Finite Group Databasebeta

← #11738448

Subgroups — #11738448

Order 896.

334 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90424C211yesmaximal, characteristic
2#90274C211yesmaximal, characteristic, complemented
3C7×#475C211yesmaximal, characteristic, complemented
4#89863C211yesmaximal, characteristic, complemented
5#89800C211yesmaximal, characteristic, complemented
6#89643C211yesmaximal, characteristic, complemented
7#89640C211yesmaximal, characteristic, complemented
8#89636C211yesmaximal, characteristic
9#89629C211yesmaximal, characteristic, complemented
10#89617C211yesmaximal, characteristic, complemented
11#89611C211yesmaximal, characteristic, complemented
12#89590C211yesmaximal, characteristic, complemented
13#89585C211yesmaximal, characteristic, complemented
14#89584C211yesmaximal, characteristic
15#89565C211yesmaximal, characteristic, complemented
16C7×C4⋊Q8C2211yescharacteristic
17C4×C7×Q8C2211yescharacteristic
18#6510—21no—
19#6509—21no—
20#6509C2211yescharacteristic
21#6507C2211yescharacteristic
22C2×#1144C2211yescharacteristic
23C2×#1143C2211yescharacteristic
24C2×Dic28C2211yescharacteristic
25C2×#1132C2211yescharacteristic
26C2×#1131C2211yescharacteristic
27C2×C8×D7C2211yescharacteristic
28#6461—21no—
29#6461—21no—
30#6460C2211yescharacteristic
31#6460C2211yescharacteristic
32#6458C2211yescharacteristic
33#6457—21no—
34#6456C2211yescharacteristic
35#6453C2211yescharacteristic
36#6452—21no—
37#6450C2211yescharacteristic
38#6439—21no—
39#6438C2211yescharacteristic
40#6435C2211yescharacteristic
41#6433—21no—
42C4×Dic14C2211yescharacteristic
43C7×#106C2211yescharacteristic
44C7×#106C2211yescharacteristic
45C7×C4⋊C8C2211yescharacteristic
46C7×#103C2211yescharacteristic
47C7×#103C2211yescharacteristic
48#6395C2211yescharacteristic
49#6390C2211yescharacteristic
50Dic7⋊C8C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52#6384C2211yescharacteristic
53#6384C2211yescharacteristic
54#6384C2211yescharacteristic
55#6383C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3258—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×D28—21no—
63C2×D28C2311yescharacteristic
64C2×C4×D7—21no—
65C2×C4×D7C2311yescharacteristic
66C2×Dic14C2311yescharacteristic
67C2×Dic14C2311yescharacteristic
68C2×C7×C8C2311yescharacteristic
69C2×C7×C8C2311yescharacteristic
70C7×C4⋊C4—21no—
71C7×C4⋊C4—21no—
72C7×C4⋊C4C2311yescharacteristic
73C7×C4⋊C4C2311yescharacteristic
74C7×C4⋊C4C2311yescharacteristic
75C42×C7—21no—
76C42×C7C2311yescharacteristic
77#1144—41no—
78#1143—41no—
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)⋊C4—21no—
86(C2×D7)⋊C4D411yescharacteristic
87(C2×D7)⋊C4D411yescharacteristic
88C4⋊Dic7—21no—
89C4⋊Dic7—21no—
90C4⋊Dic7—21no—
91C4⋊Dic7C2311yescharacteristic
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4D411yescharacteristic
99Dic7⋊C4D411yescharacteristic
100C4×Dic7—21no—
101C4×Dic7—21no—
102C4×Dic7C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105Dic28—41no—
106#1132—41no—
107#1131—41no—
108C8×D7—41no—
109#542—71no—
110#541—71no—
111#479—71no—
112#478—71no—
113#477—71no—
114#475D711yescharacteristic, complemented
115#470—71no—
116#468—71no—
117#467—71no—
118#465—71no—
119#464—71no—
120#464—71no—
121#462—71no—
122#461—71no—
123#424—71no—
124C22×D7—21no—
125C22×D7C2×D411yescharacteristic
126C7×Q8—21no—
127C7×Q8—21no—
128C7×Q8—81no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7C2×D411yescharacteristic
134C2×C4×C7C2×D411yescharacteristic
135C2×C4×C7C2411yescharacteristic
136C2×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7—21no—
139C2×Dic7C2×D411yescharacteristic
140C2×Dic7C2×D411yescharacteristic
141C2×Dic7C2×D411yescharacteristic
142D28—21no—
143D28—21no—
144D28—81no—
145C4×D7—21no—
146C4×D7—21no—
147C4×D7—81no—
148Dic14—21no—
149Dic14—21no—
150Dic14—21no—
151Dic14—21no—
152C7×C8—21no—
153C7×C8—21no—
154C7⋊C8—21no—
155C7⋊C8—21no—
156C2×Q16—71no—
157C2×Q16—71no—
158C2×SD16—71no—
159C2×SD16—71no—
160C2×M16—71no—
161C22×C8—71no—
162C4⋊Q8—71no—
163C4⋊Q8C2×D711yescharacteristic
164#126—141no—
165#126—141no—
166#126—141no—
167#125—71no—
168#125—141no—
169#125—141no—
170#124—71no—
171#124—141no—
172#123—141no—
173C22⋊Q8—71no—
174C22⋊Q8—71no—
175C22⋊Q8—71no—
176C22⋊Q8—141no—
177#121—71no—
178C4×Q8—71no—
179C4×Q8—71no—
180C4×Q8C2×D711yescharacteristic
181C4×D4—71no—
182#117—71no—
183#107—71no—
184#107—71no—
185#106C2×D711yescharacteristic
186#106C2×D711yescharacteristic
187C4⋊C8—71no—
188C4⋊C8—71no—
189C4⋊C8—71no—
190C4⋊C8C2×D711yescharacteristic
191#103—71no—
192#103—71no—
193#103—71no—
194#103—71no—
195#103C2×D711yescharacteristic
196#103C2×D711yescharacteristic
197#102—71no—
198#102—71no—
199C22×C7C22×D411yescharacteristic
200C2×D7—21no—
201C2×D7—21no—
202C2×D7—21no—
203C2×D7—41no—
204C2×D7—41no—
205C2×D7—41no—
206C4×C7—21no—
207C4×C7—21no—
208C4×C7—41no—
209C4×C7—41no—
210C4×C7—41no—
211C4×C7—41no—
212C4×C7H(5,𝔽2)11yescharacteristic
213C4×C72−1+411yescharacteristic
214Dic7—21no—
215Dic7—21no—
216Dic7—21no—
217Dic7—41no—
218Dic7—41no—
219Dic7—41no—
220C2×Q8—21no—
221C2×Q8—71no—
222C2×Q8—71no—
223C2×Q8C22×D711yescharacteristic
224C2×D4—71no—
225C2×D4—141no—
226C22×C4—71no—
227C22×C4—141no—
228Q16—281no—
229Q16—281no—
230SD16—281no—
231SD16—281no—
232M16—281no—
233C2×C8—71no—
234C2×C8—71no—
235C2×C8—281no—
236C2×C8C22×D711yescharacteristic
237C2×C8C22×D711yescharacteristic
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—71no—
241C4⋊C4—71no—
242C4⋊C4—71no—
243C4⋊C4—141no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4—141no—
247C4⋊C4—141no—
248C4⋊C4—141no—
249C4⋊C4—141no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4C22×D711yescharacteristic
253C4⋊C4C22×D711yescharacteristic
254C4⋊C4C22×D711yescharacteristic
255C22⋊C4—71no—
256C22⋊C4—71no—
257C22⋊C4—141no—
258C22⋊C4—141no—
259C22⋊C4—141no—
260C22⋊C4—141no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C42—21no—
265C42—71no—
266C42—141no—
267C42—141no—
268C42C22×D711yescharacteristic
269C2×C7#53711yescharacteristic
270C2×C7#57711yescharacteristic
271C2×C7#57811yescharacteristic
272D7—41no—
273D7—81no—
274C23—71no—
275C23—141no—
276Q8—21no—
277Q8—21no—
278Q8—81no—
279Q8—141no—
280Q8—141no—
281Q8—141no—
282Q8—141no—
283D4—141no—
284D4—141no—
285D4—561no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—71no—
291C2×C4—71no—
292C2×C4—71no—
293C2×C4—141no—
294C2×C4—141no—
295C2×C4—141no—
296C2×C4—141no—
297C2×C4—141no—
298C2×C4—561no—
299C2×C4D4×D711yescharacteristic
300C2×C4D4×D711yescharacteristic
301C2×C4C23×D711yescharacteristic
302C8—21no—
303C8—21no—
304C8—141no—
305C8—141no—
306C7#325811yescharacteristic, complemented, Sylow
307C22—141no—
308C22—141no—
309C22—141no—
310C22—281no—
311C22—281no—
312C22—281no—
313C22C2×D4×D711yescharacteristic
314C4—21no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—141no—
321C4—141no—
322C4—141no—
323C4—281no—
324C4—281no—
325C4—281no—
326C4#655111yescharacteristic
327C4#655311yescharacteristic
328C2—281no—
329C2—561no—
330C2#9032011yescharacteristic
331C2#9045311yescharacteristic
332C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#475C2
socleC22×C7C22×D4
prime radical#6458C22
hypercenter#475D7
hypocenterC7#3258
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
2. #90274C2semidirect
3. C7×#475C2semidirect
4. #89863C2semidirect
5. #89800C2semidirect
6. #89643C2semidirect
7. #89640C2semidirect
9. #89629C2semidirect
10. #89617C2semidirect
11. #89611C2semidirect
12. #89590C2semidirect
13. #89585C2semidirect
15. #89565C2semidirect
114. #475D7semidirect
306. C7#3258semidirect

Maximal subgroups

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

subgroupindex
#904242
#902742
C7×#4752
#898632
#898002
#896432
#896402
#896362
#896292
#896172
#896112
#895902
#895852
#895842
#895652
#32587