Finite Group Databasebeta

← #11739366

Subgroups — #11739366

Order 896.

332 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90382C211yesmaximal, characteristic
2#90379C211yesmaximal, characteristic, complemented
3C7×#499C211yesmaximal, characteristic, complemented
4#89840C211yesmaximal, characteristic, complemented
5#89836C211yesmaximal, characteristic, complemented
6#89662C211yesmaximal, characteristic, complemented
7#89660C211yesmaximal, characteristic, complemented
8#89657C211yesmaximal, characteristic
9#89647C211yesmaximal, characteristic, complemented
10#89642C211yesmaximal, characteristic, complemented
11#89639C211yesmaximal, characteristic, complemented
12#89638C211yesmaximal, characteristic
13#89626C211yesmaximal, characteristic, complemented
14#89469C211yesmaximal, characteristic, complemented
15#89457C211yesmaximal, characteristic, complemented
16C7×#125C2211yescharacteristic
17C7×#125C2211yescharacteristic
18#6461—21no—
19#6460C2211yescharacteristic
20#6459—21no—
21#6459—21no—
22#6458C2211yescharacteristic
23#6457—21no—
24#6456—21no—
25#6456C2211yescharacteristic
26#6455—21no—
27#6455C2211yescharacteristic
28#6455C2211yescharacteristic
29#6451C2211yescharacteristic
30#6451C2211yescharacteristic
31#6450—21no—
32#6450—21no—
33#6450C2211yescharacteristic
34#6438C2211yescharacteristic
35C42×D7C2211yescharacteristic
36#6433C2211yescharacteristic
37C7×#107C2211yescharacteristic
38C7×#107C2211yescharacteristic
39C7×#106C2211yescharacteristic
40C7×#106C2211yescharacteristic
41C4×C7×C8C2211yescharacteristic
42#6395C2211yescharacteristic
43#6395C2211yescharacteristic
44(C2×D7)⋊C8C2211yescharacteristic
45(C2×D7)⋊C8C2211yescharacteristic
46#6390C2211yescharacteristic
47#6390C2211yescharacteristic
48Dic7⋊C8C2211yescharacteristic
49Dic7⋊C8C2211yescharacteristic
50#6384C2211yescharacteristic
51#6384C2211yescharacteristic
52#6383C2211yescharacteristic
53#6383C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58C4×C7⋊C8C2211yescharacteristic
59#3233—71nomaximal, Sylow
60C2×D28—21no—
61C2×D28C2311yescharacteristic
62C2×C4×D7—21no—
63C2×C4×D7D411yescharacteristic
64C2×C4×D7D411yescharacteristic
65C2×C4×D7C2311yescharacteristic
66C2×Dic14—21no—
67C2×Dic14—21no—
68C2×Dic14C2311yescharacteristic
69C2×C7×C8C2311yescharacteristic
70C2×C7×C8C2311yescharacteristic
71C7×C4⋊C4—21no—
72C7×C4⋊C4—21no—
73C7×C4⋊C4—21no—
74C7×C4⋊C4—21no—
75C7×C4⋊C4C2311yescharacteristic
76C7×C4⋊C4C2311yescharacteristic
77C7×C4⋊C4C2311yescharacteristic
78C7×C4⋊C4C2311yescharacteristic
79C42×C7C2311yescharacteristic
80(C2×D7)⋊C4—21no—
81(C2×D7)⋊C4—21no—
82(C2×D7)⋊C4—21no—
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4—21no—
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4—21no—
87C4⋊Dic7—21no—
88C4⋊Dic7C2311yescharacteristic
89C4⋊Dic7C2311yescharacteristic
90Dic7⋊C4—21no—
91Dic7⋊C4—21no—
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95C4×Dic7—21no—
96C4×Dic7—21no—
97C4×Dic7—21no—
98C4×Dic7D411yescharacteristic
99C4×Dic7D411yescharacteristic
100C4×Dic7C2311yescharacteristic
101C2×C7⋊C8C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103#533—71no—
104#529—71no—
105#499—71no—
106#499D711yescharacteristic, complemented
107#488—71no—
108#488—71no—
109#485—71no—
110#485—71no—
111#482—71no—
112#482—71no—
113#477—71no—
114#476—71no—
115#475—71no—
116#474—71no—
117#431—71no—
118C22×D7—21no—
119C22×D7C2×D411yescharacteristic
120C2×C4×C7—21no—
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7C2×D411yescharacteristic
125C2×C4×C7C2×D411yescharacteristic
126C2×C4×C7C2411yescharacteristic
127C2×Dic7—21no—
128C2×Dic7—21no—
129C2×Dic7—21no—
130C2×Dic7C2×D411yescharacteristic
131C2×Dic7C2×D411yescharacteristic
132C2×Dic7C2×D411yescharacteristic
133D28—21no—
134D28—21no—
135D28—42no—
136C4×D7—21no—
137C4×D7—21no—
138C4×D7—22no—
139C4×D7—41no—
140C4×D7—42no—
141Dic14—21no—
142Dic14—21no—
143Dic14—42no—
144Dic14—42no—
145C7×C8—21no—
146C7×C8—21no—
147C7⋊C8—21no—
148C7⋊C8—21no—
149C4⋊Q8—71no—
150C4⋊Q8—71no—
151#126—141no—
152#125—71no—
153#125C2×D711yescharacteristic
154#125C2×D711yescharacteristic
155#124—71no—
156#123—141no—
157C22⋊Q8—71no—
158C22⋊Q8—141no—
159C22⋊Q8—141no—
160#121—71no—
161C4×Q8—71no—
162C4×Q8—141no—
163C4×Q8—141no—
164C4×D4—71no—
165C4×D4—141no—
166#117—71no—
167#117—71no—
168#117—141no—
169C2×C42—71no—
170#107—71no—
171#107—71no—
172#107C2×D711yescharacteristic
173#107C2×D711yescharacteristic
174#106—71no—
175#106—71no—
176#106C2×D711yescharacteristic
177#106C2×D711yescharacteristic
178C4⋊C8—71no—
179C4⋊C8—71no—
180#103—71no—
181#103—71no—
182#103—71no—
183#103—71no—
184#102—71no—
185#102—71no—
186#102—71no—
187#102—71no—
188C22⋊C8—71no—
189C22⋊C8—71no—
190C4×C8—71no—
191C4×C8C2×D711yescharacteristic
192C22×C7C22×D411yescharacteristic
193C2×D7—21no—
194C2×D7—21no—
195C2×D7—21no—
196C2×D7—41no—
197C2×D7—41no—
198C2×D7—41no—
199C4×C7—21no—
200C4×C7—21no—
201C4×C7—41no—
202C4×C7—41no—
203C4×C7—41no—
204C4×C7—41no—
205C4×C72−1+411yescharacteristic
206C4×C72−1+411yescharacteristic
207Dic7—21no—
208Dic7—21no—
209Dic7—41no—
210Dic7—41no—
211Dic7—41no—
212Dic7#13512yes—
213C2×Q8—71no—
214C2×Q8—141no—
215C2×Q8—141no—
216C2×D4—71no—
217C2×D4—141no—
218C22×C4—71no—
219C22×C4—71no—
220C22×C4—71no—
221C22×C4—141no—
222C2×C8—71no—
223C2×C8—71no—
224C2×C8C22×D711yescharacteristic
225C2×C8C22×D711yescharacteristic
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
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⋊C4C22×D711yescharacteristic
239C4⋊C4C22×D711yescharacteristic
240C4⋊C4C22×D711yescharacteristic
241C4⋊C4C22×D711yescharacteristic
242C22⋊C4—141no—
243C22⋊C4—141no—
244C22⋊C4—141no—
245C22⋊C4—141no—
246C22⋊C4—141no—
247C22⋊C4—141no—
248C22⋊C4—141no—
249C42—71no—
250C42—71no—
251C42—71no—
252C42—141no—
253C42—141no—
254C42—141no—
255C42C22×D711yescharacteristic
256C2×C7#53611yescharacteristic
257C2×C7C2×#13511yescharacteristic
258C2×C7#57511yescharacteristic
259D7—41no—
260D7—81no—
261C23—71no—
262C23—141no—
263Q8—141no—
264Q8—141no—
265Q8—282no—
266Q8—282no—
267D4—141no—
268D4—141no—
269D4—282no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—71no—
275C2×C4—71no—
276C2×C4—71no—
277C2×C4—142no—
278C2×C4—141no—
279C2×C4—141no—
280C2×C4—141no—
281C2×C4—141no—
282C2×C4—141no—
283C2×C4—281no—
284C2×C4—282no—
285C2×C4D4×D711yescharacteristic
286C2×C4D4×D711yescharacteristic
287C2×C4C23×D711yescharacteristic
288C8—21no—
289C8—21no—
290C8—141no—
291C8—141no—
292C7#323311yescharacteristic, complemented, Sylow
293C22—141no—
294C22—141no—
295C22—141no—
296C22—281no—
297C22—281no—
298C22—281no—
299C22C2×D4×D711yescharacteristic
300C4—21no—
301C4—21no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—72no—
307C4—141no—
308C4—141no—
309C4—281no—
310C4—281no—
311C4—281no—
312C4#655111yescharacteristic
313C4#655111yescharacteristic
314C2—281no—
315C2—561no—
316C2#9041511yescharacteristic
317C2D7×#13511yescharacteristic
318C2#9045911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#499C2
socleC22×C7C22×D4
prime radical#6458C22
hypercenter#499D7
hypocenterC7#3233
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
2. #90379C2semidirect
3. C7×#499C2semidirect
4. #89840C2semidirect
5. #89836C2semidirect
6. #89662C2semidirect
7. #89660C2semidirect
9. #89647C2semidirect
10. #89642C2semidirect
11. #89639C2semidirect
13. #89626C2semidirect
14. #89469C2semidirect
15. #89457C2semidirect
106. #499D7semidirect
292. C7#3233semidirect

Maximal subgroups

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

subgroupindex
#903822
#903792
C7×#4992
#898402
#898362
#896622
#896602
#896572
#896472
#896422
#896392
#896382
#896262
#894692
#894572
#32337