Finite Group Databasebeta

← #21433338

Subgroups — #21433338

Order 7552.

758 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14273758C211yesmaximal, characteristic, complemented
2C2×#14273753C211yesmaximal, characteristic, complemented
3#16837913C211yesmaximal, characteristic, complemented
4#16837564C214yesmaximal, complemented
5#16837547C214yesmaximal, complemented
6#16837411C211yesmaximal, characteristic, complemented
7#16837407C211yesmaximal, characteristic, complemented
8#16837366C211yesmaximal, characteristic, complemented
9C22×#41×C59C211yesmaximal, characteristic, complemented
10C2×#11745685C2211yescharacteristic
11#14273758—24no—
12#14273753—24no—
13C2×#11745672C2212yescomplemented
14C2×#11745670C2212yescomplemented
15C2×#11745671C2212yescomplemented
16C2×#11745669C2212yescomplemented
17C2×#11745664C2211yescharacteristic
18C2×#11745664C2211yescharacteristic
19#14273710—21no—
20#14273706C2212yescomplemented
21#14273701C2212yescomplemented
22#14273696—21no—
23C4×C22⋊D59—22no—
24#14273686C2211yescharacteristic
25#14273651C2214yescomplemented
26#14273648C2214yescomplemented
27C22⋊(C59⋊C8)C2214yescomplemented
28C2×#41×C59—24no—
29C2×#41×C59—24no—
30C2×#41×C59—22no—
31C2×#41×C59C2214yes—
32C22×Q8×C59C2211yescharacteristic
33C22×D4×C59—21no—
34C22×D4×C59C2211yescharacteristic
35C23×C4×C59—21no—
36C23×C4×C59C2211yescharacteristic
37C24×C59—21no—
38C24×C59D411yescharacteristic
39#41×C59—216no—
40#41×C59—28no—
41#41×C59—28no—
42C2×Q8×C59—22no—
43C2×Q8×C59—21no—
44C2×Q8×C59D414yescomplemented
45C2×Q8×C59C2312yes—
46C2×D4×C59—28no—
47C2×D4×C59—24no—
48C2×D4×C59—21no—
49C2×D4×C59—22no—
50C2×D4×C59D414yescomplemented
51C2×D4×C59C2312yes—
52C22×C4×C59—28no—
53C22×C4×C59—21no—
54C22×C4×C59—24no—
55C22×C4×C59—22no—
56C22×C4×C59—22no—
57C22×C4×C59D411yescharacteristic
58C22×C4×C59D412yes—
59C22×C4×C59C2312yes—
60C22×C4×C59C2311yescharacteristic
61C2×C22⋊D59—22no—
62C2×C22⋊D59—21no—
63#11745685—22no—
64#11745685—22no—
65C2×D236C2311yescharacteristic
66C2×C4×D59—21no—
67C2×Dic118C2311yescharacteristic
68C22⋊Dic59—21no—
69C22⋊Dic59—22no—
70#11745672—24no—
71#11745671—24no—
72#11745670—24no—
73#11745669—24no—
74(C2×D59)⋊C4—22no—
75C4⋊Dic59C2312yes—
76Dic59⋊C4—22no—
77C4×Dic59—21no—
78#11745664—22no—
79#11745664—22no—
80C2×C59⋊C8C2312yes—
81C2×C59⋊C8C2312yes—
82C23×C59—22no—
83C23×C59—24no—
84C23×C59—22no—
85C23×C59—42no—
86C23×C59—44no—
87C23×C59C2×D412yes—
88C23×C59C2×D411yescharacteristic
89C22×D59—21no—
90Q8×C59—24no—
91Q8×C59—22no—
92Q8×C59C2×D414yes—
93D4×C59—216no—
94D4×C59—24no—
95D4×C59—22no—
96D4×C59C2×D414yes—
97C2×C4×C59—216no—
98C2×C4×C59—22no—
99C2×C4×C59—24no—
100C2×C4×C59—22no—
101C2×C4×C59—24no—
102C2×C4×C59—22no—
103C2×C4×C59—22no—
104C2×C4×C59C2×D412yes—
105C2×C4×C59C2×D411yescharacteristic
106C2×C4×C59C2×D412yes—
107C2×C4×C59C2×D412yes—
108C2×C4×C59C2411yescharacteristic
109C22⋊D59—44no—
110C22⋊D59—42no—
111C2×Dic59—22no—
112C2×Dic59—21no—
113D236—21no—
114D236—21no—
115C4×D59—42no—
116Dic118—21no—
117Dic118—21no—
118C59⋊C8—22no—
119C59⋊C8—22no—
120C22×C59—22no—
121C22×C59—22no—
122C22×C59—24no—
123C22×C59—22no—
124C22×C59—22no—
125C22×C59—44no—
126C22×C59—48no—
127C22×C59—44no—
128C22×C59#12012yes—
129C22×C59C22×D411yescharacteristic
130C2×D59—41no—
131C2×D59—42no—
132C4×C59—22no—
133C4×C59—24no—
134C4×C59#12012yes—
135C4×C59C22×D411yescharacteristic
136C4×C59C22×D411yescharacteristic
137Dic59—41no—
138Dic59—42no—
139#3005—591nomaximal, Sylow
140C2×C59—22no—
141C2×C59—42no—
142C2×C59—44no—
143C2×C59C2×#12011yescharacteristic
144C2×C59#57512yes—
145D59—81no—
146C22×#41D5911yescharacteristic, complemented
147C2×#137—591no—
148C2×AGL(1,ℤ/8)—591no—
149#525—591no—
150#452—594no—
151#449—594no—
152#419—591no—
153#418—591no—
154#407—591no—
155C59#300511yescharacteristic, complemented, Sylow
156C2×#41—24no—
157C2×#41—24no—
158C2×#41—22no—
159C2×#41—591no—
160C2×#41C2×D5914yes—
161C22×Q8C2×D5911yescharacteristic
162C22×D4—21no—
163C22×D4C2×D5911yescharacteristic
164C23×C4—21no—
165C23×C4C2×D5911yescharacteristic
166#137—1184no—
167AGL(1,ℤ/8)—1184no—
168C2×Q16—592no—
169C2×SD16—592no—
170C2×SD16—592no—
171C2×D8—592no—
172C2×M16—591no—
173C2×M16—591no—
174#123—1181no—
175C22⋊Q8—592no—
176#121—592no—
177#120—1181no—
178C4×D4—1182no—
179#117—591no—
180#103—594no—
181#102—594no—
182C22⋊C8—594no—
183C24—21no—
184C24C22⋊D5911yescharacteristic
185#41—216no—
186#41—28no—
187#41—28no—
188#41—1182no—
189#41—1182no—
190C2×Q8—22no—
191C2×Q8—21no—
192C2×Q8—591no—
193C2×Q8C22⋊D5914yescomplemented
194C2×Q8C22×D5912yes—
195C2×D4—28no—
196C2×D4—24no—
197C2×D4—21no—
198C2×D4—22no—
199C2×D4—591no—
200C2×D4—1182no—
201C2×D4—1181no—
202C2×D4C22⋊D5914yescomplemented
203C2×D4C22×D5912yes—
204C22×C4—28no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—1181no—
210C22×C4C22⋊D5912yes—
211C22×C4C22⋊D5911yescharacteristic
212C22×C4C22×D5912yes—
213C22×C4C22×D5911yescharacteristic
214Q16—1184no—
215SD16—1184no—
216SD16—1184no—
217D8—1184no—
218M16—1182no—
219M16—1182no—
220C2×C8—592no—
221C2×C8—592no—
222C4⋊C4—592no—
223C4⋊C4—1182no—
224C22⋊C4—1181no—
225C22⋊C4—1182no—
226C22⋊C4—1182no—
227C42—1181no—
228C23—22no—
229C23—24no—
230C23—22no—
231C23—42no—
232C23—44no—
233C23—1181no—
234C23C2×C22⋊D5912yes—
235C23C2×C22⋊D5911yescharacteristic
236Q8—24no—
237Q8—22no—
238Q8—1181no—
239Q8—1181no—
240Q8C2×C22⋊D5914yes—
241D4—216no—
242D4—24no—
243D4—22no—
244D4—1181no—
245D4—1181no—
246D4—2364no—
247D4—2362no—
248D4C2×C22⋊D5914yes—
249C2×C4—216no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—1182no—
257C2×C4—1181no—
258C2×C4—2362no—
259C2×C4C2×C22⋊D5912yes—
260C2×C4C2×C22⋊D5911yescharacteristic
261C2×C4C2×C22⋊D5912yes—
262C2×C4C2×C22⋊D5912yes—
263C2×C4C23×D5911yescharacteristic
264C8—1182no—
265C8—1182no—
266C22—22no—
267C22—22no—
268C22—24no—
269C22—22no—
270C22—22no—
271C22—44no—
272C22—48no—
273C22—44no—
274C22—2361no—
275C22—2362no—
276C22#1427369612yes—
277C22C22×C22⋊D5911yescharacteristic
278C4—22no—
279C4—24no—
280C4—2361no—
281C4—2362no—
282C4#1427369612yes—
283C4C22×C22⋊D5911yescharacteristic
284C4C22×C22⋊D5911yescharacteristic
285C2—22no—
286C2—42no—
287C2—44no—
288C2—4721no—
289C2C2×#1427369611yescharacteristic
290C2#1683820012yes—
subgroupG/H
centerC22C22×C22⋊D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC22×#41×C59C2
socleC22×C59C22×D4
hypercenterC22×#41D59
hypocenterC59#3005
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#14273758C2semidirect
2. C2×#14273753C2semidirect
3. #16837913C2semidirect
4. #16837564C2semidirect
5. #16837547C2semidirect
6. #16837411C2semidirect
7. #16837407C2semidirect
8. #16837366C2semidirect
9. C22×#41×C59C2semidirect
13. C2×#11745672C22semidirect
14. C2×#11745670C22semidirect
15. C2×#11745671C22semidirect
16. C2×#11745669C22semidirect
20. #14273706C22semidirect
21. #14273701C22semidirect
25. #14273651C22semidirect
26. #14273648C22semidirect
27. C22⋊(C59⋊C8)C22semidirect
44. C2×Q8×C59D4semidirect
50. C2×D4×C59D4semidirect
146. C22×#41D59semidirect
155. C59#3005semidirect
193. C2×Q8C22⋊D59semidirect
202. C2×D4C22⋊D59semidirect