Finite Group Databasebeta

← #11738804

Subgroups — #11738804

Order 896.

328 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90291C211yesmaximal, characteristic
2#90208C211yesmaximal, characteristic, complemented
3C7×#482C211yesmaximal, characteristic
4#89807C211yesmaximal, characteristic
5#89768C211yesmaximal, characteristic, complemented
6#89649C211yesmaximal, characteristic, complemented
7#89648C211yesmaximal, characteristic, complemented
8#89629C211yesmaximal, characteristic, complemented
9#89628C211yesmaximal, characteristic, complemented
10#89538C211yesmaximal, characteristic, complemented
11#89537C211yesmaximal, characteristic, complemented
12#89537C211yesmaximal, characteristic, complemented
13#89535C211yesmaximal, characteristic, complemented
14#89495C211yesmaximal, characteristic, complemented
15#89491C211yesmaximal, characteristic, complemented
16C7×#121C2211yescharacteristic
17C7×#117C2211yescharacteristic
18#6499C2211yescharacteristic
19#6498—21no—
20D4×Dic7—21no—
21D4×Dic7C2211yescharacteristic
22#6489C2211yescharacteristic
23#6487C2211yescharacteristic
24C2×C4×Dic7C2211yescharacteristic
25#6453C2211yescharacteristic
26#6453C2211yescharacteristic
27#6453C2211yescharacteristic
28#6451C2211yescharacteristic
29#6451C2211yescharacteristic
30#6450C2211yescharacteristic
31#6442—21no—
32#6441—21no—
33#6441—21no—
34#6440—21no—
35C4×Dic14—21no—
36C4×Dic14—21no—
37C7×#107C2211yescharacteristic
38C7×#106C2211yescharacteristic
39C7×#102C2211yescharacteristic
40C7×#102C2211yescharacteristic
41C7×C22⋊C8C2211yescharacteristic
42#6406C2211yescharacteristic
43#6406C2211yescharacteristic
44C22⋊(C7⋊C8)C2211yescharacteristic
45#6392C2211yescharacteristic
46#6391C2211yescharacteristic
47#6390C2211yescharacteristic
48#6390C2211yescharacteristic
49#6389C2211yescharacteristic
50#6389C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52Dic7⋊C8C2211yescharacteristic
53#6384C2211yescharacteristic
54#6384C2211yescharacteristic
55#6382C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3240—71nomaximal, Sylow
60C2×C7×D4—21no—
61C2×C7×D4C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×Dic7—21no—
64C22×Dic7D411yescharacteristic
65C22×Dic7D411yescharacteristic
66C2×Dic14—21no—
67C2×Dic14—21no—
68C2×Dic14C2311yescharacteristic
69C2×C7×C8C2311yescharacteristic
70C2×C7×C8C2311yescharacteristic
71C7×C4⋊C4C2311yescharacteristic
72C7×C4⋊C4C2311yescharacteristic
73C7×C4⋊C4C2311yescharacteristic
74C7×C22⋊C4—21no—
75C7×C22⋊C4—21no—
76C42×C7—21no—
77C22⋊Dic7—21no—
78C22⋊Dic7—21no—
79C22⋊Dic7—21no—
80C22⋊Dic7—21no—
81C22⋊Dic7—21no—
82C4⋊Dic7—21no—
83C4⋊Dic7—21no—
84C4⋊Dic7—21no—
85C4⋊Dic7C2311yescharacteristic
86C4⋊Dic7C2311yescharacteristic
87C4⋊Dic7C2311yescharacteristic
88Dic7⋊C4—21no—
89Dic7⋊C4—21no—
90Dic7⋊C4—21no—
91Dic7⋊C4—21no—
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95C4×Dic7—21no—
96C4×Dic7—21no—
97C4×Dic7D411yescharacteristic
98C4×Dic7D411yescharacteristic
99C4×Dic7C2311yescharacteristic
100C4×Dic7C2311yescharacteristic
101C2×C7⋊C8C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103#533—71no—
104#529—71no—
105#501—71no—
106#501—71no—
107#489—71no—
108#489—71no—
109#485—71no—
110#485—71no—
111#482—71no—
112#482D711yescharacteristic
113#479—71no—
114#479—71no—
115#478—71no—
116#478—71no—
117#432—71no—
118C23×C7—21no—
119C23×C7C2×D411yescharacteristic
120C7×D4—21no—
121C7×D4—21no—
122C7×D4—42no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7C2×D411yescharacteristic
129C2×C4×C7C2411yescharacteristic
130C2×Dic7—21no—
131C2×Dic7—21no—
132C2×Dic7—21no—
133C2×Dic7—21no—
134C2×Dic7—41no—
135C2×Dic7—41no—
136C2×Dic7—42no—
137C2×Dic7C2×D411yescharacteristic
138C2×Dic7C2×D411yescharacteristic
139C2×Dic7C2×D411yescharacteristic
140C2×Dic7C2×D411yescharacteristic
141Dic14—21no—
142Dic14—21no—
143Dic14—42no—
144Dic14—42no—
145C7×C8—21no—
146C7×C8—21no—
147C7⋊C8—21no—
148C7⋊C8—21no—
149C4⋊Q8—71no—
150C4⋊Q8—71no—
151#126—141no—
152#125—71no—
153#125—71no—
154#125—71no—
155#124—71no—
156#123—141no—
157C22⋊Q8—71no—
158C22⋊Q8—141no—
159C22⋊Q8—141no—
160#121C2×D711yescharacteristic
161C4×Q8—71no—
162C4×Q8—141no—
163C4×Q8—141no—
164C4×D4—71no—
165C4×D4—141no—
166#117—71no—
167#117—141no—
168#117C2×D711yescharacteristic
169C2×C42—71no—
170#107—71no—
171#107—71no—
172#107—71no—
173#107C2×D711yescharacteristic
174#106—71no—
175#106—71no—
176#106—71no—
177#106C2×D711yescharacteristic
178C4⋊C8—71no—
179C4⋊C8—71no—
180#103—71no—
181#103—71no—
182#103—71no—
183#103—71no—
184#102—71no—
185#102—71no—
186#102C2×D711yescharacteristic
187#102C2×D711yescharacteristic
188C22⋊C8—71no—
189C22⋊C8C2×D711yescharacteristic
190#97—71no—
191#97—71no—
192C22×C7—21no—
193C22×C7—21no—
194C22×C7—21no—
195C22×C7—41no—
196C22×C7—41no—
197C22×C7—41no—
198C22×C7C22×D411yescharacteristic
199C4×C7—21no—
200C4×C7—41no—
201C4×C7—41no—
202C4×C7—41no—
203C4×C72−1+411yescharacteristic
204C4×C72−1+411yescharacteristic
205Dic7—21no—
206Dic7—21no—
207Dic7—21no—
208Dic7—21no—
209Dic7—41no—
210Dic7—41no—
211Dic7—41no—
212Dic7—41no—
213C2×Q8—71no—
214C2×Q8—141no—
215C2×Q8—141no—
216C2×D4—21no—
217C2×D4C22×D711yescharacteristic
218C22×C4—71no—
219C22×C4—71no—
220C22×C4—141no—
221C22×C4C22×D711yescharacteristic
222C2×C8—71no—
223C2×C8—71no—
224C2×C8C22×D711yescharacteristic
225C2×C8C22×D711yescharacteristic
226C4⋊C4—71no—
227C4⋊C4—71no—
228C4⋊C4—71no—
229C4⋊C4—141no—
230C4⋊C4—141no—
231C4⋊C4—141no—
232C4⋊C4—141no—
233C4⋊C4—141no—
234C4⋊C4—141no—
235C4⋊C4—141no—
236C4⋊C4—141no—
237C4⋊C4—141no—
238C4⋊C4—141no—
239C4⋊C4C22×D711yescharacteristic
240C4⋊C4C22×D711yescharacteristic
241C4⋊C4C22×D711yescharacteristic
242C22⋊C4—21no—
243C22⋊C4—21no—
244C22⋊C4—141no—
245C22⋊C4—141no—
246C22⋊C4—141no—
247C22⋊C4—141no—
248C22⋊C4—141no—
249C42—21no—
250C42—71no—
251C42—71no—
252C42—71no—
253C42—71no—
254C42—141no—
255C42—141no—
256C2×C7—41no—
257C2×C7—81no—
258C2×C7#53611yescharacteristic
259C2×C7#57511yescharacteristic
260C2×C7#57511yescharacteristic
261C23—21no—
262C23D4×D711yescharacteristic
263Q8—141no—
264Q8—141no—
265Q8—282no—
266Q8—282no—
267D4—21no—
268D4—21no—
269D4—42no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—71no—
276C2×C4—71no—
277C2×C4—71no—
278C2×C4—71no—
279C2×C4—141no—
280C2×C4—141no—
281C2×C4—141no—
282C2×C4—141no—
283C2×C4—281no—
284C2×C4—281no—
285C2×C4—282no—
286C2×C4D4×D711yescharacteristic
287C2×C4C23×D711yescharacteristic
288C8—21no—
289C8—21no—
290C8—141no—
291C8—141no—
292C7#324011yescharacteristic, complemented, Sylow
293C22—21no—
294C22—21no—
295C22—21no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22C2×D4×D711yescharacteristic
300C4—21no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—141no—
305C4—141no—
306C4—141no—
307C4—141no—
308C4—281no—
309C4—281no—
310C4—281no—
311C4—281no—
312C4#655411yescharacteristic
313C4#655411yescharacteristic
314C2—41no—
315C2—81no—
316C2#9033911yescharacteristic
317C2#9045911yescharacteristic
318C2#9046411yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#482C2
socleC22×C7C22×D4
prime radicalC7×#121C22
hypercenter#482D7
hypocenterC7#3240
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
2. #90208C2semidirect
5. #89768C2semidirect
6. #89649C2semidirect
7. #89648C2semidirect
8. #89629C2semidirect
9. #89628C2semidirect
10. #89538C2semidirect
11. #89537C2semidirect
12. #89537C2semidirect
13. #89535C2semidirect
14. #89495C2semidirect
15. #89491C2semidirect
292. C7#3240semidirect

Maximal subgroups

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

subgroupindex
#902912
#902082
C7×#4822
#898072
#897682
#896492
#896482
#896292
#896282
#895382
#895372
#895352
#894952
#894912
#32407