Finite Group Databasebeta

← #21433727

Subgroups — #21433727

Order 7552.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 143 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×#14273760C211yesmaximal, characteristic, complemented
2C2×M16×D59C211yesmaximal, characteristic, complemented
3#16837914C211yesmaximal, characteristic
4C2×#14273707C211yesmaximal, characteristic, complemented
5#16837661C212yesmaximal, complemented
6#16837649C212yesmaximal, complemented
7#16837644C212yesmaximal, complemented
8#16837632C212yesmaximal, complemented
9#16837410C211yesmaximal, characteristic, complemented
10#16837401C211yesmaximal, characteristic, complemented
11C59×#428C211yesmaximal, characteristic, complemented
12C2×#11745685C2211yescharacteristic
13C22×Dic118C2211yescharacteristic
14C22×C4×D59C2211yescharacteristic
15#14273760—24no—
16C2×Dic236C2212yescomplemented
17C2×#11745660C2212yescomplemented
18C2×#11745664C2211yescharacteristic
19C2×#11745659C2212yes—
20M16×D59—24no—
21C2×C8×D59C2212yes—
22#14273711—21no—
23#14273707—21no—
24#14273707—21no—
25#14273707C2212yes—
26#14273707C2211yescharacteristic
27#14273707C2212yes—
28#14273707C2211yescharacteristic
29#14273702C2211yescharacteristic
30#14273702C2211yescharacteristic
31#14273697—21no—
32C4×D236—21no—
33C4×D236—21no—
34C2×C4⋊Dic59—21no—
35C2×(C2×D59)⋊C4—21no—
36#14273650C2212yescomplemented
37#14273650C2212yescomplemented
38#14273650C2212yescomplemented
39#14273647C2212yescomplemented
40C2×M16×C59C2211yescharacteristic
41#117×C59C2211yescharacteristic
42C2×C4⋊C4×C59C2211yescharacteristic
43#107×C59C2212yescomplemented
44#106×C59C2212yescomplemented
45C23×D59D411yescharacteristic
46C22×C4×C59—21no—
47C22×C4×C59C2311yescharacteristic
48C2×C22⋊D59—21no—
49C22×Dic59—21no—
50C22×Dic59D411yescharacteristic
51#11745685—22no—
52#11745685—22no—
53C2×D236—21no—
54C2×D236—21no—
55C2×D236C2311yescharacteristic
56C2×C4×D59—22no—
57C2×C4×D59—22no—
58C2×C4×D59—21no—
59C2×C4×D59D411yescharacteristic
60C2×C4×D59D411yescharacteristic
61C2×C4×D59C2311yescharacteristic
62C2×C4×D59C2311yescharacteristic
63C2×Dic118—22no—
64C2×Dic118—22no—
65C2×Dic118—21no—
66C2×Dic118C2312yes—
67C2×Dic118C2311yescharacteristic
68M16×C59D414yescomplemented
69C2×C8×C59C2312yes—
70C4⋊C4×C59—21no—
71C4⋊C4×C59C2311yescharacteristic
72C4⋊C4×C59C2312yes—
73C4⋊C4×C59C2311yescharacteristic
74C22⋊C4×C59—21no—
75C42×C59—21no—
76(C2×D59)⋊C4—22no—
77(C2×D59)⋊C4—22no—
78(C2×D59)⋊C4—21no—
79(C2×D59)⋊C4—21no—
80(C2×D59)⋊C4—21no—
81(C2×D59)⋊C4—21no—
82C4⋊Dic59—22no—
83C4⋊Dic59—22no—
84C4⋊Dic59—21no—
85C4⋊Dic59—21no—
86#11745664—22no—
87C2×C59⋊C8C2312yes—
88Dic236—24no—
89#11745660—24no—
90#11745659—24no—
91C8×D59—24no—
92C23×C59C2×D411yescharacteristic
93C22×D59—22no—
94C22×D59—22no—
95C22×D59—21no—
96C22×D59—41no—
97C22×D59C2×D411yescharacteristic
98C22×D59C2×D411yescharacteristic
99C2×C4×C59—22no—
100C2×C4×C59—21no—
101C2×C4×C59—21no—
102C2×C4×C59—42no—
103C2×C4×C59C2×D412yes—
104C2×C4×C59C2×D411yescharacteristic
105C2×C4×C59#4112yes—
106C2×C4×C59C2411yescharacteristic
107C22⋊D59—42no—
108C2×Dic59—22no—
109C2×Dic59—22no—
110C2×Dic59—21no—
111C2×Dic59—42no—
112C2×Dic59C2×D411yescharacteristic
113C2×Dic59C2×D411yescharacteristic
114D236—21no—
115D236—21no—
116D236—42no—
117D236—42no—
118C4×D59—22no—
119C4×D59—22no—
120C4×D59—21no—
121C4×D59—21no—
122C4×D59—21no—
123C4×D59—21no—
124C4×D59—42no—
125Dic118—22no—
126Dic118—21no—
127Dic118—22no—
128Dic118—21no—
129Dic118—42no—
130C8×C59C2×D414yes—
131C59⋊C8—22no—
132C22×C59—21no—
133C22×C59—21no—
134C22×C59#12112yes—
135C22×C59C22×D411yescharacteristic
136C2×D59—21no—
137C2×D59—21no—
138C2×D59—21no—
139C2×D59—21no—
140C2×D59—21no—
141C2×D59—44no—
142C2×D59—41no—
143C2×D59—41no—
144C2×D59—41no—
145C2×D59#13712yescomplemented
146C4×C59—41no—
147C4×C59—41no—
148C4×C59—42no—
149C4×C59#12112yes—
150C4×C59C22×D411yescharacteristic
151C4×C59C2×#4111yescharacteristic
152Dic59—21no—
153Dic59—21no—
154Dic59—42no—
155Dic59—41no—
156#3056—591nomaximal, Sylow
157C2×C59—22no—
158C2×C59C2×#12111yescharacteristic
159C2×C59C2×#13711yescharacteristic
160C2×C59#57511yescharacteristic
161D59—22no—
162D59—41no—
163D59—81no—
164C2×#137—591no—
165C22×M16—591no—
166#525—591no—
167C2×C22⋊Q8—591no—
168#470—592no—
169#468—592no—
170#467—592no—
171#465—592no—
172#428D5911yescharacteristic, complemented
173#419—591no—
174#417—591no—
175C59#305611yescharacteristic, complemented, Sylow
176C2×#41—591no—
177C22×Q8—591no—
178C23×C4—591no—
179#137—1184no—
180C2×Q16—592no—
181C2×SD16—592no—
182C2×M16—591no—
183C2×M16—592no—
184C2×M16—1184no—
185C2×M16C2×D5911yescharacteristic
186C22×C8—592no—
187#123—1181no—
188C22⋊Q8—592no—
189C22⋊Q8—592no—
190C22⋊Q8—591no—
191C22⋊Q8—591no—
192C22⋊Q8—1181no—
193C22⋊Q8—1181no—
194#121—591no—
195#121—591no—
196#120—1181no—
197C4×D4—1181no—
198C4×D4—1181no—
199#117C2×D5911yescharacteristic
200C2×C4⋊C4—1181no—
201C2×C4⋊C4C2×D5911yescharacteristic
202C2×C22⋊C4—1181no—
203#107C2×D5912yescomplemented
204#106C2×D5912yescomplemented
205#103—592no—
206#103—592no—
207#103—592no—
208#102—592no—
209C24—591no—
210#41—1182no—
211#41—1182no—
212C2×Q8—592no—
213C2×Q8—591no—
214C2×Q8—1181no—
215C2×Q8—1182no—
216C2×Q8—1182no—
217C2×D4—591no—
218C2×D4—1181no—
219C2×D4—1181no—
220C2×D4—1181no—
221C22×C4—21no—
222C22×C4—591no—
223C22×C4—591no—
224C22×C4—591no—
225C22×C4—591no—
226C22×C4—591no—
227C22×C4—1182no—
228C22×C4—1182no—
229C22×C4—1181no—
230C22×C4—1181no—
231C22×C4C22×D5911yescharacteristic
232Q16—1184no—
233SD16—1184no—
234M16—1182no—
235M16—1184no—
236M16D23614yescomplemented
237C2×C8—592no—
238C2×C8—1184no—
239C2×C8C22×D5912yes—
240C4⋊C4—21no—
241C4⋊C4—1182no—
242C4⋊C4—1182no—
243C4⋊C4—1181no—
244C4⋊C4—1181no—
245C4⋊C4C22×D5912yes—
246C4⋊C4C22×D5911yescharacteristic
247C4⋊C4C22×D5911yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—1181no—
250C22⋊C4—1181no—
251C22⋊C4—1182no—
252C22⋊C4—1182no—
253C22⋊C4—1181no—
254C22⋊C4—1181no—
255C42—21no—
256C23—591no—
257C23—591no—
258C23—1182no—
259C23—1182no—
260C23—1181no—
261C23—2361no—
262C23D4×D5911yescharacteristic
263Q8—1182no—
264Q8—1181no—
265Q8—1182no—
266Q8—1181no—
267Q8—2362no—
268D4—1181no—
269D4—1181no—
270D4—2362no—
271D4—2362no—
272D4—2362no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—591no—
278C2×C4—591no—
279C2×C4—1182no—
280C2×C4—1182no—
281C2×C4—1182no—
282C2×C4—1181no—
283C2×C4—1181no—
284C2×C4—1181no—
285C2×C4—1181no—
286C2×C4—1181no—
287C2×C4—1182no—
288C2×C4—2362no—
289C2×C4—2362no—
290C2×C4C2×D23612yes—
291C2×C4D4×D5911yescharacteristic
292C2×C4#1174568912yes—
293C2×C4C23×D5911yescharacteristic
294C8—1182no—
295C8C2×D23614yes—
296C22—21no—
297C22—21no—
298C22—592no—
299C22—1181no—
300C22—1181no—
301C22—1181no—
302C22—1181no—
303C22—1181no—
304C22—2364no—
305C22—2361no—
306C22—2361no—
307C22—2361no—
308C22#1427370212yes—
309C22C2×D4×D5911yescharacteristic
310C4—41no—
311C4—42no—
312C4—41no—
313C4—1181no—
314C4—1181no—
315C4—2361no—
316C4—2362no—
317C4#1427370212yes—
318C4C22×D23611yescharacteristic
319C4C2×#1174568911yescharacteristic
320C2—22no—
321C2—1182no—
322C2—4721no—
323C2—2361no—
324C2C2×#1427370211yescharacteristic
325C2#137×D5911yescharacteristic
326C2#1683819811yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#428C2
socleC22×C59C22×D4
hypercenter#428D59
hypocenterC59#3056
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
1. C2×#14273760C2semidirect
2. C2×M16×D59C2semidirect
4. C2×#14273707C2semidirect
5. #16837661C2semidirect
6. #16837649C2semidirect
7. #16837644C2semidirect
8. #16837632C2semidirect
9. #16837410C2semidirect
10. #16837401C2semidirect
11. C59×#428C2semidirect
16. C2×Dic236C22semidirect
17. C2×#11745660C22semidirect
36. #14273650C22semidirect
37. #14273650C22semidirect
38. #14273650C22semidirect
39. #14273647C22semidirect
43. #107×C59C22semidirect
44. #106×C59C22semidirect
68. M16×C59D4semidirect
145. C2×D59#137semidirect
172. #428D59semidirect
175. C59#3056semidirect
203. #107C2×D59semidirect
204. #106C2×D59semidirect
236. M16D236semidirect