Finite Group Databasebeta

← #21434030

Subgroups — #21434030

Order 7552.

414 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 125 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#16837914C211yesmaximal, characteristic
2C2×#14273685C211yesmaximal, characteristic, complemented
3#16837746C212yesmaximal, complemented
4#16837745C212yesmaximal, complemented
5#16837729C212yesmaximal, complemented
6#16837728C212yesmaximal, complemented
7#16837445C211yesmaximal, characteristic, complemented
8#16837404C211yesmaximal, characteristic, complemented
9#16837401C211yesmaximal, characteristic, complemented
10#16837365C211yesmaximal, characteristic, complemented
11C59×#428C211yesmaximal, characteristic, complemented
12C2×#11745685C2211yescharacteristic
13C22×C4×D59C2211yescharacteristic
14C2×#11745664C2211yescharacteristic
15#14273711—21no—
16#14273707C2211yescharacteristic
17#14273707C2211yescharacteristic
18#14273702C2211yescharacteristic
19#14273702C2211yescharacteristic
20#14273697—21no—
21C4×D236—21no—
22C4×D236—21no—
23#14273685—21no—
24#14273685—21no—
25#14273685C2212yes—
26#14273685C2212yes—
27C2×C4⋊Dic59C2211yescharacteristic
28C2×(C2×D59)⋊C4—21no—
29C2×C4×Dic59—21no—
30#14273660C2212yescomplemented
31#14273658C2212yescomplemented
32#14273652C2212yescomplemented
33#14273650C2212yescomplemented
34#14273649C2212yescomplemented
35#14273647C2212yescomplemented
36(C2×D59)⋊C8C2212yes—
37(C2×D59)⋊C8C2212yes—
38C2×M16×C59C2211yescharacteristic
39#117×C59C2211yescharacteristic
40C2×C4⋊C4×C59C2211yescharacteristic
41#107×C59C2212yescomplemented
42#106×C59C2212yescomplemented
43C23×D59D411yescharacteristic
44C22×C4×C59—21no—
45C22×C4×C59C2311yescharacteristic
46C2×C22⋊D59—21no—
47C22×Dic59—21no—
48C22×Dic59D411yescharacteristic
49#11745685—22no—
50#11745685—22no—
51C2×D236—21no—
52C2×D236—21no—
53C2×D236C2311yescharacteristic
54C2×C4×D59—22no—
55C2×C4×D59—22no—
56C2×C4×D59—21no—
57C2×C4×D59D411yescharacteristic
58C2×C4×D59D411yescharacteristic
59C2×C4×D59C2311yescharacteristic
60C2×C4×D59C2311yescharacteristic
61C2×Dic118C2311yescharacteristic
62M16×C59—22no—
63C2×C8×C59C2312yes—
64C4⋊C4×C59—21no—
65C4⋊C4×C59C2311yescharacteristic
66C4⋊C4×C59C2312yes—
67C4⋊C4×C59C2311yescharacteristic
68C22⋊C4×C59—21no—
69C42×C59—21no—
70(C2×D59)⋊C4—22no—
71(C2×D59)⋊C4—22no—
72(C2×D59)⋊C4—21no—
73(C2×D59)⋊C4—21no—
74(C2×D59)⋊C4—21no—
75(C2×D59)⋊C4—21no—
76C4⋊Dic59—21no—
77C4⋊Dic59—21no—
78C4⋊Dic59—21no—
79C4⋊Dic59C2312yes—
80C4×Dic59—22no—
81C4×Dic59—22no—
82#11745664—22no—
83C2×C59⋊C8C2312yes—
84C23×C59C2×D411yescharacteristic
85C22×D59—22no—
86C22×D59—21no—
87C22×D59—41no—
88C22×D59—41no—
89C22×D59C2×D411yescharacteristic
90C22×D59C2×D411yescharacteristic
91C2×C4×C59—22no—
92C2×C4×C59—21no—
93C2×C4×C59—21no—
94C2×C4×C59—42no—
95C2×C4×C59C2×D411yescharacteristic
96C2×C4×C59#4112yes—
97C2×C4×C59#4112yes—
98C2×C4×C59C2411yescharacteristic
99C22⋊D59—42no—
100C2×Dic59—22no—
101C2×Dic59—22no—
102C2×Dic59—21no—
103C2×Dic59—42no—
104C2×Dic59C2×D411yescharacteristic
105C2×Dic59C2×D411yescharacteristic
106D236—21no—
107D236—21no—
108D236—42no—
109D236—42no—
110C4×D59—22no—
111C4×D59—22no—
112C4×D59—21no—
113C4×D59—21no—
114C4×D59—21no—
115C4×D59—21no—
116C4×D59—42no—
117Dic118—21no—
118Dic118—21no—
119C8×C59—22no—
120C59⋊C8—22no—
121C22×C59—21no—
122C22×C59—21no—
123C22×C59#12312yes—
124C22×C59C22×D411yescharacteristic
125C2×D59—21no—
126C2×D59—21no—
127C2×D59—21no—
128C2×D59—21no—
129C2×D59—21no—
130C2×D59—21no—
131C2×D59—44no—
132C2×D59—41no—
133C2×D59—41no—
134C2×D59—41no—
135C4×C59—41no—
136C4×C59—41no—
137C4×C59—42no—
138C4×C59#12312yes—
139C4×C59C2×#4111yescharacteristic
140C4×C59C2×#4111yescharacteristic
141Dic59—21no—
142Dic59—21no—
143Dic59—41no—
144Dic59—42no—
145#3093—591nomaximal, Sylow
146C2×C59—22no—
147C2×C59C2×#12311yescharacteristic
148C2×C59#57511yescharacteristic
149C2×C59#57511yescharacteristic
150D59—41no—
151D59—41no—
152D59—81no—
153#525—591no—
154C2×#117—591no—
155#485—592no—
156#485—592no—
157#482—592no—
158#482—592no—
159#428—591no—
160#428D5911yescharacteristic, complemented
161#417—591no—
162#417—591no—
163#407—591no—
164C59#309311yescharacteristic, complemented, Sylow
165C2×#41—591no—
166C23×C4—591no—
167C2×M16—591no—
168C2×M16C2×D5911yescharacteristic
169#123—1181no—
170C22⋊Q8—591no—
171C22⋊Q8—591no—
172#121—591no—
173#121—591no—
174#120—1181no—
175C4×D4—1181no—
176C4×D4—1181no—
177#117—592no—
178#117—592no—
179#117—1181no—
180#117—1181no—
181#117C2×D5911yescharacteristic
182C2×C4⋊C4—591no—
183C2×C4⋊C4C2×D5911yescharacteristic
184C2×C22⋊C4—1181no—
185C2×C42—1181no—
186#107—592no—
187#107C2×D5912yescomplemented
188#106—592no—
189#106C2×D5912yescomplemented
190#103—592no—
191#103—592no—
192#102—592no—
193#102—592no—
194C22⋊C8—592no—
195C22⋊C8—592no—
196C24—591no—
197#41—1182no—
198#41—1182no—
199C2×Q8—591no—
200C2×D4—591no—
201C2×D4—1181no—
202C2×D4—1181no—
203C2×D4—1181no—
204C22×C4—21no—
205C22×C4—591no—
206C22×C4—591no—
207C22×C4—591no—
208C22×C4—591no—
209C22×C4—591no—
210C22×C4—1182no—
211C22×C4—1182no—
212C22×C4—1181no—
213C22×C4—1181no—
214C22×C4C22×D5911yescharacteristic
215M16—22no—
216M16—1182no—
217C2×C8—592no—
218C2×C8C22×D5912yes—
219C4⋊C4—21no—
220C4⋊C4—592no—
221C4⋊C4—1181no—
222C4⋊C4—1181no—
223C4⋊C4—1181no—
224C4⋊C4C22×D5912yes—
225C4⋊C4C22×D5911yescharacteristic
226C4⋊C4C22×D5911yescharacteristic
227C22⋊C4—21no—
228C22⋊C4—1181no—
229C22⋊C4—1181no—
230C22⋊C4—1182no—
231C22⋊C4—1182no—
232C22⋊C4—1181no—
233C22⋊C4—1181no—
234C42—21no—
235C42—1182no—
236C42—1182no—
237C23—591no—
238C23—591no—
239C23—1182no—
240C23—1181no—
241C23—2361no—
242C23—2361no—
243C23D4×D5911yescharacteristic
244Q8—1181no—
245Q8—1181no—
246D4—1181no—
247D4—1181no—
248D4—2362no—
249D4—2362no—
250D4—2362no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—42no—
255C2×C4—591no—
256C2×C4—591no—
257C2×C4—1182no—
258C2×C4—1182no—
259C2×C4—1182no—
260C2×C4—1181no—
261C2×C4—1181no—
262C2×C4—1181no—
263C2×C4—1181no—
264C2×C4—1182no—
265C2×C4—1181no—
266C2×C4—2362no—
267C2×C4—2362no—
268C2×C4#1174568512yes—
269C2×C4D4×D5911yescharacteristic
270C2×C4#1174568912yes—
271C2×C4C23×D5911yescharacteristic
272C8—22no—
273C8—1182no—
274C22—21no—
275C22—21no—
276C22—1181no—
277C22—1181no—
278C22—1181no—
279C22—1181no—
280C22—1181no—
281C22—1181no—
282C22—2364no—
283C22—2361no—
284C22—2361no—
285C22—2361no—
286C22#1427371212yes—
287C22C2×D4×D5911yescharacteristic
288C4—42no—
289C4—41no—
290C4—41no—
291C4—1181no—
292C4—1181no—
293C4—2361no—
294C4—2362no—
295C4#1427371212yes—
296C4C2×#1174568511yescharacteristic
297C4C2×#1174568911yescharacteristic
298C2—22no—
299C2—4721no—
300C2—2361no—
301C2—2361no—
302C2C2×#1427371211yescharacteristic
303C2#1683819811yescharacteristic
304C2#1683820111yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#428C2
socleC22×C59C22×D4
hypercenter#428D59
hypocenterC59#3093
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. C2×#14273685C2semidirect
3. #16837746C2semidirect
4. #16837745C2semidirect
5. #16837729C2semidirect
6. #16837728C2semidirect
7. #16837445C2semidirect
8. #16837404C2semidirect
9. #16837401C2semidirect
10. #16837365C2semidirect
11. C59×#428C2semidirect
30. #14273660C22semidirect
31. #14273658C22semidirect
32. #14273652C22semidirect
33. #14273650C22semidirect
34. #14273649C22semidirect
35. #14273647C22semidirect
41. #107×C59C22semidirect
42. #106×C59C22semidirect
160. #428D59semidirect
164. C59#3093semidirect
187. #107C2×D59semidirect
189. #106C2×D59semidirect