Finite Group Databasebeta

← #22600416

Subgroups — #22600416

Order 9088.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986707C211yesmaximal, characteristic, complemented
2#17986646C211yesmaximal, characteristic
3#17986357C211yesmaximal, characteristic, complemented
4#17986336C211yesmaximal, characteristic, complemented
5#17986310C211yesmaximal, characteristic, complemented
6#17986278C211yesmaximal, characteristic, complemented
7#17986239C211yesmaximal, characteristic, complemented
8#17986219C211yesmaximal, characteristic, complemented
9#17986159C211yesmaximal, characteristic, complemented
10#17986148C211yesmaximal, characteristic, complemented
11#17986102C211yesmaximal, characteristic, complemented
12#17986068C211yesmaximal, characteristic, complemented
13#17986001C211yesmaximal, characteristic, complemented
14#102×D71C211yesmaximal, characteristic, complemented
15#468×C71C211yesmaximal, characteristic, complemented
16C2×#11765865C2211yescharacteristic
17C2×D4×D71C2211yescharacteristic
18C2×#11765855C2211yescharacteristic, complemented
19C2×#11765852C2211yescharacteristic
20C2×C8×D71C2211yescharacteristic
21#14556127C2211yescharacteristic, complemented
22#14556118—21no—
23#14556115C2211yescharacteristic
24#14556110—21no—
25#14556108—21no—
26#14556108C2211yescharacteristic
27#14556105—21no—
28#14556104C2211yescharacteristic
29#14556099—21no—
30#14556096—21no—
31Q8×Dic71C2211yescharacteristic, complemented
32#14556090C2211yescharacteristic
33C4×C22⋊D71C2211yescharacteristic
34(C22⋊D71)⋊C4—21no—
35C2×C4⋊Dic71C2211yescharacteristic, complemented
36C4⋊C4×D71C2211yescharacteristic
37C22⋊C4×D71—21no—
38#14556061C2211yescharacteristic, complemented
39#14556061C2211yescharacteristic, complemented
40Dic71⋊C8C2211yescharacteristic, complemented
41#14556053C2211yescharacteristic, complemented
42#14556053C2211yescharacteristic, complemented
43#14556051C2211yescharacteristic, complemented
44#14556050C2211yescharacteristic
45#14556050C2211yescharacteristic
46#14556049C2211yescharacteristic
47#14556049C2211yescharacteristic
48#14556048C2211yescharacteristic
49C22⋊(C71⋊C8)C2211yescharacteristic, complemented
50(C2×D71)⋊C8C2211yescharacteristic
51#14556038C2211yescharacteristic, complemented
52C2×SD16×C71C2211yescharacteristic
53C2×M16×C71C2211yescharacteristic, complemented
54C22⋊Q8×C71C2211yescharacteristic, complemented
55#121×C71C2211yescharacteristic
56#107×C71C2211yescharacteristic, complemented
57#103×C71C2211yescharacteristic, complemented
58#102×C71C2211yescharacteristic
59C23×D71—21no—
60#11765865—22no—
61#11765865—41no—
62C2×Dic142—21no—
63C2×Dic142C2311yescharacteristic
64C2×D284C2311yescharacteristic
65C2×C22⋊D71—21no—
66C2×C22⋊D71—21no—
67C2×C22⋊D71D411yescharacteristic
68D4×D71—22no—
69D4×D71—41no—
70C22×Dic71—21no—
71C2×C4×D71—21no—
72C2×C4×D71—21no—
73C2×C4×D71—21no—
74C2×C4×D71C2311yescharacteristic
75#11765855—41no—
76#11765852—22no—
77C2×C71⋊C8C2311yescharacteristic
78C2×C71⋊C8C2311yescharacteristic
79C8×D71—41no—
80C4⋊Dic71—21no—
81C4⋊Dic71—21no—
82C4⋊Dic71—21no—
83C4⋊Dic71C2311yescharacteristic
84C4⋊Dic71C2311yescharacteristic
85Dic71⋊C4—21no—
86Dic71⋊C4—21no—
87Dic71⋊C4—21no—
88Dic71⋊C4D411yescharacteristic
89C22⋊Dic71—21no—
90C22⋊Dic71—21no—
91C22⋊Dic71D411yescharacteristic
92(C2×D71)⋊C4—21no—
93(C2×D71)⋊C4—21no—
94(C2×D71)⋊C4—21no—
95(C2×D71)⋊C4D411yescharacteristic
96C4×Dic71—21no—
97C4×Dic71C2311yescharacteristic
98C2×Q8×C71C2311yescharacteristic
99C2×D4×C71—21no—
100C2×D4×C71C2311yescharacteristic
101C22×C4×C71C2311yescharacteristic
102SD16×C71—41no—
103M16×C71—41no—
104C2×C8×C71C2311yescharacteristic
105C2×C8×C71C2311yescharacteristic
106C4⋊C4×C71—21no—
107C4⋊C4×C71C2311yescharacteristic
108C4⋊C4×C71C2311yescharacteristic
109C22⋊C4×C71—21no—
110C22⋊C4×C71—21no—
111C23×C71—21no—
112C23×C71C2×D411yescharacteristic
113C22×D71—21no—
114C22×D71—42no—
115C22×D71—42no—
116C22×D71—81no—
117C22×D71C2×D411yescharacteristic
118Q8×C71—21no—
119Q8×C71—21no—
120D4×C71—21no—
121D4×C71—21no—
122D4×C71—81no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71C2×D411yescharacteristic
129C2×C4×C71C2411yescharacteristic
130C22⋊D71—22no—
131C22⋊D71—42no—
132C22⋊D71—42no—
133C2×Dic71—21no—
134C2×Dic71—21no—
135C2×Dic71—21no—
136C2×Dic71—81no—
137C2×Dic71C2×D411yescharacteristic
138C2×Dic71C2×D411yescharacteristic
139C2×Dic71C2×D411yescharacteristic
140D284—21no—
141D284#4112yes—
142C4×D71—21no—
143C4×D71—21no—
144C4×D71—42no—
145C4×D71—81no—
146C4×D71—81no—
147Dic142—21no—
148Dic142—21no—
149Dic142—81no—
150C8×C71—21no—
151C8×C71—21no—
152C71⋊C8—21no—
153C71⋊C8—21no—
154C22×C71—21no—
155C22×C71—21no—
156C22×C71—21no—
157C22×C71—41no—
158C22×C71—41no—
159C22×C71—41no—
160C22×C71C22×D411yescharacteristic
161C2×D71—21no—
162C2×D71—21no—
163C2×D71—22no—
164C2×D71—41no—
165C2×D71—41no—
166C2×D71—84no—
167C2×D71#13712yes—
168C4×C71—21no—
169C4×C71—41no—
170C4×C71—41no—
171C4×C71—41no—
172C4×C71C2×#4111yescharacteristic
173C4×C71H(5,𝔽2)11yescharacteristic
174Dic71—21no—
175Dic71—21no—
176Dic71—21no—
177Dic71—41no—
178Dic71—41no—
179Dic71—41no—
180C2×C71—41no—
181C2×C71—81no—
182C2×C71#54611yescharacteristic
183C2×C71C2×#13711yescharacteristic
184C2×C71#57611yescharacteristic
185D71—22no—
186D71—42no—
187#3314—711nomaximal, Sylow
188C71#331411yescharacteristic, complemented, Sylow
189#554—711no—
190#546—711no—
191#489—711no—
192#484—711no—
193#480—711no—
194#474—711no—
195#468D7111yescharacteristic, complemented
196#467—711no—
197#464—711no—
198#452—711no—
199#450—711no—
200#440—711no—
201#435—711no—
202#417—711no—
203C2×#102—711no—
204C2×#41—711no—
205C22×D4—711no—
206C2×Q16—711no—
207C2×SD16—711no—
208C2×SD16C2×D7111yescharacteristic
209C2×M16C2×D7111yescharacteristic, complemented
210C22×C8—711no—
211C4⋊Q8—711no—
212#125—1421no—
213#124—711no—
214#123—1421no—
215C22⋊Q8—711no—
216C22⋊Q8—711no—
217C22⋊Q8—1421no—
218C22⋊Q8—1421no—
219C22⋊Q8C2×D7111yescharacteristic, complemented
220#121—1421no—
221#121C2×D7111yescharacteristic
222#120—1421no—
223C4×Q8—711no—
224C4×D4—711no—
225C4×D4—711no—
226C4×D4—1421no—
227C2×C4⋊C4—711no—
228C2×C4⋊C4—711no—
229C2×C22⋊C4—1421no—
230#107—711no—
231#107—711no—
232#107C2×D7111yescharacteristic, complemented
233C4⋊C8—711no—
234#103—711no—
235#103—711no—
236#103—711no—
237#103C2×D7111yescharacteristic, complemented
238#102—711no—
239#102—711no—
240#102—711no—
241#102—711no—
242#102—711no—
243#102C2×D7111yescharacteristic
244C22⋊C8—711no—
245C22⋊C8—711no—
246#97—711no—
247C24—1421no—
248#41—2841no—
249#41—1422no—
250C2×Q8—711no—
251C2×Q8—1421no—
252C2×Q8C22×D7111yescharacteristic
253C2×D4—21no—
254C2×D4—2841no—
255C2×D4—711no—
256C2×D4—711no—
257C2×D4—1421no—
258C2×D4—1422no—
259C2×D4—1421no—
260C2×D4C22×D7111yescharacteristic
261C22×C4—711no—
262C22×C4—1421no—
263C22×C4—1421no—
264C22×C4—1421no—
265C22×C4—1421no—
266C22×C4C22×D7111yescharacteristic
267Q16—2841no—
268SD16—41no—
269SD16—1422no—
270M16—41no—
271C2×C8—2841no—
272C2×C8—711no—
273C2×C8—711no—
274C2×C8C22×D7111yescharacteristic
275C2×C8C22×D7111yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—711no—
278C4⋊C4—711no—
279C4⋊C4—711no—
280C4⋊C4—1421no—
281C4⋊C4—1421no—
282C4⋊C4—1421no—
283C4⋊C4—1421no—
284C4⋊C4—1421no—
285C4⋊C4—1421no—
286C4⋊C4C22×D7111yescharacteristic
287C4⋊C4C22×D7111yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—711no—
291C22⋊C4—711no—
292C22⋊C4—1421no—
293C22⋊C4—1421no—
294C22⋊C4—1421no—
295C22⋊C4—1421no—
296C22⋊C4—1421no—
297C42—711no—
298C42—1421no—
299C23—21no—
300C23—2842no—
301C23—2842no—
302C23—5681no—
303C23—711no—
304C23—1421no—
305C23D4×D7111yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—5681no—
309Q8—1421no—
310Q8—1421no—
311D4—21no—
312D4—21no—
313D4—81no—
314D4—2842no—
315D4—2842no—
316D4—712no—
317D4—1422no—
318D4—1421no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—2842no—
325C2×C4—5681no—
326C2×C4—5681no—
327C2×C4—5681no—
328C2×C4—711no—
329C2×C4—711no—
330C2×C4—711no—
331C2×C4—1421no—
332C2×C4—1421no—
333C2×C4—1421no—
334C2×C4—1421no—
335C2×C4—1421no—
336C2×C4D4×D7111yescharacteristic
337C2×C4C23×D7111yescharacteristic
338C8—21no—
339C8—21no—
340C8—1421no—
341C8—1421no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—2841no—
349C22—2841no—
350C22—5684no—
351C22—712no—
352C22—1422no—
353C22—1421no—
354C22—1421no—
355C22C2×D4×D7111yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—2841no—
361C4—2841no—
362C4—2841no—
363C4—1421no—
364C4—1421no—
365C4—1421no—
366C4#41×D7111yescharacteristic
367C4#1455617311yescharacteristic
368C2—41no—
369C2—81no—
370C2—2842no—
371C2—1422no—
372C2#1798663811yescharacteristic
373C2#137×D7111yescharacteristic
374C2#1798680311yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#468×C71C2
socleC22×C71C22×D4
hypercenter#468D71
hypocenterC71#3314
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #17986707C2semidirect
3. #17986357C2semidirect
4. #17986336C2semidirect
5. #17986310C2semidirect
6. #17986278C2semidirect
7. #17986239C2semidirect
8. #17986219C2semidirect
9. #17986159C2semidirect
10. #17986148C2semidirect
11. #17986102C2semidirect
12. #17986068C2semidirect
13. #17986001C2semidirect
14. #102×D71C2semidirect
15. #468×C71C2semidirect
18. C2×#11765855C22semidirect
21. #14556127C22semidirect
31. Q8×Dic71C22semidirect
35. C2×C4⋊Dic71C22semidirect
38. #14556061C22semidirect
39. #14556061C22semidirect
40. Dic71⋊C8C22semidirect
41. #14556053C22semidirect
42. #14556053C22semidirect
43. #14556051C22semidirect
49. C22⋊(C71⋊C8)C22semidirect
51. #14556038C22semidirect
53. C2×M16×C71C22semidirect
54. C22⋊Q8×C71C22semidirect
56. #107×C71C22semidirect
57. #103×C71C22semidirect
188. C71#3314semidirect
195. #468D71semidirect
209. C2×M16C2×D71semidirect
219. C22⋊Q8C2×D71semidirect
232. #107C2×D71semidirect
237. #103C2×D71semidirect