Finite Group Databasebeta

← #22599237

Subgroups — #22599237

Order 9088.

402 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986577C211yesmaximal, characteristic, complemented
2#17986334C211yesmaximal, characteristic, complemented
3#17986332C211yesmaximal, characteristic, complemented
4#17986312C211yesmaximal, characteristic, complemented
5#17986310C211yesmaximal, characteristic, complemented
6#17986247C211yesmaximal, characteristic, complemented
7#17986243C211yesmaximal, characteristic, complemented
8#17986238C211yesmaximal, characteristic, complemented
9#17986233C211yesmaximal, characteristic, complemented
10#17986161C211yesmaximal, characteristic, complemented
11#17986159C211yesmaximal, characteristic, complemented
12#17986144C211yesmaximal, characteristic, complemented
13#17986142C211yesmaximal, characteristic, complemented
14#17985966C211yesmaximal, characteristic, complemented
15#537×C71C211yesmaximal, characteristic, complemented
16C2×#11765865C2211yescharacteristic
17C2×#11765855C2211yescharacteristic
18C2×#11765853C2211yescharacteristic
19C2×#11765852C2211yescharacteristic
20C2×#11765850C2211yescharacteristic
21C2×#11765849C2211yescharacteristic
22C22×C71⋊C8C2211yescharacteristic
23#14556107C2211yescharacteristic
24#14556107C2211yescharacteristic
25#14556104—21no—
26#14556102C2211yescharacteristic
27#14556102C2211yescharacteristic
28#14556100—21no—
29#14556100—21no—
30#14556099—21no—
31C2×C4⋊Dic71C2211yescharacteristic
32#14556061C2211yescharacteristic
33#14556061C2211yescharacteristic
34#14556061C2211yescharacteristic
35#14556061C2211yescharacteristic
36#14556052C2211yescharacteristic
37#14556052C2211yescharacteristic
38#14556051C2211yescharacteristic
39#14556051C2211yescharacteristic
40#14556049C2211yescharacteristic
41#14556049C2211yescharacteristic
42#14556048C2211yescharacteristic
43#14556048C2211yescharacteristic
44C22⋊(C71⋊C8)C2211yescharacteristic
45C22⋊(C71⋊C8)C2211yescharacteristic
46C22⋊(C71⋊C8)C2211yescharacteristic
47C22⋊(C71⋊C8)C2211yescharacteristic
48C2×#41×C71C2211yescharacteristic
49C2×#41×C71C2211yescharacteristic
50C22⋊Q8×C71—21no—
51C22⋊Q8×C71C2211yescharacteristic
52C22⋊Q8×C71C2211yescharacteristic
53#121×C71—21no—
54#121×C71—21no—
55#121×C71—21no—
56#121×C71C2211yescharacteristic
57#121×C71C2211yescharacteristic
58C2×C4⋊C4×C71C2211yescharacteristic
59#11765865—41no—
60#11765865—41no—
61C2×Dic142C2311yescharacteristic
62C2×D284C2311yescharacteristic
63C2×C22⋊D71—21no—
64C2×C22⋊D71—21no—
65C2×C22⋊D71—21no—
66C22×Dic71—21no—
67C2×C4×D71—21no—
68#11765855—41no—
69#11765853—41no—
70#11765852—41no—
71#11765850—41no—
72#11765849—41no—
73C2×C71⋊C8—41no—
74C2×C71⋊C8C2311yescharacteristic
75C2×C71⋊C8C2311yescharacteristic
76C2×C71⋊C8C2311yescharacteristic
77C2×C71⋊C8C2311yescharacteristic
78C4⋊Dic71—21no—
79C4⋊Dic71C2311yescharacteristic
80C4⋊Dic71C2311yescharacteristic
81Dic71⋊C4—21no—
82Dic71⋊C4—21no—
83C22⋊Dic71—21no—
84C22⋊Dic71—21no—
85(C2×D71)⋊C4—21no—
86(C2×D71)⋊C4—21no—
87#41×C71—41no—
88#41×C71—41no—
89#41×C71—41no—
90#41×C71—41no—
91C2×Q8×C71D411yescharacteristic
92C2×Q8×C71C2311yescharacteristic
93C2×D4×C71—21no—
94C2×D4×C71—21no—
95C2×D4×C71—21no—
96C2×D4×C71D411yescharacteristic
97C2×D4×C71D411yescharacteristic
98C2×D4×C71D411yescharacteristic
99C2×D4×C71C2311yescharacteristic
100C22×C4×C71—21no—
101C22×C4×C71—21no—
102C22×C4×C71C2311yescharacteristic
103C22×C4×C71C2311yescharacteristic
104C22×C4×C71C2311yescharacteristic
105C4⋊C4×C71—21no—
106C4⋊C4×C71—21no—
107C4⋊C4×C71—21no—
108C4⋊C4×C71C2311yescharacteristic
109C4⋊C4×C71C2311yescharacteristic
110C22⋊C4×C71—21no—
111C22⋊C4×C71—21no—
112C22⋊C4×C71—21no—
113C22⋊C4×C71—21no—
114C23×C71—21no—
115C23×C71C2×D411yescharacteristic
116C23×C71C2×D411yescharacteristic
117C23×C71C2×D411yescharacteristic
118C22×D71—21no—
119Q8×C71—21no—
120Q8×C71—21no—
121Q8×C71—41no—
122D4×C71—21no—
123D4×C71—21no—
124D4×C71—41no—
125D4×C71—41no—
126D4×C71—41no—
127D4×C71—42no—
128D4×C71—81no—
129D4×C71—81no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71—21no—
134C2×C4×C71—21no—
135C2×C4×C71—21no—
136C2×C4×C71—21no—
137C2×C4×C71—21no—
138C2×C4×C71—21no—
139C2×C4×C71—42no—
140C2×C4×C71—81no—
141C2×C4×C71C2×D411yescharacteristic
142C2×C4×C71C2×D411yescharacteristic
143C2×C4×C71C2×D411yescharacteristic
144C2×C4×C71C2411yescharacteristic
145C22⋊D71—42no—
146C22⋊D71—81no—
147C22⋊D71—81no—
148C2×Dic71—21no—
149C2×Dic71—21no—
150C2×Dic71—21no—
151C2×Dic71—81no—
152D284—21no—
153D284—21no—
154C4×D71—42no—
155Dic142—21no—
156Dic142—21no—
157C71⋊C8—21no—
158C71⋊C8—21no—
159C71⋊C8—21no—
160C71⋊C8—21no—
161C22×C71—21no—
162C22×C71—21no—
163C22×C71—21no—
164C22×C71—21no—
165C22×C71—21no—
166C22×C71—21no—
167C22×C71—21no—
168C22×C71—21no—
169C22×C71—21no—
170C22×C71—41no—
171C22×C71—41no—
172C22×C71—41no—
173C22×C71C22×D411yescharacteristic
174C2×D71—41no—
175C2×D71—41no—
176C2×D71—41no—
177C4×C71—21no—
178C4×C71—21no—
179C4×C71—21no—
180C4×C71—41no—
181C4×C71—41no—
182C4×C71—41no—
183C4×C71H(5,𝔽2)11yescharacteristic
184C4×C71H(5,𝔽2)11yescharacteristic
185Dic71—41no—
186Dic71—41no—
187Dic71—41no—
188C2×C71—41no—
189C2×C71—41no—
190C2×C71—41no—
191C2×C71—81no—
192C2×C71#53411yescharacteristic
193C2×C71#57611yescharacteristic
194C2×C71#57811yescharacteristic
195D71—81no—
196#3203—711nomaximal, Sylow
197C71#320311yescharacteristic, complemented, Sylow
198#537—711no—
199#537D7111yescharacteristic, complemented
200#484—711no—
201#484—711no—
202#480—711no—
203#480—711no—
204#468—711no—
205#468—711no—
206#467—711no—
207#466—711no—
208#452—711no—
209#452—711no—
210#449—711no—
211#449—711no—
212#408—711no—
213C2×#41—711no—
214C2×#41C2×D7111yescharacteristic
215C2×#41C2×D7111yescharacteristic
216C2×Q16—711no—
217C2×SD16—711no—
218C2×SD16—711no—
219C2×D8—711no—
220C2×M16—711no—
221C22×C8—711no—
222C22⋊Q8—21no—
223C22⋊Q8—711no—
224C22⋊Q8—711no—
225C22⋊Q8—1421no—
226C22⋊Q8C2×D7111yescharacteristic
227C22⋊Q8C2×D7111yescharacteristic
228#121—21no—
229#121—21no—
230#121—21no—
231#121—711no—
232#121—711no—
233#121—1421no—
234#121—1421no—
235#121—1421no—
236#121C2×D7111yescharacteristic
237#121C2×D7111yescharacteristic
238C2×C4⋊C4—711no—
239C2×C4⋊C4C2×D7111yescharacteristic
240#107—711no—
241#107—711no—
242#107—711no—
243#107—711no—
244#103—711no—
245#103—711no—
246#103—711no—
247#103—711no—
248#102—711no—
249#102—711no—
250#102—711no—
251#102—711no—
252C22⋊C8—711no—
253C22⋊C8—711no—
254C22⋊C8—711no—
255C22⋊C8—711no—
256#41—41no—
257#41—41no—
258#41—41no—
259#41—41no—
260#41—2841no—
261#41—2841no—
262C2×Q8—711no—
263C2×Q8C22⋊D7111yescharacteristic
264C2×Q8C22×D7111yescharacteristic
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—711no—
269C2×D4—1421no—
270C2×D4—1421no—
271C2×D4—1421no—
272C2×D4C22⋊D7111yescharacteristic
273C2×D4C22⋊D7111yescharacteristic
274C2×D4C22⋊D7111yescharacteristic
275C2×D4C22×D7111yescharacteristic
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—1421no—
279C22×C4—1421no—
280C22×C4C22×D7111yescharacteristic
281C22×C4C22×D7111yescharacteristic
282C22×C4C22×D7111yescharacteristic
283Q16—2841no—
284SD16—2841no—
285SD16—2841no—
286D8—2841no—
287M16—2841no—
288C2×C8—2841no—
289C2×C8—711no—
290C2×C8—711no—
291C2×C8—711no—
292C2×C8—711no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—711no—
297C4⋊C4—711no—
298C4⋊C4—1421no—
299C4⋊C4—1421no—
300C4⋊C4—1421no—
301C4⋊C4C22×D7111yescharacteristic
302C4⋊C4C22×D7111yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—1421no—
308C22⋊C4—1421no—
309C22⋊C4—1421no—
310C22⋊C4—1421no—
311C23—21no—
312C23—1421no—
313C23C2×C22⋊D7111yescharacteristic
314C23C2×C22⋊D7111yescharacteristic
315C23C2×C22⋊D7111yescharacteristic
316Q8—21no—
317Q8—21no—
318Q8—41no—
319Q8—1421no—
320Q8—1421no—
321D4—21no—
322D4—21no—
323D4—41no—
324D4—41no—
325D4—41no—
326D4—42no—
327D4—81no—
328D4—81no—
329D4—2842no—
330D4—5681no—
331D4—5681no—
332D4—1421no—
333D4—1421no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—81no—
345C2×C4—2842no—
346C2×C4—5681no—
347C2×C4—1421no—
348C2×C4—1421no—
349C2×C4—1421no—
350C2×C4C2×C22⋊D7111yescharacteristic
351C2×C4C2×C22⋊D7111yescharacteristic
352C2×C4C2×C22⋊D7111yescharacteristic
353C2×C4C23×D7111yescharacteristic
354C8—1421no—
355C8—1421no—
356C8—1421no—
357C8—1421no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—2841no—
371C22—2841no—
372C22—2841no—
373C22C22×C22⋊D7111yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—2841no—
381C4—2841no—
382C4—2841no—
383C4#1455617311yescharacteristic
384C4#1455617311yescharacteristic
385C2—41no—
386C2—41no—
387C2—41no—
388C2—81no—
389C2—5681no—
390C2#1798655811yescharacteristic
391C2#1798680211yescharacteristic
392C2#1798681411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#537×C71C2
socleC22×C71C22×D4
hypercenter#537D71
hypocenterC71#3203
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×#41 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #17986577C2semidirect
2. #17986334C2semidirect
3. #17986332C2semidirect
4. #17986312C2semidirect
5. #17986310C2semidirect
6. #17986247C2semidirect
7. #17986243C2semidirect
8. #17986238C2semidirect
9. #17986233C2semidirect
10. #17986161C2semidirect
11. #17986159C2semidirect
12. #17986144C2semidirect
13. #17986142C2semidirect
14. #17985966C2semidirect
15. #537×C71C2semidirect
197. C71#3203semidirect
199. #537D71semidirect