Finite Group Databasebeta

← #21433405

Subgroups — #21433405

Order 7552.

764 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14273772C211yesmaximal, characteristic, complemented
2C2×#14273753C211yesmaximal, characteristic, complemented
3C2×#14273749C211yesmaximal, characteristic, complemented
4#16837983C211yesmaximal, characteristic, complemented
5#16837565C212yesmaximal, complemented
6#16837554C212yesmaximal, complemented
7#16837548C212yesmaximal, complemented
8#16837537C212yesmaximal, complemented
9#16837401C211yesmaximal, characteristic, complemented
10#16837368C211yesmaximal, characteristic, complemented
11C2×#102×C59C211yesmaximal, characteristic, complemented
12#14273772—24no—
13#14273772—24no—
14C2×#11745685C2211yescharacteristic
15C2×#11745685C2211yescharacteristic
16C2×#11745685C2211yescharacteristic, complemented
17C2×#11745687—21no—
18C2×#11745687C2212yes—
19C22×C22⋊D59—22no—
20C2×D4×D59—21no—
21C2×D4×D59C2212yes—
22#14273753—24no—
23#14273749—24no—
24C2×Dic236C2211yescharacteristic, complemented
25C2×#11745660C2211yescharacteristic, complemented
26C2×#11745660C2211yescharacteristic, complemented
27C2×#11745670C2212yescomplemented
28C2×D472C2211yescharacteristic, complemented
29C2×#11745669C2212yescomplemented
30C2×#11745664C2211yescharacteristic, complemented
31#14273707—21no—
32#14273707C2212yescomplemented
33#14273702—21no—
34#14273702C2212yescomplemented
35#14273701—22no—
36#14273650C2212yescomplemented
37#14273647C2212yescomplemented
38(C2×D59)⋊C8C2212yescomplemented
39(C2×D59)⋊C8C2212yescomplemented
40C22×D4×C59C2211yescharacteristic
41C22×C8×C59C2211yescharacteristic, complemented
42C2×C4⋊C4×C59C2211yescharacteristic, complemented
43#102×C59C2212yescomplemented
44#102×C59C2212yescomplemented
45C24×C59—21no—
46C23×D59—22no—
47C2×D4×C59—21no—
48C2×D4×C59—22no—
49C2×D4×C59D414yescomplemented
50C2×D4×C59C2312yes—
51C22×C4×C59—21no—
52C22×C4×C59C2311yescharacteristic
53C2×C22⋊D59—216no—
54C2×C22⋊D59—21no—
55C2×C22⋊D59—22no—
56C2×C22⋊D59—21no—
57C2×C22⋊D59—22no—
58C2×C22⋊D59D412yes—
59C22×Dic59—22no—
60#11745687—28no—
61#11745687—24no—
62#11745687—24no—
63D4×D59—28no—
64D4×D59—24no—
65D4×D59—24no—
66#11745685—22no—
67#11745685—22no—
68#11745685—22no—
69#11745685—22no—
70#11745685—22no—
71#11745685D414yescomplemented
72C2×D236—22no—
73C2×D236D411yescharacteristic
74C2×D236C2311yescharacteristic
75C2×D236C2311yescharacteristic
76C2×C4×D59—21no—
77C2×C4×D59—21no—
78C2×C4×D59C2312yes—
79C2×Dic118D411yescharacteristic
80C2×Dic118C2311yescharacteristic
81C2×Dic118C2311yescharacteristic
82C2×C8×C59—22no—
83C2×C8×C59C2311yescharacteristic
84C2×C8×C59C2311yescharacteristic
85C4⋊C4×C59—21no—
86C4⋊C4×C59C2312yes—
87#11745670—24no—
88#11745669—24no—
89(C2×D59)⋊C4—22no—
90(C2×D59)⋊C4—22no—
91C4⋊Dic59—22no—
92#11745664—22no—
93C2×C59⋊C8C2312yes—
94Dic236—22no—
95D472—22no—
96#11745660—22no—
97#11745660—22no—
98C23×C59—22no—
99C23×C59—24no—
100C23×C59—81no—
101C23×C59—81no—
102C23×C59C2×D411yescharacteristic
103C22×D59—21no—
104C22×D59—28no—
105C22×D59—21no—
106C22×D59—82no—
107C22×D59—82no—
108C22×D59C2×D412yes—
109D4×C59—24no—
110D4×C59—22no—
111D4×C59C2×D414yes—
112C2×C4×C59—22no—
113C2×C4×C59—81no—
114C2×C4×C59C2×D412yes—
115C2×C4×C59C2×D411yescharacteristic
116C2×C4×C59C2×D412yes—
117C2×C4×C59C2411yescharacteristic
118C22⋊D59—28no—
119C22⋊D59—28no—
120C22⋊D59—24no—
121C22⋊D59—24no—
122C22⋊D59—42no—
123C2×Dic59—28no—
124C2×Dic59—21no—
125C2×Dic59—21no—
126C2×Dic59C2×D412yes—
127D236—22no—
128D236—21no—
129D236—21no—
130D236—21no—
131D236—82no—
132D236C2×D412yes—
133C4×D59—24no—
134C4×D59—22no—
135C4×D59—22no—
136C4×D59—42no—
137Dic118—22no—
138Dic118—21no—
139Dic118—21no—
140Dic118—21no—
141Dic118C2×D412yes—
142C8×C59—21no—
143C8×C59—21no—
144C59⋊C8—22no—
145C22×C59—21no—
146C22×C59—21no—
147C22×C59—24no—
148C22×C59—42no—
149C22×C59—42no—
150C22×C59—84no—
151C22×C59#12012yes—
152C22×C59C22×D411yescharacteristic
153C2×D59—22no—
154C2×D59—22no—
155C2×D59—22no—
156C2×D59—22no—
157C2×D59—41no—
158C2×D59—41no—
159C2×D59—41no—
160C2×D59—41no—
161C2×D59—41no—
162C2×D59—88no—
163C4×C59—42no—
164C4×C59#12012yes—
165C4×C59C22×D411yescharacteristic
166C4×C59C22×D411yescharacteristic
167Dic59—22no—
168Dic59—22no—
169Dic59—41no—
170#3012—591nomaximal, Sylow
171C2×C59—22no—
172C2×C59—44no—
173C2×C59C2×#12011yescharacteristic
174C2×C59#57611yescharacteristic
175C2×C59#57711yescharacteristic
176D59—42no—
177D59—42no—
178D59—81no—
179C2×H(5,𝔽2)—591no—
180C2×AGL(1,ℤ/8)—591no—
181C2×#135—591no—
182#537—591no—
183#452—592no—
184#450—592no—
185#449—592no—
186#447—592no—
187#417—591no—
188C2×#102D5911yescharacteristic, complemented
189#408—591no—
190C59#301211yescharacteristic, complemented, Sylow
191H(5,𝔽2)—1184no—
192H(5,𝔽2)—1184no—
193C2×#41—591no—
194C2×#41—592no—
195C2×#41—591no—
196C2×#41—591no—
197C2×#41—1181no—
198C22×D4—592no—
199C22×D4—1181no—
200C22×D4—1182no—
201C22×D4C2×D5911yescharacteristic
202AGL(1,ℤ/8)—1184no—
203#135—1184no—
204C2×Q16—591no—
205C2×SD16—592no—
206C2×SD16—591no—
207C2×SD16—591no—
208C2×D8—592no—
209C2×D8—591no—
210C2×M16—591no—
211C22×C8C2×D5911yescharacteristic, complemented
212C22⋊Q8—592no—
213C22⋊Q8—1181no—
214#121—592no—
215#121—1181no—
216#121—1182no—
217C2×C4⋊C4C2×D5911yescharacteristic, complemented
218#103—592no—
219#102—592no—
220#102C2×D5912yescomplemented
221#102C2×D5912yescomplemented
222C22⋊C8—592no—
223C22⋊C8—592no—
224C24—21no—
225C24—1182no—
226#41—594no—
227#41—1184no—
228#41—1188no—
229#41—1184no—
230#41—1182no—
231#41—1182no—
232#41—1182no—
233#41—1182no—
234#41—1182no—
235C2×Q8—591no—
236C2×Q8—591no—
237C2×Q8—591no—
238C2×D4—21no—
239C2×D4—22no—
240C2×D4—592no—
241C2×D4—591no—
242C2×D4—591no—
243C2×D4—591no—
244C2×D4—1182no—
245C2×D4—1182no—
246C2×D4—1184no—
247C2×D4—1188no—
248C2×D4—1182no—
249C2×D4—11816no—
250C2×D4—1181no—
251C2×D4—1184no—
252C2×D4—1181no—
253C2×D4D23614yescomplemented
254C2×D4C22×D5912yes—
255C22×C4—21no—
256C22×C4—592no—
257C22×C4—1182no—
258C22×C4—1181no—
259C22×C4—1181no—
260C22×C4C22×D5911yescharacteristic
261Q16—1182no—
262SD16—1182no—
263SD16—1182no—
264SD16—1184no—
265D8—1182no—
266D8—1184no—
267M16—1182no—
268C2×C8—22no—
269C2×C8—592no—
270C2×C8C22×D5911yescharacteristic
271C2×C8C22×D5911yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—1182no—
274C4⋊C4C22×D5912yes—
275C22⋊C4—1182no—
276C22⋊C4—1182no—
277C23—22no—
278C23—24no—
279C23—81no—
280C23—81no—
281C23—592no—
282C23—1188no—
283C23—1181no—
284C23—1181no—
285C23—4722no—
286C23—4722no—
287C23D4×D5911yescharacteristic
288Q8—592no—
289Q8—1182no—
290Q8—1181no—
291Q8—1181no—
292Q8—1181no—
293D4—24no—
294D4—22no—
295D4—592no—
296D4—1188no—
297D4—1188no—
298D4—1184no—
299D4—1184no—
300D4—1182no—
301D4—1181no—
302D4—1181no—
303D4—1181no—
304D4—4722no—
305D4—2362no—
306D4C2×D23614yes—
307C2×C4—22no—
308C2×C4—81no—
309C2×C4—592no—
310C2×C4—1188no—
311C2×C4—1181no—
312C2×C4—1184no—
313C2×C4—1182no—
314C2×C4—1182no—
315C2×C4—1181no—
316C2×C4—2362no—
317C2×C4C2×D23612yes—
318C2×C4D4×D5911yescharacteristic
319C2×C4D4×D5912yes—
320C2×C4C23×D5911yescharacteristic
321C8—21no—
322C8—21no—
323C8—1182no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—84no—
330C22—1182no—
331C22—1182no—
332C22—1182no—
333C22—1182no—
334C22—4728no—
335C22—2361no—
336C22—2361no—
337C22—2361no—
338C22—2361no—
339C22—2361no—
340C22#1427369712yes—
341C22C2×D4×D5911yescharacteristic
342C4—42no—
343C4—1182no—
344C4—1182no—
345C4—2361no—
346C4#1427369712yes—
347C4C2×D4×D5911yescharacteristic
348C4C22×D23611yescharacteristic
349C2—22no—
350C2—44no—
351C2—4721no—
352C2—2362no—
353C2—2362no—
354C2C2×#1427369711yescharacteristic
355C2#1683820311yescharacteristic
356C2#1683820811yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC2×#102×C59C2
socleC22×C59C22×D4
hypercenterC2×#102D59
hypocenterC59#3012
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#14273772C2semidirect
2. C2×#14273753C2semidirect
3. C2×#14273749C2semidirect
4. #16837983C2semidirect
5. #16837565C2semidirect
6. #16837554C2semidirect
7. #16837548C2semidirect
8. #16837537C2semidirect
9. #16837401C2semidirect
10. #16837368C2semidirect
11. C2×#102×C59C2semidirect
16. C2×#11745685C22semidirect
24. C2×Dic236C22semidirect
25. C2×#11745660C22semidirect
26. C2×#11745660C22semidirect
27. C2×#11745670C22semidirect
28. C2×D472C22semidirect
29. C2×#11745669C22semidirect
30. C2×#11745664C22semidirect
32. #14273707C22semidirect
34. #14273702C22semidirect
36. #14273650C22semidirect
37. #14273647C22semidirect
38. (C2×D59)⋊C8C22semidirect
39. (C2×D59)⋊C8C22semidirect
41. C22×C8×C59C22semidirect
42. C2×C4⋊C4×C59C22semidirect
43. #102×C59C22semidirect
44. #102×C59C22semidirect
49. C2×D4×C59D4semidirect
71. #11745685D4semidirect
188. C2×#102D59semidirect
190. C59#3012semidirect
211. C22×C8C2×D59semidirect
217. C2×C4⋊C4C2×D59semidirect
220. #102C2×D59semidirect
221. #102C2×D59semidirect
253. C2×D4D236semidirect