Finite Group Databasebeta

← #21427490

Subgroups — #21427490

Order 7552.

814 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14273711C211yesmaximal, characteristic, complemented
2C2×#14273697C211yesmaximal, characteristic
3C22×(C2×D59)⋊C4C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D59C211yesmaximal, characteristic, complemented
5#16837329C212yesmaximal, complemented
6#16837319C212yesmaximal, complemented
7#16837281C211yesmaximal, characteristic, complemented
8#16837280C212yesmaximal, complemented
9#16837263C211yesmaximal, characteristic, complemented
10#16837259C212yesmaximal, complemented
11C59×#385C211yesmaximal, characteristic, complemented
12C24×D59C2211yescharacteristic
13C22×D236C2211yescharacteristic
14C22×D236C2211yescharacteristic
15C22×C22⋊D59C2211yescharacteristic
16C23×Dic59C2211yescharacteristic
17C22×C4×D59C2212yes—
18#14273711—24no—
19#14273697—24no—
20C2×C4⋊Dic59C2212yescomplemented
21C2×C22⋊Dic59C2211yescharacteristic
22C2×(C2×D59)⋊C4—24no—
23C2×(C2×D59)⋊C4C2211yescharacteristic
24C2×(C2×D59)⋊C4C2211yescharacteristic
25C2×(C2×D59)⋊C4C2212yes—
26C2×(C2×D59)⋊C4C2212yes—
27C2×(C2×D59)⋊C4C2212yes—
28C2×(C2×D59)⋊C4C2212yescomplemented
29C2×(C2×D59)⋊C4C2212yescomplemented
30C22⋊C4×D59—24no—
31C59⋊#95C2212yescomplemented
32C59⋊#95C2212yescomplemented
33C59⋊#95C2212yescomplemented
34C23×C4×C59C2211yescharacteristic
35C2×C4⋊C4×C59C2212yescomplemented
36C2×C22⋊C4×C59C2211yescharacteristic
37C2×C22⋊C4×C59C2211yescharacteristic
38#95×C59C2212yescomplemented
39C24×C59C2311yescharacteristic
40C23×D59—22no—
41C23×D59—22no—
42C23×D59—24no—
43C23×D59—42no—
44C23×D59D414yes—
45C23×D59C2311yescharacteristic
46C23×D59C2311yescharacteristic
47C23×D59C2311yescharacteristic
48C22×C4×C59—22no—
49C22×C4×C59—42no—
50C22×C4×C59C2311yescharacteristic
51C22×C4×C59C2312yes—
52C22×C4×C59C2311yescharacteristic
53C22×C4×C59C2312yes—
54C2×C22⋊D59—22no—
55C2×C22⋊D59—22no—
56C2×C22⋊D59—22no—
57C22×Dic59—22no—
58C22×Dic59—22no—
59C22×Dic59—22no—
60C22×Dic59C2312yes—
61C22×Dic59C2312yes—
62C22×Dic59C2311yescharacteristic
63C2×D236—22no—
64C2×D236—22no—
65C2×D236—24no—
66C2×D236—42no—
67C2×C4×D59—24no—
68C2×C4×D59—24no—
69C2×C4×D59—42no—
70C4⋊C4×C59—42no—
71C22⋊C4×C59—22no—
72C22⋊C4×C59D414yescomplemented
73C22⋊Dic59—22no—
74(C2×D59)⋊C4—24no—
75(C2×D59)⋊C4—24no—
76(C2×D59)⋊C4—22no—
77(C2×D59)⋊C4—22no—
78(C2×D59)⋊C4—24no—
79(C2×D59)⋊C4—24no—
80(C2×D59)⋊C4—42no—
81C4⋊Dic59—24no—
82C23×C59—22no—
83C23×C59—22no—
84C23×C59C2×D412yes—
85C23×C59C2×D412yes—
86C23×C59#4112yes—
87C23×C59C2411yescharacteristic
88C22×D59—21no—
89C22×D59—21no—
90C22×D59—22no—
91C22×D59—28no—
92C22×D59—21no—
93C22×D59—22no—
94C22×D59—21no—
95C22×D59—22no—
96C22×D59—28no—
97C22×D59—21no—
98C22×D59—21no—
99C22×D59—22no—
100C22×D59—21no—
101C22×D59—22no—
102C22×D59—44no—
103C22×D59—48no—
104C22×D59—48no—
105C22×D59C2×D412yes—
106C22×D59C2×D412yes—
107C22×D59#4114yes—
108C2×C4×C59—22no—
109C2×C4×C59—22no—
110C2×C4×C59—22no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
113C2×C4×C59—22no—
114C2×C4×C59—22no—
115C2×C4×C59—22no—
116C2×C4×C59—22no—
117C2×C4×C59—44no—
118C2×C4×C59C2×D414yes—
119C22⋊D59—44no—
120C2×Dic59—21no—
121C2×Dic59—22no—
122C2×Dic59—22no—
123C2×Dic59—21no—
124C2×Dic59—22no—
125C2×Dic59—22no—
126C2×Dic59—22no—
127C2×Dic59—21no—
128C2×Dic59—22no—
129C2×Dic59—44no—
130D236—44no—
131D236—44no—
132C4×D59—48no—
133C22×C59—22no—
134C22×C59—22no—
135C22×C59—24no—
136C22×C59—24no—
137C22×C59#12114yes—
138C22×C59C22×D411yescharacteristic
139C22×C59C22×D411yescharacteristic
140C22×C59C2×#4112yes—
141C22×C59C2×#4111yescharacteristic
142C22×C59H(5,𝔽2)12yes—
143C2×D59—22no—
144C2×D59—22no—
145C2×D59—24no—
146C2×D59—24no—
147C2×D59—22no—
148C2×D59—22no—
149C2×D59—22no—
150C2×D59—416no—
151C2×D59—42no—
152C2×D59—42no—
153C2×D59—41no—
154C2×D59—41no—
155C2×D59—41no—
156C2×D59—42no—
157C2×D59—42no—
158C2×D59#12314yescomplemented
159C4×C59—24no—
160C4×C59—41no—
161C4×C59—41no—
162C4×C59—42no—
163Dic59—42no—
164Dic59—41no—
165Dic59—42no—
166#2412—591nomaximal, Sylow
167C2×C59—24no—
168C2×C59C2×#12111yescharacteristic
169C2×C59C2×#12311yescharacteristic
170C2×C59#53411yescharacteristic
171C2×C59#54612yes—
172C2×C59#55112yes—
173D59—24no—
174D59—42no—
175D59—81no—
176C2×#123—591no—
177C2×#120—591no—
178C22×C22⋊C4—591no—
179C22×C22⋊C4—591no—
180#397—592no—
181#394—592no—
182#386—591no—
183#386—592no—
184#385D5911yescharacteristic, complemented
185#381—591no—
186#380—592no—
187C59#241211yescharacteristic, complemented, Sylow
188C25—591no—
189C22×D4—591no—
190C22×D4—591no—
191C22×D4—591no—
192C23×C4—591no—
193C23×C4—592no—
194C23×C4C2×D5911yescharacteristic
195#123—1184no—
196#120—1184no—
197C2×C4⋊C4—592no—
198C2×C4⋊C4C2×D5912yescomplemented
199C2×C22⋊C4—591no—
200C2×C22⋊C4—592no—
201C2×C22⋊C4—592no—
202C2×C22⋊C4—592no—
203C2×C22⋊C4—591no—
204C2×C22⋊C4—591no—
205C2×C22⋊C4—592no—
206C2×C22⋊C4—592no—
207C2×C22⋊C4—1184no—
208C2×C22⋊C4—1184no—
209C2×C22⋊C4C2×D5911yescharacteristic
210C2×C22⋊C4C2×D5911yescharacteristic
211#95—592no—
212#95—592no—
213#95—592no—
214#95C2×D5912yescomplemented
215C24—591no—
216C24—591no—
217C24—594no—
218C24—591no—
219C24—1184no—
220C24—1182no—
221C24—1182no—
222C24—2362no—
223C24C22×D5911yescharacteristic
224C2×D4—1182no—
225C2×D4—1182no—
226C2×D4—1184no—
227C2×D4—1182no—
228C2×D4—1182no—
229C2×D4—1182no—
230C2×D4—2362no—
231C22×C4—22no—
232C22×C4—42no—
233C22×C4—592no—
234C22×C4—592no—
235C22×C4—591no—
236C22×C4—1184no—
237C22×C4—1182no—
238C22×C4—1182no—
239C22×C4—1182no—
240C22×C4—1184no—
241C22×C4—2362no—
242C22×C4C22×D5911yescharacteristic
243C22×C4C22×D5911yescharacteristic
244C22×C4C22×D5912yes—
245C22×C4C22×D5912yes—
246C4⋊C4—42no—
247C4⋊C4—1184no—
248C22⋊C4—22no—
249C22⋊C4—1184no—
250C22⋊C4—1184no—
251C22⋊C4—1182no—
252C22⋊C4—1182no—
253C22⋊C4—1182no—
254C22⋊C4—1184no—
255C22⋊C4—1184no—
256C22⋊C4—2362no—
257C22⋊C4D23614yescomplemented
258C23—22no—
259C23—22no—
260C23—592no—
261C23—594no—
262C23—592no—
263C23—1188no—
264C23—1181no—
265C23—1181no—
266C23—1182no—
267C23—1188no—
268C23—1181no—
269C23—1182no—
270C23—1181no—
271C23—1182no—
272C23—1181no—
273C23—1181no—
274C23—1182no—
275C23—1181no—
276C23—1182no—
277C23—2364no—
278C23—2368no—
279C23—2368no—
280C23C2×D23612yes—
281C23D4×D5912yes—
282C23#1174568912yes—
283C23C23×D5911yescharacteristic
284D4—2364no—
285D4—2364no—
286D4—2364no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—44no—
297C2×C4—1182no—
298C2×C4—1181no—
299C2×C4—1182no—
300C2×C4—1182no—
301C2×C4—1181no—
302C2×C4—1182no—
303C2×C4—1182no—
304C2×C4—1181no—
305C2×C4—1182no—
306C2×C4—2368no—
307C2×C4—2364no—
308C2×C4C2×D23614yes—
309C22—22no—
310C22—22no—
311C22—24no—
312C22—24no—
313C22—594no—
314C22—1182no—
315C22—1182no—
316C22—1184no—
317C22—1184no—
318C22—1182no—
319C22—1182no—
320C22—1182no—
321C22—23616no—
322C22—2361no—
323C22—2361no—
324C22—2361no—
325C22—2362no—
326C22—2362no—
327C22—2362no—
328C22—2362no—
329C22#1427370214yes—
330C22C2×D4×D5911yescharacteristic
331C22C22×D23611yescharacteristic
332C22#41×D5912yes—
333C22C2×#1174568911yescharacteristic
334C22#1427377212yes—
335C4—24no—
336C4—41no—
337C4—42no—
338C4—41no—
339C4—2362no—
340C4—2361no—
341C4—2362no—
342C2—24no—
343C2—1184no—
344C2—4721no—
345C2—2362no—
346C2C2×#1427370211yescharacteristic
347C2#123×D5911yescharacteristic
348C2#1683796211yescharacteristic
349C2#1683804712yes—
350C2#1683807912yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#385C2
hypercenter#385D59
hypocenterC59#2412
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#14273711C2semidirect
3. C22×(C2×D59)⋊C4C2semidirect
4. C2×C22⋊C4×D59C2semidirect
5. #16837329C2semidirect
6. #16837319C2semidirect
7. #16837281C2semidirect
8. #16837280C2semidirect
9. #16837263C2semidirect
10. #16837259C2semidirect
11. C59×#385C2semidirect
20. C2×C4⋊Dic59C22semidirect
28. C2×(C2×D59)⋊C4C22semidirect
29. C2×(C2×D59)⋊C4C22semidirect
31. C59⋊#95C22semidirect
32. C59⋊#95C22semidirect
33. C59⋊#95C22semidirect
35. C2×C4⋊C4×C59C22semidirect
38. #95×C59C22semidirect
72. C22⋊C4×C59D4semidirect
158. C2×D59#123semidirect
184. #385D59semidirect
187. C59#2412semidirect
198. C2×C4⋊C4C2×D59semidirect
214. #95C2×D59semidirect
257. C22⋊C4D236semidirect