Finite Group Databasebeta

← #22600445

Subgroups — #22600445

Order 9088.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986658C211yesmaximal, characteristic
2#17986361C211yesmaximal, characteristic, complemented
3#17986329C211yesmaximal, characteristic, complemented
4#17986317C211yesmaximal, characteristic, complemented
5#17986281C211yesmaximal, characteristic, complemented
6#17986255C211yesmaximal, characteristic, complemented
7#17986230C211yesmaximal, characteristic, complemented
8#17986221C211yesmaximal, characteristic, complemented
9#17986159C211yesmaximal, characteristic, complemented
10#17986154C211yesmaximal, characteristic, complemented
11#17986104C211yesmaximal, characteristic, complemented
12#17986071C211yesmaximal, characteristic, complemented
13#17985998C211yesmaximal, characteristic, complemented
14C2×#14556051C211yesmaximal, characteristic, complemented
15#554×C71C211yesmaximal, characteristic, complemented
16C2×#11765865C2211yescharacteristic
17C22×Dic142C2211yescharacteristic
18C2×#11765855C2211yescharacteristic
19C2×#11765855C2211yescharacteristic
20C2×#11765852C2211yescharacteristic
21C2×#11765849C2211yescharacteristic
22C22×C71⋊C8C2211yescharacteristic
23#14556116C2211yescharacteristic
24#14556114—21no—
25#14556107C2211yescharacteristic
26#14556106—21no—
27#14556102C2211yescharacteristic
28#14556100—21no—
29C4×Dic142C2211yescharacteristic
30(C22⋊D71)⋊C4—21no—
31#14556059C2211yescharacteristic
32#14556059C2211yescharacteristic
33#14556059C2211yescharacteristic
34C4⋊(C71⋊C8)C2211yescharacteristic
35#14556052C2211yescharacteristic
36#14556051C2211yescharacteristic
37#14556051C2211yescharacteristic
38#14556051C2211yescharacteristic
39#14556051C2211yescharacteristic
40#14556051C2211yescharacteristic
41#14556051C2211yescharacteristic
42#14556048C2211yescharacteristic
43#14556048C2211yescharacteristic
44#14556048C2211yescharacteristic
45C22⋊(C71⋊C8)C2211yescharacteristic
46C22⋊(C71⋊C8)C2211yescharacteristic
47#14556039C2211yescharacteristic
48C4⋊Q8×C71C2211yescharacteristic
49#125×C71—21no—
50C22⋊Q8×C71—21no—
51C22⋊Q8×C71—21no—
52C22⋊Q8×C71C2211yescharacteristic
53C22⋊Q8×C71C2211yescharacteristic
54C4×Q8×C71C2211yescharacteristic
55C4×D4×C71—21no—
56C4×D4×C71C2211yescharacteristic
57C2×C4⋊C4×C71C2211yescharacteristic
58C2×C4⋊C4×C71C2211yescharacteristic
59#11765865—22no—
60#11765865—41no—
61C2×Dic142—22no—
62C2×Dic142—21no—
63C2×Dic142—41no—
64C2×Dic142C2311yescharacteristic
65C2×Dic142C2311yescharacteristic
66C2×D284C2311yescharacteristic
67C2×C22⋊D71—21no—
68C2×C22⋊D71—21no—
69C22×Dic71—21no—
70C2×C4×D71—21no—
71#11765855—22no—
72#11765855—41no—
73#11765852—41no—
74#11765849—41no—
75C2×C71⋊C8—41no—
76C2×C71⋊C8C2311yescharacteristic
77C2×C71⋊C8C2311yescharacteristic
78C2×C71⋊C8C2311yescharacteristic
79C2×C71⋊C8C2311yescharacteristic
80C4⋊Dic71C2311yescharacteristic
81Dic71⋊C4—21no—
82Dic71⋊C4—21no—
83C22⋊Dic71—21no—
84(C2×D71)⋊C4—21no—
85(C2×D71)⋊C4—21no—
86(C2×D71)⋊C4—21no—
87C4×Dic71—21no—
88C2×Q8×C71—21no—
89C2×Q8×C71C2311yescharacteristic
90C2×D4×C71D411yescharacteristic
91C22×C4×C71—21no—
92C22×C4×C71—21no—
93C22×C4×C71C2311yescharacteristic
94C22×C4×C71C2311yescharacteristic
95C4⋊C4×C71—21no—
96C4⋊C4×C71—21no—
97C4⋊C4×C71—21no—
98C4⋊C4×C71—21no—
99C4⋊C4×C71—21no—
100C4⋊C4×C71—21no—
101C4⋊C4×C71D411yescharacteristic
102C4⋊C4×C71C2311yescharacteristic
103C4⋊C4×C71C2311yescharacteristic
104C4⋊C4×C71C2311yescharacteristic
105C22⋊C4×C71—21no—
106C22⋊C4×C71—21no—
107C22⋊C4×C71D411yescharacteristic
108C22⋊C4×C71D411yescharacteristic
109C42×C71—21no—
110C42×C71C2311yescharacteristic
111C23×C71C2×D411yescharacteristic
112C23×C71C2×D411yescharacteristic
113C22×D71—21no—
114Q8×C71—21no—
115Q8×C71—21no—
116Q8×C71—81no—
117D4×C71—22no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—81no—
127C2×C4×C71—81no—
128C2×C4×C71C2×D411yescharacteristic
129C2×C4×C71C2×D411yescharacteristic
130C2×C4×C71C2×D411yescharacteristic
131C2×C4×C71C2×D411yescharacteristic
132C2×C4×C71C2411yescharacteristic
133C22⋊D71—42no—
134C22⋊D71—42no—
135C2×Dic71—21no—
136C2×Dic71—21no—
137C2×Dic71—21no—
138C2×Dic71—42no—
139D284—21no—
140D284—21no—
141C4×D71—42no—
142Dic142—21no—
143Dic142—21no—
144Dic142—21no—
145Dic142—42no—
146Dic142#4112yes—
147C71⋊C8—21no—
148C71⋊C8—21no—
149C71⋊C8—21no—
150C71⋊C8—21no—
151C22×C71—21no—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71—21no—
155C22×C71—21no—
156C22×C71#13712yes—
157C22×C71C22×D411yescharacteristic
158C2×D71—41no—
159C2×D71—41no—
160C2×D71—41no—
161C4×C71—21no—
162C4×C71—21no—
163C4×C71—21no—
164C4×C71—21no—
165C4×C71—41no—
166C4×C71—41no—
167C4×C71—41no—
168C4×C71—41no—
169C4×C71C2×#4111yescharacteristic
170C4×C71H(5,𝔽2)11yescharacteristic
171Dic71—22no—
172Dic71—41no—
173Dic71—41no—
174C2×C71—22no—
175C2×C71—41no—
176C2×C71#54611yescharacteristic
177C2×C71C2×#13711yescharacteristic
178C2×C71#57711yescharacteristic
179D71—81no—
180#3316—711nomaximal, Sylow
181C71#331611yescharacteristic, complemented, Sylow
182#554D7111yescharacteristic, complemented
183#548—711no—
184#489—711no—
185#483—711no—
186#481—711no—
187#475—711no—
188#470—711no—
189#465—711no—
190#464—711no—
191#452—711no—
192#451—711no—
193#441—711no—
194#435—711no—
195#417—711no—
196C2×#103—711no—
197C2×#41—711no—
198C22×Q8—711no—
199C2×Q16—711no—
200C2×Q16—711no—
201C2×SD16—711no—
202C2×M16—711no—
203C22×C8—711no—
204C4⋊Q8C2×D7111yescharacteristic
205#125—21no—
206#124—711no—
207#124—1421no—
208C22⋊Q8—21no—
209C22⋊Q8—21no—
210C22⋊Q8—711no—
211C22⋊Q8—1421no—
212C22⋊Q8C2×D7111yescharacteristic
213C22⋊Q8C2×D7111yescharacteristic
214#121—711no—
215#121—1421no—
216C4×Q8—711no—
217C4×Q8C2×D7111yescharacteristic
218C4×D4—21no—
219C4×D4—1421no—
220C4×D4C2×D7111yescharacteristic
221C2×C4⋊C4C2×D7111yescharacteristic
222C2×C4⋊C4C2×D7111yescharacteristic
223#106—711no—
224#106—711no—
225#106—711no—
226C4⋊C8—711no—
227#103—711no—
228#103—711no—
229#103—711no—
230#103—711no—
231#103—711no—
232#103—711no—
233#103—711no—
234#102—711no—
235#102—711no—
236#102—711no—
237C22⋊C8—711no—
238C22⋊C8—711no—
239#97—711no—
240#41—2841no—
241#41—1422no—
242C2×Q8—21no—
243C2×Q8—2841no—
244C2×Q8—711no—
245C2×Q8—711no—
246C2×Q8—1422no—
247C2×Q8—1421no—
248C2×Q8C22×D7111yescharacteristic
249C2×D4—711no—
250C2×D4—1421no—
251C2×D4—1421no—
252C2×D4C22⋊D7111yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—1421no—
256C22×C4—1421no—
257C22×C4C22×D7111yescharacteristic
258C22×C4C22×D7111yescharacteristic
259Q16—2841no—
260Q16—1422no—
261SD16—2841no—
262M16—2841no—
263C2×C8—2841no—
264C2×C8—711no—
265C2×C8—711no—
266C2×C8—711no—
267C2×C8—711no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—711no—
275C4⋊C4—1421no—
276C4⋊C4—1421no—
277C4⋊C4C22⋊D7111yescharacteristic
278C4⋊C4C22×D7111yescharacteristic
279C4⋊C4C22×D7111yescharacteristic
280C4⋊C4C22×D7111yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—1421no—
284C22⋊C4—1421no—
285C22⋊C4—1421no—
286C22⋊C4—1421no—
287C22⋊C4C22⋊D7111yescharacteristic
288C22⋊C4C22⋊D7111yescharacteristic
289C42—21no—
290C42—1421no—
291C42C22×D7111yescharacteristic
292C23—1421no—
293C23C2×C22⋊D7111yescharacteristic
294C23C2×C22⋊D7111yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—81no—
298Q8—2842no—
299Q8—712no—
300Q8—1421no—
301Q8—1421no—
302Q8—1421no—
303D4—22no—
304D4—2842no—
305D4—2842no—
306D4—1421no—
307D4—1421no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—2842no—
319C2×C4—2842no—
320C2×C4—1421no—
321C2×C4—1421no—
322C2×C4—1421no—
323C2×C4C2×C22⋊D7111yescharacteristic
324C2×C4C2×C22⋊D7111yescharacteristic
325C2×C4C2×C22⋊D7111yescharacteristic
326C2×C4C2×C22⋊D7111yescharacteristic
327C2×C4C23×D7111yescharacteristic
328C8—1421no—
329C8—1421no—
330C8—1421no—
331C8—1421no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—2841no—
338C22—2841no—
339C22—2841no—
340C22#1455615912yes—
341C22C22×C22⋊D7111yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—2841no—
351C4—2841no—
352C4—1422no—
353C4#41×D7111yescharacteristic
354C4#1455617411yescharacteristic
355C2—22no—
356C2—41no—
357C2—5681no—
358C2#1798663711yescharacteristic
359C2C2×#1455615911yescharacteristic
360C2#1798681011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#554×C71C2
socleC22×C71C22×D4
hypercenter#554D71
hypocenterC71#3316
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #17986361C2semidirect
3. #17986329C2semidirect
4. #17986317C2semidirect
5. #17986281C2semidirect
6. #17986255C2semidirect
7. #17986230C2semidirect
8. #17986221C2semidirect
9. #17986159C2semidirect
10. #17986154C2semidirect
11. #17986104C2semidirect
12. #17986071C2semidirect
13. #17985998C2semidirect
14. C2×#14556051C2semidirect
15. #554×C71C2semidirect
181. C71#3316semidirect
182. #554D71semidirect