Finite Group Databasebeta

← #11738647

Subgroups — #11738647

Order 896.

358 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90381C211yesmaximal, characteristic, complemented
2#90250C211yesmaximal, characteristic, complemented
3C7×#478C211yesmaximal, characteristic, complemented
4#89840C211yesmaximal, characteristic, complemented
5C4×#1141C211yesmaximal, characteristic, complemented
6#89662C211yesmaximal, characteristic, complemented
7#89661C211yesmaximal, characteristic, complemented
8#89641C211yesmaximal, characteristic, complemented
9#89629C211yesmaximal, characteristic, complemented
10#89616C211yesmaximal, characteristic, complemented
11#89603C211yesmaximal, characteristic, complemented
12#89559C211yesmaximal, characteristic, complemented
13#89558C211yesmaximal, characteristic
14#89537C211yesmaximal, characteristic, complemented
15#89536C211yesmaximal, characteristic, complemented
16C7×#125C2211yescharacteristic
17C4×C7×D4C2211yescharacteristic
18#6503C2211yescharacteristic
19#6499C2211yescharacteristic
20C2×#1141C2211yescharacteristic
21#6492—21no—
22C4×C22⋊D7—21no—
23C2×D56C2211yescharacteristic
24C2×#1132C2211yescharacteristic
25#6460—21no—
26#6459—21no—
27#6459—21no—
28#6456C2211yescharacteristic
29#6456C2211yescharacteristic
30#6456C2211yescharacteristic
31D7×C4⋊C4—21no—
32#6453C2211yescharacteristic
33#6451C2211yescharacteristic
34#6450C2211yescharacteristic
35#6448—21no—
36#6446—21no—
37#6438C2211yescharacteristic
38C4×D28C2211yescharacteristic
39C4×Dic14C2211yescharacteristic
40C7×#107C2211yescharacteristic
41C7×#106C2211yescharacteristic
42C7×C4⋊C8C2211yescharacteristic
43C7×#102C2211yescharacteristic
44C7×#102C2211yescharacteristic
45#6406C2211yescharacteristic
46#6395C2211yescharacteristic
47#6390C2211yescharacteristic
48#6389C2211yescharacteristic
49#6389C2211yescharacteristic
50Dic7⋊C8C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52#6384C2211yescharacteristic
53#6384C2211yescharacteristic
54#6383C2211yescharacteristic
55#6383C2211yescharacteristic
56#6383C2211yescharacteristic
57#6381C2211yescharacteristic
58C4×C7⋊C8C2211yescharacteristic
59#3370—71nomaximal, Sylow
60C2×C7×D4C2311yescharacteristic
61C22×C4×C7—21no—
62C2×C22⋊D7—21no—
63C2×C22⋊D7—21no—
64C2×D28C2311yescharacteristic
65C2×D28C2311yescharacteristic
66C2×C4×D7—21no—
67C2×C4×D7—21no—
68C2×C4×D7—21no—
69C2×C4×D7—21no—
70C2×Dic14—21no—
71C2×Dic14D411yescharacteristic
72C2×Dic14C2311yescharacteristic
73C2×C7×C8C2311yescharacteristic
74C2×C7×C8C2311yescharacteristic
75C7×C4⋊C4—21no—
76C7×C4⋊C4—21no—
77C7×C4⋊C4C2311yescharacteristic
78C7×C4⋊C4C2311yescharacteristic
79C7×C4⋊C4C2311yescharacteristic
80C7×C22⋊C4—21no—
81C42×C7C2311yescharacteristic
82C22⋊Dic7—21no—
83C22⋊Dic7—21no—
84#1141—22no—
85(C2×D7)⋊C4—21no—
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—
91C4⋊Dic7—21no—
92C4⋊Dic7D411yescharacteristic
93C4⋊Dic7C2311yescharacteristic
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4D411yescharacteristic
98Dic7⋊C4D411yescharacteristic
99C4×Dic7—21no—
100C4×Dic7C2311yescharacteristic
101C4×Dic7C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104D56—22no—
105#1132—22no—
106#559—71no—
107#554—71no—
108#490—71no—
109#489—71no—
110#488—71no—
111#479—71no—
112#478D711yescharacteristic, complemented
113#474—71no—
114#463—71no—
115#463—71no—
116#459—71no—
117#446—71no—
118#442—71no—
119#440—71no—
120C4×D8—71no—
121C23×C7—21no—
122C22×D7—21no—
123C22×D7—21no—
124C7×D4—21no—
125C7×D4—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—81no—
130C2×C4×C7C2×D411yescharacteristic
131C2×C4×C7C2×D411yescharacteristic
132C2×C4×C7C2411yescharacteristic
133C22⋊D7—42no—
134C22⋊D7—81no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7C2×D411yescharacteristic
138C2×Dic7C2×D411yescharacteristic
139C2×Dic7C2×D411yescharacteristic
140C2×Dic7C2×D411yescharacteristic
141D28—21no—
142D28—21no—
143D28—21no—
144D28#4112yes—
145C4×D7—42no—
146C4×D7—81no—
147C4×D7—81no—
148C4×D7—81no—
149Dic14—21no—
150Dic14—21no—
151Dic14—41no—
152Dic14—81no—
153C7×C8—21no—
154C7×C8—21no—
155C7⋊C8—21no—
156C7⋊C8—21no—
157C2×SD16—71no—
158C2×D8—71no—
159C2×D8—71no—
160C4⋊Q8—71no—
161#127—71no—
162#125—71no—
163#125C2×D711yescharacteristic
164#124—71no—
165#124—71no—
166#124—141no—
167#123—141no—
168#123—141no—
169C22⋊Q8—141no—
170C22⋊Q8—141no—
171C22⋊Q8—141no—
172C4×Q8—71no—
173C4×Q8—71no—
174C4×D4—71no—
175C4×D4—71no—
176C4×D4—71no—
177C4×D4—71no—
178C4×D4—141no—
179C4×D4C2×D711yescharacteristic
180C2×C4⋊C4—141no—
181#107—71no—
182#107C2×D711yescharacteristic
183#106C2×D711yescharacteristic
184C4⋊C8—71no—
185C4⋊C8—71no—
186C4⋊C8C2×D711yescharacteristic
187#103—71no—
188#103—71no—
189#103—71no—
190#102—71no—
191#102—71no—
192#102—71no—
193#102—71no—
194#102—71no—
195#102C2×D711yescharacteristic
196#102C2×D711yescharacteristic
197#97—71no—
198#97—71no—
199C4×C8—71no—
200C22×C7—41no—
201C22×C7—41no—
202C22×C7—41no—
203C22×C7C22×D411yescharacteristic
204C2×D7—22no—
205C2×D7—41no—
206C2×D7—41no—
207C2×D7—41no—
208C2×D7—41no—
209C2×D7—41no—
210C4×C7—21no—
211C4×C7—21no—
212C4×C7—41no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7C2×#4111yescharacteristic
216C4×C72−1+411yescharacteristic
217Dic7—21no—
218Dic7—21no—
219Dic7—21no—
220Dic7—21no—
221Dic7—41no—
222Dic7—41no—
223C2×Q8—71no—
224C2×Q8—71no—
225C2×Q8—141no—
226C2×D4—71no—
227C2×D4—71no—
228C2×D4—141no—
229C2×D4—141no—
230C2×D4C22×D711yescharacteristic
231C22×C4—21no—
232C22×C4—141no—
233C22×C4—141no—
234C22×C4—141no—
235C22×C4—141no—
236SD16—142no—
237D8—142no—
238D8—142no—
239C2×C8—71no—
240C2×C8—71no—
241C2×C8C22×D711yescharacteristic
242C2×C8C22×D711yescharacteristic
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—71no—
246C4⋊C4—71no—
247C4⋊C4—71no—
248C4⋊C4—71no—
249C4⋊C4—141no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4C22×D711yescharacteristic
254C4⋊C4C22×D711yescharacteristic
255C4⋊C4C22×D711yescharacteristic
256C22⋊C4—21no—
257C22⋊C4—141no—
258C22⋊C4—141no—
259C22⋊C4—141no—
260C22⋊C4—141no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C22⋊C4—141no—
265C42—71no—
266C42—71no—
267C42—141no—
268C42C22×D711yescharacteristic
269C2×C7—81no—
270C2×C7#54811yescharacteristic
271C2×C7#57511yescharacteristic
272C2×C7#57611yescharacteristic
273D7—42no—
274D7—81no—
275C23—21no—
276C23—141no—
277C23—141no—
278Q8—141no—
279Q8—141no—
280Q8—281no—
281Q8—561no—
282D4—21no—
283D4—21no—
284D4—72no—
285D4—141no—
286D4—141no—
287D4—141no—
288D4—282no—
289D4—561no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—71no—
294C2×C4—71no—
295C2×C4—71no—
296C2×C4—71no—
297C2×C4—81no—
298C2×C4—141no—
299C2×C4—141no—
300C2×C4—282no—
301C2×C4—561no—
302C2×C4—561no—
303C2×C4—561no—
304C2×C4D4×D711yescharacteristic
305C2×C4D4×D711yescharacteristic
306C2×C4C23×D711yescharacteristic
307C8—21no—
308C8—21no—
309C8—141no—
310C8—141no—
311C7#337011yescharacteristic, complemented, Sylow
312C22—41no—
313C22—41no—
314C22—41no—
315C22—142no—
316C22—281no—
317C22—281no—
318C22—281no—
319C22—281no—
320C22—281no—
321C22C2×D4×D711yescharacteristic
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—141no—
328C4—141no—
329C4—141no—
330C4—141no—
331C4—281no—
332C4—281no—
333C4#655111yescharacteristic
334C4D7×#4111yescharacteristic
335C2—81no—
336C2—282no—
337C2—561no—
338C2#9032411yescharacteristic
339C2#9045911yescharacteristic
340C2#9046511yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#478C2
socleC22×C7C22×D4
prime radical#89558C2
hypercenter#478D7
hypocenterC7#3370
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #90381C2semidirect
2. #90250C2semidirect
3. C7×#478C2semidirect
4. #89840C2semidirect
5. C4×#1141C2semidirect
6. #89662C2semidirect
7. #89661C2semidirect
8. #89641C2semidirect
9. #89629C2semidirect
10. #89616C2semidirect
11. #89603C2semidirect
12. #89559C2semidirect
14. #89537C2semidirect
15. #89536C2semidirect
112. #478D7semidirect
311. C7#3370semidirect

Maximal subgroups

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

subgroupindex
#903812
#902502
C7×#4782
#898402
C4×#11412
#896622
#896612
#896412
#896292
#896162
#896032
#895592
#895582
#895372
#895362
#33707