Finite Group Databasebeta

← #11738688

Subgroups — #11738688

Order 896.

320 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#90380C211yesmaximal, characteristic, complemented
2#90273C211yesmaximal, characteristic
3C7×#479C211yesmaximal, characteristic, complemented
4#89838C211yesmaximal, characteristic, complemented
5#89792C211yesmaximal, characteristic, complemented
6#89657C211yesmaximal, characteristic
7#89648C211yesmaximal, characteristic, complemented
8#89635C211yesmaximal, characteristic
9#89627C211yesmaximal, characteristic, complemented
10#89614C211yesmaximal, characteristic, complemented
11#89612C211yesmaximal, characteristic, complemented
12#89587C211yesmaximal, characteristic, complemented
13#89577C211yesmaximal, characteristic, complemented
14#89572C211yesmaximal, characteristic, complemented
15#89567C211yesmaximal, characteristic, complemented
16C7×#125C2211yescharacteristic
17C4×C7×Q8C2211yescharacteristic
18#6510C2211yescharacteristic
19#6509C2211yescharacteristic
20#6460—21no—
21#6459—21no—
22#6459—21no—
23#6458C2211yescharacteristic
24#6457—21no—
25#6455C2211yescharacteristic
26D7×C4⋊C4C2211yescharacteristic
27#6453C2211yescharacteristic
28#6452—21no—
29#6451—21no—
30#6451C2211yescharacteristic
31#6451C2211yescharacteristic
32#6439—21no—
33#6438—21no—
34C4×D28C2211yescharacteristic
35#6435C2211yescharacteristic
36#6432C2211yescharacteristic
37C7×#107C2211yescharacteristic
38C7×#106C2211yescharacteristic
39C7×C4⋊C8C2211yescharacteristic
40C7×#103C2211yescharacteristic
41C7×#103C2211yescharacteristic
42#6409C2211yescharacteristic
43#6409C2211yescharacteristic
44#6395C2211yescharacteristic
45#6395C2211yescharacteristic
46(C2×D7)⋊C8C2211yescharacteristic
47(C2×D7)⋊C8C2211yescharacteristic
48#6391C2211yescharacteristic
49#6391C2211yescharacteristic
50#6389C2211yescharacteristic
51C8×Dic7C2211yescharacteristic
52#6383C2211yescharacteristic
53#6383C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3244—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×D28—21no—
62C2×D28C2311yescharacteristic
63C2×C4×D7—21no—
64C2×C4×D7—21no—
65C2×C4×D7C2311yescharacteristic
66C2×Dic14—21no—
67C2×Dic14—21no—
68C2×Dic14—21no—
69C2×C7×C8C2311yescharacteristic
70C2×C7×C8C2311yescharacteristic
71C7×C4⋊C4—21no—
72C7×C4⋊C4—21no—
73C7×C4⋊C4—21no—
74C7×C4⋊C4C2311yescharacteristic
75C7×C4⋊C4C2311yescharacteristic
76C7×C4⋊C4C2311yescharacteristic
77C42×C7—21no—
78C42×C7C2311yescharacteristic
79(C2×D7)⋊C4—21no—
80(C2×D7)⋊C4—21no—
81(C2×D7)⋊C4—21no—
82(C2×D7)⋊C4—21no—
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4—21no—
85(C2×D7)⋊C4D411yescharacteristic
86(C2×D7)⋊C4D411yescharacteristic
87C4⋊Dic7—21no—
88C4⋊Dic7—21no—
89C4⋊Dic7C2311yescharacteristic
90C4⋊Dic7C2311yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4D411yescharacteristic
98Dic7⋊C4D411yescharacteristic
99C4×Dic7—21no—
100C4×Dic7C2311yescharacteristic
101C2×C7⋊C8C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103#544—71no—
104#542—71no—
105#501—71no—
106#498—71no—
107#489—71no—
108#488—71no—
109#485—71no—
110#483—71no—
111#482—71no—
112#481—71no—
113#479D711yescharacteristic, complemented
114#478—71no—
115#476—71no—
116#475—71no—
117#433—71no—
118C22×D7—21no—
119C22×D7C2×D411yescharacteristic
120C7×Q8—21no—
121C7×Q8—21no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7C2×D411yescharacteristic
127C2×C4×C7C2×D411yescharacteristic
128C2×C4×C7C2411yescharacteristic
129C2×Dic7—21no—
130C2×Dic7—21no—
131C2×Dic7—21no—
132C2×Dic7C2×D411yescharacteristic
133C2×Dic7C2×D411yescharacteristic
134C2×Dic7C2×D411yescharacteristic
135D28—21no—
136D28—21no—
137D28—81no—
138C4×D7—21no—
139C4×D7—21no—
140C4×D7—81no—
141C4×D7—81no—
142Dic14—81no—
143Dic14—81no—
144Dic14—81no—
145C7×C8—21no—
146C7×C8—21no—
147C7⋊C8—21no—
148C7⋊C8—21no—
149C4⋊Q8—71no—
150C4⋊Q8—71no—
151C4⋊Q8—71no—
152C4⋊Q8—141no—
153#126—141no—
154#125—71no—
155#125—141no—
156#125C2×D711yescharacteristic
157#124—71no—
158#124—141no—
159#123—141no—
160C22⋊Q8—71no—
161C22⋊Q8—141no—
162C22⋊Q8—141no—
163C22⋊Q8—141no—
164#121—71no—
165C4×Q8C2×D711yescharacteristic
166C4×D4—71no—
167#117—71no—
168#117—71no—
169C2×C4⋊C4—71no—
170#107—71no—
171#107C2×D711yescharacteristic
172#106—71no—
173#106—71no—
174#106—71no—
175#106—71no—
176#106—71no—
177#106C2×D711yescharacteristic
178C4⋊C8—71no—
179C4⋊C8C2×D711yescharacteristic
180#103—71no—
181#103—71no—
182#103C2×D711yescharacteristic
183#103C2×D711yescharacteristic
184#102—71no—
185#102—71no—
186#102—71no—
187#102—71no—
188C22⋊C8—71no—
189C22⋊C8—71no—
190#97—71no—
191C4×C8—71no—
192C22×C7C22×D411yescharacteristic
193C2×D7—21no—
194C2×D7—21no—
195C2×D7—21no—
196C2×D7—41no—
197C2×D7—41no—
198C2×D7—41no—
199C4×C7—21no—
200C4×C7—21no—
201C4×C7—41no—
202C4×C7—41no—
203C4×C7—41no—
204C4×C7—41no—
205C4×C72−1+411yescharacteristic
206C4×C72−1+411yescharacteristic
207Dic7—21no—
208Dic7—21no—
209Dic7—21no—
210Dic7—41no—
211Dic7—41no—
212Dic7—41no—
213C2×Q8—141no—
214C2×Q8—141no—
215C2×Q8—141no—
216C2×Q8C22×D711yescharacteristic
217C2×D4—71no—
218C2×D4—141no—
219C22×C4—71no—
220C22×C4—141no—
221C22×C4—141no—
222C2×C8—71no—
223C2×C8—71no—
224C2×C8C22×D711yescharacteristic
225C2×C8C22×D711yescharacteristic
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—71no—
230C4⋊C4—71no—
231C4⋊C4—71no—
232C4⋊C4—71no—
233C4⋊C4—71no—
234C4⋊C4—141no—
235C4⋊C4—141no—
236C4⋊C4—141no—
237C4⋊C4—141no—
238C4⋊C4—141no—
239C4⋊C4—141no—
240C4⋊C4—141no—
241C4⋊C4C22×D711yescharacteristic
242C4⋊C4C22×D711yescharacteristic
243C4⋊C4C22×D711yescharacteristic
244C22⋊C4—71no—
245C22⋊C4—71no—
246C22⋊C4—141no—
247C22⋊C4—141no—
248C22⋊C4—141no—
249C22⋊C4—141no—
250C22⋊C4—141no—
251C22⋊C4—141no—
252C42—21no—
253C42—71no—
254C42—141no—
255C42C22×D711yescharacteristic
256C2×C7#53611yescharacteristic
257C2×C7#57711yescharacteristic
258C2×C7#57711yescharacteristic
259D7—41no—
260D7—81no—
261C23—71no—
262C23—141no—
263Q8—21no—
264Q8—21no—
265Q8—561no—
266Q8—561no—
267Q8—561no—
268D4—141no—
269D4—141no—
270D4—561no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—71no—
276C2×C4—71no—
277C2×C4—71no—
278C2×C4—141no—
279C2×C4—141no—
280C2×C4—141no—
281C2×C4—141no—
282C2×C4—141no—
283C2×C4—561no—
284C2×C4—561no—
285C2×C4D4×D711yescharacteristic
286C2×C4D4×D711yescharacteristic
287C2×C4C23×D711yescharacteristic
288C8—21no—
289C8—21no—
290C8—141no—
291C8—141no—
292C7#324411yescharacteristic, complemented, Sylow
293C22—141no—
294C22—141no—
295C22—141no—
296C22—281no—
297C22—281no—
298C22—281no—
299C22C2×D4×D711yescharacteristic
300C4—21no—
301C4—21no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—141no—
307C4—141no—
308C4—141no—
309C4—281no—
310C4—281no—
311C4—281no—
312C4#655111yescharacteristic
313C4#655411yescharacteristic
314C2—281no—
315C2—561no—
316C2#9031911yescharacteristic
317C2#9046111yescharacteristic
318C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#479C2
socleC22×C7C22×D4
prime radical#6458C22
hypercenter#479D7
hypocenterC7#3244
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
1. #90380C2semidirect
3. C7×#479C2semidirect
4. #89838C2semidirect
5. #89792C2semidirect
7. #89648C2semidirect
9. #89627C2semidirect
10. #89614C2semidirect
11. #89612C2semidirect
12. #89587C2semidirect
13. #89577C2semidirect
14. #89572C2semidirect
15. #89567C2semidirect
113. #479D7semidirect
292. C7#3244semidirect

Maximal subgroups

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

subgroupindex
#903802
#902732
C7×#4792
#898382
#897922
#896572
#896482
#896352
#896272
#896142
#896122
#895872
#895772
#895722
#895672
#32447