Finite Group Databasebeta

← #11739107

Subgroups — #11739107

Order 896.

344 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#90376C211yesmaximal, characteristic, complemented
2#90372C211yesmaximal, characteristic
3C7×#488C211yesmaximal, characteristic, complemented
4#89838C211yesmaximal, characteristic, complemented
5#89826C211yesmaximal, characteristic, complemented
6#89571C211yesmaximal, characteristic, complemented
7#89570C211yesmaximal, characteristic, complemented
8#89567C211yesmaximal, characteristic, complemented
9#89561C211yesmaximal, characteristic, complemented
10#89536C211yesmaximal, characteristic
11#89534C211yesmaximal, characteristic
12#89533C211yesmaximal, characteristic, complemented
13#89529C211yesmaximal, characteristic, complemented
14C4×#1132C211yesmaximal, characteristic, complemented
15#89451C211yesmaximal, characteristic, complemented
16C7×#125C2211yescharacteristic
17C7×#124C2211yescharacteristic
18#6510C2211yescharacteristic
19Q8×Dic7C2211yescharacteristic
20C2×#1144C2211yescharacteristic
21#6503C2211yescharacteristic
22D4×Dic7C2211yescharacteristic
23C2×#1142C2211yescharacteristic
24C2×#1132C2211yescharacteristic
25#6453C2211yescharacteristic
26#6452—21no—
27#6452—21no—
28#6451—21no—
29#6451C2211yescharacteristic
30#6450—21no—
31#6450C2211yescharacteristic
32#6450C2211yescharacteristic
33#6449—21no—
34#6448—21no—
35#6446—21no—
36(C22⋊D7)⋊C4—21no—
37C4×D28C2211yescharacteristic
38C4×Dic14C2211yescharacteristic
39C4×Dic14C2211yescharacteristic
40C7×#103C2211yescharacteristic
41C7×#103C2211yescharacteristic
42C7×#102C2211yescharacteristic
43C7×#102C2211yescharacteristic
44C4×C7×C8C2211yescharacteristic
45#6409C2211yescharacteristic
46#6406C2211yescharacteristic
47#6395C2211yescharacteristic
48#6391C2211yescharacteristic
49#6390C2211yescharacteristic
50#6389C2211yescharacteristic
51#6389C2211yescharacteristic
52Dic7⋊C8C2211yescharacteristic
53Dic7⋊C8C2211yescharacteristic
54#6383C2211yescharacteristic
55#6383C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3371—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×D4C2311yescharacteristic
62C2×C22⋊D7—21no—
63C2×C22⋊D7—21no—
64C22×Dic7—21no—
65C2×D28C2311yescharacteristic
66C2×C4×D7—21no—
67C2×Dic14—21no—
68C2×Dic14D411yescharacteristic
69C2×Dic14C2311yescharacteristic
70C2×C7×C8C2311yescharacteristic
71C2×C7×C8C2311yescharacteristic
72C7×C4⋊C4—21no—
73C7×C4⋊C4—21no—
74C7×C4⋊C4C2311yescharacteristic
75C7×C4⋊C4C2311yescharacteristic
76C7×C22⋊C4—21no—
77C7×C22⋊C4—21no—
78C42×C7C2311yescharacteristic
79C22⋊Dic7—21no—
80#1144—22no—
81#1142—22no—
82(C2×D7)⋊C4—21no—
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4—21no—
85C4⋊Dic7—21no—
86C4⋊Dic7—21no—
87C4⋊Dic7D411yescharacteristic
88C4⋊Dic7C2311yescharacteristic
89C4⋊Dic7C2311yescharacteristic
90Dic7⋊C4—21no—
91Dic7⋊C4—21no—
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4D411yescharacteristic
96Dic7⋊C4D411yescharacteristic
97C4×Dic7—21no—
98C4×Dic7—21no—
99C4×Dic7—21no—
100C4×Dic7—21no—
101C4×Dic7C2311yescharacteristic
102C4×Dic7C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105#1132—22no—
106#559—71no—
107#557—71no—
108#490—71no—
109#489—71no—
110#488D711yescharacteristic, complemented
111#479—71no—
112#478—71no—
113#476—71no—
114#464—71no—
115#464—71no—
116#460—71no—
117#446—71no—
118#441—71no—
119#440—71no—
120C4×SD16—71no—
121C23×C7—21no—
122C22×D7—21no—
123C7×Q8—21no—
124C7×Q8—21no—
125C7×D4—21no—
126C7×D4—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7C2×D411yescharacteristic
131C2×C4×C7C2×D411yescharacteristic
132C2×C4×C7C2411yescharacteristic
133C22⋊D7—42no—
134C22⋊D7—81no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7—81no—
139C2×Dic7C2×D411yescharacteristic
140C2×Dic7C2×D411yescharacteristic
141C2×Dic7C2×D411yescharacteristic
142C2×Dic7C2×D411yescharacteristic
143D28—21no—
144D28—21no—
145C4×D7—81no—
146Dic14—21no—
147Dic14—41no—
148Dic14—42no—
149Dic14#4112yes—
150C7×C8—21no—
151C7×C8—21no—
152C7⋊C8—21no—
153C7⋊C8—21no—
154C2×Q16—71no—
155C2×SD16—71no—
156C2×SD16—71no—
157C4⋊Q8—71no—
158C4⋊Q8—141no—
159#127—71no—
160#125—71no—
161#125—141no—
162#125—141no—
163#125C2×D711yescharacteristic
164#124—71no—
165#124—141no—
166#124C2×D711yescharacteristic
167#123—141no—
168#123—141no—
169C4×Q8—71no—
170C4×Q8—71no—
171C4×Q8—71no—
172C4×Q8—71no—
173C4×Q8—71no—
174C4×Q8—141no—
175C4×D4—71no—
176C4×D4—71no—
177C4×D4—141no—
178#107—71no—
179#106—71no—
180#106—71no—
181C4⋊C8—71no—
182C4⋊C8—71no—
183C4⋊C8—71no—
184#103—71no—
185#103—71no—
186#103C2×D711yescharacteristic
187#103C2×D711yescharacteristic
188#102—71no—
189#102—71no—
190#102—71no—
191#102—71no—
192#102C2×D711yescharacteristic
193#102C2×D711yescharacteristic
194#97—71no—
195#97—71no—
196C4×C8C2×D711yescharacteristic
197C22×C7—41no—
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—22no—
212Dic7—21no—
213Dic7—21no—
214Dic7—21no—
215Dic7—21no—
216Dic7—41no—
217Dic7—41no—
218C2×Q8—71no—
219C2×Q8—71no—
220C2×Q8—141no—
221C2×Q8C22×D711yescharacteristic
222C2×D4—71no—
223C2×D4—141no—
224C2×D4—141no—
225C2×D4C22×D711yescharacteristic
226C22×C4—141no—
227C22×C4—141no—
228Q16—142no—
229SD16—142no—
230SD16—142no—
231C2×C8—71no—
232C2×C8—71no—
233C2×C8C22×D711yescharacteristic
234C2×C8C22×D711yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—71no—
238C4⋊C4—71no—
239C4⋊C4—71no—
240C4⋊C4—71no—
241C4⋊C4—71no—
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—21no—
253C22⋊C4—141no—
254C22⋊C4—141no—
255C22⋊C4—141no—
256C22⋊C4—141no—
257C42—71no—
258C42—71no—
259C42—141no—
260C42—141no—
261C42—141no—
262C42—141no—
263C42C22×D711yescharacteristic
264C2×C7—81no—
265C2×C7#54811yescharacteristic
266C2×C7#57511yescharacteristic
267C2×C7#57711yescharacteristic
268D7—81no—
269C23—21no—
270C23—141no—
271Q8—21no—
272Q8—21no—
273Q8—72no—
274Q8—141no—
275Q8—281no—
276Q8—282no—
277D4—21no—
278D4—21no—
279D4—141no—
280D4—141no—
281D4—282no—
282D4—561no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—71no—
287C2×C4—71no—
288C2×C4—71no—
289C2×C4—71no—
290C2×C4—141no—
291C2×C4—141no—
292C2×C4—141no—
293C2×C4—561no—
294C2×C4—561no—
295C2×C4D4×D711yescharacteristic
296C2×C4D4×D711yescharacteristic
297C2×C4C23×D711yescharacteristic
298C8—21no—
299C8—21no—
300C8—141no—
301C8—141no—
302C7#337111yescharacteristic, complemented, Sylow
303C22—41no—
304C22—41no—
305C22—41no—
306C22—281no—
307C22—281no—
308C22—281no—
309C22C2×D4×D711yescharacteristic
310C4—21no—
311C4—21no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—142no—
316C4—141no—
317C4—141no—
318C4—141no—
319C4—141no—
320C4—281no—
321C4—281no—
322C4D7×#4111yescharacteristic
323C4#655411yescharacteristic
324C2—81no—
325C2—561no—
326C2#9036811yescharacteristic
327C2#9045911yescharacteristic
328C2#9046111yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#488C2
socleC22×C7C22×D4
prime radical#6503C22
hypercenter#488D7
hypocenterC7#3371
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #90376C2semidirect
3. C7×#488C2semidirect
4. #89838C2semidirect
5. #89826C2semidirect
6. #89571C2semidirect
7. #89570C2semidirect
8. #89567C2semidirect
9. #89561C2semidirect
12. #89533C2semidirect
13. #89529C2semidirect
14. C4×#1132C2semidirect
15. #89451C2semidirect
110. #488D7semidirect
302. C7#3371semidirect

Maximal subgroups

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

subgroupindex
#903762
#903722
C7×#4882
#898382
#898262
#895712
#895702
#895672
#895612
#895362
#895342
#895332
#895292
C4×#11322
#894512
#33717