Finite Group Databasebeta

← #11738125

Subgroups — #11738125

Order 896.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90326C211yesmaximal, characteristic, complemented
2#90300C211yesmaximal, characteristic
3C7×#465C211yesmaximal, characteristic, complemented
4#89941C211yesmaximal, characteristic, complemented
5#89935C211yesmaximal, characteristic, complemented
6C2×#6395C211yesmaximal, characteristic, complemented
7C8×C22⋊D7C211yesmaximal, characteristic, complemented
8#89818C211yesmaximal, characteristic, complemented
9#89806C211yesmaximal, characteristic, complemented
10#89642C211yesmaximal, characteristic, complemented
11#89635C211yesmaximal, characteristic, complemented
12#89579C211yesmaximal, characteristic, complemented
13#89560C211yesmaximal, characteristic, complemented
14D7×#102C211yesmaximal, characteristic, complemented
15#89532C211yesmaximal, 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×C8×D7C2211yescharacteristic, complemented
32#6460—21no—
33#6458C2211yescharacteristic
34#6456C2211yescharacteristic
35D7×C4⋊C4C2211yescharacteristic, complemented
36#6452—21no—
37#6451C2211yescharacteristic, complemented
38#6447—21no—
39#6445—21no—
40(C22⋊D7)⋊C4—21no—
41#6441—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
53Dic7⋊C8C2211yescharacteristic, complemented
54C8×Dic7C2211yescharacteristic, complemented
55#6383C2211yescharacteristic
56#6383C2211yescharacteristic, complemented
57#6382C2211yescharacteristic, complemented
58#6382C2211yescharacteristic, complemented
59#3295—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×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⋊Dic7D411yescharacteristic
94#1143—22no—
95#1143—41no—
96(C2×D7)⋊C4—21no—
97(C2×D7)⋊C4—21no—
98(C2×D7)⋊C4—21no—
99(C2×D7)⋊C4D411yescharacteristic
100C4⋊Dic7—21no—
101C4⋊Dic7—21no—
102C4⋊Dic7C2311yescharacteristic
103C4⋊Dic7C2311yescharacteristic
104Dic7⋊C4—21no—
105Dic7⋊C4—21no—
106Dic7⋊C4—21no—
107Dic7⋊C4D411yescharacteristic
108C4×Dic7—21no—
109C4×Dic7C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111C2×C7⋊C8C2311yescharacteristic
112C8×D7—41no—
113#554—71no—
114D42—71no—
115#486—71no—
116#481—71no—
117#481—71no—
118#476—71no—
119#465—71no—
120#465D711yescharacteristic, complemented
121#460—71no—
122#450—71no—
123#450—71no—
124C4×SD16—71no—
125C8×D4—71no—
126C2×#102—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—42no—
152C22⋊D7—42no—
153C22⋊D7—81no—
154C22⋊D7SD1614yescomplemented
155C2×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7—81no—
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—
170C4×D7—81no—
171Dic14—81no—
172C7×C8—21no—
173C7×C8—21no—
174C7⋊C8—21no—
175C7⋊C8—21no—
176C22×D4—71no—
177C22×D4—71no—
178C22×D4—141no—
179C2×SD16—71no—
180C2×SD16—71no—
181C2×SD16C2×D711yescharacteristic, complemented
182C22×C8—71no—
183C22×C8C2×D711yescharacteristic, complemented
184C4⋊Q8—71no—
185#127—71no—
186#125—141no—
187C22⋊Q8—71no—
188C22⋊Q8—141no—
189C22⋊Q8—141no—
190C22⋊Q8C2×D711yescharacteristic, complemented
191#121—71no—
192#121—141no—
193#121C2×D711yescharacteristic
194#120—141no—
195#120—141no—
196C4×Q8—71no—
197C4×D4—71no—
198C4×D4—71no—
199C4×D4—141no—
200C2×C4⋊C4—71no—
201C2×C4⋊C4—71no—
202#106—71no—
203#106—71no—
204#106C2×D711yescharacteristic, complemented
205C4⋊C8—71no—
206#103C2×D711yescharacteristic, complemented
207#102—71no—
208#102—71no—
209#102—71no—
210#102—71no—
211#102—71no—
212#102—71no—
213#102—71no—
214#102—71no—
215#102C2×D711yescharacteristic, complemented
216C22⋊C8—71no—
217C22⋊C8—71no—
218C4×C8—71no—
219C22×C7—21no—
220C22×C7—21no—
221C22×C7—41no—
222C22×C7—41no—
223C22×C7—41no—
224C22×C7C2×SD1612yes—
225C22×C7C22×D411yescharacteristic
226C2×D7—21no—
227C2×D7—21no—
228C2×D7—22no—
229C2×D7—41no—
230C2×D7—41no—
231C2×D7—41no—
232C2×D7—41no—
233C2×D7—41no—
234C2×D7—84no—
235C2×D7—84no—
236C2×D7C2×SD1612yes—
237C4×C7—21no—
238C4×C7—41no—
239C4×C7—41no—
240C4×C7—41no—
241C4×C7C2×#4111yescharacteristic
242C4×C7H(5,𝔽2)11yescharacteristic
243Dic7—21no—
244Dic7—21no—
245Dic7—41no—
246Dic7—41no—
247Dic7C2×SD1612yes—
248C24—141no—
249C24—141no—
250C2×Q8—141no—
251C2×Q8C22×D711yescharacteristic
252C2×D4—21no—
253C2×D4—71no—
254C2×D4—71no—
255C2×D4—71no—
256C2×D4—142no—
257C2×D4—141no—
258C2×D4—141no—
259C2×D4—141no—
260C2×D4—141no—
261C2×D4—142no—
262C2×D4—141no—
263C2×D4—281no—
264C2×D4—281no—
265C2×D4—284no—
266C2×D4C22×D711yescharacteristic
267C22×C4—71no—
268C22×C4—141no—
269C22×C4—141no—
270C22×C4—141no—
271C22×C4C22×D711yescharacteristic
272SD16—41no—
273SD16—142no—
274SD16—281no—
275C2×C8—41no—
276C2×C8—71no—
277C2×C8—71no—
278C2×C8—281no—
279C2×C8C22×D711yescharacteristic
280C2×C8C22×D711yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—71no—
283C4⋊C4—71no—
284C4⋊C4—71no—
285C4⋊C4—141no—
286C4⋊C4—141no—
287C4⋊C4—141no—
288C4⋊C4—141no—
289C4⋊C4—141no—
290C4⋊C4C22×D711yescharacteristic
291C4⋊C4C22×D711yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—71no—
295C22⋊C4—71no—
296C22⋊C4—141no—
297C22⋊C4—141no—
298C22⋊C4—141no—
299C42—71no—
300C42—141no—
301C2×C7—22no—
302C2×C7—81no—
303C2×C7#54611yescharacteristic
304C2×C7C22×SD1611yescharacteristic
305C2×C7#57611yescharacteristic
306D7—22no—
307D7—42no—
308D7—81no—
309C23—21no—
310C23—71no—
311C23—141no—
312C23—141no—
313C23—282no—
314C23—282no—
315C23—282no—
316C23—282no—
317C23—561no—
318C23—561no—
319C23D4×D711yescharacteristic
320Q8—21no—
321Q8—21no—
322Q8—561no—
323D4—21no—
324D4—21no—
325D4—42no—
326D4—74no—
327D4—72no—
328D4—141no—
329D4—141no—
330D4—141no—
331D4—282no—
332D4—282no—
333D4—282no—
334D4—282no—
335D4—561no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—71no—
342C2×C4—71no—
343C2×C4—71no—
344C2×C4—141no—
345C2×C4—141no—
346C2×C4—141no—
347C2×C4—141no—
348C2×C4—282no—
349C2×C4—561no—
350C2×C4—561no—
351C2×C4D4×D711yescharacteristic
352C2×C4C23×D711yescharacteristic
353C8—21no—
354C8—21no—
355C8—141no—
356C8—141no—
357C7#329511yescharacteristic, complemented, Sylow
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—72no—
364C22—141no—
365C22—141no—
366C22—142no—
367C22—281no—
368C22—281no—
369C22—281no—
370C22—281no—
371C22—281no—
372C22—564no—
373C22—564no—
374C22D7×SD1612yes—
375C22C2×D4×D711yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—72no—
381C4—141no—
382C4—141no—
383C4—281no—
384C4—281no—
385C4#654811yescharacteristic
386C4D7×#4111yescharacteristic
387C2—22no—
388C2—81no—
389C2—142no—
390C2—282no—
391C2—561no—
392C2#9030111yescharacteristic
393C2C2×D7×SD1611yescharacteristic
394C2#9046011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#465C2
socleC22×C7C22×D4
prime radical#90300C2
hypercenter#465D7
hypocenterC7#3295
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. #90326C2semidirect
3. C7×#465C2semidirect
4. #89941C2semidirect
5. #89935C2semidirect
6. C2×#6395C2semidirect
7. C8×C22⋊D7C2semidirect
8. #89818C2semidirect
9. #89806C2semidirect
10. #89642C2semidirect
11. #89635C2semidirect
12. #89579C2semidirect
13. #89560C2semidirect
14. D7×#102C2semidirect
15. #89532C2semidirect
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×C8×D7C22semidirect
35. D7×C4⋊C4C22semidirect
37. #6451C22semidirect
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. Dic7⋊C8C22semidirect
54. C8×Dic7C22semidirect
56. #6383C22semidirect
57. #6382C22semidirect
58. #6382C22semidirect
120. #465D7semidirect
154. C22⋊D7SD16semidirect
181. C2×SD16C2×D7semidirect
183. C22×C8C2×D7semidirect
190. C22⋊Q8C2×D7semidirect
204. #106C2×D7semidirect
206. #103C2×D7semidirect
215. #102C2×D7semidirect
357. C7#3295semidirect

Maximal subgroups

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

subgroupindex
#903262
#903002
C7×#4652
#899412
#899352
C2×#63952
C8×C22⋊D72
#898182
#898062
#896422
#896352
#895792
#895602
D7×#1022
#895322
#32957