Finite Group Databasebeta

← #21433376

Subgroups — #21433376

Order 7552.

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

normal subgroups only

This group has 752 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×#14273773C211yesmaximal, characteristic
2C2×#14273757C211yesmaximal, characteristic, complemented
3C22×#11745660C211yesmaximal, characteristic, complemented
4#16837978C211yesmaximal, characteristic, complemented
5#16837567C214yesmaximal, complemented
6#16837554C214yesmaximal, complemented
7#16837406C211yesmaximal, characteristic, complemented
8#16837368C211yesmaximal, characteristic, complemented
9C59×#416C211yesmaximal, characteristic, complemented
10#14273773—24no—
11#14273773—24no—
12C2×#11745689C2212yes—
13C2×#11745685—21no—
14C2×#11745685C2211yescharacteristic
15C22×Dic118C2211yescharacteristic, complemented
16C22×D236—21no—
17C22×D236C2211yescharacteristic
18C2×D4×D59—22no—
19C2×D4×D59C2212yes—
20#14273757—24no—
21C2×#11745672C2212yescomplemented
22C2×#11745660—24no—
23C2×#11745660C2214yescomplemented
24C2×#11745670C2212yescomplemented
25C2×#11745664C2211yescharacteristic, complemented
26#14273727—21no—
27#14273715—21no—
28#14273711—22no—
29#14273707C2214yescomplemented
30#14273647C2214yescomplemented
31(C2×D59)⋊C8C2214yescomplemented
32C2×#41×C59C2211yescharacteristic
33C22×C8×C59C2211yescharacteristic, complemented
34#117×C59C2211yescharacteristic, complemented
35#103×C59C2212yescomplemented
36#102×C59C2212yescomplemented
37C23×D59—21no—
38C23×D59—22no—
39#41×C59—22no—
40#41×C59D414yes—
41C2×Q8×C59C2311yescharacteristic
42C2×D4×C59—21no—
43C2×D4×C59C2311yescharacteristic
44C22×C4×C59—21no—
45C22×C4×C59C2311yescharacteristic
46C2×C22⋊D59—22no—
47C2×C22⋊D59D412yes—
48C22×Dic59—21no—
49#11745689—24no—
50#11745689—24no—
51D4×D59—216no—
52D4×D59—24no—
53D4×D59—24no—
54#11745685—28no—
55#11745685—22no—
56#11745685—22no—
57C2×D236—28no—
58C2×D236—22no—
59C2×D236—21no—
60C2×D236—22no—
61C2×D236D411yescharacteristic
62C2×D236D414yes—
63C2×D236C2312yes—
64C2×C4×D59—22no—
65C2×C4×D59C2312yes—
66C2×Dic118—22no—
67C2×Dic118—22no—
68C2×Dic118—21no—
69C2×Dic118D411yescharacteristic
70C2×Dic118C2312yes—
71C2×C8×C59—22no—
72C2×C8×C59C2312yes—
73C4⋊C4×C59C2312yes—
74C22⋊C4×C59—21no—
75C42×C59—21no—
76#11745672—24no—
77#11745670—24no—
78(C2×D59)⋊C4—24no—
79C4⋊Dic59—24no—
80#11745664—22no—
81C2×C59⋊C8C2312yes—
82#11745660—28no—
83C23×C59—21no—
84C23×C59C2×D411yescharacteristic
85C22×D59—22no—
86C22×D59—28no—
87C22×D59—24no—
88C22×D59—84no—
89C22×D59—82no—
90C22×D59C2×D412yes—
91Q8×C59—21no—
92Q8×C59C2×D412yes—
93D4×C59—24no—
94D4×C59—21no—
95D4×C59C2×D412yes—
96C2×C4×C59—21no—
97C2×C4×C59—24no—
98C2×C4×C59—22no—
99C2×C4×C59C2×D412yes—
100C2×C4×C59C2×D411yescharacteristic
101C2×C4×C59C2×D412yes—
102C2×C4×C59C2411yescharacteristic
103C22⋊D59—28no—
104C22⋊D59—24no—
105C2×Dic59—22no—
106C2×Dic59—42no—
107C2×Dic59C2×D412yes—
108D236—28no—
109D236—22no—
110D236—24no—
111D236—22no—
112D236C2×D414yes—
113C4×D59—28no—
114C4×D59—22no—
115C4×D59—22no—
116Dic118—22no—
117Dic118—22no—
118Dic118—22no—
119Dic118—42no—
120C8×C59—22no—
121C59⋊C8—22no—
122C22×C59—22no—
123C22×C59—22no—
124C22×C59—42no—
125C22×C59#12012yes—
126C22×C59C22×D411yescharacteristic
127C2×D59—22no—
128C2×D59—24no—
129C2×D59—24no—
130C2×D59—44no—
131C2×D59—84no—
132C2×D59—88no—
133C4×C59—22no—
134C4×C59—42no—
135C4×C59#12012yes—
136C4×C59C22×D411yescharacteristic
137C4×C59C22×D411yescharacteristic
138Dic59—22no—
139Dic59—42no—
140#3009—591nomaximal, Sylow
141C2×C59—22no—
142C2×C59—42no—
143C2×C59C2×#12011yescharacteristic
144C2×C59#57712yes—
145D59—42no—
146D59—44no—
147C2×H(5,𝔽2)—591no—
148C2×#137—591no—
149C22×SD16—591no—
150#536—591no—
151#452—594no—
152#450—594no—
153#418—591no—
154#416D5911yescharacteristic, complemented
155#408—591no—
156C59#300911yescharacteristic, complemented, Sylow
157H(5,𝔽2)—1184no—
158H(5,𝔽2)—1184no—
159C2×#41—591no—
160C2×#41—592no—
161C2×#41—1181no—
162C2×#41C2×D5911yescharacteristic
163C22×Q8—591no—
164C22×D4—592no—
165C22×D4—591no—
166C22×D4—1181no—
167C22×D4—1182no—
168#137—1184no—
169C2×Q16—592no—
170C2×SD16—592no—
171C2×SD16—594no—
172C2×SD16—1184no—
173C2×M16—591no—
174C22×C8C2×D5911yescharacteristic, complemented
175C4⋊Q8—1181no—
176#124—1181no—
177#123—1182no—
178C22⋊Q8—594no—
179#117C2×D5911yescharacteristic, complemented
180#103C2×D5912yescomplemented
181#102—594no—
182#102C2×D5912yescomplemented
183C22⋊C8—594no—
184C24—1182no—
185C24—1181no—
186#41—22no—
187#41—1188no—
188#41—1184no—
189#41—1184no—
190#41—1182no—
191#41—1182no—
192#41D23614yes—
193C2×Q8—591no—
194C2×Q8—592no—
195C2×Q8—1181no—
196C2×Q8—1182no—
197C2×Q8—1182no—
198C2×Q8C22×D5911yescharacteristic
199C2×D4—21no—
200C2×D4—592no—
201C2×D4—594no—
202C2×D4—591no—
203C2×D4—592no—
204C2×D4—1182no—
205C2×D4—1188no—
206C2×D4—1182no—
207C2×D4—1184no—
208C2×D4—11816no—
209C2×D4—1184no—
210C2×D4—1182no—
211C2×D4—1181no—
212C2×D4C22×D5911yescharacteristic
213C22×C4—21no—
214C22×C4—592no—
215C22×C4—1182no—
216C22×C4—1181no—
217C22×C4C22×D5911yescharacteristic
218Q16—1184no—
219SD16—1188no—
220SD16—1184no—
221M16—1182no—
222C2×C8—22no—
223C2×C8—592no—
224C2×C8C22×D5912yes—
225C4⋊C4—1184no—
226C4⋊C4C22×D5912yes—
227C22⋊C4—21no—
228C22⋊C4—1184no—
229C42—21no—
230C23—21no—
231C23—592no—
232C23—1188no—
233C23—1184no—
234C23—1182no—
235C23—4722no—
236C23—4724no—
237C23D4×D5911yescharacteristic
238Q8—21no—
239Q8—1182no—
240Q8—1182no—
241Q8—1182no—
242Q8—2362no—
243Q8C2×D23612yes—
244D4—24no—
245D4—21no—
246D4—594no—
247D4—1188no—
248D4—1188no—
249D4—1184no—
250D4—1184no—
251D4—1182no—
252D4—1182no—
253D4C2×D23612yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—592no—
258C2×C4—1188no—
259C2×C4—1182no—
260C2×C4—1182no—
261C2×C4—1182no—
262C2×C4—2362no—
263C2×C4C2×D23612yes—
264C2×C4D4×D5911yescharacteristic
265C2×C4D4×D5912yes—
266C2×C4C23×D5911yescharacteristic
267C8—22no—
268C8—1182no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—1184no—
273C22—1182no—
274C22—1184no—
275C22—4724no—
276C22—4728no—
277C22—2364no—
278C22#1427369712yes—
279C22C2×D4×D5911yescharacteristic
280C4—22no—
281C4—42no—
282C4—1182no—
283C4—2362no—
284C4#1427369712yes—
285C4C2×D4×D5911yescharacteristic
286C4C22×D23611yescharacteristic
287C2—22no—
288C2—42no—
289C2—2362no—
290C2—2364no—
291C2C2×#1427369711yescharacteristic
292C2#1683820912yes—
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#416C2
socleC22×C59C22×D4
hypercenter#416D59
hypocenterC59#3009
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C2×#14273757C2semidirect
3. C22×#11745660C2semidirect
4. #16837978C2semidirect
5. #16837567C2semidirect
6. #16837554C2semidirect
7. #16837406C2semidirect
8. #16837368C2semidirect
9. C59×#416C2semidirect
15. C22×Dic118C22semidirect
21. C2×#11745672C22semidirect
23. C2×#11745660C22semidirect
24. C2×#11745670C22semidirect
25. C2×#11745664C22semidirect
29. #14273707C22semidirect
30. #14273647C22semidirect
31. (C2×D59)⋊C8C22semidirect
33. C22×C8×C59C22semidirect
34. #117×C59C22semidirect
35. #103×C59C22semidirect
36. #102×C59C22semidirect
154. #416D59semidirect
156. C59#3009semidirect
174. C22×C8C2×D59semidirect
179. #117C2×D59semidirect
180. #103C2×D59semidirect
182. #102C2×D59semidirect