Finite Group Databasebeta

← #11736944

Subgroups — #11736944

Order 896.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90246C211yesmaximal, characteristic, complemented
2#90241C211yesmaximal, characteristic, complemented
3C7×#435C211yesmaximal, characteristic, complemented
4#89907C211yesmaximal, characteristic, complemented
5#89905C211yesmaximal, characteristic, complemented
6#89886C211yesmaximal, characteristic, complemented
7C2×#6395C211yesmaximal, characteristic, complemented
8#89616C211yesmaximal, characteristic, complemented
9#89614C211yesmaximal, characteristic, complemented
10#89507C211yesmaximal, characteristic, complemented
11#89506C211yesmaximal, characteristic, complemented
12#89504C211yesmaximal, characteristic, complemented
13#89493C211yesmaximal, characteristic, complemented
14#89486C211yesmaximal, characteristic, complemented
15#89483C211yesmaximal, characteristic, complemented
16C2×#1159—21no—
17C2×#1157C2211yescharacteristic
18C22×D28C2211yescharacteristic
19C2×C7×M16C2211yescharacteristic, complemented
20C22×C7×C8C2211yescharacteristic
21C4×C7×D4C2211yescharacteristic
22#6500—21no—
23#6493C2211yescharacteristic
24#6493C2211yescharacteristic
25#6493C2211yescharacteristic, complemented
26C2×C4⋊Dic7C2211yescharacteristic
27C2×C4⋊Dic7C2211yescharacteristic, complemented
28#6487C2211yescharacteristic
29C2×D56C2211yescharacteristic
30C2×D56C2211yescharacteristic
31C2×#1132C2211yescharacteristic, complemented
32#6460—21no—
33#6457—21no—
34#6449—21no—
35#6449—21no—
36#6447—21no—
37D7×C22⋊C4—21no—
38#6438C2211yescharacteristic
39C4×D28C2211yescharacteristic
40C4×D28C2211yescharacteristic, complemented
41#6432C2211yescharacteristic, complemented
42C7×C4⋊C8C2211yescharacteristic, complemented
43C7×C22⋊C8C2211yescharacteristic
44C7×C22⋊C8C2211yescharacteristic, complemented
45C7×#97C2211yescharacteristic, complemented
46#6395C2211yescharacteristic
47#6395C2211yescharacteristic
48#6395C2211yescharacteristic
49#6395C2211yescharacteristic
50#6395C2211yescharacteristic
51#6395C2211yescharacteristic
52#6395C2211yescharacteristic, complemented
53#6391C2211yescharacteristic, complemented
54#6391C2211yescharacteristic, complemented
55#6391C2211yescharacteristic, complemented
56#6390C2211yescharacteristic, complemented
57#6390C2211yescharacteristic, complemented
58#6390C2211yescharacteristic, complemented
59#3306—71nomaximal, Sylow
60C23×D7—21no—
61C2×C7×D4D411yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C2×C22⋊D7—21no—
65C2×C22⋊D7—21no—
66C2×C22⋊D7—21no—
67C2×C22⋊D7—21no—
68C22×Dic7—21no—
69C22×Dic7—21no—
70#1159—44no—
71#1157—22no—
72#1157—41no—
73C2×D28—22no—
74C2×D28—21no—
75C2×D28—41no—
76C2×D28C2311yescharacteristic
77C2×D28C2311yescharacteristic
78C2×D28C2311yescharacteristic
79C2×C4×D7—21no—
80C2×C4×D7—21no—
81C2×C4×D7—21no—
82C2×Dic14—21no—
83C2×Dic14C2311yescharacteristic
84C7×M16—41no—
85C2×C7×C8—41no—
86C2×C7×C8C2311yescharacteristic
87C2×C7×C8C2311yescharacteristic
88C2×C7×C8C2311yescharacteristic
89C2×C7×C8C2311yescharacteristic
90C7×C4⋊C4D411yescharacteristic
91C7×C22⋊C4D411yescharacteristic
92C7×C22⋊C4D411yescharacteristic
93C42×C7C2311yescharacteristic
94C22⋊Dic7—21no—
95(C2×D7)⋊C4—21no—
96(C2×D7)⋊C4—21no—
97(C2×D7)⋊C4—21no—
98(C2×D7)⋊C4—21no—
99(C2×D7)⋊C4—21no—
100(C2×D7)⋊C4—21no—
101(C2×D7)⋊C4—21no—
102C4⋊Dic7—21no—
103C4⋊Dic7—21no—
104C4⋊Dic7—21no—
105C4⋊Dic7C2311yescharacteristic
106C4⋊Dic7C2311yescharacteristic
107C4⋊Dic7C2311yescharacteristic
108C4⋊Dic7C2311yescharacteristic
109Dic7⋊C4—21no—
110D56—22no—
111D56—41no—
112#1132—41no—
113#547—71no—
114#546—71no—
115#489—71no—
116#483—71no—
117#481—71no—
118#476—71no—
119#469—71no—
120#465—71no—
121#463—71no—
122#449—71no—
123#447—71no—
124#442—71no—
125#435D711yescharacteristic, complemented
126#417—71no—
127C2×#102—71no—
128C23×C7C2×D411yescharacteristic
129C23×C7C2×D411yescharacteristic
130C22×D7—21no—
131C22×D7—21no—
132C22×D7—21no—
133C22×D7—42no—
134C22×D7—42no—
135C22×D7—81no—
136C7×D4—22no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7C2×D411yescharacteristic
142C2×C4×C7C2×D411yescharacteristic
143C2×C4×C7C2×D411yescharacteristic
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2411yescharacteristic
146C22⋊D7—42no—
147C22⋊D7—42no—
148C22⋊D7—42no—
149C22⋊D7—81no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7—21no—
153C2×Dic7—21no—
154C2×Dic7—21no—
155C2×Dic7—42no—
156C2×Dic7—42no—
157D28—21no—
158D28—21no—
159D28—21no—
160D28—21no—
161D28—21no—
162D28—42no—
163D28#4112yescomplemented
164C4×D7—42no—
165C4×D7—42no—
166C4×D7—81no—
167Dic14—21no—
168Dic14—21no—
169Dic14—42no—
170C7×C8—21no—
171C7×C8—21no—
172C7×C8—21no—
173C7×C8—21no—
174C2×#41—71no—
175C2×#41—141no—
176C22×D4—71no—
177C2×SD16—71no—
178C2×D8—71no—
179C2×D8—71no—
180C2×M16C2×D711yescharacteristic, complemented
181C22×C8C2×D711yescharacteristic
182C4⋊Q8—71no—
183#124—71no—
184#123—141no—
185#123—141no—
186#123—141no—
187C22⋊Q8—71no—
188C22⋊Q8—141no—
189#121—71no—
190#121—71no—
191#121—71no—
192#121—141no—
193#120—141no—
194C4×D4—71no—
195C4×D4—71no—
196C4×D4C2×D711yescharacteristic
197C2×C4⋊C4—71no—
198C2×C4⋊C4—71no—
199C2×C22⋊C4—141no—
200#106—71no—
201#106—71no—
202#106—71no—
203C4⋊C8C2×D711yescharacteristic, complemented
204#103—71no—
205#103—71no—
206#103—71no—
207#102—71no—
208#102—71no—
209#102—71no—
210#102—71no—
211#102—71no—
212#102—71no—
213#102—71no—
214C22⋊C8C2×D711yescharacteristic
215C22⋊C8C2×D711yescharacteristic, complemented
216#97C2×D711yescharacteristic, complemented
217C22×C7—21no—
218C22×C7—21no—
219C22×C7—21no—
220C22×C7—21no—
221C22×C7—21no—
222C22×C7AGL(1,ℤ/8)12yes—
223C22×C7C22×D411yescharacteristic
224C2×D7—22no—
225C2×D7—41no—
226C2×D7—41no—
227C2×D7—41no—
228C2×D7—41no—
229C2×D7—41no—
230C2×D7—41no—
231C2×D7—41no—
232C2×D7—41no—
233C2×D7—84no—
234C4×C7—21no—
235C4×C7—21no—
236C4×C7—21no—
237C4×C7—21no—
238C4×C7C2×#4111yescharacteristic
239C4×C7H(5,𝔽2)11yescharacteristic
240Dic7—41no—
241Dic7—41no—
242Dic7—41no—
243Dic7—41no—
244Dic7—41no—
245C24—141no—
246#41—142no—
247#41—281no—
248#41—284no—
249C2×Q8—71no—
250C2×Q8—141no—
251C2×D4—71no—
252C2×D4—71no—
253C2×D4—71no—
254C2×D4—142no—
255C2×D4—141no—
256C2×D4—141no—
257C2×D4—141no—
258C2×D4—141no—
259C2×D4—141no—
260C2×D4—281no—
261C2×D4D2811yescharacteristic
262C22×C4—141no—
263C22×C4—141no—
264C22×C4—141no—
265C22×C4—141no—
266C22×C4—141no—
267C22×C4C22×D711yescharacteristic
268C22×C4C22×D711yescharacteristic
269SD16—281no—
270D8—142no—
271D8—281no—
272M16—41no—
273C2×C8—41no—
274C2×C8C22×D711yescharacteristic
275C2×C8C22×D711yescharacteristic
276C2×C8C22×D711yescharacteristic
277C2×C8C22×D711yescharacteristic
278C4⋊C4—71no—
279C4⋊C4—71no—
280C4⋊C4—71no—
281C4⋊C4—71no—
282C4⋊C4—141no—
283C4⋊C4—141no—
284C4⋊C4—141no—
285C4⋊C4—141no—
286C4⋊C4D2811yescharacteristic
287C22⋊C4—141no—
288C22⋊C4—141no—
289C22⋊C4—141no—
290C22⋊C4—141no—
291C22⋊C4—141no—
292C22⋊C4—141no—
293C22⋊C4—141no—
294C22⋊C4—141no—
295C22⋊C4D2811yescharacteristic
296C22⋊C4D2811yescharacteristic
297C42C22×D711yescharacteristic
298C2×C7—22no—
299C2×C7—41no—
300C2×C7#54611yescharacteristic
301C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
302C2×C7#57711yescharacteristic
303D7—42no—
304D7—81no—
305D7—81no—
306C23—141no—
307C23—141no—
308C23—141no—
309C23—282no—
310C23—282no—
311C23—561no—
312C23C2×D2811yescharacteristic
313C23C2×D2811yescharacteristic
314Q8—141no—
315Q8—141no—
316Q8—282no—
317D4—22no—
318D4—72no—
319D4—141no—
320D4—141no—
321D4—141no—
322D4—141no—
323D4—141no—
324D4—282no—
325D4—282no—
326D4—282no—
327D4—282no—
328D4—561no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—141no—
334C2×C4—141no—
335C2×C4—141no—
336C2×C4—141no—
337C2×C4—141no—
338C2×C4—282no—
339C2×C4—282no—
340C2×C4—282no—
341C2×C4—282no—
342C2×C4—561no—
343C2×C4C2×D2811yescharacteristic
344C2×C4C2×D2811yescharacteristic
345C2×C4C2×D2811yescharacteristic
346C2×C4C2×D2811yescharacteristic
347C2×C4C23×D711yescharacteristic
348C8—21no—
349C8—21no—
350C8—21no—
351C8—21no—
352C7#330611yescharacteristic, complemented, Sylow
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—142no—
359C22—281no—
360C22—281no—
361C22—281no—
362C22—281no—
363C22—281no—
364C22—281no—
365C22—281no—
366C22—281no—
367C22—564no—
368C22#647112yes—
369C22C22×D2811yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—281no—
375C4—281no—
376C4—281no—
377C4—281no—
378C4—281no—
379C4#654811yescharacteristic
380C4D7×#4111yescharacteristic
381C2—22no—
382C2—41no—
383C2—282no—
384C2—561no—
385C2—561no—
386C2#9024511yescharacteristic
387C2C2×#647111yescharacteristic
388C2#9044211yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#435C2
socleC22×C7C22×D4
hypercenter#435D7
hypocenterC7#3306
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #90246C2semidirect
2. #90241C2semidirect
3. C7×#435C2semidirect
4. #89907C2semidirect
5. #89905C2semidirect
6. #89886C2semidirect
7. C2×#6395C2semidirect
8. #89616C2semidirect
9. #89614C2semidirect
10. #89507C2semidirect
11. #89506C2semidirect
12. #89504C2semidirect
13. #89493C2semidirect
14. #89486C2semidirect
15. #89483C2semidirect
19. C2×C7×M16C22semidirect
25. #6493C22semidirect
27. C2×C4⋊Dic7C22semidirect
31. C2×#1132C22semidirect
40. C4×D28C22semidirect
41. #6432C22semidirect
42. C7×C4⋊C8C22semidirect
44. C7×C22⋊C8C22semidirect
45. C7×#97C22semidirect
52. #6395C22semidirect
53. #6391C22semidirect
54. #6391C22semidirect
55. #6391C22semidirect
56. #6390C22semidirect
57. #6390C22semidirect
58. #6390C22semidirect
125. #435D7semidirect
163. D28#41semidirect
180. C2×M16C2×D7semidirect
203. C4⋊C8C2×D7semidirect
215. C22⋊C8C2×D7semidirect
216. #97C2×D7semidirect
352. C7#3306semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#902462
#902412
C7×#4352
#899072
#899052
#898862
C2×#63952
#896162
#896142
#895072
#895062
#895042
#894932
#894862
#894832
#33067