Finite Group Databasebeta

← #22600385

Subgroups — #22600385

Order 9088.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×Dic142C211yesmaximal, characteristic
2#17986369C211yesmaximal, characteristic, complemented
3#17986340C211yesmaximal, characteristic, complemented
4#17986329C211yesmaximal, characteristic, complemented
5#17986303C211yesmaximal, characteristic, complemented
6#17986255C211yesmaximal, characteristic, complemented
7#17986238C211yesmaximal, characteristic, complemented
8#17986209C211yesmaximal, characteristic, complemented
9#17986154C211yesmaximal, characteristic, complemented
10#17986154C211yesmaximal, characteristic, complemented
11#17986104C211yesmaximal, characteristic, complemented
12#17986071C211yesmaximal, characteristic
13#17986007C211yesmaximal, characteristic, complemented
14C2×#14556051C211yesmaximal, characteristic, complemented
15#552×C71C211yesmaximal, characteristic
16C22×Dic142C2211yescharacteristic
17C22×Dic142C2211yescharacteristic
18C2×#11765855C2211yescharacteristic
19C2×#11765855C2211yescharacteristic
20C2×#11765855C2211yescharacteristic
21C2×#11765849C2211yescharacteristic
22C22×C71⋊C8C2211yescharacteristic
23#14556128C2211yescharacteristic
24#14556127—21no—
25#14556107C2211yescharacteristic
26#14556107C2211yescharacteristic
27#14556105—21no—
28#14556105—21no—
29C4×Dic142C2211yescharacteristic
30D4×Dic71—21no—
31#14556061C2211yescharacteristic
32#14556059C2211yescharacteristic
33#14556059C2211yescharacteristic
34C4⋊(C71⋊C8)C2211yescharacteristic
35#14556052C2211yescharacteristic
36#14556051C2211yescharacteristic
37#14556051C2211yescharacteristic
38#14556051C2211yescharacteristic
39#14556051C2211yescharacteristic
40#14556051C2211yescharacteristic
41#14556051C2211yescharacteristic
42#14556051C2211yescharacteristic
43#14556051C2211yescharacteristic
44#14556051C2211yescharacteristic
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
59C2×Dic142—22no—
60C2×Dic142—22no—
61C2×Dic142—21no—
62C2×Dic142—21no—
63C2×Dic142—41no—
64C2×Dic142—41no—
65C2×Dic142C2311yescharacteristic
66C2×Dic142C2311yescharacteristic
67C2×Dic142C2311yescharacteristic
68C22×Dic71—21no—
69C22×Dic71—21no—
70#11765855—22no—
71#11765855—41no—
72#11765855—41no—
73#11765849—41no—
74C2×C71⋊C8—41no—
75C2×C71⋊C8C2311yescharacteristic
76C2×C71⋊C8C2311yescharacteristic
77C2×C71⋊C8C2311yescharacteristic
78C2×C71⋊C8C2311yescharacteristic
79C4⋊Dic71—21no—
80C4⋊Dic71—21no—
81C4⋊Dic71C2311yescharacteristic
82Dic71⋊C4—21no—
83Dic71⋊C4—21no—
84Dic71⋊C4—21no—
85C22⋊Dic71—21no—
86C22⋊Dic71—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
113Q8×C71—21no—
114Q8×C71—21no—
115D4×C71—22no—
116C2×C4×C71—21no—
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—42no—
125C2×C4×C71C2×D411yescharacteristic
126C2×C4×C71C2×D411yescharacteristic
127C2×C4×C71C2×D411yescharacteristic
128C2×C4×C71C2×D411yescharacteristic
129C2×C4×C71C2411yescharacteristic
130C2×Dic71—21no—
131C2×Dic71—21no—
132C2×Dic71—21no—
133C2×Dic71—21no—
134C2×Dic71—42no—
135C2×Dic71—42no—
136Dic142—21no—
137Dic142—21no—
138Dic142—21no—
139Dic142—21no—
140Dic142—21no—
141Dic142—42no—
142Dic142—42no—
143Dic142#4112yes—
144C71⋊C8—21no—
145C71⋊C8—21no—
146C71⋊C8—21no—
147C71⋊C8—21no—
148C22×C71—21no—
149C22×C71—21no—
150C22×C71—21no—
151C22×C71—21no—
152C22×C71—21no—
153C22×C71#13712yes—
154C22×C71C22×D411yescharacteristic
155C4×C71—21no—
156C4×C71—21no—
157C4×C71—21no—
158C4×C71—21no—
159C4×C71—41no—
160C4×C71—41no—
161C4×C71—41no—
162C4×C71—41no—
163C4×C71C2×#4111yescharacteristic
164C4×C71H(5,𝔽2)11yescharacteristic
165Dic71—22no—
166Dic71—41no—
167Dic71—41no—
168Dic71—41no—
169C2×C71—22no—
170C2×C71—41no—
171C2×C71#54611yescharacteristic
172C2×C71C2×#13711yescharacteristic
173C2×C71#57811yescharacteristic
174#3312—711nomaximal, Sylow
175C71#331211yescharacteristic, complemented, Sylow
176#552D7111yescharacteristic
177D4×Q8—711no—
178#491—711no—
179#485—711no—
180#483—711no—
181#479—711no—
182#470—711no—
183#467—711no—
184#462—711no—
185#451—711no—
186#451—711no—
187#441—711no—
188#435—711no—
189#419—711no—
190C2×#103—711no—
191C22×Q8—711no—
192C22×Q8—711no—
193C2×Q16—711no—
194C2×Q16—711no—
195C2×Q16—711no—
196C2×M16—711no—
197C22×C8—711no—
198C4⋊Q8—711no—
199C4⋊Q8—1421no—
200#126—21no—
201#125—21no—
202#125C2×D7111yescharacteristic
203#123—21no—
204C22⋊Q8—711no—
205C22⋊Q8—711no—
206C22⋊Q8—1421no—
207C22⋊Q8—1421no—
208C22⋊Q8C2×D7111yescharacteristic
209C22⋊Q8C2×D7111yescharacteristic
210C4×Q8—711no—
211C4×Q8C2×D7111yescharacteristic
212C4×D4—1421no—
213C4×D4C2×D7111yescharacteristic
214#117—21no—
215#117C2×D7111yescharacteristic
216C2×C4⋊C4C2×D7111yescharacteristic
217#107—711no—
218#106—711no—
219#106—711no—
220C4⋊C8—711no—
221#103—711no—
222#103—711no—
223#103—711no—
224#103—711no—
225#103—711no—
226#103—711no—
227#103—711no—
228#103—711no—
229#103—711no—
230#103—711no—
231C22⋊C8—711no—
232C22⋊C8—711no—
233#97—711no—
234C2×Q8—2841no—
235C2×Q8—2841no—
236C2×Q8—711no—
237C2×Q8—711no—
238C2×Q8—711no—
239C2×Q8—1422no—
240C2×Q8—1422no—
241C2×Q8—1421no—
242C2×Q8—1421no—
243C2×Q8C22×D7111yescharacteristic
244C2×D4C22⋊D7111yescharacteristic
245C22×C4—21no—
246C22×C4—1421no—
247C22×C4—1421no—
248C22×C4C22×D7111yescharacteristic
249C22×C4C22×D7111yescharacteristic
250Q16—2841no—
251Q16—2841no—
252Q16—1422no—
253M16—2841no—
254C2×C8—2841no—
255C2×C8—711no—
256C2×C8—711no—
257C2×C8—711no—
258C2×C8—711no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—711no—
266C4⋊C4—1421no—
267C4⋊C4—1421no—
268C4⋊C4—1421no—
269C4⋊C4—1421no—
270C4⋊C4—1421no—
271C4⋊C4C22⋊D7111yescharacteristic
272C4⋊C4C22×D7111yescharacteristic
273C4⋊C4C22×D7111yescharacteristic
274C4⋊C4C22×D7111yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—1421no—
279C22⋊C4—1421no—
280C22⋊C4C22⋊D7111yescharacteristic
281C22⋊C4C22⋊D7111yescharacteristic
282C42—21no—
283C42—21no—
284C42—1421no—
285C42C22×D7111yescharacteristic
286C23C2×C22⋊D7111yescharacteristic
287C23C2×C22⋊D7111yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—2842no—
291Q8—2842no—
292Q8—712no—
293Q8—1421no—
294Q8—1421no—
295Q8—1421no—
296Q8—1421no—
297Q8—1421no—
298D4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—2842no—
309C2×C4—2842no—
310C2×C4—1421no—
311C2×C4—1421no—
312C2×C4—1421no—
313C2×C4—1421no—
314C2×C4C2×C22⋊D7111yescharacteristic
315C2×C4C2×C22⋊D7111yescharacteristic
316C2×C4C2×C22⋊D7111yescharacteristic
317C2×C4C2×C22⋊D7111yescharacteristic
318C2×C4C23×D7111yescharacteristic
319C8—1421no—
320C8—1421no—
321C8—1421no—
322C8—1421no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22#1455615812yes—
329C22C22×C22⋊D7111yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—2841no—
339C4—2841no—
340C4—2841no—
341C4—1422no—
342C4#41×D7111yescharacteristic
343C4#1455617411yescharacteristic
344C2—22no—
345C2—41no—
346C2#1798663711yescharacteristic
347C2C2×#1455615811yescharacteristic
348C2#1798681411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#552×C71C2
socleC22×C71C22×D4
hypercenter#552D71
hypocenterC71#3312
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. #17986369C2semidirect
3. #17986340C2semidirect
4. #17986329C2semidirect
5. #17986303C2semidirect
6. #17986255C2semidirect
7. #17986238C2semidirect
8. #17986209C2semidirect
9. #17986154C2semidirect
10. #17986154C2semidirect
11. #17986104C2semidirect
13. #17986007C2semidirect
14. C2×#14556051C2semidirect
175. C71#3312semidirect