Finite Group Databasebeta

← #22591800

Subgroups — #22591800

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×#14556112C211yesmaximal, characteristic, complemented
2C2×#14556096C211yesmaximal, characteristic
3C2×D4×Dic71C211yesmaximal, characteristic, complemented
4C22×(C2×D71)⋊C4C211yesmaximal, characteristic, complemented
5#17985936C211yesmaximal, characteristic, complemented
6#17985926C211yesmaximal, characteristic, complemented
7#17985915C211yesmaximal, characteristic, complemented
8#17985915C211yesmaximal, characteristic, complemented
9#17985891C211yesmaximal, characteristic, complemented
10#17985891C211yesmaximal, characteristic, complemented
11#17985876C211yesmaximal, characteristic, complemented
12#17985874C211yesmaximal, characteristic, complemented
13#17985859C211yesmaximal, characteristic, complemented
14#17985854C211yesmaximal, characteristic, complemented
15#386×C71C211yesmaximal, characteristic, complemented
16C24×D71C2211yescharacteristic
17C22×C22⋊D71C2211yescharacteristic
18C22×C22⋊D71C2211yescharacteristic
19C23×Dic71C2211yescharacteristic
20C23×Dic71C2211yescharacteristic, complemented
21#14556112—24no—
22#14556096—24no—
23D4×Dic71—24no—
24C2×C4⋊Dic71C2211yescharacteristic, complemented
25C2×C4⋊Dic71C2211yescharacteristic, complemented
26C2×C4⋊Dic71C2211yescharacteristic, complemented
27C2×C4⋊Dic71C2211yescharacteristic, complemented
28C2×C4⋊Dic71C2211yescharacteristic, complemented
29C2×C22⋊Dic71C2211yescharacteristic
30C2×C22⋊Dic71C2211yescharacteristic
31C2×C22⋊Dic71C2211yescharacteristic, complemented
32C2×(C2×D71)⋊C4—24no—
33C2×(C2×D71)⋊C4C2211yescharacteristic
34C2×(C2×D71)⋊C4C2211yescharacteristic
35C2×(C2×D71)⋊C4C2211yescharacteristic
36C2×(C2×D71)⋊C4C2211yescharacteristic
37C2×(C2×D71)⋊C4C2211yescharacteristic
38C2×(C2×D71)⋊C4C2211yescharacteristic
39C2×(C2×D71)⋊C4C2211yescharacteristic
40C2×(C2×D71)⋊C4C2211yescharacteristic
41C2×(C2×D71)⋊C4C2211yescharacteristic
42C2×(C2×D71)⋊C4C2211yescharacteristic
43C2×C4×Dic71C2211yescharacteristic, complemented
44C71⋊#95C2211yescharacteristic, complemented
45C71⋊#95C2211yescharacteristic, complemented
46C71⋊#95C2211yescharacteristic, complemented
47C71⋊#95C2211yescharacteristic, complemented
48C22×D4×C71C2211yescharacteristic
49C23×C4×C71C2211yescharacteristic
50C2×C22⋊C4×C71C2211yescharacteristic
51C2×C22⋊C4×C71C2211yescharacteristic, complemented
52C2×C22⋊C4×C71C2211yescharacteristic, complemented
53#95×C71C2211yescharacteristic, complemented
54#95×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⋊D71—41no—
70C22×Dic71—22no—
71C22×Dic71—22no—
72C22×Dic71—22no—
73C22×Dic71—22no—
74C22×Dic71—41no—
75C22×Dic71—41no—
76C22×Dic71C2311yescharacteristic
77C22×Dic71C2311yescharacteristic
78C22×Dic71C2311yescharacteristic
79C22×Dic71C2311yescharacteristic
80C22×Dic71C2311yescharacteristic
81C22×Dic71C2311yescharacteristic
82C4⋊Dic71—22no—
83C4⋊Dic71—22no—
84C4⋊Dic71—22no—
85C4⋊Dic71—41no—
86C4⋊Dic71—41no—
87C22⋊Dic71—22no—
88C22⋊Dic71—22no—
89C22⋊Dic71—22no—
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—
98(C2×D71)⋊C4—41no—
99(C2×D71)⋊C4D414yescomplemented
100C4×Dic71—22no—
101C24×C71C2311yescharacteristic
102C24×C71C2311yescharacteristic
103C2×D4×C71—22no—
104C2×D4×C71—22no—
105C2×D4×C71D414yes—
106C22×C4×C71—22no—
107C22×C4×C71—22no—
108C22×C4×C71—41no—
109C22×C4×C71C2311yescharacteristic
110C22×C4×C71C2311yescharacteristic
111C22×C4×C71C2311yescharacteristic
112C22×C4×C71C2311yescharacteristic
113C22×C4×C71C2311yescharacteristic
114C22⋊C4×C71—22no—
115C22⋊C4×C71—22no—
116C22⋊C4×C71—41no—
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×C71C2×D412yes—
128C23×C71C2×D412yes—
129C23×C71#4112yes—
130C23×C71#4112yes—
131C23×C71C2411yescharacteristic
132C22×D71—28no—
133C22×D71—21no—
134C22×D71—21no—
135C22×D71—21no—
136C22×D71—21no—
137C22×D71—21no—
138C22×D71—21no—
139C22×D71—21no—
140C22×D71—21no—
141C22×D71—21no—
142C22×D71—21no—
143C22×D71—21no—
144C22×D71—21no—
145C22×D71—44no—
146C22×D71—44no—
147C22×D71—44no—
148C22×D71—44no—
149C22×D71—44no—
150C22×D71—44no—
151C22×D71C2×D412yes—
152C22×D71#4112yes—
153D4×C71—28no—
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—44no—
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×Dic71—44no—
190C2×Dic71—44no—
191C2×Dic71C2×D412yes—
192C22×C71—22no—
193C22×C71—22no—
194C22×C71—22no—
195C22×C71—22no—
196C22×C71—22no—
197C22×C71—22no—
198C22×C71—22no—
199C22×C71—22no—
200C22×C71—22no—
201C22×C71—41no—
202C22×C71—41no—
203C22×C71—41no—
204C22×C71—41no—
205C22×C71#12314yescomplemented
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—416no—
220C2×D71—41no—
221C2×D71—41no—
222C2×D71—41no—
223C2×D71—41no—
224C2×D71—41no—
225C2×D71—41no—
226C2×D71—41no—
227C2×D71—41no—
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—24no—
240C2×C71—42no—
241C2×C71C2×#12311yescharacteristic
242C2×C71#52511yescharacteristic
243C2×C71#53811yescharacteristic
244C2×C71D4211yescharacteristic
245C2×C71#54611yescharacteristic
246C2×C71#54611yescharacteristic
247C2×C71#55111yescharacteristic
248D71—42no—
249D71—42no—
250#2418—711nomaximal, Sylow
251C71#241811yescharacteristic, complemented, Sylow
252C2×#123—711no—
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#386D7111yescharacteristic, complemented
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
274#123—1424no—
275#120—1424no—
276C4×D4—1424no—
277C2×C4⋊C4—711no—
278C2×C4⋊C4—711no—
279C2×C4⋊C4—711no—
280C2×C4⋊C4—711no—
281C2×C4⋊C4—711no—
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, complemented
298C2×C22⋊C4C2×D7111yescharacteristic, complemented
299C2×C42—711no—
300#95—711no—
301#95—711no—
302#95—711no—
303#95—711no—
304#95C2×D7111yescharacteristic, complemented
305#95C2×D7111yescharacteristic, complemented
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—2841no—
320C2×D4—2841no—
321C2×D4—1422no—
322C2×D4—1422no—
323C2×D4—1422no—
324C2×D4—1422no—
325C2×D4D28414yes—
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—2841no—
347C4⋊C4—2841no—
348C4⋊C4—1422no—
349C4⋊C4—1422no—
350C4⋊C4—1422no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—41no—
354C22⋊C4—2841no—
355C22⋊C4—2841no—
356C22⋊C4—714no—
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⋊C4—1422no—
367C42—1422no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—2844no—
379C23—2844no—
380C23—2844no—
381C23—2844no—
382C23—2844no—
383C23—2844no—
384C23—712no—
385C23—712no—
386C23—1428no—
387C23—1421no—
388C23—1421no—
389C23—1421no—
390C23—1421no—
391C23—1421no—
392C23—1421no—
393C23—1421no—
394C23—1421no—
395C23—1421no—
396C23—1421no—
397C23—1421no—
398C23—1421no—
399C23C2×D28412yes—
400C23C2×D28412yes—
401C23#1176586412yes—
402C23#1176586412yes—
403C23C23×D7111yescharacteristic
404D4—28no—
405D4—2844no—
406D4—2844no—
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×C4D4×D7112yes—
442C2×C4C2×D28412yes—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—28416no—
457C22—2841no—
458C22—2841no—
459C22—2841no—
460C22—2841no—
461C22—2841no—
462C22—2841no—
463C22—2841no—
464C22—2841no—
465C22—1422no—
466C22—1422no—
467C22—1422no—
468C22—1422no—
469C22—1422no—
470C22—1422no—
471C22#1455611214yescomplemented
472C22C2×D4×D7111yescharacteristic
473C22C22×D28411yescharacteristic
474C22#41×D7111yescharacteristic
475C22C2×#1176586411yescharacteristic
476C22C2×#1176586411yescharacteristic
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—24no—
491C2—42no—
492C2—2842no—
493C2—2842no—
494C2C2×#1455611211yescharacteristic
495C2#1798650911yescharacteristic
496C2#1798658611yescharacteristic
497C2D4×D28411yescharacteristic
498C2#1798663611yescharacteristic
499C2#1798664411yescharacteristic
500C2#1798667211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#386×C71C2
hypercenter#386D71
hypocenterC71#2418
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#14556112C2semidirect
3. C2×D4×Dic71C2semidirect
4. C22×(C2×D71)⋊C4C2semidirect
5. #17985936C2semidirect
6. #17985926C2semidirect
7. #17985915C2semidirect
8. #17985915C2semidirect
9. #17985891C2semidirect
10. #17985891C2semidirect
11. #17985876C2semidirect
12. #17985874C2semidirect
13. #17985859C2semidirect
14. #17985854C2semidirect
15. #386×C71C2semidirect
20. C23×Dic71C22semidirect
24. C2×C4⋊Dic71C22semidirect
25. C2×C4⋊Dic71C22semidirect
26. C2×C4⋊Dic71C22semidirect
27. C2×C4⋊Dic71C22semidirect
28. C2×C4⋊Dic71C22semidirect
31. C2×C22⋊Dic71C22semidirect
43. C2×C4×Dic71C22semidirect
44. C71⋊#95C22semidirect
45. C71⋊#95C22semidirect
46. C71⋊#95C22semidirect
47. C71⋊#95C22semidirect
51. C2×C22⋊C4×C71C22semidirect
52. C2×C22⋊C4×C71C22semidirect
53. #95×C71C22semidirect
54. #95×C71C22semidirect
99. (C2×D71)⋊C4D4semidirect
205. C22×C71#123semidirect
251. C71#2418semidirect
263. #386D71semidirect
297. C2×C22⋊C4C2×D71semidirect
298. C2×C22⋊C4C2×D71semidirect
304. #95C2×D71semidirect
305. #95C2×D71semidirect
471. C22#14556112semidirect