Finite Group Databasebeta

← #11736639

Subgroups — #11736639

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#90224C211yesmaximal, characteristic
2#90220C211yesmaximal, characteristic
3C7×#427C211yesmaximal, characteristic, complemented
4#89888C211yesmaximal, characteristic, complemented
5#89883C211yesmaximal, characteristic
6#89768C211yesmaximal, characteristic, complemented
7#89768C211yesmaximal, characteristic, complemented
8#89660C211yesmaximal, characteristic, complemented
9#89656C211yesmaximal, characteristic, complemented
10#89649C211yesmaximal, characteristic, complemented
11#89647C211yesmaximal, characteristic, complemented
12#89640C211yesmaximal, characteristic, complemented
13#89633C211yesmaximal, characteristic, complemented
14#89628C211yesmaximal, characteristic, complemented
15#89625C211yesmaximal, characteristic, complemented
16C22×C7×C8C2211yescharacteristic
17C7×#117C2211yescharacteristic
18C7×#117C2211yescharacteristic
19C4×C22⋊D7C2211yescharacteristic
20#6487C2211yescharacteristic
21#6487C2211yescharacteristic
22C2×Dic28C2211yescharacteristic
23C2×#1131C2211yescharacteristic
24#6460C2211yescharacteristic
25#6460C2211yescharacteristic
26#6455C2211yescharacteristic
27D7×C4⋊C4C2211yescharacteristic
28#6453C2211yescharacteristic
29#6451C2211yescharacteristic
30#6450C2211yescharacteristic
31#6450C2211yescharacteristic
32#6448—21no—
33#6446—21no—
34#6442—21no—
35#6442—21no—
36#6439—21no—
37#6435—21no—
38#6433—21no—
39C4×Dic14—21no—
40C7×#107C2211yescharacteristic
41C7×#107C2211yescharacteristic
42C7×#106C2211yescharacteristic
43C7×#106C2211yescharacteristic
44C22⋊(C7⋊C8)C2211yescharacteristic
45(C2×D7)⋊C8C2211yescharacteristic
46#6392C2211yescharacteristic
47#6390C2211yescharacteristic
48#6390C2211yescharacteristic
49#6389C2211yescharacteristic
50Dic7⋊C8C2211yescharacteristic
51#6384C2211yescharacteristic
52#6384C2211yescharacteristic
53#6384C2211yescharacteristic
54#6384C2211yescharacteristic
55#6382C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3349—71nomaximal, Sylow
60C22×C4×C7C2311yescharacteristic
61C2×C22⋊D7D411yescharacteristic
62C2×C4×D7—21no—
63C2×C4×D7C2311yescharacteristic
64C2×Dic14—21no—
65C2×Dic14C2311yescharacteristic
66C2×Dic14C2311yescharacteristic
67C2×C7×C8—22no—
68C2×C7×C8C2311yescharacteristic
69C2×C7×C8C2311yescharacteristic
70C7×C4⋊C4C2311yescharacteristic
71C7×C4⋊C4C2311yescharacteristic
72C7×C4⋊C4C2311yescharacteristic
73C7×C4⋊C4C2311yescharacteristic
74C7×C22⋊C4—21no—
75C7×C22⋊C4—21no—
76C42×C7—21no—
77C42×C7—21no—
78C22⋊Dic7—21no—
79C22⋊Dic7—21no—
80C22⋊Dic7D411yescharacteristic
81(C2×D7)⋊C4—21no—
82(C2×D7)⋊C4—21no—
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4D411yescharacteristic
85C4⋊Dic7—21no—
86C4⋊Dic7—21no—
87C4⋊Dic7—21no—
88C4⋊Dic7C2311yescharacteristic
89C4⋊Dic7C2311yescharacteristic
90Dic7⋊C4—21no—
91Dic7⋊C4—21no—
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4D411yescharacteristic
98C4×Dic7—21no—
99C4×Dic7—21no—
100C4×Dic7—21no—
101C4×Dic7C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104Dic28—22no—
105#1131—22no—
106#556—71no—
107#552—71no—
108#501—71no—
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#427D711yescharacteristic, complemented
121C23×C7C2×D411yescharacteristic
122C22×D7C2×D411yescharacteristic
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7C2×D411yescharacteristic
130C2×C4×C7C2411yescharacteristic
131C22⋊D7—41no—
132C2×Dic7—21no—
133C2×Dic7—21no—
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7C2×D411yescharacteristic
137C2×Dic7C2×D411yescharacteristic
138C2×Dic7C2×D411yescharacteristic
139C4×D7—21no—
140C4×D7—21no—
141C4×D7—42no—
142Dic14—21no—
143Dic14—21no—
144Dic14—21no—
145Dic14—21no—
146Dic14—42no—
147C7×C8—21no—
148C7×C8#4112yes—
149C7⋊C8—21no—
150C7⋊C8—21no—
151C2×Q16—71no—
152C2×M16—71no—
153C22×C8C2×D711yescharacteristic
154C4⋊Q8—71no—
155#126—141no—
156#126—141no—
157#126—141no—
158#125—71no—
159#125—141no—
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—141no—
172#117C2×D711yescharacteristic
173#117C2×D711yescharacteristic
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#97—71no—
194C22×C7—21no—
195C22×C7—21no—
196C22×C7—21no—
197C22×C7C22×D411yescharacteristic
198C2×D7—21no—
199C2×D7—21no—
200C2×D7—21no—
201C4×C7—21no—
202C4×C7—41no—
203C4×C7—41no—
204C4×C7—41no—
205C4×C7—41no—
206C4×C7C2×#4111yescharacteristic
207C4×C72−1+411yescharacteristic
208Dic7—21no—
209Dic7—21no—
210Dic7—21no—
211Dic7—41no—
212Dic7—41no—
213Dic7—41no—
214Dic7—41no—
215C2×Q8—71no—
216C2×Q8—71no—
217C2×Q8—141no—
218C2×D4—71no—
219C22×C4—71no—
220C22×C4—141no—
221C22×C4C22×D711yescharacteristic
222Q16—142no—
223M16—142no—
224C2×C8—22no—
225C2×C8—71no—
226C2×C8—71no—
227C2×C8C22×D711yescharacteristic
228C2×C8C22×D711yescharacteristic
229C4⋊C4—71no—
230C4⋊C4—71no—
231C4⋊C4—71no—
232C4⋊C4—141no—
233C4⋊C4—141no—
234C4⋊C4—141no—
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—21no—
247C22⋊C4—21no—
248C22⋊C4—71no—
249C22⋊C4—71no—
250C22⋊C4—141no—
251C22⋊C4—141no—
252C22⋊C4—141no—
253C22⋊C4—141no—
254C22⋊C4—141no—
255C42—21no—
256C42—21no—
257C42—71no—
258C42—141no—
259C42—141no—
260C42—141no—
261C2×C7—41no—
262C2×C7#54711yescharacteristic
263C2×C7#57511yescharacteristic
264C2×C7#57811yescharacteristic
265D7—41no—
266C23—71no—
267C23D4×D711yescharacteristic
268Q8—141no—
269Q8—141no—
270Q8—141no—
271Q8—141no—
272Q8—282no—
273D4—281no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—71no—
281C2×C4—71no—
282C2×C4—71no—
283C2×C4—141no—
284C2×C4—141no—
285C2×C4—141no—
286C2×C4—141no—
287C2×C4—141no—
288C2×C4—141no—
289C2×C4—282no—
290C2×C4D4×D711yescharacteristic
291C2×C4C23×D711yescharacteristic
292C8—21no—
293C8—141no—
294C8—141no—
295C8#115712yes—
296C7#334911yescharacteristic, complemented, Sylow
297C22—21no—
298C22—21no—
299C22—21no—
300C22—141no—
301C22—141no—
302C22—141no—
303C22C2×D4×D711yescharacteristic
304C4—21no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—141no—
310C4—141no—
311C4—141no—
312C4—281no—
313C4—281no—
314C4—281no—
315C4—281no—
316C4C2×#115711yescharacteristic
317C4#655111yescharacteristic
318C2—41no—
319C2—281no—
320C2#9019811yescharacteristic
321C2#9045911yescharacteristic
322C2#9046211yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#427C2
socleC22×C7C22×D4
prime radicalC2×C22⋊D7D4
hypercenter#427D7
hypocenterC7#3349
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
3. C7×#427C2semidirect
4. #89888C2semidirect
6. #89768C2semidirect
7. #89768C2semidirect
8. #89660C2semidirect
9. #89656C2semidirect
10. #89649C2semidirect
11. #89647C2semidirect
12. #89640C2semidirect
13. #89633C2semidirect
14. #89628C2semidirect
15. #89625C2semidirect
120. #427D7semidirect
296. C7#3349semidirect

Maximal subgroups

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

subgroupindex
#902242
#902202
C7×#4272
#898882
#898832
#897682
#896602
#896562
#896492
#896472
#896402
#896332
#896282
#896252
#33497