Finite Group Databasebeta

← #11738567

Subgroups — #11738567

Order 896.

380 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90423C211yesmaximal, characteristic, complemented
2#90267C211yesmaximal, characteristic
3C7×#477C211yesmaximal, characteristic, complemented
4#89861C211yesmaximal, characteristic, complemented
5#89802C211yesmaximal, characteristic, complemented
6#89660C211yesmaximal, characteristic, complemented
7#89659C211yesmaximal, characteristic, complemented
8#89658C211yesmaximal, characteristic, complemented
9#89657C211yesmaximal, characteristic, complemented
10#89617C211yesmaximal, characteristic, complemented
11#89608C211yesmaximal, characteristic, complemented
12#89582C211yesmaximal, characteristic, complemented
13#89580C211yesmaximal, characteristic, complemented
14#89576C211yesmaximal, characteristic, complemented
15#89574C211yesmaximal, characteristic, complemented
16C2×#1161C2211yescharacteristic
17C2×Q8×D7C2211yescharacteristic
18C7×C4⋊Q8C2211yescharacteristic
19C4×C7×Q8C2211yescharacteristic
20#6510—21no—
21Q8×Dic7—21no—
22C2×#1144C2211yescharacteristic
23C2×Dic28C2211yescharacteristic
24C2×#1132C2211yescharacteristic
25C2×#1131C2211yescharacteristic
26C2×C8×D7C2211yescharacteristic
27#6461—21no—
28#6461—21no—
29#6460—21no—
30#6460C2211yescharacteristic
31#6460C2211yescharacteristic
32#6460C2211yescharacteristic
33#6458—21no—
34#6458C2211yescharacteristic
35#6455C2211yescharacteristic
36#6455C2211yescharacteristic
37#6438—21no—
38#6438C2211yescharacteristic
39C4×D28—21no—
40C4×D28C2211yescharacteristic
41C4×Dic14C2211yescharacteristic
42C7×#107C2211yescharacteristic
43C7×#107C2211yescharacteristic
44C7×C4⋊C8C2211yescharacteristic
45C7×#103C2211yescharacteristic
46C7×#103C2211yescharacteristic
47#6409C2211yescharacteristic
48#6395C2211yescharacteristic
49(C2×D7)⋊C8C2211yescharacteristic
50(C2×D7)⋊C8C2211yescharacteristic
51#6390C2211yescharacteristic
52#6384C2211yescharacteristic
53#6384C2211yescharacteristic
54#6384C2211yescharacteristic
55#6383C2211yescharacteristic
56#6383C2211yescharacteristic
57#6382C2211yescharacteristic
58#6377C2211yescharacteristic
59#3313—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8C2311yescharacteristic
62#1161—22no—
63#1161—41no—
64Q8×D7—22no—
65Q8×D7—41no—
66C2×D28—21no—
67C2×D28—21no—
68C2×D28D411yescharacteristic
69C2×D28C2311yescharacteristic
70C2×C4×D7—21no—
71C2×C4×D7—21no—
72C2×C4×D7C2311yescharacteristic
73C2×C4×D7C2311yescharacteristic
74C2×Dic14—21no—
75C2×Dic14C2311yescharacteristic
76C2×Dic14C2311yescharacteristic
77C2×C7×C8C2311yescharacteristic
78C2×C7×C8C2311yescharacteristic
79C7×C4⋊C4—21no—
80C7×C4⋊C4—21no—
81C7×C4⋊C4C2311yescharacteristic
82C7×C4⋊C4C2311yescharacteristic
83C7×C4⋊C4C2311yescharacteristic
84C42×C7—21no—
85C42×C7C2311yescharacteristic
86#1144—22no—
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)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4D411yescharacteristic
96(C2×D7)⋊C4D411yescharacteristic
97C4⋊Dic7—21no—
98C4⋊Dic7—21no—
99C4⋊Dic7—21no—
100C4⋊Dic7D411yescharacteristic
101C4⋊Dic7C2311yescharacteristic
102Dic7⋊C4—21no—
103C4×Dic7—21no—
104C4×Dic7—21no—
105C4×Dic7—21no—
106C2×C7⋊C8C2311yescharacteristic
107C2×C7⋊C8C2311yescharacteristic
108Dic28—41no—
109#1132—41no—
110#1131—41no—
111C8×D7—41no—
112#556—71no—
113#548—71no—
114#489—71no—
115#485—71no—
116#482—71no—
117#477D711yescharacteristic, complemented
118#468—71no—
119#467—71no—
120#464—71no—
121#452—71no—
122#451—71no—
123#441—71no—
124#435—71no—
125#419—71no—
126#416—71no—
127C22×D7—21no—
128C22×D7C2×D411yescharacteristic
129C22×D7C2×D411yescharacteristic
130C7×Q8—21no—
131C7×Q8—42no—
132C7×Q8#4112yes—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7C2×D411yescharacteristic
138C2×C4×C7C2×D411yescharacteristic
139C2×C4×C7C2411yescharacteristic
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7C2×D411yescharacteristic
144C2×Dic7C2×D411yescharacteristic
145D28—21no—
146D28—21no—
147D28—41no—
148D28—42no—
149D28—42no—
150C4×D7—21no—
151C4×D7—21no—
152C4×D7—21no—
153C4×D7—21no—
154C4×D7—42no—
155C4×D7—42no—
156Dic14—21no—
157Dic14—21no—
158Dic14—21no—
159Dic14—21no—
160Dic14—42no—
161C7×C8—21no—
162C7×C8—21no—
163C7⋊C8—21no—
164C7⋊C8—21no—
165C2×#41—71no—
166C22×Q8—71no—
167C2×Q16—71no—
168C2×Q16—71no—
169C2×SD16—71no—
170C2×M16—71no—
171C22×C8—71no—
172C4⋊Q8C2×D711yescharacteristic
173#126—141no—
174#126—141no—
175#124—71no—
176#124—141no—
177#124—141no—
178C22⋊Q8—71no—
179C22⋊Q8—71no—
180C22⋊Q8—71no—
181C22⋊Q8—141no—
182#121—71no—
183#121—141no—
184C4×Q8—71no—
185C4×Q8—141no—
186C4×Q8C2×D711yescharacteristic
187C4×D4—71no—
188C4×D4—141no—
189#117—71no—
190#117—71no—
191#107C2×D711yescharacteristic
192#107C2×D711yescharacteristic
193#106—71no—
194C4⋊C8C2×D711yescharacteristic
195#103—71no—
196#103—71no—
197#103—71no—
198#103—71no—
199#103—71no—
200#103C2×D711yescharacteristic
201#103C2×D711yescharacteristic
202#102—71no—
203#102—71no—
204#102—71no—
205C22⋊C8—71no—
206C22⋊C8—71no—
207#97—71no—
208C22×C7C22×D411yescharacteristic
209C2×D7—21no—
210C2×D7—21no—
211C2×D7—21no—
212C2×D7—21no—
213C2×D7—21no—
214C2×D7—21no—
215C2×D7—41no—
216C2×D7—41no—
217C2×D7—41no—
218C4×C7—22no—
219C4×C7—21no—
220C4×C7—21no—
221C4×C7—41no—
222C4×C7—41no—
223C4×C7—41no—
224C4×C7C2×#4111yescharacteristic
225C4×C7H(5,𝔽2)11yescharacteristic
226Dic7—21no—
227Dic7—21no—
228Dic7—41no—
229Dic7—41no—
230Dic7—41no—
231#41—142no—
232#41—281no—
233C2×Q8—21no—
234C2×Q8—71no—
235C2×Q8—71no—
236C2×Q8—141no—
237C2×Q8—142no—
238C2×Q8—281no—
239C2×Q8C22×D711yescharacteristic
240C2×D4—71no—
241C2×D4—71no—
242C2×D4—141no—
243C2×D4—141no—
244C22×C4—71no—
245C22×C4—71no—
246C22×C4—141no—
247C22×C4—141no—
248Q16—142no—
249Q16—281no—
250SD16—281no—
251M16—281no—
252C2×C8—71no—
253C2×C8—71no—
254C2×C8—281no—
255C2×C8C22×D711yescharacteristic
256C2×C8C22×D711yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—71no—
260C4⋊C4—71no—
261C4⋊C4—141no—
262C4⋊C4—141no—
263C4⋊C4—141no—
264C4⋊C4—141no—
265C4⋊C4C22×D711yescharacteristic
266C4⋊C4C22×D711yescharacteristic
267C4⋊C4C22×D711yescharacteristic
268C22⋊C4—71no—
269C22⋊C4—71no—
270C22⋊C4—141no—
271C22⋊C4—141no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C22⋊C4—141no—
277C22⋊C4—141no—
278C42—21no—
279C42—141no—
280C42—141no—
281C42—141no—
282C42C22×D711yescharacteristic
283C2×C7#54611yescharacteristic
284C2×C7#57511yescharacteristic
285C2×C7#57811yescharacteristic
286D7—41no—
287D7—41no—
288D7—81no—
289C23—71no—
290C23—71no—
291C23—141no—
292Q8—21no—
293Q8—42no—
294Q8—141no—
295Q8—141no—
296Q8—141no—
297Q8—141no—
298Q8—282no—
299Q8#116112yes—
300D4—141no—
301D4—141no—
302D4—281no—
303D4—282no—
304D4—282no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—71no—
310C2×C4—71no—
311C2×C4—141no—
312C2×C4—141no—
313C2×C4—141no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—141no—
317C2×C4—141no—
318C2×C4—282no—
319C2×C4—282no—
320C2×C4D4×D711yescharacteristic
321C2×C4D4×D711yescharacteristic
322C2×C4C23×D711yescharacteristic
323C8—21no—
324C8—21no—
325C8—141no—
326C8—141no—
327C7#331311yescharacteristic, complemented, Sylow
328C22—141no—
329C22—141no—
330C22—141no—
331C22—141no—
332C22—141no—
333C22—141no—
334C22—281no—
335C22—281no—
336C22—281no—
337C22C2×D4×D711yescharacteristic
338C4—21no—
339C4—21no—
340C4—22no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—141no—
345C4—141no—
346C4—281no—
347C4—281no—
348C4—281no—
349C4C2×#116111yescharacteristic
350C4#655311yescharacteristic
351C2—281no—
352C2—281no—
353C2—561no—
354C2#9032111yescharacteristic
355C2#9045711yescharacteristic
356C2#9046411yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#477C2
socleC22×C7C22×D4
prime radical#90267C2
hypercenter#477D7
hypocenterC7#3313
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
1. #90423C2semidirect
3. C7×#477C2semidirect
4. #89861C2semidirect
5. #89802C2semidirect
6. #89660C2semidirect
7. #89659C2semidirect
8. #89658C2semidirect
9. #89657C2semidirect
10. #89617C2semidirect
11. #89608C2semidirect
12. #89582C2semidirect
13. #89580C2semidirect
14. #89576C2semidirect
15. #89574C2semidirect
117. #477D7semidirect
327. C7#3313semidirect

Maximal subgroups

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

subgroupindex
#904232
#902672
C7×#4772
#898612
#898022
#896602
#896592
#896582
#896572
#896172
#896082
#895822
#895802
#895762
#895742
#33137