Finite Group Databasebeta

← #21429902

Subgroups — #21429902

Order 7552.

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

normal subgroups only

This group has 668 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×#14273707C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D59C211yesmaximal, characteristic
3#16837337C211yesmaximal, characteristic, complemented
4#16837325C212yesmaximal, complemented
5#16837324C212yesmaximal, complemented
6#16837315C211yesmaximal, characteristic, complemented
7#16837314C211yesmaximal, characteristic, complemented
8#16837273C212yesmaximal, complemented
9#16837261C211yesmaximal, characteristic, complemented
10#16837257C212yesmaximal
11C59×#399C211yesmaximal, characteristic, complemented
12C24×D59C2211yescharacteristic
13C22×Dic118C2211yescharacteristic, complemented
14C22×C4×D59C2211yescharacteristic
15C22×C4×D59C2211yescharacteristic
16#14273707—24no—
17C2×C4⋊Dic59C2212yescomplemented
18C2×Dic59⋊C4C2212yescomplemented
19C2×Dic59⋊C4C2212yescomplemented
20C2×C22⋊Dic59C2211yescharacteristic
21C2×C22⋊Dic59C2212yes—
22C2×(C2×D59)⋊C4C2212yes—
23C2×(C2×D59)⋊C4C2212yes—
24C2×(C2×D59)⋊C4C2211yescharacteristic
25C2×(C2×D59)⋊C4C2211yescharacteristic
26C2×(C2×D59)⋊C4C2212yes—
27C2×(C2×D59)⋊C4C2212yes—
28C22⋊C4×D59—24no—
29C59⋊#95C2211yescharacteristic, complemented
30C59⋊#95C2212yescomplemented
31C59⋊#95C2212yescomplemented
32C2×C4⋊C4×C59C2211yescharacteristic, complemented
33C2×C4⋊C4×C59C2212yescomplemented
34C2×C22⋊C4×C59C2211yescharacteristic
35C2×C22⋊C4×C59C2212yes—
36#95×C59C2211yescharacteristic, complemented
37C24×C59C2311yescharacteristic
38C23×D59—22no—
39C23×D59—22no—
40C23×D59—42no—
41C23×D59—42no—
42C23×D59Q814yescomplemented
43C23×D59C2311yescharacteristic
44C23×D59C2311yescharacteristic
45C22×C4×C59C2312yes—
46C22×C4×C59C2311yescharacteristic
47C22×C4×C59C2312yes—
48C22×C4×C59C2311yescharacteristic
49C22×Dic59C2312yes—
50C22×Dic59C2312yes—
51C22×Dic59C2311yescharacteristic
52C22×Dic59C2311yescharacteristic
53C2×C4×D59—22no—
54C2×C4×D59—42no—
55C2×C4×D59—42no—
56C2×C4×D59D414yes—
57C2×Dic118—22no—
58C2×Dic118—42no—
59C4⋊C4×C59—22no—
60C4⋊C4×C59—42no—
61C22⋊C4×C59—22no—
62C22⋊C4×C59—42no—
63C22⋊Dic59—22no—
64C22⋊Dic59—42no—
65(C2×D59)⋊C4—24no—
66(C2×D59)⋊C4—24no—
67(C2×D59)⋊C4—22no—
68(C2×D59)⋊C4—22no—
69(C2×D59)⋊C4—42no—
70(C2×D59)⋊C4—42no—
71C4⋊Dic59—24no—
72Dic59⋊C4—42no—
73Dic59⋊C4—42no—
74C23×C59—21no—
75C23×C59—21no—
76C23×C59—22no—
77C23×C59—22no—
78C23×C59C2×Q812yes—
79C23×C59C2411yescharacteristic
80C22×D59—22no—
81C22×D59—22no—
82C22×D59—21no—
83C22×D59—21no—
84C22×D59—22no—
85C22×D59—22no—
86C22×D59—28no—
87C22×D59—48no—
88C22×D59—44no—
89C22×D59—44no—
90C22×D59—48no—
91C22×D59C2×D412yes—
92C22×D59C2×Q812yes—
93C22×D59C2×Q812yes—
94C22×D59#4112yes—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59—21no—
98C2×C4×C59—22no—
99C2×C4×C59—22no—
100C2×C4×C59—22no—
101C2×C4×C59—22no—
102C2×C4×C59—22no—
103C2×C4×C59—22no—
104C2×C4×C59C2×D412yes—
105C2×Dic59—22no—
106C2×Dic59—22no—
107C2×Dic59—21no—
108C2×Dic59—22no—
109C2×Dic59—22no—
110C2×Dic59—22no—
111C2×Dic59—22no—
112C2×Dic59—22no—
113C2×Dic59—22no—
114C2×Dic59C2×D412yes—
115C4×D59—44no—
116C4×D59—44no—
117Dic118—44no—
118C22×C59—22no—
119C22×C59—24no—
120C22×C59—41no—
121C22×C59—41no—
122C22×C59—42no—
123C22×C59C22×D411yescharacteristic
124C22×C59C22×Q811yescharacteristic
125C22×C59C2×#4111yescharacteristic
126C22×C59H(5,𝔽2)12yes—
127C22×C59H(5,𝔽2)12yes—
128C2×D59—22no—
129C2×D59—22no—
130C2×D59—24no—
131C2×D59—24no—
132C2×D59—22no—
133C2×D59—24no—
134C2×D59—416no—
135C2×D59—41no—
136C2×D59—41no—
137C2×D59—42no—
138C2×D59C22⋊Q814yes—
139C4×C59—22no—
140C4×C59—41no—
141C4×C59—42no—
142C4×C59—42no—
143Dic59—22no—
144Dic59—42no—
145Dic59—41no—
146Dic59—42no—
147#2626—591nomaximal, Sylow
148C2×C59—42no—
149C2×C59C2×C22⋊Q811yescharacteristic
150C2×C59#53411yescharacteristic
151C2×C59#53511yescharacteristic
152C2×C59#53812yes—
153C2×C59#54312yes—
154D59—24no—
155D59—42no—
156C2×C22⋊Q8—591no—
157C22×C22⋊C4—591no—
158#399—591no—
159#399D5911yescharacteristic, complemented
160#396—592no—
161#396—592no—
162#393—591no—
163#393—591no—
164#385—592no—
165#380—591no—
166#379—592no—
167C59#262611yescharacteristic, complemented, Sylow
168C25—591no—
169C22×Q8—591no—
170C23×C4—591no—
171C23×C4—591no—
172C22⋊Q8—1184no—
173C2×C4⋊C4—592no—
174C2×C4⋊C4—592no—
175C2×C4⋊C4—592no—
176C2×C4⋊C4C2×D5911yescharacteristic, complemented
177C2×C4⋊C4C2×D5912yescomplemented
178C2×C22⋊C4—591no—
179C2×C22⋊C4—591no—
180C2×C22⋊C4—591no—
181C2×C22⋊C4—592no—
182C2×C22⋊C4—592no—
183C2×C22⋊C4—592no—
184C2×C22⋊C4—592no—
185C2×C22⋊C4—592no—
186C2×C22⋊C4—1184no—
187C2×C22⋊C4C2×D5911yescharacteristic
188C2×C22⋊C4C2×D5912yes—
189#95—591no—
190#95—592no—
191#95—592no—
192#95C2×D5911yescharacteristic, complemented
193C24—591no—
194C24—591no—
195C24—594no—
196C24—1182no—
197C24—1182no—
198C24—2362no—
199C24—2362no—
200C24C22×D5911yescharacteristic
201C2×Q8—1182no—
202C2×Q8—2362no—
203C22×C4—591no—
204C22×C4—594no—
205C22×C4—592no—
206C22×C4—592no—
207C22×C4—591no—
208C22×C4—1182no—
209C22×C4—2362no—
210C22×C4—2362no—
211C22×C4C22×D5911yescharacteristic
212C22×C4C22×D5912yes—
213C22×C4C22×D5912yes—
214C22×C4C22×D5911yescharacteristic
215C4⋊C4—22no—
216C4⋊C4—42no—
217C4⋊C4—1184no—
218C4⋊C4—2362no—
219C4⋊C4—2362no—
220C22⋊C4—22no—
221C22⋊C4—42no—
222C22⋊C4—1182no—
223C22⋊C4—1184no—
224C22⋊C4—1184no—
225C22⋊C4—1182no—
226C22⋊C4—1182no—
227C22⋊C4—2362no—
228C22⋊C4—2362no—
229C22⋊C4—2362no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—22no—
234C23—592no—
235C23—592no—
236C23—592no—
237C23—592no—
238C23—1188no—
239C23—1182no—
240C23—1182no—
241C23—1181no—
242C23—1181no—
243C23—1182no—
244C23—1182no—
245C23—2368no—
246C23—2364no—
247C23—2364no—
248C23—2368no—
249C23Q8×D5912yes—
250C23C23×D5911yescharacteristic
251Q8—2364no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—592no—
262C2×C4—1182no—
263C2×C4—1181no—
264C2×C4—1182no—
265C2×C4—1182no—
266C2×C4—1182no—
267C2×C4—1182no—
268C2×C4—1182no—
269C2×C4—1182no—
270C2×C4—1182no—
271C2×C4—2364no—
272C2×C4—2364no—
273C2×C4D4×D5912yes—
274C22—22no—
275C22—24no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—594no—
280C22—1182no—
281C22—1182no—
282C22—1184no—
283C22—1184no—
284C22—1182no—
285C22—1184no—
286C22—23616no—
287C22—2361no—
288C22—2361no—
289C22—2362no—
290C22C2×D4×D5911yescharacteristic
291C22C2×Q8×D5911yescharacteristic
292C22#41×D5911yescharacteristic
293C22#1427377212yes—
294C22#1427377312yes—
295C4—22no—
296C4—41no—
297C4—42no—
298C4—42no—
299C4—1182no—
300C4—2362no—
301C4—2361no—
302C4—2362no—
303C2—42no—
304C2—1184no—
305C2—2362no—
306C2C22⋊Q8×D5911yescharacteristic
307C2#1683796311yescharacteristic
308C2#1683796511yescharacteristic
309C2#1683798812yes—
310C2#1683802712yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#399C2
hypercenter#399D59
hypocenterC59#2626
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#14273707C2semidirect
3. #16837337C2semidirect
4. #16837325C2semidirect
5. #16837324C2semidirect
6. #16837315C2semidirect
7. #16837314C2semidirect
8. #16837273C2semidirect
9. #16837261C2semidirect
11. C59×#399C2semidirect
13. C22×Dic118C22semidirect
17. C2×C4⋊Dic59C22semidirect
18. C2×Dic59⋊C4C22semidirect
19. C2×Dic59⋊C4C22semidirect
29. C59⋊#95C22semidirect
30. C59⋊#95C22semidirect
31. C59⋊#95C22semidirect
32. C2×C4⋊C4×C59C22semidirect
33. C2×C4⋊C4×C59C22semidirect
36. #95×C59C22semidirect
42. C23×D59Q8semidirect
159. #399D59semidirect
167. C59#2626semidirect
176. C2×C4⋊C4C2×D59semidirect
177. C2×C4⋊C4C2×D59semidirect
192. #95C2×D59semidirect