Finite Group Databasebeta

← #11739111

Subgroups — #11739111

Order 896.

362 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90424C211yesmaximal, characteristic, complemented
2#90373C211yesmaximal, characteristic, complemented
3C7×#489C211yesmaximal, characteristic, complemented
4#89851C211yesmaximal, characteristic, complemented
5#89831C211yesmaximal, characteristic, complemented
6#89588C211yesmaximal, characteristic, complemented
7#89579C211yesmaximal, characteristic, complemented
8#89561C211yesmaximal, characteristic, complemented
9#89560C211yesmaximal, characteristic, complemented
10#89557C211yesmaximal, characteristic, complemented
11#89554C211yesmaximal, characteristic
12#89550C211yesmaximal, characteristic, complemented
13#89530C211yesmaximal, characteristic, complemented
14#89486C211yesmaximal, characteristic, complemented
15#89480C211yesmaximal, characteristic, complemented
16C7×C4⋊Q8C2211yescharacteristic
17C7×#124C2211yescharacteristic
18#6510—21no—
19#6509—21no—
20#6509C2211yescharacteristic
21Q8×Dic7C2211yescharacteristic
22C2×#1144C2211yescharacteristic
23C2×#1143C2211yescharacteristic
24#6501C2211yescharacteristic
25D4×Dic7C2211yescharacteristic
26C2×#1142C2211yescharacteristic
27C2×#1141C2211yescharacteristic
28C2×#1131C2211yescharacteristic
29C2×C8×D7C2211yescharacteristic
30#6461—21no—
31#6460C2211yescharacteristic
32#6458C2211yescharacteristic
33#6457—21no—
34#6456C2211yescharacteristic
35#6450C2211yescharacteristic
36#6449—21no—
37#6448—21no—
38#6442—21no—
39#6441—21no—
40#6438C2211yescharacteristic
41#6435C2211yescharacteristic
42#6432C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×#103C2211yescharacteristic
45C7×#102C2211yescharacteristic
46C7×#102C2211yescharacteristic
47C7×#97C2211yescharacteristic
48#6395C2211yescharacteristic
49#6395C2211yescharacteristic
50#6390C2211yescharacteristic
51#6390C2211yescharacteristic
52#6389C2211yescharacteristic
53C8×Dic7C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6382C2211yescharacteristic
57#6382C2211yescharacteristic
58C4×C7⋊C8C2211yescharacteristic
59#3173—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4C2311yescharacteristic
63C2×C22⋊D7—21no—
64C22×Dic7—21no—
65C2×D28—21no—
66C2×D28C2311yescharacteristic
67C2×C4×D7—21no—
68C2×C4×D7C2311yescharacteristic
69C2×Dic14—21no—
70C2×Dic14C2311yescharacteristic
71C2×C7×C8C2311yescharacteristic
72C2×C7×C8C2311yescharacteristic
73C7×C4⋊C4—21no—
74C7×C4⋊C4C2311yescharacteristic
75C7×C4⋊C4C2311yescharacteristic
76C7×C22⋊C4—21no—
77C7×C22⋊C4—21no—
78C42×C7C2311yescharacteristic
79C22⋊Dic7—21no—
80C22⋊Dic7—21no—
81#1144—22no—
82#1143—22no—
83#1142—22no—
84#1141—22no—
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4—21no—
87(C2×D7)⋊C4—21no—
88(C2×D7)⋊C4—21no—
89(C2×D7)⋊C4—21no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4D411yescharacteristic
92(C2×D7)⋊C4D411yescharacteristic
93C4⋊Dic7—21no—
94C4⋊Dic7—21no—
95C4⋊Dic7—21no—
96C4⋊Dic7C2311yescharacteristic
97C4⋊Dic7C2311yescharacteristic
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4D411yescharacteristic
101Dic7⋊C4D411yescharacteristic
102C4×Dic7—21no—
103C4×Dic7—21no—
104C4×Dic7C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107#1131—22no—
108C8×D7—22no—
109#542—71no—
110#542—71no—
111#498—71no—
112#495—71no—
113#489—71no—
114#489D711yescharacteristic, complemented
115#470—71no—
116#469—71no—
117#465—71no—
118#465—71no—
119#442—71no—
120#441—71no—
121C4×SD16—71no—
122C4×SD16—71no—
123#405—71no—
124C23×C7—21no—
125C22×D7—21no—
126C22×D7C2×D411yescharacteristic
127C7×Q8—21no—
128C7×Q8—21no—
129C7×Q8—81no—
130C7×D4—21no—
131C7×D4—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7C2×D411yescharacteristic
136C2×C4×C7C2×D411yescharacteristic
137C2×C4×C7C2411yescharacteristic
138C22⋊D7—81no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—81no—
143C2×Dic7C2×D411yescharacteristic
144C2×Dic7C2×D411yescharacteristic
145C2×Dic7C2×D411yescharacteristic
146D28—21no—
147D28—21no—
148D28—81no—
149C4×D7—21no—
150C4×D7—21no—
151C4×D7—81no—
152Dic14—21no—
153Dic14—21no—
154Dic14—81no—
155C7×C8—21no—
156C7×C8—21no—
157C7⋊C8#4112yes—
158C7⋊C8#4112yes—
159C2×Q16—71no—
160C2×SD16—71no—
161C2×SD16—71no—
162C2×D8—71no—
163C2×M16—71no—
164C22×C8—71no—
165C4⋊Q8—71no—
166C4⋊Q8C2×D711yescharacteristic
167#126—141no—
168#126—141no—
169#124—71no—
170#124—141no—
171#124—141no—
172#124C2×D711yescharacteristic
173#123—141no—
174#123—141no—
175C22⋊Q8—71no—
176C22⋊Q8—71no—
177C22⋊Q8—141no—
178C22⋊Q8—141no—
179#121—71no—
180#121—71no—
181C4×Q8—71no—
182C4×Q8—71no—
183C4×D4—71no—
184C4×D4—71no—
185#117—71no—
186#106—71no—
187#106—71no—
188#106—71no—
189#106—71no—
190#103—71no—
191#103—71no—
192#103C2×D711yescharacteristic
193#103C2×D711yescharacteristic
194#102—71no—
195#102—71no—
196#102C2×D711yescharacteristic
197#102C2×D711yescharacteristic
198#97—71no—
199#97C2×D711yescharacteristic
200C4×C8—71no—
201C4×C8—71no—
202C22×C7—41no—
203C22×C7—41no—
204C22×C7—41no—
205C22×C7C22×D411yescharacteristic
206C2×D7—21no—
207C2×D7—21no—
208C2×D7—21no—
209C2×D7—41no—
210C2×D7—41no—
211C2×D7—41no—
212C4×C7—21no—
213C4×C7—21no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7—41no—
217C4×C7C2×#4111yescharacteristic
218C4×C7C2×#4111yescharacteristic
219Dic7—21no—
220Dic7—21no—
221Dic7—21no—
222Dic7—41no—
223Dic7—41no—
224Dic7—41no—
225C2×Q8—21no—
226C2×Q8—71no—
227C2×Q8—141no—
228C2×Q8C22×D711yescharacteristic
229C2×D4—71no—
230C2×D4—141no—
231C2×D4—141no—
232C2×D4C22×D711yescharacteristic
233C22×C4—71no—
234C22×C4—141no—
235C22×C4—141no—
236Q16—142no—
237SD16—142no—
238SD16—142no—
239D8—142no—
240M16—142no—
241C2×C8—71no—
242C2×C8—71no—
243C2×C8—142no—
244C2×C8C22×D711yescharacteristic
245C2×C8C22×D711yescharacteristic
246C4⋊C4—21no—
247C4⋊C4—71no—
248C4⋊C4—71no—
249C4⋊C4—71no—
250C4⋊C4—71no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4—141no—
254C4⋊C4—141no—
255C4⋊C4—141no—
256C4⋊C4C22×D711yescharacteristic
257C4⋊C4C22×D711yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—71no—
261C22⋊C4—71no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C22⋊C4—141no—
265C22⋊C4—141no—
266C22⋊C4—141no—
267C22⋊C4—141no—
268C22⋊C4—141no—
269C22⋊C4—141no—
270C42—71no—
271C42—141no—
272C42—141no—
273C42C22×D711yescharacteristic
274C2×C7—81no—
275C2×C7#53211yescharacteristic
276C2×C7#57711yescharacteristic
277C2×C7#57711yescharacteristic
278D7—41no—
279D7—81no—
280C23—21no—
281C23—71no—
282C23—141no—
283Q8—21no—
284Q8—21no—
285Q8—81no—
286Q8—141no—
287Q8—141no—
288Q8—561no—
289D4—21no—
290D4—21no—
291D4—141no—
292D4—141no—
293D4—561no—
294D4—561no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—71no—
299C2×C4—71no—
300C2×C4—71no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—141no—
305C2×C4—141no—
306C2×C4—561no—
307C2×C4—561no—
308C2×C4D4×D711yescharacteristic
309C2×C4D4×D711yescharacteristic
310C2×C4C23×D711yescharacteristic
311C8—21no—
312C8—21no—
313C8—72no—
314C8—72no—
315C7#317311yescharacteristic, complemented, Sylow
316C22—41no—
317C22—41no—
318C22—41no—
319C22—141no—
320C22—141no—
321C22—141no—
322C22—281no—
323C22—281no—
324C22—281no—
325C22C2×D4×D711yescharacteristic
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—141no—
332C4—141no—
333C4—141no—
334C4—281no—
335C4—281no—
336C4—281no—
337C4D7×#4111yescharacteristic
338C4D7×#4111yescharacteristic
339C2—81no—
340C2—281no—
341C2—561no—
342C2#9035911yescharacteristic
343C2#9046611yescharacteristic
344C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#489C2
socleC22×C7C22×D4
prime radical#89554C2
hypercenter#489D7
hypocenterC7#3173
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #90424C2semidirect
2. #90373C2semidirect
3. C7×#489C2semidirect
4. #89851C2semidirect
5. #89831C2semidirect
6. #89588C2semidirect
7. #89579C2semidirect
8. #89561C2semidirect
9. #89560C2semidirect
10. #89557C2semidirect
12. #89550C2semidirect
13. #89530C2semidirect
14. #89486C2semidirect
15. #89480C2semidirect
114. #489D7semidirect
315. C7#3173semidirect

Maximal subgroups

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

subgroupindex
#904242
#903732
C7×#4892
#898512
#898312
#895882
#895792
#895612
#895602
#895572
#895542
#895502
#895302
#894862
#894802
#31737