Finite Group Databasebeta

← #11738128

Subgroups — #11738128

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#90321C211yesmaximal, characteristic, complemented
2#90302C211yesmaximal, characteristic, complemented
3C7×#465C211yesmaximal, characteristic, complemented
4#89944C211yesmaximal, characteristic, complemented
5#89937C211yesmaximal, characteristic, complemented
6C2×#6395C211yesmaximal, characteristic, complemented
7#89883C211yesmaximal, characteristic, complemented
8#89815C211yesmaximal, characteristic, complemented
9#89808C211yesmaximal, characteristic, complemented
10#89642C211yesmaximal, characteristic, complemented
11#89635C211yesmaximal, characteristic, complemented
12#89575C211yesmaximal, characteristic, complemented
13#89567C211yesmaximal, characteristic, complemented
14#89557C211yesmaximal, characteristic, complemented
15#89554C211yesmaximal, characteristic, complemented
16C2×#1161C2211yescharacteristic
17C2×#1159—21no—
18C22×D28C2211yescharacteristic
19C2×C7×SD16C2211yescharacteristic, complemented
20C22×C7×C8C2211yescharacteristic
21C7×C22⋊Q8C2211yescharacteristic
22C7×#121C2211yescharacteristic
23#6510C2211yescharacteristic
24#6501C2211yescharacteristic, complemented
25#6498—21no—
26D4×Dic7C2211yescharacteristic, complemented
27C2×#1141C2211yescharacteristic
28C2×#1141C2211yescharacteristic
29C4×C22⋊D7C2211yescharacteristic
30C2×C4⋊Dic7C2211yescharacteristic
31C2×#1131C2211yescharacteristic, complemented
32#6458—21no—
33#6458C2211yescharacteristic
34#6457—21no—
35#6456C2211yescharacteristic
36D7×C4⋊C4C2211yescharacteristic, complemented
37#6451C2211yescharacteristic, complemented
38#6446—21no—
39#6445—21no—
40D7×C22⋊C4—21no—
41#6441—21no—
42C7×#106C2211yescharacteristic, complemented
43C7×#103C2211yescharacteristic, complemented
44C7×#102C2211yescharacteristic, complemented
45#6409C2211yescharacteristic, complemented
46#6409C2211yescharacteristic, complemented
47C22⋊(C7⋊C8)C2211yescharacteristic
48#6395C2211yescharacteristic
49#6395C2211yescharacteristic
50#6395C2211yescharacteristic
51#6395C2211yescharacteristic
52(C2×D7)⋊C8C2211yescharacteristic, complemented
53#6389C2211yescharacteristic, complemented
54Dic7⋊C8C2211yescharacteristic, complemented
55#6383C2211yescharacteristic
56#6383C2211yescharacteristic
57#6382C2211yescharacteristic, complemented
58#6382C2211yescharacteristic, complemented
59#3306—71nomaximal, Sylow
60C23×D7—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4—21no—
63C2×C7×D4C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C2×C22⋊D7—21no—
66C2×C22⋊D7D411yescharacteristic
67C22×Dic7—21no—
68C22×Dic7—21no—
69#1161—22no—
70#1161—41no—
71#1159—44no—
72C2×D28—22no—
73C2×D28—21no—
74C2×D28—21no—
75C2×D28—21no—
76C2×D28—41no—
77C2×D28C2311yescharacteristic
78C2×D28C2311yescharacteristic
79C2×C4×D7—21no—
80C2×C4×D7—21no—
81C2×C4×D7—21no—
82C2×C4×D7C2311yescharacteristic
83C2×Dic14—21no—
84C7×SD16—41no—
85C2×C7×C8—41no—
86C2×C7×C8C2311yescharacteristic
87C2×C7×C8C2311yescharacteristic
88C7×C4⋊C4—21no—
89C7×C4⋊C4C2311yescharacteristic
90C7×C4⋊C4C2311yescharacteristic
91C7×C22⋊C4—21no—
92C7×C22⋊C4—21no—
93C22⋊Dic7—21no—
94C22⋊Dic7—21no—
95C22⋊Dic7D411yescharacteristic
96#1141—22no—
97#1141—41no—
98(C2×D7)⋊C4—21no—
99(C2×D7)⋊C4—21no—
100(C2×D7)⋊C4—21no—
101(C2×D7)⋊C4—21no—
102(C2×D7)⋊C4D411yescharacteristic
103C4⋊Dic7—21no—
104C4⋊Dic7C2311yescharacteristic
105C4⋊Dic7C2311yescharacteristic
106Dic7⋊C4—21no—
107Dic7⋊C4—21no—
108Dic7⋊C4D411yescharacteristic
109C4×Dic7C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111C2×C7⋊C8C2311yescharacteristic
112#1131—41no—
113#547—71no—
114#546—71no—
115#489—71no—
116#483—71no—
117#481—71no—
118#476—71no—
119#469—71no—
120#465D711yescharacteristic, complemented
121#463—71no—
122#449—71no—
123#447—71no—
124#442—71no—
125#435—71no—
126#417—71no—
127C2×#102—71no—
128C23×C7—21no—
129C23×C7C2×D411yescharacteristic
130C22×D7—21no—
131C22×D7—21no—
132C22×D7—42no—
133C22×D7—42no—
134C22×D7—81no—
135C22×D7C2×D411yescharacteristic
136C7×Q8—21no—
137C7×Q8—21no—
138C7×D4—21no—
139C7×D4—21no—
140C7×D4—42no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7C2×D411yescharacteristic
147C2×C4×C7C2411yescharacteristic
148C22⋊D7—22no—
149C22⋊D7—42no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7—42no—
153C2×Dic7—42no—
154C2×Dic7C2×D411yescharacteristic
155C2×Dic7C2×D411yescharacteristic
156C2×Dic7C2×D411yescharacteristic
157D28—21no—
158D28—21no—
159D28—21no—
160D28—42no—
161D28—42no—
162D28—81no—
163D28#4112yescomplemented
164C4×D7—21no—
165C4×D7—21no—
166C4×D7—42no—
167C4×D7—42no—
168C4×D7—81no—
169Dic14—42no—
170C7×C8—21no—
171C7×C8—21no—
172C7⋊C8—21no—
173C7⋊C8—21no—
174C2×#41—71no—
175C2×#41—141no—
176C22×D4—71no—
177C2×SD16C2×D711yescharacteristic, complemented
178C2×D8—71no—
179C2×D8—71no—
180C2×M16—71no—
181C22×C8C2×D711yescharacteristic
182C4⋊Q8—71no—
183#124—71no—
184#123—141no—
185#123—141no—
186#123—141no—
187C22⋊Q8—141no—
188C22⋊Q8C2×D711yescharacteristic
189#121—71no—
190#121—71no—
191#121—141no—
192#121C2×D711yescharacteristic
193#120—141no—
194C4×D4—71no—
195C4×D4—71no—
196C4×D4—71no—
197C2×C4⋊C4—71no—
198C2×C4⋊C4—71no—
199C2×C22⋊C4—141no—
200#106—71no—
201#106—71no—
202#106C2×D711yescharacteristic, complemented
203C4⋊C8—71no—
204#103—71no—
205#103—71no—
206#103C2×D711yescharacteristic, complemented
207#102—71no—
208#102—71no—
209#102—71no—
210#102—71no—
211#102—71no—
212#102—71no—
213#102C2×D711yescharacteristic, complemented
214C22⋊C8—71no—
215C22⋊C8—71no—
216#97—71no—
217C22×C7—21no—
218C22×C7—21no—
219C22×C7—41no—
220C22×C7—41no—
221C22×C7—41no—
222C22×C7AGL(1,ℤ/8)12yes—
223C22×C7C22×D411yescharacteristic
224C2×D7—21no—
225C2×D7—22no—
226C2×D7—21no—
227C2×D7—21no—
228C2×D7—41no—
229C2×D7—41no—
230C2×D7—41no—
231C2×D7—41no—
232C2×D7—41no—
233C2×D7—84no—
234C4×C7—21no—
235C4×C7—41no—
236C4×C7—41no—
237C4×C7—41no—
238C4×C7C2×#4111yescharacteristic
239C4×C7H(5,𝔽2)11yescharacteristic
240Dic7—21no—
241Dic7—21no—
242Dic7—21no—
243Dic7—41no—
244Dic7—41no—
245C24—141no—
246#41—142no—
247#41—281no—
248#41—284no—
249C2×Q8—141no—
250C2×Q8C22×D711yescharacteristic
251C2×D4—21no—
252C2×D4—71no—
253C2×D4—71no—
254C2×D4—71no—
255C2×D4—142no—
256C2×D4—141no—
257C2×D4—141no—
258C2×D4—141no—
259C2×D4—141no—
260C2×D4—281no—
261C2×D4C22×D711yescharacteristic
262C22×C4—71no—
263C22×C4—141no—
264C22×C4—141no—
265C22×C4—141no—
266C22×C4—141no—
267C22×C4—141no—
268C22×C4C22×D711yescharacteristic
269SD16—41no—
270D8—142no—
271D8—281no—
272M16—281no—
273C2×C8—41no—
274C2×C8—71no—
275C2×C8—71no—
276C2×C8C22×D711yescharacteristic
277C2×C8C22×D711yescharacteristic
278C4⋊C4—21no—
279C4⋊C4—71no—
280C4⋊C4—71no—
281C4⋊C4—71no—
282C4⋊C4—141no—
283C4⋊C4—141no—
284C4⋊C4—141no—
285C4⋊C4C22×D711yescharacteristic
286C4⋊C4C22×D711yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—71no—
290C22⋊C4—71no—
291C22⋊C4—141no—
292C22⋊C4—141no—
293C22⋊C4—141no—
294C22⋊C4—141no—
295C22⋊C4—141no—
296C22⋊C4—141no—
297C42—71no—
298C2×C7—22no—
299C2×C7—81no—
300C2×C7#54611yescharacteristic
301C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
302C2×C7#57711yescharacteristic
303D7—41no—
304D7—42no—
305D7—81no—
306C23—21no—
307C23—71no—
308C23—141no—
309C23—141no—
310C23—282no—
311C23—282no—
312C23—561no—
313C23D4×D711yescharacteristic
314Q8—21no—
315Q8—21no—
316Q8—282no—
317D4—21no—
318D4—21no—
319D4—42no—
320D4—72no—
321D4—142no—
322D4—141no—
323D4—141no—
324D4—141no—
325D4—282no—
326D4—282no—
327D4—282no—
328D4—561no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—71no—
335C2×C4—71no—
336C2×C4—71no—
337C2×C4—141no—
338C2×C4—141no—
339C2×C4—141no—
340C2×C4—141no—
341C2×C4—282no—
342C2×C4—282no—
343C2×C4—282no—
344C2×C4—282no—
345C2×C4—561no—
346C2×C4D4×D711yescharacteristic
347C2×C4C23×D711yescharacteristic
348C8—21no—
349C8—21no—
350C8—141no—
351C8—141no—
352C7#330611yescharacteristic, complemented, Sylow
353C22—21no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—141no—
359C22—141no—
360C22—141no—
361C22—142no—
362C22—281no—
363C22—281no—
364C22—281no—
365C22—281no—
366C22—281no—
367C22—564no—
368C22#647712yes—
369C22C2×D4×D711yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—141no—
375C4—141no—
376C4—141no—
377C4—281no—
378C4—281no—
379C4#654811yescharacteristic
380C4D7×#4111yescharacteristic
381C2—22no—
382C2—81no—
383C2—281no—
384C2—282no—
385C2—561no—
386C2#9030111yescharacteristic
387C2C2×#647711yescharacteristic
388C2#9046111yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#465C2
socleC22×C7C22×D4
hypercenter#465D7
hypocenterC7#3306
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #90321C2semidirect
2. #90302C2semidirect
3. C7×#465C2semidirect
4. #89944C2semidirect
5. #89937C2semidirect
6. C2×#6395C2semidirect
7. #89883C2semidirect
8. #89815C2semidirect
9. #89808C2semidirect
10. #89642C2semidirect
11. #89635C2semidirect
12. #89575C2semidirect
13. #89567C2semidirect
14. #89557C2semidirect
15. #89554C2semidirect
19. C2×C7×SD16C22semidirect
24. #6501C22semidirect
26. D4×Dic7C22semidirect
31. C2×#1131C22semidirect
36. D7×C4⋊C4C22semidirect
37. #6451C22semidirect
42. C7×#106C22semidirect
43. C7×#103C22semidirect
44. C7×#102C22semidirect
45. #6409C22semidirect
46. #6409C22semidirect
52. (C2×D7)⋊C8C22semidirect
53. #6389C22semidirect
54. Dic7⋊C8C22semidirect
57. #6382C22semidirect
58. #6382C22semidirect
120. #465D7semidirect
163. D28#41semidirect
177. C2×SD16C2×D7semidirect
202. #106C2×D7semidirect
206. #103C2×D7semidirect
213. #102C2×D7semidirect
352. C7#3306semidirect

Maximal subgroups

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

subgroupindex
#903212
#903022
C7×#4652
#899442
#899372
C2×#63952
#898832
#898152
#898082
#896422
#896352
#895752
#895672
#895572
#895542
#33067