Finite Group Databasebeta

← #11738682

Subgroups — #11738682

Order 896.

334 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
1D7×#125C211yesmaximal, characteristic, complemented
2#90271C211yesmaximal, characteristic
3C7×#479C211yesmaximal, characteristic, complemented
4#89838C211yesmaximal, characteristic, complemented
5#89794C211yesmaximal, characteristic, complemented
6#89654C211yesmaximal, characteristic
7#89650C211yesmaximal, characteristic, complemented
8#89635C211yesmaximal, characteristic
9#89630C211yesmaximal, characteristic, complemented
10#89619C211yesmaximal, characteristic, complemented
11#89605C211yesmaximal, characteristic, complemented
12#89581C211yesmaximal, characteristic, complemented
13#89577C211yesmaximal, characteristic, complemented
14#89572C211yesmaximal, characteristic, complemented
15#89572C211yesmaximal, characteristic, complemented
16C7×#125C2211yescharacteristic
17C4×C7×Q8C2211yescharacteristic
18#6510C2211yescharacteristic
19#6509C2211yescharacteristic
20#6461—21no—
21#6458C2211yescharacteristic
22#6457—21no—
23D7×C4⋊C4—21no—
24D7×C4⋊C4—21no—
25D7×C4⋊C4C2211yescharacteristic
26D7×C4⋊C4C2211yescharacteristic
27#6453C2211yescharacteristic
28#6453C2211yescharacteristic
29#6453C2211yescharacteristic
30#6452—21no—
31#6452—21no—
32C4×D28—21no—
33C4×D28C2211yescharacteristic
34#6435—21no—
35C42×D7C2211yescharacteristic
36#6433C2211yescharacteristic
37C7×#107C2211yescharacteristic
38C7×#106C2211yescharacteristic
39C7×C4⋊C8C2211yescharacteristic
40C7×#103C2211yescharacteristic
41C7×#103C2211yescharacteristic
42#6409C2211yescharacteristic
43#6409C2211yescharacteristic
44#6395C2211yescharacteristic
45#6395C2211yescharacteristic
46(C2×D7)⋊C8C2211yescharacteristic
47(C2×D7)⋊C8C2211yescharacteristic
48#6392C2211yescharacteristic
49#6391C2211yescharacteristic
50C8×Dic7C2211yescharacteristic
51C8×Dic7C2211yescharacteristic
52#6383C2211yescharacteristic
53#6383C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3232—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×D28—21no—
62C2×D28C2311yescharacteristic
63C2×C4×D7—21no—
64C2×C4×D7—21no—
65C2×C4×D7—21no—
66C2×C4×D7D411yescharacteristic
67C2×C4×D7D411yescharacteristic
68C2×C4×D7C2311yescharacteristic
69C2×C7×C8C2311yescharacteristic
70C2×C7×C8C2311yescharacteristic
71C7×C4⋊C4—21no—
72C7×C4⋊C4—21no—
73C7×C4⋊C4—21no—
74C7×C4⋊C4C2311yescharacteristic
75C7×C4⋊C4C2311yescharacteristic
76C7×C4⋊C4C2311yescharacteristic
77C42×C7—21no—
78C42×C7C2311yescharacteristic
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—
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⋊C4—21no—
98C4×Dic7D411yescharacteristic
99C4×Dic7D411yescharacteristic
100C4×Dic7C2311yescharacteristic
101C2×C7⋊C8C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103#529—71no—
104C2×#125—71no—
105#499—71no—
106#499—71no—
107#488—71no—
108#488—71no—
109#484—71no—
110#483—71no—
111#481—71no—
112#480—71no—
113#479—71no—
114#479D711yescharacteristic, complemented
115#478—71no—
116#478—71no—
117#431—71no—
118C22×D7—21no—
119C22×D7C2×D411yescharacteristic
120C7×Q8—21no—
121C7×Q8—21no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7C2×D411yescharacteristic
127C2×C4×C7C2×D411yescharacteristic
128C2×C4×C7C2411yescharacteristic
129C2×Dic7—21no—
130C2×Dic7—21no—
131C2×Dic7—21no—
132C2×Dic7C2×D411yescharacteristic
133C2×Dic7C2×D411yescharacteristic
134C2×Dic7C2×D411yescharacteristic
135D28—21no—
136D28—21no—
137D28—42no—
138C4×D7—22no—
139C4×D7—21no—
140C4×D7—21no—
141C4×D7—41no—
142C4×D7—42no—
143C4×D7—42no—
144C4×D7—42no—
145C7×C8—21no—
146C7×C8—21no—
147C7⋊C8—21no—
148C7⋊C8—21no—
149#126—141no—
150#125—71no—
151#125—71no—
152#125—71no—
153#125—71no—
154#125—141no—
155#125—141no—
156#125C2×D711yescharacteristic
157#124—71no—
158#123—141no—
159C22⋊Q8—71no—
160#121—71no—
161C4×Q8C2×D711yescharacteristic
162C4×D4—71no—
163C4×D4—141no—
164#117—141no—
165C2×C4⋊C4—71no—
166C2×C4⋊C4—71no—
167C2×C4⋊C4—141no—
168C2×C4⋊C4—141no—
169C2×C42—71no—
170#107—71no—
171#107—71no—
172#107—71no—
173#107C2×D711yescharacteristic
174#106—71no—
175#106—71no—
176#106—71no—
177#106C2×D711yescharacteristic
178C4⋊C8—71no—
179C4⋊C8C2×D711yescharacteristic
180#103—71no—
181#103—71no—
182#103C2×D711yescharacteristic
183#103C2×D711yescharacteristic
184#102—71no—
185#102—71no—
186#102—71no—
187#102—71no—
188C22⋊C8—71no—
189C22⋊C8—71no—
190C4×C8—71no—
191C4×C8—71no—
192C22×C7C22×D411yescharacteristic
193C2×D7—21no—
194C2×D7—21no—
195C2×D7—41no—
196C2×D7—41no—
197C2×D7—41no—
198C2×D7#13512yes—
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—21no—
210Dic7—41no—
211Dic7—41no—
212Dic7—41no—
213C2×Q8C22×D711yescharacteristic
214C2×D4—71no—
215C2×D4—141no—
216C22×C4—71no—
217C22×C4—71no—
218C22×C4—71no—
219C22×C4—141no—
220C22×C4—141no—
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—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⋊C4—141no—
243C4⋊C4C22×D711yescharacteristic
244C4⋊C4C22×D711yescharacteristic
245C4⋊C4C22×D711yescharacteristic
246C22⋊C4—141no—
247C22⋊C4—141no—
248C22⋊C4—141no—
249C22⋊C4—141no—
250C22⋊C4—141no—
251C42—21no—
252C42—71no—
253C42—71no—
254C42—71no—
255C42C22×D711yescharacteristic
256C2×C7#53611yescharacteristic
257C2×C7C2×#13511yescharacteristic
258C2×C7#57511yescharacteristic
259D7—22no—
260D7—81no—
261C23—71no—
262C23—141no—
263Q8—21no—
264Q8—21no—
265D4—141no—
266D4—141no—
267D4—282no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—71no—
273C2×C4—71no—
274C2×C4—71no—
275C2×C4—142no—
276C2×C4—141no—
277C2×C4—141no—
278C2×C4—141no—
279C2×C4—141no—
280C2×C4—141no—
281C2×C4—281no—
282C2×C4—282no—
283C2×C4—282no—
284C2×C4—282no—
285C2×C4D4×D711yescharacteristic
286C2×C4D4×D711yescharacteristic
287C2×C4C23×D711yescharacteristic
288C8—21no—
289C8—21no—
290C8—141no—
291C8—141no—
292C7#323211yescharacteristic, complemented, Sylow
293C22—72no—
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—141no—
307C4—141no—
308C4—141no—
309C4—281no—
310C4—281no—
311C4—281no—
312C4#655111yescharacteristic
313C4#655411yescharacteristic
314C2—142no—
315C2—561no—
316C2#9031911yescharacteristic
317C2D7×#13511yescharacteristic
318C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#479C2
socleC22×C7C22×D4
prime radical#6458C22
hypercenter#479D7
hypocenterC7#3232
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
1. D7×#125C2semidirect
3. C7×#479C2semidirect
4. #89838C2semidirect
5. #89794C2semidirect
7. #89650C2semidirect
9. #89630C2semidirect
10. #89619C2semidirect
11. #89605C2semidirect
12. #89581C2semidirect
13. #89577C2semidirect
14. #89572C2semidirect
15. #89572C2semidirect
114. #479D7semidirect
292. C7#3232semidirect

Maximal subgroups

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

subgroupindex
D7×#1252
#902712
C7×#4792
#898382
#897942
#896542
#896502
#896352
#896302
#896192
#896052
#895812
#895772
#895722
#32327