Finite Group Databasebeta

← #11736957

Subgroups — #11736957

Order 896.

380 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90243C211yesmaximal, characteristic
2#90233C211yesmaximal, characteristic, complemented
3C7×#435C211yesmaximal, characteristic, complemented
4#89906C211yesmaximal, characteristic, complemented
5#89892C211yesmaximal, characteristic, complemented
6#89888C211yesmaximal, characteristic, complemented
7#89885C211yesmaximal, characteristic, complemented
8#89622C211yesmaximal, characteristic, complemented
9#89617C211yesmaximal, characteristic, complemented
10#89511C211yesmaximal, characteristic, complemented
11#89509C211yesmaximal, characteristic, complemented
12#89505C211yesmaximal, characteristic, complemented
13#89495C211yesmaximal, 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—
22#6499—21no—
23#6493C2211yescharacteristic
24#6489C2211yescharacteristic
25#6489C2211yescharacteristic
26#6487C2211yescharacteristic
27#6487C2211yescharacteristic
28#6487C2211yescharacteristic
29C2×Dic28C2211yescharacteristic
30C2×Dic28C2211yescharacteristic
31C2×#1132C2211yescharacteristic
32#6459—21no—
33#6450—21no—
34#6448—21no—
35(C22⋊D7)⋊C4—21no—
36#6442—21no—
37#6442—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#6392C2211yescharacteristic
50#6392C2211yescharacteristic
51#6391C2211yescharacteristic
52#6390C2211yescharacteristic
53#6390C2211yescharacteristic
54#6390C2211yescharacteristic
55#6390C2211yescharacteristic
56#6390C2211yescharacteristic
57#6390C2211yescharacteristic
58#6390C2211yescharacteristic
59#3313—71nomaximal, Sylow
60C2×C7×D4D411yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C2×C22⋊D7—21no—
64C2×C22⋊D7—21no—
65C22×Dic7—21no—
66#1157—22no—
67#1157—41no—
68C2×D28C2311yescharacteristic
69C2×C4×D7—21no—
70C2×Dic14—22no—
71C2×Dic14—21no—
72C2×Dic14—21no—
73C2×Dic14—41no—
74C2×Dic14C2311yescharacteristic
75C2×Dic14C2311yescharacteristic
76C2×Dic14C2311yescharacteristic
77C7×M16—41no—
78C2×C7×C8—41no—
79C2×C7×C8C2311yescharacteristic
80C2×C7×C8C2311yescharacteristic
81C2×C7×C8C2311yescharacteristic
82C2×C7×C8C2311yescharacteristic
83C7×C4⋊C4D411yescharacteristic
84C7×C22⋊C4D411yescharacteristic
85C7×C22⋊C4D411yescharacteristic
86C42×C7C2311yescharacteristic
87C22⋊Dic7—21no—
88C22⋊Dic7—21no—
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⋊Dic7C2311yescharacteristic
97C4⋊Dic7C2311yescharacteristic
98C4⋊Dic7C2311yescharacteristic
99C4⋊Dic7C2311yescharacteristic
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104Dic7⋊C4—21no—
105C4×Dic7—21no—
106C4×Dic7—21no—
107C4×Dic7—21no—
108C4×Dic7—21no—
109Dic28—22no—
110Dic28—41no—
111#1132—41no—
112#556—71no—
113#548—71no—
114#489—71no—
115#485—71no—
116#482—71no—
117#477—71no—
118#468—71no—
119#467—71no—
120#464—71no—
121#452—71no—
122#451—71no—
123#441—71no—
124#435D711yescharacteristic, complemented
125#419—71no—
126#416—71no—
127C23×C7C2×D411yescharacteristic
128C23×C7C2×D411yescharacteristic
129C22×D7—21no—
130C7×D4—41no—
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—
150D28—21no—
151D28—21no—
152C4×D7—42no—
153Dic14—21no—
154Dic14—21no—
155Dic14—21no—
156Dic14—21no—
157Dic14—21no—
158Dic14—42no—
159Dic14—42no—
160Dic14#4112yes—
161C7×C8—21no—
162C7×C8—21no—
163C7×C8—21no—
164C7×C8—21no—
165C2×#41—71no—
166C22×Q8—71no—
167C2×Q16—71no—
168C2×Q16—71no—
169C2×SD16—71no—
170C2×M16C2×D711yescharacteristic
171C22×C8C2×D711yescharacteristic
172C4⋊Q8—71no—
173#126—141no—
174#126—141no—
175#124—71no—
176#124—141no—
177#124—141no—
178C22⋊Q8—71no—
179C22⋊Q8—71no—
180C22⋊Q8—71no—
181C22⋊Q8—141no—
182#121—71no—
183#121—141no—
184C4×Q8—71no—
185C4×Q8—71no—
186C4×Q8—141no—
187C4×D4—141no—
188C4×D4C2×D711yescharacteristic
189#117—71no—
190#117—71no—
191#107—71no—
192#107—71no—
193#106—71no—
194C4⋊C8C2×D711yescharacteristic
195#103—71no—
196#103—71no—
197#103—71no—
198#103—71no—
199#103—71no—
200#103—71no—
201#103—71no—
202#102—71no—
203#102—71no—
204#102—71no—
205C22⋊C8C2×D711yescharacteristic
206C22⋊C8C2×D711yescharacteristic
207#97C2×D711yescharacteristic
208C22×C7—21no—
209C22×C7—21no—
210C22×C7—21no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7C22×D411yescharacteristic
215C2×D7—41no—
216C2×D7—41no—
217C2×D7—41no—
218C4×C7—21no—
219C4×C7—21no—
220C4×C7—21no—
221C4×C7—21no—
222C4×C7C2×#4111yescharacteristic
223C4×C7H(5,𝔽2)11yescharacteristic
224Dic7—22no—
225Dic7—41no—
226Dic7—41no—
227Dic7—41no—
228Dic7—41no—
229Dic7—41no—
230Dic7—41no—
231#41—142no—
232#41—281no—
233C2×Q8—71no—
234C2×Q8—71no—
235C2×Q8—71no—
236C2×Q8—142no—
237C2×Q8—141no—
238C2×Q8—141no—
239C2×Q8—281no—
240C2×D4—71no—
241C2×D4—141no—
242C2×D4—141no—
243C2×D4D2811yescharacteristic
244C22×C4—141no—
245C22×C4—141no—
246C22×C4C22×D711yescharacteristic
247C22×C4C22×D711yescharacteristic
248Q16—142no—
249Q16—281no—
250SD16—281no—
251M16—41no—
252C2×C8—41no—
253C2×C8C22×D711yescharacteristic
254C2×C8C22×D711yescharacteristic
255C2×C8C22×D711yescharacteristic
256C2×C8C22×D711yescharacteristic
257C4⋊C4—71no—
258C4⋊C4—71no—
259C4⋊C4—71no—
260C4⋊C4—71no—
261C4⋊C4—141no—
262C4⋊C4—141no—
263C4⋊C4—141no—
264C4⋊C4—141no—
265C4⋊C4—141no—
266C4⋊C4—141no—
267C4⋊C4D2811yescharacteristic
268C22⋊C4—141no—
269C22⋊C4—141no—
270C22⋊C4—141no—
271C22⋊C4—141no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C22⋊C4D2811yescharacteristic
277C22⋊C4D2811yescharacteristic
278C42—141no—
279C42—141no—
280C42—141no—
281C42—141no—
282C42C22×D711yescharacteristic
283C2×C7—41no—
284C2×C7—41no—
285C2×C7#54611yescharacteristic
286C2×C7#57511yescharacteristic
287C2×C7#57811yescharacteristic
288D7—81no—
289C23—141no—
290C23C2×D2811yescharacteristic
291C23C2×D2811yescharacteristic
292Q8—72no—
293Q8—141no—
294Q8—141no—
295Q8—141no—
296Q8—141no—
297Q8—141no—
298Q8—282no—
299Q8—282no—
300D4—41no—
301D4—141no—
302D4—141no—
303D4—282no—
304D4—282no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—141no—
310C2×C4—141no—
311C2×C4—141no—
312C2×C4—141no—
313C2×C4—141no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—282no—
317C2×C4—282no—
318C2×C4C2×D2811yescharacteristic
319C2×C4C2×D2811yescharacteristic
320C2×C4C2×D2811yescharacteristic
321C2×C4C2×D2811yescharacteristic
322C2×C4C23×D711yescharacteristic
323C8—21no—
324C8—21no—
325C8—21no—
326C8—21no—
327C7#331311yescharacteristic, complemented, Sylow
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—281no—
335C22—281no—
336C22—281no—
337C22C22×D2811yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—142no—
343C4—281no—
344C4—281no—
345C4—281no—
346C4—281no—
347C4—281no—
348C4—281no—
349C4#654811yescharacteristic
350C4D7×#4111yescharacteristic
351C2—41no—
352C2—41no—
353C2—561no—
354C2#9024511yescharacteristic
355C2#9043911yescharacteristic
356C2#9044411yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#435C2
socleC22×C7C22×D4
prime radical#90243C2
hypercenter#435D7
hypocenterC7#3313
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. #90233C2semidirect
3. C7×#435C2semidirect
4. #89906C2semidirect
5. #89892C2semidirect
6. #89888C2semidirect
7. #89885C2semidirect
8. #89622C2semidirect
9. #89617C2semidirect
10. #89511C2semidirect
11. #89509C2semidirect
12. #89505C2semidirect
13. #89495C2semidirect
14. #89488C2semidirect
15. #89486C2semidirect
124. #435D7semidirect
327. C7#3313semidirect

Maximal subgroups

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

subgroupindex
#902432
#902332
C7×#4352
#899062
#898922
#898882
#898852
#896222
#896172
#895112
#895092
#895052
#894952
#894882
#894862
#33137