Finite Group Databasebeta

← #11736638

Subgroups — #11736638

Order 896.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90223C211yesmaximal, characteristic
2#90220C211yesmaximal, characteristic, complemented
3C7×#427C211yesmaximal, characteristic, complemented
4#89886C211yesmaximal, characteristic, complemented
5#89883C211yesmaximal, characteristic, complemented
6#89775C211yesmaximal, characteristic, complemented
7#89768C211yesmaximal, characteristic, complemented
8#89661C211yesmaximal, characteristic, complemented
9#89658C211yesmaximal, characteristic, complemented
10#89653C211yesmaximal, characteristic, complemented
11#89648C211yesmaximal, characteristic, complemented
12#89643C211yesmaximal, characteristic, complemented
13#89639C211yesmaximal, characteristic, complemented
14#89635C211yesmaximal, characteristic, complemented
15#89626C211yesmaximal, characteristic, complemented
16C22×C7×C8C2211yescharacteristic
17C7×#117C2211yescharacteristic
18C7×#117C2211yescharacteristic
19#6493C2211yescharacteristic
20C4×C22⋊D7C2211yescharacteristic
21#6487C2211yescharacteristic
22C2×#1132C2211yescharacteristic
23C2×#1131C2211yescharacteristic
24#6460C2211yescharacteristic
25#6458C2211yescharacteristic
26#6456C2211yescharacteristic
27#6455C2211yescharacteristic
28D7×C4⋊C4C2211yescharacteristic
29#6453C2211yescharacteristic
30#6451C2211yescharacteristic
31#6450C2211yescharacteristic
32#6448—21no—
33#6447—21no—
34#6446—21no—
35#6442—21no—
36#6438—21no—
37#6435—21no—
38#6435—21no—
39#6433—21no—
40C7×#107C2211yescharacteristic
41C7×#107C2211yescharacteristic
42C7×#106C2211yescharacteristic
43C7×#106C2211yescharacteristic
44C22⋊(C7⋊C8)C2211yescharacteristic
45#6395C2211yescharacteristic
46(C2×D7)⋊C8C2211yescharacteristic
47#6391C2211yescharacteristic
48#6390C2211yescharacteristic
49#6389C2211yescharacteristic
50Dic7⋊C8C2211yescharacteristic
51#6384C2211yescharacteristic
52#6384C2211yescharacteristic
53#6383C2211yescharacteristic
54#6383C2211yescharacteristic
55#6382C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3348—71nomaximal, Sylow
60C22×C4×C7C2311yescharacteristic
61C2×C22⋊D7—21no—
62C2×C22⋊D7D411yescharacteristic
63C2×D28—21no—
64C2×D28C2311yescharacteristic
65C2×C4×D7—21no—
66C2×C4×D7—21no—
67C2×C4×D7C2311yescharacteristic
68C2×Dic14—21no—
69C2×Dic14C2311yescharacteristic
70C2×C7×C8—22no—
71C2×C7×C8C2311yescharacteristic
72C2×C7×C8C2311yescharacteristic
73C7×C4⋊C4C2311yescharacteristic
74C7×C4⋊C4C2311yescharacteristic
75C7×C4⋊C4C2311yescharacteristic
76C7×C4⋊C4C2311yescharacteristic
77C7×C22⋊C4—21no—
78C7×C22⋊C4—21no—
79C42×C7—21no—
80C42×C7—21no—
81C22⋊Dic7—21no—
82C22⋊Dic7D411yescharacteristic
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4—21no—
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4—21no—
87(C2×D7)⋊C4—21no—
88(C2×D7)⋊C4D411yescharacteristic
89C4⋊Dic7—21no—
90C4⋊Dic7—21no—
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4D411yescharacteristic
99C4×Dic7—21no—
100C4×Dic7—21no—
101C4×Dic7C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104#1132—22no—
105#1131—22no—
106#555—71no—
107#552—71no—
108#501—71no—
109#485—71no—
110#485—71no—
111#482—71no—
112#481—71no—
113#478—71no—
114#475—71no—
115#468—71no—
116#465—71no—
117#440—71no—
118#435—71no—
119#428—71no—
120#427D711yescharacteristic, complemented
121C23×C7C2×D411yescharacteristic
122C22×D7—21no—
123C22×D7C2×D411yescharacteristic
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7C2×D411yescharacteristic
131C2×C4×C7C2411yescharacteristic
132C22⋊D7—41no—
133C22⋊D7—81no—
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7C2×D411yescharacteristic
138C2×Dic7C2×D411yescharacteristic
139C2×Dic7C2×D411yescharacteristic
140D28—21no—
141D28—21no—
142D28—81no—
143C4×D7—21no—
144C4×D7—21no—
145C4×D7—42no—
146C4×D7—42no—
147Dic14—21no—
148Dic14—21no—
149Dic14—81no—
150C7×C8—21no—
151C7×C8#4112yes—
152C7⋊C8—21no—
153C7⋊C8—21no—
154C2×SD16—71no—
155C2×M16—71no—
156C22×C8C2×D711yescharacteristic
157C4⋊Q8—71no—
158#126—141no—
159#125—71no—
160#125—141no—
161#124—141no—
162#124—141no—
163#123—141no—
164C22⋊Q8—71no—
165C22⋊Q8—71no—
166#121—71no—
167#121—71no—
168#121—141no—
169C4×Q8—71no—
170C4×D4—71no—
171C4×D4—71no—
172#117—71no—
173#117—141no—
174#117—141no—
175#117C2×D711yescharacteristic
176#117C2×D711yescharacteristic
177C2×C4⋊C4—71no—
178#107—71no—
179#107—71no—
180#107C2×D711yescharacteristic
181#107C2×D711yescharacteristic
182#106—71no—
183#106—71no—
184#106—71no—
185#106C2×D711yescharacteristic
186#106C2×D711yescharacteristic
187C4⋊C8—71no—
188#103—71no—
189#103—71no—
190#103—71no—
191#102—71no—
192#102—71no—
193#102—71no—
194C22⋊C8—71no—
195C22⋊C8—71no—
196#97—71no—
197C22×C7—21no—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7C22×D411yescharacteristic
201C2×D7—21no—
202C2×D7—21no—
203C2×D7—21no—
204C2×D7—41no—
205C2×D7—41no—
206C2×D7—41no—
207C4×C7—21no—
208C4×C7—41no—
209C4×C7—41no—
210C4×C7—41no—
211C4×C7—41no—
212C4×C7C2×#4111yescharacteristic
213C4×C72−1+411yescharacteristic
214Dic7—21no—
215Dic7—21no—
216Dic7—21no—
217Dic7—41no—
218Dic7—41no—
219Dic7—41no—
220C2×Q8—71no—
221C2×Q8—141no—
222C2×D4—71no—
223C2×D4—71no—
224C2×D4—141no—
225C2×D4—141no—
226C22×C4—71no—
227C22×C4—141no—
228C22×C4—141no—
229C22×C4C22×D711yescharacteristic
230SD16—142no—
231M16—142no—
232C2×C8—22no—
233C2×C8—71no—
234C2×C8—71no—
235C2×C8C22×D711yescharacteristic
236C2×C8C22×D711yescharacteristic
237C4⋊C4—71no—
238C4⋊C4—71no—
239C4⋊C4—71no—
240C4⋊C4—141no—
241C4⋊C4—141no—
242C4⋊C4—141no—
243C4⋊C4—141no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4—141no—
247C4⋊C4C22×D711yescharacteristic
248C4⋊C4C22×D711yescharacteristic
249C4⋊C4C22×D711yescharacteristic
250C4⋊C4C22×D711yescharacteristic
251C22⋊C4—21no—
252C22⋊C4—21no—
253C22⋊C4—71no—
254C22⋊C4—71no—
255C22⋊C4—141no—
256C22⋊C4—141no—
257C22⋊C4—141no—
258C22⋊C4—141no—
259C22⋊C4—141no—
260C22⋊C4—141no—
261C42—21no—
262C42—21no—
263C42—71no—
264C42—141no—
265C42—141no—
266C2×C7—41no—
267C2×C7#54711yescharacteristic
268C2×C7#57511yescharacteristic
269C2×C7#57711yescharacteristic
270D7—41no—
271D7—81no—
272C23—71no—
273C23—141no—
274C23D4×D711yescharacteristic
275Q8—141no—
276Q8—141no—
277Q8—561no—
278D4—141no—
279D4—141no—
280D4—281no—
281D4—561no—
282D4—561no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—71no—
290C2×C4—71no—
291C2×C4—71no—
292C2×C4—141no—
293C2×C4—141no—
294C2×C4—141no—
295C2×C4—141no—
296C2×C4—141no—
297C2×C4—282no—
298C2×C4—282no—
299C2×C4D4×D711yescharacteristic
300C2×C4C23×D711yescharacteristic
301C8—21no—
302C8—141no—
303C8—141no—
304C8#115712yes—
305C7#334811yescharacteristic, complemented, Sylow
306C22—21no—
307C22—21no—
308C22—21no—
309C22—141no—
310C22—141no—
311C22—141no—
312C22—281no—
313C22—281no—
314C22—281no—
315C22C2×D4×D711yescharacteristic
316C4—21no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—141no—
322C4—141no—
323C4—141no—
324C4—281no—
325C4—281no—
326C4—281no—
327C4C2×#115711yescharacteristic
328C4#655111yescharacteristic
329C2—41no—
330C2—281no—
331C2—561no—
332C2#9019811yescharacteristic
333C2#9045911yescharacteristic
334C2#9046111yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#427C2
socleC22×C7C22×D4
prime radical#90223C2
hypercenter#427D7
hypocenterC7#3348
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
2. #90220C2semidirect
3. C7×#427C2semidirect
4. #89886C2semidirect
5. #89883C2semidirect
6. #89775C2semidirect
7. #89768C2semidirect
8. #89661C2semidirect
9. #89658C2semidirect
10. #89653C2semidirect
11. #89648C2semidirect
12. #89643C2semidirect
13. #89639C2semidirect
14. #89635C2semidirect
15. #89626C2semidirect
120. #427D7semidirect
305. C7#3348semidirect

Maximal subgroups

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

subgroupindex
#902232
#902202
C7×#4272
#898862
#898832
#897752
#897682
#896612
#896582
#896532
#896482
#896432
#896392
#896352
#896262
#33487