Finite Group Databasebeta

← #21433561

Subgroups — #21433561

Order 7552.

548 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 131 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×#14273754C212yesmaximal, complemented
2#16837953C211yesmaximal, characteristic, complemented
3C4×D4×D59C211yesmaximal, characteristic, complemented
4#16837620C212yesmaximal, complemented
5#16837619C212yesmaximal, complemented
6#16837604C211yesmaximal, characteristic, complemented
7#16837598C211yesmaximal, characteristic, complemented
8#16837597C211yesmaximal, characteristic, complemented
9#16837424C211yesmaximal, characteristic, complemented
10#16837402C212yesmaximal, complemented
11C59×#461C211yesmaximal, characteristic, complemented
12C2×#11745689C2212yes—
13C2×D4×D59C2211yescharacteristic
14C22×C4×D59—21no—
15#14273754—28no—
16C2×#11745671C2212yes—
17C2×D472C2212yes—
18C2×#11745669C2212yes—
19C2×#11745659C2212yes—
20#14273724C2211yescharacteristic, complemented
21#14273716C2212yescomplemented
22#14273702—21no—
23#14273702—21no—
24C4×D236C2211yescharacteristic
25D4×Dic59C2211yescharacteristic
26#14273689—21no—
27#14273689—21no—
28#14273689C2211yescharacteristic
29C4×C22⋊D59—21no—
30(C22⋊D59)⋊C4—21no—
31C4⋊C4×D59C2211yescharacteristic
32C22⋊C4×D59—21no—
33C42×D59C2211yescharacteristic
34C4⋊(C59⋊C8)C2211yescharacteristic, complemented
35Dic59⋊C8C2212yescomplemented
36#14273651C2212yescomplemented
37#14273649C2212yescomplemented
38#14273647C2212yescomplemented
39C2×SD16×C59C2212yes—
40C4⋊Q8×C59C2211yescharacteristic, complemented
41C4×D4×C59C2211yescharacteristic
42C4⋊C8×C59C2211yescharacteristic, complemented
43#103×C59C2212yescomplemented
44C23×D59—21no—
45C2×Q8×C59C2312yes—
46C2×D4×C59C2311yescharacteristic
47C22×C4×C59—21no—
48C2×C22⋊D59—21no—
49C22×Dic59—21no—
50#11745689—24no—
51#11745689—24no—
52D4×D59—22no—
53D4×D59D414yescomplemented
54C2×D236—22no—
55C2×D236—21no—
56C2×D236—21no—
57C2×D236C2312yes—
58C2×D236C2311yescharacteristic
59C2×C4×D59—21no—
60C2×C4×D59—21no—
61C2×C4×D59—21no—
62C2×C4×D59—22no—
63C2×C4×D59—44no—
64C2×C4×D59D411yescharacteristic
65C2×C4×D59D411yescharacteristic
66C2×C4×D59C2311yescharacteristic
67SD16×C59—24no—
68C2×C8×C59C2312yes—
69C4⋊C4×C59—21no—
70C4⋊C4×C59—21no—
71C4⋊C4×C59C2311yescharacteristic
72C22⋊C4×C59—21no—
73C42×C59C2311yescharacteristic
74C22⋊Dic59—21no—
75#11745671—24no—
76#11745669—24no—
77(C2×D59)⋊C4—21no—
78(C2×D59)⋊C4—21no—
79(C2×D59)⋊C4—22no—
80(C2×D59)⋊C4—22no—
81C4⋊Dic59C2311yescharacteristic
82Dic59⋊C4—21no—
83C4×Dic59D411yescharacteristic
84C4×Dic59D411yescharacteristic
85C4×Dic59C2311yescharacteristic
86C2×C59⋊C8C2312yes—
87D472—24no—
88#11745659—24no—
89C23×C59—21no—
90C22×D59—21no—
91C22×D59—24no—
92C22×D59—22no—
93C22×D59—42no—
94C22×D59—42no—
95C22×D59C2×D411yescharacteristic
96Q8×C59—22no—
97Q8×C59—22no—
98D4×C59—21no—
99D4×C59C2×D412yes—
100C2×C4×C59—21no—
101C2×C4×C59—22no—
102C2×C4×C59—42no—
103C2×C4×C59C2×D411yescharacteristic
104C2×C4×C59C2×D411yescharacteristic
105C2×C4×C59C2411yescharacteristic
106C22⋊D59—24no—
107C2×Dic59—21no—
108C2×Dic59—42no—
109C2×Dic59C2×D411yescharacteristic
110C2×Dic59C2×D411yescharacteristic
111C2×Dic59C2×D411yescharacteristic
112D236—22no—
113D236—21no—
114D236—22no—
115D236—42no—
116D236—44no—
117D236—42no—
118D236C2×D412yes—
119C4×D59—22no—
120C4×D59—41no—
121C4×D59—42no—
122C4×D59—42no—
123C4×D59—42no—
124C4×D59—44no—
125C4×D59C2×D412yes—
126C4×D59#4112yes—
127C8×C59—22no—
128C59⋊C8—22no—
129C22×C59—22no—
130C22×C59—41no—
131C22×C59—41no—
132C22×C59C22×D411yescharacteristic
133C2×D59—21no—
134C2×D59—21no—
135C2×D59—22no—
136C2×D59—48no—
137C2×D59—41no—
138C2×D59—41no—
139C2×D59—42no—
140C2×D59—42no—
141C2×D59—42no—
142C2×D59#12112yes—
143C4×C59—21no—
144C4×C59—42no—
145C4×C59—41no—
146C4×C59AGL(1,ℤ/8)12yescomplemented
147C4×C59C22×D411yescharacteristic
148C4×C59C2×#4111yescharacteristic
149Dic59—21no—
150Dic59—21no—
151Dic59—41no—
152Dic59#12112yes—
153#3036—591nomaximal, Sylow
154C2×C59—42no—
155C2×C59C2×#12111yescharacteristic
156C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
157C2×C59#57511yescharacteristic
158D59—22no—
159D59—42no—
160D59—82no—
161C2×AGL(1,ℤ/8)—592no—
162#532—591no—
163C2×C4×D4—591no—
164#463—592no—
165#463—592no—
166#461—591no—
167#461D5911yescharacteristic, complemented
168#459—591no—
169#459—591no—
170#423—591no—
171#417—592no—
172C59#303611yescharacteristic, complemented, Sylow
173C2×#41—592no—
174C22×D4—591no—
175C23×C4—1181no—
176AGL(1,ℤ/8)—1188no—
177C2×SD16—592no—
178C2×SD16C2×D5912yes—
179C2×D8—592no—
180C2×D8—592no—
181C2×M16—592no—
182C4⋊Q8C2×D5911yescharacteristic, complemented
183#127—591no—
184#124—592no—
185#121—1181no—
186#121—1181no—
187C4×D4—591no—
188C4×D4—591no—
189C4×D4—591no—
190C4×D4—1181no—
191C4×D4—1181no—
192C4×D4—1181no—
193C4×D4—1181no—
194C4×D4C2×D5911yescharacteristic
195C2×C4⋊C4—591no—
196C2×C22⋊C4—1181no—
197C2×C42—591no—
198C4⋊C8—592no—
199C4⋊C8—591no—
200C4⋊C8C2×D5911yescharacteristic, complemented
201#103—592no—
202#103C2×D5912yescomplemented
203#102—592no—
204#102—592no—
205C24—1181no—
206#41—1184no—
207#41—1184no—
208C2×Q8C22×D5912yes—
209C2×D4—594no—
210C2×D4—591no—
211C2×D4—592no—
212C2×D4—1182no—
213C2×D4—1182no—
214C2×D4—1181no—
215C2×D4—1181no—
216C2×D4—1181no—
217C2×D4C22×D5911yescharacteristic
218C22×C4—21no—
219C22×C4—591no—
220C22×C4—591no—
221C22×C4—591no—
222C22×C4—1181no—
223C22×C4—1182no—
224C22×C4—1181no—
225C22×C4—1181no—
226C22×C4—1181no—
227C22×C4—2364no—
228SD16—24no—
229SD16—1184no—
230D8—1184no—
231D8—1184no—
232M16—1184no—
233C2×C8—592no—
234C2×C8C22×D5912yes—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—591no—
238C4⋊C4—1181no—
239C4⋊C4C22×D5911yescharacteristic
240C22⋊C4—21no—
241C22⋊C4—1181no—
242C22⋊C4—1182no—
243C22⋊C4—1182no—
244C22⋊C4—1181no—
245C22⋊C4—1181no—
246C42—591no—
247C42—591no—
248C42—591no—
249C42C22×D5911yescharacteristic
250C23—21no—
251C23—591no—
252C23—1184no—
253C23—1181no—
254C23—1182no—
255C23—2362no—
256C23—2362no—
257Q8—22no—
258Q8—22no—
259D4—21no—
260D4—592no—
261D4—1184no—
262D4—1182no—
263D4—1181no—
264D4—1182no—
265D4—2362no—
266D4—2364no—
267D4—2362no—
268D4D4×D5912yes—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—42no—
272C2×C4—591no—
273C2×C4—592no—
274C2×C4—592no—
275C2×C4—591no—
276C2×C4—591no—
277C2×C4—1182no—
278C2×C4—1181no—
279C2×C4—2361no—
280C2×C4—2364no—
281C2×C4—2362no—
282C2×C4—2362no—
283C2×C4—2362no—
284C2×C4—2362no—
285C2×C4D4×D5911yescharacteristic
286C2×C4D4×D5911yescharacteristic
287C2×C4C23×D5911yescharacteristic
288C8—22no—
289C8—1182no—
290C22—22no—
291C22—41no—
292C22—41no—
293C22—592no—
294C22—1181no—
295C22—1181no—
296C22—1182no—
297C22—2368no—
298C22—2362no—
299C22—2362no—
300C22—2362no—
301C22—2361no—
302C22—2361no—
303C22C2×D4×D5911yescharacteristic
304C4—21no—
305C4—42no—
306C4—41no—
307C4—592no—
308C4—1181no—
309C4—1181no—
310C4—2361no—
311C4#1427375412yescomplemented
312C4C2×D4×D5911yescharacteristic
313C4#41×D5911yescharacteristic
314C2—42no—
315C2—1182no—
316C2—4722no—
317C2—2362no—
318C2#121×D5911yescharacteristic
319C2C2×#1427375411yescharacteristic
320C2#1683820111yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#461C2
socleC22×C59C22×D4
hypercenter#461D59
hypocenterC59#3036
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#14273754C2semidirect
2. #16837953C2semidirect
3. C4×D4×D59C2semidirect
4. #16837620C2semidirect
5. #16837619C2semidirect
6. #16837604C2semidirect
7. #16837598C2semidirect
8. #16837597C2semidirect
9. #16837424C2semidirect
10. #16837402C2semidirect
11. C59×#461C2semidirect
20. #14273724C22semidirect
21. #14273716C22semidirect
34. C4⋊(C59⋊C8)C22semidirect
35. Dic59⋊C8C22semidirect
36. #14273651C22semidirect
37. #14273649C22semidirect
38. #14273647C22semidirect
40. C4⋊Q8×C59C22semidirect
42. C4⋊C8×C59C22semidirect
43. #103×C59C22semidirect
53. D4×D59D4semidirect
146. C4×C59AGL(1,ℤ/8)semidirect
167. #461D59semidirect
172. C59#3036semidirect
182. C4⋊Q8C2×D59semidirect
200. C4⋊C8C2×D59semidirect
202. #103C2×D59semidirect
311. C4#14273754semidirect