Finite Group Databasebeta

← #11736837

Subgroups — #11736837

Order 896.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90225C211yesmaximal, characteristic, complemented
2#90216C211yesmaximal, characteristic, complemented
3C7×#433C211yesmaximal, characteristic, complemented
4#89620C211yesmaximal, characteristic, complemented
5#89617C211yesmaximal, characteristic, complemented
6#89616C211yesmaximal, characteristic, complemented
7#89615C211yesmaximal, characteristic, complemented
8#89506C211yesmaximal, characteristic, complemented
9#89505C211yesmaximal, characteristic, complemented
10#89504C211yesmaximal, characteristic, complemented
11#89503C211yesmaximal, characteristic, complemented
12#89486C211yesmaximal, characteristic, complemented
13#89484C211yesmaximal, characteristic, complemented
14#89468C211yesmaximal, characteristic, complemented
15#89466C211yesmaximal, characteristic, complemented
16C2×#1157C2211yescharacteristic
17C22×D28C2211yescharacteristic
18C7×#117C2211yescharacteristic
19#6493C2211yescharacteristic
20#6487C2211yescharacteristic
21C2×Dic28C2211yescharacteristic, complemented
22C2×D56C2211yescharacteristic
23C2×D56C2211yescharacteristic
24C2×D56C2211yescharacteristic, complemented
25C2×#1132C2211yescharacteristic
26C2×#1132C2211yescharacteristic
27C2×#1132C2211yescharacteristic, complemented
28C2×#1132C2211yescharacteristic, complemented
29#6461—21no—
30#6459—21no—
31#6458—21no—
32#6458—21no—
33#6448—21no—
34#6446—21no—
35#6445—21no—
36#6445—21no—
37#6438C2211yescharacteristic
38#6438C2211yescharacteristic
39#6438C2211yescharacteristic, complemented
40#6437C2211yescharacteristic
41#6437C2211yescharacteristic
42C4×D28C2211yescharacteristic, complemented
43#6432C2211yescharacteristic, complemented
44C4×Dic14C2211yescharacteristic, complemented
45C7×C4⋊C8C2211yescharacteristic, complemented
46C7×C4⋊C8C2211yescharacteristic, complemented
47C7×C22⋊C8C2211yescharacteristic
48C7×C22⋊C8C2211yescharacteristic
49C7×#97C2211yescharacteristic, complemented
50C4×C7×C8C2211yescharacteristic, complemented
51#6395C2211yescharacteristic
52#6395C2211yescharacteristic
53#6395C2211yescharacteristic
54#6395C2211yescharacteristic
55#6395C2211yescharacteristic, complemented
56#6395C2211yescharacteristic, complemented
57#6390C2211yescharacteristic, complemented
58#6390C2211yescharacteristic, complemented
59#3224—71nomaximal, Sylow
60C23×D7—21no—
61C22×C4×C7C2311yescharacteristic
62C2×C22⋊D7—21no—
63C2×C22⋊D7—21no—
64#1157—41no—
65#1157—41no—
66C2×D28—21no—
67C2×D28—21no—
68C2×D28—21no—
69C2×D28—21no—
70C2×D28—21no—
71C2×D28—41no—
72C2×D28—41no—
73C2×D28C2311yescharacteristic
74C2×D28C2311yescharacteristic
75C2×D28C2311yescharacteristic
76C2×D28C2311yescharacteristic
77C2×C4×D7—21no—
78C2×C4×D7—21no—
79C2×Dic14—21no—
80C2×Dic14C2311yescharacteristic
81C2×Dic14C2311yescharacteristic
82C2×C7×C8C2311yescharacteristic
83C2×C7×C8C2311yescharacteristic
84C2×C7×C8C2311yescharacteristic
85C2×C7×C8C2311yescharacteristic
86C7×C4⋊C4D411yescharacteristic
87C7×C4⋊C4D411yescharacteristic
88C7×C22⋊C4D411yescharacteristic
89C7×C22⋊C4D411yescharacteristic
90C42×C7C2311yescharacteristic
91C42×C7C2311yescharacteristic
92C22⋊Dic7—21no—
93(C2×D7)⋊C4—21no—
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—
101C4⋊Dic7—21no—
102C4⋊Dic7C2311yescharacteristic
103C4⋊Dic7C2311yescharacteristic
104Dic7⋊C4—21no—
105Dic7⋊C4—21no—
106C4×Dic7—21no—
107Dic28—41no—
108D56—41no—
109D56—41no—
110D56—41no—
111#1132—41no—
112#1132—41no—
113#1132—41no—
114#1132—41no—
115#542—71no—
116#535—71no—
117#496—71no—
118#495—71no—
119#489—71no—
120#486—71no—
121#464—71no—
122#463—71no—
123#460—71no—
124#459—71no—
125#452—71no—
126#450—71no—
127#449—71no—
128#447—71no—
129#433D711yescharacteristic, complemented
130C23×C7C2×D411yescharacteristic
131C22×D7—21no—
132C22×D7—21no—
133C22×D7—21no—
134C22×D7—21no—
135C22×D7—42no—
136C22×D7—81no—
137C22×D7—81no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7C2×D411yescharacteristic
141C2×C4×C7C2×D411yescharacteristic
142C2×C4×C7C2×D411yescharacteristic
143C2×C4×C7C2×D411yescharacteristic
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2411yescharacteristic
146C22⋊D7—42no—
147C22⋊D7—81no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152D28—21no—
153D28—21no—
154D28—21no—
155D28—21no—
156D28—21no—
157D28—21no—
158D28—21no—
159D28—21no—
160D28—42no—
161D28—81no—
162D28—81no—
163D28—81no—
164D28—81no—
165C4×D7—42no—
166C4×D7—81no—
167Dic14—21no—
168Dic14—21no—
169Dic14—21no—
170Dic14—21no—
171Dic14—81no—
172C7×C8—21no—
173C7×C8—21no—
174C7×C8—21no—
175C7×C8—21no—
176C2×#41—71no—
177C22×D4—71no—
178C2×Q16—71no—
179C2×SD16—71no—
180C2×SD16—71no—
181C2×SD16—71no—
182C2×SD16—71no—
183C2×D8—71no—
184C2×D8—71no—
185C2×D8—71no—
186C4⋊Q8—71no—
187#127—71no—
188#127—71no—
189#126—141no—
190#124—71no—
191#124—71no—
192#124—71no—
193#124—141no—
194#123—141no—
195C22⋊Q8—71no—
196C22⋊Q8—141no—
197#121—71no—
198#121—141no—
199#121—141no—
200#120—141no—
201#120—141no—
202C4×Q8—71no—
203C4×D4—71no—
204#117C2×D711yescharacteristic
205C4⋊C8C2×D711yescharacteristic, complemented
206C4⋊C8C2×D711yescharacteristic, complemented
207#103—71no—
208#103—71no—
209#102—71no—
210#102—71no—
211#102—71no—
212#102—71no—
213#102—71no—
214#102—71no—
215C22⋊C8C2×D711yescharacteristic
216C22⋊C8C2×D711yescharacteristic
217#97C2×D711yescharacteristic, complemented
218C4×C8C2×D711yescharacteristic, complemented
219C22×C7—21no—
220C22×C7—21no—
221C22×C7—21no—
222C22×C7C22×D411yescharacteristic
223C2×D7—41no—
224C2×D7—41no—
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—41no—
234C2×D7—41no—
235C2×D7—84no—
236C4×C7—21no—
237C4×C7—21no—
238C4×C7—21no—
239C4×C7—21no—
240C4×C7—21no—
241C4×C7H(5,𝔽2)11yescharacteristic
242C4×C7H(5,𝔽2)11yescharacteristic
243Dic7—41no—
244Dic7—41no—
245Dic7—41no—
246Dic7—41no—
247C24—141no—
248#41—281no—
249#41—281no—
250C2×Q8—71no—
251C2×Q8—71no—
252C2×Q8—141no—
253C2×D4—71no—
254C2×D4—71no—
255C2×D4—71no—
256C2×D4—71no—
257C2×D4—141no—
258C2×D4—141no—
259C2×D4—141no—
260C2×D4—141no—
261C2×D4—141no—
262C2×D4—141no—
263C2×D4—141no—
264C2×D4—281no—
265C2×D4—281no—
266C22×C4—141no—
267C22×C4—141no—
268C22×C4C22×D711yescharacteristic
269Q16—281no—
270SD16—281no—
271SD16—281no—
272SD16—281no—
273SD16—281no—
274D8—281no—
275D8—281no—
276D8—281no—
277C2×C8C22×D711yescharacteristic
278C2×C8C22×D711yescharacteristic
279C2×C8C22×D711yescharacteristic
280C2×C8C22×D711yescharacteristic
281C4⋊C4—71no—
282C4⋊C4—71no—
283C4⋊C4—141no—
284C4⋊C4—141no—
285C4⋊C4—141no—
286C4⋊C4D2811yescharacteristic
287C4⋊C4D2811yescharacteristic
288C22⋊C4—141no—
289C22⋊C4—141no—
290C22⋊C4—141no—
291C22⋊C4—141no—
292C22⋊C4—141no—
293C22⋊C4—141no—
294C22⋊C4—141no—
295C22⋊C4—141no—
296C22⋊C4—141no—
297C22⋊C4D2811yescharacteristic
298C22⋊C4D2811yescharacteristic
299C42—141no—
300C42C22×D711yescharacteristic
301C42C22×D711yescharacteristic
302C2×C7—41no—
303C2×C7#53511yescharacteristic
304C2×C7#57611yescharacteristic
305C2×C7#57711yescharacteristic
306D7—81no—
307D7—81no—
308D7—81no—
309D7—81no—
310C23—141no—
311C23—141no—
312C23—141no—
313C23—141no—
314C23—282no—
315C23—561no—
316C23—561no—
317C23C2×D2811yescharacteristic
318Q8—141no—
319Q8—141no—
320Q8—141no—
321Q8—141no—
322Q8—561no—
323D4—141no—
324D4—141no—
325D4—141no—
326D4—141no—
327D4—141no—
328D4—141no—
329D4—141no—
330D4—141no—
331D4—282no—
332D4—282no—
333D4—561no—
334D4—561no—
335D4—561no—
336D4—561no—
337D4—561no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—141no—
341C2×C4—141no—
342C2×C4—141no—
343C2×C4—141no—
344C2×C4—282no—
345C2×C4—561no—
346C2×C4C2×D2811yescharacteristic
347C2×C4C2×D2811yescharacteristic
348C2×C4C2×D2811yescharacteristic
349C2×C4C2×D2811yescharacteristic
350C2×C4C2×D2811yescharacteristic
351C2×C4C23×D711yescharacteristic
352C8—21no—
353C8—21no—
354C8—21no—
355C8—21no—
356C7#322411yescharacteristic, complemented, Sylow
357C22—21no—
358C22—21no—
359C22—21no—
360C22—281no—
361C22—281no—
362C22—281no—
363C22—281no—
364C22—281no—
365C22—281no—
366C22—281no—
367C22—281no—
368C22—281no—
369C22—281no—
370C22—281no—
371C22—281no—
372C22—564no—
373C22C22×D2811yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—281no—
380C4—281no—
381C4—281no—
382C4—281no—
383C4#655311yescharacteristic
384C4#655311yescharacteristic
385C2—41no—
386C2—561no—
387C2—561no—
388C2—561no—
389C2—561no—
390C2#9021611yescharacteristic
391C2#9044211yescharacteristic
392C2#9044311yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#433C2
socleC22×C7C22×D4
hypercenter#433D7
hypocenterC7#3224
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #90225C2semidirect
2. #90216C2semidirect
3. C7×#433C2semidirect
4. #89620C2semidirect
5. #89617C2semidirect
6. #89616C2semidirect
7. #89615C2semidirect
8. #89506C2semidirect
9. #89505C2semidirect
10. #89504C2semidirect
11. #89503C2semidirect
12. #89486C2semidirect
13. #89484C2semidirect
14. #89468C2semidirect
15. #89466C2semidirect
21. C2×Dic28C22semidirect
24. C2×D56C22semidirect
27. C2×#1132C22semidirect
28. C2×#1132C22semidirect
39. #6438C22semidirect
42. C4×D28C22semidirect
43. #6432C22semidirect
44. C4×Dic14C22semidirect
45. C7×C4⋊C8C22semidirect
46. C7×C4⋊C8C22semidirect
49. C7×#97C22semidirect
50. C4×C7×C8C22semidirect
55. #6395C22semidirect
56. #6395C22semidirect
57. #6390C22semidirect
58. #6390C22semidirect
129. #433D7semidirect
205. C4⋊C8C2×D7semidirect
206. C4⋊C8C2×D7semidirect
217. #97C2×D7semidirect
218. C4×C8C2×D7semidirect
356. C7#3224semidirect

Maximal subgroups

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

subgroupindex
#902252
#902162
C7×#4332
#896202
#896172
#896162
#896152
#895062
#895052
#895042
#895032
#894862
#894842
#894682
#894662
#32247