Finite Group Databasebeta

← #11738883

Subgroups — #11738883

Order 896.

344 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90330C211yesmaximal, characteristic
2#90202C211yesmaximal, characteristic
3C7×#483C211yesmaximal, characteristic, complemented
4#89824C211yesmaximal, characteristic, complemented
5#89736C211yesmaximal, characteristic, complemented
6#89634C211yesmaximal, characteristic, complemented
7D7×#106C211yesmaximal, characteristic, complemented
8#89629C211yesmaximal, characteristic, complemented
9#89628C211yesmaximal, characteristic, complemented
10#89585C211yesmaximal, characteristic, complemented
11#89581C211yesmaximal, characteristic, complemented
12#89563C211yesmaximal, characteristic, complemented
13#89561C211yesmaximal, characteristic, complemented
14#89500C211yesmaximal, characteristic
15#89495C211yesmaximal, characteristic, complemented
16C7×C22⋊Q8C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18#6509C2211yescharacteristic
19Q8×Dic7C2211yescharacteristic
20C2×#1144C2211yescharacteristic
21C4×C22⋊D7—21no—
22C4×C22⋊D7C2211yescharacteristic
23#6489C2211yescharacteristic
24#6487—21no—
25#6487C2211yescharacteristic
26C2×#1136C2211yescharacteristic
27C2×C8×D7C2211yescharacteristic
28#6460C2211yescharacteristic
29#6459—21no—
30#6455—21no—
31D7×C4⋊C4C2211yescharacteristic
32D7×C4⋊C4C2211yescharacteristic
33#6453C2211yescharacteristic
34#6452—21no—
35#6452—21no—
36#6451C2211yescharacteristic
37#6450C2211yescharacteristic
38#6446—21no—
39#6442—21no—
40C7×#106C2211yescharacteristic
41C7×#106C2211yescharacteristic
42C7×#103C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×C22⋊C8C2211yescharacteristic
45#6409C2211yescharacteristic
46(C2×D7)⋊C8C2211yescharacteristic
47#6392C2211yescharacteristic
48#6391C2211yescharacteristic
49#6390C2211yescharacteristic
50#6390C2211yescharacteristic
51#6389C2211yescharacteristic
52Dic7⋊C8C2211yescharacteristic
53#6384C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3344—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C22×C4×C7—21no—
62C22×C4×C7C2311yescharacteristic
63C2×C22⋊D7D411yescharacteristic
64C2×C4×D7—21no—
65C2×C4×D7—21no—
66C2×C4×D7C2311yescharacteristic
67C2×Dic14—21no—
68C2×Dic14C2311yescharacteristic
69C2×C7×C8C2311yescharacteristic
70C2×C7×C8C2311yescharacteristic
71C7×C4⋊C4—21no—
72C7×C4⋊C4—21no—
73C7×C4⋊C4C2311yescharacteristic
74C7×C4⋊C4C2311yescharacteristic
75C7×C4⋊C4C2311yescharacteristic
76C7×C22⋊C4—21no—
77C22⋊Dic7—21no—
78C22⋊Dic7—21no—
79C22⋊Dic7D411yescharacteristic
80#1144—22no—
81(C2×D7)⋊C4—21no—
82(C2×D7)⋊C4—21no—
83(C2×D7)⋊C4D411yescharacteristic
84C4⋊Dic7—21no—
85C4⋊Dic7—21no—
86C4⋊Dic7—21no—
87C4⋊Dic7C2311yescharacteristic
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
102#1136—22no—
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105C8×D7—22no—
106#554—71no—
107#552—71no—
108#501—71no—
109#485—71no—
110#484—71no—
111#483—71no—
112#483D711yescharacteristic, complemented
113#479—71no—
114#475—71no—
115#470—71no—
116#467—71no—
117#441—71no—
118#435—71no—
119#428—71no—
120C2×#106—71no—
121C23×C7C2×D411yescharacteristic
122C22×D7C2×D411yescharacteristic
123C7×Q8—21no—
124C7×Q8—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—81no—
132C2×C4×C7C2×D411yescharacteristic
133C2×C4×C7C2411yescharacteristic
134C22⋊D7—22no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7—21no—
139C2×Dic7C2×D411yescharacteristic
140C2×Dic7C2×D411yescharacteristic
141C2×Dic7C2×D411yescharacteristic
142C4×D7—21no—
143C4×D7—21no—
144C4×D7—42no—
145C4×D7—81no—
146Dic14—21no—
147Dic14—21no—
148Dic14—81no—
149C7×C8—21no—
150C7×C8—21no—
151C7⋊C8—21no—
152C7⋊C8#4112yes—
153C2×Q16—71no—
154C2×M16—71no—
155C22×C8—71no—
156C4⋊Q8—71no—
157#126—141no—
158#125—71no—
159#125—141no—
160#125—141no—
161#123—141no—
162C22⋊Q8—71no—
163C22⋊Q8—71no—
164C22⋊Q8—71no—
165C22⋊Q8—141no—
166C22⋊Q8—141no—
167C22⋊Q8C2×D711yescharacteristic
168C4×Q8—71no—
169C4×Q8—71no—
170C4×D4—71no—
171C4×D4—141no—
172#117—71no—
173#117—141no—
174C2×C4⋊C4—71no—
175C2×C4⋊C4—71no—
176C2×C4⋊C4C2×D711yescharacteristic
177#107—71no—
178#107—71no—
179#107—71no—
180#106—71no—
181#106—71no—
182#106—71no—
183#106—71no—
184#106C2×D711yescharacteristic
185#106C2×D711yescharacteristic
186C4⋊C8—71no—
187#103—71no—
188#103—71no—
189#103—71no—
190#103—71no—
191#103C2×D711yescharacteristic
192#103C2×D711yescharacteristic
193C22⋊C8—71no—
194C22⋊C8C2×D711yescharacteristic
195#97—71no—
196C22×C7—21no—
197C22×C7—21no—
198C22×C7—21no—
199C22×C7C22×D411yescharacteristic
200C2×D7—21no—
201C2×D7—21no—
202C2×D7#13712yes—
203C4×C7—21no—
204C4×C7—41no—
205C4×C7—41no—
206C4×C7—41no—
207C4×C7—41no—
208C4×C7C2×#4111yescharacteristic
209C4×C72−1+411yescharacteristic
210Dic7—21no—
211Dic7—21no—
212Dic7—21no—
213Dic7—41no—
214Dic7—41no—
215Dic7—41no—
216Dic7—41no—
217C2×Q8—71no—
218C2×Q8—141no—
219C2×Q8C22×D711yescharacteristic
220C2×D4—71no—
221C22×C4—21no—
222C22×C4—71no—
223C22×C4—141no—
224C22×C4—141no—
225C22×C4C22×D711yescharacteristic
226Q16—142no—
227M16—142no—
228C2×C8—71no—
229C2×C8—71no—
230C2×C8—142no—
231C2×C8C22×D711yescharacteristic
232C2×C8C22×D711yescharacteristic
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—71no—
236C4⋊C4—71no—
237C4⋊C4—71no—
238C4⋊C4—71no—
239C4⋊C4—141no—
240C4⋊C4—141no—
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
251C4⋊C4C22×D711yescharacteristic
252C22⋊C4—21no—
253C22⋊C4—71no—
254C22⋊C4—71no—
255C22⋊C4—141no—
256C22⋊C4—141no—
257C22⋊C4—141no—
258C22⋊C4—141no—
259C42—71no—
260C42—141no—
261C42—141no—
262C42—141no—
263C2×C7—41no—
264C2×C7#54711yescharacteristic
265C2×C7C2×#13711yescharacteristic
266C2×C7#57711yescharacteristic
267D7—22no—
268C23—71no—
269C23D4×D711yescharacteristic
270Q8—21no—
271Q8—21no—
272Q8—141no—
273Q8—141no—
274Q8—561no—
275D4—142no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—71no—
283C2×C4—71no—
284C2×C4—71no—
285C2×C4—81no—
286C2×C4—141no—
287C2×C4—141no—
288C2×C4—141no—
289C2×C4—141no—
290C2×C4—141no—
291C2×C4—141no—
292C2×C4—282no—
293C2×C4—561no—
294C2×C4D4×D711yescharacteristic
295C2×C4C23×D711yescharacteristic
296C8—21no—
297C8—21no—
298C8—72no—
299C8—141no—
300C7#334411yescharacteristic, complemented, Sylow
301C22—21no—
302C22—21no—
303C22—21no—
304C22—72no—
305C22—141no—
306C22—141no—
307C22C2×D4×D711yescharacteristic
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—141no—
314C4—141no—
315C4—141no—
316C4—281no—
317C4—281no—
318C4—281no—
319C4—281no—
320C4D7×#4111yescharacteristic
321C4#655411yescharacteristic
322C2—41no—
323C2—142no—
324C2#9034611yescharacteristic
325C2#9045311yescharacteristic
326C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#483C2
socleC22×C7C22×D4
prime radicalC2×C22⋊D7D4
hypercenter#483D7
hypocenterC7#3344
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
3. C7×#483C2semidirect
4. #89824C2semidirect
5. #89736C2semidirect
6. #89634C2semidirect
7. D7×#106C2semidirect
8. #89629C2semidirect
9. #89628C2semidirect
10. #89585C2semidirect
11. #89581C2semidirect
12. #89563C2semidirect
13. #89561C2semidirect
15. #89495C2semidirect
112. #483D7semidirect
300. C7#3344semidirect

Maximal subgroups

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

subgroupindex
#903302
#902022
C7×#4832
#898242
#897362
#896342
D7×#1062
#896292
#896282
#895852
#895812
#895632
#895612
#895002
#894952
#33447