Finite Group Databasebeta

← #11739439

Subgroups — #11739439

Order 896.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90425C211yesmaximal, characteristic, complemented
2#90386C211yesmaximal, characteristic, complemented
3C7×#501C211yesmaximal, characteristic, complemented
4#89859C211yesmaximal, characteristic, complemented
5#89838C211yesmaximal, characteristic, complemented
6#89657C211yesmaximal, characteristic, complemented
7#89655C211yesmaximal, characteristic, complemented
8#89654C211yesmaximal, characteristic, complemented
9#89653C211yesmaximal, characteristic, complemented
10#89638C211yesmaximal, characteristic, complemented
11#89638C211yesmaximal, characteristic, complemented
12#89637C211yesmaximal, characteristic, complemented
13#89632C211yesmaximal, characteristic, complemented
14#89483C211yesmaximal, characteristic, complemented
15#89478C211yesmaximal, characteristic, complemented
16C7×C4⋊Q8C2211yescharacteristic
17C7×#125C2211yescharacteristic
18#6510—21no—
19#6510—21no—
20C2×D56C2211yescharacteristic
21C2×#1131C2211yescharacteristic
22C2×C8×D7C2211yescharacteristic
23#6461—21no—
24#6458—21no—
25#6458—21no—
26#6458C2211yescharacteristic
27#6458C2211yescharacteristic
28#6458C2211yescharacteristic
29#6458C2211yescharacteristic
30#6457—21no—
31#6456—21no—
32#6455—21no—
33#6455C2211yescharacteristic
34#6455C2211yescharacteristic
35#6455C2211yescharacteristic
36D7×C4⋊C4C2211yescharacteristic
37C4×D28C2211yescharacteristic
38C4×D28C2211yescharacteristic
39C4×D28C2211yescharacteristic
40C7×#107C2211yescharacteristic
41C7×#107C2211yescharacteristic
42C7×#106C2211yescharacteristic
43C7×#106C2211yescharacteristic
44C7×#97C2211yescharacteristic
45#6395C2211yescharacteristic
46#6395C2211yescharacteristic
47(C2×D7)⋊C8C2211yescharacteristic
48(C2×D7)⋊C8C2211yescharacteristic
49#6391C2211yescharacteristic
50#6383C2211yescharacteristic
51#6383C2211yescharacteristic
52#6383C2211yescharacteristic
53#6383C2211yescharacteristic
54#6382C2211yescharacteristic
55#6381C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3347—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×D28—21no—
62C2×D28—21no—
63C2×D28—21no—
64C2×D28—21no—
65C2×D28D411yescharacteristic
66C2×D28C2311yescharacteristic
67C2×D28C2311yescharacteristic
68C2×C4×D7—21no—
69C2×C4×D7—21no—
70C2×C4×D7—21no—
71C2×C4×D7C2311yescharacteristic
72C2×C4×D7C2311yescharacteristic
73C2×C7×C8C2311yescharacteristic
74C2×C7×C8C2311yescharacteristic
75C7×C4⋊C4—21no—
76C7×C4⋊C4—21no—
77C7×C4⋊C4—21no—
78C7×C4⋊C4C2311yescharacteristic
79C7×C4⋊C4C2311yescharacteristic
80C7×C4⋊C4C2311yescharacteristic
81C7×C4⋊C4C2311yescharacteristic
82C42×C7C2311yescharacteristic
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4—21no—
85(C2×D7)⋊C4—21no—
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—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4D411yescharacteristic
93(C2×D7)⋊C4D411yescharacteristic
94C4⋊Dic7D411yescharacteristic
95C4⋊Dic7C2311yescharacteristic
96C4⋊Dic7C2311yescharacteristic
97Dic7⋊C4—21no—
98C4×Dic7—21no—
99C4×Dic7—21no—
100C4×Dic7—21no—
101C2×C7⋊C8C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103D56—22no—
104#1131—22no—
105C8×D7—22no—
106#555—71no—
107#553—71no—
108#501D711yescharacteristic, complemented
109#482—71no—
110#482—71no—
111#482—71no—
112#480—71no—
113#478—71no—
114#474—71no—
115#469—71no—
116#466—71no—
117#442—71no—
118#435—71no—
119#428—71no—
120#427—71no—
121C22×D7—21no—
122C22×D7—21no—
123C22×D7C2×D411yescharacteristic
124C22×D7C2×D411yescharacteristic
125C7×Q8—81no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7C2×D411yescharacteristic
131C2×C4×C7C2×D411yescharacteristic
132C2×C4×C7C2411yescharacteristic
133C2×Dic7—21no—
134C2×Dic7—21no—
135C2×Dic7C2×D411yescharacteristic
136C2×Dic7C2×D411yescharacteristic
137D28—21no—
138D28—21no—
139D28—21no—
140D28—21no—
141D28—41no—
142D28—42no—
143D28—81no—
144D28—81no—
145D28—81no—
146C4×D7—21no—
147C4×D7—21no—
148C4×D7—21no—
149C4×D7—21no—
150C4×D7—42no—
151C4×D7—81no—
152C4×D7—81no—
153C7×C8—21no—
154C7×C8#4112yes—
155C7⋊C8—21no—
156C7⋊C8—21no—
157C2×D8—71no—
158C2×M16—71no—
159C22×C8—71no—
160C4⋊Q8C2×D711yescharacteristic
161#126—141no—
162#125C2×D711yescharacteristic
163#124—141no—
164#124—141no—
165#123—141no—
166#121—71no—
167#121—71no—
168#121—71no—
169#121—71no—
170#121—141no—
171#121—141no—
172C4×D4—71no—
173C4×D4—71no—
174C4×D4—71no—
175C4×D4—141no—
176#117—71no—
177#117—71no—
178#117—71no—
179#117—141no—
180C2×C4⋊C4—71no—
181#107—71no—
182#107—71no—
183#107—71no—
184#107C2×D711yescharacteristic
185#107C2×D711yescharacteristic
186#106—71no—
187#106—71no—
188#106C2×D711yescharacteristic
189#106C2×D711yescharacteristic
190C4⋊C8—71no—
191#102—71no—
192#102—71no—
193#102—71no—
194#102—71no—
195#102—71no—
196#102—71no—
197C22⋊C8—71no—
198C22⋊C8—71no—
199#97C2×D711yescharacteristic
200C22×C7C22×D411yescharacteristic
201C2×D7—21no—
202C2×D7—21no—
203C2×D7—21no—
204C2×D7—21no—
205C2×D7—21no—
206C2×D7—21no—
207C2×D7—41no—
208C2×D7—41no—
209C2×D7—41no—
210C2×D7—41no—
211C2×D7—41no—
212C2×D7—41no—
213C4×C7—21no—
214C4×C7—21no—
215C4×C7—41no—
216C4×C7—41no—
217C4×C7—41no—
218C4×C7—41no—
219C4×C7C2×#4111yescharacteristic
220C4×C72−1+411yescharacteristic
221Dic7—21no—
222Dic7—21no—
223Dic7—41no—
224Dic7—41no—
225C2×Q8—21no—
226C2×D4—71no—
227C2×D4—71no—
228C2×D4—71no—
229C2×D4—141no—
230C2×D4—141no—
231C2×D4—141no—
232C2×D4—141no—
233C22×C4—71no—
234C22×C4—71no—
235C22×C4—141no—
236C22×C4—141no—
237C22×C4—141no—
238D8—142no—
239M16—142no—
240C2×C8—71no—
241C2×C8—71no—
242C2×C8—142no—
243C2×C8C22×D711yescharacteristic
244C2×C8C22×D711yescharacteristic
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—71no—
249C4⋊C4—71no—
250C4⋊C4—71no—
251C4⋊C4—141no—
252C4⋊C4C22×D711yescharacteristic
253C4⋊C4C22×D711yescharacteristic
254C4⋊C4C22×D711yescharacteristic
255C4⋊C4C22×D711yescharacteristic
256C22⋊C4—71no—
257C22⋊C4—71no—
258C22⋊C4—141no—
259C22⋊C4—141no—
260C22⋊C4—141no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C22⋊C4—141no—
265C22⋊C4—141no—
266C22⋊C4—141no—
267C42—141no—
268C42—141no—
269C42—141no—
270C42C22×D711yescharacteristic
271C2×C7#54711yescharacteristic
272C2×C7#57511yescharacteristic
273C2×C7#57611yescharacteristic
274D7—41no—
275D7—41no—
276D7—81no—
277D7—81no—
278C23—71no—
279C23—71no—
280C23—141no—
281C23—141no—
282Q8—81no—
283D4—141no—
284D4—141no—
285D4—141no—
286D4—141no—
287D4—281no—
288D4—282no—
289D4—561no—
290D4—561no—
291D4—561no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—71no—
297C2×C4—71no—
298C2×C4—141no—
299C2×C4—141no—
300C2×C4—141no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—282no—
305C2×C4—561no—
306C2×C4—561no—
307C2×C4D4×D711yescharacteristic
308C2×C4D4×D711yescharacteristic
309C2×C4C23×D711yescharacteristic
310C8—21no—
311C8—141no—
312C8—141no—
313C8#116112yes—
314C7#334711yescharacteristic, complemented, Sylow
315C22—141no—
316C22—141no—
317C22—141no—
318C22—141no—
319C22—141no—
320C22—141no—
321C22—281no—
322C22—281no—
323C22—281no—
324C22—281no—
325C22—281no—
326C22—281no—
327C22C2×D4×D711yescharacteristic
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—141no—
335C4—141no—
336C4—281no—
337C4—281no—
338C4C2×#116111yescharacteristic
339C4#655111yescharacteristic
340C2—281no—
341C2—281no—
342C2—561no—
343C2—561no—
344C2#9041711yescharacteristic
345C2#9046511yescharacteristic
346C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#501C2
socleC22×C7C22×D4
hypercenter#501D7
hypocenterC7#3347
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
1. #90425C2semidirect
2. #90386C2semidirect
3. C7×#501C2semidirect
4. #89859C2semidirect
5. #89838C2semidirect
6. #89657C2semidirect
7. #89655C2semidirect
8. #89654C2semidirect
9. #89653C2semidirect
10. #89638C2semidirect
11. #89638C2semidirect
12. #89637C2semidirect
13. #89632C2semidirect
14. #89483C2semidirect
15. #89478C2semidirect
108. #501D7semidirect
314. C7#3347semidirect

Maximal subgroups

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

subgroupindex
#904252
#903862
C7×#5012
#898592
#898382
#896572
#896552
#896542
#896532
#896382
#896372
#896322
#894832
#894782
#33477