Finite Group Databasebeta

← #21427699

Subgroups — #21427699

Order 7552.

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

normal subgroups only

This group has 914 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
1C22×#11745687C211yesmaximal, characteristic, complemented
2C2×#14273709C211yesmaximal, characteristic, complemented
3C2×#14273698C212yesmaximal, complemented
4#16837323C211yesmaximal, characteristic, complemented
5#16837316C211yesmaximal, characteristic, complemented
6#16837314C211yesmaximal, characteristic, complemented
7#16837282C212yesmaximal, complemented
8#16837278C212yesmaximal, complemented
9#16837277C212yesmaximal, complemented
10#95×D59C211yesmaximal, characteristic, complemented
11C59×#392C211yesmaximal, characteristic, complemented
12C2×#11745687—28no—
13C2×#11745687—24no—
14C22×Dic118C2211yescharacteristic
15C22×C22⋊D59C2211yescharacteristic
16C22×C22⋊D59C2212yes—
17C23×Dic59C2212yes—
18C22×C4×D59C2212yes—
19C22×C4×D59C2211yescharacteristic
20#14273709—24no—
21#14273698—28no—
22C2×C4⋊Dic59C2212yes—
23C2×C4⋊Dic59C2211yescharacteristic
24C2×Dic59⋊C4C2211yescharacteristic
25C2×C22⋊Dic59C2212yes—
26C2×C22⋊Dic59C2212yes—
27C2×C22⋊Dic59C2211yescharacteristic
28C2×C22⋊Dic59C2212yes—
29C2×(C2×D59)⋊C4C2211yescharacteristic
30C59⋊#95C2212yes—
31C59⋊#95C2211yescharacteristic
32C59⋊#95C2212yes—
33C59⋊#95C2212yes—
34C22×D4×C59C2211yescharacteristic
35C22×D4×C59C2212yes—
36C2×C22⋊C4×C59C2211yescharacteristic
37C2×C22⋊C4×C59C2212yes—
38#95×C59C2211yescharacteristic
39C24×C59C2311yescharacteristic
40C24×C59C2312yes—
41C23×D59C2311yescharacteristic
42C2×D4×C59—24no—
43C2×D4×C59—24no—
44C2×D4×C59—22no—
45C2×D4×C59—24no—
46C2×D4×C59—42no—
47C22×C4×C59C2311yescharacteristic
48C22×C4×C59C2311yescharacteristic
49C22×C4×C59C2312yes—
50C2×C22⋊D59—24no—
51C2×C22⋊D59—24no—
52C2×C22⋊D59—24no—
53C2×C22⋊D59—22no—
54C2×C22⋊D59D418yescomplemented
55C22×Dic59—24no—
56C22×Dic59—24no—
57C22×Dic59—24no—
58C22×Dic59C2312yes—
59C22×Dic59C2311yescharacteristic
60C22×Dic59C2311yescharacteristic
61C22×Dic59C2311yescharacteristic
62C22×Dic59C2312yes—
63#11745687—232no—
64C2×C4×D59—24no—
65C2×C4×D59—24no—
66C2×C4×D59—24no—
67C2×C4×D59—42no—
68C2×C4×D59D414yescomplemented
69C2×Dic118—22no—
70C2×Dic118—24no—
71C22⋊C4×C59—22no—
72C22⋊C4×C59—24no—
73C22⋊Dic59—24no—
74C22⋊Dic59—24no—
75C22⋊Dic59—22no—
76C22⋊Dic59—42no—
77(C2×D59)⋊C4—22no—
78C4⋊Dic59—24no—
79C4⋊Dic59—22no—
80Dic59⋊C4—22no—
81C23×C59—22no—
82C23×C59—22no—
83C23×C59—22no—
84C23×C59—22no—
85C23×C59—22no—
86C23×C59—22no—
87C23×C59—21no—
88C23×C59—22no—
89C23×C59—22no—
90C23×C59—21no—
91C23×C59—21no—
92C23×C59C2×D414yes—
93C23×C59C2411yescharacteristic
94C22×D59—22no—
95C22×D59—21no—
96C22×D59—21no—
97C22×D59C2×D412yes—
98C22×D59C2×D414yes—
99D4×C59—28no—
100D4×C59—48no—
101C2×C4×C59—22no—
102C2×C4×C59—22no—
103C2×C4×C59—22no—
104C2×C4×C59—21no—
105C2×C4×C59—22no—
106C2×C4×C59C2×D412yes—
107C2×C4×C59#4114yescomplemented
108C22⋊D59—216no—
109C22⋊D59—44no—
110C2×Dic59—216no—
111C2×Dic59—22no—
112C2×Dic59—22no—
113C2×Dic59—22no—
114C2×Dic59—21no—
115C2×Dic59—21no—
116C2×Dic59—22no—
117C2×Dic59—22no—
118C2×Dic59—22no—
119C2×Dic59—21no—
120C2×Dic59—22no—
121C2×Dic59—21no—
122C2×Dic59C2×D414yes—
123C2×Dic59C2×D412yes—
124C4×D59—28no—
125C4×D59—48no—
126Dic118—28no—
127C22×C59—24no—
128C22×C59—24no—
129C22×C59—24no—
130C22×C59—42no—
131C22×C59—42no—
132C22×C59—42no—
133C22×C59—42no—
134C22×C59—41no—
135C22×C59—41no—
136C22×C59—41no—
137C22×C59—42no—
138C22×C59—42no—
139C22×C59C22×D411yescharacteristic
140C22×C59C22×D412yes—
141C22×C59C2×#4112yes—
142C22×C59H(5,𝔽2)11yescharacteristic
143C22×C592−1+411yescharacteristic
144C2×D59—22no—
145C2×D59—22no—
146C2×D59—24no—
147C2×D59—24no—
148C2×D59#12014yes—
149C4×C59—22no—
150C4×C59—24no—
151C4×C59—41no—
152Dic59—24no—
153Dic59—22no—
154Dic59—41no—
155Dic59—41no—
156Dic59—42no—
157#2430—591nomaximal, Sylow
158C2×C59—44no—
159C2×C59—81no—
160C2×C59C2×#12011yescharacteristic
161C2×C59#53211yescharacteristic
162C2×C59#53711yescharacteristic
163C2×C59#54612yes—
164C2×C59#54712yes—
165D59—24no—
166C22×#41—591no—
167C2×#123—591no—
168C2×#121—592no—
169#395—591no—
170#394—591no—
171#393—591no—
172#392D5911yescharacteristic, complemented
173#386—592no—
174#386—592no—
175#385—592no—
176C2×#95—591no—
177C59#243011yescharacteristic, complemented, Sylow
178C2×#41—1184no—
179C2×#41—1188no—
180C22×Q8—591no—
181C22×D4—592no—
182C22×D4—591no—
183C22×D4C2×D5911yescharacteristic
184C22×D4C2×D5912yes—
185C23×C4—591no—
186C23×C4—592no—
187C23×C4—592no—
188#123—1184no—
189#121—1188no—
190C2×C4⋊C4—592no—
191C2×C4⋊C4—591no—
192C2×C4⋊C4—591no—
193C2×C22⋊C4—592no—
194C2×C22⋊C4—592no—
195C2×C22⋊C4—591no—
196C2×C22⋊C4—591no—
197C2×C22⋊C4—592no—
198C2×C22⋊C4C2×D5911yescharacteristic
199C2×C22⋊C4C2×D5912yes—
200#95—592no—
201#95—591no—
202#95—592no—
203#95—592no—
204#95C2×D5911yescharacteristic
205C24—591no—
206C24C22×D5911yescharacteristic
207C24C22×D5912yes—
208#41—11832no—
209C2×Q8—1184no—
210C2×Q8—1182no—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—22no—
214C2×D4—24no—
215C2×D4—42no—
216C2×D4—598no—
217C2×D4—1184no—
218C2×D4—1184no—
219C2×D4—1184no—
220C2×D4—1182no—
221C22×C4—591no—
222C22×C4—592no—
223C22×C4—594no—
224C22×C4—592no—
225C22×C4—591no—
226C22×C4—591no—
227C22×C4—1184no—
228C22×C4—1184no—
229C22×C4—1184no—
230C22×C4—1184no—
231C22×C4—1184no—
232C22×C4—1184no—
233C22×C4—2362no—
234C22×C4C22×D5911yescharacteristic
235C22×C4C22×D5911yescharacteristic
236C22×C4C22×D5912yes—
237C4⋊C4—1184no—
238C4⋊C4—1182no—
239C4⋊C4—1182no—
240C22⋊C4—22no—
241C22⋊C4—24no—
242C22⋊C4—1182no—
243C22⋊C4—1184no—
244C22⋊C4—1182no—
245C22⋊C4—1184no—
246C22⋊C4—2362no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—21no—
254C23—22no—
255C23—22no—
256C23—21no—
257C23—21no—
258C23—592no—
259C23—594no—
260C23—1182no—
261C23—1181no—
262C23—1181no—
263C23D4×D5914yes—
264C23C23×D5911yescharacteristic
265Q8—1188no—
266D4—28no—
267D4—48no—
268D4—11816no—
269D4—2364no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—592no—
276C2×C4—594no—
277C2×C4—1188no—
278C2×C4—11816no—
279C2×C4—1182no—
280C2×C4—1182no—
281C2×C4—1182no—
282C2×C4—1181no—
283C2×C4—1182no—
284C2×C4—1182no—
285C2×C4—1181no—
286C2×C4—1182no—
287C2×C4—1181no—
288C2×C4—1182no—
289C2×C4—1181no—
290C2×C4—2368no—
291C2×C4D4×D5912yes—
292C2×C4#1174568714yescomplemented
293C22—24no—
294C22—24no—
295C22—24no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—41no—
301C22—41no—
302C22—42no—
303C22—42no—
304C22—41no—
305C22—594no—
306C22—1182no—
307C22—1182no—
308C22—1184no—
309C22—1184no—
310C22C2×D4×D5911yescharacteristic
311C22C2×D4×D5912yes—
312C22C2×#1174568712yes—
313C22#1427377211yescharacteristic
314C22#1427377411yescharacteristic
315C4—24no—
316C4—22no—
317C4—41no—
318C4—1182no—
319C4—1184no—
320C4—2361no—
321C4—2362no—
322C4—2361no—
323C2—44no—
324C2—81no—
325C2—1184no—
326C2#120×D5911yescharacteristic
327C2#1683795011yescharacteristic
328C2#1683798111yescharacteristic
329C2#1683803912yes—
330C2#1683805212yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#392C2
hypercenter#392D59
hypocenterC59#2430
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×#11745687C2semidirect
2. C2×#14273709C2semidirect
3. C2×#14273698C2semidirect
4. #16837323C2semidirect
5. #16837316C2semidirect
6. #16837314C2semidirect
7. #16837282C2semidirect
8. #16837278C2semidirect
9. #16837277C2semidirect
10. #95×D59C2semidirect
11. C59×#392C2semidirect
54. C2×C22⋊D59D4semidirect
68. C2×C4×D59D4semidirect
107. C2×C4×C59#41semidirect
172. #392D59semidirect
177. C59#2430semidirect
292. C2×C4#11745687semidirect