Finite Group Databasebeta

← #21427481

Subgroups — #21427481

Order 7552.

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

normal subgroups only

This group has 786 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×#14273710C211yesmaximal, characteristic, complemented
2C2×#14273696C211yesmaximal, characteristic
3C2×#14273686C211yesmaximal, characteristic, complemented
4#16837325C212yesmaximal, complemented
5#16837318C212yesmaximal, complemented
6#16837302C212yesmaximal, complemented
7#16837296C212yesmaximal, complemented
8#16837281C211yesmaximal, characteristic, complemented
9#16837275C211yesmaximal, characteristic, complemented
10#16837258C211yesmaximal, characteristic, complemented
11C22×C22⋊C4×C59C211yesmaximal, characteristic, complemented
12C22×C22⋊D59C2211yescharacteristic
13C22×C22⋊D59C2212yes—
14#14273710—24no—
15#14273696—24no—
16#14273686—24no—
17C2×C4⋊Dic59C2212yescomplemented
18C2×Dic59⋊C4C2211yescharacteristic, complemented
19C2×Dic59⋊C4C2211yescharacteristic, complemented
20C2×Dic59⋊C4C2212yescomplemented
21C2×C22⋊Dic59C2212yes—
22C2×C22⋊Dic59C2211yescharacteristic
23C2×C22⋊Dic59C2211yescharacteristic
24C2×C22⋊Dic59C2212yes—
25C2×C22⋊Dic59C2212yes—
26C2×C22⋊Dic59C2211yescharacteristic
27C2×(C2×D59)⋊C4C2211yescharacteristic, complemented
28C2×(C2×D59)⋊C4C2211yescharacteristic, complemented
29C2×(C2×D59)⋊C4C2212yescomplemented
30C2×C4×Dic59C2212yescomplemented
31C59⋊#95C2212yescomplemented
32C59⋊#95C2212yescomplemented
33C25×C59C2211yescharacteristic
34C23×C4×C59C2211yescharacteristic
35C23×C4×C59C2211yescharacteristic
36C2×C22⋊C4×C59—24no—
37C2×C22⋊C4×C59C2212yes—
38C2×C22⋊C4×C59C2212yes—
39C24×C59—22no—
40C24×C59—24no—
41C24×C59—22no—
42C24×C59—22no—
43C24×C59—42no—
44C24×C59C2312yes—
45C24×C59C2311yescharacteristic
46C23×D59C2311yescharacteristic
47C22×C4×C59—22no—
48C22×C4×C59—22no—
49C22×C4×C59—22no—
50C22×C4×C59—42no—
51C22×C4×C59D414yescomplemented
52C22×C4×C59C2312yes—
53C22×C4×C59C2311yescharacteristic
54C22×C4×C59C2311yescharacteristic
55C2×C22⋊D59—22no—
56C2×C22⋊D59—22no—
57C2×C22⋊D59—24no—
58C2×C22⋊D59—22no—
59C2×C22⋊D59—24no—
60C2×C22⋊D59—42no—
61C22×Dic59C2311yescharacteristic
62C22×Dic59C2311yescharacteristic
63C22×Dic59C2312yes—
64C22×Dic59C2311yescharacteristic
65C22×Dic59C2312yes—
66C22⋊C4×C59—24no—
67C22⋊C4×C59—24no—
68C22⋊Dic59—22no—
69C22⋊Dic59—22no—
70C22⋊Dic59—24no—
71C22⋊Dic59—24no—
72C22⋊Dic59—42no—
73C22⋊Dic59D414yes—
74(C2×D59)⋊C4—22no—
75(C2×D59)⋊C4—22no—
76(C2×D59)⋊C4—42no—
77C4⋊Dic59—24no—
78Dic59⋊C4—22no—
79Dic59⋊C4—22no—
80Dic59⋊C4—42no—
81C4×Dic59—24no—
82C23×C59—22no—
83C23×C59—22no—
84C23×C59—28no—
85C23×C59—22no—
86C23×C59—22no—
87C23×C59—22no—
88C23×C59—22no—
89C23×C59—22no—
90C23×C59—22no—
91C23×C59—48no—
92C23×C59—48no—
93C23×C59—44no—
94C23×C59—44no—
95C23×C59C2×D412yes—
96C23×C59C2×D412yes—
97C23×C59#4114yes—
98C23×C59#4112yes—
99C23×C59C2411yescharacteristic
100C22×D59—21no—
101C22×D59—21no—
102C22×D59—21no—
103C22×D59—22no—
104C22×D59—22no—
105C2×C4×C59—28no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—22no—
111C2×C4×C59—22no—
112C2×C4×C59—22no—
113C2×C4×C59—44no—
114C2×C4×C59C2×D412yes—
115C2×C4×C59C2×D412yes—
116C22⋊D59—48no—
117C22⋊D59—44no—
118C2×Dic59—22no—
119C2×Dic59—22no—
120C2×Dic59—22no—
121C2×Dic59—21no—
122C2×Dic59—21no—
123C2×Dic59—21no—
124C2×Dic59—21no—
125C2×Dic59—22no—
126C2×Dic59—22no—
127C2×Dic59—21no—
128C2×Dic59—22no—
129C2×Dic59—22no—
130C2×Dic59C2×D414yes—
131C22×C59—22no—
132C22×C59—22no—
133C22×C59—24no—
134C22×C59—24no—
135C22×C59—24no—
136C22×C59—24no—
137C22×C59—24no—
138C22×C59—416no—
139C22×C59—42no—
140C22×C59—42no—
141C22×C59—42no—
142C22×C59—42no—
143C22×C59#12114yes—
144C22×C59C22×D411yescharacteristic
145C22×C59C22×D411yescharacteristic
146C22×C59C2×#4111yescharacteristic
147C22×C59C2×#4112yes—
148C22×C59H(5,𝔽2)12yes—
149C2×D59—41no—
150C2×D59—41no—
151C2×D59—41no—
152C2×D59—42no—
153C2×D59—42no—
154C4×C59—22no—
155C4×C59—22no—
156C4×C59—42no—
157Dic59—24no—
158Dic59—41no—
159Dic59—41no—
160Dic59—41no—
161Dic59—42no—
162#2411—591nomaximal, Sylow
163C2×C59—24no—
164C2×C59—44no—
165C2×C59C2×#12111yescharacteristic
166C2×C59#52511yescharacteristic
167C2×C59#53511yescharacteristic
168C2×C59#54612yes—
169C2×C59#55112yes—
170D59—81no—
171C2×#123—591no—
172C2×#120—591no—
173C2×#117—591no—
174C22×C22⋊C4D5911yescharacteristic, complemented
175#396—592no—
176#394—592no—
177#390—592no—
178#388—592no—
179#386—591no—
180#385—591no—
181#379—591no—
182C59#241111yescharacteristic, complemented, Sylow
183C25C2×D5911yescharacteristic
184C22×D4—591no—
185C22×D4—592no—
186C23×C4C2×D5911yescharacteristic
187C23×C4C2×D5911yescharacteristic
188#123—1184no—
189#120—1184no—
190#117—1184no—
191C2×C4⋊C4—592no—
192C2×C4⋊C4—591no—
193C2×C4⋊C4—591no—
194C2×C4⋊C4—592no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—591no—
197C2×C22⋊C4—591no—
198C2×C22⋊C4—592no—
199C2×C22⋊C4—591no—
200C2×C22⋊C4—592no—
201C2×C22⋊C4—591no—
202C2×C22⋊C4—591no—
203C2×C22⋊C4—592no—
204C2×C22⋊C4—592no—
205C2×C22⋊C4C2×D5912yes—
206C2×C22⋊C4C2×D5912yes—
207C2×C42—592no—
208#95—592no—
209#95—592no—
210C24—24no—
211C24—22no—
212C24—22no—
213C24—22no—
214C24—42no—
215C24—591no—
216C24C22×D5912yes—
217C24C22×D5911yescharacteristic
218C2×D4—1184no—
219C2×D4—1184no—
220C2×D4—1182no—
221C2×D4—1182no—
222C2×D4—1182no—
223C2×D4—2362no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—42no—
228C22×C4—592no—
229C22×C4—591no—
230C22×C4—591no—
231C22×C4—592no—
232C22×C4—591no—
233C22×C4C22⋊D5914yescomplemented
234C22×C4C22×D5912yes—
235C22×C4C22×D5911yescharacteristic
236C22×C4C22×D5911yescharacteristic
237C4⋊C4—1184no—
238C4⋊C4—1182no—
239C4⋊C4—1182no—
240C4⋊C4—2362no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C22⋊C4—594no—
244C22⋊C4—1184no—
245C22⋊C4—1184no—
246C22⋊C4—1182no—
247C22⋊C4—1182no—
248C22⋊C4—1182no—
249C22⋊C4—1182no—
250C22⋊C4—2362no—
251C22⋊C4—2362no—
252C42—1184no—
253C23—22no—
254C23—22no—
255C23—28no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—48no—
263C23—48no—
264C23—44no—
265C23—44no—
266C23—1181no—
267C23—1181no—
268C23—1181no—
269C23—1182no—
270C23—1182no—
271C23#1174568514yes—
272C23D4×D5912yes—
273C23#1174568712yes—
274C23C2×C22⋊D5912yes—
275C23C23×D5911yescharacteristic
276D4—2368no—
277D4—2364no—
278C2×C4—28no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—44no—
287C2×C4—594no—
288C2×C4—1182no—
289C2×C4—1182no—
290C2×C4—1182no—
291C2×C4—1181no—
292C2×C4—1181no—
293C2×C4—1181no—
294C2×C4—1181no—
295C2×C4—1181no—
296C2×C4—1182no—
297C2×C4—1182no—
298C2×C4—1182no—
299C2×C4—1182no—
300C2×C4C2×C22⋊D5912yes—
301C2×C4C2×C22⋊D5912yes—
302C22—22no—
303C22—22no—
304C22—24no—
305C22—24no—
306C22—24no—
307C22—24no—
308C22—24no—
309C22—416no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—2361no—
315C22—2361no—
316C22—2361no—
317C22—2362no—
318C22—2362no—
319C22#1427369914yes—
320C22C2×D4×D5911yescharacteristic
321C22C22×C22⋊D5911yescharacteristic
322C22C2×#1174568711yescharacteristic
323C22C2×#1174568512yes—
324C22#1427377212yes—
325C4—22no—
326C4—22no—
327C4—42no—
328C4—1184no—
329C4—2361no—
330C4—2361no—
331C4—2361no—
332C4—2362no—
333C2—24no—
334C2—44no—
335C2—4721no—
336C2C2×#1427369911yescharacteristic
337C2#1683791311yescharacteristic
338C2#1683796911yescharacteristic
339C2#1683804212yes—
340C2#1683807812yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC22×C22⋊C4×C59C2
hypercenterC22×C22⋊C4D59
hypocenterC59#2411
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#14273710C2semidirect
3. C2×#14273686C2semidirect
4. #16837325C2semidirect
5. #16837318C2semidirect
6. #16837302C2semidirect
7. #16837296C2semidirect
8. #16837281C2semidirect
9. #16837275C2semidirect
10. #16837258C2semidirect
11. C22×C22⋊C4×C59C2semidirect
17. C2×C4⋊Dic59C22semidirect
18. C2×Dic59⋊C4C22semidirect
19. C2×Dic59⋊C4C22semidirect
20. C2×Dic59⋊C4C22semidirect
27. C2×(C2×D59)⋊C4C22semidirect
28. C2×(C2×D59)⋊C4C22semidirect
29. C2×(C2×D59)⋊C4C22semidirect
30. C2×C4×Dic59C22semidirect
31. C59⋊#95C22semidirect
32. C59⋊#95C22semidirect
51. C22×C4×C59D4semidirect
174. C22×C22⋊C4D59semidirect
182. C59#2411semidirect
233. C22×C4C22⋊D59semidirect