Finite Group Databasebeta

← #22591733

Subgroups — #22591733

Order 9088.

814 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14556113C211yesmaximal, characteristic, complemented
2C2×#14556098C211yesmaximal, characteristic
3C22×(C2×D71)⋊C4C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D71C211yesmaximal, characteristic, complemented
5#17985928C211yesmaximal, characteristic, complemented
6#17985925C211yesmaximal, characteristic, complemented
7#17985917C211yesmaximal, characteristic, complemented
8#17985916C211yesmaximal, characteristic, complemented
9#17985878C211yesmaximal, characteristic, complemented
10#17985877C211yesmaximal, characteristic, complemented
11#17985876C211yesmaximal, characteristic, complemented
12#17985870C211yesmaximal, characteristic, complemented
13#17985861C211yesmaximal, characteristic, complemented
14#17985857C211yesmaximal, characteristic, complemented
15#380×C71C211yesmaximal, characteristic, complemented
16C24×D71C2211yescharacteristic
17C22×D284C2211yescharacteristic
18C22×D284C2211yescharacteristic
19C22×C22⋊D71C2211yescharacteristic
20C23×Dic71C2211yescharacteristic
21C22×C4×D71C2211yescharacteristic
22C22×C4×D71C2211yescharacteristic
23#14556113—24no—
24#14556098—24no—
25C2×Dic71⋊C4C2211yescharacteristic, complemented
26C2×Dic71⋊C4C2211yescharacteristic, complemented
27C2×C22⋊Dic71C2211yescharacteristic
28C2×(C2×D71)⋊C4—24no—
29C2×(C2×D71)⋊C4C2211yescharacteristic
30C2×(C2×D71)⋊C4C2211yescharacteristic
31C2×(C2×D71)⋊C4C2211yescharacteristic
32C2×(C2×D71)⋊C4C2211yescharacteristic
33C2×(C2×D71)⋊C4C2211yescharacteristic
34C2×(C2×D71)⋊C4C2211yescharacteristic
35C2×(C2×D71)⋊C4C2211yescharacteristic
36C2×(C2×D71)⋊C4C2211yescharacteristic
37C2×(C2×D71)⋊C4C2211yescharacteristic, complemented
38C2×(C2×D71)⋊C4C2211yescharacteristic, complemented
39C2×(C2×D71)⋊C4C2211yescharacteristic, complemented
40C2×(C2×D71)⋊C4C2211yescharacteristic, complemented
41C22⋊C4×D71—24no—
42C71⋊#95C2211yescharacteristic, complemented
43C71⋊#95C2211yescharacteristic, complemented
44C71⋊#95C2211yescharacteristic, complemented
45C71⋊#95C2211yescharacteristic, complemented
46C71⋊#95C2211yescharacteristic, complemented
47C71⋊#95C2211yescharacteristic, complemented
48C23×C4×C71C2211yescharacteristic
49C2×C4⋊C4×C71C2211yescharacteristic, complemented
50C2×C4⋊C4×C71C2211yescharacteristic, complemented
51C2×C22⋊C4×C71C2211yescharacteristic
52C2×C22⋊C4×C71C2211yescharacteristic
53#95×C71C2211yescharacteristic, complemented
54#95×C71C2211yescharacteristic, complemented
55C23×D71—22no—
56C23×D71—22no—
57C23×D71—22no—
58C23×D71—22no—
59C23×D71—41no—
60C23×D71—41no—
61C23×D71D414yes—
62C23×D71C2311yescharacteristic
63C23×D71C2311yescharacteristic
64C23×D71C2311yescharacteristic
65C2×D284—22no—
66C2×D284—22no—
67C2×D284—22no—
68C2×D284—22no—
69C2×D284—41no—
70C2×D284—41no—
71C2×C22⋊D71—22no—
72C2×C22⋊D71—22no—
73C2×C22⋊D71—22no—
74C22×Dic71—22no—
75C22×Dic71—22no—
76C22×Dic71—41no—
77C22×Dic71C2311yescharacteristic
78C22×Dic71C2311yescharacteristic
79C22×Dic71C2311yescharacteristic
80C22×Dic71C2311yescharacteristic
81C22×Dic71C2311yescharacteristic
82C2×C4×D71—22no—
83C2×C4×D71—22no—
84C2×C4×D71—22no—
85C2×C4×D71—22no—
86C2×C4×D71—41no—
87C2×C4×D71—41no—
88Dic71⋊C4—22no—
89Dic71⋊C4—41no—
90C22⋊Dic71—22no—
91(C2×D71)⋊C4—22no—
92(C2×D71)⋊C4—22no—
93(C2×D71)⋊C4—22no—
94(C2×D71)⋊C4—22no—
95(C2×D71)⋊C4—22no—
96(C2×D71)⋊C4—22no—
97(C2×D71)⋊C4—22no—
98(C2×D71)⋊C4—22no—
99(C2×D71)⋊C4—22no—
100(C2×D71)⋊C4—41no—
101(C2×D71)⋊C4—41no—
102(C2×D71)⋊C4D414yescomplemented
103C24×C71C2311yescharacteristic
104C22×C4×C71—22no—
105C22×C4×C71—22no—
106C22×C4×C71—41no—
107C22×C4×C71C2311yescharacteristic
108C22×C4×C71C2311yescharacteristic
109C22×C4×C71C2311yescharacteristic
110C22×C4×C71C2311yescharacteristic
111C22×C4×C71C2311yescharacteristic
112C22×C4×C71C2311yescharacteristic
113C4⋊C4×C71—22no—
114C4⋊C4×C71—41no—
115C22⋊C4×C71—22no—
116C22⋊C4×C71—22no—
117C23×C71—21no—
118C23×C71—21no—
119C23×C71—21no—
120C23×C71—21no—
121C23×C71C2×D412yes—
122C23×C71#4112yes—
123C23×C71#4112yes—
124C23×C71C2411yescharacteristic
125C22×D71—21no—
126C22×D71—21no—
127C22×D71—21no—
128C22×D71—21no—
129C22×D71—28no—
130C22×D71—21no—
131C22×D71—21no—
132C22×D71—21no—
133C22×D71—21no—
134C22×D71—21no—
135C22×D71—21no—
136C22×D71—28no—
137C22×D71—21no—
138C22×D71—21no—
139C22×D71—21no—
140C22×D71—21no—
141C22×D71—21no—
142C22×D71—21no—
143C22×D71—21no—
144C22×D71—44no—
145C22×D71—44no—
146C22×D71—44no—
147C22×D71—44no—
148C22×D71—44no—
149C22×D71C2×D412yes—
150C22×D71C2×D412yes—
151C22×D71C2×D412yes—
152C22×D71#4112yes—
153C2×C4×C71—21no—
154C2×C4×C71—21no—
155C2×C4×C71—21no—
156C2×C4×C71—21no—
157C2×C4×C71—21no—
158C2×C4×C71—21no—
159C2×C4×C71—21no—
160C2×C4×C71—21no—
161C2×C4×C71—21no—
162C2×C4×C71—21no—
163C2×C4×C71—21no—
164C2×C4×C71—21no—
165C2×C4×C71—21no—
166C2×C4×C71—21no—
167C2×C4×C71—21no—
168C2×C4×C71—21no—
169C2×C4×C71—21no—
170C2×C4×C71—44no—
171C2×C4×C71C2×D412yes—
172C22⋊D71—44no—
173C2×Dic71—21no—
174C2×Dic71—21no—
175C2×Dic71—21no—
176C2×Dic71—21no—
177C2×Dic71—21no—
178C2×Dic71—21no—
179C2×Dic71—21no—
180C2×Dic71—21no—
181C2×Dic71—21no—
182C2×Dic71—21no—
183C2×Dic71—21no—
184C2×Dic71—21no—
185C2×Dic71—21no—
186C2×Dic71—21no—
187C2×Dic71—44no—
188C2×Dic71C2×D412yes—
189D284—44no—
190D284—44no—
191C4×D71—44no—
192C4×D71—44no—
193C22×C71—22no—
194C22×C71—22no—
195C22×C71—22no—
196C22×C71—22no—
197C22×C71—22no—
198C22×C71—22no—
199C22×C71#12314yescomplemented
200C22×C71C22×D411yescharacteristic
201C22×C71C22×D411yescharacteristic
202C22×C71C2×#4111yescharacteristic
203C22×C71C2×#4111yescharacteristic
204C22×C71C2×#4111yescharacteristic
205C22×C71H(5,𝔽2)11yescharacteristic
206C22×C71H(5,𝔽2)11yescharacteristic
207C2×D71—22no—
208C2×D71—22no—
209C2×D71—22no—
210C2×D71—22no—
211C2×D71—22no—
212C2×D71—22no—
213C2×D71—22no—
214C2×D71—22no—
215C2×D71—22no—
216C2×D71—416no—
217C2×D71—41no—
218C2×D71—41no—
219C2×D71—41no—
220C2×D71—41no—
221C2×D71—41no—
222C2×D71—41no—
223C2×D71—41no—
224C2×D71—41no—
225C2×D71—41no—
226C2×D71—41no—
227C2×D71—41no—
228C2×D71#12114yes—
229C4×C71—22no—
230C4×C71—41no—
231C4×C71—41no—
232C4×C71—41no—
233C4×C71—41no—
234C4×C71—41no—
235Dic71—22no—
236Dic71—41no—
237Dic71—41no—
238Dic71—41no—
239Dic71—41no—
240C2×C71—24no—
241C2×C71C2×#12111yescharacteristic
242C2×C71C2×#12311yescharacteristic
243C2×C71#53411yescharacteristic
244C2×C71#54611yescharacteristic
245C2×C71#54611yescharacteristic
246C2×C71#55111yescharacteristic
247C2×C71#55111yescharacteristic
248D71—24no—
249D71—42no—
250D71—81no—
251#2412—711nomaximal, Sylow
252C71#241211yescharacteristic, complemented, Sylow
253C2×#123—711no—
254C2×#120—711no—
255C22×C22⋊C4—711no—
256C22×C22⋊C4—711no—
257#397—711no—
258#397—711no—
259#394—711no—
260#394—711no—
261#386—711no—
262#386—711no—
263#386—711no—
264#385—711no—
265#381—711no—
266#380—711no—
267#380D7111yescharacteristic, complemented
268C25—711no—
269C22×D4—711no—
270C22×D4—711no—
271C22×D4—711no—
272C23×C4—711no—
273C23×C4—711no—
274C23×C4—711no—
275C23×C4C2×D7111yescharacteristic
276#123—1424no—
277#120—1424no—
278C2×C4⋊C4—711no—
279C2×C4⋊C4—711no—
280C2×C4⋊C4C2×D7111yescharacteristic, complemented
281C2×C4⋊C4C2×D7111yescharacteristic, complemented
282C2×C22⋊C4—711no—
283C2×C22⋊C4—711no—
284C2×C22⋊C4—711no—
285C2×C22⋊C4—711no—
286C2×C22⋊C4—711no—
287C2×C22⋊C4—711no—
288C2×C22⋊C4—711no—
289C2×C22⋊C4—711no—
290C2×C22⋊C4—711no—
291C2×C22⋊C4—711no—
292C2×C22⋊C4—711no—
293C2×C22⋊C4—711no—
294C2×C22⋊C4—711no—
295C2×C22⋊C4—1424no—
296C2×C22⋊C4—1424no—
297C2×C22⋊C4C2×D7111yescharacteristic
298C2×C22⋊C4C2×D7111yescharacteristic
299#95—711no—
300#95—711no—
301#95—711no—
302#95—711no—
303#95—711no—
304#95—711no—
305#95C2×D7111yescharacteristic, complemented
306#95C2×D7111yescharacteristic, complemented
307C24—2841no—
308C24—2841no—
309C24—714no—
310C24—711no—
311C24—711no—
312C24—711no—
313C24—1422no—
314C24—1422no—
315C24—1422no—
316C24—1422no—
317C24C22×D7111yescharacteristic
318C2×D4—2841no—
319C2×D4—2841no—
320C2×D4—1422no—
321C2×D4—1422no—
322C2×D4—1422no—
323C2×D4—1422no—
324C2×D4—1422no—
325C2×D4—1422no—
326C2×D4—1422no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—41no—
330C22×C4—2841no—
331C22×C4—2841no—
332C22×C4—2841no—
333C22×C4—711no—
334C22×C4—711no—
335C22×C4—711no—
336C22×C4—711no—
337C22×C4—711no—
338C22×C4—1422no—
339C22×C4—1422no—
340C22×C4—1422no—
341C22×C4—1422no—
342C22×C4—1422no—
343C22×C4—1422no—
344C22×C4C22×D7111yescharacteristic
345C22×C4C22×D7111yescharacteristic
346C22×C4C22×D7111yescharacteristic
347C22×C4C22×D7111yescharacteristic
348C22×C4C22×D7111yescharacteristic
349C22×C4C22×D7111yescharacteristic
350C4⋊C4—22no—
351C4⋊C4—41no—
352C4⋊C4—2841no—
353C4⋊C4—1422no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—2841no—
357C22⋊C4—2841no—
358C22⋊C4—714no—
359C22⋊C4—1422no—
360C22⋊C4—1422no—
361C22⋊C4—1422no—
362C22⋊C4—1422no—
363C22⋊C4—1422no—
364C22⋊C4—1422no—
365C22⋊C4—1422no—
366C22⋊C4—1422no—
367C22⋊C4—1422no—
368C22⋊C4—1422no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—2844no—
374C23—2844no—
375C23—2844no—
376C23—2844no—
377C23—2844no—
378C23—712no—
379C23—712no—
380C23—712no—
381C23—712no—
382C23—1428no—
383C23—1421no—
384C23—1421no—
385C23—1421no—
386C23—1421no—
387C23—1421no—
388C23—1421no—
389C23—1421no—
390C23—1421no—
391C23—1421no—
392C23—1421no—
393C23—1421no—
394C23—1421no—
395C23—1421no—
396C23—1428no—
397C23—1421no—
398C23—1421no—
399C23—1421no—
400C23—1421no—
401C23D4×D7112yes—
402C23#1176586512yes—
403C23#1176586612yes—
404C23C23×D7111yescharacteristic
405D4—2844no—
406D4—2844no—
407D4—2844no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—44no—
426C2×C4—2844no—
427C2×C4—2844no—
428C2×C4—2844no—
429C2×C4—712no—
430C2×C4—1421no—
431C2×C4—1421no—
432C2×C4—1421no—
433C2×C4—1421no—
434C2×C4—1421no—
435C2×C4—1421no—
436C2×C4—1421no—
437C2×C4—1421no—
438C2×C4—1421no—
439C2×C4—1421no—
440C2×C4—1421no—
441C2×C4—1421no—
442C2×C4—1421no—
443C2×C4—1421no—
444C2×C4D4×D7112yes—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—28416no—
452C22—2841no—
453C22—2841no—
454C22—2841no—
455C22—2841no—
456C22—2841no—
457C22—2841no—
458C22—2841no—
459C22—2841no—
460C22—2841no—
461C22—2841no—
462C22—2841no—
463C22—714no—
464C22—1422no—
465C22—1422no—
466C22—1422no—
467C22—1422no—
468C22—1422no—
469C22—1422no—
470C22—1422no—
471C22—1422no—
472C22—1422no—
473C22#1455611314yescomplemented
474C22C2×D4×D7111yescharacteristic
475C22C2×D4×D7111yescharacteristic
476C22#41×D7111yescharacteristic
477C22C2×#1176586511yescharacteristic
478C22C2×#1176586611yescharacteristic
479C22#1455617311yescharacteristic
480C22#1455617411yescharacteristic
481C4—22no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—2841no—
488C4—2841no—
489C4—2841no—
490C4—2841no—
491C4—1422no—
492C2—24no—
493C2—2842no—
494C2—5681no—
495C2—1424no—
496C2#121×D7111yescharacteristic
497C2C2×#1455611311yescharacteristic
498C2#1798655711yescharacteristic
499C2#1798663911yescharacteristic
500C2#1798664011yescharacteristic
501C2#1798667311yescharacteristic
502C2#1798667611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#380×C71C2
hypercenter#380D71
hypocenterC71#2412
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C2×#14556113C2semidirect
3. C22×(C2×D71)⋊C4C2semidirect
4. C2×C22⋊C4×D71C2semidirect
5. #17985928C2semidirect
6. #17985925C2semidirect
7. #17985917C2semidirect
8. #17985916C2semidirect
9. #17985878C2semidirect
10. #17985877C2semidirect
11. #17985876C2semidirect
12. #17985870C2semidirect
13. #17985861C2semidirect
14. #17985857C2semidirect
15. #380×C71C2semidirect
25. C2×Dic71⋊C4C22semidirect
26. C2×Dic71⋊C4C22semidirect
37. C2×(C2×D71)⋊C4C22semidirect
38. C2×(C2×D71)⋊C4C22semidirect
39. C2×(C2×D71)⋊C4C22semidirect
40. C2×(C2×D71)⋊C4C22semidirect
42. C71⋊#95C22semidirect
43. C71⋊#95C22semidirect
44. C71⋊#95C22semidirect
45. C71⋊#95C22semidirect
46. C71⋊#95C22semidirect
47. C71⋊#95C22semidirect
49. C2×C4⋊C4×C71C22semidirect
50. C2×C4⋊C4×C71C22semidirect
53. #95×C71C22semidirect
54. #95×C71C22semidirect
102. (C2×D71)⋊C4D4semidirect
199. C22×C71#123semidirect
252. C71#2412semidirect
267. #380D71semidirect
280. C2×C4⋊C4C2×D71semidirect
281. C2×C4⋊C4C2×D71semidirect
305. #95C2×D71semidirect
306. #95C2×D71semidirect
473. C22#14556113semidirect