C2×C71—81no—
153C2×C71#53411yescharacteristic
154C2×C71C22×SD1611yescharacteristic
155C2×C71#57511yescharacteristic
156D71—22no—
157D71—22no—
158D71—81no—
159#3187—711nomaximal, Sylow
160C71#318711yescharacteristic, complemented, Sylow
161C2×C22⋊Q8—711no—
162C2×#121—711no—
163#483—712no—
164#481—711no—
165#481—711no—
166#465—712no—
167#465—712no—
168#450—712no—
169#448—711no—
170#448D7111yescharacteristic, complemented
171C2×C22⋊C8—711no—
172C22×Q8C2×D7111yescharacteristic, complemented
173C22×D4—711no—
174C22×D4—1421no—
175C23×C4—711no—
176C2×SD16—712no—
177C2×SD16C2×D7112yescomplemented
178C22×C8—712no—
179C22⋊Q8—712no—
180C22⋊Q8—712no—
181C22⋊Q8—1421no—
182C22⋊Q8—1421no—
183#121—711no—
184#121—711no—
185#121—711no—
186#121—1421no—
187#121—1421no—
188#121C2×D7111yescharacteristic
189C2×C4⋊C4—711no—
190C2×C4⋊C4—711no—
191C2×C4⋊C4—1421no—
192C2×C22⋊C4—1421no—
193C2×C22⋊C4—1421no—
194#106—712no—
195#106—712no—
196#103—712no—
197#103C2×D7112yescomplemented
198#102—712no—
199#102—712no—
200C22⋊C8—711no—
201C22⋊C8—712no—
202C22⋊C8C2×D7111yescharacteristic, complemented
203C24—711no—
204C24—1421no—
205C2×Q8—21no—
206C2×Q8—42no—
207C2×Q8C22×D7112yes—
208C2×D4—21no—
209C2×D4—2844no—
210C2×D4—2842no—
211C2×D4—711no—
212C2×D4—1421no—
213C2×D4—1421no—
214C2×D4—1421no—
215C2×D4—1421no—
216C2×D4C22×D7111yescharacteristic
217C22×C4—21no—
218C22×C4—2842no—
219C22×C4—711no—
220C22×C4—711no—
221C22×C4—711no—
222C22×C4—711no—
223C22×C4—711no—
224C22×C4—1421no—
225C22×C4—1421no—
226C22×C4C22×D7111yescharacteristic
227SD16—42no—
228SD16—2842no—
229C2×C8—2842no—
230C2×C8—712no—
231C2×C8C22×D7112yes—
232C4⋊C4—712no—
233C4⋊C4—711no—
234C4⋊C4—1421no—
235C4⋊C4—1422no—
236C4⋊C4—1422no—
237C4⋊C4—1421no—
238C4⋊C4C22×D7111yescharacteristic
239C22⋊C4—21no—
240C22⋊C4—1421no—
241C22⋊C4—1422no—
242C22⋊C4—1422no—
243C22⋊C4—1421no—
244C22⋊C4—1421no—
245C23—21no—
246C23—2841no—
247C23—2841no—
248C23—2842no—
249C23—2842no—
250C23—2842no—
251C23—714no—
252C23—711no—
253C23—711no—
254C23—1421no—
255C23D4×D7111yescharacteristic
256Q8—22no—
257Q8—22no—
258Q8—42no—
259D4—22no—
260D4—42no—
261D4—2842no—
262D4—2842no—
263D4—2842no—
264D4—2842no—
265D4—1422no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—42no—
270C2×C4—2841no—
271C2×C4—2841no—
272C2×C4—2842no—
273C2×C4—2842no—
274C2×C4—2841no—
275C2×C4—711no—
276C2×C4—711no—
277C2×C4—1422no—
278C2×C4—1421no—
279C2×C4—1422no—
280C2×C4—1422no—
281C2×C4D4×D7111yescharacteristic
282C2×C4C23×D7111yescharacteristic
283C8—22no—
284C8—1422no—
285C22—21no—
286C22—21no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—2844no—
291C22—2841no—
292C22—2841no—
293C22—2841no—
294C22—5684no—
295C22—712no—
296C22—712no—
297C22—1421no—
298C22—1421no—
299C22—1421no—
300C22—1421no—
301C22SD16×D7112yes—
302C22C2×D4×D7111yescharacteristic
303C4—21no—
304C4—41no—
305C4—42no—
306C4—2841no—
307C4—2842no—
308C4—1421no—
309C4—1421no—
310C4#1455617312yes—
311C2—22no—
312C2—81no—
313C2—5681no—
314C2—1422no—
315C2—1422no—
316C2#1798655911yescharacteristic
317C2C2×SD16×D7111yescharacteristic
318C2#1798679811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#448×C71C2
socleC22×C71C22×D4
hypercenter#448D71
hypocenterC71#3187
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#14556104C2semidirect
3. #17986328C2semidirect
4. #17986318C2semidirect
5. #17986315C2semidirect
6. #17986231C2semidirect
7. #17986225C2semidirect
8. #17986148C2semidirect
9. #17986138C2semidirect
10. C22⋊C8×D71C2semidirect
11. #448×C71C2semidirect
15. C2×#11765852C22semidirect
16. C2×C8×D71C22semidirect
20. #14556104C22semidirect
26. C2×C4⋊Dic71C22semidirect
31. #14556060C22semidirect
32. #14556059C22semidirect
33. #14556052C22semidirect
34. #14556050C22semidirect
35. #14556049C22semidirect
36. C22⋊(C71⋊C8)C22semidirect
38. C22×Q8×C71C22semidirect
39. C2×SD16×C71C22semidirect
41. #103×C71C22semidirect
42. C22⋊C8×C71C22semidirect
93. C22×D71SD16semidirect
160. C71#3187semidirect
170. #448D71semidirect
172. C22×Q8C2×D71semidirect
177. C2×SD16C2×D71semidirect
197. #103C2×D71semidirect
202. C22⋊C8C2×D71semidirect