Finite Group Databasebeta

← #22600593

Subgroups — #22600593

Order 9088.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986695C211yesmaximal, characteristic, complemented
2#17986353C211yesmaximal, characteristic, complemented
3#17986327C211yesmaximal, characteristic, complemented
4#17986320C211yesmaximal, characteristic, complemented
5#17986306C211yesmaximal, characteristic, complemented
6#17986250C211yesmaximal, characteristic, complemented
7#17986247C211yesmaximal, characteristic, complemented
8#17986211C211yesmaximal, characteristic, complemented
9#17986144C211yesmaximal, characteristic, complemented
10#17986138C211yesmaximal, characteristic, complemented
11C4×#11765852C211yesmaximal, characteristic, complemented
12#17986077C211yesmaximal, characteristic, complemented
13#17986008C211yesmaximal, characteristic, complemented
14#17986000C211yesmaximal, characteristic, complemented
15#548×C71C211yesmaximal, characteristic, complemented
16C2×#11765852C2211yescharacteristic
17C2×#11765852C2211yescharacteristic
18C2×#11765850C2211yescharacteristic
19C2×#11765849C2211yescharacteristic
20C2×#11765849C2211yescharacteristic
21#14556123—21no—
22#14556120C2211yescharacteristic
23#14556110—21no—
24#14556102C2211yescharacteristic
25#14556102C2211yescharacteristic
26#14556100—21no—
27C4×D284C2211yescharacteristic
28(C22⋊D71)⋊C4—21no—
29#14556087C2211yescharacteristic
30C2×C4⋊Dic71C2211yescharacteristic
31#14556061C2211yescharacteristic
32#14556059C2211yescharacteristic
33#14556059C2211yescharacteristic
34C4⋊(C71⋊C8)C2211yescharacteristic
35#14556052C2211yescharacteristic
36#14556052C2211yescharacteristic
37#14556052C2211yescharacteristic
38#14556052C2211yescharacteristic
39#14556052C2211yescharacteristic
40#14556052C2211yescharacteristic
41#14556049C2211yescharacteristic
42#14556049C2211yescharacteristic
43#14556049C2211yescharacteristic
44#14556048C2211yescharacteristic
45C22⋊(C71⋊C8)C2211yescharacteristic
46C22⋊(C71⋊C8)C2211yescharacteristic
47C4×C71⋊C8C2211yescharacteristic
48C2×#41×C71C2211yescharacteristic
49C22×Q8×C71C2211yescharacteristic
50#124×C71—21no—
51#124×C71C2211yescharacteristic
52C22⋊Q8×C71—21no—
53C22⋊Q8×C71C2211yescharacteristic
54#121×C71—21no—
55#121×C71C2211yescharacteristic
56C4×Q8×C71C2211yescharacteristic
57C4×D4×C71—21no—
58C4×D4×C71C2211yescharacteristic
59C2×D284C2311yescharacteristic
60C2×C22⋊D71—21no—
61C2×C22⋊D71—21no—
62C22×Dic71—21no—
63C2×C4×D71—21no—
64#11765852—22no—
65#11765852—41no—
66#11765850—41no—
67#11765849—41no—
68#11765849—41no—
69C2×C71⋊C8C2311yescharacteristic
70C2×C71⋊C8C2311yescharacteristic
71C2×C71⋊C8C2311yescharacteristic
72C2×C71⋊C8C2311yescharacteristic
73C4⋊Dic71—21no—
74C4⋊Dic71C2311yescharacteristic
75C4⋊Dic71C2311yescharacteristic
76C4⋊Dic71C2311yescharacteristic
77Dic71⋊C4—21no—
78Dic71⋊C4—21no—
79C22⋊Dic71—21no—
80(C2×D71)⋊C4—21no—
81(C2×D71)⋊C4—21no—
82(C2×D71)⋊C4—21no—
83C4×Dic71—21no—
84#41×C71—22no—
85#41×C71—41no—
86C2×Q8×C71—21no—
87C2×Q8×C71—22no—
88C2×Q8×C71—41no—
89C2×Q8×C71C2311yescharacteristic
90C2×Q8×C71C2311yescharacteristic
91C2×D4×C71—21no—
92C2×D4×C71—21no—
93C2×D4×C71D411yescharacteristic
94C2×D4×C71C2311yescharacteristic
95C22×C4×C71—21no—
96C22×C4×C71—21no—
97C22×C4×C71C2311yescharacteristic
98C22×C4×C71C2311yescharacteristic
99C4⋊C4×C71—21no—
100C4⋊C4×C71—21no—
101C4⋊C4×C71D411yescharacteristic
102C4⋊C4×C71C2311yescharacteristic
103C22⋊C4×C71—21no—
104C22⋊C4×C71—21no—
105C22⋊C4×C71—21no—
106C22⋊C4×C71—21no—
107C22⋊C4×C71D411yescharacteristic
108C22⋊C4×C71D411yescharacteristic
109C42×C71—21no—
110C42×C71C2311yescharacteristic
111C23×C71—21no—
112C23×C71C2×D411yescharacteristic
113C23×C71C2×D411yescharacteristic
114C22×D71—21no—
115Q8×C71—21no—
116Q8×C71—21no—
117Q8×C71—21no—
118Q8×C71—42no—
119Q8×C71#4112yes—
120D4×C71—21no—
121D4×C71—21no—
122D4×C71—41no—
123D4×C71—42no—
124D4×C71—42no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—42no—
133C2×C4×C71—42no—
134C2×C4×C71C2×D411yescharacteristic
135C2×C4×C71C2×D411yescharacteristic
136C2×C4×C71C2×D411yescharacteristic
137C2×C4×C71C2×D411yescharacteristic
138C2×C4×C71C2411yescharacteristic
139C22⋊D71—42no—
140C22⋊D71—81no—
141C2×Dic71—21no—
142C2×Dic71—21no—
143C2×Dic71—21no—
144C2×Dic71—81no—
145D284—21no—
146D284—21no—
147C4×D71—81no—
148C71⋊C8—21no—
149C71⋊C8—21no—
150C71⋊C8—21no—
151C71⋊C8—21no—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71—21no—
155C22×C71—21no—
156C22×C71—21no—
157C22×C71—21no—
158C22×C71—41no—
159C22×C71—41no—
160C22×C71—41no—
161C22×C71C22×D411yescharacteristic
162C2×D71—41no—
163C2×D71—41no—
164C2×D71—41no—
165C4×C71—21no—
166C4×C71—21no—
167C4×C71—22no—
168C4×C71—21no—
169C4×C71—21no—
170C4×C71—41no—
171C4×C71—41no—
172C4×C71C2×#4111yescharacteristic
173C4×C71H(5,𝔽2)11yescharacteristic
174Dic71—41no—
175Dic71—41no—
176Dic71—41no—
177C2×C71—41no—
178C2×C71—41no—
179C2×C71—81no—
180C2×C71#54611yescharacteristic
181C2×C71#57511yescharacteristic
182C2×C71#57711yescharacteristic
183D71—81no—
184#3326—711nomaximal, Sylow
185C71#332611yescharacteristic, complemented, Sylow
186#553—711no—
187#548D7111yescharacteristic, complemented
188#488—711no—
189#483—711no—
190#482—711no—
191#479—711no—
192#469—711no—
193#468—711no—
194#463—711no—
195#449—711no—
196#448—711no—
197C4×SD16—711no—
198#436—711no—
199#419—711no—
200#417—711no—
201C2×#41C2×D7111yescharacteristic
202C22×Q8C2×D7111yescharacteristic
203C2×SD16—711no—
204C2×SD16—711no—
205C2×D8—711no—
206C2×M16—711no—
207C2×M16—711no—
208#126—1421no—
209#125—711no—
210#124—21no—
211#124C2×D7111yescharacteristic
212#123—1421no—
213C22⋊Q8—21no—
214C22⋊Q8C2×D7111yescharacteristic
215#121—21no—
216#121—711no—
217#121—711no—
218#121—1421no—
219#121C2×D7111yescharacteristic
220C4×Q8C2×D7111yescharacteristic
221C4×D4—21no—
222C4×D4—711no—
223C4×D4—1421no—
224C4×D4C2×D7111yescharacteristic
225#117—711no—
226C2×C4⋊C4—711no—
227#107—711no—
228#106—711no—
229#106—711no—
230C4⋊C8—711no—
231#103—711no—
232#103—711no—
233#103—711no—
234#103—711no—
235#103—711no—
236#103—711no—
237#102—711no—
238#102—711no—
239#102—711no—
240#102—711no—
241C22⋊C8—711no—
242C22⋊C8—711no—
243C4×C8—711no—
244#41—22no—
245#41—41no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8C22×D7111yescharacteristic
250C2×Q8C22×D7111yescharacteristic
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—711no—
254C2×D4—1421no—
255C2×D4—1421no—
256C2×D4C22⋊D7111yescharacteristic
257C2×D4C22×D7111yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—1421no—
261C22×C4—1421no—
262C22×C4C22×D7111yescharacteristic
263C22×C4C22×D7111yescharacteristic
264SD16—2841no—
265SD16—1422no—
266D8—2841no—
267M16—2841no—
268M16—2841no—
269C2×C8—711no—
270C2×C8—711no—
271C2×C8—711no—
272C2×C8—711no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—711no—
276C4⋊C4—711no—
277C4⋊C4—711no—
278C4⋊C4—1421no—
279C4⋊C4—1421no—
280C4⋊C4—1421no—
281C4⋊C4C22⋊D7111yescharacteristic
282C4⋊C4C22×D7111yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—1421no—
288C22⋊C4—1421no—
289C22⋊C4—1421no—
290C22⋊C4—1421no—
291C22⋊C4C22⋊D7111yescharacteristic
292C22⋊C4C22⋊D7111yescharacteristic
293C42—21no—
294C42—1421no—
295C42C22×D7111yescharacteristic
296C23—21no—
297C23—1421no—
298C23C2×C22⋊D7111yescharacteristic
299C23C2×C22⋊D7111yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8#1176586412yes—
305D4—21no—
306D4—21no—
307D4—41no—
308D4—42no—
309D4—42no—
310D4—2842no—
311D4—5681no—
312D4—1421no—
313D4—1421no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—5681no—
324C2×C4—5681no—
325C2×C4—1421no—
326C2×C4—1421no—
327C2×C4—1421no—
328C2×C4C2×C22⋊D7111yescharacteristic
329C2×C4C2×C22⋊D7111yescharacteristic
330C2×C4C2×C22⋊D7111yescharacteristic
331C2×C4C2×C22⋊D7111yescharacteristic
332C2×C4C23×D7111yescharacteristic
333C8—1421no—
334C8—1421no—
335C8—1421no—
336C8—1421no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—2841no—
347C22—2841no—
348C22—2841no—
349C22C22×C22⋊D7111yescharacteristic
350C4—21no—
351C4—21no—
352C4—22no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—2841no—
358C4—2841no—
359C4—2841no—
360C4C2×#1176586411yescharacteristic
361C4#1455617311yescharacteristic
362C2—41no—
363C2—41no—
364C2—81no—
365C2—5681no—
366C2#1798664311yescharacteristic
367C2#1798679711yescharacteristic
368C2#1798680711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#548×C71C2
socleC22×C71C22×D4
hypercenter#548D71
hypocenterC71#3326
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #17986695C2semidirect
2. #17986353C2semidirect
3. #17986327C2semidirect
4. #17986320C2semidirect
5. #17986306C2semidirect
6. #17986250C2semidirect
7. #17986247C2semidirect
8. #17986211C2semidirect
9. #17986144C2semidirect
10. #17986138C2semidirect
11. C4×#11765852C2semidirect
12. #17986077C2semidirect
13. #17986008C2semidirect
14. #17986000C2semidirect
15. #548×C71C2semidirect
185. C71#3326semidirect
187. #548D71semidirect