Finite Group Databasebeta

← #11736958

Subgroups — #11736958

Order 896.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90243C211yesmaximal, characteristic
2#90234C211yesmaximal, characteristic, complemented
3C7×#435C211yesmaximal, characteristic, complemented
4#89909C211yesmaximal, characteristic, complemented
5#89905C211yesmaximal, characteristic, complemented
6#89886C211yesmaximal, characteristic, complemented
7C2×#6390C211yesmaximal, characteristic, complemented
8#89623C211yesmaximal, characteristic, complemented
9#89617C211yesmaximal, characteristic, complemented
10#89511C211yesmaximal, characteristic, complemented
11#89507C211yesmaximal, characteristic, complemented
12#89505C211yesmaximal, characteristic, complemented
13#89493C211yesmaximal, characteristic, complemented
14#89488C211yesmaximal, characteristic, complemented
15#89486C211yesmaximal, characteristic, complemented
16C2×#1157C2211yescharacteristic
17C22×Dic14C2211yescharacteristic
18C2×C7×M16C2211yescharacteristic
19C22×C7×C8C2211yescharacteristic
20C4×C7×D4C2211yescharacteristic
21#6502—21no—
22D4×Dic7—21no—
23#6493C2211yescharacteristic
24C2×C4⋊Dic7C2211yescharacteristic
25C2×C4⋊Dic7C2211yescharacteristic
26#6487C2211yescharacteristic
27#6487C2211yescharacteristic
28#6487C2211yescharacteristic
29C2×Dic28C2211yescharacteristic
30C2×Dic28C2211yescharacteristic
31C2×#1132C2211yescharacteristic
32#6459—21no—
33#6453—21no—
34#6448—21no—
35(C22⋊D7)⋊C4—21no—
36#6441—21no—
37#6441—21no—
38#6438C2211yescharacteristic
39#6432C2211yescharacteristic
40C4×Dic14C2211yescharacteristic
41C4×Dic14C2211yescharacteristic
42C7×C4⋊C8C2211yescharacteristic
43C7×C22⋊C8C2211yescharacteristic
44C7×C22⋊C8C2211yescharacteristic
45C7×#97C2211yescharacteristic
46#6395C2211yescharacteristic
47#6395C2211yescharacteristic
48#6395C2211yescharacteristic
49#6391C2211yescharacteristic
50#6391C2211yescharacteristic
51#6391C2211yescharacteristic
52#6390C2211yescharacteristic
53#6390C2211yescharacteristic
54#6390C2211yescharacteristic
55#6390C2211yescharacteristic
56#6390C2211yescharacteristic
57#6390C2211yescharacteristic
58#6390C2211yescharacteristic
59#3316—71nomaximal, Sylow
60C2×C7×D4D411yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C2×C22⋊D7—21no—
64C2×C22⋊D7—21no—
65C22×Dic7—21no—
66C22×Dic7—21no—
67C22×Dic7—21no—
68#1157—22no—
69#1157—41no—
70C2×D28C2311yescharacteristic
71C2×C4×D7—21no—
72C2×Dic14—22no—
73C2×Dic14—21no—
74C2×Dic14—21no—
75C2×Dic14—41no—
76C2×Dic14C2311yescharacteristic
77C2×Dic14C2311yescharacteristic
78C2×Dic14C2311yescharacteristic
79C7×M16—41no—
80C2×C7×C8—41no—
81C2×C7×C8C2311yescharacteristic
82C2×C7×C8C2311yescharacteristic
83C2×C7×C8C2311yescharacteristic
84C2×C7×C8C2311yescharacteristic
85C7×C4⋊C4D411yescharacteristic
86C7×C22⋊C4D411yescharacteristic
87C7×C22⋊C4D411yescharacteristic
88C42×C7C2311yescharacteristic
89C22⋊Dic7—21no—
90C22⋊Dic7—21no—
91C22⋊Dic7—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95C4⋊Dic7—21no—
96C4⋊Dic7—21no—
97C4⋊Dic7—21no—
98C4⋊Dic7C2311yescharacteristic
99C4⋊Dic7C2311yescharacteristic
100C4⋊Dic7C2311yescharacteristic
101C4⋊Dic7C2311yescharacteristic
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104Dic7⋊C4—21no—
105Dic7⋊C4—21no—
106Dic7⋊C4—21no—
107C4×Dic7—21no—
108C4×Dic7—21no—
109Dic28—22no—
110Dic28—41no—
111#1132—41no—
112#554—71no—
113#548—71no—
114#489—71no—
115#483—71no—
116#481—71no—
117#475—71no—
118#470—71no—
119#465—71no—
120#464—71no—
121#452—71no—
122#451—71no—
123#441—71no—
124#435D711yescharacteristic, complemented
125#417—71no—
126C2×#103—71no—
127C23×C7C2×D411yescharacteristic
128C23×C7C2×D411yescharacteristic
129C22×D7—21no—
130C7×D4—22no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7C2×D411yescharacteristic
136C2×C4×C7C2×D411yescharacteristic
137C2×C4×C7C2×D411yescharacteristic
138C2×C4×C7C2×D411yescharacteristic
139C2×C4×C7C2411yescharacteristic
140C22⋊D7—42no—
141C22⋊D7—42no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7—42no—
150C2×Dic7—81no—
151C2×Dic7—81no—
152D28—21no—
153D28—21no—
154C4×D7—42no—
155Dic14—21no—
156Dic14—21no—
157Dic14—21no—
158Dic14—21no—
159Dic14—21no—
160Dic14—42no—
161Dic14—81no—
162Dic14#4112yes—
163C7×C8—21no—
164C7×C8—21no—
165C7×C8—21no—
166C7×C8—21no—
167C2×#41—71no—
168C22×Q8—71no—
169C2×Q16—71no—
170C2×Q16—71no—
171C2×SD16—71no—
172C2×M16C2×D711yescharacteristic
173C22×C8C2×D711yescharacteristic
174C4⋊Q8—71no—
175#125—141no—
176#124—71no—
177#124—141no—
178C22⋊Q8—71no—
179C22⋊Q8—71no—
180C22⋊Q8—71no—
181C22⋊Q8—141no—
182C22⋊Q8—141no—
183C22⋊Q8—141no—
184#121—71no—
185#121—141no—
186C4×Q8—71no—
187C4×Q8—71no—
188C4×D4—141no—
189C4×D4—141no—
190C4×D4C2×D711yescharacteristic
191C2×C4⋊C4—71no—
192C2×C4⋊C4—71no—
193#106—71no—
194#106—71no—
195#106—71no—
196C4⋊C8C2×D711yescharacteristic
197#103—71no—
198#103—71no—
199#103—71no—
200#103—71no—
201#103—71no—
202#103—71no—
203#103—71no—
204#102—71no—
205#102—71no—
206#102—71no—
207C22⋊C8C2×D711yescharacteristic
208C22⋊C8C2×D711yescharacteristic
209#97C2×D711yescharacteristic
210C22×C7—21no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7—21no—
215C22×C7#13712yes—
216C22×C7C22×D411yescharacteristic
217C2×D7—41no—
218C2×D7—41no—
219C2×D7—41no—
220C4×C7—21no—
221C4×C7—21no—
222C4×C7—21no—
223C4×C7—21no—
224C4×C7C2×#4111yescharacteristic
225C4×C7H(5,𝔽2)11yescharacteristic
226Dic7—22no—
227Dic7—41no—
228Dic7—41no—
229Dic7—41no—
230Dic7—41no—
231Dic7—41no—
232Dic7—41no—
233#41—142no—
234#41—281no—
235C2×Q8—71no—
236C2×Q8—71no—
237C2×Q8—71no—
238C2×Q8—142no—
239C2×Q8—141no—
240C2×Q8—141no—
241C2×Q8—281no—
242C2×D4—71no—
243C2×D4—141no—
244C2×D4—141no—
245C2×D4D2811yescharacteristic
246C22×C4—141no—
247C22×C4—141no—
248C22×C4—141no—
249C22×C4—141no—
250C22×C4C22×D711yescharacteristic
251C22×C4C22×D711yescharacteristic
252Q16—142no—
253Q16—281no—
254SD16—281no—
255M16—41no—
256C2×C8—41no—
257C2×C8C22×D711yescharacteristic
258C2×C8C22×D711yescharacteristic
259C2×C8C22×D711yescharacteristic
260C2×C8C22×D711yescharacteristic
261C4⋊C4—71no—
262C4⋊C4—71no—
263C4⋊C4—71no—
264C4⋊C4—71no—
265C4⋊C4—141no—
266C4⋊C4—141no—
267C4⋊C4—141no—
268C4⋊C4—141no—
269C4⋊C4—141no—
270C4⋊C4—141no—
271C4⋊C4—141no—
272C4⋊C4—141no—
273C4⋊C4D2811yescharacteristic
274C22⋊C4—141no—
275C22⋊C4—141no—
276C22⋊C4—141no—
277C22⋊C4—141no—
278C22⋊C4—141no—
279C22⋊C4—141no—
280C22⋊C4D2811yescharacteristic
281C22⋊C4D2811yescharacteristic
282C42—141no—
283C42—141no—
284C42C22×D711yescharacteristic
285C2×C7—22no—
286C2×C7—41no—
287C2×C7#54611yescharacteristic
288C2×C7C2×#13711yescharacteristic
289C2×C7#57711yescharacteristic
290D7—81no—
291C23—141no—
292C23C2×D2811yescharacteristic
293C23C2×D2811yescharacteristic
294Q8—72no—
295Q8—141no—
296Q8—141no—
297Q8—141no—
298Q8—141no—
299Q8—141no—
300Q8—282no—
301Q8—561no—
302D4—22no—
303D4—141no—
304D4—141no—
305D4—282no—
306D4—282no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—141no—
312C2×C4—141no—
313C2×C4—141no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—141no—
317C2×C4—141no—
318C2×C4—282no—
319C2×C4—282no—
320C2×C4—561no—
321C2×C4—561no—
322C2×C4C2×D2811yescharacteristic
323C2×C4C2×D2811yescharacteristic
324C2×C4C2×D2811yescharacteristic
325C2×C4C2×D2811yescharacteristic
326C2×C4C23×D711yescharacteristic
327C8—21no—
328C8—21no—
329C8—21no—
330C8—21no—
331C7#331611yescharacteristic, complemented, Sylow
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—281no—
338C22—281no—
339C22—281no—
340C22#647212yes—
341C22C22×D2811yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—142no—
347C4—281no—
348C4—281no—
349C4—281no—
350C4—281no—
351C4—281no—
352C4—281no—
353C4#654811yescharacteristic
354C4D7×#4111yescharacteristic
355C2—22no—
356C2—41no—
357C2—561no—
358C2#9024511yescharacteristic
359C2C2×#647211yescharacteristic
360C2#9044211yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#435C2
socleC22×C7C22×D4
prime radical#90243C2
hypercenter#435D7
hypocenterC7#3316
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
2. #90234C2semidirect
3. C7×#435C2semidirect
4. #89909C2semidirect
5. #89905C2semidirect
6. #89886C2semidirect
7. C2×#6390C2semidirect
8. #89623C2semidirect
9. #89617C2semidirect
10. #89511C2semidirect
11. #89507C2semidirect
12. #89505C2semidirect
13. #89493C2semidirect
14. #89488C2semidirect
15. #89486C2semidirect
124. #435D7semidirect
331. C7#3316semidirect

Maximal subgroups

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

subgroupindex
#902432
#902342
C7×#4352
#899092
#899052
#898862
C2×#63902
#896232
#896172
#895112
#895072
#895052
#894932
#894882
#894862
#33167