Finite Group Databasebeta

← #11739087

Subgroups — #11739087

Order 896.

410 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90382C211yesmaximal, characteristic, complemented
2D7×#124C211yesmaximal, characteristic, complemented
3C7×#488C211yesmaximal, characteristic, complemented
4#89840C211yesmaximal, characteristic, complemented
5#89831C211yesmaximal, characteristic, complemented
6#89591C211yesmaximal, characteristic, complemented
7#89580C211yesmaximal, characteristic, complemented
8#89578C211yesmaximal, characteristic, complemented
9#89570C211yesmaximal, characteristic, complemented
10#89559C211yesmaximal, characteristic, complemented
11#89548C211yesmaximal, characteristic, complemented
12#89545C211yesmaximal, characteristic
13#89539C211yesmaximal, characteristic, complemented
14#89468C211yesmaximal, characteristic, complemented
15#89457C211yesmaximal, characteristic, complemented
16C2×Q8×D7C2211yescharacteristic
17C2×D4×D7C2211yescharacteristic
18C7×#125C2211yescharacteristic, complemented
19C7×#124C2211yescharacteristic
20#6510C2211yescharacteristic
21C2×#1144C2211yescharacteristic, complemented
22C2×#1143C2211yescharacteristic, complemented
23#6499C2211yescharacteristic
24C2×#1142C2211yescharacteristic
25C2×#1141C2211yescharacteristic
26C2×Dic28C2211yescharacteristic, complemented
27C2×D56C2211yescharacteristic
28C2×#1132C2211yescharacteristic
29C2×#1132C2211yescharacteristic, complemented
30#6461—21no—
31#6460C2211yescharacteristic
32#6458C2211yescharacteristic
33#6457—21no—
34#6456—21no—
35#6456C2211yescharacteristic, complemented
36#6455—21no—
37#6450C2211yescharacteristic, complemented
38#6448—21no—
39#6448—21no—
40D7×C22⋊C4—21no—
41D7×C22⋊C4—21no—
42#6438C2211yescharacteristic
43#6438C2211yescharacteristic, complemented
44C42×D7C2211yescharacteristic
45C7×#103C2211yescharacteristic, complemented
46C7×#103C2211yescharacteristic, complemented
47C7×#102C2211yescharacteristic
48C7×#102C2211yescharacteristic
49C4×C7×C8C2211yescharacteristic, complemented
50(C2×D7)⋊C8C2211yescharacteristic
51(C2×D7)⋊C8C2211yescharacteristic
52Dic7⋊C8C2211yescharacteristic, complemented
53Dic7⋊C8C2211yescharacteristic, complemented
54#6384C2211yescharacteristic, complemented
55#6384C2211yescharacteristic, complemented
56#6383C2211yescharacteristic
57#6383C2211yescharacteristic
58C4×C7⋊C8C2211yescharacteristic, complemented
59#3211—71nomaximal, Sylow
60C23×D7—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4C2311yescharacteristic
63C2×C22⋊D7—21no—
64Q8×D7—41no—
65Q8×D7—41no—
66D4×D7—41no—
67D4×D7—41no—
68C2×D28—21no—
69C2×D28C2311yescharacteristic
70C2×D28C2311yescharacteristic
71C2×C4×D7—21no—
72C2×C4×D7—21no—
73C2×C4×D7D411yescharacteristic
74C2×C4×D7D411yescharacteristic
75C2×C4×D7C2311yescharacteristic
76C2×Dic14—21no—
77C2×Dic14C2311yescharacteristic
78C2×Dic14C2311yescharacteristic
79C2×C7×C8C2311yescharacteristic
80C2×C7×C8C2311yescharacteristic
81C7×C4⋊C4—21no—
82C7×C4⋊C4—21no—
83C7×C4⋊C4C2311yescharacteristic
84C7×C4⋊C4C2311yescharacteristic
85C7×C22⋊C4—21no—
86C7×C22⋊C4—21no—
87C42×C7C2311yescharacteristic
88C22⋊Dic7—21no—
89C22⋊Dic7—21no—
90#1144—41no—
91#1143—41no—
92#1142—41no—
93#1141—41no—
94(C2×D7)⋊C4—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—
102(C2×D7)⋊C4—21no—
103C4⋊Dic7—21no—
104Dic7⋊C4—21no—
105C4×Dic7—21no—
106C4×Dic7D411yescharacteristic
107C4×Dic7D411yescharacteristic
108C4×Dic7C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111Dic28—41no—
112D56—41no—
113#1132—41no—
114#1132—41no—
115#529—71no—
116C2×#124—71no—
117#495—71no—
118#495—71no—
119#488—71no—
120#488D711yescharacteristic, complemented
121#464—71no—
122#464—71no—
123#463—71no—
124#463—71no—
125#451—71no—
126#450—71no—
127#448—71no—
128#447—71no—
129#431—71no—
130C23×C7—21no—
131C22×D7—21no—
132C22×D7—21no—
133C22×D7—42no—
134C22×D7—42no—
135C22×D7—42no—
136C22×D7C2×D411yescharacteristic
137C7×Q8—21no—
138C7×Q8—21no—
139C7×D4—21no—
140C7×D4—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2×D411yescharacteristic
146C2×C4×C7C2411yescharacteristic
147C22⋊D7—42no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7C2×D411yescharacteristic
151C2×Dic7C2×D411yescharacteristic
152C2×Dic7C2×D411yescharacteristic
153D28—21no—
154D28—21no—
155D28—21no—
156D28—21no—
157D28—42no—
158C4×D7—22no—
159C4×D7—21no—
160C4×D7—21no—
161C4×D7—41no—
162C4×D7—42no—
163C4×D7—42no—
164Dic14—21no—
165Dic14—21no—
166Dic14—21no—
167Dic14—21no—
168Dic14—42no—
169C7×C8—21no—
170C7×C8—21no—
171C7⋊C8—21no—
172C7⋊C8—21no—
173C22×Q8—71no—
174C22×D4—71no—
175C2×Q16—71no—
176C2×Q16—71no—
177C2×SD16—71no—
178C2×SD16—71no—
179C2×SD16—71no—
180C2×SD16—71no—
181C2×D8—71no—
182C2×D8—71no—
183#126—141no—
184#125C2×D711yescharacteristic, complemented
185#124—71no—
186#124—71no—
187#124—71no—
188#124—71no—
189#124—141no—
190#124—141no—
191#124C2×D711yescharacteristic
192#123—141no—
193C22⋊Q8—71no—
194#121—71no—
195C4×Q8—71no—
196C4×D4—71no—
197C4×D4—141no—
198#117—141no—
199C2×C22⋊C4—141no—
200C2×C22⋊C4—141no—
201C2×C42—71no—
202C4⋊C8—71no—
203C4⋊C8—71no—
204#103—71no—
205#103—71no—
206#103C2×D711yescharacteristic, complemented
207#103C2×D711yescharacteristic, complemented
208#102—71no—
209#102—71no—
210#102C2×D711yescharacteristic
211#102C2×D711yescharacteristic
212C22⋊C8—71no—
213C22⋊C8—71no—
214C4×C8—71no—
215C4×C8C2×D711yescharacteristic, complemented
216C22×C7—41no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7C22×D411yescharacteristic
220C2×D7—21no—
221C2×D7—21no—
222C2×D7—41no—
223C2×D7—41no—
224C2×D7—41no—
225C2×D7—41no—
226C2×D7—41no—
227C2×D7—41no—
228C2×D7—84no—
229C2×D7#13512yes—
230C4×C7—21no—
231C4×C7—21no—
232C4×C7—41no—
233C4×C7—41no—
234C4×C7—41no—
235C4×C7H(5,𝔽2)11yescharacteristic
236C4×C7H(5,𝔽2)11yescharacteristic
237Dic7—21no—
238Dic7—21no—
239Dic7—21no—
240Dic7—41no—
241Dic7—41no—
242C24—141no—
243C2×Q8—71no—
244C2×Q8—71no—
245C2×Q8—141no—
246C2×Q8—281no—
247C2×Q8—281no—
248C2×Q8C22×D711yescharacteristic
249C2×D4—71no—
250C2×D4—71no—
251C2×D4—141no—
252C2×D4—141no—
253C2×D4—281no—
254C2×D4—281no—
255C2×D4C22×D711yescharacteristic
256C22×C4—71no—
257C22×C4—71no—
258C22×C4—71no—
259C22×C4—141no—
260C22×C4—141no—
261Q16—281no—
262Q16—281no—
263SD16—281no—
264SD16—281no—
265SD16—281no—
266SD16—281no—
267D8—281no—
268D8—281no—
269C2×C8—71no—
270C2×C8—71no—
271C2×C8C22×D711yescharacteristic
272C2×C8C22×D711yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—141no—
276C4⋊C4—141no—
277C4⋊C4C22×D711yescharacteristic
278C4⋊C4C22×D711yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—141no—
282C22⋊C4—141no—
283C22⋊C4—141no—
284C22⋊C4—141no—
285C22⋊C4—141no—
286C22⋊C4—141no—
287C22⋊C4—141no—
288C22⋊C4—141no—
289C22⋊C4—141no—
290C22⋊C4—141no—
291C22⋊C4—141no—
292C42—71no—
293C42—71no—
294C42—71no—
295C42—141no—
296C42C22×D711yescharacteristic
297C2×C7—81no—
298C2×C7#53511yescharacteristic
299C2×C7C2×#13511yescharacteristic
300C2×C7#57511yescharacteristic
301D7—22no—
302D7—81no—
303D7—81no—
304C23—21no—
305C23—71no—
306C23—141no—
307C23—141no—
308C23—282no—
309C23—282no—
310C23—282no—
311Q8—21no—
312Q8—21no—
313Q8—141no—
314Q8—141no—
315Q8—141no—
316Q8—141no—
317Q8—282no—
318D4—21no—
319D4—21no—
320D4—141no—
321D4—141no—
322D4—141no—
323D4—141no—
324D4—282no—
325D4—282no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—71no—
330C2×C4—71no—
331C2×C4—71no—
332C2×C4—142no—
333C2×C4—141no—
334C2×C4—141no—
335C2×C4—141no—
336C2×C4—141no—
337C2×C4—281no—
338C2×C4—282no—
339C2×C4—282no—
340C2×C4D4×D711yescharacteristic
341C2×C4D4×D711yescharacteristic
342C2×C4C23×D711yescharacteristic
343C8—21no—
344C8—21no—
345C8—141no—
346C8—141no—
347C7#321111yescharacteristic, complemented, Sylow
348C22—41no—
349C22—41no—
350C22—41no—
351C22—72no—
352C22—141no—
353C22—141no—
354C22—281no—
355C22—281no—
356C22—281no—
357C22—281no—
358C22—281no—
359C22—281no—
360C22—564no—
361C22C2×D4×D711yescharacteristic
362C4—21no—
363C4—21no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—141no—
368C4—141no—
369C4—141no—
370C4—281no—
371C4—281no—
372C4#655311yescharacteristic
373C4#655311yescharacteristic
374C2—81no—
375C2—142no—
376C2—561no—
377C2—561no—
378C2#9036511yescharacteristic
379C2D7×#13511yescharacteristic
380C2#9045911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#488C2
socleC22×C7C22×D4
prime radical#89545C2
hypercenter#488D7
hypocenterC7#3211
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #90382C2semidirect
2. D7×#124C2semidirect
3. C7×#488C2semidirect
4. #89840C2semidirect
5. #89831C2semidirect
6. #89591C2semidirect
7. #89580C2semidirect
8. #89578C2semidirect
9. #89570C2semidirect
10. #89559C2semidirect
11. #89548C2semidirect
13. #89539C2semidirect
14. #89468C2semidirect
15. #89457C2semidirect
18. C7×#125C22semidirect
21. C2×#1144C22semidirect
22. C2×#1143C22semidirect
26. C2×Dic28C22semidirect
29. C2×#1132C22semidirect
35. #6456C22semidirect
37. #6450C22semidirect
43. #6438C22semidirect
45. C7×#103C22semidirect
46. C7×#103C22semidirect
49. C4×C7×C8C22semidirect
52. Dic7⋊C8C22semidirect
53. Dic7⋊C8C22semidirect
54. #6384C22semidirect
55. #6384C22semidirect
58. C4×C7⋊C8C22semidirect
120. #488D7semidirect
184. #125C2×D7semidirect
206. #103C2×D7semidirect
207. #103C2×D7semidirect
215. C4×C8C2×D7semidirect
347. C7#3211semidirect

Maximal subgroups

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

subgroupindex
#903822
D7×#1242
C7×#4882
#898402
#898312
#895912
#895802
#895782
#895702
#895592
#895482
#895452
#895392
#894682
#894572
#32117