Finite Group Databasebeta

← #21427416

Subgroups — #21427416

Order 7552.

1056 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 159 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, complemented
2C2×#14273697C212yesmaximal, complemented
3C2×C22⋊C4×D59C211yesmaximal, characteristic, complemented
4#16837313C211yesmaximal, characteristic, complemented
5#16837312C211yesmaximal, characteristic, complemented
6#16837311C211yesmaximal, characteristic, complemented
7#16837280C212yesmaximal, complemented
8#16837280C212yesmaximal, complemented
9#16837278C212yesmaximal, complemented
10#16837264C211yesmaximal, characteristic, complemented
11C59×#393C211yesmaximal, characteristic, complemented
12C24×D59C2211yescharacteristic
13C2×#11745689—28no—
14C2×#11745689—24no—
15C22×D236C2212yes—
16C22×D236C2212yes—
17C22×D236C2211yescharacteristic, complemented
18C22×C22⋊D59C2212yes—
19C22×C4×D59C2211yescharacteristic
20C22×C4×D59C2211yescharacteristic
21C22×C4×D59C2212yescomplemented
22#14273697—28no—
23C2×C22⋊Dic59C2211yescharacteristic
24C2×(C2×D59)⋊C4C2212yes—
25C2×(C2×D59)⋊C4C2212yes—
26C2×(C2×D59)⋊C4C2211yescharacteristic
27C2×(C2×D59)⋊C4C2211yescharacteristic
28C2×(C2×D59)⋊C4C2212yescomplemented
29C2×(C2×D59)⋊C4C2212yescomplemented
30C22⋊C4×D59—24no—
31C59⋊#95C2212yescomplemented
32C59⋊#95C2211yescharacteristic, complemented
33C59⋊#95C2212yescomplemented
34C22×Q8×C59C2211yescharacteristic, complemented
35C2×C22⋊C4×C59C2212yes—
36C2×C22⋊C4×C59C2211yescharacteristic
37#95×C59C2211yescharacteristic, complemented
38#95×C59C2212yescomplemented
39C24×C59C2311yescharacteristic
40C23×D59—22no—
41C23×D59—22no—
42C23×D59—22no—
43C23×D59—24no—
44C23×D59—42no—
45C23×D59C2312yes—
46C23×D59C2311yescharacteristic
47C23×D59C2311yescharacteristic
48C2×Q8×C59—24no—
49C2×Q8×C59—22no—
50C22×C4×C59C2312yes—
51C22×C4×C59C2311yescharacteristic
52C22×C4×C59C2311yescharacteristic
53C22×C4×C59C2312yes—
54C2×C22⋊D59—24no—
55C2×C22⋊D59—24no—
56C2×C22⋊D59—42no—
57C22×Dic59C2312yes—
58C22×Dic59C2311yescharacteristic
59C22×Dic59C2311yescharacteristic
60#11745689—232no—
61C2×D236—24no—
62C2×D236—24no—
63C2×D236—24no—
64C2×D236—22no—
65C2×D236—24no—
66C2×D236—24no—
67C2×D236—42no—
68C2×D236D418yescomplemented
69C2×C4×D59—24no—
70C2×C4×D59—24no—
71C2×C4×D59—24no—
72C2×C4×D59—24no—
73C2×C4×D59—22no—
74C2×C4×D59—42no—
75C2×C4×D59D414yescomplemented
76C22⋊C4×C59—22no—
77C22⋊C4×C59—24no—
78C22⋊Dic59—22no—
79(C2×D59)⋊C4—24no—
80(C2×D59)⋊C4—22no—
81(C2×D59)⋊C4—24no—
82(C2×D59)⋊C4—22no—
83(C2×D59)⋊C4—42no—
84(C2×D59)⋊C4—42no—
85C23×C59—21no—
86C23×C59—22no—
87C23×C59—21no—
88C23×C59—22no—
89C23×C59#4112yescomplemented
90C23×C59C2411yescharacteristic
91C22×D59—21no—
92C22×D59—21no—
93C22×D59—22no—
94C22×D59—28no—
95C22×D59—21no—
96C22×D59—22no—
97C22×D59—22no—
98C22×D59—22no—
99C22×D59—22no—
100C22×D59—22no—
101C22×D59—21no—
102C22×D59—22no—
103C22×D59—22no—
104C22×D59—22no—
105C22×D59—48no—
106C22×D59—48no—
107C22×D59—44no—
108C22×D59—44no—
109C22×D59C2×D414yes—
110C22×D59C2×D414yes—
111C22×D59C2×D412yes—
112C22×D59#4112yescomplemented
113Q8×C59—28no—
114C2×C4×C59—22no—
115C2×C4×C59—22no—
116C2×C4×C59—22no—
117C2×C4×C59—21no—
118C2×C4×C59—22no—
119C2×C4×C59—22no—
120C2×C4×C59—22no—
121C2×C4×C59—22no—
122C2×C4×C59C2×D414yes—
123C2×C4×C59C2×D412yes—
124C22⋊D59—48no—
125C2×Dic59—22no—
126C2×Dic59—22no—
127C2×Dic59—21no—
128C2×Dic59—22no—
129C2×Dic59—22no—
130C2×Dic59—22no—
131C2×Dic59C2×D412yes—
132D236—28no—
133D236—216no—
134D236—48no—
135C4×D59—216no—
136C4×D59—28no—
137C4×D59—44no—
138C22×C59—22no—
139C22×C59—24no—
140C22×C59—41no—
141C22×C59—41no—
142C22×C59—42no—
143C22×C59C22×D411yescharacteristic
144C22×C59C22×D412yes—
145C22×C59C2×#4111yescharacteristic
146C22×C59C2×#4111yescharacteristic
147C22×C59H(5,𝔽2)12yes—
148C2×D59—22no—
149C2×D59—22no—
150C2×D59—24no—
151C2×D59—24no—
152C2×D59—24no—
153C2×D59—22no—
154C2×D59—24no—
155C2×D59—24no—
156C2×D59—24no—
157C2×D59—416no—
158C2×D59—41no—
159C2×D59—42no—
160C2×D59—42no—
161C2×D59—41no—
162C2×D59—42no—
163C2×D59—42no—
164C2×D59—42no—
165C2×D59#12014yes—
166C4×C59—24no—
167C4×C59—22no—
168C4×C59—41no—
169C4×C59—42no—
170Dic59—22no—
171Dic59—41no—
172Dic59—42no—
173#2404—591nomaximal, Sylow
174C2×C59—42no—
175C2×C59C2×#12011yescharacteristic
176C2×C59#52511yescharacteristic
177C2×C59#53511yescharacteristic
178C2×C59#54612yes—
179C2×C59#54612yes—
180D59—24no—
181D59—44no—
182D59—42no—
183C22×#41—591no—
184C2×#120—592no—
185C22×C22⋊C4—591no—
186#393—591no—
187#393D5911yescharacteristic, complemented
188#392—591no—
189#392—591no—
190#386—592no—
191#386—592no—
192#386—592no—
193#381—591no—
194C59#240411yescharacteristic, complemented, Sylow
195C25—591no—
196C2×#41—1184no—
197C2×#41—1188no—
198C22×Q8C2×D5911yescharacteristic, complemented
199C22×D4—592no—
200C22×D4—592no—
201C22×D4—591no—
202C22×D4—592no—
203C23×C4—591no—
204C23×C4—592no—
205C23×C4—591no—
206#120—1188no—
207C2×C22⋊C4—591no—
208C2×C22⋊C4—591no—
209C2×C22⋊C4—591no—
210C2×C22⋊C4—592no—
211C2×C22⋊C4—592no—
212C2×C22⋊C4—592no—
213C2×C22⋊C4—592no—
214C2×C22⋊C4—1184no—
215C2×C22⋊C4C2×D5911yescharacteristic
216C2×C22⋊C4C2×D5912yes—
217#95—592no—
218#95—591no—
219#95—592no—
220#95C2×D5911yescharacteristic, complemented
221#95C2×D5912yescomplemented
222C24—591no—
223C24—591no—
224C24—592no—
225C24—1182no—
226C24—1182no—
227C24—1184no—
228C24—1182no—
229C24—2362no—
230C24C22×D5911yescharacteristic
231#41—11832no—
232C2×Q8—22no—
233C2×Q8—24no—
234C2×D4—598no—
235C2×D4—1184no—
236C2×D4—1184no—
237C2×D4—1184no—
238C2×D4—1184no—
239C2×D4—1184no—
240C2×D4—1184no—
241C2×D4—1184no—
242C2×D4—1182no—
243C2×D4—2362no—
244C2×D4—2362no—
245C22×C4—591no—
246C22×C4—594no—
247C22×C4—592no—
248C22×C4—591no—
249C22×C4—1184no—
250C22×C4—1184no—
251C22×C4—1184no—
252C22×C4—1182no—
253C22×C4—1184no—
254C22×C4—2362no—
255C22×C4C22×D5912yes—
256C22×C4C22×D5911yescharacteristic
257C22×C4C22×D5911yescharacteristic
258C22×C4C22×D5912yes—
259C22⋊C4—22no—
260C22⋊C4—24no—
261C22⋊C4—1182no—
262C22⋊C4—1184no—
263C22⋊C4—1182no—
264C22⋊C4—1182no—
265C22⋊C4—1184no—
266C22⋊C4—2362no—
267C22⋊C4—2362no—
268C23—21no—
269C23—22no—
270C23—21no—
271C23—22no—
272C23—594no—
273C23—592no—
274C23—594no—
275C23—592no—
276C23—1188no—
277C23—1181no—
278C23—1181no—
279C23—1182no—
280C23—1181no—
281C23—1182no—
282C23—1182no—
283C23—1182no—
284C23—1182no—
285C23—1182no—
286C23—1181no—
287C23—1182no—
288C23—1182no—
289C23—1182no—
290C23—2368no—
291C23—2364no—
292C23—2368no—
293C23—2364no—
294C23#1174568912yescomplemented
295C23C23×D5911yescharacteristic
296Q8—28no—
297D4—11816no—
298D4—1188no—
299D4—2368no—
300D4—2368no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—592no—
310C2×C4—1188no—
311C2×C4—11816no—
312C2×C4—1182no—
313C2×C4—1181no—
314C2×C4—1182no—
315C2×C4—1182no—
316C2×C4—1182no—
317C2×C4—1182no—
318C2×C4—2364no—
319C2×C4D4×D5912yes—
320C2×C4D4×D5914yes—
321C22—22no—
322C22—24no—
323C22—41no—
324C22—42no—
325C22—41no—
326C22—594no—
327C22—1182no—
328C22—1182no—
329C22—1184no—
330C22—1184no—
331C22—1182no—
332C22—1184no—
333C22—1184no—
334C22—1184no—
335C22—1184no—
336C22—23616no—
337C22—2362no—
338C22—2361no—
339C22—2362no—
340C22—2362no—
341C22—2362no—
342C22—2362no—
343C22—2361no—
344C22C2×D4×D5911yescharacteristic
345C22C2×D4×D5912yes—
346C22#41×D5911yescharacteristic
347C22C2×#1174568911yescharacteristic
348C22#1427377312yes—
349C4—22no—
350C4—24no—
351C4—41no—
352C4—42no—
353C4—1182no—
354C4—2361no—
355C4—2362no—
356C2—42no—
357C2—1184no—
358C2—2362no—
359C2—2364no—
360C2#120×D5911yescharacteristic
361C2#1683791511yescharacteristic
362C2#1683797011yescharacteristic
363C2#1683804312yes—
364C2#1683804512yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#393C2
hypercenter#393D59
hypocenterC59#2404
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×#11745689C2semidirect
2. C2×#14273697C2semidirect
3. C2×C22⋊C4×D59C2semidirect
4. #16837313C2semidirect
5. #16837312C2semidirect
6. #16837311C2semidirect
7. #16837280C2semidirect
8. #16837280C2semidirect
9. #16837278C2semidirect
10. #16837264C2semidirect
11. C59×#393C2semidirect
17. C22×D236C22semidirect
21. C22×C4×D59C22semidirect
28. C2×(C2×D59)⋊C4C22semidirect
29. C2×(C2×D59)⋊C4C22semidirect
31. C59⋊#95C22semidirect
32. C59⋊#95C22semidirect
33. C59⋊#95C22semidirect
34. C22×Q8×C59C22semidirect
37. #95×C59C22semidirect
38. #95×C59C22semidirect
68. C2×D236D4semidirect
75. C2×C4×D59D4semidirect
89. C23×C59#41semidirect
112. C22×D59#41semidirect
187. #393D59semidirect
194. C59#2404semidirect
198. C22×Q8C2×D59semidirect
220. #95C2×D59semidirect
221. #95C2×D59semidirect
294. C23#11745689semidirect