Finite Group Databasebeta

← #11737298

Subgroups — #11737298

Order 896.

384 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90250C211yesmaximal, characteristic, complemented
2D4×Dic14C211yesmaximal, characteristic, complemented
3C7×#442C211yesmaximal, characteristic, complemented
4#89925C211yesmaximal, characteristic, complemented
5#89922C211yesmaximal, characteristic
6#89788C211yesmaximal, characteristic, complemented
7#89780C211yesmaximal, characteristic, complemented
8#89642C211yesmaximal, characteristic, complemented
9#89626C211yesmaximal, characteristic, complemented
10#89557C211yesmaximal, characteristic, complemented
11#89537C211yesmaximal, characteristic, complemented
12#89532C211yesmaximal, characteristic, complemented
13#89530C211yesmaximal, characteristic, complemented
14#89486C211yesmaximal, characteristic, complemented
15#89473C211yesmaximal, characteristic, complemented
16C22×Dic14—21no—
17C2×C7×D8C2211yescharacteristic
18C4×C7×D4C2211yescharacteristic
19C4×C7×D4C2211yescharacteristic
20#6503C2211yescharacteristic
21#6499C2211yescharacteristic
22D4×Dic7C2211yescharacteristic
23D4×Dic7C2211yescharacteristic
24C2×#1142C2211yescharacteristic
25C2×#1141C2211yescharacteristic
26#6492—21no—
27C4×C22⋊D7—21no—
28#6487—21no—
29#6456C2211yescharacteristic
30#6451C2211yescharacteristic
31#6451C2211yescharacteristic
32#6450C2211yescharacteristic
33#6448—21no—
34#6446—21no—
35#6441—21no—
36#6441—21no—
37#6438C2211yescharacteristic
38#6432C2211yescharacteristic
39C4×Dic14C2211yescharacteristic
40C7×#106C2211yescharacteristic
41C7×#102C2211yescharacteristic
42C7×#102C2211yescharacteristic
43C7×#97C2211yescharacteristic
44#6406C2211yescharacteristic
45#6406C2211yescharacteristic
46#6395C2211yescharacteristic
47#6395C2211yescharacteristic
48#6390C2211yescharacteristic
49#6390C2211yescharacteristic
50#6389C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52Dic7⋊C8C2211yescharacteristic
53C8×Dic7C2211yescharacteristic
54#6384C2211yescharacteristic
55#6383C2211yescharacteristic
56#6382C2211yescharacteristic
57#6382C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3373—71nomaximal, Sylow
60C2×C7×D4C2311yescharacteristic
61C2×C7×D4C2311yescharacteristic
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C2×C22⋊D7—21no—
65C2×C22⋊D7—21no—
66C22×Dic7—21no—
67C22×Dic7—21no—
68C2×D28C2311yescharacteristic
69C2×C4×D7—21no—
70C2×Dic14—21no—
71C2×Dic14—21no—
72C2×Dic14—21no—
73C2×Dic14—44no—
74C2×Dic14D411yescharacteristic
75C2×Dic14C2311yescharacteristic
76C7×D8—22no—
77C2×C7×C8C2311yescharacteristic
78C2×C7×C8C2311yescharacteristic
79C7×C4⋊C4C2311yescharacteristic
80C7×C4⋊C4C2311yescharacteristic
81C7×C22⋊C4—21no—
82C7×C22⋊C4—21no—
83C42×C7C2311yescharacteristic
84C22⋊Dic7—21no—
85C22⋊Dic7—21no—
86C22⋊Dic7—21no—
87C22⋊Dic7—21no—
88#1142—22no—
89#1141—22no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93C4⋊Dic7—21no—
94C4⋊Dic7D411yescharacteristic
95C4⋊Dic7C2311yescharacteristic
96C4⋊Dic7C2311yescharacteristic
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4D411yescharacteristic
101Dic7⋊C4D411yescharacteristic
102C4×Dic7—21no—
103C4×Dic7C2311yescharacteristic
104C4×Dic7C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107#559—71no—
108D4×Q8—71no—
109#489—71no—
110#489—71no—
111#486—71no—
112#476—71no—
113#476—71no—
114#475—71no—
115#463—71no—
116#463—71no—
117#459—71no—
118#446—71no—
119#442—71no—
120#442D711yescharacteristic, complemented
121C4×SD16—71no—
122C23×C7—21no—
123C23×C7—21no—
124C22×D7—21no—
125C7×D4—21no—
126C7×D4—21no—
127C7×D4—21no—
128C7×D4#4112yes—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—42no—
132C2×C4×C7—81no—
133C2×C4×C7C2×D411yescharacteristic
134C2×C4×C7C2×D411yescharacteristic
135C2×C4×C7C2411yescharacteristic
136C22⋊D7—42no—
137C22⋊D7—81no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—42no—
142C2×Dic7—42no—
143C2×Dic7C2×D411yescharacteristic
144C2×Dic7C2×D411yescharacteristic
145C2×Dic7C2×D411yescharacteristic
146C2×Dic7C2×D411yescharacteristic
147D28—21no—
148D28—21no—
149C4×D7—81no—
150Dic14—22no—
151Dic14—21no—
152Dic14—21no—
153Dic14—42no—
154Dic14—42no—
155Dic14—42no—
156C7×C8—21no—
157C7×C8—21no—
158C7⋊C8—21no—
159C7⋊C8—21no—
160C22×Q8—141no—
161C2×SD16—71no—
162C2×D8—71no—
163C2×D8C2×D711yescharacteristic
164C4⋊Q8—71no—
165C4⋊Q8—71no—
166C4⋊Q8—71no—
167#127—71no—
168#124—71no—
169#124—71no—
170#124—141no—
171#123—141no—
172#123—141no—
173C22⋊Q8—141no—
174C22⋊Q8—141no—
175C22⋊Q8—141no—
176C4×Q8—71no—
177C4×Q8—71no—
178C4×D4—71no—
179C4×D4—71no—
180C4×D4—71no—
181C4×D4—141no—
182C4×D4C2×D711yescharacteristic
183C4×D4C2×D711yescharacteristic
184#106—71no—
185#106—71no—
186#106C2×D711yescharacteristic
187C4⋊C8—71no—
188C4⋊C8—71no—
189C4⋊C8—71no—
190#103—71no—
191#103—71no—
192#103—71no—
193#102—71no—
194#102—71no—
195#102—71no—
196#102—71no—
197#102—71no—
198#102C2×D711yescharacteristic
199#102C2×D711yescharacteristic
200#97—71no—
201#97C2×D711yescharacteristic
202C4×C8—71no—
203C22×C7—22no—
204C22×C7—41no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7—41no—
208C22×C7—41no—
209C22×C7C22×D411yescharacteristic
210C2×D7—41no—
211C2×D7—41no—
212C2×D7—41no—
213C4×C7—21no—
214C4×C7—21no—
215C4×C7—41no—
216C4×C7—41no—
217C4×C7C2×#4111yescharacteristic
218C4×C72−1+411yescharacteristic
219Dic7—21no—
220Dic7—21no—
221Dic7—21no—
222Dic7—41no—
223Dic7—41no—
224Dic7—41no—
225Dic7AGL(1,ℤ/8)12yes—
226C2×Q8—71no—
227C2×Q8—71no—
228C2×Q8—141no—
229C2×Q8—141no—
230C2×Q8—141no—
231C2×Q8—284no—
232C2×D4—71no—
233C2×D4—141no—
234C2×D4—141no—
235C2×D4C22×D711yescharacteristic
236C2×D4C22×D711yescharacteristic
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—141no—
240C22×C4—141no—
241C22×C4—141no—
242SD16—142no—
243D8—22no—
244D8—142no—
245C2×C8—71no—
246C2×C8—71no—
247C2×C8C22×D711yescharacteristic
248C2×C8C22×D711yescharacteristic
249C4⋊C4—71no—
250C4⋊C4—71no—
251C4⋊C4—71no—
252C4⋊C4—71no—
253C4⋊C4—71no—
254C4⋊C4—141no—
255C4⋊C4—141no—
256C4⋊C4—141no—
257C4⋊C4—141no—
258C4⋊C4C22×D711yescharacteristic
259C4⋊C4C22×D711yescharacteristic
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C22⋊C4—141no—
265C22⋊C4—141no—
266C22⋊C4—141no—
267C22⋊C4—141no—
268C22⋊C4—141no—
269C42—71no—
270C42—71no—
271C42—141no—
272C42C22×D711yescharacteristic
273C2×C7—42no—
274C2×C7—81no—
275C2×C7#54811yescharacteristic
276C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
277C2×C7#57711yescharacteristic
278D7—81no—
279C23—21no—
280C23—21no—
281C23—141no—
282Q8—142no—
283Q8—141no—
284Q8—141no—
285Q8—282no—
286Q8—282no—
287Q8—282no—
288D4—21no—
289D4—21no—
290D4—21no—
291D4—141no—
292D4—141no—
293D4—282no—
294D4—561no—
295D4#115712yes—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—42no—
299C2×C4—71no—
300C2×C4—71no—
301C2×C4—71no—
302C2×C4—71no—
303C2×C4—81no—
304C2×C4—141no—
305C2×C4—141no—
306C2×C4—141no—
307C2×C4—282no—
308C2×C4—282no—
309C2×C4—561no—
310C2×C4D4×D711yescharacteristic
311C2×C4D4×D711yescharacteristic
312C2×C4C23×D711yescharacteristic
313C8—21no—
314C8—21no—
315C8—141no—
316C8—141no—
317C7#337311yescharacteristic, complemented, Sylow
318C22—22no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—281no—
325C22—281no—
326C22—281no—
327C22C2×D4×D711yescharacteristic
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—72no—
333C4—141no—
334C4—141no—
335C4—141no—
336C4—281no—
337C4—281no—
338C4—281no—
339C4C2×#115711yescharacteristic
340C4#655411yescharacteristic
341C2—42no—
342C2—81no—
343C2—561no—
344C2#9024311yescharacteristic
345C2D7×AGL(1,ℤ/8)11yescharacteristic
346C2#9046611yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#442C2
socleC22×C7C22×D4
prime radical#89922C2
hypercenter#442D7
hypocenterC7#3373
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #90250C2semidirect
2. D4×Dic14C2semidirect
3. C7×#442C2semidirect
4. #89925C2semidirect
6. #89788C2semidirect
7. #89780C2semidirect
8. #89642C2semidirect
9. #89626C2semidirect
10. #89557C2semidirect
11. #89537C2semidirect
12. #89532C2semidirect
13. #89530C2semidirect
14. #89486C2semidirect
15. #89473C2semidirect
120. #442D7semidirect
317. C7#3373semidirect

Maximal subgroups

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

subgroupindex
#902502
D4×Dic142
C7×#4422
#899252
#899222
#897882
#897802
#896422
#896262
#895572
#895372
#895322
#895302
#894862
#894732
#33737