Finite Group Databasebeta

← #11737202

Subgroups — #11737202

Order 896.

346 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90259C211yesmaximal, characteristic, complemented
2#90234C211yesmaximal, characteristic, complemented
3C7×#440C211yesmaximal, characteristic, complemented
4#89937C211yesmaximal, characteristic
5#89934C211yesmaximal, characteristic, complemented
6#89792C211yesmaximal, characteristic, complemented
7#89788C211yesmaximal, characteristic
8#89663C211yesmaximal, characteristic, complemented
9#89649C211yesmaximal, characteristic, complemented
10#89572C211yesmaximal, characteristic, complemented
11#89569C211yesmaximal, characteristic, complemented
12#89557C211yesmaximal, characteristic
13#89530C211yesmaximal, characteristic, complemented
14#89486C211yesmaximal, characteristic, complemented
15#89473C211yesmaximal, characteristic, complemented
16C2×C7×SD16C2211yescharacteristic
17C4×C7×Q8C2211yescharacteristic
18C4×C7×D4C2211yescharacteristic
19#6510C2211yescharacteristic
20#6507C2211yescharacteristic
21D4×Dic7C2211yescharacteristic
22D4×Dic7C2211yescharacteristic
23C2×#1142C2211yescharacteristic
24C2×#1141C2211yescharacteristic
25C2×C4⋊Dic7—21no—
26#6487—21no—
27#6461—21no—
28#6459—21no—
29#6456C2211yescharacteristic
30#6453C2211yescharacteristic
31#6453C2211yescharacteristic
32#6450C2211yescharacteristic
33#6441—21no—
34#6441—21no—
35#6439—21no—
36#6438C2211yescharacteristic
37#6435—21no—
38#6432C2211yescharacteristic
39C4×Dic14C2211yescharacteristic
40C7×#107C2211yescharacteristic
41C7×#103C2211yescharacteristic
42C7×#102C2211yescharacteristic
43C7×#97C2211yescharacteristic
44#6409C2211yescharacteristic
45#6409C2211yescharacteristic
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#3376—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×D4C2311yescharacteristic
62C22×C4×C7—21no—
63C22×Dic7—21no—
64C22×Dic7—21no—
65C2×D28C2311yescharacteristic
66C2×C4×D7—21no—
67C2×Dic14—21no—
68C2×Dic14D411yescharacteristic
69C2×Dic14C2311yescharacteristic
70C7×SD16—22no—
71C2×C7×C8C2311yescharacteristic
72C2×C7×C8C2311yescharacteristic
73C7×C4⋊C4—21no—
74C7×C4⋊C4C2311yescharacteristic
75C7×C4⋊C4C2311yescharacteristic
76C7×C22⋊C4—21no—
77C42×C7—21no—
78C42×C7C2311yescharacteristic
79C22⋊Dic7—21no—
80C22⋊Dic7—21no—
81#1142—22no—
82#1141—22no—
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—
88C4⋊Dic7—21no—
89C4⋊Dic7—21no—
90C4⋊Dic7—21no—
91C4⋊Dic7D411yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93C4⋊Dic7C2311yescharacteristic
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4D411yescharacteristic
100Dic7⋊C4D411yescharacteristic
101C4×Dic7—21no—
102C4×Dic7C2311yescharacteristic
103C4×Dic7C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106#556—71no—
107#554—71no—
108#491—71no—
109#489—71no—
110#488—71no—
111#479—71no—
112#478—71no—
113#475—71no—
114#463—71no—
115#463—71no—
116#461—71no—
117#446—71no—
118#442—71no—
119#440D711yescharacteristic, complemented
120C4×SD16—71no—
121C23×C7—21no—
122C22×D7—21no—
123C7×Q8—21no—
124C7×Q8—21no—
125C7×D4—21no—
126C7×D4#4112yes—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—42no—
131C2×C4×C7C2×D411yescharacteristic
132C2×C4×C7C2×D411yescharacteristic
133C2×C4×C7C2411yescharacteristic
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7—81no—
138C2×Dic7—81no—
139C2×Dic7C2×D411yescharacteristic
140C2×Dic7C2×D411yescharacteristic
141C2×Dic7C2×D411yescharacteristic
142C2×Dic7C2×D411yescharacteristic
143D28—21no—
144D28—21no—
145C4×D7—42no—
146Dic14—21no—
147Dic14—21no—
148Dic14—41no—
149Dic14—81no—
150C7×C8—21no—
151C7×C8—21no—
152C7⋊C8—21no—
153C7⋊C8—21no—
154C2×SD16—71no—
155C2×SD16C2×D711yescharacteristic
156C2×D8—71no—
157C4⋊Q8—71no—
158C4⋊Q8—71no—
159#126—141no—
160#126—141no—
161#125—71no—
162#125—71no—
163#124—71no—
164#124—71no—
165C22⋊Q8—141no—
166C22⋊Q8—141no—
167C22⋊Q8—141no—
168C22⋊Q8—141no—
169C4×Q8—71no—
170C4×Q8—71no—
171C4×Q8C2×D711yescharacteristic
172C4×D4—71no—
173C4×D4—71no—
174C4×D4—71no—
175C4×D4C2×D711yescharacteristic
176#117—141no—
177C2×C4⋊C4—141no—
178#107C2×D711yescharacteristic
179#106—71no—
180#106—71no—
181C4⋊C8—71no—
182C4⋊C8—71no—
183C4⋊C8—71no—
184#103—71no—
185#103—71no—
186#103—71no—
187#103—71no—
188#103—71no—
189#103C2×D711yescharacteristic
190#102—71no—
191#102—71no—
192#102—71no—
193#102C2×D711yescharacteristic
194#97—71no—
195#97C2×D711yescharacteristic
196C4×C8—71no—
197C22×C7—22no—
198C22×C7—41no—
199C22×C7—41no—
200C22×C7C22×D411yescharacteristic
201C2×D7—41no—
202C2×D7—41no—
203C2×D7—41no—
204C4×C7—21no—
205C4×C7—21no—
206C4×C7—41no—
207C4×C7—41no—
208C4×C7—41no—
209C4×C7C2×#4111yescharacteristic
210C4×C72−1+411yescharacteristic
211Dic7—21no—
212Dic7—21no—
213Dic7—21no—
214Dic7—21no—
215Dic7—41no—
216Dic7—41no—
217Dic7—41no—
218C2×Q8—71no—
219C2×Q8—71no—
220C2×Q8—141no—
221C2×Q8C22×D711yescharacteristic
222C2×D4—71no—
223C2×D4C22×D711yescharacteristic
224C22×C4—21no—
225C22×C4—141no—
226C22×C4—141no—
227C22×C4—141no—
228SD16—22no—
229SD16—142no—
230D8—142no—
231C2×C8—71no—
232C2×C8—71no—
233C2×C8C22×D711yescharacteristic
234C2×C8C22×D711yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—71no—
237C4⋊C4—71no—
238C4⋊C4—71no—
239C4⋊C4—71no—
240C4⋊C4—71no—
241C4⋊C4—141no—
242C4⋊C4—141no—
243C4⋊C4—141no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4—141no—
247C4⋊C4—141no—
248C4⋊C4—141no—
249C4⋊C4C22×D711yescharacteristic
250C4⋊C4C22×D711yescharacteristic
251C22⋊C4—21no—
252C22⋊C4—141no—
253C22⋊C4—141no—
254C22⋊C4—141no—
255C22⋊C4—141no—
256C22⋊C4—141no—
257C22⋊C4—141no—
258C22⋊C4—141no—
259C42—21no—
260C42—71no—
261C42—71no—
262C42—141no—
263C42C22×D711yescharacteristic
264C2×C7—42no—
265C2×C7#54811yescharacteristic
266C2×C7#57511yescharacteristic
267C2×C7#57711yescharacteristic
268D7—81no—
269C23—21no—
270C23—141no—
271Q8—21no—
272Q8—21no—
273Q8—141no—
274Q8—141no—
275Q8—281no—
276Q8—561no—
277D4—21no—
278D4—141no—
279D4—141no—
280D4#115712yes—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—71no—
286C2×C4—71no—
287C2×C4—71no—
288C2×C4—71no—
289C2×C4—141no—
290C2×C4—141no—
291C2×C4—141no—
292C2×C4—282no—
293C2×C4—561no—
294C2×C4—561no—
295C2×C4D4×D711yescharacteristic
296C2×C4D4×D711yescharacteristic
297C2×C4C23×D711yescharacteristic
298C8—21no—
299C8—21no—
300C8—141no—
301C8—141no—
302C7#337611yescharacteristic, complemented, Sylow
303C22—22no—
304C22—41no—
305C22—41no—
306C22—281no—
307C22—281no—
308C22—281no—
309C22C2×D4×D711yescharacteristic
310C4—21no—
311C4—21no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—141no—
316C4—141no—
317C4—141no—
318C4—141no—
319C4—281no—
320C4—281no—
321C4—281no—
322C4C2×#115711yescharacteristic
323C4#655411yescharacteristic
324C2—42no—
325C2—561no—
326C2#9024311yescharacteristic
327C2#9046411yescharacteristic
328C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#440C2
socleC22×C7C22×D4
prime radicalC2×#1141C22
hypercenter#440D7
hypocenterC7#3376
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #90259C2semidirect
2. #90234C2semidirect
3. C7×#440C2semidirect
5. #89934C2semidirect
6. #89792C2semidirect
8. #89663C2semidirect
9. #89649C2semidirect
10. #89572C2semidirect
11. #89569C2semidirect
13. #89530C2semidirect
14. #89486C2semidirect
15. #89473C2semidirect
119. #440D7semidirect
302. C7#3376semidirect

Maximal subgroups

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

subgroupindex
#902592
#902342
C7×#4402
#899372
#899342
#897922
#897882
#896632
#896492
#895722
#895692
#895572
#895302
#894862
#894732
#33767