Finite Group Databasebeta

← #11738220

Subgroups — #11738220

Order 896.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90333C211yesmaximal, characteristic, complemented
2#90321C211yesmaximal, characteristic
3C7×#467C211yesmaximal, characteristic, complemented
4#89959C211yesmaximal, characteristic, complemented
5#89957C211yesmaximal, characteristic, complemented
6C2×#6395C211yesmaximal, characteristic, complemented
7C8×C22⋊D7C211yesmaximal, characteristic, complemented
8#89818C211yesmaximal, characteristic, complemented
9#89815C211yesmaximal, characteristic, complemented
10#89663C211yesmaximal, characteristic, complemented
11#89657C211yesmaximal, characteristic, complemented
12#89579C211yesmaximal, characteristic, complemented
13#89576C211yesmaximal, characteristic, complemented
14#89572C211yesmaximal, characteristic, complemented
15#89560C211yesmaximal, characteristic, complemented
16C2×#1161C2211yescharacteristic
17C22×D28C2211yescharacteristic
18C2×C7×Q16C2211yescharacteristic, complemented
19C22×C7×C8C2211yescharacteristic
20C7×C22⋊Q8C2211yescharacteristic
21C7×C22⋊Q8C2211yescharacteristic
22#6510C2211yescharacteristic
23#6509C2211yescharacteristic, complemented
24Q8×Dic7C2211yescharacteristic, complemented
25C2×#1143C2211yescharacteristic
26C2×#1143C2211yescharacteristic
27C4×C22⋊D7C2211yescharacteristic
28C2×C4⋊Dic7C2211yescharacteristic
29C2×C8×D7C2211yescharacteristic, complemented
30#6461—21no—
31#6458—21no—
32#6458C2211yescharacteristic
33#6457—21no—
34#6456C2211yescharacteristic
35#6455C2211yescharacteristic, complemented
36#6453—21no—
37#6453C2211yescharacteristic, complemented
38#6449—21no—
39#6445—21no—
40D7×C22⋊C4—21no—
41#6440—21no—
42C7×#107C2211yescharacteristic, complemented
43C7×#103C2211yescharacteristic, complemented
44C7×#103C2211yescharacteristic, complemented
45#6409C2211yescharacteristic, complemented
46#6409C2211yescharacteristic, complemented
47C22⋊(C7⋊C8)C2211yescharacteristic
48#6395C2211yescharacteristic
49#6395C2211yescharacteristic
50#6395C2211yescharacteristic
51#6395C2211yescharacteristic
52(C2×D7)⋊C8C2211yescharacteristic, complemented
53Dic7⋊C8C2211yescharacteristic, complemented
54C8×Dic7C2211yescharacteristic, complemented
55#6383C2211yescharacteristic
56#6383C2211yescharacteristic
57#6382C2211yescharacteristic, complemented
58#6382C2211yescharacteristic, complemented
59#3296—71nomaximal, Sylow
60C23×D7—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×C7×Q8C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C2×C22⋊D7D411yescharacteristic
65C22×Dic7—21no—
66#1161—22no—
67#1161—41no—
68C2×D28—22no—
69C2×D28—21no—
70C2×D28—21no—
71C2×D28—21no—
72C2×D28—41no—
73C2×D28C2311yescharacteristic
74C2×D28C2311yescharacteristic
75C2×C4×D7—21no—
76C2×C4×D7—21no—
77C2×C4×D7C2311yescharacteristic
78C7×Q16—41no—
79C2×C7×C8—41no—
80C2×C7×C8C2311yescharacteristic
81C2×C7×C8C2311yescharacteristic
82C7×C4⋊C4—21no—
83C7×C4⋊C4—21no—
84C7×C4⋊C4C2311yescharacteristic
85C7×C4⋊C4C2311yescharacteristic
86C7×C22⋊C4—21no—
87C7×C22⋊C4—21no—
88C22⋊Dic7D411yescharacteristic
89#1143—22no—
90#1143—41no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4—21no—
96(C2×D7)⋊C4—21no—
97(C2×D7)⋊C4D411yescharacteristic
98C4⋊Dic7—21no—
99C4⋊Dic7—21no—
100C4⋊Dic7—21no—
101C4⋊Dic7C2311yescharacteristic
102C4⋊Dic7C2311yescharacteristic
103Dic7⋊C4—21no—
104Dic7⋊C4—21no—
105Dic7⋊C4D411yescharacteristic
106C4×Dic7—21no—
107C4×Dic7—21no—
108C4×Dic7C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111C8×D7—41no—
112#552—71no—
113#546—71no—
114#488—71no—
115#483—71no—
116#482—71no—
117#478—71no—
118#467D711yescharacteristic, complemented
119#465—71no—
120#464—71no—
121#452—71no—
122#450—71no—
123C4×SD16—71no—
124C8×D4—71no—
125#416—71no—
126C2×#102—71no—
127C23×C7C2×D411yescharacteristic
128C22×D7—21no—
129C22×D7—21no—
130C22×D7—42no—
131C22×D7—42no—
132C22×D7—81no—
133C22×D7C2×D411yescharacteristic
134C7×Q8—21no—
135C7×Q8—21no—
136C7×Q8—21no—
137C7×Q8—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2411yescharacteristic
146C22⋊D7—22no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7—42no—
150C2×Dic7C2×D411yescharacteristic
151C2×Dic7C2×D411yescharacteristic
152C2×Dic7C2×D411yescharacteristic
153D28—21no—
154D28—21no—
155D28—21no—
156D28—42no—
157D28—42no—
158D28—81no—
159D28#4112yes—
160C4×D7—21no—
161C4×D7—21no—
162C4×D7—42no—
163C4×D7—81no—
164C7×C8—21no—
165C7×C8—21no—
166C7⋊C8—21no—
167C7⋊C8—21no—
168C2×#41—71no—
169C22×D4—71no—
170C2×Q16C2×D711yescharacteristic, complemented
171C2×SD16—71no—
172C2×SD16—71no—
173C22×C8—71no—
174C22×C8C2×D711yescharacteristic
175#126—141no—
176#125—71no—
177#125—141no—
178#124—71no—
179#123—141no—
180#123—141no—
181C22⋊Q8—71no—
182C22⋊Q8C2×D711yescharacteristic
183C22⋊Q8C2×D711yescharacteristic
184#121—71no—
185#121—141no—
186#120—141no—
187C4×Q8—71no—
188C4×D4—71no—
189C4×D4—71no—
190#117—71no—
191#117—141no—
192C2×C4⋊C4—71no—
193C2×C22⋊C4—141no—
194#107C2×D711yescharacteristic, complemented
195#106—71no—
196#106—71no—
197C4⋊C8—71no—
198#103—71no—
199#103—71no—
200#103C2×D711yescharacteristic, complemented
201#103C2×D711yescharacteristic, complemented
202#102—71no—
203#102—71no—
204#102—71no—
205#102—71no—
206#102—71no—
207#102—71no—
208C22⋊C8—71no—
209C22⋊C8—71no—
210C4×C8—71no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7#13512yes—
214C22×C7C22×D411yescharacteristic
215C2×D7—21no—
216C2×D7—22no—
217C2×D7—21no—
218C2×D7—21no—
219C2×D7—41no—
220C2×D7—41no—
221C2×D7—41no—
222C2×D7—41no—
223C2×D7—41no—
224C2×D7—84no—
225C4×C7—21no—
226C4×C7—41no—
227C4×C7—41no—
228C4×C7—41no—
229C4×C7—41no—
230C4×C7C2×#4111yescharacteristic
231C4×C7H(5,𝔽2)11yescharacteristic
232Dic7—21no—
233Dic7—21no—
234Dic7—21no—
235Dic7—41no—
236Dic7—41no—
237C24—141no—
238#41—142no—
239#41—281no—
240C2×Q8C22×D711yescharacteristic
241C2×Q8C22×D711yescharacteristic
242C2×D4—71no—
243C2×D4—71no—
244C2×D4—71no—
245C2×D4—142no—
246C2×D4—141no—
247C2×D4—141no—
248C2×D4—141no—
249C2×D4—281no—
250C22×C4—71no—
251C22×C4—141no—
252C22×C4—141no—
253C22×C4—141no—
254C22×C4C22×D711yescharacteristic
255Q16—41no—
256SD16—142no—
257SD16—281no—
258C2×C8—41no—
259C2×C8—71no—
260C2×C8—71no—
261C2×C8—281no—
262C2×C8C22×D711yescharacteristic
263C2×C8C22×D711yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—71no—
267C4⋊C4—71no—
268C4⋊C4—71no—
269C4⋊C4—141no—
270C4⋊C4—141no—
271C4⋊C4—141no—
272C4⋊C4—141no—
273C4⋊C4—141no—
274C4⋊C4C22×D711yescharacteristic
275C4⋊C4C22×D711yescharacteristic
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—71no—
279C22⋊C4—71no—
280C22⋊C4—141no—
281C22⋊C4—141no—
282C22⋊C4—141no—
283C22⋊C4—141no—
284C22⋊C4—141no—
285C22⋊C4—141no—
286C42—71no—
287C42—141no—
288C42—141no—
289C2×C7—22no—
290C2×C7#54611yescharacteristic
291C2×C7C2×#13511yescharacteristic
292C2×C7#57711yescharacteristic
293D7—41no—
294D7—42no—
295D7—81no—
296C23—71no—
297C23—141no—
298C23—141no—
299C23—282no—
300C23—282no—
301C23—561no—
302C23D4×D711yescharacteristic
303Q8—21no—
304Q8—21no—
305Q8—21no—
306Q8—21no—
307D4—72no—
308D4—142no—
309D4—141no—
310D4—141no—
311D4—141no—
312D4—282no—
313D4—282no—
314D4—561no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—71no—
322C2×C4—71no—
323C2×C4—71no—
324C2×C4—141no—
325C2×C4—141no—
326C2×C4—141no—
327C2×C4—141no—
328C2×C4—282no—
329C2×C4—282no—
330C2×C4—561no—
331C2×C4D4×D711yescharacteristic
332C2×C4C23×D711yescharacteristic
333C8—21no—
334C8—21no—
335C8—141no—
336C8—141no—
337C7#329611yescharacteristic, complemented, Sylow
338C22—21no—
339C22—21no—
340C22—141no—
341C22—141no—
342C22—141no—
343C22—142no—
344C22—281no—
345C22—281no—
346C22—281no—
347C22—281no—
348C22—281no—
349C22—564no—
350C22#648212yes—
351C22C2×D4×D711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—141no—
358C4—141no—
359C4—141no—
360C4—281no—
361C4—281no—
362C4#654811yescharacteristic
363C4D7×#4111yescharacteristic
364C2—22no—
365C2—281no—
366C2—282no—
367C2—561no—
368C2#9030111yescharacteristic
369C2C2×#648211yescharacteristic
370C2#9046111yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#467C2
socleC22×C7C22×D4
prime radical#90321C2
hypercenter#467D7
hypocenterC7#3296
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #90333C2semidirect
3. C7×#467C2semidirect
4. #89959C2semidirect
5. #89957C2semidirect
6. C2×#6395C2semidirect
7. C8×C22⋊D7C2semidirect
8. #89818C2semidirect
9. #89815C2semidirect
10. #89663C2semidirect
11. #89657C2semidirect
12. #89579C2semidirect
13. #89576C2semidirect
14. #89572C2semidirect
15. #89560C2semidirect
18. C2×C7×Q16C22semidirect
23. #6509C22semidirect
24. Q8×Dic7C22semidirect
29. C2×C8×D7C22semidirect
35. #6455C22semidirect
37. #6453C22semidirect
42. C7×#107C22semidirect
43. C7×#103C22semidirect
44. C7×#103C22semidirect
45. #6409C22semidirect
46. #6409C22semidirect
52. (C2×D7)⋊C8C22semidirect
53. Dic7⋊C8C22semidirect
54. C8×Dic7C22semidirect
57. #6382C22semidirect
58. #6382C22semidirect
118. #467D7semidirect
170. C2×Q16C2×D7semidirect
194. #107C2×D7semidirect
200. #103C2×D7semidirect
201. #103C2×D7semidirect
337. C7#3296semidirect

Maximal subgroups

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

subgroupindex
#903332
#903212
C7×#4672
#899592
#899572
C2×#63952
C8×C22⋊D72
#898182
#898152
#896632
#896572
#895792
#895762
#895722
#895602
#32967