Finite Group Databasebeta

← #11738446

Subgroups — #11738446

Order 896.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90424C211yesmaximal, characteristic, complemented
2#90265C211yesmaximal, characteristic, complemented
3C7×#475C211yesmaximal, characteristic, complemented
4#89856C211yesmaximal, characteristic, complemented
5#89800C211yesmaximal, characteristic, complemented
6#89641C211yesmaximal, characteristic, complemented
7#89635C211yesmaximal, characteristic, complemented
8#89634C211yesmaximal, characteristic, complemented
9#89624C211yesmaximal, characteristic, complemented
10#89616C211yesmaximal, characteristic, complemented
11#89612C211yesmaximal, characteristic, complemented
12#89583C211yesmaximal, characteristic
13#89582C211yesmaximal, characteristic, complemented
14#89561C211yesmaximal, characteristic, complemented
15#89560C211yesmaximal, characteristic, complemented
16C2×#1161C2211yescharacteristic
17C7×C4⋊Q8C2211yescharacteristic
18C4×C7×Q8C2211yescharacteristic
19#6510—21no—
20#6509—21no—
21Q8×Dic7C2211yescharacteristic
22C2×#1144C2211yescharacteristic
23C2×#1143C2211yescharacteristic
24C2×D56C2211yescharacteristic
25C2×#1132C2211yescharacteristic
26#6461—21no—
27#6460C2211yescharacteristic
28#6458C2211yescharacteristic
29#6457—21no—
30#6456—21no—
31#6456C2211yescharacteristic
32#6456C2211yescharacteristic
33D7×C4⋊C4—21no—
34D7×C4⋊C4C2211yescharacteristic
35#6450C2211yescharacteristic
36#6438C2211yescharacteristic
37C4×D28—21no—
38C4×D28C2211yescharacteristic
39#6435—21no—
40#6435C2211yescharacteristic
41C7×#106C2211yescharacteristic
42C7×#106C2211yescharacteristic
43C7×C4⋊C8C2211yescharacteristic
44C7×#103C2211yescharacteristic
45C7×#103C2211yescharacteristic
46#6395C2211yescharacteristic
47(C2×D7)⋊C8C2211yescharacteristic
48(C2×D7)⋊C8C2211yescharacteristic
49#6390C2211yescharacteristic
50#6389C2211yescharacteristic
51C8×Dic7C2211yescharacteristic
52#6384C2211yescharacteristic
53#6383C2211yescharacteristic
54#6383C2211yescharacteristic
55#6383C2211yescharacteristic
56#6382C2211yescharacteristic
57#6382C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3125—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8C2311yescharacteristic
62#1161—22no—
63#1161—22no—
64C2×D28—21no—
65C2×D28—21no—
66C2×D28C2311yescharacteristic
67C2×D28C2311yescharacteristic
68C2×C4×D7—21no—
69C2×C4×D7—21no—
70C2×C4×D7—21no—
71C2×C4×D7—21no—
72C2×C4×D7—21no—
73C2×C4×D7C2311yescharacteristic
74C2×Dic14C2311yescharacteristic
75C2×C7×C8C2311yescharacteristic
76C2×C7×C8C2311yescharacteristic
77C7×C4⋊C4—21no—
78C7×C4⋊C4—21no—
79C7×C4⋊C4C2311yescharacteristic
80C7×C4⋊C4C2311yescharacteristic
81C7×C4⋊C4C2311yescharacteristic
82C42×C7—21no—
83C42×C7C2311yescharacteristic
84#1144—22no—
85#1143—22no—
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)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4D411yescharacteristic
94(C2×D7)⋊C4D411yescharacteristic
95C4⋊Dic7—21no—
96C4⋊Dic7—21no—
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
107D56—22no—
108#1132—22no—
109#542—71no—
110#520—71no—
111#483—71no—
112#481—71no—
113#476—71no—
114#475D711yescharacteristic, complemented
115#464—71no—
116#463—71no—
117#452—71no—
118#449—71no—
119#442—71no—
120#441—71no—
121C4×SD16—71no—
122C4×SD16—71no—
123#433—71no—
124C22×D7—21no—
125C22×D7—21no—
126C22×D7C2×D411yescharacteristic
127C7×Q8—21no—
128C7×Q8—81no—
129C7×Q8#4112yes—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7C2×D411yescharacteristic
135C2×C4×C7C2×D411yescharacteristic
136C2×C4×C7C2411yescharacteristic
137C2×Dic7—21no—
138C2×Dic7—21no—
139C2×Dic7C2×D411yescharacteristic
140C2×Dic7C2×D411yescharacteristic
141C2×Dic7C2×D411yescharacteristic
142D28—24no—
143D28—21no—
144D28—21no—
145D28—21no—
146D28—81no—
147D28#4112yes—
148C4×D7—24no—
149C4×D7—21no—
150C4×D7—21no—
151C4×D7—81no—
152C4×D7—81no—
153C4×D7—81no—
154C4×D7—81no—
155Dic14—21no—
156Dic14—21no—
157C7×C8—21no—
158C7×C8—21no—
159C7⋊C8—21no—
160C7⋊C8—21no—
161C2×#41—71no—
162C2×Q16—71no—
163C2×SD16—71no—
164C2×SD16—71no—
165C2×D8—71no—
166C4⋊Q8C2×D711yescharacteristic
167#126—141no—
168#124—71no—
169#124—141no—
170#123—141no—
171C22⋊Q8—71no—
172C22⋊Q8—141no—
173#121—71no—
174C4×Q8—71no—
175C4×Q8—71no—
176C4×Q8C2×D711yescharacteristic
177C4×D4—71no—
178C4×D4—71no—
179C4×D4—71no—
180C4×D4—141no—
181C4×D4—141no—
182#117—71no—
183#117—141no—
184C2×C4⋊C4—71no—
185C2×C4⋊C4—141no—
186#106—71no—
187#106—71no—
188#106C2×D711yescharacteristic
189#106C2×D711yescharacteristic
190C4⋊C8—71no—
191C4⋊C8C2×D711yescharacteristic
192#103—71no—
193#103—71no—
194#103C2×D711yescharacteristic
195#103C2×D711yescharacteristic
196#102—71no—
197#102—71no—
198#102—71no—
199#102—71no—
200C22⋊C8—71no—
201C22⋊C8—71no—
202#97—71no—
203C4×C8—71no—
204C22×C7C22×D411yescharacteristic
205C2×D7—21no—
206C2×D7—22no—
207C2×D7—21no—
208C2×D7—21no—
209C2×D7—41no—
210C2×D7—41no—
211C2×D7—41no—
212C2×D7—41no—
213C2×D7—41no—
214C4×C7—22no—
215C4×C7—21no—
216C4×C7—21no—
217C4×C7—41no—
218C4×C7—41no—
219C4×C7—41no—
220C4×C7C2×#4111yescharacteristic
221C4×C7C2×#4111yescharacteristic
222Dic7—21no—
223Dic7—21no—
224Dic7—21no—
225Dic7—41no—
226Dic7—41no—
227#41—142no—
228#41—142no—
229C2×Q8—21no—
230C2×Q8—71no—
231C2×Q8C22×D711yescharacteristic
232C2×D4—71no—
233C2×D4—71no—
234C2×D4—141no—
235C2×D4—141no—
236C22×C4—71no—
237C22×C4—141no—
238C22×C4—141no—
239C22×C4—141no—
240C22×C4—141no—
241C22×C4—141no—
242Q16—142no—
243SD16—142no—
244SD16—142no—
245D8—142no—
246C2×C8—71no—
247C2×C8—71no—
248C2×C8C22×D711yescharacteristic
249C2×C8C22×D711yescharacteristic
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—71no—
253C4⋊C4—71no—
254C4⋊C4—71no—
255C4⋊C4—141no—
256C4⋊C4—141no—
257C4⋊C4—141no—
258C4⋊C4—141no—
259C4⋊C4C22×D711yescharacteristic
260C4⋊C4C22×D711yescharacteristic
261C4⋊C4C22×D711yescharacteristic
262C22⋊C4—71no—
263C22⋊C4—71no—
264C22⋊C4—141no—
265C22⋊C4—141no—
266C22⋊C4—141no—
267C22⋊C4—141no—
268C22⋊C4—141no—
269C22⋊C4—141no—
270C22⋊C4—141no—
271C42—21no—
272C42—71no—
273C42—141no—
274C42—141no—
275C42C22×D711yescharacteristic
276C2×C7#52511yescharacteristic
277C2×C7#57711yescharacteristic
278C2×C7#57711yescharacteristic
279D7—41no—
280D7—42no—
281D7—81no—
282C23—71no—
283C23—141no—
284C23—141no—
285Q8—21no—
286Q8—81no—
287Q8—141no—
288Q8—141no—
289Q8#116112yes—
290D4—72no—
291D4—144no—
292D4—141no—
293D4—141no—
294D4—141no—
295D4—561no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—71no—
301C2×C4—71no—
302C2×C4—71no—
303C2×C4—144no—
304C2×C4—141no—
305C2×C4—141no—
306C2×C4—141no—
307C2×C4—141no—
308C2×C4—561no—
309C2×C4—561no—
310C2×C4—561no—
311C2×C4—561no—
312C2×C4D4×D711yescharacteristic
313C2×C4D4×D711yescharacteristic
314C2×C4C23×D711yescharacteristic
315C8—21no—
316C8—21no—
317C8—141no—
318C8—141no—
319C7#312511yescharacteristic, complemented, Sylow
320C22—141no—
321C22—141no—
322C22—142no—
323C22—141no—
324C22—281no—
325C22—281no—
326C22—281no—
327C22—281no—
328C22—281no—
329C22C2×D4×D711yescharacteristic
330C4—21no—
331C4—21no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—141no—
337C4—141no—
338C4—141no—
339C4—281no—
340C4—281no—
341C4C2×#116111yescharacteristic
342C4D7×#4111yescharacteristic
343C2—281no—
344C2—282no—
345C2—561no—
346C2#9031811yescharacteristic
347C2#9045311yescharacteristic
348C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#475C2
socleC22×C7C22×D4
prime radical#89583C2
hypercenter#475D7
hypocenterC7#3125
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
1. #90424C2semidirect
2. #90265C2semidirect
3. C7×#475C2semidirect
4. #89856C2semidirect
5. #89800C2semidirect
6. #89641C2semidirect
7. #89635C2semidirect
8. #89634C2semidirect
9. #89624C2semidirect
10. #89616C2semidirect
11. #89612C2semidirect
13. #89582C2semidirect
14. #89561C2semidirect
15. #89560C2semidirect
114. #475D7semidirect
319. C7#3125semidirect

Maximal subgroups

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

subgroupindex
#904242
#902652
C7×#4752
#898562
#898002
#896412
#896352
#896342
#896242
#896162
#896122
#895832
#895822
#895612
#895602
#31257