Finite Group Databasebeta

← #11738652

Subgroups — #11738652

Order 896.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90382C211yesmaximal, characteristic
2C4×Q8×D7C211yesmaximal, characteristic, complemented
3C7×#479C211yesmaximal, characteristic, complemented
4#89835C211yesmaximal, characteristic, complemented
5#89800C211yesmaximal, characteristic, complemented
6#89661C211yesmaximal, characteristic, complemented
7#89656C211yesmaximal, characteristic, complemented
8#89635C211yesmaximal, characteristic
9#89625C211yesmaximal, characteristic, complemented
10#89617C211yesmaximal, characteristic, complemented
11#89607C211yesmaximal, characteristic, complemented
12#89589C211yesmaximal, characteristic, complemented
13#89580C211yesmaximal, characteristic, complemented
14#89578C211yesmaximal, characteristic
15#89561C211yesmaximal, characteristic, complemented
16C2×Q8×D7C2211yescharacteristic
17C7×#125C2211yescharacteristic
18C4×C7×Q8C2211yescharacteristic
19Q8×Dic7C2211yescharacteristic
20C2×#1144C2211yescharacteristic
21C2×#1143C2211yescharacteristic
22C2×Dic28C2211yescharacteristic
23C2×#1132C2211yescharacteristic
24#6461—21no—
25#6460C2211yescharacteristic
26#6458C2211yescharacteristic
27#6457—21no—
28#6456—21no—
29#6456C2211yescharacteristic
30#6455—21no—
31D7×C4⋊C4—21no—
32D7×C4⋊C4C2211yescharacteristic
33#6450—21no—
34#6450C2211yescharacteristic
35#6450C2211yescharacteristic
36#6438C2211yescharacteristic
37C42×D7—21no—
38C42×D7C2211yescharacteristic
39C4×Dic14—21no—
40C4×Dic14C2211yescharacteristic
41C7×#107C2211yescharacteristic
42C7×#106C2211yescharacteristic
43C7×C4⋊C8C2211yescharacteristic
44C7×#103C2211yescharacteristic
45C7×#103C2211yescharacteristic
46#6395C2211yescharacteristic
47(C2×D7)⋊C8C2211yescharacteristic
48(C2×D7)⋊C8C2211yescharacteristic
49#6390C2211yescharacteristic
50#6389C2211yescharacteristic
51#6389C2211yescharacteristic
52#6384C2211yescharacteristic
53#6384C2211yescharacteristic
54#6384C2211yescharacteristic
55#6383C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3117—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61Q8×D7—22no—
62Q8×D7—22no—
63C2×D28—21no—
64C2×D28C2311yescharacteristic
65C2×C4×D7—21no—
66C2×C4×D7—21no—
67C2×C4×D7—21no—
68C2×C4×D7D411yescharacteristic
69C2×C4×D7D411yescharacteristic
70C2×C4×D7C2311yescharacteristic
71C2×Dic14—21no—
72C2×Dic14C2311yescharacteristic
73C2×Dic14C2311yescharacteristic
74C2×C7×C8C2311yescharacteristic
75C2×C7×C8C2311yescharacteristic
76C7×C4⋊C4—21no—
77C7×C4⋊C4—21no—
78C7×C4⋊C4—21no—
79C7×C4⋊C4C2311yescharacteristic
80C7×C4⋊C4C2311yescharacteristic
81C7×C4⋊C4C2311yescharacteristic
82C42×C7—21no—
83C42×C7C2311yescharacteristic
84#1144—22no—
85#1143—22no—
86(C2×D7)⋊C4—21no—
87(C2×D7)⋊C4—21no—
88(C2×D7)⋊C4—21no—
89(C2×D7)⋊C4—21no—
90(C2×D7)⋊C4—21no—
91C4⋊Dic7—21no—
92C4⋊Dic7—21no—
93C4⋊Dic7C2311yescharacteristic
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98C4×Dic7—21no—
99C4×Dic7—21no—
100C4×Dic7—21no—
101C4×Dic7—21no—
102C4×Dic7D411yescharacteristic
103C4×Dic7D411yescharacteristic
104C4×Dic7C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107Dic28—22no—
108#1132—22no—
109#529—71no—
110C2×C4×Q8—71no—
111#484—71no—
112#481—71no—
113#479—71no—
114#479D711yescharacteristic, complemented
115#464—71no—
116#464—71no—
117#451—71no—
118#448—71no—
119#441—71no—
120#441—71no—
121#440—71no—
122#440—71no—
123#432—71no—
124C22×D7—21no—
125C22×D7C2×D411yescharacteristic
126C7×Q8—21no—
127C7×Q8#4112yes—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7C2×D411yescharacteristic
133C2×C4×C7C2×D411yescharacteristic
134C2×C4×C7C2411yescharacteristic
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7C2×D411yescharacteristic
139C2×Dic7C2×D411yescharacteristic
140C2×Dic7C2×D411yescharacteristic
141D28—21no—
142D28—21no—
143D28—42no—
144C4×D7—24no—
145C4×D7—22no—
146C4×D7—21no—
147C4×D7—21no—
148C4×D7—41no—
149C4×D7—42no—
150C4×D7—42no—
151Dic14—24no—
152Dic14—21no—
153Dic14—21no—
154Dic14—21no—
155Dic14#4112yes—
156C7×C8—21no—
157C7×C8—21no—
158C7⋊C8—21no—
159C7⋊C8—21no—
160C22×Q8—71no—
161C2×Q16—71no—
162C2×Q16—71no—
163C2×SD16—71no—
164C2×SD16—71no—
165#126—141no—
166#125C2×D711yescharacteristic
167#124—71no—
168#123—141no—
169C22⋊Q8—71no—
170#121—71no—
171C4×Q8—71no—
172C4×Q8—71no—
173C4×Q8—71no—
174C4×Q8—71no—
175C4×Q8—141no—
176C4×Q8—141no—
177C4×Q8C2×D711yescharacteristic
178C4×D4—71no—
179C4×D4—141no—
180#117—141no—
181C2×C4⋊C4—71no—
182C2×C4⋊C4—141no—
183C2×C42—71no—
184C2×C42—141no—
185#107—71no—
186#107C2×D711yescharacteristic
187#106—71no—
188#106C2×D711yescharacteristic
189C4⋊C8—71no—
190C4⋊C8C2×D711yescharacteristic
191#103—71no—
192#103—71no—
193#103—71no—
194#103—71no—
195#103C2×D711yescharacteristic
196#103C2×D711yescharacteristic
197#102—71no—
198#102—71no—
199C22⋊C8—71no—
200C22⋊C8—71no—
201#97—71no—
202#97—71no—
203C22×C7C22×D411yescharacteristic
204C2×D7—21no—
205C2×D7—21no—
206C2×D7—41no—
207C2×D7—41no—
208C2×D7—41no—
209C2×D7#13712yes—
210C4×C7—22no—
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7C2×#4111yescharacteristic
217C4×C7C2×#4111yescharacteristic
218Dic7—21no—
219Dic7—22no—
220Dic7—21no—
221Dic7—21no—
222Dic7—41no—
223Dic7—41no—
224C2×Q8—71no—
225C2×Q8—71no—
226C2×Q8—141no—
227C2×Q8—142no—
228C2×Q8—142no—
229C2×Q8C22×D711yescharacteristic
230C2×D4—71no—
231C2×D4—141no—
232C22×C4—71no—
233C22×C4—71no—
234C22×C4—71no—
235C22×C4—141no—
236C22×C4—141no—
237C22×C4—141no—
238Q16—142no—
239Q16—142no—
240SD16—142no—
241SD16—142no—
242C2×C8—71no—
243C2×C8—71no—
244C2×C8C22×D711yescharacteristic
245C2×C8C22×D711yescharacteristic
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—71no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4—141no—
254C4⋊C4—141no—
255C4⋊C4—141no—
256C4⋊C4C22×D711yescharacteristic
257C4⋊C4C22×D711yescharacteristic
258C4⋊C4C22×D711yescharacteristic
259C22⋊C4—141no—
260C22⋊C4—141no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C42—21no—
265C42—71no—
266C42—71no—
267C42—71no—
268C42—141no—
269C42—141no—
270C42—141no—
271C42—141no—
272C42C22×D711yescharacteristic
273C2×C7#52511yescharacteristic
274C2×C7C2×#13711yescharacteristic
275C2×C7#57511yescharacteristic
276D7—22no—
277D7—81no—
278C23—71no—
279C23—141no—
280Q8—21no—
281Q8—72no—
282Q8—144no—
283Q8—141no—
284Q8—141no—
285Q8—141no—
286Q8#116112yes—
287D4—141no—
288D4—141no—
289D4—282no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—71no—
295C2×C4—71no—
296C2×C4—71no—
297C2×C4—142no—
298C2×C4—144no—
299C2×C4—141no—
300C2×C4—141no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—281no—
305C2×C4—282no—
306C2×C4—282no—
307C2×C4D4×D711yescharacteristic
308C2×C4D4×D711yescharacteristic
309C2×C4C23×D711yescharacteristic
310C8—21no—
311C8—21no—
312C8—141no—
313C8—141no—
314C7#311711yescharacteristic, complemented, Sylow
315C22—72no—
316C22—141no—
317C22—141no—
318C22—281no—
319C22—281no—
320C22—281no—
321C22C2×D4×D711yescharacteristic
322C4—21no—
323C4—21no—
324C4—22no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—141no—
329C4—141no—
330C4—141no—
331C4—142no—
332C4—281no—
333C4—281no—
334C4C2×#116111yescharacteristic
335C4D7×#4111yescharacteristic
336C2—142no—
337C2—561no—
338C2#9031811yescharacteristic
339C2#9045911yescharacteristic
340C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#479C2
socleC22×C7C22×D4
prime radical#6458C22
hypercenter#479D7
hypocenterC7#3117
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
2. C4×Q8×D7C2semidirect
3. C7×#479C2semidirect
4. #89835C2semidirect
5. #89800C2semidirect
6. #89661C2semidirect
7. #89656C2semidirect
9. #89625C2semidirect
10. #89617C2semidirect
11. #89607C2semidirect
12. #89589C2semidirect
13. #89580C2semidirect
15. #89561C2semidirect
114. #479D7semidirect
314. C7#3117semidirect

Maximal subgroups

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

subgroupindex
#903822
C4×Q8×D72
C7×#4792
#898352
#898002
#896612
#896562
#896352
#896252
#896172
#896072
#895892
#895802
#895782
#895612
#31177