Finite Group Databasebeta

← #21433709

Subgroups — #21433709

Order 7552.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 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×M16×D59C211yesmaximal, characteristic, complemented
2#16837909C211yesmaximal, characteristic, complemented
3C2×#14273698C211yesmaximal, characteristic, complemented
4#16837652C212yesmaximal, complemented
5#16837645C212yesmaximal, complemented
6#16837635C212yesmaximal, complemented
7#16837628C212yesmaximal, complemented
8#16837444C211yesmaximal, characteristic, complemented
9#16837407C211yesmaximal, characteristic, complemented
10#16837403C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C59C211yesmaximal, characteristic, complemented
12C22×C22⋊D59—21no—
13C22×C4×D59C2211yescharacteristic
14C2×#11745664C2211yescharacteristic
15C2×#11745659C2212yescomplemented
16M16×D59—24no—
17C2×C8×D59C2212yescomplemented
18#14273708—21no—
19#14273703C2211yescharacteristic, complemented
20#14273703C2211yescharacteristic, complemented
21#14273698—21no—
22#14273698—21no—
23#14273698C2212yes—
24#14273698C2212yes—
25#14273698C2211yescharacteristic, complemented
26#14273698C2211yescharacteristic, complemented
27#14273695—21no—
28C4×C22⋊D59—21no—
29C4×C22⋊D59—21no—
30#14273686C2211yescharacteristic
31C2×C4⋊Dic59C2211yescharacteristic
32C2×C22⋊Dic59—21no—
33#14273661C2212yescomplemented
34#14273659C2212yescomplemented
35#14273651C2212yescomplemented
36#14273648C2212yescomplemented
37#14273648C2212yescomplemented
38#14273648C2212yescomplemented
39C2×#41×C59C2211yescharacteristic
40C22×D4×C59C2211yescharacteristic
41AGL(1,ℤ/8)×C59—24no—
42C2×SD16×C59C2212yescomplemented
43C2×D8×C59C2212yescomplemented
44C2×M16×C59C2211yescharacteristic
45C24×C59—21no—
46C23×D59D411yescharacteristic
47#41×C59—22no—
48#41×C59—22no—
49C2×Q8×C59C2311yescharacteristic
50C2×D4×C59—21no—
51C2×D4×C59—22no—
52C2×D4×C59—21no—
53C2×D4×C59—22no—
54C2×D4×C59C2311yescharacteristic
55C2×D4×C59C2312yes—
56C22×C4×C59—21no—
57C22×C4×C59C2311yescharacteristic
58C2×C22⋊D59—21no—
59C2×C22⋊D59—21no—
60C2×C22⋊D59—22no—
61C2×C22⋊D59—22no—
62C2×C22⋊D59—44no—
63C22×Dic59—21no—
64C22×Dic59D411yescharacteristic
65C2×C4×D59—22no—
66C2×C4×D59—22no—
67C2×C4×D59D411yescharacteristic
68C2×C4×D59D411yescharacteristic
69C2×C4×D59C2311yescharacteristic
70C2×C4×D59C2311yescharacteristic
71SD16×C59—24no—
72D8×C59—24no—
73M16×C59D414yescomplemented
74C2×C8×C59C2312yes—
75C22⋊Dic59—21no—
76C22⋊Dic59—22no—
77C22⋊Dic59—22no—
78C22⋊Dic59—21no—
79C22⋊Dic59—21no—
80(C2×D59)⋊C4—21no—
81(C2×D59)⋊C4—21no—
82C4⋊Dic59—21no—
83C4⋊Dic59C2311yescharacteristic
84C4⋊Dic59C2311yescharacteristic
85C4⋊Dic59C2312yes—
86Dic59⋊C4—21no—
87Dic59⋊C4—21no—
88C4×Dic59—21no—
89#11745664—22no—
90C2×C59⋊C8C2312yes—
91#11745659—24no—
92C8×D59—24no—
93C23×C59—21no—
94C23×C59—22no—
95C23×C59—42no—
96C23×C59—42no—
97C23×C59—42no—
98C23×C59C2×D411yescharacteristic
99C22×D59—22no—
100C22×D59—22no—
101C22×D59—41no—
102C22×D59C2×D411yescharacteristic
103C22×D59C2×D411yescharacteristic
104Q8×C59—21no—
105Q8×C59—21no—
106D4×C59—21no—
107D4×C59—22no—
108D4×C59—22no—
109D4×C59—21no—
110D4×C59—42no—
111D4×C59—42no—
112C2×C4×C59—21no—
113C2×C4×C59—42no—
114C2×C4×C59C2×D411yescharacteristic
115C2×C4×C59C2×D412yes—
116C2×C4×C59#4112yes—
117C2×C4×C59C2411yescharacteristic
118C22⋊D59—42no—
119C22⋊D59—44no—
120C22⋊D59—42no—
121C22⋊D59—44no—
122C2×Dic59—22no—
123C2×Dic59—21no—
124C2×Dic59—21no—
125C2×Dic59—22no—
126C2×Dic59—42no—
127C2×Dic59C2×D411yescharacteristic
128C2×Dic59C2×D411yescharacteristic
129C4×D59—22no—
130C4×D59—22no—
131C4×D59—21no—
132C4×D59—21no—
133C4×D59—21no—
134C4×D59—21no—
135C8×C59C2×D414yescomplemented
136C59⋊C8—22no—
137C22×C59—21no—
138C22×C59—21no—
139C22×C59—41no—
140C22×C59—42no—
141C22×C59—42no—
142C22×C59—41no—
143C22×C59—41no—
144C22×C59—42no—
145C22×C59—84no—
146C22×C59#12112yes—
147C22×C59C22×D411yescharacteristic
148C2×D59—21no—
149C2×D59—21no—
150C2×D59—21no—
151C2×D59—21no—
152C2×D59—21no—
153C2×D59—44no—
154C2×D59AGL(1,ℤ/8)12yescomplemented
155C4×C59—41no—
156C4×C59#12112yes—
157C4×C59C22×D411yescharacteristic
158C4×C59C2×#4111yescharacteristic
159Dic59—21no—
160Dic59—21no—
161Dic59—42no—
162Dic59—41no—
163Dic59—41no—
164#3055—591nomaximal, Sylow
165C2×C59—22no—
166C2×C59—81no—
167C2×C59—82no—
168C2×C59C2×#12111yescharacteristic
169C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
170C2×C59#57511yescharacteristic
171D59—22no—
172D59—41no—
173C2×AGL(1,ℤ/8)D5911yescharacteristic, complemented
174C22×M16—591no—
175#525—591no—
176C2×#121—591no—
177#469—592no—
178#468—592no—
179#466—592no—
180#465—592no—
181#428—591no—
182#418—591no—
183#417—591no—
184C59#305511yescharacteristic, complemented, Sylow
185C2×#41C2×D5911yescharacteristic
186C22×D4—1181no—
187C22×D4C2×D5911yescharacteristic
188C23×C4—591no—
189AGL(1,ℤ/8)—24no—
190C2×SD16C2×D5912yescomplemented
191C2×D8C2×D5912yescomplemented
192C2×M16—591no—
193C2×M16—592no—
194C2×M16—1184no—
195C2×M16C2×D5911yescharacteristic
196C22×C8—592no—
197#123—1181no—
198C22⋊Q8—591no—
199C22⋊Q8—591no—
200#121—592no—
201#121—592no—
202#121—591no—
203#121—591no—
204#121—1181no—
205#121—1181no—
206#120—1181no—
207C4×D4—1181no—
208C4×D4—1181no—
209#117—591no—
210C2×C4⋊C4—591no—
211C2×C22⋊C4—1181no—
212#107—592no—
213#106—592no—
214#103—592no—
215#102—592no—
216#102—592no—
217#102—592no—
218C24—21no—
219C24—591no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D5911yescharacteristic
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—1181no—
228C2×D4—1181no—
229C2×D4—1182no—
230C2×D4—1182no—
231C2×D4—2364no—
232C2×D4C22×D5911yescharacteristic
233C2×D4C22×D5912yes—
234C22×C4—21no—
235C22×C4—591no—
236C22×C4—591no—
237C22×C4—591no—
238C22×C4—591no—
239C22×C4—591no—
240C22×C4—1182no—
241C22×C4—1182no—
242C22×C4—1181no—
243C22×C4C22×D5911yescharacteristic
244SD16—24no—
245D8—24no—
246M16—1182no—
247M16—1184no—
248M16C22⋊D5914yescomplemented
249C2×C8—592no—
250C2×C8—1184no—
251C2×C8C22×D5912yes—
252C4⋊C4—591no—
253C4⋊C4—591no—
254C4⋊C4—592no—
255C4⋊C4—1181no—
256C4⋊C4—1181no—
257C4⋊C4—1181no—
258C22⋊C4—1181no—
259C22⋊C4—1181no—
260C22⋊C4—1182no—
261C22⋊C4—1182no—
262C22⋊C4—1181no—
263C22⋊C4—1181no—
264C22⋊C4—1181no—
265C42—1181no—
266C23—21no—
267C23—22no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—591no—
272C23—591no—
273C23—1182no—
274C23—1182no—
275C23—2361no—
276C23D4×D5911yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—22no—
281D4—22no—
282D4—21no—
283D4—42no—
284D4—42no—
285D4—2362no—
286D4—2364no—
287D4—2362no—
288D4—2364no—
289C2×C4—21no—
290C2×C4—42no—
291C2×C4—591no—
292C2×C4—591no—
293C2×C4—1182no—
294C2×C4—1182no—
295C2×C4—1182no—
296C2×C4—1181no—
297C2×C4—1181no—
298C2×C4—1181no—
299C2×C4—1181no—
300C2×C4—1181no—
301C2×C4—1181no—
302C2×C4—1182no—
303C2×C4—2362no—
304C2×C4D4×D5911yescharacteristic
305C2×C4#1174568712yes—
306C2×C4C2×C22⋊D5912yes—
307C2×C4C23×D5911yescharacteristic
308C8—1182no—
309C8C2×C22⋊D5914yescomplemented
310C22—21no—
311C22—21no—
312C22—41no—
313C22—42no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—84no—
319C22—592no—
320C22—1181no—
321C22—1181no—
322C22—1181no—
323C22—1181no—
324C22—1181no—
325C22—2364no—
326C22#1427369812yes—
327C22C2×D4×D5911yescharacteristic
328C4—41no—
329C4—1181no—
330C4—1181no—
331C4—2362no—
332C4—2361no—
333C4—2361no—
334C4#1427369812yes—
335C4C22×C22⋊D5911yescharacteristic
336C4C2×#1174568711yescharacteristic
337C2—22no—
338C2—82no—
339C2—81no—
340C2—1182no—
341C2—2361no—
342C2C2×#1427369811yescharacteristic
343C2AGL(1,ℤ/8)×D5911yescharacteristic
344C2#1683819811yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC2×AGL(1,ℤ/8)×C59C2
socleC22×C59C22×D4
hypercenterC2×AGL(1,ℤ/8)D59
hypocenterC59#3055
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×M16×D59C2semidirect
2. #16837909C2semidirect
3. C2×#14273698C2semidirect
4. #16837652C2semidirect
5. #16837645C2semidirect
6. #16837635C2semidirect
7. #16837628C2semidirect
8. #16837444C2semidirect
9. #16837407C2semidirect
10. #16837403C2semidirect
11. C2×AGL(1,ℤ/8)×C59C2semidirect
15. C2×#11745659C22semidirect
17. C2×C8×D59C22semidirect
19. #14273703C22semidirect
20. #14273703C22semidirect
25. #14273698C22semidirect
26. #14273698C22semidirect
33. #14273661C22semidirect
34. #14273659C22semidirect
35. #14273651C22semidirect
36. #14273648C22semidirect
37. #14273648C22semidirect
38. #14273648C22semidirect
42. C2×SD16×C59C22semidirect
43. C2×D8×C59C22semidirect
73. M16×C59D4semidirect
135. C8×C59C2×D4semidirect
154. C2×D59AGL(1,ℤ/8)semidirect
173. C2×AGL(1,ℤ/8)D59semidirect
184. C59#3055semidirect
190. C2×SD16C2×D59semidirect
191. C2×D8C2×D59semidirect
248. M16C22⋊D59semidirect
309. C8C2×C22⋊D59semidirect