Finite Group Databasebeta

← #22600221

Subgroups — #22600221

Order 9088.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986658C211yesmaximal, characteristic
2#17986356C211yesmaximal, characteristic, complemented
3#17986340C211yesmaximal, characteristic, complemented
4#17986312C211yesmaximal, characteristic, complemented
5#17986303C211yesmaximal, characteristic, complemented
6#17986238C211yesmaximal, characteristic, complemented
7#17986230C211yesmaximal, characteristic, complemented
8#17986221C211yesmaximal, characteristic, complemented
9#17986159C211yesmaximal, characteristic, complemented
10#17986154C211yesmaximal, characteristic, complemented
11C4×#11765855C211yesmaximal, characteristic, complemented
12D4×C71⋊C8C211yesmaximal, characteristic, complemented
13#17985993C211yesmaximal, characteristic, complemented
14C2×#14556051C211yesmaximal, characteristic, complemented
15#552×C71C211yesmaximal, characteristic, complemented
16C2×#11765865C2211yescharacteristic
17C22×Dic142C2211yescharacteristic
18C2×#11765855C2211yescharacteristic
19C2×#11765855C2211yescharacteristic
20C2×#11765852C2211yescharacteristic
21C22×C71⋊C8C2211yescharacteristic
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#14556061C2211yescharacteristic
32#14556061C2211yescharacteristic
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
47C4×C71⋊C8C2211yescharacteristic
48#126×C71—21no—
49#125×C71—21no—
50#125×C71C2211yescharacteristic
51#123×C71—21no—
52C22⋊Q8×C71C2211yescharacteristic
53C22⋊Q8×C71C2211yescharacteristic
54C4×Q8×C71C2211yescharacteristic
55C4×D4×C71C2211yescharacteristic
56#117×C71—21no—
57#117×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—
74C2×C71⋊C8—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×C71C2311yescharacteristic
89C2×D4×C71D411yescharacteristic
90C22×C4×C71—21no—
91C22×C4×C71C2311yescharacteristic
92C22×C4×C71C2311yescharacteristic
93C4⋊C4×C71—21no—
94C4⋊C4×C71—21no—
95C4⋊C4×C71—21no—
96C4⋊C4×C71—21no—
97C4⋊C4×C71—21no—
98C4⋊C4×C71—21no—
99C4⋊C4×C71D411yescharacteristic
100C4⋊C4×C71C2311yescharacteristic
101C4⋊C4×C71C2311yescharacteristic
102C4⋊C4×C71C2311yescharacteristic
103C22⋊C4×C71—21no—
104C22⋊C4×C71—21no—
105C22⋊C4×C71—21no—
106C22⋊C4×C71D411yescharacteristic
107C22⋊C4×C71D411yescharacteristic
108C42×C71—21no—
109C42×C71—21no—
110C42×C71C2311yescharacteristic
111C23×C71C2×D411yescharacteristic
112C23×C71C2×D411yescharacteristic
113C22×D71—21no—
114Q8×C71—21no—
115Q8×C71—21no—
116D4×C71—22no—
117C2×C4×C71—21no—
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—42no—
126C2×C4×C71C2×D411yescharacteristic
127C2×C4×C71C2×D411yescharacteristic
128C2×C4×C71C2×D411yescharacteristic
129C2×C4×C71C2×D411yescharacteristic
130C2×C4×C71C2411yescharacteristic
131C22⋊D71—42no—
132C22⋊D71—42no—
133C2×Dic71—21no—
134C2×Dic71—21no—
135C2×Dic71—21no—
136C2×Dic71—42no—
137D284—21no—
138D284—21no—
139C4×D71—42no—
140Dic142—21no—
141Dic142—21no—
142Dic142—21no—
143Dic142—42no—
144Dic142#4112yes—
145C71⋊C8—21no—
146C71⋊C8—21no—
147C71⋊C8—21no—
148C71⋊C8—21no—
149C22×C71—21no—
150C22×C71—21no—
151C22×C71—21no—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71#13512yes—
155C22×C71C22×D411yescharacteristic
156C2×D71—41no—
157C2×D71—41no—
158C2×D71—41no—
159C4×C71—21no—
160C4×C71—21no—
161C4×C71—21no—
162C4×C71—21no—
163C4×C71—41no—
164C4×C71—41no—
165C4×C71—41no—
166C4×C71—41no—
167C4×C71C2×#4111yescharacteristic
168C4×C71H(5,𝔽2)11yescharacteristic
169Dic71—22no—
170Dic71—41no—
171Dic71—41no—
172C2×C71—22no—
173C2×C71—41no—
174C2×C71#54611yescharacteristic
175C2×C71C2×#13511yescharacteristic
176C2×C71#57811yescharacteristic
177D71—81no—
178#3300—711nomaximal, Sylow
179C71#330011yescharacteristic, complemented, Sylow
180#552D7111yescharacteristic, complemented
181#548—711no—
182#488—711no—
183#485—711no—
184#480—711no—
185#479—711no—
186#467—711no—
187#465—711no—
188#464—711no—
189#452—711no—
190#451—711no—
191C4×Q16—711no—
192C8×D4—711no—
193#416—711no—
194C2×#103—711no—
195C2×#41—711no—
196C22×Q8—711no—
197C2×Q16—711no—
198C2×Q16—711no—
199C2×SD16—711no—
200C22×C8—711no—
201C22×C8—711no—
202#126—21no—
203#125—21no—
204#125C2×D7111yescharacteristic
205#124—711no—
206#124—1421no—
207#123—21no—
208C22⋊Q8—711no—
209C22⋊Q8—1421no—
210C22⋊Q8C2×D7111yescharacteristic
211C22⋊Q8C2×D7111yescharacteristic
212#121—711no—
213#121—1421no—
214C4×Q8—711no—
215C4×Q8C2×D7111yescharacteristic
216C4×D4—1421no—
217C4×D4C2×D7111yescharacteristic
218#117—21no—
219#117C2×D7111yescharacteristic
220C2×C4⋊C4C2×D7111yescharacteristic
221#107—711no—
222#107—711no—
223#106—711no—
224C4⋊C8—711no—
225#103—711no—
226#103—711no—
227#103—711no—
228#103—711no—
229#103—711no—
230#103—711no—
231#103—711no—
232#102—711no—
233#102—711no—
234#102—711no—
235C22⋊C8—711no—
236C22⋊C8—711no—
237C4×C8—711no—
238#41—2841no—
239#41—1422no—
240C2×Q8—2841no—
241C2×Q8—711no—
242C2×Q8—711no—
243C2×Q8—1422no—
244C2×Q8—1421no—
245C2×Q8C22×D7111yescharacteristic
246C2×D4—711no—
247C2×D4—1421no—
248C2×D4—1421no—
249C2×D4C22⋊D7111yescharacteristic
250C22×C4—21no—
251C22×C4—1421no—
252C22×C4—1421no—
253C22×C4C22×D7111yescharacteristic
254C22×C4C22×D7111yescharacteristic
255Q16—2841no—
256Q16—1422no—
257SD16—2841no—
258C2×C8—2841no—
259C2×C8—2841no—
260C2×C8—711no—
261C2×C8—711no—
262C2×C8—711no—
263C2×C8—711no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—711no—
271C4⋊C4—1421no—
272C4⋊C4—1421no—
273C4⋊C4C22⋊D7111yescharacteristic
274C4⋊C4C22×D7111yescharacteristic
275C4⋊C4C22×D7111yescharacteristic
276C4⋊C4C22×D7111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—1421no—
281C22⋊C4—1421no—
282C22⋊C4—1421no—
283C22⋊C4—1421no—
284C22⋊C4C22⋊D7111yescharacteristic
285C22⋊C4C22⋊D7111yescharacteristic
286C42—21no—
287C42—21no—
288C42—1421no—
289C42C22×D7111yescharacteristic
290C23—1421no—
291C23C2×C22⋊D7111yescharacteristic
292C23C2×C22⋊D7111yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—2842no—
296Q8—712no—
297Q8—1421no—
298Q8—1421no—
299Q8—1421no—
300D4—22no—
301D4—2842no—
302D4—2842no—
303D4—1421no—
304D4—1421no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—2842no—
315C2×C4—2842no—
316C2×C4—1421no—
317C2×C4—1421no—
318C2×C4—1421no—
319C2×C4C2×C22⋊D7111yescharacteristic
320C2×C4C2×C22⋊D7111yescharacteristic
321C2×C4C2×C22⋊D7111yescharacteristic
322C2×C4C2×C22⋊D7111yescharacteristic
323C2×C4C23×D7111yescharacteristic
324C8—1421no—
325C8—1421no—
326C8—1421no—
327C8—1421no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—2841no—
334C22—2841no—
335C22—2841no—
336C22#1455614812yes—
337C22C22×C22⋊D7111yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—2841no—
347C4—2841no—
348C4—1422no—
349C4#41×D7111yescharacteristic
350C4#1455617411yescharacteristic
351C2—22no—
352C2—41no—
353C2—5681no—
354C2#1798663711yescharacteristic
355C2C2×#1455614811yescharacteristic
356C2#1798681411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#552×C71C2
socleC22×C71C22×D4
hypercenter#552D71
hypocenterC71#3300
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #17986356C2semidirect
3. #17986340C2semidirect
4. #17986312C2semidirect
5. #17986303C2semidirect
6. #17986238C2semidirect
7. #17986230C2semidirect
8. #17986221C2semidirect
9. #17986159C2semidirect
10. #17986154C2semidirect
11. C4×#11765855C2semidirect
12. D4×C71⋊C8C2semidirect
13. #17985993C2semidirect
14. C2×#14556051C2semidirect
15. #552×C71C2semidirect
179. C71#3300semidirect
180. #552D71semidirect