Finite Group Databasebeta

← #11738053

Subgroups — #11738053

Order 896.

396 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90365C211yesmaximal, characteristic
2#90274C211yesmaximal, characteristic, complemented
3C7×#464C211yesmaximal, characteristic, complemented
4#89962C211yesmaximal, characteristic, complemented
5#89959C211yesmaximal, characteristic, complemented
6#89947C211yesmaximal, characteristic, complemented
7#89941C211yesmaximal, characteristic, complemented
8#89826C211yesmaximal, characteristic, complemented
9#89799C211yesmaximal, characteristic, complemented
10#89620C211yesmaximal, characteristic, complemented
11#89612C211yesmaximal, characteristic, complemented
12#89582C211yesmaximal, characteristic, complemented
13#89572C211yesmaximal, characteristic, complemented
14#89548C211yesmaximal, characteristic, complemented
15#89536C211yesmaximal, characteristic, complemented
16C2×#1161C2211yescharacteristic
17C2×D4×D7C2211yescharacteristic
18C2×C7×Q16C2211yescharacteristic, complemented
19C2×C7×SD16C2211yescharacteristic
20C7×#124C2211yescharacteristic
21C4×C7×Q8C2211yescharacteristic, complemented
22#6510C2211yescharacteristic
23#6509C2211yescharacteristic
24#6507C2211yescharacteristic, complemented
25C2×#1144C2211yescharacteristic, complemented
26C2×#1143C2211yescharacteristic
27C2×#1143C2211yescharacteristic, complemented
28#6503C2211yescharacteristic
29C2×#1142C2211yescharacteristic
30C2×#1132C2211yescharacteristic
31C2×#1132C2211yescharacteristic, complemented
32#6461—21no—
33#6460C2211yescharacteristic
34#6453C2211yescharacteristic, complemented
35#6452—21no—
36#6447—21no—
37#6447—21no—
38#6445—21no—
39#6445—21no—
40#6439—21no—
41#6437C2211yescharacteristic
42#6435C2211yescharacteristic
43#6433—21no—
44C4×Dic14C2211yescharacteristic, complemented
45C7×C4⋊C8C2211yescharacteristic, complemented
46C7×#103C2211yescharacteristic, complemented
47C7×#102C2211yescharacteristic
48#6409C2211yescharacteristic, complemented
49#6406C2211yescharacteristic
50#6395C2211yescharacteristic
51#6395C2211yescharacteristic, complemented
52(C2×D7)⋊C8C2211yescharacteristic
53(C2×D7)⋊C8C2211yescharacteristic
54#6389C2211yescharacteristic, complemented
55C8×Dic7C2211yescharacteristic, complemented
56#6383C2211yescharacteristic
57#6383C2211yescharacteristic, complemented
58C4⋊(C7⋊C8)C2211yescharacteristic, complemented
59#3222—71nomaximal, Sylow
60C23×D7—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×C7×Q8C2311yescharacteristic
63C2×C7×D4C2311yescharacteristic
64C2×C22⋊D7—21no—
65C2×C22⋊D7—21no—
66C2×C22⋊D7—21no—
67#1161—41no—
68#1161—41no—
69D4×D7—41no—
70D4×D7—41no—
71C2×D28—21no—
72C2×D28—21no—
73C2×D28—21no—
74C2×D28C2311yescharacteristic
75C2×D28C2311yescharacteristic
76C2×C4×D7—21no—
77C2×C4×D7C2311yescharacteristic
78C2×Dic14C2311yescharacteristic
79C7×Q16—41no—
80C7×SD16—41no—
81C2×C7×C8C2311yescharacteristic
82C2×C7×C8C2311yescharacteristic
83C7×C4⋊C4—21no—
84C7×C4⋊C4C2311yescharacteristic
85C7×C22⋊C4—21no—
86C7×C22⋊C4—21no—
87C42×C7—21no—
88C42×C7C2311yescharacteristic
89#1144—41no—
90#1143—41no—
91#1143—41no—
92#1142—41no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4—21no—
96(C2×D7)⋊C4—21no—
97(C2×D7)⋊C4D411yescharacteristic
98(C2×D7)⋊C4D411yescharacteristic
99C4⋊Dic7—21no—
100C4⋊Dic7—21no—
101C4⋊Dic7C2311yescharacteristic
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104Dic7⋊C4—21no—
105Dic7⋊C4—21no—
106Dic7⋊C4—21no—
107Dic7⋊C4D411yescharacteristic
108Dic7⋊C4D411yescharacteristic
109C4×Dic7—21no—
110C4×Dic7C2311yescharacteristic
111C2×C7⋊C8C2311yescharacteristic
112C2×C7⋊C8C2311yescharacteristic
113#1132—41no—
114#1132—41no—
115#541—71no—
116#535—71no—
117#497—71no—
118#492—71no—
119#490—71no—
120#488—71no—
121#464—71no—
122#464D711yescharacteristic, complemented
123#460—71no—
124#460—71no—
125#452—71no—
126#452—71no—
127#450—71no—
128#450—71no—
129#433—71no—
130C23×C7—21no—
131C22×D7—21no—
132C22×D7—21no—
133C22×D7—42no—
134C22×D7—81no—
135C22×D7—81no—
136C22×D7C2×D411yescharacteristic
137C7×Q8—21no—
138C7×Q8—21no—
139C7×Q8—21no—
140C7×Q8—21no—
141C7×D4—21no—
142C7×D4—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7C2×D411yescharacteristic
147C2×C4×C7C2×D411yescharacteristic
148C2×C4×C7C2411yescharacteristic
149C22⋊D7—42no—
150C22⋊D7—81no—
151C22⋊D7—81no—
152C2×Dic7—21no—
153C2×Dic7—21no—
154C2×Dic7C2×D411yescharacteristic
155C2×Dic7C2×D411yescharacteristic
156C2×Dic7C2×D411yescharacteristic
157D28—21no—
158D28—21no—
159D28—21no—
160D28—21no—
161D28—42no—
162D28—81no—
163D28—81no—
164C4×D7—21no—
165C4×D7—21no—
166C4×D7—42no—
167Dic14—21no—
168Dic14—21no—
169C7×C8—21no—
170C7×C8—21no—
171C7⋊C8—21no—
172C7⋊C8—21no—
173C2×#41—71no—
174C22×D4—71no—
175C2×Q16—71no—
176C2×Q16C2×D711yescharacteristic, complemented
177C2×SD16—71no—
178C2×SD16—71no—
179C2×SD16—71no—
180C2×SD16—71no—
181C2×SD16—71no—
182C2×SD16C2×D711yescharacteristic
183C4⋊Q8—71no—
184#127—71no—
185#127—71no—
186#126—141no—
187#126—141no—
188#125—71no—
189#125—141no—
190#125—141no—
191#124—71no—
192#124C2×D711yescharacteristic
193C22⋊Q8—71no—
194C22⋊Q8—71no—
195#121—141no—
196#121—141no—
197#120—141no—
198#120—141no—
199C4×Q8—71no—
200C4×Q8C2×D711yescharacteristic, complemented
201#117—71no—
202C4⋊C8—71no—
203C4⋊C8C2×D711yescharacteristic, complemented
204#103—71no—
205#103C2×D711yescharacteristic, complemented
206#102—71no—
207#102—71no—
208#102—71no—
209#102—71no—
210#102—71no—
211#102C2×D711yescharacteristic
212C22⋊C8—71no—
213C22⋊C8—71no—
214#97—71no—
215C4×C8—71no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7C22×D411yescharacteristic
220C2×D7—21no—
221C2×D7—21no—
222C2×D7—21no—
223C2×D7—41no—
224C2×D7—41no—
225C2×D7—41no—
226C2×D7—41no—
227C2×D7—41no—
228C2×D7—41no—
229C2×D7—84no—
230C4×C7—21no—
231C4×C7—21no—
232C4×C7—41no—
233C4×C7—41no—
234C4×C7—41no—
235C4×C7H(5,𝔽2)11yescharacteristic
236C4×C7H(5,𝔽2)11yescharacteristic
237Dic7—21no—
238Dic7—21no—
239Dic7—21no—
240Dic7—41no—
241Dic7—41no—
242C24—141no—
243#41—281no—
244#41—281no—
245C2×Q8—71no—
246C2×Q8C22×D711yescharacteristic
247C2×Q8C22×D711yescharacteristic
248C2×D4—71no—
249C2×D4—71no—
250C2×D4—141no—
251C2×D4—141no—
252C2×D4—141no—
253C2×D4—141no—
254C2×D4—141no—
255C2×D4—141no—
256C2×D4—281no—
257C2×D4—281no—
258C2×D4C22×D711yescharacteristic
259C22×C4—71no—
260C22×C4—141no—
261Q16—41no—
262Q16—281no—
263SD16—41no—
264SD16—281no—
265SD16—281no—
266SD16—281no—
267SD16—281no—
268SD16—281no—
269C2×C8—71no—
270C2×C8—71no—
271C2×C8C22×D711yescharacteristic
272C2×C8C22×D711yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—71no—
275C4⋊C4—71no—
276C4⋊C4—71no—
277C4⋊C4—141no—
278C4⋊C4—141no—
279C4⋊C4—141no—
280C4⋊C4—141no—
281C4⋊C4—141no—
282C4⋊C4—141no—
283C4⋊C4—141no—
284C4⋊C4C22×D711yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—71no—
288C22⋊C4—71no—
289C22⋊C4—141no—
290C22⋊C4—141no—
291C22⋊C4—141no—
292C22⋊C4—141no—
293C42—21no—
294C42—71no—
295C42—141no—
296C42C22×D711yescharacteristic
297C2×C7—81no—
298C2×C7#53511yescharacteristic
299C2×C7#57711yescharacteristic
300C2×C7#57711yescharacteristic
301D7—41no—
302D7—81no—
303D7—81no—
304C23—21no—
305C23—71no—
306C23—141no—
307C23—141no—
308C23—282no—
309C23—561no—
310C23—561no—
311Q8—21no—
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8—141no—
316Q8—141no—
317D4—21no—
318D4—21no—
319D4—141no—
320D4—141no—
321D4—141no—
322D4—141no—
323D4—282no—
324D4—282no—
325D4—561no—
326D4—561no—
327D4—561no—
328D4—561no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—71no—
333C2×C4—71no—
334C2×C4—71no—
335C2×C4—141no—
336C2×C4—141no—
337C2×C4—141no—
338C2×C4—141no—
339C2×C4—282no—
340C2×C4D4×D711yescharacteristic
341C2×C4D4×D711yescharacteristic
342C2×C4C23×D711yescharacteristic
343C8—21no—
344C8—21no—
345C8—141no—
346C8—141no—
347C7#322211yescharacteristic, complemented, Sylow
348C22—41no—
349C22—41no—
350C22—41no—
351C22—141no—
352C22—141no—
353C22—141no—
354C22—281no—
355C22—281no—
356C22—281no—
357C22—281no—
358C22—281no—
359C22—281no—
360C22—564no—
361C22C2×D4×D711yescharacteristic
362C4—21no—
363C4—21no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—141no—
368C4—141no—
369C4—141no—
370C4—281no—
371C4—281no—
372C4#654811yescharacteristic
373C4#655311yescharacteristic
374C2—81no—
375C2—281no—
376C2—561no—
377C2—561no—
378C2#9029811yescharacteristic
379C2#9046111yescharacteristic
380C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#464C2
socleC22×C7C22×D4
prime radical#90365C2
hypercenter#464D7
hypocenterC7#3222
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. #90274C2semidirect
3. C7×#464C2semidirect
4. #89962C2semidirect
5. #89959C2semidirect
6. #89947C2semidirect
7. #89941C2semidirect
8. #89826C2semidirect
9. #89799C2semidirect
10. #89620C2semidirect
11. #89612C2semidirect
12. #89582C2semidirect
13. #89572C2semidirect
14. #89548C2semidirect
15. #89536C2semidirect
18. C2×C7×Q16C22semidirect
21. C4×C7×Q8C22semidirect
24. #6507C22semidirect
25. C2×#1144C22semidirect
27. C2×#1143C22semidirect
31. C2×#1132C22semidirect
34. #6453C22semidirect
44. C4×Dic14C22semidirect
45. C7×C4⋊C8C22semidirect
46. C7×#103C22semidirect
48. #6409C22semidirect
51. #6395C22semidirect
54. #6389C22semidirect
55. C8×Dic7C22semidirect
57. #6383C22semidirect
58. C4⋊(C7⋊C8)C22semidirect
122. #464D7semidirect
176. C2×Q16C2×D7semidirect
200. C4×Q8C2×D7semidirect
203. C4⋊C8C2×D7semidirect
205. #103C2×D7semidirect
347. C7#3222semidirect

Maximal subgroups

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

subgroupindex
#903652
#902742
C7×#4642
#899622
#899592
#899472
#899412
#898262
#897992
#896202
#896122
#895822
#895722
#895482
#895362
#32227