Finite Group Databasebeta

← #22591799

Subgroups — #22591799

Order 9088.

800 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14556096C211yesmaximal, characteristic
2C2×C4×C22⋊D71C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D71C211yesmaximal, characteristic, complemented
4#17985934C211yesmaximal, characteristic, complemented
5#17985927C211yesmaximal, characteristic, complemented
6#17985915C211yesmaximal, characteristic, complemented
7#17985913C211yesmaximal, characteristic, complemented
8#17985892C211yesmaximal, characteristic, complemented
9#17985889C211yesmaximal, characteristic, complemented
10#17985878C211yesmaximal, characteristic, complemented
11#17985875C211yesmaximal, characteristic, complemented
12#17985870C211yesmaximal, characteristic, complemented
13#17985856C211yesmaximal, characteristic, complemented
14#17985854C211yesmaximal, characteristic, complemented
15C2×#123×C71C211yesmaximal, characteristic, complemented
16C24×D71C2211yescharacteristic
17C22×C22⋊D71C2211yescharacteristic
18C22×C22⋊D71C2211yescharacteristic
19C22×C4×D71C2211yescharacteristic
20C22×C4×D71C2211yescharacteristic
21#14556096—24no—
22C4×C22⋊D71—24no—
23C2×C4⋊Dic71C2211yescharacteristic, complemented
24C2×Dic71⋊C4C2211yescharacteristic, complemented
25C2×Dic71⋊C4C2211yescharacteristic, complemented
26C2×C22⋊Dic71C2211yescharacteristic
27C2×C22⋊Dic71C2211yescharacteristic
28C2×C22⋊Dic71C2211yescharacteristic
29C2×C22⋊Dic71C2211yescharacteristic, complemented
30C2×C22⋊Dic71C2211yescharacteristic, complemented
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
38C2×(C2×D71)⋊C4C2211yescharacteristic
39C22⋊C4×D71—24no—
40C2×C4×Dic71C2211yescharacteristic, complemented
41C71⋊#95C2211yescharacteristic, complemented
42C71⋊#95C2211yescharacteristic, complemented
43C71⋊#95C2211yescharacteristic, complemented
44C71⋊#95C2211yescharacteristic, complemented
45C71⋊#95C2211yescharacteristic, complemented
46C71⋊#95C2211yescharacteristic, complemented
47C22×D4×C71C2211yescharacteristic
48C23×C4×C71C2211yescharacteristic, complemented
49#123×C71—24no—
50C2×C4⋊C4×C71C2211yescharacteristic, complemented
51C2×C4⋊C4×C71C2211yescharacteristic, complemented
52C2×C22⋊C4×C71C2211yescharacteristic
53C2×C22⋊C4×C71C2211yescharacteristic
54C2×C22⋊C4×C71C2211yescharacteristic, complemented
55C23×D71—22no—
56C23×D71—22no—
57C23×D71—22no—
58C23×D71—22no—
59C23×D71—22no—
60C23×D71—41no—
61C23×D71—41no—
62C23×D71C2311yescharacteristic
63C23×D71C2311yescharacteristic
64C2×C22⋊D71—22no—
65C2×C22⋊D71—22no—
66C2×C22⋊D71—22no—
67C2×C22⋊D71—22no—
68C2×C22⋊D71—41no—
69C2×C22⋊D71D414yes—
70C22×Dic71C2311yescharacteristic
71C22×Dic71C2311yescharacteristic
72C22×Dic71C2311yescharacteristic
73C22×Dic71C2311yescharacteristic
74C22×Dic71C2311yescharacteristic
75C22×Dic71C2311yescharacteristic
76C2×C4×D71—22no—
77C2×C4×D71—22no—
78C2×C4×D71—22no—
79C2×C4×D71—22no—
80C2×C4×D71—41no—
81C2×C4×D71—41no—
82C4⋊Dic71—41no—
83Dic71⋊C4—22no—
84Dic71⋊C4—41no—
85C22⋊Dic71—22no—
86C22⋊Dic71—22no—
87C22⋊Dic71—22no—
88C22⋊Dic71—41no—
89C22⋊Dic71—41no—
90(C2×D71)⋊C4—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—41no—
98C4×Dic71—22no—
99C24×C71C2311yescharacteristic
100C24×C71C2311yescharacteristic
101C2×D4×C71—22no—
102C2×D4×C71—22no—
103C2×D4×C71—41no—
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
112C4⋊C4×C71—22no—
113C4⋊C4×C71—22no—
114C22⋊C4×C71—22no—
115C22⋊C4×C71—22no—
116C22⋊C4×C71D414yescomplemented
117C23×C71—21no—
118C23×C71—21no—
119C23×C71—21no—
120C23×C71—21no—
121C23×C71—21no—
122C23×C71—21no—
123C23×C71—21no—
124C23×C71—21no—
125C23×C71—21no—
126C23×C71—21no—
127C23×C71—21no—
128C23×C71—21no—
129C23×C71C2×D412yes—
130C23×C71C2×D412yes—
131C23×C71C2411yescharacteristic
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—44no—
144C22×D71—44no—
145C22×D71—44no—
146C22×D71—44no—
147C22×D71—44no—
148C22×D71—44no—
149C22×D71C2×D412yes—
150C22×D71#4112yes—
151C22×D71#4112yes—
152C22×D71#4112yes—
153D4×C71—44no—
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—44no—
168C2×C4×C71C2×D412yes—
169C2×C4×C71C2×D412yes—
170C22⋊D71—28no—
171C22⋊D71—44no—
172C2×Dic71—21no—
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—21no—
188C2×Dic71—21no—
189C2×Dic71C2×D412yes—
190C4×D71—44no—
191C4×D71—44no—
192C22×C71—22no—
193C22×C71—22no—
194C22×C71—22no—
195C22×C71—22no—
196C22×C71—22no—
197C22×C71—22no—
198C22×C71—41no—
199C22×C71—41no—
200C22×C71—41no—
201C22×C71—41no—
202C22×C71—41no—
203C22×C71—41no—
204C22×C71—41no—
205C22×C71—41no—
206C22×C71C22×D411yescharacteristic
207C22×C71C22×D411yescharacteristic
208C22×C71C2×#4111yescharacteristic
209C22×C71C2×#4111yescharacteristic
210C22×C71C2×#4111yescharacteristic
211C22×C71H(5,𝔽2)11yescharacteristic
212C22×C71H(5,𝔽2)11yescharacteristic
213C2×D71—22no—
214C2×D71—22no—
215C2×D71—22no—
216C2×D71—22no—
217C2×D71—22no—
218C2×D71—22no—
219C2×D71—22no—
220C2×D71—22no—
221C2×D71—22no—
222C2×D71—416no—
223C2×D71—41no—
224C2×D71—41no—
225C2×D71—41no—
226C2×D71—41no—
227C2×D71#12314yescomplemented
228C4×C71—22no—
229C4×C71—22no—
230C4×C71—41no—
231C4×C71—41no—
232C4×C71—41no—
233Dic71—22no—
234Dic71—41no—
235Dic71—41no—
236Dic71—41no—
237Dic71—41no—
238Dic71—41no—
239C2×C71—42no—
240C2×C71—42no—
241C2×C71C2×#12311yescharacteristic
242C2×C71#52511yescharacteristic
243C2×C71#53811yescharacteristic
244C2×C71D4211yescharacteristic
245C2×C71#54611yescharacteristic
246C2×C71#54611yescharacteristic
247C2×C71#55111yescharacteristic
248D71—24no—
249D71—42no—
250#2418—711nomaximal, Sylow
251C71#241811yescharacteristic, complemented, Sylow
252C2×#123D7111yescharacteristic, complemented
253C2×#120—711no—
254C2×C4×D4—711no—
255C22×C22⋊C4—711no—
256#399—711no—
257#397—711no—
258#394—711no—
259#394—711no—
260#388—711no—
261#388—711no—
262#386—711no—
263#386—711no—
264#385—711no—
265#380—711no—
266#379—711no—
267C25—711no—
268C22×D4—711no—
269C22×D4—711no—
270C22×D4C2×D7111yescharacteristic
271C23×C4—711no—
272C23×C4—711no—
273C23×C4C2×D7111yescharacteristic, complemented
274#123—24no—
275#120—1424no—
276C4×D4—1424no—
277C2×C4⋊C4—711no—
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⋊C4C2×D7111yescharacteristic
297C2×C22⋊C4C2×D7111yescharacteristic
298C2×C22⋊C4C2×D7111yescharacteristic, complemented
299C2×C42—711no—
300#95—711no—
301#95—711no—
302#95—711no—
303#95—711no—
304#95—711no—
305#95—711no—
306C24—2841no—
307C24—2841no—
308C24—711no—
309C24—711no—
310C24—1422no—
311C24—1422no—
312C24—1422no—
313C24—1422no—
314C24—1422no—
315C24C22×D7111yescharacteristic
316C24C22×D7111yescharacteristic
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—41no—
320C2×D4—2841no—
321C2×D4—714no—
322C2×D4—1422no—
323C2×D4—1422no—
324C2×D4—1422no—
325C2×D4—1422no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—41no—
329C22×C4—2841no—
330C22×C4—2841no—
331C22×C4—711no—
332C22×C4—711no—
333C22×C4—711no—
334C22×C4—711no—
335C22×C4—711no—
336C22×C4—711no—
337C22×C4—1422no—
338C22×C4—1422no—
339C22×C4—1422no—
340C22×C4—1422no—
341C22×C4C22×D7111yescharacteristic
342C22×C4C22×D7111yescharacteristic
343C22×C4C22×D7111yescharacteristic
344C22×C4C22×D7111yescharacteristic
345C22×C4C22×D7111yescharacteristic
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—2841no—
349C4⋊C4—2841no—
350C4⋊C4—1422no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—2841no—
354C22⋊C4—2841no—
355C22⋊C4—2841no—
356C22⋊C4—1422no—
357C22⋊C4—1422no—
358C22⋊C4—1422no—
359C22⋊C4—1422no—
360C22⋊C4—1422no—
361C22⋊C4—1422no—
362C22⋊C4—1422no—
363C22⋊C4—1422no—
364C22⋊C4—1422no—
365C22⋊C4—1422no—
366C22⋊C4C22⋊D7114yescomplemented
367C42—1422no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—2844no—
381C23—2844no—
382C23—2844no—
383C23—2844no—
384C23—2844no—
385C23—2844no—
386C23—712no—
387C23—712no—
388C23—712no—
389C23—712no—
390C23—1428no—
391C23—1421no—
392C23—1421no—
393C23—1421no—
394C23—1421no—
395C23—1421no—
396C23—1421no—
397C23—1421no—
398C23—1421no—
399C23—1421no—
400C23—1421no—
401C23D4×D7112yes—
402C23C2×C22⋊D7112yes—
403C23C23×D7111yescharacteristic
404D4—44no—
405D4—2844no—
406D4—1428no—
407C2×C4—21no—
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—44no—
421C2×C4—2844no—
422C2×C4—2844no—
423C2×C4—712no—
424C2×C4—1421no—
425C2×C4—1421no—
426C2×C4—1421no—
427C2×C4—1421no—
428C2×C4—1421no—
429C2×C4—1421no—
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×C4C2×C22⋊D7112yes—
442C2×C4C2×C22⋊D7112yes—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—28416no—
458C22—2841no—
459C22—2841no—
460C22—2841no—
461C22—2841no—
462C22—714no—
463C22—1422no—
464C22—1422no—
465C22—1422no—
466C22—1422no—
467C22—1422no—
468C22—1422no—
469C22—1422no—
470C22—1422no—
471C22—1422no—
472C22C2×D4×D7111yescharacteristic
473C22C22×C22⋊D7111yescharacteristic
474C22#41×D7111yescharacteristic
475C22#41×D7111yescharacteristic
476C22#41×D7111yescharacteristic
477C22#1455617311yescharacteristic
478C22#1455617411yescharacteristic
479C4—22no—
480C4—22no—
481C4—41no—
482C4—41no—
483C4—41no—
484C4—2841no—
485C4—2841no—
486C4—2841no—
487C4—2841no—
488C4—2841no—
489C4—1422no—
490C2—42no—
491C2—42no—
492C2—2842no—
493C2—1424no—
494C2#123×D7111yescharacteristic
495C2#1798650611yescharacteristic
496C2#1798658511yescharacteristic
497C2D4×C22⋊D7111yescharacteristic
498C2#1798663711yescharacteristic
499C2#1798664111yescharacteristic
500C2#1798667711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
FittingC2×#123×C71C2
hypercenterC2×#123D71
hypocenterC71#2418
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D71C2semidirect
3. C2×C22⋊C4×D71C2semidirect
4. #17985934C2semidirect
5. #17985927C2semidirect
6. #17985915C2semidirect
7. #17985913C2semidirect
8. #17985892C2semidirect
9. #17985889C2semidirect
10. #17985878C2semidirect
11. #17985875C2semidirect
12. #17985870C2semidirect
13. #17985856C2semidirect
14. #17985854C2semidirect
15. C2×#123×C71C2semidirect
23. C2×C4⋊Dic71C22semidirect
24. C2×Dic71⋊C4C22semidirect
25. C2×Dic71⋊C4C22semidirect
29. C2×C22⋊Dic71C22semidirect
30. C2×C22⋊Dic71C22semidirect
40. C2×C4×Dic71C22semidirect
41. C71⋊#95C22semidirect
42. C71⋊#95C22semidirect
43. C71⋊#95C22semidirect
44. C71⋊#95C22semidirect
45. C71⋊#95C22semidirect
46. C71⋊#95C22semidirect
48. C23×C4×C71C22semidirect
50. C2×C4⋊C4×C71C22semidirect
51. C2×C4⋊C4×C71C22semidirect
54. C2×C22⋊C4×C71C22semidirect
116. C22⋊C4×C71D4semidirect
227. C2×D71#123semidirect
251. C71#2418semidirect
252. C2×#123D71semidirect
273. C23×C4C2×D71semidirect
280. C2×C4⋊C4C2×D71semidirect
281. C2×C4⋊C4C2×D71semidirect
298. C2×C22⋊C4C2×D71semidirect
366. C22⋊C4C22⋊D71semidirect