Finite Group Databasebeta

← #21434017

Subgroups — #21434017

Order 7552.

458 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 129 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
1#16837909C211yesmaximal, characteristic
2C2×C4⋊C4×D59C211yesmaximal, characteristic, complemented
3#16837736C212yesmaximal, complemented
4#16837731C212yesmaximal, complemented
5#16837719C212yesmaximal, complemented
6#16837714C212yesmaximal, complemented
7#16837444C211yesmaximal, characteristic, complemented
8#16837442C211yesmaximal, characteristic, complemented
9#16837403C211yesmaximal, characteristic, complemented
10#16837365C211yesmaximal, characteristic, complemented
11C59×#417C211yesmaximal, characteristic, complemented
12C22×C4×D59—21no—
13C22×C4×D59C2211yescharacteristic
14C2×#11745664C2211yescharacteristic
15#14273708—21no—
16#14273703C2211yescharacteristic
17#14273703C2211yescharacteristic
18#14273698C2211yescharacteristic
19#14273698C2211yescharacteristic
20#14273695—21no—
21C4×C22⋊D59—21no—
22C4×C22⋊D59—21no—
23#14273686C2211yescharacteristic
24C2×C4⋊Dic59C2211yescharacteristic
25C2×Dic59⋊C4—21no—
26C4⋊C4×D59—21no—
27C4⋊C4×D59—21no—
28C4⋊C4×D59C2212yes—
29C4⋊C4×D59C2212yes—
30#14273661C2212yescomplemented
31#14273660C2212yescomplemented
32#14273659C2212yescomplemented
33#14273658C2212yescomplemented
34#14273651C2212yescomplemented
35#14273648C2212yescomplemented
36(C2×D59)⋊C8C2212yes—
37(C2×D59)⋊C8C2212yes—
38C2×#41×C59C2211yescharacteristic
39C2×M16×C59C2211yescharacteristic
40C2×C4⋊C4×C59C2211yescharacteristic
41#103×C59C2212yescomplemented
42#102×C59C2212yescomplemented
43C23×D59D411yescharacteristic
44#41×C59—22no—
45#41×C59—22no—
46C2×Q8×C59C2311yescharacteristic
47C2×D4×C59—21no—
48C2×D4×C59C2311yescharacteristic
49C22×C4×C59—21no—
50C22×C4×C59—21no—
51C22×C4×C59C2311yescharacteristic
52C2×C22⋊D59—21no—
53C2×C22⋊D59—21no—
54C22×Dic59—21no—
55C22×Dic59D411yescharacteristic
56C2×C4×D59—22no—
57C2×C4×D59—22no—
58C2×C4×D59—22no—
59C2×C4×D59—22no—
60C2×C4×D59—44no—
61C2×C4×D59D411yescharacteristic
62C2×C4×D59D411yescharacteristic
63C2×C4×D59C2311yescharacteristic
64C2×C4×D59C2311yescharacteristic
65M16×C59—22no—
66C2×C8×C59C2312yes—
67C4⋊C4×C59—21no—
68C4⋊C4×C59C2312yes—
69C22⋊Dic59—21no—
70C22⋊Dic59—21no—
71C22⋊Dic59—21no—
72(C2×D59)⋊C4—21no—
73(C2×D59)⋊C4—21no—
74C4⋊Dic59—21no—
75C4⋊Dic59C2312yes—
76C4⋊Dic59C2311yescharacteristic
77C4⋊Dic59C2311yescharacteristic
78Dic59⋊C4—22no—
79Dic59⋊C4—22no—
80Dic59⋊C4—21no—
81Dic59⋊C4—21no—
82C4×Dic59—21no—
83#11745664—22no—
84C2×C59⋊C8C2312yes—
85C23×C59—21no—
86C23×C59C2×D411yescharacteristic
87C22×D59—22no—
88C22×D59—22no—
89C22×D59—22no—
90C22×D59C2×D411yescharacteristic
91C22×D59C2×D411yescharacteristic
92Q8×C59—21no—
93Q8×C59—21no—
94D4×C59—21no—
95D4×C59—21no—
96D4×C59—42no—
97C2×C4×C59—21no—
98C2×C4×C59—22no—
99C2×C4×C59—42no—
100C2×C4×C59—42no—
101C2×C4×C59C2×D411yescharacteristic
102C2×C4×C59#4112yes—
103C2×C4×C59#4112yes—
104C2×C4×C59C2411yescharacteristic
105C22⋊D59—42no—
106C22⋊D59—42no—
107C2×Dic59—22no—
108C2×Dic59—21no—
109C2×Dic59—21no—
110C2×Dic59—22no—
111C2×Dic59—42no—
112C2×Dic59C2×D411yescharacteristic
113C2×Dic59C2×D411yescharacteristic
114C4×D59—22no—
115C4×D59—22no—
116C4×D59—21no—
117C4×D59—21no—
118C4×D59—21no—
119C4×D59—21no—
120C4×D59—44no—
121C4×D59—44no—
122C8×C59—22no—
123C59⋊C8—22no—
124C22×C59—21no—
125C22×C59—21no—
126C22×C59—41no—
127C22×C59—41no—
128C22×C59—41no—
129C22×C59#12312yes—
130C22×C59C22×D411yescharacteristic
131C2×D59—28no—
132C2×D59—21no—
133C2×D59—21no—
134C2×D59—21no—
135C2×D59—21no—
136C2×D59AGL(1,ℤ/8)12yes—
137C2×D59#13712yes—
138C4×C59—41no—
139C4×C59—42no—
140C4×C59#12312yes—
141C4×C59C2×#4111yescharacteristic
142C4×C59C2×#4111yescharacteristic
143Dic59—21no—
144Dic59—21no—
145Dic59—42no—
146Dic59—41no—
147Dic59—41no—
148#3092—591nomaximal, Sylow
149C2×C59—22no—
150C2×C59—81no—
151C2×C59C2×#12311yescharacteristic
152C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
153C2×C59C2×#13711yescharacteristic
154D59—22no—
155D59—22no—
156#525—591no—
157C22×C4⋊C4—591no—
158#484—592no—
159#483—592no—
160#481—592no—
161#480—592no—
162#428—591no—
163#428—591no—
164#417—591no—
165#417D5911yescharacteristic, complemented
166#407—591no—
167C59#309211yescharacteristic, complemented, Sylow
168C2×#41C2×D5911yescharacteristic
169C23×C4—591no—
170C23×C4—1181no—
171C2×M16—591no—
172C2×M16C2×D5911yescharacteristic
173#123—1181no—
174C22⋊Q8—591no—
175C22⋊Q8—591no—
176#121—591no—
177#121—591no—
178#120—1181no—
179C4×D4—1181no—
180C4×D4—1181no—
181#117—591no—
182C2×C4⋊C4—591no—
183C2×C4⋊C4—592no—
184C2×C4⋊C4—592no—
185C2×C4⋊C4—1181no—
186C2×C4⋊C4—1181no—
187C2×C4⋊C4—1181no—
188C2×C4⋊C4C2×D5911yescharacteristic
189#107—592no—
190#107—592no—
191#106—592no—
192#106—592no—
193#103—592no—
194#103C2×D5912yescomplemented
195#102—592no—
196#102C2×D5912yescomplemented
197C22⋊C8—592no—
198C22⋊C8—592no—
199C24—591no—
200#41—22no—
201#41—22no—
202C2×Q8C22×D5911yescharacteristic
203C2×D4—21no—
204C2×D4—1181no—
205C2×D4—1181no—
206C2×D4C22×D5911yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—591no—
210C22×C4—591no—
211C22×C4—591no—
212C22×C4—591no—
213C22×C4—591no—
214C22×C4—1182no—
215C22×C4—1182no—
216C22×C4—1182no—
217C22×C4—1182no—
218C22×C4—1181no—
219C22×C4—2364no—
220C22×C4C22×D5911yescharacteristic
221M16—22no—
222M16—1182no—
223C2×C8—592no—
224C2×C8C22×D5912yes—
225C4⋊C4—21no—
226C4⋊C4—592no—
227C4⋊C4—591no—
228C4⋊C4—591no—
229C4⋊C4—1182no—
230C4⋊C4—1182no—
231C4⋊C4—1181no—
232C4⋊C4—1181no—
233C4⋊C4—1181no—
234C4⋊C4C22×D5912yes—
235C22⋊C4—1181no—
236C22⋊C4—1181no—
237C22⋊C4—1181no—
238C22⋊C4—1181no—
239C22⋊C4—1181no—
240C42—1181no—
241C23—21no—
242C23—591no—
243C23—591no—
244C23—1182no—
245C23—1182no—
246C23—1182no—
247C23D4×D5911yescharacteristic
248Q8—21no—
249Q8—21no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—2362no—
254D4—2362no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—591no—
260C2×C4—591no—
261C2×C4—1182no—
262C2×C4—1182no—
263C2×C4—1182no—
264C2×C4—1181no—
265C2×C4—1181no—
266C2×C4—1181no—
267C2×C4—1181no—
268C2×C4—1181no—
269C2×C4—1181no—
270C2×C4—1182no—
271C2×C4—2364no—
272C2×C4—2364no—
273C2×C4—2362no—
274C2×C4#1174568512yes—
275C2×C4D4×D5911yescharacteristic
276C2×C4#1174568712yes—
277C2×C4C23×D5911yescharacteristic
278C8—22no—
279C8—1182no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—592no—
286C22—592no—
287C22—1188no—
288C22—1181no—
289C22—1181no—
290C22—1181no—
291C22—1181no—
292C22#1427370912yes—
293C22C2×D4×D5911yescharacteristic
294C4—41no—
295C4—42no—
296C4—1181no—
297C4—1181no—
298C4—2362no—
299C4—2361no—
300C4—2361no—
301C4#1427370912yes—
302C4C2×#1174568711yescharacteristic
303C4C2×#1174568511yescharacteristic
304C2—22no—
305C2—81no—
306C2—1182no—
307C2—1182no—
308C2C2×#1427370911yescharacteristic
309C2AGL(1,ℤ/8)×D5911yescharacteristic
310C2#137×D5911yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#417C2
socleC22×C59C22×D4
hypercenter#417D59
hypocenterC59#3092
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D59C2semidirect
3. #16837736C2semidirect
4. #16837731C2semidirect
5. #16837719C2semidirect
6. #16837714C2semidirect
7. #16837444C2semidirect
8. #16837442C2semidirect
9. #16837403C2semidirect
10. #16837365C2semidirect
11. C59×#417C2semidirect
30. #14273661C22semidirect
31. #14273660C22semidirect
32. #14273659C22semidirect
33. #14273658C22semidirect
34. #14273651C22semidirect
35. #14273648C22semidirect
41. #103×C59C22semidirect
42. #102×C59C22semidirect
165. #417D59semidirect
167. C59#3092semidirect
194. #103C2×D59semidirect
196. #102C2×D59semidirect