Finite Group Databasebeta

← #22600422

Subgroups — #22600422

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#17986704C211yesmaximal, characteristic, complemented
2#17986637C211yesmaximal, characteristic
3#17986360C211yesmaximal, characteristic, complemented
4#17986333C211yesmaximal, characteristic, complemented
5#17986312C211yesmaximal, characteristic, complemented
6#17986277C211yesmaximal, characteristic, complemented
7#17986245C211yesmaximal, characteristic, complemented
8#17986239C211yesmaximal, characteristic, complemented
9#17986224C211yesmaximal, characteristic, complemented
10#17986161C211yesmaximal, characteristic, complemented
11#17986151C211yesmaximal, characteristic, complemented
12#17986096C211yesmaximal, characteristic, complemented
13#17986068C211yesmaximal, characteristic, complemented
14#102×D71C211yesmaximal, characteristic, complemented
15#417×C71C211yesmaximal, characteristic, complemented
16C2×D4×D71C2211yescharacteristic
17C2×#11765855C2211yescharacteristic, complemented
18C2×#11765854C2211yescharacteristic
19C2×#11765853C2211yescharacteristic
20C2×C8×D71C2211yescharacteristic
21#14556127C2211yescharacteristic, complemented
22#14556118—21no—
23#14556117C2211yescharacteristic
24#14556111—21no—
25#14556108C2211yescharacteristic
26#14556107—21no—
27#14556107C2211yescharacteristic, complemented
28#14556106—21no—
29#14556104C2211yescharacteristic
30#14556102C2211yescharacteristic
31#14556100—21no—
32#14556096—21no—
33#14556093C2211yescharacteristic, complemented
34D4×Dic71C2211yescharacteristic
35C4×C22⋊D71—21no—
36C4×C22⋊D71C2211yescharacteristic
37C4⋊C4×D71C2211yescharacteristic
38C2×(C2×D71)⋊C4—21no—
39#14556062C2211yescharacteristic, complemented
40#14556061C2211yescharacteristic, complemented
41#14556061C2211yescharacteristic, complemented
42Dic71⋊C8C2211yescharacteristic, complemented
43#14556053C2211yescharacteristic, complemented
44#14556052C2211yescharacteristic, complemented
45#14556050C2211yescharacteristic
46#14556049C2211yescharacteristic
47#14556048C2211yescharacteristic
48#14556048C2211yescharacteristic
49C22⋊(C71⋊C8)C2211yescharacteristic, complemented
50(C2×D71)⋊C8C2211yescharacteristic
51#14556038C2211yescharacteristic, complemented
52C2×#41×C71C2211yescharacteristic
53C2×M16×C71C2211yescharacteristic, complemented
54C2×C4⋊C4×C71C2211yescharacteristic, complemented
55#103×C71C2211yescharacteristic, complemented
56#103×C71C2211yescharacteristic, complemented
57#102×C71C2211yescharacteristic
58#102×C71C2211yescharacteristic
59C23×D71—21no—
60C2×Dic142—21no—
61C2×Dic142C2311yescharacteristic
62C2×D284C2311yescharacteristic
63C2×C22⋊D71—21no—
64C2×C22⋊D71—21no—
65C2×C22⋊D71D411yescharacteristic
66D4×D71—22no—
67D4×D71—41no—
68C22×Dic71—21no—
69C2×C4×D71—21no—
70C2×C4×D71C2311yescharacteristic
71#11765855—41no—
72#11765854—41no—
73#11765853—22no—
74C2×C71⋊C8C2311yescharacteristic
75C2×C71⋊C8C2311yescharacteristic
76C8×D71—41no—
77C4⋊Dic71—21no—
78C4⋊Dic71C2311yescharacteristic
79C4⋊Dic71C2311yescharacteristic
80Dic71⋊C4—21no—
81Dic71⋊C4—21no—
82Dic71⋊C4—21no—
83Dic71⋊C4—21no—
84Dic71⋊C4—21no—
85Dic71⋊C4D411yescharacteristic
86C22⋊Dic71—21no—
87C22⋊Dic71—21no—
88C22⋊Dic71D411yescharacteristic
89(C2×D71)⋊C4—21no—
90(C2×D71)⋊C4—21no—
91(C2×D71)⋊C4—21no—
92(C2×D71)⋊C4—21no—
93(C2×D71)⋊C4—21no—
94(C2×D71)⋊C4D411yescharacteristic
95C4×Dic71—21no—
96C4×Dic71C2311yescharacteristic
97#41×C71—22no—
98#41×C71—41no—
99C2×Q8×C71C2311yescharacteristic
100C2×D4×C71—21no—
101C2×D4×C71C2311yescharacteristic
102C22×C4×C71—21no—
103C22×C4×C71—21no—
104C22×C4×C71C2311yescharacteristic
105M16×C71—41no—
106C2×C8×C71C2311yescharacteristic
107C2×C8×C71C2311yescharacteristic
108C4⋊C4×C71—21no—
109C4⋊C4×C71C2311yescharacteristic
110C4⋊C4×C71C2311yescharacteristic
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—42no—
122D4×C71#4112yes—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—42no—
129C2×C4×C71—81no—
130C2×C4×C71C2×D411yescharacteristic
131C2×C4×C71C2411yescharacteristic
132C22⋊D71—22no—
133C22⋊D71—42no—
134C22⋊D71—81no—
135C2×Dic71—21no—
136C2×Dic71—21no—
137C2×Dic71—21no—
138C2×Dic71—81no—
139C2×Dic71C2×D411yescharacteristic
140C2×Dic71C2×D411yescharacteristic
141C2×Dic71C2×D411yescharacteristic
142D284—21no—
143D284—21no—
144C4×D71—21no—
145C4×D71—21no—
146C4×D71—81no—
147Dic142—21no—
148Dic142—21no—
149Dic142—81no—
150C8×C71—21no—
151C8×C71—21no—
152C71⋊C8—21no—
153C71⋊C8—21no—
154C22×C71—22no—
155C22×C71—21no—
156C22×C71—21no—
157C22×C71—21no—
158C22×C71—41no—
159C22×C71—41no—
160C22×C71C22×D411yescharacteristic
161C2×D71—21no—
162C2×D71—21no—
163C2×D71—41no—
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—42no—
181C2×C71—41no—
182C2×C71#54611yescharacteristic
183C2×C71C2×#13711yescharacteristic
184C2×C71#57611yescharacteristic
185D71—22no—
186D71—81no—
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#468—711no—
196#467—711no—
197#464—711no—
198#452—711no—
199#450—711no—
200#440—711no—
201#435—711no—
202#417D7111yescharacteristic, complemented
203C2×#102—711no—
204C2×#41C2×D7111yescharacteristic
205C22×D4—711no—
206C2×Q16—711no—
207C2×SD16—711no—
208C2×SD16—711no—
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—711no—
218C22⋊Q8—1421no—
219C22⋊Q8—1421no—
220#121—711no—
221#121—1421no—
222#120—1421no—
223C4×Q8—711no—
224C4×D4—711no—
225C4×D4—711no—
226C4×D4—1421no—
227C2×C4⋊C4—711no—
228C2×C4⋊C4C2×D7111yescharacteristic, complemented
229C2×C22⋊C4—1421no—
230#107—711no—
231#107—711no—
232#107—711no—
233C4⋊C8—711no—
234#103—711no—
235#103—711no—
236#103C2×D7111yescharacteristic, complemented
237#103C2×D7111yescharacteristic, complemented
238#102—711no—
239#102—711no—
240#102—711no—
241#102—711no—
242#102C2×D7111yescharacteristic
243#102C2×D7111yescharacteristic
244C22⋊C8—711no—
245C22⋊C8—711no—
246#97—711no—
247C24—1421no—
248#41—22no—
249#41—41no—
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—21no—
262C22×C4—21no—
263C22×C4—711no—
264C22×C4—1421no—
265C22×C4—1421no—
266C22×C4C22×D7111yescharacteristic
267Q16—2841no—
268SD16—2841no—
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—711no—
289C22⋊C4—711no—
290C22⋊C4—1421no—
291C22⋊C4—1421no—
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—42no—
313D4—2842no—
314D4—5681no—
315D4—1422no—
316D4—1421no—
317D4—1421no—
318D4#1176586512yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—81no—
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—22no—
344C22—21no—
345C22—21no—
346C22—41no—
347C22—41no—
348C22—2841no—
349C22—2841no—
350C22—2841no—
351C22—5684no—
352C22—712no—
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—
366C4C2×#1176586511yescharacteristic
367C4#1455617311yescharacteristic
368C2—41no—
369C2—42no—
370C2—5681no—
371C2—1422no—
372C2#1798663911yescharacteristic
373C2#137×D7111yescharacteristic
374C2#1798680311yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#417×C71C2
socleC22×C71C22×D4
hypercenter#417D71
hypocenterC71#3314
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #17986704C2semidirect
3. #17986360C2semidirect
4. #17986333C2semidirect
5. #17986312C2semidirect
6. #17986277C2semidirect
7. #17986245C2semidirect
8. #17986239C2semidirect
9. #17986224C2semidirect
10. #17986161C2semidirect
11. #17986151C2semidirect
12. #17986096C2semidirect
13. #17986068C2semidirect
14. #102×D71C2semidirect
15. #417×C71C2semidirect
17. C2×#11765855C22semidirect
21. #14556127C22semidirect
27. #14556107C22semidirect
33. #14556093C22semidirect
39. #14556062C22semidirect
40. #14556061C22semidirect
41. #14556061C22semidirect
42. Dic71⋊C8C22semidirect
43. #14556053C22semidirect
44. #14556052C22semidirect
49. C22⋊(C71⋊C8)C22semidirect
51. #14556038C22semidirect
53. C2×M16×C71C22semidirect
54. C2×C4⋊C4×C71C22semidirect
55. #103×C71C22semidirect
56. #103×C71C22semidirect
188. C71#3314semidirect
202. #417D71semidirect
209. C2×M16C2×D71semidirect
228. C2×C4⋊C4C2×D71semidirect
236. #103C2×D71semidirect
237. #103C2×D71semidirect