Finite Group Databasebeta

← #21433355

Subgroups — #21433355

Order 7552.

754 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#11745689C211yesmaximal, characteristic
2C2×#14273759C212yesmaximal, complemented
3C22⋊Q8×D59C211yesmaximal, characteristic, complemented
4#16837567C212yesmaximal, complemented
5#16837563C212yesmaximal, complemented
6#16837558C211yesmaximal, characteristic, complemented
7#16837555C211yesmaximal, characteristic, complemented
8#16837552C211yesmaximal, characteristic, complemented
9#16837402C212yesmaximal, complemented
10#16837364C211yesmaximal, characteristic, complemented
11C59×#451C211yesmaximal, characteristic, complemented
12C2×#11745689—28no—
13C2×#11745689—21no—
14C2×#11745689—21no—
15C2×#11745689C2212yes—
16C2×#11745689C2212yes—
17C2×Q8×D59C2211yescharacteristic, complemented
18C22×D236—21no—
19C22×D236C2211yescharacteristic
20C22×C4×D59—21no—
21C22×C4×D59C2211yescharacteristic
22#14273759—28no—
23C2×#11745672C2212yescomplemented
24C2×#11745660C2212yescomplemented
25C2×#11745671C2212yescomplemented
26C2×#11745659C2212yescomplemented
27#14273707C2211yescharacteristic, complemented
28#14273706C2211yescharacteristic
29#14273705—21no—
30#14273704—21no—
31#14273703C2211yescharacteristic, complemented
32C4⋊C4×D59—21no—
33C4⋊C4×D59C2211yescharacteristic, complemented
34C22⋊C4×D59—21no—
35#14273651C2212yescomplemented
36#14273649C2212yescomplemented
37#14273647C2212yescomplemented
38C22⋊(C59⋊C8)C2211yescharacteristic
39(C2×D59)⋊C8C2212yescomplemented
40C22×Q8×C59C2211yescharacteristic
41C2×Q16×C59C2212yescomplemented
42C22⋊Q8×C59C2211yescharacteristic
43#103×C59C2212yescomplemented
44C22⋊C8×C59C2211yescharacteristic
45C23×D59—21no—
46C23×D59D411yescharacteristic
47C2×Q8×C59—24no—
48C2×Q8×C59—21no—
49C2×Q8×C59C2311yescharacteristic
50C2×Q8×C59C2312yes—
51C22×C4×C59—21no—
52C22×C4×C59C2311yescharacteristic
53C22×Dic59D411yescharacteristic
54#11745689—28no—
55#11745689—28no—
56#11745689—24no—
57#11745689—24no—
58#11745689—24no—
59#11745689D418yes—
60Q8×D59—24no—
61C2×D236—28no—
62C2×D236—22no—
63C2×D236—22no—
64C2×D236—21no—
65C2×D236—24no—
66C2×D236C2312yes—
67C2×C4×D59—28no—
68C2×C4×D59—22no—
69C2×C4×D59—22no—
70C2×C4×D59—24no—
71C2×C4×D59—21no—
72C2×C4×D59—21no—
73C2×C4×D59D411yescharacteristic
74C2×C4×D59D411yescharacteristic
75C2×C4×D59C2311yescharacteristic
76C2×C4×D59C2311yescharacteristic
77C2×Dic118—21no—
78C2×Dic118C2311yescharacteristic
79Q16×C59—24no—
80C2×C8×C59C2312yes—
81C4⋊C4×C59—21no—
82C4⋊C4×C59C2311yescharacteristic
83C22⋊C4×C59—21no—
84C22⋊Dic59—21no—
85#11745672—24no—
86#11745671—24no—
87(C2×D59)⋊C4—21no—
88(C2×D59)⋊C4—21no—
89C4⋊Dic59—21no—
90C4⋊Dic59C2311yescharacteristic
91Dic59⋊C4—21no—
92Dic59⋊C4—21no—
93Dic59⋊C4—21no—
94C2×C59⋊C8C2312yes—
95#11745660—24no—
96#11745659—24no—
97C23×C59C2×D411yescharacteristic
98C22×D59—24no—
99C22×D59—22no—
100C22×D59—22no—
101C22×D59—22no—
102C22×D59—41no—
103C22×D59—42no—
104C22×D59—42no—
105C22×D59C2×D411yescharacteristic
106C22×D59C2×D411yescharacteristic
107Q8×C59—24no—
108Q8×C59—22no—
109Q8×C59—22no—
110Q8×C59C2×D414yes—
111C2×C4×C59—24no—
112C2×C4×C59—22no—
113C2×C4×C59—22no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59C2×D411yescharacteristic
117C2×C4×C59C2411yescharacteristic
118C2×Dic59—22no—
119C2×Dic59—21no—
120C2×Dic59—21no—
121C2×Dic59C2×D411yescharacteristic
122C2×Dic59C2×D411yescharacteristic
123D236—28no—
124D236—24no—
125D236—28no—
126D236—22no—
127D236C2×D414yes—
128C4×D59—28no—
129C4×D59—22no—
130C4×D59—22no—
131C4×D59—28no—
132C4×D59—22no—
133C4×D59—42no—
134C4×D59—42no—
135C4×D59C2×D414yes—
136Dic118—22no—
137Dic118—42no—
138C8×C59—22no—
139C59⋊C8—22no—
140C22×C59—21no—
141C22×C59—21no—
142C22×C59#13712yescomplemented
143C22×C59C22×D411yescharacteristic
144C2×D59—21no—
145C2×D59—21no—
146C2×D59—21no—
147C2×D59—24no—
148C2×D59—21no—
149C2×D59—21no—
150C2×D59—44no—
151C2×D59—48no—
152C2×D59—42no—
153C2×D59—42no—
154C2×D59#12012yes—
155C4×C59—24no—
156C4×C59—21no—
157C4×C59—41no—
158C4×C59—41no—
159C4×C59C22×D412yes—
160Dic59—21no—
161Dic59—41no—
162Dic59—41no—
163Dic59#12012yes—
164#3007—591nomaximal, Sylow
165C2×C59—22no—
166C2×C59C2×#12011yescharacteristic
167C2×C59C2×#13711yescharacteristic
168C2×C59#57511yescharacteristic
169D59—22no—
170D59—41no—
171D59—44no—
172C22×#41—591no—
173C2×#137—592no—
174C2×C22⋊Q8—591no—
175#452—592no—
176#452—592no—
177#451—591no—
178#451D5911yescharacteristic, complemented
179#450—591no—
180#450—591no—
181#417—592no—
182#407—591no—
183C59#300711yescharacteristic, complemented, Sylow
184C2×#41—592no—
185C2×#41—592no—
186C2×#41—1181no—
187C2×#41—1181no—
188C2×#41—1188no—
189C22×Q8—591no—
190C22×Q8C2×D5911yescharacteristic
191C22×D4—591no—
192C22×D4—1181no—
193C23×C4—591no—
194C23×C4—1181no—
195#137—1188no—
196C2×Q16—592no—
197C2×Q16C2×D5912yescomplemented
198C2×SD16—592no—
199C2×SD16—592no—
200C2×M16—592no—
201C22⋊Q8—591no—
202C22⋊Q8—591no—
203C22⋊Q8—591no—
204C22⋊Q8—1181no—
205C22⋊Q8—1181no—
206C22⋊Q8C2×D5911yescharacteristic
207C2×C4⋊C4—591no—
208C2×C4⋊C4—1181no—
209C2×C22⋊C4—1181no—
210#103—592no—
211#103C2×D5912yescomplemented
212#102—592no—
213#102—592no—
214C22⋊C8—591no—
215C22⋊C8—592no—
216C22⋊C8C2×D5911yescharacteristic
217C24—591no—
218C24—1181no—
219#41—598no—
220#41—1184no—
221#41—1184no—
222#41—1188no—
223#41—1188no—
224#41—1184no—
225C2×Q8—24no—
226C2×Q8—21no—
227C2×Q8—591no—
228C2×Q8—1181no—
229C2×Q8—1184no—
230C2×Q8C22×D5911yescharacteristic
231C2×Q8C22×D5912yes—
232C2×D4—592no—
233C2×D4—1184no—
234C2×D4—1182no—
235C2×D4—1182no—
236C2×D4—1188no—
237C2×D4—1181no—
238C22×C4—21no—
239C22×C4—591no—
240C22×C4—591no—
241C22×C4—591no—
242C22×C4—591no—
243C22×C4—591no—
244C22×C4—1184no—
245C22×C4—1182no—
246C22×C4—1182no—
247C22×C4—1188no—
248C22×C4—1181no—
249C22×C4—1181no—
250C22×C4C22×D5911yescharacteristic
251Q16—24no—
252Q16—1184no—
253SD16—1184no—
254SD16—1184no—
255M16—1184no—
256C2×C8—592no—
257C2×C8C22×D5912yes—
258C4⋊C4—21no—
259C4⋊C4—591no—
260C4⋊C4—1181no—
261C4⋊C4—1181no—
262C4⋊C4—1181no—
263C4⋊C4—1181no—
264C4⋊C4C22×D5911yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—1181no—
267C22⋊C4—1181no—
268C22⋊C4—1181no—
269C23—591no—
270C23—591no—
271C23—1182no—
272C23—1184no—
273C23—1182no—
274C23—1182no—
275C23—2361no—
276C23—2362no—
277C23—2362no—
278C23D4×D5911yescharacteristic
279Q8—24no—
280Q8—22no—
281Q8—22no—
282Q8—1182no—
283Q8—2362no—
284Q8D4×D5914yes—
285D4—594no—
286D4—1188no—
287D4—1188no—
288D4—1184no—
289D4—1182no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—591no—
296C2×C4—594no—
297C2×C4—591no—
298C2×C4—1182no—
299C2×C4—1188no—
300C2×C4—1182no—
301C2×C4—1188no—
302C2×C4—1182no—
303C2×C4—1182no—
304C2×C4—1181no—
305C2×C4—1181no—
306C2×C4—2362no—
307C2×C4—2362no—
308C2×C4D4×D5911yescharacteristic
309C2×C4C23×D5911yescharacteristic
310C8—22no—
311C8—1182no—
312C22—21no—
313C22—21no—
314C22—592no—
315C22—1181no—
316C22—1181no—
317C22—1181no—
318C22—1181no—
319C22—1184no—
320C22—1181no—
321C22—2364no—
322C22—2368no—
323C22—2362no—
324C22—2362no—
325C22#1427375912yescomplemented
326C22C2×D4×D5911yescharacteristic
327C4—24no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—592no—
332C4—1181no—
333C4—2361no—
334C4—2361no—
335C4C2×D4×D5912yes—
336C2—22no—
337C2—1182no—
338C2—2361no—
339C2—2364no—
340C2#120×D5911yescharacteristic
341C2C2×#1427375911yescharacteristic
342C2#1683820111yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#451C2
socleC22×C59C22×D4
hypercenter#451D59
hypocenterC59#3007
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×#14273759C2semidirect
3. C22⋊Q8×D59C2semidirect
4. #16837567C2semidirect
5. #16837563C2semidirect
6. #16837558C2semidirect
7. #16837555C2semidirect
8. #16837552C2semidirect
9. #16837402C2semidirect
10. #16837364C2semidirect
11. C59×#451C2semidirect
17. C2×Q8×D59C22semidirect
23. C2×#11745672C22semidirect
24. C2×#11745660C22semidirect
25. C2×#11745671C22semidirect
26. C2×#11745659C22semidirect
27. #14273707C22semidirect
31. #14273703C22semidirect
33. C4⋊C4×D59C22semidirect
35. #14273651C22semidirect
36. #14273649C22semidirect
37. #14273647C22semidirect
39. (C2×D59)⋊C8C22semidirect
41. C2×Q16×C59C22semidirect
43. #103×C59C22semidirect
142. C22×C59#137semidirect
178. #451D59semidirect
183. C59#3007semidirect
197. C2×Q16C2×D59semidirect
211. #103C2×D59semidirect
325. C22#14273759semidirect