Finite Group Databasebeta

← #11738765

Subgroups — #11738765

Order 896.

388 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90302C211yesmaximal, characteristic
2#90202C211yesmaximal, characteristic, complemented
3C7×#481C211yesmaximal, characteristic, complemented
4#89813C211yesmaximal, characteristic, complemented
5C2×#6382C211yesmaximal, characteristic, complemented
6#89634C211yesmaximal, characteristic, complemented
7D7×#106C211yesmaximal, characteristic, complemented
8#89627C211yesmaximal, characteristic, complemented
9#89626C211yesmaximal, characteristic, complemented
10#89550C211yesmaximal, characteristic, complemented
11#89549C211yesmaximal, characteristic, complemented
12#89533C211yesmaximal, characteristic, complemented
13#89530C211yesmaximal, characteristic, complemented
14(C22⋊D7)⋊C8C211yesmaximal, characteristic, complemented
15#89493C211yesmaximal, characteristic, complemented
16C2×#1159—21no—
17C7×#121C2211yescharacteristic
18C2×C7×C4⋊C4C2211yescharacteristic
19#6501C2211yescharacteristic
20#6498—21no—
21D4×Dic7C2211yescharacteristic
22C2×#1142C2211yescharacteristic
23C4×C22⋊D7—21no—
24C4×C22⋊D7C2211yescharacteristic
25C2×C4⋊Dic7C2211yescharacteristic
26#6487—21no—
27#6487C2211yescharacteristic
28C22×C7⋊C8C2211yescharacteristic
29C2×C8×D7C2211yescharacteristic
30#6460C2211yescharacteristic
31#6459—21no—
32D7×C4⋊C4C2211yescharacteristic
33D7×C4⋊C4C2211yescharacteristic
34#6452—21no—
35#6451C2211yescharacteristic
36#6451C2211yescharacteristic
37#6450C2211yescharacteristic
38#6446—21no—
39#6441—21no—
40C7×#106C2211yescharacteristic
41C7×#106C2211yescharacteristic
42C7×#102C2211yescharacteristic
43C7×#102C2211yescharacteristic
44C7×C22⋊C8C2211yescharacteristic
45#6406C2211yescharacteristic
46(C2×D7)⋊C8C2211yescharacteristic
47#6391C2211yescharacteristic
48#6391C2211yescharacteristic
49#6390C2211yescharacteristic
50#6390C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52C8×Dic7C2211yescharacteristic
53#6384C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6382C2211yescharacteristic
57#6382C2211yescharacteristic
58#6382C2211yescharacteristic
59#3335—71nomaximal, Sylow
60C2×C7×D4—21no—
61C2×C7×D4C2311yescharacteristic
62C22×C4×C7—21no—
63C22×C4×C7C2311yescharacteristic
64C2×C22⋊D7—21no—
65C2×C22⋊D7D411yescharacteristic
66C22×Dic7—21no—
67C22×Dic7—21no—
68#1159—44no—
69C2×C4×D7—21no—
70C2×C4×D7—21no—
71C2×C4×D7C2311yescharacteristic
72C2×Dic14—21no—
73C2×Dic14—21no—
74C2×Dic14C2311yescharacteristic
75C2×C7×C8C2311yescharacteristic
76C2×C7×C8C2311yescharacteristic
77C7×C4⋊C4—21no—
78C7×C4⋊C4C2311yescharacteristic
79C7×C4⋊C4C2311yescharacteristic
80C7×C4⋊C4C2311yescharacteristic
81C7×C22⋊C4—21no—
82C22⋊Dic7—21no—
83C22⋊Dic7—21no—
84C22⋊Dic7—21no—
85C22⋊Dic7D411yescharacteristic
86#1142—22no—
87(C2×D7)⋊C4—21no—
88(C2×D7)⋊C4D411yescharacteristic
89C4⋊Dic7—21no—
90C4⋊Dic7—21no—
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93C4⋊Dic7C2311yescharacteristic
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4D411yescharacteristic
101C4×Dic7—21no—
102C4×Dic7C2311yescharacteristic
103C2×C7⋊C8—22no—
104C2×C7⋊C8C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106C8×D7—22no—
107#554—71no—
108#547—71no—
109#498—71no—
110#483—71no—
111#483—71no—
112#481—71no—
113#481D711yescharacteristic, complemented
114#476—71no—
115#475—71no—
116#465—71no—
117#465—71no—
118C4×SD16—71no—
119C8×D4—71no—
120C2×#106—71no—
121C2×#106—71no—
122C23×C7—21no—
123C23×C7C2×D411yescharacteristic
124C22×D7C2×D411yescharacteristic
125C7×D4—21no—
126C7×D4—21no—
127C7×D4—42no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—81no—
134C2×C4×C7C2×D411yescharacteristic
135C2×C4×C7C2411yescharacteristic
136C22⋊D7—42no—
137C22⋊D7SD1614yescomplemented
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—42no—
143C2×Dic7—42no—
144C2×Dic7C2×D411yescharacteristic
145C2×Dic7C2×D411yescharacteristic
146C2×Dic7C2×D411yescharacteristic
147C4×D7—21no—
148C4×D7—21no—
149C4×D7—42no—
150C4×D7—81no—
151Dic14—21no—
152Dic14—21no—
153Dic14—42no—
154Dic14—81no—
155C7×C8—21no—
156C7×C8—21no—
157C7⋊C8—21no—
158C7⋊C8#4112yescomplemented
159C2×#41—141no—
160C2×SD16—71no—
161C22×C8—71no—
162C22×C8—71no—
163C4⋊Q8—71no—
164C4⋊Q8—71no—
165#125—141no—
166#123—141no—
167#123—141no—
168C22⋊Q8—71no—
169C22⋊Q8—71no—
170C22⋊Q8—141no—
171C22⋊Q8—141no—
172C22⋊Q8—141no—
173#121—71no—
174#121C2×D711yescharacteristic
175C4×Q8—71no—
176C4×D4—71no—
177C4×D4—71no—
178C4×D4—141no—
179C2×C4⋊C4—71no—
180C2×C4⋊C4—71no—
181C2×C4⋊C4—71no—
182C2×C4⋊C4C2×D711yescharacteristic
183#106—71no—
184#106—71no—
185#106—71no—
186#106—71no—
187#106—71no—
188#106—71no—
189#106—71no—
190#106C2×D711yescharacteristic
191#106C2×D711yescharacteristic
192C4⋊C8—71no—
193#103—71no—
194#103—71no—
195#103—71no—
196#102—71no—
197#102C2×D711yescharacteristic
198#102C2×D711yescharacteristic
199C22⋊C8—71no—
200C22⋊C8C2×D711yescharacteristic
201C4×C8—71no—
202C22×C7—21no—
203C22×C7—21no—
204C22×C7—41no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7C2×SD1612yes—
208C22×C7C22×D411yescharacteristic
209C2×D7—21no—
210C2×D7—21no—
211C2×D7C2×SD1612yes—
212C4×C7—21no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7C2×#4111yescharacteristic
217C4×C72−1+411yescharacteristic
218Dic7—21no—
219Dic7—21no—
220Dic7—41no—
221Dic7—41no—
222Dic7—41no—
223Dic7—41no—
224Dic7C2×SD1612yes—
225#41—284no—
226C2×Q8—71no—
227C2×Q8—141no—
228C2×Q8—141no—
229C2×D4—21no—
230C2×D4—71no—
231C2×D4—141no—
232C2×D4C22×D711yescharacteristic
233C22×C4—21no—
234C22×C4—71no—
235C22×C4—141no—
236C22×C4—141no—
237C22×C4—141no—
238C22×C4—141no—
239C22×C4C22×D711yescharacteristic
240SD16—142no—
241C2×C8—71no—
242C2×C8—71no—
243C2×C8—142no—
244C2×C8—142no—
245C2×C8C22×D711yescharacteristic
246C2×C8C22×D711yescharacteristic
247C4⋊C4—21no—
248C4⋊C4—71no—
249C4⋊C4—71no—
250C4⋊C4—71no—
251C4⋊C4—71no—
252C4⋊C4—141no—
253C4⋊C4—141no—
254C4⋊C4—141no—
255C4⋊C4—141no—
256C4⋊C4—141no—
257C4⋊C4—141no—
258C4⋊C4—141no—
259C4⋊C4—141no—
260C4⋊C4C22×D711yescharacteristic
261C4⋊C4C22×D711yescharacteristic
262C4⋊C4C22×D711yescharacteristic
263C22⋊C4—21no—
264C22⋊C4—71no—
265C22⋊C4—71no—
266C22⋊C4—141no—
267C22⋊C4—141no—
268C22⋊C4—141no—
269C22⋊C4—141no—
270C42—71no—
271C42—141no—
272C2×C7—22no—
273C2×C7—81no—
274C2×C7#54711yescharacteristic
275C2×C7C22×SD1611yescharacteristic
276C2×C7#57711yescharacteristic
277D7—22no—
278C23—21no—
279C23—71no—
280C23D4×D711yescharacteristic
281Q8—141no—
282Q8—141no—
283Q8—282no—
284Q8—561no—
285D4—21no—
286D4—21no—
287D4—42no—
288D4—74no—
289D4—282no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—71no—
296C2×C4—71no—
297C2×C4—71no—
298C2×C4—81no—
299C2×C4—141no—
300C2×C4—141no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—141no—
305C2×C4—282no—
306C2×C4—282no—
307C2×C4—282no—
308C2×C4—561no—
309C2×C4D4×D711yescharacteristic
310C2×C4C23×D711yescharacteristic
311C8—21no—
312C8—21no—
313C8—72no—
314C8—141no—
315C7#333511yescharacteristic, complemented, Sylow
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—72no—
322C22—141no—
323C22—141no—
324C22D7×SD1612yes—
325C22C2×D4×D711yescharacteristic
326C4—21no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—72no—
331C4—141no—
332C4—141no—
333C4—281no—
334C4—281no—
335C4—281no—
336C4—281no—
337C4D7×#4111yescharacteristic
338C4#655411yescharacteristic
339C2—22no—
340C2—81no—
341C2—142no—
342C2#9034611yescharacteristic
343C2C2×D7×SD1611yescharacteristic
344C2#9046611yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#481C2
socleC22×C7C22×D4
prime radical#90302C2
hypercenter#481D7
hypocenterC7#3335
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. #90202C2semidirect
3. C7×#481C2semidirect
4. #89813C2semidirect
5. C2×#6382C2semidirect
6. #89634C2semidirect
7. D7×#106C2semidirect
8. #89627C2semidirect
9. #89626C2semidirect
10. #89550C2semidirect
11. #89549C2semidirect
12. #89533C2semidirect
13. #89530C2semidirect
14. (C22⋊D7)⋊C8C2semidirect
15. #89493C2semidirect
113. #481D7semidirect
137. C22⋊D7SD16semidirect
158. C7⋊C8#41semidirect
315. C7#3335semidirect

Maximal subgroups

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

subgroupindex
#903022
#902022
C7×#4812
#898132
C2×#63822
#896342
D7×#1062
#896272
#896262
#895502
#895492
#895332
#895302
(C22⋊D7)⋊C82
#894932
#33357