Finite Group Databasebeta

← #11739368

Subgroups — #11739368

Order 896.

308 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90388C211yesmaximal, characteristic
2#90380C211yesmaximal, characteristic
3C7×#499C211yesmaximal, characteristic, complemented
4#89841C211yesmaximal, characteristic, complemented
5#89837C211yesmaximal, characteristic, complemented
6#89660C211yesmaximal, characteristic
7#89656C211yesmaximal, characteristic
8#89649C211yesmaximal, characteristic, complemented
9#89647C211yesmaximal, characteristic, complemented
10#89639C211yesmaximal, characteristic
11#89634C211yesmaximal, characteristic
12#89629C211yesmaximal, characteristic, complemented
13#89626C211yesmaximal, characteristic, complemented
14#89462C211yesmaximal, characteristic
15#89453C211yesmaximal, characteristic, complemented
16C7×#125C2211yescharacteristic
17C7×#125C2211yescharacteristic
18#6461—21no—
19#6461—21no—
20#6460—21no—
21#6460C2211yescharacteristic
22#6460C2211yescharacteristic
23#6459—21no—
24#6455C2211yescharacteristic
25D7×C4⋊C4C2211yescharacteristic
26#6453—21no—
27#6453—21no—
28#6453C2211yescharacteristic
29#6452—21no—
30#6451—21no—
31#6451C2211yescharacteristic
32#6450C2211yescharacteristic
33#6450C2211yescharacteristic
34#6435C2211yescharacteristic
35#6432C2211yescharacteristic
36#6432C2211yescharacteristic
37C7×#107C2211yescharacteristic
38C7×#107C2211yescharacteristic
39C7×#106C2211yescharacteristic
40C7×#106C2211yescharacteristic
41C4×C7×C8C2211yescharacteristic
42(C2×D7)⋊C8C2211yescharacteristic
43(C2×D7)⋊C8C2211yescharacteristic
44#6390C2211yescharacteristic
45#6390C2211yescharacteristic
46#6390C2211yescharacteristic
47#6390C2211yescharacteristic
48Dic7⋊C8C2211yescharacteristic
49Dic7⋊C8C2211yescharacteristic
50#6384C2211yescharacteristic
51#6384C2211yescharacteristic
52#6384C2211yescharacteristic
53#6384C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58#6377C2211yescharacteristic
59#3242—71nomaximal, Sylow
60C2×C4×D7—21no—
61C2×C4×D7C2311yescharacteristic
62C2×Dic14—21no—
63C2×Dic14—21no—
64C2×Dic14C2311yescharacteristic
65C2×Dic14C2311yescharacteristic
66C2×C7×C8C2311yescharacteristic
67C2×C7×C8C2311yescharacteristic
68C7×C4⋊C4—21no—
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)⋊C4D411yescharacteristic
81(C2×D7)⋊C4D411yescharacteristic
82C4⋊Dic7—21no—
83C4⋊Dic7—21no—
84C4⋊Dic7—21no—
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⋊C4D411yescharacteristic
96Dic7⋊C4D411yescharacteristic
97C4×Dic7—21no—
98C4×Dic7—21no—
99C4×Dic7—21no—
100C4×Dic7C2311yescharacteristic
101C2×C7⋊C8C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103#544—71no—
104#541—71no—
105#501—71no—
106#499D711yescharacteristic, complemented
107#491—71no—
108#487—71no—
109#485—71no—
110#485—71no—
111#484—71no—
112#483—71no—
113#479—71no—
114#479—71no—
115#477—71no—
116#475—71no—
117#433—71no—
118C22×D7C2×D411yescharacteristic
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7C2×D411yescharacteristic
124C2×C4×C7C2×D411yescharacteristic
125C2×C4×C7C2411yescharacteristic
126C2×Dic7—21no—
127C2×Dic7—21no—
128C2×Dic7—21no—
129C2×Dic7—21no—
130C2×Dic7C2×D411yescharacteristic
131C2×Dic7C2×D411yescharacteristic
132C2×Dic7C2×D411yescharacteristic
133C4×D7—21no—
134C4×D7—21no—
135C4×D7—81no—
136Dic14—21no—
137Dic14—21no—
138Dic14—21no—
139Dic14—21no—
140Dic14—81no—
141Dic14—81no—
142C7×C8—21no—
143C7×C8—21no—
144C7⋊C8—21no—
145C7⋊C8—21no—
146C4⋊Q8—71no—
147C4⋊Q8—71no—
148C4⋊Q8—71no—
149C4⋊Q8—141no—
150#126—141no—
151#126—141no—
152#125—71no—
153#125—141no—
154#125—141no—
155#125—141no—
156#125C2×D711yescharacteristic
157#125C2×D711yescharacteristic
158C22⋊Q8—71no—
159C22⋊Q8—71no—
160C22⋊Q8—141no—
161C22⋊Q8—141no—
162C4×Q8—71no—
163C4×Q8—71no—
164#117—71no—
165#117—71no—
166C2×C4⋊C4—71no—
167#107—71no—
168#107—71no—
169#107C2×D711yescharacteristic
170#107C2×D711yescharacteristic
171#106—71no—
172#106—71no—
173#106C2×D711yescharacteristic
174#106C2×D711yescharacteristic
175C4⋊C8—71no—
176C4⋊C8—71no—
177#103—71no—
178#103—71no—
179#103—71no—
180#103—71no—
181#103—71no—
182#103—71no—
183#103—71no—
184#103—71no—
185C22⋊C8—71no—
186C22⋊C8—71no—
187#97—71no—
188C4×C8C2×D711yescharacteristic
189C22×C7C22×D411yescharacteristic
190C2×D7—21no—
191C2×D7—21no—
192C2×D7—21no—
193C4×C7—21no—
194C4×C7—21no—
195C4×C7—41no—
196C4×C7—41no—
197C4×C7—41no—
198C4×C7—41no—
199C4×C72−1+411yescharacteristic
200C4×C72−1+411yescharacteristic
201Dic7—21no—
202Dic7—21no—
203Dic7—21no—
204Dic7—41no—
205Dic7—41no—
206Dic7—41no—
207Dic7—41no—
208C2×Q8—71no—
209C2×Q8—71no—
210C2×Q8—141no—
211C2×Q8—141no—
212C22×C4—71no—
213C22×C4—141no—
214C2×C8—71no—
215C2×C8—71no—
216C2×C8C22×D711yescharacteristic
217C2×C8C22×D711yescharacteristic
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—71no—
223C4⋊C4—71no—
224C4⋊C4—71no—
225C4⋊C4—71no—
226C4⋊C4—141no—
227C4⋊C4—141no—
228C4⋊C4—141no—
229C4⋊C4—141no—
230C4⋊C4—141no—
231C4⋊C4—141no—
232C4⋊C4—141no—
233C4⋊C4—141no—
234C4⋊C4—141no—
235C4⋊C4—141no—
236C4⋊C4—141no—
237C4⋊C4C22×D711yescharacteristic
238C4⋊C4C22×D711yescharacteristic
239C4⋊C4C22×D711yescharacteristic
240C4⋊C4C22×D711yescharacteristic
241C22⋊C4—71no—
242C22⋊C4—71no—
243C22⋊C4—141no—
244C22⋊C4—141no—
245C22⋊C4—141no—
246C42—71no—
247C42—141no—
248C42—141no—
249C42—141no—
250C42C22×D711yescharacteristic
251C2×C7#53611yescharacteristic
252C2×C7#57711yescharacteristic
253C2×C7#57811yescharacteristic
254D7—41no—
255C23—71no—
256Q8—141no—
257Q8—141no—
258Q8—141no—
259Q8—141no—
260Q8—561no—
261Q8—561no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—71no—
267C2×C4—71no—
268C2×C4—71no—
269C2×C4—141no—
270C2×C4—141no—
271C2×C4—141no—
272C2×C4—141no—
273C2×C4—141no—
274C2×C4—141no—
275C2×C4—561no—
276C2×C4D4×D711yescharacteristic
277C2×C4D4×D711yescharacteristic
278C2×C4C23×D711yescharacteristic
279C8—21no—
280C8—21no—
281C8—141no—
282C8—141no—
283C7#324211yescharacteristic, complemented, Sylow
284C22—141no—
285C22—141no—
286C22—141no—
287C22C2×D4×D711yescharacteristic
288C4—21no—
289C4—21no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—41no—
294C4—141no—
295C4—141no—
296C4—141no—
297C4—281no—
298C4—281no—
299C4—281no—
300C4—281no—
301C4#655111yescharacteristic
302C4#655111yescharacteristic
303C2—281no—
304C2#9041511yescharacteristic
305C2#9046111yescharacteristic
306C2#9046211yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#499C2
socleC22×C7C22×D4
prime radicalC22×D7C2×D4
hypercenter#499D7
hypocenterC7#3242
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #499

Complements

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

NH ≅ G/Ntype
3. C7×#499C2semidirect
4. #89841C2semidirect
5. #89837C2semidirect
8. #89649C2semidirect
9. #89647C2semidirect
12. #89629C2semidirect
13. #89626C2semidirect
15. #89453C2semidirect
106. #499D7semidirect
283. C7#3242semidirect

Maximal subgroups

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

subgroupindex
#903882
#903802
C7×#4992
#898412
#898372
#896602
#896562
#896492
#896472
#896392
#896342
#896292
#896262
#894622
#894532
#32427