Finite Group Databasebeta

← #11738113

Subgroups — #11738113

Order 896.

410 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#90319C211yesmaximal, characteristic, complemented
2#90298C211yesmaximal, characteristic
3C7×#465C211yesmaximal, characteristic, complemented
4#89942C211yesmaximal, characteristic, complemented
5#89941C211yesmaximal, characteristic, complemented
6#89886C211yesmaximal, characteristic, complemented
7#89882C211yesmaximal, characteristic, complemented
8#89821C211yesmaximal, characteristic, complemented
9#89814C211yesmaximal, characteristic, complemented
10#89639C211yesmaximal, characteristic, complemented
11#89638C211yesmaximal, characteristic, complemented
12#89587C211yesmaximal, characteristic, complemented
13#89578C211yesmaximal, characteristic, complemented
14#89551C211yesmaximal, characteristic, complemented
15#89548C211yesmaximal, characteristic, complemented
16C2×Q8×D7C2211yescharacteristic, complemented
17C2×D4×D7C2211yescharacteristic
18C2×#1157C2211yescharacteristic
19C2×C7×SD16C2211yescharacteristic
20C22×C7×C8C2211yescharacteristic, complemented
21C7×C22⋊Q8C2211yescharacteristic, complemented
22C7×#121C2211yescharacteristic
23#6509C2211yescharacteristic
24C2×#1143C2211yescharacteristic
25#6503—21no—
26#6501C2211yescharacteristic
27#6500—21no—
28C2×#1142C2211yescharacteristic
29#6493C2211yescharacteristic
30#6487C2211yescharacteristic, complemented
31C2×#1136C2211yescharacteristic, complemented
32C2×#1132C2211yescharacteristic
33#6460C2211yescharacteristic
34#6458C2211yescharacteristic
35#6457—21no—
36#6455C2211yescharacteristic
37#6455C2211yescharacteristic, complemented
38#6451—21no—
39#6448—21no—
40#6448—21no—
41#6446—21no—
42#6445—21no—
43C7×#106C2211yescharacteristic, complemented
44C7×#103C2211yescharacteristic, complemented
45C7×#102C2211yescharacteristic
46#6409C2211yescharacteristic, complemented
47#6406C2211yescharacteristic
48#6395C2211yescharacteristic
49(C2×D7)⋊C8C2211yescharacteristic
50(C2×D7)⋊C8C2211yescharacteristic
51(C2×D7)⋊C8C2211yescharacteristic, complemented
52(C2×D7)⋊C8C2211yescharacteristic, complemented
53#6391C2211yescharacteristic, complemented
54#6390C2211yescharacteristic, complemented
55#6384C2211yescharacteristic, complemented
56#6383C2211yescharacteristic
57#6381C2211yescharacteristic, complemented
58#6381C2211yescharacteristic, complemented
59#3200—71nomaximal, Sylow
60C23×D7—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4—21no—
63C2×C7×D4C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C2×C22⋊D7—21no—
66C2×C22⋊D7—21no—
67C2×C22⋊D7—21no—
68C2×C22⋊D7D411yescharacteristic
69C2×C22⋊D7D411yescharacteristic
70Q8×D7—41no—
71Q8×D7—41no—
72D4×D7—41no—
73D4×D7—41no—
74#1157—41no—
75#1157—41no—
76C2×D28—21no—
77C2×D28D411yescharacteristic
78C2×D28C2311yescharacteristic
79C2×C4×D7—21no—
80C2×C4×D7C2311yescharacteristic
81C2×C4×D7C2311yescharacteristic
82C2×Dic14—21no—
83C2×Dic14D411yescharacteristic
84C2×Dic14C2311yescharacteristic
85C7×SD16—41no—
86C2×C7×C8—41no—
87C2×C7×C8C2311yescharacteristic
88C2×C7×C8C2311yescharacteristic
89C7×C4⋊C4—21no—
90C7×C4⋊C4C2311yescharacteristic
91C7×C4⋊C4C2311yescharacteristic
92C7×C22⋊C4—21no—
93C7×C22⋊C4—21no—
94C22⋊Dic7—21no—
95C22⋊Dic7—21no—
96#1143—41no—
97#1142—41no—
98(C2×D7)⋊C4—21no—
99(C2×D7)⋊C4—21no—
100(C2×D7)⋊C4—21no—
101(C2×D7)⋊C4—21no—
102(C2×D7)⋊C4—21no—
103(C2×D7)⋊C4—21no—
104C4⋊Dic7—21no—
105C4⋊Dic7C2311yescharacteristic
106C4⋊Dic7C2311yescharacteristic
107Dic7⋊C4—21no—
108Dic7⋊C4—21no—
109C4×Dic7—21no—
110C4×Dic7—21no—
111#1136—41no—
112C2×C7⋊C8C2311yescharacteristic
113C2×C7⋊C8C2311yescharacteristic
114#1132—41no—
115#536—71no—
116#535—71no—
117#485—71no—
118#485—71no—
119#482—71no—
120#482—71no—
121#468—71no—
122#468—71no—
123#465—71no—
124#465D711yescharacteristic, complemented
125#450—71no—
126#450—71no—
127#448—71no—
128#448—71no—
129#408—71no—
130C23×C7—21no—
131C23×C7C2×D411yescharacteristic
132C22×D7—21no—
133C22×D7—42no—
134C22×D7—81no—
135C22×D7—81no—
136C22×D7C2×D411yescharacteristic
137C22×D7C2×D411yescharacteristic
138C7×Q8—21no—
139C7×Q8—21no—
140C7×D4—21no—
141C7×D4—21no—
142C7×D4—81no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7C2×D411yescharacteristic
149C2×C4×C7C2411yescharacteristic
150C22⋊D7—41no—
151C22⋊D7—41no—
152C22⋊D7—42no—
153C22⋊D7—81no—
154C22⋊D7—81no—
155C2×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7—21no—
158C2×Dic7C2×D411yescharacteristic
159C2×Dic7C2×D411yescharacteristic
160D28—21no—
161D28—21no—
162D28—41no—
163D28—81no—
164C4×D7—21no—
165C4×D7—21no—
166C4×D7—21no—
167C4×D7—21no—
168C4×D7—42no—
169Dic14—21no—
170Dic14—21no—
171Dic14—41no—
172Dic14—42no—
173C7×C8—21no—
174C7×C8—21no—
175C7⋊C8—21no—
176C7⋊C8—21no—
177C2×#41—71no—
178C22×Q8—71no—
179C22×D4—71no—
180C2×SD16—71no—
181C2×SD16—71no—
182C2×SD16—71no—
183C2×SD16C2×D711yescharacteristic
184C2×M16—71no—
185C22×C8C2×D711yescharacteristic, complemented
186C4⋊Q8—141no—
187#127—141no—
188#124—141no—
189#124—141no—
190#123—141no—
191#123—141no—
192C22⋊Q8—71no—
193C22⋊Q8—71no—
194C22⋊Q8—71no—
195C22⋊Q8C2×D711yescharacteristic, complemented
196#121—71no—
197#121—71no—
198#121—71no—
199#121C2×D711yescharacteristic
200#120—141no—
201#120—141no—
202#117—71no—
203#117—71no—
204#107—71no—
205#107—71no—
206#106—71no—
207#106C2×D711yescharacteristic, complemented
208#103—71no—
209#103—71no—
210#103—71no—
211#103C2×D711yescharacteristic, complemented
212#102—71no—
213#102—71no—
214#102—71no—
215#102C2×D711yescharacteristic
216C22⋊C8—71no—
217C22⋊C8—71no—
218C22⋊C8—71no—
219C22⋊C8—71no—
220C22×C7—21no—
221C22×C7—21no—
222C22×C7—21no—
223C22×C7—41no—
224C22×C7—41no—
225C22×C7—41no—
226C22×C7C22×D411yescharacteristic
227C2×D7—21no—
228C2×D7—21no—
229C2×D7—21no—
230C2×D7—21no—
231C2×D7—21no—
232C2×D7—21no—
233C2×D7—41no—
234C2×D7—41no—
235C2×D7—41no—
236C2×D7—84no—
237C4×C7—21no—
238C4×C7—41no—
239C4×C7—41no—
240C4×C7—41no—
241C4×C7H(5,𝔽2)11yescharacteristic
242C4×C7H(5,𝔽2)11yescharacteristic
243Dic7—21no—
244Dic7—21no—
245Dic7—41no—
246Dic7—41no—
247Dic7—41no—
248C24—141no—
249#41—281no—
250#41—281no—
251C2×Q8—71no—
252C2×Q8—71no—
253C2×Q8—141no—
254C2×Q8—281no—
255C2×Q8—281no—
256C2×Q8C22×D711yescharacteristic
257C2×D4—21no—
258C2×D4—71no—
259C2×D4—71no—
260C2×D4—71no—
261C2×D4—71no—
262C2×D4—141no—
263C2×D4—141no—
264C2×D4—141no—
265C2×D4—141no—
266C2×D4—281no—
267C2×D4—281no—
268C2×D4C22×D711yescharacteristic
269C22×C4—71no—
270C22×C4—71no—
271C22×C4—141no—
272C22×C4C22×D711yescharacteristic
273SD16—41no—
274SD16—281no—
275SD16—281no—
276SD16—281no—
277M16—281no—
278C2×C8—41no—
279C2×C8—71no—
280C2×C8—71no—
281C2×C8C22×D711yescharacteristic
282C2×C8C22×D711yescharacteristic
283C4⋊C4—21no—
284C4⋊C4—71no—
285C4⋊C4—71no—
286C4⋊C4—141no—
287C4⋊C4—141no—
288C4⋊C4—141no—
289C4⋊C4C22×D711yescharacteristic
290C4⋊C4C22×D711yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—141no—
294C22⋊C4—141no—
295C22⋊C4—141no—
296C22⋊C4—141no—
297C22⋊C4—141no—
298C22⋊C4—141no—
299C22⋊C4—141no—
300C22⋊C4—141no—
301C42—141no—
302C42—141no—
303C2×C7—41no—
304C2×C7—81no—
305C2×C7#53411yescharacteristic
306C2×C7#57711yescharacteristic
307C2×C7#57711yescharacteristic
308D7—41no—
309D7—41no—
310D7—81no—
311C23—21no—
312C23—71no—
313C23—71no—
314C23—141no—
315C23—282no—
316C23—561no—
317C23—561no—
318C23D4×D711yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—141no—
322Q8—141no—
323Q8—281no—
324Q8—282no—
325D4—21no—
326D4—21no—
327D4—81no—
328D4—141no—
329D4—141no—
330D4—281no—
331D4—281no—
332D4—281no—
333D4—282no—
334D4—561no—
335D4—561no—
336D4—561no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—71no—
343C2×C4—71no—
344C2×C4—141no—
345C2×C4—141no—
346C2×C4—141no—
347C2×C4—141no—
348C2×C4—141no—
349C2×C4—141no—
350C2×C4—141no—
351C2×C4—282no—
352C2×C4D4×D711yescharacteristic
353C2×C4C23×D711yescharacteristic
354C8—21no—
355C8—21no—
356C8—141no—
357C8—141no—
358C7#320011yescharacteristic, complemented, Sylow
359C22—21no—
360C22—21no—
361C22—21no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—141no—
366C22—141no—
367C22—141no—
368C22—141no—
369C22—141no—
370C22—141no—
371C22—281no—
372C22—281no—
373C22—281no—
374C22—564no—
375C22C2×D4×D711yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—141no—
381C4—141no—
382C4—281no—
383C4—281no—
384C4—281no—
385C4#654811yescharacteristic
386C4#655311yescharacteristic
387C2—41no—
388C2—81no—
389C2—281no—
390C2—281no—
391C2—561no—
392C2#9029611yescharacteristic
393C2#9045311yescharacteristic
394C2#9046111yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#465C2
socleC22×C7C22×D4
prime radical#90298C2
hypercenter#465D7
hypocenterC7#3200
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. #90319C2semidirect
3. C7×#465C2semidirect
4. #89942C2semidirect
5. #89941C2semidirect
6. #89886C2semidirect
7. #89882C2semidirect
8. #89821C2semidirect
9. #89814C2semidirect
10. #89639C2semidirect
11. #89638C2semidirect
12. #89587C2semidirect
13. #89578C2semidirect
14. #89551C2semidirect
15. #89548C2semidirect
16. C2×Q8×D7C22semidirect
20. C22×C7×C8C22semidirect
21. C7×C22⋊Q8C22semidirect
30. #6487C22semidirect
31. C2×#1136C22semidirect
37. #6455C22semidirect
43. C7×#106C22semidirect
44. C7×#103C22semidirect
46. #6409C22semidirect
51. (C2×D7)⋊C8C22semidirect
52. (C2×D7)⋊C8C22semidirect
53. #6391C22semidirect
54. #6390C22semidirect
55. #6384C22semidirect
57. #6381C22semidirect
58. #6381C22semidirect
124. #465D7semidirect
185. C22×C8C2×D7semidirect
195. C22⋊Q8C2×D7semidirect
207. #106C2×D7semidirect
211. #103C2×D7semidirect
358. C7#3200semidirect

Maximal subgroups

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

subgroupindex
#903192
#902982
C7×#4652
#899422
#899412
#898862
#898822
#898212
#898142
#896392
#896382
#895872
#895782
#895512
#895482
#32007