Finite Group Databasebeta

← #11739441

Subgroups — #11739441

Order 896.

336 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90423C211yesmaximal, characteristic
2#90385C211yesmaximal, characteristic
3C7×#501C211yesmaximal, characteristic, complemented
4#89863C211yesmaximal, characteristic, complemented
5#89835C211yesmaximal, characteristic, complemented
6#89660C211yesmaximal, characteristic, complemented
7#89659C211yesmaximal, characteristic, complemented
8#89656C211yesmaximal, characteristic, complemented
9#89653C211yesmaximal, characteristic, complemented
10#89640C211yesmaximal, characteristic, complemented
11#89639C211yesmaximal, characteristic, complemented
12#89639C211yesmaximal, characteristic, complemented
13#89632C211yesmaximal, characteristic, complemented
14#89488C211yesmaximal, characteristic, complemented
15#89478C211yesmaximal, characteristic
16C7×C4⋊Q8C2211yescharacteristic
17C7×#125C2211yescharacteristic
18#6510—21no—
19Q8×Dic7—21no—
20C2×Dic28C2211yescharacteristic
21C2×#1131C2211yescharacteristic
22C2×C8×D7C2211yescharacteristic
23#6461—21no—
24#6461—21no—
25#6461—21no—
26#6460C2211yescharacteristic
27#6460C2211yescharacteristic
28#6460C2211yescharacteristic
29#6460C2211yescharacteristic
30#6457—21no—
31#6455—21no—
32#6455C2211yescharacteristic
33#6455C2211yescharacteristic
34#6455C2211yescharacteristic
35D7×C4⋊C4C2211yescharacteristic
36#6453—21no—
37C4×D28C2211yescharacteristic
38C4×Dic14C2211yescharacteristic
39C4×Dic14C2211yescharacteristic
40C7×#107C2211yescharacteristic
41C7×#107C2211yescharacteristic
42C7×#106C2211yescharacteristic
43C7×#106C2211yescharacteristic
44C7×#97C2211yescharacteristic
45(C2×D7)⋊C8C2211yescharacteristic
46(C2×D7)⋊C8C2211yescharacteristic
47#6391C2211yescharacteristic
48#6390C2211yescharacteristic
49#6390C2211yescharacteristic
50#6384C2211yescharacteristic
51#6384C2211yescharacteristic
52#6384C2211yescharacteristic
53#6384C2211yescharacteristic
54#6382C2211yescharacteristic
55#6381C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3349—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×D28D411yescharacteristic
62C2×C4×D7—21no—
63C2×C4×D7C2311yescharacteristic
64C2×C4×D7C2311yescharacteristic
65C2×Dic14C2311yescharacteristic
66C2×Dic14C2311yescharacteristic
67C2×C7×C8C2311yescharacteristic
68C2×C7×C8C2311yescharacteristic
69C7×C4⋊C4—21no—
70C7×C4⋊C4—21no—
71C7×C4⋊C4—21no—
72C7×C4⋊C4C2311yescharacteristic
73C7×C4⋊C4C2311yescharacteristic
74C7×C4⋊C4C2311yescharacteristic
75C7×C4⋊C4C2311yescharacteristic
76C42×C7C2311yescharacteristic
77(C2×D7)⋊C4—21no—
78(C2×D7)⋊C4—21no—
79(C2×D7)⋊C4—21no—
80(C2×D7)⋊C4—21no—
81(C2×D7)⋊C4—21no—
82(C2×D7)⋊C4—21no—
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4D411yescharacteristic
85(C2×D7)⋊C4D411yescharacteristic
86C4⋊Dic7—21no—
87C4⋊Dic7—21no—
88C4⋊Dic7—21no—
89C4⋊Dic7—21no—
90C4⋊Dic7D411yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96C4×Dic7—21no—
97C4×Dic7—21no—
98C4×Dic7—21no—
99C4×Dic7—21no—
100C4×Dic7—21no—
101C2×C7⋊C8C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103Dic28—22no—
104#1131—22no—
105C8×D7—22no—
106#556—71no—
107#552—71no—
108#501D711yescharacteristic, complemented
109#485—71no—
110#485—71no—
111#485—71no—
112#484—71no—
113#479—71no—
114#477—71no—
115#470—71no—
116#467—71no—
117#441—71no—
118#435—71no—
119#428—71no—
120#427—71no—
121C22×D7C2×D411yescharacteristic
122C22×D7C2×D411yescharacteristic
123C7×Q8—42no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7C2×D411yescharacteristic
129C2×C4×C7C2×D411yescharacteristic
130C2×C4×C7C2411yescharacteristic
131C2×Dic7—21no—
132C2×Dic7—21no—
133C2×Dic7—21no—
134C2×Dic7—21no—
135C2×Dic7C2×D411yescharacteristic
136C2×Dic7C2×D411yescharacteristic
137D28—41no—
138C4×D7—21no—
139C4×D7—21no—
140C4×D7—21no—
141C4×D7—21no—
142C4×D7—42no—
143Dic14—21no—
144Dic14—21no—
145Dic14—21no—
146Dic14—21no—
147C7×C8—21no—
148C7×C8#4112yes—
149C7⋊C8—21no—
150C7⋊C8—21no—
151C2×Q16—71no—
152C2×M16—71no—
153C22×C8—71no—
154C4⋊Q8C2×D711yescharacteristic
155#126—141no—
156#126—141no—
157#126—141no—
158#125—141no—
159#125C2×D711yescharacteristic
160#124—141no—
161#123—141no—
162C22⋊Q8—71no—
163C22⋊Q8—71no—
164C22⋊Q8—71no—
165C22⋊Q8—71no—
166C4×Q8—71no—
167C4×Q8—71no—
168C4×Q8—141no—
169C4×D4—71no—
170#117—71no—
171#117—71no—
172#117—71no—
173#117—141no—
174C2×C4⋊C4—71no—
175#107—71no—
176#107—71no—
177#107—71no—
178#107C2×D711yescharacteristic
179#107C2×D711yescharacteristic
180#106—71no—
181#106—71no—
182#106C2×D711yescharacteristic
183#106C2×D711yescharacteristic
184C4⋊C8—71no—
185#103—71no—
186#103—71no—
187#103—71no—
188#103—71no—
189#103—71no—
190#103—71no—
191C22⋊C8—71no—
192C22⋊C8—71no—
193#97C2×D711yescharacteristic
194C22×C7C22×D411yescharacteristic
195C2×D7—21no—
196C2×D7—21no—
197C2×D7—21no—
198C2×D7—21no—
199C2×D7—21no—
200C2×D7—21no—
201C4×C7—21no—
202C4×C7—21no—
203C4×C7—41no—
204C4×C7—41no—
205C4×C7—41no—
206C4×C7—41no—
207C4×C7C2×#4111yescharacteristic
208C4×C72−1+411yescharacteristic
209Dic7—21no—
210Dic7—21no—
211Dic7—41no—
212Dic7—41no—
213Dic7—41no—
214Dic7—41no—
215C2×Q8—21no—
216C2×Q8—71no—
217C2×Q8—71no—
218C2×D4—71no—
219C22×C4—71no—
220C22×C4—71no—
221C22×C4—141no—
222Q16—142no—
223M16—142no—
224C2×C8—71no—
225C2×C8—71no—
226C2×C8—142no—
227C2×C8C22×D711yescharacteristic
228C2×C8C22×D711yescharacteristic
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—71no—
233C4⋊C4—71no—
234C4⋊C4—71no—
235C4⋊C4—141no—
236C4⋊C4—141no—
237C4⋊C4—141no—
238C4⋊C4—141no—
239C4⋊C4—141no—
240C4⋊C4—141no—
241C4⋊C4—141no—
242C4⋊C4C22×D711yescharacteristic
243C4⋊C4C22×D711yescharacteristic
244C4⋊C4C22×D711yescharacteristic
245C4⋊C4C22×D711yescharacteristic
246C22⋊C4—71no—
247C22⋊C4—71no—
248C22⋊C4—141no—
249C22⋊C4—141no—
250C22⋊C4—141no—
251C22⋊C4—141no—
252C22⋊C4—141no—
253C22⋊C4—141no—
254C22⋊C4—141no—
255C42—141no—
256C42—141no—
257C42—141no—
258C42—141no—
259C42—141no—
260C42C22×D711yescharacteristic
261C2×C7#54711yescharacteristic
262C2×C7#57511yescharacteristic
263C2×C7#57811yescharacteristic
264D7—41no—
265D7—41no—
266C23—71no—
267C23—71no—
268Q8—42no—
269Q8—141no—
270Q8—141no—
271Q8—141no—
272Q8—141no—
273D4—281no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—71no—
279C2×C4—71no—
280C2×C4—141no—
281C2×C4—141no—
282C2×C4—141no—
283C2×C4—141no—
284C2×C4—141no—
285C2×C4—141no—
286C2×C4—141no—
287C2×C4—141no—
288C2×C4—282no—
289C2×C4D4×D711yescharacteristic
290C2×C4D4×D711yescharacteristic
291C2×C4C23×D711yescharacteristic
292C8—21no—
293C8—141no—
294C8—141no—
295C8#116112yes—
296C7#334911yescharacteristic, complemented, Sylow
297C22—141no—
298C22—141no—
299C22—141no—
300C22—141no—
301C22—141no—
302C22—141no—
303C22C2×D4×D711yescharacteristic
304C4—21no—
305C4—21no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—141no—
311C4—141no—
312C4—281no—
313C4—281no—
314C4—281no—
315C4—281no—
316C4C2×#116111yescharacteristic
317C4#655111yescharacteristic
318C2—281no—
319C2—281no—
320C2#9041711yescharacteristic
321C2#9046411yescharacteristic
322C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#501C2
socleC22×C7C22×D4
prime radicalC2×D28D4
hypercenter#501D7
hypocenterC7#3349
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
3. C7×#501C2semidirect
4. #89863C2semidirect
5. #89835C2semidirect
6. #89660C2semidirect
7. #89659C2semidirect
8. #89656C2semidirect
9. #89653C2semidirect
10. #89640C2semidirect
11. #89639C2semidirect
12. #89639C2semidirect
13. #89632C2semidirect
14. #89488C2semidirect
108. #501D7semidirect
296. C7#3349semidirect

Maximal subgroups

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

subgroupindex
#904232
#903852
C7×#5012
#898632
#898352
#896602
#896592
#896562
#896532
#896402
#896392
#896322
#894882
#894782
#33497