Finite Group Databasebeta

← #11739159

Subgroups — #11739159

Order 896.

404 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90406C211yesmaximal, characteristic
2#90380C211yesmaximal, characteristic, complemented
3C7×#490C211yesmaximal, characteristic, complemented
4#89842C211yesmaximal, characteristic, complemented
5#89840C211yesmaximal, characteristic, complemented
6#89544C211yesmaximal, characteristic, complemented
7#89543C211yesmaximal, characteristic, complemented
8#89542C211yesmaximal, characteristic, complemented
9D7×#102C211yesmaximal, characteristic, complemented
10#89540C211yesmaximal, characteristic, complemented
11#89537C211yesmaximal, characteristic, complemented
12#89536C211yesmaximal, characteristic, complemented
13#89532C211yesmaximal, characteristic, complemented
14#89486C211yesmaximal, characteristic, complemented
15#89480C211yesmaximal, characteristic, complemented
16C2×#1159C2211yescharacteristic
17C2×D4×D7C2211yescharacteristic
18C7×#127C2211yescharacteristic
19C7×#125C2211yescharacteristic, complemented
20#6503C2211yescharacteristic
21#6502—21no—
22#6502—21no—
23#6500—21no—
24#6500—21no—
25#6499C2211yescharacteristic
26C2×#1131C2211yescharacteristic
27C2×C8×D7C2211yescharacteristic
28#6460—21no—
29#6459—21no—
30#6455C2211yescharacteristic
31D7×C4⋊C4C2211yescharacteristic
32#6453C2211yescharacteristic, complemented
33#6452—21no—
34#6451—21no—
35#6451C2211yescharacteristic, complemented
36#6438C2211yescharacteristic
37#6435C2211yescharacteristic
38#6432C2211yescharacteristic, complemented
39C7×#102C2211yescharacteristic
40C7×#102C2211yescharacteristic
41C7×#102C2211yescharacteristic, complemented
42C7×#102C2211yescharacteristic, complemented
43C7×#97C2211yescharacteristic, complemented
44#6406C2211yescharacteristic
45#6406C2211yescharacteristic
46#6406C2211yescharacteristic, complemented
47#6406C2211yescharacteristic, complemented
48#6395C2211yescharacteristic
49#6395C2211yescharacteristic
50#6390C2211yescharacteristic, complemented
51#6390C2211yescharacteristic, complemented
52#6389C2211yescharacteristic, complemented
53C8×Dic7C2211yescharacteristic, complemented
54#6384C2211yescharacteristic, complemented
55#6384C2211yescharacteristic, complemented
56#6383C2211yescharacteristic
57#6383C2211yescharacteristic
58C4×C7⋊C8C2211yescharacteristic, complemented
59#3136—71nomaximal, Sylow
60C23×D7—21no—
61C2×C7×D4—21no—
62C2×C7×D4—21no—
63C2×C7×D4C2311yescharacteristic
64C2×C7×D4C2311yescharacteristic
65C2×C22⋊D7—21no—
66C2×C22⋊D7—21no—
67C2×C22⋊D7—21no—
68C2×C22⋊D7—21no—
69C22×Dic7—21no—
70#1159—22no—
71#1159—22no—
72D4×D7—22no—
73D4×D7—22no—
74C2×D28C2311yescharacteristic
75C2×C4×D7—21no—
76C2×C4×D7C2311yescharacteristic
77C2×Dic14—21no—
78C2×Dic14—21no—
79C2×Dic14C2311yescharacteristic
80C2×C7×C8C2311yescharacteristic
81C2×C7×C8C2311yescharacteristic
82C7×C4⋊C4—21no—
83C7×C4⋊C4—21no—
84C7×C4⋊C4C2311yescharacteristic
85C7×C4⋊C4C2311yescharacteristic
86C42×C7C2311yescharacteristic
87C22⋊Dic7—21no—
88C22⋊Dic7—21no—
89(C2×D7)⋊C4—21no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4D411yescharacteristic
93(C2×D7)⋊C4D411yescharacteristic
94C4⋊Dic7—21no—
95C4⋊Dic7—21no—
96C4⋊Dic7C2311yescharacteristic
97C4⋊Dic7C2311yescharacteristic
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4D411yescharacteristic
102Dic7⋊C4D411yescharacteristic
103C4×Dic7—21no—
104C4×Dic7C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107#1131—22no—
108C8×D7—22no—
109#544—71no—
110#535—71no—
111#490—71no—
112#490D711yescharacteristic, complemented
113#489—71no—
114#489—71no—
115#488—71no—
116#488—71no—
117#486—71no—
118#486—71no—
119#418—71no—
120#417—71no—
121#416—71no—
122C2×#102—71no—
123#405—71no—
124C23×C7—21no—
125C23×C7—21no—
126C22×D7—21no—
127C22×D7—42no—
128C22×D7—81no—
129C22×D7—81no—
130C22×D7C2×D411yescharacteristic
131C7×D4—21no—
132C7×D4—21no—
133C7×D4—21no—
134C7×D4—21no—
135C7×D4—81no—
136C7×D4—81no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7C2×D411yescharacteristic
140C2×C4×C7C2×D411yescharacteristic
141C2×C4×C7C2411yescharacteristic
142C22⋊D7—42no—
143C22⋊D7—42no—
144C22⋊D7—81no—
145C22⋊D7—81no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7—42no—
150C2×Dic7C2×D411yescharacteristic
151C2×Dic7C2×D411yescharacteristic
152C2×Dic7C2×D411yescharacteristic
153D28—21no—
154D28—21no—
155C4×D7—81no—
156C4×D7#4112yes—
157C4×D7#4112yes—
158Dic14—21no—
159Dic14—21no—
160Dic14—81no—
161Dic14—81no—
162C7×C8—21no—
163C7×C8—21no—
164C7⋊C8—21no—
165C7⋊C8—21no—
166C2×#41—71no—
167C22×D4—71no—
168C2×M16—71no—
169C22×C8—71no—
170C4⋊Q8—71no—
171C4⋊Q8—71no—
172C4⋊Q8—141no—
173#127—71no—
174#127C2×D711yescharacteristic
175#125—71no—
176#125—141no—
177#125C2×D711yescharacteristic, complemented
178#124—71no—
179#124—71no—
180C22⋊Q8—141no—
181C22⋊Q8—141no—
182#121—141no—
183#121—141no—
184#120—141no—
185#120—141no—
186#117—71no—
187#117—71no—
188C2×C4⋊C4—71no—
189#103—71no—
190#103—71no—
191#103—71no—
192#103—71no—
193#102—71no—
194#102—71no—
195#102—71no—
196#102—71no—
197#102—71no—
198#102—71no—
199#102—71no—
200#102—71no—
201#102C2×D711yescharacteristic
202#102C2×D711yescharacteristic
203#102C2×D711yescharacteristic, complemented
204#102C2×D711yescharacteristic, complemented
205#97—71no—
206#97C2×D711yescharacteristic, complemented
207C4×C8—71no—
208C4×C8—71no—
209C22×C7—41no—
210C22×C7—41no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7C22×D411yescharacteristic
216C2×D7—21no—
217C2×D7—21no—
218C2×D7—41no—
219C2×D7—41no—
220C2×D7—41no—
221C2×D7—84no—
222C2×D7#12412yes—
223C4×C7—21no—
224C4×C7—21no—
225C4×C7—41no—
226C4×C7—41no—
227C4×C7C2×#4111yescharacteristic
228C4×C7C2×#4111yescharacteristic
229Dic7—21no—
230Dic7—21no—
231Dic7—41no—
232Dic7—41no—
233Dic7—41no—
234Dic7#12412yes—
235C24—141no—
236#41—142no—
237#41—142no—
238C2×Q8—71no—
239C2×Q8—141no—
240C2×Q8—141no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—71no—
244C2×D4—142no—
245C2×D4—142no—
246C2×D4—141no—
247C2×D4—141no—
248C2×D4—141no—
249C2×D4—141no—
250C2×D4C22×D711yescharacteristic
251C2×D4C22×D711yescharacteristic
252C22×C4—71no—
253C22×C4—141no—
254C22×C4—141no—
255M16—142no—
256C2×C8—71no—
257C2×C8—71no—
258C2×C8—142no—
259C2×C8C22×D711yescharacteristic
260C2×C8C22×D711yescharacteristic
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—71no—
264C4⋊C4—71no—
265C4⋊C4—71no—
266C4⋊C4—71no—
267C4⋊C4—141no—
268C4⋊C4—141no—
269C4⋊C4—141no—
270C4⋊C4—141no—
271C4⋊C4—141no—
272C4⋊C4C22×D711yescharacteristic
273C4⋊C4C22×D711yescharacteristic
274C22⋊C4—71no—
275C22⋊C4—71no—
276C22⋊C4—141no—
277C22⋊C4—141no—
278C22⋊C4—141no—
279C22⋊C4—141no—
280C22⋊C4—141no—
281C42—71no—
282C42—141no—
283C42C22×D711yescharacteristic
284C2×C7—81no—
285C2×C7—81no—
286C2×C7C2×#12411yescharacteristic
287C2×C7#57611yescharacteristic
288C2×C7#57711yescharacteristic
289D7—22no—
290D7—81no—
291C23—21no—
292C23—21no—
293C23—71no—
294C23—141no—
295C23—282no—
296C23—561no—
297C23—561no—
298Q8—141no—
299Q8—141no—
300Q8—561no—
301Q8—561no—
302D4—21no—
303D4—21no—
304D4—21no—
305D4—21no—
306D4—81no—
307D4—81no—
308D4—141no—
309D4—141no—
310D4—282no—
311D4—282no—
312D4—561no—
313D4—561no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—71no—
317C2×C4—72no—
318C2×C4—72no—
319C2×C4—71no—
320C2×C4—71no—
321C2×C4—141no—
322C2×C4—141no—
323C2×C4—141no—
324C2×C4—282no—
325C2×C4—561no—
326C2×C4D4×D711yescharacteristic
327C2×C4D4×D711yescharacteristic
328C2×C4C23×D711yescharacteristic
329C8—21no—
330C8—21no—
331C8—141no—
332C8—141no—
333C7#313611yescharacteristic, complemented, Sylow
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—72no—
341C22—141no—
342C22—141no—
343C22—281no—
344C22—281no—
345C22—281no—
346C22—564no—
347C22C2×D4×D711yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—72no—
353C4—141no—
354C4—141no—
355C4—281no—
356C4—281no—
357C4—281no—
358C4D7×#4111yescharacteristic
359C4D7×#4111yescharacteristic
360C2—81no—
361C2—81no—
362C2—142no—
363C2—561no—
364C2D7×#12411yescharacteristic
365C2#9046511yescharacteristic
366C2#9046611yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#490C2
socleC22×C7C22×D4
prime radical#90406C2
hypercenter#490D7
hypocenterC7#3136
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
2. #90380C2semidirect
3. C7×#490C2semidirect
4. #89842C2semidirect
5. #89840C2semidirect
6. #89544C2semidirect
7. #89543C2semidirect
8. #89542C2semidirect
9. D7×#102C2semidirect
10. #89540C2semidirect
11. #89537C2semidirect
12. #89536C2semidirect
13. #89532C2semidirect
14. #89486C2semidirect
15. #89480C2semidirect
19. C7×#125C22semidirect
32. #6453C22semidirect
35. #6451C22semidirect
38. #6432C22semidirect
41. C7×#102C22semidirect
42. C7×#102C22semidirect
43. C7×#97C22semidirect
46. #6406C22semidirect
47. #6406C22semidirect
50. #6390C22semidirect
51. #6390C22semidirect
52. #6389C22semidirect
53. C8×Dic7C22semidirect
54. #6384C22semidirect
55. #6384C22semidirect
58. C4×C7⋊C8C22semidirect
112. #490D7semidirect
177. #125C2×D7semidirect
203. #102C2×D7semidirect
204. #102C2×D7semidirect
206. #97C2×D7semidirect
333. C7#3136semidirect

Maximal subgroups

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

subgroupindex
#904062
#903802
C7×#4902
#898422
#898402
#895442
#895432
#895422
D7×#1022
#895402
#895372
#895362
#895322
#894862
#894802
#31367