Finite Group Databasebeta

← #21428985

Subgroups — #21428985

Order 7552.

800 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 161 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×#14273695C211yesmaximal, characteristic
2C2×D4×Dic59C211yesmaximal, characteristic, complemented
3C22×(C2×D59)⋊C4C211yesmaximal, characteristic, complemented
4#16837324C212yesmaximal, complemented
5#16837317C212yesmaximal, complemented
6#16837292C212yesmaximal, complemented
7#16837279C212yesmaximal, complemented
8#16837265C211yesmaximal, characteristic, complemented
9#16837260C211yesmaximal, characteristic, complemented
10#16837257C211yesmaximal, characteristic, complemented
11C2×#123×C59C211yesmaximal, characteristic, complemented
12C24×D59C2211yescharacteristic
13C22×C22⋊D59C2212yes—
14C23×Dic59C2211yescharacteristic
15C23×Dic59C2211yescharacteristic, complemented
16#14273695—24no—
17D4×Dic59—24no—
18C2×C4⋊Dic59C2211yescharacteristic, complemented
19C2×Dic59⋊C4C2212yescomplemented
20C2×C22⋊Dic59C2211yescharacteristic
21C2×C22⋊Dic59C2211yescharacteristic
22C2×C22⋊Dic59C2211yescharacteristic, complemented
23C2×(C2×D59)⋊C4—24no—
24C2×(C2×D59)⋊C4C2212yes—
25C2×(C2×D59)⋊C4C2212yes—
26C2×(C2×D59)⋊C4C2212yes—
27C2×(C2×D59)⋊C4C2212yes—
28C2×(C2×D59)⋊C4C2212yes—
29C2×C4×Dic59C2211yescharacteristic, complemented
30C59⋊#95C2212yescomplemented
31C59⋊#95C2212yescomplemented
32C59⋊#95C2212yescomplemented
33C22×D4×C59C2211yescharacteristic
34C23×C4×C59C2211yescharacteristic
35#123×C59—24no—
36C2×C4⋊C4×C59C2212yescomplemented
37C2×C22⋊C4×C59C2211yescharacteristic
38C2×C22⋊C4×C59C2212yescomplemented
39C24×C59C2311yescharacteristic
40C24×C59C2311yescharacteristic
41C23×D59—22no—
42C23×D59—24no—
43C23×D59—22no—
44C23×D59—22no—
45C23×D59—42no—
46C23×D59C2312yes—
47C2×D4×C59—22no—
48C2×D4×C59—22no—
49C2×D4×C59D414yes—
50C22×C4×C59—22no—
51C22×C4×C59—22no—
52C22×C4×C59—22no—
53C22×C4×C59C2312yes—
54C22×C4×C59C2312yes—
55C22×C4×C59C2311yescharacteristic
56C2×C22⋊D59—24no—
57C2×C22⋊D59—24no—
58C2×C22⋊D59—42no—
59C22×Dic59—22no—
60C22×Dic59—42no—
61C22×Dic59—42no—
62C22×Dic59D414yescomplemented
63C22×Dic59C2312yes—
64C22×Dic59C2311yescharacteristic
65C22×Dic59C2311yescharacteristic
66C22×Dic59C2311yescharacteristic
67C22×Dic59C2311yescharacteristic
68C4⋊C4×C59—24no—
69C22⋊C4×C59—22no—
70C22⋊C4×C59—24no—
71C22⋊Dic59—22no—
72C22⋊Dic59—22no—
73C22⋊Dic59—22no—
74(C2×D59)⋊C4—24no—
75(C2×D59)⋊C4—24no—
76(C2×D59)⋊C4—24no—
77(C2×D59)⋊C4—24no—
78(C2×D59)⋊C4—42no—
79C4⋊Dic59—22no—
80Dic59⋊C4—42no—
81C4×Dic59—22no—
82C23×C59—22no—
83C23×C59—22no—
84C23×C59—21no—
85C23×C59—22no—
86C23×C59—21no—
87C23×C59—22no—
88C23×C59C2×D412yes—
89C23×C59C2×D412yes—
90C23×C59C2×D412yes—
91C23×C59#4112yes—
92C23×C59C2411yescharacteristic
93C22×D59—28no—
94C22×D59—22no—
95C22×D59—22no—
96C22×D59—22no—
97C22×D59—22no—
98C22×D59—22no—
99C22×D59—22no—
100C22×D59—48no—
101C22×D59—48no—
102C22×D59—44no—
103C22×D59—44no—
104C22×D59#4114yes—
105D4×C59—28no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—22no—
109C2×C4×C59—22no—
110C2×C4×C59—22no—
111C2×C4×C59—22no—
112C2×C4×C59—22no—
113C2×C4×C59—22no—
114C2×C4×C59—44no—
115C2×C4×C59C2×D412yes—
116C22⋊D59—48no—
117C2×Dic59—22no—
118C2×Dic59—22no—
119C2×Dic59—21no—
120C2×Dic59—21no—
121C2×Dic59—22no—
122C2×Dic59—22no—
123C2×Dic59—22no—
124C2×Dic59—22no—
125C2×Dic59—22no—
126C2×Dic59—44no—
127C2×Dic59—44no—
128C2×Dic59C2×D412yes—
129C2×Dic59C2×D412yes—
130C22×C59—22no—
131C22×C59—22no—
132C22×C59—24no—
133C22×C59—24no—
134C22×C59—22no—
135C22×C59—24no—
136C22×C59—41no—
137C22×C59—41no—
138C22×C59—42no—
139C22×C59#12114yes—
140C22×C59C22×D411yescharacteristic
141C22×C59C22×D411yescharacteristic
142C22×C59C2×#4112yes—
143C22×C59C2×#4111yescharacteristic
144C22×C59H(5,𝔽2)12yes—
145C2×D59—24no—
146C2×D59—24no—
147C2×D59—24no—
148C2×D59—416no—
149C2×D59—42no—
150C2×D59—42no—
151C2×D59—42no—
152C2×D59—42no—
153C4×C59—22no—
154C4×C59—42no—
155C4×C59—42no—
156Dic59—22no—
157Dic59—22no—
158Dic59—41no—
159Dic59—41no—
160Dic59—42no—
161#2534—591nomaximal, Sylow
162C2×C59—24no—
163C2×C59—42no—
164C2×C59C2×#12111yescharacteristic
165C2×C59#52511yescharacteristic
166C2×C59#53411yescharacteristic
167C2×C59#54612yes—
168C2×C59#55112yes—
169D59—44no—
170C2×#123D5911yescharacteristic, complemented
171C2×#120—591no—
172C2×C4×D4—591no—
173C22×C22⋊C4—591no—
174#396—592no—
175#394—592no—
176#388—592no—
177#386—592no—
178#381—591no—
179#380—591no—
180#379—591no—
181C59#253411yescharacteristic, complemented, Sylow
182C25—591no—
183C22×D4—592no—
184C22×D4C2×D5911yescharacteristic
185C23×C4—591no—
186C23×C4—591no—
187C23×C4C2×D5911yescharacteristic
188#123—24no—
189#120—1184no—
190C4×D4—1184no—
191C2×C4⋊C4—591no—
192C2×C4⋊C4—592no—
193C2×C4⋊C4C2×D5912yescomplemented
194C2×C22⋊C4—591no—
195C2×C22⋊C4—591no—
196C2×C22⋊C4—591no—
197C2×C22⋊C4—592no—
198C2×C22⋊C4—592no—
199C2×C22⋊C4—592no—
200C2×C22⋊C4—592no—
201C2×C22⋊C4—592no—
202C2×C22⋊C4—1184no—
203C2×C22⋊C4C2×D5911yescharacteristic
204C2×C22⋊C4C2×D5912yescomplemented
205C2×C42—591no—
206#95—592no—
207#95—592no—
208#95—592no—
209C24—592no—
210C24—1184no—
211C24—1182no—
212C24—1182no—
213C24—1182no—
214C24—2362no—
215C24C22×D5911yescharacteristic
216C24C22×D5911yescharacteristic
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—1184no—
220C2×D4—1184no—
221C2×D4—2362no—
222C2×D4C22⋊D5914yes—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—591no—
227C22×C4—591no—
228C22×C4—594no—
229C22×C4—592no—
230C22×C4—591no—
231C22×C4—591no—
232C22×C4—1182no—
233C22×C4—2362no—
234C22×C4—2362no—
235C22×C4C22×D5912yes—
236C22×C4C22×D5912yes—
237C22×C4C22×D5911yescharacteristic
238C4⋊C4—24no—
239C4⋊C4—1182no—
240C4⋊C4—2362no—
241C22⋊C4—22no—
242C22⋊C4—24no—
243C22⋊C4—1184no—
244C22⋊C4—1184no—
245C22⋊C4—1184no—
246C22⋊C4—1182no—
247C22⋊C4—1182no—
248C22⋊C4—1182no—
249C22⋊C4—1184no—
250C22⋊C4—2362no—
251C42—1182no—
252C23—22no—
253C23—22no—
254C23—21no—
255C23—22no—
256C23—21no—
257C23—22no—
258C23—594no—
259C23—1188no—
260C23—1182no—
261C23—1182no—
262C23—1182no—
263C23—1182no—
264C23—1182no—
265C23—1182no—
266C23—2368no—
267C23—2368no—
268C23—2364no—
269C23—2364no—
270C23D4×D5912yes—
271C23#1174568712yes—
272C23C2×C22⋊D5912yes—
273C23C2×C22⋊D5912yes—
274C23C23×D5911yescharacteristic
275D4—28no—
276D4—2368no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—44no—
286C2×C4—592no—
287C2×C4—592no—
288C2×C4—1182no—
289C2×C4—1182no—
290C2×C4—1181no—
291C2×C4—1181no—
292C2×C4—1182no—
293C2×C4—1182no—
294C2×C4—1182no—
295C2×C4—1182no—
296C2×C4—1182no—
297C2×C4—2364no—
298C2×C4—2364no—
299C2×C4C2×C22⋊D5912yes—
300C22—22no—
301C22—22no—
302C22—24no—
303C22—24no—
304C22—22no—
305C22—24no—
306C22—41no—
307C22—41no—
308C22—42no—
309C22—1184no—
310C22—1184no—
311C22—1184no—
312C22—23616no—
313C22—2362no—
314C22—2362no—
315C22—2362no—
316C22—2362no—
317C22#1427369914yes—
318C22C2×D4×D5911yescharacteristic
319C22C22×C22⋊D5911yescharacteristic
320C22#41×D5912yes—
321C22C2×#1174568711yescharacteristic
322C22#1427377312yes—
323C4—22no—
324C4—42no—
325C4—42no—
326C4—1182no—
327C4—1182no—
328C4—2361no—
329C4—2361no—
330C4—2362no—
331C2—24no—
332C2—42no—
333C2—2364no—
334C2C2×#1427369911yescharacteristic
335C2#1683791011yescharacteristic
336C2#1683796311yescharacteristic
337C2#1683804012yes—
338C2#1683807512yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC2×#123×C59C2
hypercenterC2×#123D59
hypocenterC59#2534
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×D4×Dic59C2semidirect
3. C22×(C2×D59)⋊C4C2semidirect
4. #16837324C2semidirect
5. #16837317C2semidirect
6. #16837292C2semidirect
7. #16837279C2semidirect
8. #16837265C2semidirect
9. #16837260C2semidirect
10. #16837257C2semidirect
11. C2×#123×C59C2semidirect
15. C23×Dic59C22semidirect
18. C2×C4⋊Dic59C22semidirect
19. C2×Dic59⋊C4C22semidirect
22. C2×C22⋊Dic59C22semidirect
29. C2×C4×Dic59C22semidirect
30. C59⋊#95C22semidirect
31. C59⋊#95C22semidirect
32. C59⋊#95C22semidirect
36. C2×C4⋊C4×C59C22semidirect
38. C2×C22⋊C4×C59C22semidirect
62. C22×Dic59D4semidirect
170. C2×#123D59semidirect
181. C59#2534semidirect
193. C2×C4⋊C4C2×D59semidirect
204. C2×C22⋊C4C2×D59semidirect