Finite Group Databasebeta

← #22600505

Subgroups — #22600505

Order 9088.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×Dic142C211yesmaximal, characteristic, complemented
2#17986369C211yesmaximal, characteristic, complemented
3#17986340C211yesmaximal, characteristic, complemented
4#17986334C211yesmaximal, characteristic, complemented
5#17986303C211yesmaximal, characteristic, complemented
6#17986243C211yesmaximal, characteristic, complemented
7#17986226C211yesmaximal, characteristic, complemented
8#17986204C211yesmaximal, characteristic, complemented
9#17986137C211yesmaximal, characteristic, complemented
10#17986137C211yesmaximal, characteristic, complemented
11#17986099C211yesmaximal, characteristic, complemented
12#17986071C211yesmaximal, characteristic, complemented
13#17986007C211yesmaximal, characteristic, complemented
14C2×#14556051C211yesmaximal, characteristic, complemented
15#553×C71C211yesmaximal, characteristic
16C22×Dic142C2211yescharacteristic
17C22×Dic142C2211yescharacteristic
18C2×#11765853C2211yescharacteristic
19C2×#11765853C2211yescharacteristic
20C2×#11765853C2211yescharacteristic
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#14556061C2211yescharacteristic
33#14556059C2211yescharacteristic
34C4⋊(C71⋊C8)C2211yescharacteristic
35#14556051C2211yescharacteristic
36#14556051C2211yescharacteristic
37#14556051C2211yescharacteristic
38#14556051C2211yescharacteristic
39#14556051C2211yescharacteristic
40#14556051C2211yescharacteristic
41#14556051C2211yescharacteristic
42#14556051C2211yescharacteristic
43#14556051C2211yescharacteristic
44#14556049C2211yescharacteristic
45C22⋊(C71⋊C8)C2211yescharacteristic
46C22⋊(C71⋊C8)C2211yescharacteristic
47#14556039C2211yescharacteristic
48#126×C71—21no—
49#125×C71C2211yescharacteristic
50#123×C71—21no—
51#121×C71—21no—
52#121×C71C2211yescharacteristic
53#121×C71C2211yescharacteristic
54C4×D4×C71—21no—
55C4×D4×C71C2211yescharacteristic
56C4×D4×C71C2211yescharacteristic
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#11765853—22no—
71#11765853—41no—
72#11765853—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×D4×C71—21no—
89C2×D4×C71—21no—
90C2×D4×C71D411yescharacteristic
91C2×D4×C71C2311yescharacteristic
92C22×C4×C71—21no—
93C22×C4×C71—21no—
94C22×C4×C71C2311yescharacteristic
95C22×C4×C71C2311yescharacteristic
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×C71—21no—
107C22⋊C4×C71D411yescharacteristic
108C22⋊C4×C71D411yescharacteristic
109C42×C71—21no—
110C42×C71C2311yescharacteristic
111C23×C71—21no—
112C23×C71C2×D411yescharacteristic
113C23×C71C2×D411yescharacteristic
114D4×C71—22no—
115D4×C71—21no—
116D4×C71—21no—
117D4×C71—42no—
118D4×C71—81no—
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—21no—
126C2×C4×C71—81no—
127C2×C4×C71—81no—
128C2×C4×C71C2×D411yescharacteristic
129C2×C4×C71C2×D411yescharacteristic
130C2×C4×C71C2×D411yescharacteristic
131C2×C4×C71C2×D411yescharacteristic
132C2×C4×C71C2411yescharacteristic
133C2×Dic71—21no—
134C2×Dic71—21no—
135C2×Dic71—21no—
136C2×Dic71—21no—
137C2×Dic71—42no—
138C2×Dic71—42no—
139Dic142—21no—
140Dic142—21no—
141Dic142—21no—
142Dic142—21no—
143Dic142—21no—
144Dic142—42no—
145Dic142—42no—
146Dic142#4112yes—
147C71⋊C8—21no—
148C71⋊C8—21no—
149C71⋊C8—21no—
150C71⋊C8—21no—
151C22×C71—21no—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71—21no—
155C22×C71—21no—
156C22×C71—41no—
157C22×C71—41no—
158C22×C71—41no—
159C22×C71#13712yes—
160C22×C71C22×D411yescharacteristic
161C4×C71—21no—
162C4×C71—21no—
163C4×C71—21no—
164C4×C71—21no—
165C4×C71—41no—
166C4×C71—41no—
167C4×C71—41no—
168C4×C71C2×#4111yescharacteristic
169C4×C71H(5,𝔽2)11yescharacteristic
170Dic71—22no—
171Dic71—41no—
172Dic71—41no—
173Dic71—41no—
174C2×C71—22no—
175C2×C71—41no—
176C2×C71—81no—
177C2×C71#54611yescharacteristic
178C2×C71C2×#13711yescharacteristic
179C2×C71#57711yescharacteristic
180#3320—711nomaximal, Sylow
181C71#332011yescharacteristic, complemented, Sylow
182#553D7111yescharacteristic
183D4×Q8—711no—
184#491—711no—
185#485—711no—
186#484—711no—
187#479—711no—
188#468—711no—
189#465—711no—
190#461—711no—
191#448—711no—
192#448—711no—
193#440—711no—
194#435—711no—
195#419—711no—
196C2×#103—711no—
197C22×Q8—711no—
198C22×Q8—711no—
199C2×SD16—711no—
200C2×SD16—711no—
201C2×SD16—711no—
202C2×M16—711no—
203C22×C8—711no—
204C4⋊Q8—711no—
205C4⋊Q8—1421no—
206#126—21no—
207#125C2×D7111yescharacteristic
208#123—21no—
209C22⋊Q8—711no—
210C22⋊Q8—711no—
211C22⋊Q8—1421no—
212C22⋊Q8—1421no—
213#121—21no—
214#121C2×D7111yescharacteristic
215#121C2×D7111yescharacteristic
216C4×Q8—711no—
217C4×D4—21no—
218C4×D4—1421no—
219C4×D4C2×D7111yescharacteristic
220C4×D4C2×D7111yescharacteristic
221#117C2×D7111yescharacteristic
222C2×C4⋊C4C2×D7111yescharacteristic
223#107—711no—
224#107—711no—
225#106—711no—
226C4⋊C8—711no—
227#103—711no—
228#103—711no—
229#103—711no—
230#103—711no—
231#103—711no—
232#103—711no—
233#103—711no—
234#103—711no—
235#103—711no—
236#102—711no—
237C22⋊C8—711no—
238C22⋊C8—711no—
239#97—711no—
240C2×Q8—2841no—
241C2×Q8—2841no—
242C2×Q8—711no—
243C2×Q8—711no—
244C2×Q8—711no—
245C2×Q8—1422no—
246C2×Q8—1422no—
247C2×Q8—1421no—
248C2×Q8—1421no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4C22⋊D7111yescharacteristic
252C2×D4C22×D7111yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—1421no—
256C22×C4—1421no—
257C22×C4C22×D7111yescharacteristic
258C22×C4C22×D7111yescharacteristic
259SD16—2841no—
260SD16—2841no—
261SD16—1422no—
262M16—2841no—
263C2×C8—2841no—
264C2×C8—711no—
265C2×C8—711no—
266C2×C8—711no—
267C2×C8—711no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—711no—
272C4⋊C4—1421no—
273C4⋊C4—1421no—
274C4⋊C4—1421no—
275C4⋊C4—1421no—
276C4⋊C4—1421no—
277C4⋊C4C22⋊D7111yescharacteristic
278C4⋊C4C22×D7111yescharacteristic
279C4⋊C4C22×D7111yescharacteristic
280C4⋊C4C22×D7111yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—1421no—
286C22⋊C4—1421no—
287C22⋊C4C22⋊D7111yescharacteristic
288C22⋊C4C22⋊D7111yescharacteristic
289C42—21no—
290C42—1421no—
291C42C22×D7111yescharacteristic
292C23—21no—
293C23C2×C22⋊D7111yescharacteristic
294C23C2×C22⋊D7111yescharacteristic
295Q8—2842no—
296Q8—2842no—
297Q8—712no—
298Q8—1421no—
299Q8—1421no—
300Q8—1421no—
301Q8—1421no—
302Q8—1421no—
303D4—22no—
304D4—21no—
305D4—21no—
306D4—42no—
307D4—81no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—2842no—
318C2×C4—2842no—
319C2×C4—1421no—
320C2×C4—1421no—
321C2×C4—1421no—
322C2×C4—1421no—
323C2×C4C2×C22⋊D7111yescharacteristic
324C2×C4C2×C22⋊D7111yescharacteristic
325C2×C4C2×C22⋊D7111yescharacteristic
326C2×C4C2×C22⋊D7111yescharacteristic
327C2×C4C23×D7111yescharacteristic
328C8—1421no—
329C8—1421no—
330C8—1421no—
331C8—1421no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22#1455615812yes—
341C22C22×C22⋊D7111yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—2841no—
350C4—2841no—
351C4—2841no—
352C4—1422no—
353C4#41×D7111yescharacteristic
354C4#1455617411yescharacteristic
355C2—22no—
356C2—41no—
357C2—81no—
358C2#1798663711yescharacteristic
359C2C2×#1455615811yescharacteristic
360C2#1798681011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#553×C71C2
socleC22×C71C22×D4
hypercenter#553D71
hypocenterC71#3320
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. D4×Dic142C2semidirect
2. #17986369C2semidirect
3. #17986340C2semidirect
4. #17986334C2semidirect
5. #17986303C2semidirect
6. #17986243C2semidirect
7. #17986226C2semidirect
8. #17986204C2semidirect
9. #17986137C2semidirect
10. #17986137C2semidirect
11. #17986099C2semidirect
12. #17986071C2semidirect
13. #17986007C2semidirect
14. C2×#14556051C2semidirect
181. C71#3320semidirect