Finite Group Databasebeta

← #11738130

Subgroups — #11738130

Order 896.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90333C211yesmaximal, characteristic, complemented
2#90300C211yesmaximal, characteristic
3C7×#465C211yesmaximal, characteristic, complemented
4#89941C211yesmaximal, characteristic, complemented
5#89935C211yesmaximal, characteristic, complemented
6C2×#6395C211yesmaximal, characteristic, complemented
7#89883C211yesmaximal, characteristic, complemented
8#89818C211yesmaximal, characteristic, complemented
9#89805C211yesmaximal, characteristic, complemented
10#89643C211yesmaximal, characteristic, complemented
11#89638C211yesmaximal, characteristic, complemented
12#89589C211yesmaximal, characteristic, complemented
13#89584C211yesmaximal, characteristic, complemented
14#89542C211yesmaximal, characteristic, complemented
15#89536C211yesmaximal, characteristic, complemented
16C2×D4×D7—21no—
17C2×D4×D7C2211yescharacteristic
18C22×D28C2211yescharacteristic
19C2×C7×SD16C2211yescharacteristic, complemented
20C22×C7×C8C2211yescharacteristic, complemented
21C7×C22⋊Q8C2211yescharacteristic, complemented
22C7×#121C2211yescharacteristic
23#6509C2211yescharacteristic, complemented
24Q8×Dic7C2211yescharacteristic, complemented
25C2×#1143C2211yescharacteristic
26C2×#1143C2211yescharacteristic, complemented
27#6503C2211yescharacteristic
28#6500—21no—
29C4×C22⋊D7C2211yescharacteristic
30C2×C4⋊Dic7C2211yescharacteristic, complemented
31C2×#1131C2211yescharacteristic, complemented
32#6461—21no—
33#6458C2211yescharacteristic
34#6456C2211yescharacteristic
35#6455C2211yescharacteristic, complemented
36#6453—21no—
37#6453C2211yescharacteristic, complemented
38#6449—21no—
39#6447—21no—
40#6445—21no—
41#6440—21no—
42C7×#106C2211yescharacteristic, complemented
43C7×#103C2211yescharacteristic, complemented
44C7×#102C2211yescharacteristic, complemented
45#6406C2211yescharacteristic, complemented
46#6406C2211yescharacteristic, complemented
47C22⋊(C7⋊C8)C2211yescharacteristic, complemented
48#6395C2211yescharacteristic
49#6395C2211yescharacteristic
50#6395C2211yescharacteristic, complemented
51#6395C2211yescharacteristic, complemented
52(C2×D7)⋊C8C2211yescharacteristic, complemented
53#6389C2211yescharacteristic, complemented
54Dic7⋊C8C2211yescharacteristic, complemented
55#6383C2211yescharacteristic
56#6383C2211yescharacteristic, complemented
57#6381C2211yescharacteristic, complemented
58#6381C2211yescharacteristic, complemented
59#3310—71nomaximal, Sylow
60C23×D7—21no—
61C23×D7—21no—
62C2×C7×Q8C2311yescharacteristic
63C2×C7×D4—21no—
64C2×C7×D4C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C2×C22⋊D7—21no—
67C2×C22⋊D7—21no—
68C2×C22⋊D7—21no—
69C2×C22⋊D7D411yescharacteristic
70C22×Dic7—21no—
71D4×D7—22no—
72D4×D7—41no—
73D4×D7—44no—
74C2×D28—22no—
75C2×D28—21no—
76C2×D28—21no—
77C2×D28—41no—
78C2×D28C2311yescharacteristic
79C2×D28C2311yescharacteristic
80C2×C4×D7—21no—
81C2×C4×D7C2311yescharacteristic
82C7×SD16—41no—
83C2×C7×C8—41no—
84C2×C7×C8C2311yescharacteristic
85C2×C7×C8C2311yescharacteristic
86C7×C4⋊C4—21no—
87C7×C4⋊C4C2311yescharacteristic
88C7×C4⋊C4C2311yescharacteristic
89C7×C22⋊C4—21no—
90C7×C22⋊C4—21no—
91C22⋊Dic7D411yescharacteristic
92#1143—22no—
93#1143—41no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4—21no—
96(C2×D7)⋊C4—21no—
97(C2×D7)⋊C4—21no—
98(C2×D7)⋊C4D411yescharacteristic
99C4⋊Dic7—21no—
100C4⋊Dic7—21no—
101C4⋊Dic7—21no—
102C4⋊Dic7C2311yescharacteristic
103C4⋊Dic7C2311yescharacteristic
104Dic7⋊C4—21no—
105Dic7⋊C4—21no—
106Dic7⋊C4D411yescharacteristic
107C4×Dic7—21no—
108C4×Dic7—21no—
109C4×Dic7C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111C2×C7⋊C8C2311yescharacteristic
112#1131—41no—
113#552—71no—
114D42—71no—
115#490—71no—
116#482—71no—
117#480—71no—
118#478—71no—
119#468—71no—
120#465D711yescharacteristic, complemented
121#460—71no—
122#450—71no—
123#450—71no—
124#440—71no—
125#435—71no—
126#418—71no—
127C2×#102—71no—
128C23×C7—21no—
129C23×C7C2×D411yescharacteristic
130C22×D7—21no—
131C22×D7—21no—
132C22×D7—42no—
133C22×D7—42no—
134C22×D7—42no—
135C22×D7—42no—
136C22×D7—81no—
137C22×D7—81no—
138C22×D7C2×D411yescharacteristic
139C7×Q8—21no—
140C7×Q8—21no—
141C7×D4—21no—
142C7×D4—21no—
143C7×D4—42no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7C2×D411yescharacteristic
150C2×C4×C7C2411yescharacteristic
151C22⋊D7—22no—
152C22⋊D7—42no—
153C22⋊D7—42no—
154C22⋊D7—81no—
155C2×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7—42no—
158C2×Dic7C2×D411yescharacteristic
159C2×Dic7C2×D411yescharacteristic
160C2×Dic7C2×D411yescharacteristic
161D28—21no—
162D28—21no—
163D28—21no—
164D28—42no—
165D28—42no—
166D28#4112yes—
167C4×D7—21no—
168C4×D7—21no—
169C4×D7—42no—
170C7×C8—21no—
171C7×C8—21no—
172C7⋊C8—21no—
173C7⋊C8—21no—
174C22×D4—71no—
175C22×D4—71no—
176C22×D4—141no—
177C2×SD16—71no—
178C2×SD16—71no—
179C2×SD16C2×D711yescharacteristic, complemented
180C2×M16—71no—
181C22×C8C2×D711yescharacteristic, complemented
182#127—71no—
183#126—141no—
184#125—71no—
185#125—141no—
186#123—141no—
187C22⋊Q8—71no—
188C22⋊Q8C2×D711yescharacteristic, complemented
189#121—71no—
190#121—141no—
191#121C2×D711yescharacteristic
192#120—141no—
193#120—141no—
194C4×Q8—71no—
195C4×D4—71no—
196C4×D4—71no—
197#117—71no—
198#117—141no—
199C2×C4⋊C4—71no—
200#107—71no—
201#107—71no—
202#106C2×D711yescharacteristic, complemented
203C4⋊C8—71no—
204#103C2×D711yescharacteristic, complemented
205#102—71no—
206#102—71no—
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—
234C2×D7—84no—
235C4×C7—21no—
236C4×C7—41no—
237C4×C7—41no—
238C4×C7—41no—
239C4×C7C2×#4111yescharacteristic
240C4×C7H(5,𝔽2)11yescharacteristic
241Dic7—21no—
242Dic7—21no—
243Dic7—21no—
244Dic7—41no—
245Dic7—41no—
246C24—141no—
247C24—141no—
248C2×Q8C22×D711yescharacteristic
249C2×D4—21no—
250C2×D4—71no—
251C2×D4—71no—
252C2×D4—71no—
253C2×D4—142no—
254C2×D4—141no—
255C2×D4—141no—
256C2×D4—141no—
257C2×D4—141no—
258C2×D4—142no—
259C2×D4—141no—
260C2×D4—281no—
261C2×D4—281no—
262C2×D4—284no—
263C2×D4C22×D711yescharacteristic
264C22×C4—71no—
265C22×C4—141no—
266C22×C4—141no—
267C22×C4C22×D711yescharacteristic
268SD16—41no—
269SD16—142no—
270SD16—281no—
271M16—281no—
272C2×C8—41no—
273C2×C8—71no—
274C2×C8—71no—
275C2×C8C22×D711yescharacteristic
276C2×C8C22×D711yescharacteristic
277C4⋊C4—21no—
278C4⋊C4—71no—
279C4⋊C4—71no—
280C4⋊C4—71no—
281C4⋊C4—141no—
282C4⋊C4—141no—
283C4⋊C4—141no—
284C4⋊C4—141no—
285C4⋊C4—141no—
286C4⋊C4C22×D711yescharacteristic
287C4⋊C4C22×D711yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—71no—
291C22⋊C4—71no—
292C22⋊C4—141no—
293C22⋊C4—141no—
294C22⋊C4—141no—
295C22⋊C4—141no—
296C42—71no—
297C42—141no—
298C42—141no—
299C2×C7—22no—
300C2×C7—81no—
301C2×C7#54611yescharacteristic
302C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
303C2×C7#57711yescharacteristic
304D7—41no—
305D7—42no—
306D7—81no—
307C23—21no—
308C23—71no—
309C23—141no—
310C23—141no—
311C23—282no—
312C23—282no—
313C23—282no—
314C23—282no—
315C23—561no—
316C23—561no—
317C23D4×D711yescharacteristic
318Q8—21no—
319Q8—21no—
320D4—21no—
321D4—21no—
322D4—42no—
323D4—72no—
324D4—142no—
325D4—141no—
326D4—141no—
327D4—141no—
328D4—282no—
329D4—282no—
330D4—282no—
331D4—282no—
332D4—561no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—71no—
339C2×C4—71no—
340C2×C4—71no—
341C2×C4—141no—
342C2×C4—141no—
343C2×C4—141no—
344C2×C4—141no—
345C2×C4—282no—
346C2×C4—282no—
347C2×C4D4×D711yescharacteristic
348C2×C4C23×D711yescharacteristic
349C8—21no—
350C8—21no—
351C8—141no—
352C8—141no—
353C7#331011yescharacteristic, complemented, Sylow
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—141no—
360C22—141no—
361C22—141no—
362C22—142no—
363C22—281no—
364C22—281no—
365C22—281no—
366C22—281no—
367C22—281no—
368C22—564no—
369C22—564no—
370C22#647712yes—
371C22C2×D4×D711yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—141no—
377C4—141no—
378C4—141no—
379C4—281no—
380C4—281no—
381C4#654811yescharacteristic
382C4D7×#4111yescharacteristic
383C2—22no—
384C2—81no—
385C2—281no—
386C2—282no—
387C2—561no—
388C2#9030111yescharacteristic
389C2C2×#647711yescharacteristic
390C2#9046111yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#465C2
socleC22×C7C22×D4
prime radical#90300C2
hypercenter#465D7
hypocenterC7#3310
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. #90333C2semidirect
3. C7×#465C2semidirect
4. #89941C2semidirect
5. #89935C2semidirect
6. C2×#6395C2semidirect
7. #89883C2semidirect
8. #89818C2semidirect
9. #89805C2semidirect
10. #89643C2semidirect
11. #89638C2semidirect
12. #89589C2semidirect
13. #89584C2semidirect
14. #89542C2semidirect
15. #89536C2semidirect
19. C2×C7×SD16C22semidirect
20. C22×C7×C8C22semidirect
21. C7×C22⋊Q8C22semidirect
23. #6509C22semidirect
24. Q8×Dic7C22semidirect
26. C2×#1143C22semidirect
30. C2×C4⋊Dic7C22semidirect
31. C2×#1131C22semidirect
35. #6455C22semidirect
37. #6453C22semidirect
42. C7×#106C22semidirect
43. C7×#103C22semidirect
44. C7×#102C22semidirect
45. #6406C22semidirect
46. #6406C22semidirect
47. C22⋊(C7⋊C8)C22semidirect
50. #6395C22semidirect
51. #6395C22semidirect
52. (C2×D7)⋊C8C22semidirect
53. #6389C22semidirect
54. Dic7⋊C8C22semidirect
56. #6383C22semidirect
57. #6381C22semidirect
58. #6381C22semidirect
120. #465D7semidirect
179. C2×SD16C2×D7semidirect
181. C22×C8C2×D7semidirect
188. C22⋊Q8C2×D7semidirect
202. #106C2×D7semidirect
204. #103C2×D7semidirect
213. #102C2×D7semidirect
353. C7#3310semidirect

Maximal subgroups

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

subgroupindex
#903332
#903002
C7×#4652
#899412
#899352
C2×#63952
#898832
#898182
#898052
#896432
#896382
#895892
#895842
#895422
#895362
#33107