Finite Group Databasebeta

← #22600780

Subgroups — #22600780

Order 9088.

376 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986653C211yesmaximal, characteristic
2#17986409C211yesmaximal, characteristic, complemented
3#17986332C211yesmaximal, characteristic, complemented
4#17986327C211yesmaximal, characteristic, complemented
5#17986323C211yesmaximal, characteristic, complemented
6#17986312C211yesmaximal, characteristic, complemented
7#17986296C211yesmaximal, characteristic, complemented
8#17986294C211yesmaximal, characteristic, complemented
9#17986247C211yesmaximal, characteristic, complemented
10#17986230C211yesmaximal, characteristic, complemented
11#17986097C211yesmaximal, characteristic, complemented
12#17986071C211yesmaximal, characteristic, complemented
13#17986042C211yesmaximal, characteristic, complemented
14C2×#14556061C211yesmaximal, characteristic, complemented
15#552×C71C211yesmaximal, characteristic, complemented
16C2×#11765864—21no—
17C2×#11765852C2211yescharacteristic
18C2×#11765849C2211yescharacteristic
19C22×C71⋊C8C2211yescharacteristic
20#14556128C2211yescharacteristic
21#14556112—21no—
22#14556112—21no—
23#14556108—21no—
24#14556102C2211yescharacteristic
25#14556102C2211yescharacteristic
26C4×D284C2211yescharacteristic
27C2×C4⋊Dic71C2211yescharacteristic
28C2×C4⋊Dic71C2211yescharacteristic
29#14556061C2211yescharacteristic
30#14556061C2211yescharacteristic
31#14556061C2211yescharacteristic
32#14556061C2211yescharacteristic
33#14556061C2211yescharacteristic
34#14556061C2211yescharacteristic
35#14556059C2211yescharacteristic
36#14556059C2211yescharacteristic
37#14556059C2211yescharacteristic
38C4⋊(C71⋊C8)C2211yescharacteristic
39#14556052C2211yescharacteristic
40#14556052C2211yescharacteristic
41#14556052C2211yescharacteristic
42#14556048C2211yescharacteristic
43#14556048C2211yescharacteristic
44#14556048C2211yescharacteristic
45C22⋊(C71⋊C8)C2211yescharacteristic
46C22⋊(C71⋊C8)C2211yescharacteristic
47#14556039C2211yescharacteristic
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#11765864—44no—
60C2×Dic142—21no—
61C2×D284C2311yescharacteristic
62C2×C22⋊D71—21no—
63C2×C22⋊D71—21no—
64C22×Dic71—21no—
65C22×Dic71—21no—
66C2×C4×D71—21no—
67#11765852—22no—
68#11765849—22no—
69C2×C71⋊C8—22no—
70C2×C71⋊C8C2311yescharacteristic
71C2×C71⋊C8C2311yescharacteristic
72C2×C71⋊C8C2311yescharacteristic
73C2×C71⋊C8C2311yescharacteristic
74C4⋊Dic71—21no—
75C4⋊Dic71—21no—
76C4⋊Dic71—21no—
77C4⋊Dic71C2311yescharacteristic
78C4⋊Dic71C2311yescharacteristic
79C4⋊Dic71C2311yescharacteristic
80(C2×D71)⋊C4—21no—
81(C2×D71)⋊C4—21no—
82(C2×D71)⋊C4—21no—
83C2×Q8×C71C2311yescharacteristic
84C2×D4×C71D411yescharacteristic
85C22×C4×C71—21no—
86C22×C4×C71C2311yescharacteristic
87C22×C4×C71C2311yescharacteristic
88C4⋊C4×C71—21no—
89C4⋊C4×C71—21no—
90C4⋊C4×C71—21no—
91C4⋊C4×C71—21no—
92C4⋊C4×C71—21no—
93C4⋊C4×C71—21no—
94C4⋊C4×C71D411yescharacteristic
95C4⋊C4×C71C2311yescharacteristic
96C4⋊C4×C71C2311yescharacteristic
97C4⋊C4×C71C2311yescharacteristic
98C22⋊C4×C71—21no—
99C22⋊C4×C71—21no—
100C22⋊C4×C71—21no—
101C22⋊C4×C71D411yescharacteristic
102C22⋊C4×C71D411yescharacteristic
103C42×C71—21no—
104C42×C71—21no—
105C42×C71C2311yescharacteristic
106C23×C71C2×D411yescharacteristic
107C23×C71C2×D411yescharacteristic
108C22×D71—21no—
109Q8×C71—21no—
110Q8×C71—21no—
111D4×C71—22no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—42no—
121C2×C4×C71C2×D411yescharacteristic
122C2×C4×C71C2×D411yescharacteristic
123C2×C4×C71C2×D411yescharacteristic
124C2×C4×C71C2×D411yescharacteristic
125C2×C4×C71C2411yescharacteristic
126C22⋊D71—42no—
127C22⋊D71—42no—
128C2×Dic71—21no—
129C2×Dic71—21no—
130C2×Dic71—21no—
131C2×Dic71—42no—
132C2×Dic71—42no—
133D284—21no—
134D284—21no—
135C4×D71—42no—
136Dic142—42no—
137C71⋊C8—21no—
138C71⋊C8—21no—
139C71⋊C8—21no—
140C71⋊C8#4112yescomplemented
141C22×C71—21no—
142C22×C71—21no—
143C22×C71—21no—
144C22×C71—21no—
145C22×C71—21no—
146C22×C71AGL(1,ℤ/8)12yes—
147C22×C71C22×D411yescharacteristic
148C2×D71—41no—
149C2×D71—41no—
150C2×D71—41no—
151C4×C71—21no—
152C4×C71—21no—
153C4×C71—21no—
154C4×C71—21no—
155C4×C71—41no—
156C4×C71—41no—
157C4×C71—41no—
158C4×C71—41no—
159C4×C71C2×#4111yescharacteristic
160C4×C712−1+411yescharacteristic
161Dic71—41no—
162Dic71—41no—
163Dic71—41no—
164C2×C71—22no—
165C2×C71—41no—
166C2×C71#54711yescharacteristic
167C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C71#57811yescharacteristic
169D71—81no—
170#3343—711nomaximal, Sylow
171C71#334311yescharacteristic, complemented, Sylow
172#552D7111yescharacteristic, complemented
173#547—711no—
174#501—711no—
175#484—711no—
176#483—711no—
177#482—711no—
178#480—711no—
179#478—711no—
180#477—711no—
181#468—711no—
182#465—711no—
183#440—711no—
184#435—711no—
185#428—711no—
186C2×#107—711no—
187C2×#41—1421no—
188C2×SD16—711no—
189C2×M16—711no—
190C22×C8—711no—
191C4⋊Q8—711no—
192#126—21no—
193#125—21no—
194#125C2×D7111yescharacteristic
195#123—21no—
196#123—1421no—
197#123—1421no—
198C22⋊Q8—1421no—
199C22⋊Q8C2×D7111yescharacteristic
200C22⋊Q8C2×D7111yescharacteristic
201#121—711no—
202#121—711no—
203C4×Q8C2×D7111yescharacteristic
204C4×D4—711no—
205C4×D4C2×D7111yescharacteristic
206#117—21no—
207#117C2×D7111yescharacteristic
208C2×C4⋊C4—711no—
209C2×C4⋊C4—711no—
210C2×C4⋊C4C2×D7111yescharacteristic
211#107—711no—
212#107—711no—
213#107—711no—
214#107—711no—
215#107—711no—
216#107—711no—
217#106—711no—
218#106—711no—
219#106—711no—
220C4⋊C8—711no—
221#103—711no—
222#103—711no—
223#103—711no—
224#102—711no—
225#102—711no—
226#102—711no—
227C22⋊C8—711no—
228C22⋊C8—711no—
229#97—711no—
230#41—2844no—
231C2×Q8—1421no—
232C2×Q8C22×D7111yescharacteristic
233C2×D4—711no—
234C2×D4—1421no—
235C2×D4—1421no—
236C2×D4C22⋊D7111yescharacteristic
237C22×C4—21no—
238C22×C4—1421no—
239C22×C4—1421no—
240C22×C4—1421no—
241C22×C4C22×D7111yescharacteristic
242C22×C4C22×D7111yescharacteristic
243SD16—1422no—
244M16—1422no—
245C2×C8—711no—
246C2×C8—711no—
247C2×C8—711no—
248C2×C8—711no—
249C2×C8—1422no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—711no—
257C4⋊C4—711no—
258C4⋊C4—711no—
259C4⋊C4—1421no—
260C4⋊C4—1421no—
261C4⋊C4—1421no—
262C4⋊C4C22⋊D7111yescharacteristic
263C4⋊C4C22×D7111yescharacteristic
264C4⋊C4C22×D7111yescharacteristic
265C4⋊C4C22×D7111yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—1421no—
270C22⋊C4—1421no—
271C22⋊C4—1421no—
272C22⋊C4C22⋊D7111yescharacteristic
273C22⋊C4C22⋊D7111yescharacteristic
274C42—21no—
275C42—21no—
276C42C22×D7111yescharacteristic
277C23—1421no—
278C23C2×C22⋊D7111yescharacteristic
279C23C2×C22⋊D7111yescharacteristic
280Q8—21no—
281Q8—21no—
282Q8—2842no—
283D4—22no—
284D4—2842no—
285D4—2842no—
286D4—1421no—
287D4—1421no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—42no—
297C2×C4—2842no—
298C2×C4—2842no—
299C2×C4—2842no—
300C2×C4—1421no—
301C2×C4—1421no—
302C2×C4—1421no—
303C2×C4C2×C22⋊D7111yescharacteristic
304C2×C4C2×C22⋊D7111yescharacteristic
305C2×C4C2×C22⋊D7111yescharacteristic
306C2×C4C2×C22⋊D7111yescharacteristic
307C2×C4C23×D7111yescharacteristic
308C8—712no—
309C8—1421no—
310C8—1421no—
311C8—1421no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—2841no—
318C22—2841no—
319C22—2841no—
320C22#1455615312yes—
321C22C22×C22⋊D7111yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—2841no—
331C4—2841no—
332C4—2841no—
333C4#41×D7111yescharacteristic
334C4#1455617511yescharacteristic
335C2—22no—
336C2—41no—
337C2—5681no—
338C2#1798664811yescharacteristic
339C2C2×#1455615311yescharacteristic
340C2#1798681411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#552×C71C2
socleC22×C71C22×D4
hypercenter#552D71
hypocenterC71#3343
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. #17986409C2semidirect
3. #17986332C2semidirect
4. #17986327C2semidirect
5. #17986323C2semidirect
6. #17986312C2semidirect
7. #17986296C2semidirect
8. #17986294C2semidirect
9. #17986247C2semidirect
10. #17986230C2semidirect
11. #17986097C2semidirect
12. #17986071C2semidirect
13. #17986042C2semidirect
14. C2×#14556061C2semidirect
15. #552×C71C2semidirect
140. C71⋊C8#41semidirect
171. C71#3343semidirect
172. #552D71semidirect