Finite Group Databasebeta

← #21428562

Subgroups — #21428562

Order 7552.

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

normal subgroups only

This group has 708 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×#14273712C212yesmaximal, complemented
2C2×#14273709C212yesmaximal, complemented
3C2×#14273700C211yesmaximal, characteristic
4C2×C4⋊C4×D59C211yesmaximal, characteristic, complemented
5#16837339C211yesmaximal, characteristic, complemented
6#16837338C211yesmaximal, characteristic, complemented
7#16837337C211yesmaximal, characteristic, complemented
8#16837302C212yesmaximal, complemented
9#16837271C212yesmaximal, complemented
10#16837249C211yesmaximal, characteristic, complemented
11C59×#399C211yesmaximal, characteristic, complemented
12C22×D236C2211yescharacteristic
13C22×C22⋊D59C2211yescharacteristic
14C22×C22⋊D59C2211yescharacteristic
15C22×C4×D59C2212yes—
16C22×C4×D59C2211yescharacteristic
17#14273712—28no—
18#14273709—28no—
19#14273700—24no—
20C2×C4⋊Dic59C2212yes—
21C2×C4⋊Dic59C2211yescharacteristic
22C2×Dic59⋊C4C2211yescharacteristic
23C2×Dic59⋊C4C2212yes—
24C2×Dic59⋊C4C2212yes—
25C2×Dic59⋊C4C2212yes—
26C4⋊C4×D59—24no—
27C2×C22⋊Dic59C2212yes—
28C2×(C2×D59)⋊C4C2211yescharacteristic
29C2×(C2×D59)⋊C4C2212yes—
30C2×(C2×D59)⋊C4C2212yes—
31C2×(C2×D59)⋊C4C2212yes—
32C2×C4×Dic59C2212yes—
33C59⋊#95C2211yescharacteristic
34C2×C4⋊C4×C59C2212yes—
35C2×C4⋊C4×C59C2211yescharacteristic
36C2×C22⋊C4×C59C2211yescharacteristic
37C2×C22⋊C4×C59C2212yes—
38#95×C59C2211yescharacteristic
39C24×C59C2311yescharacteristic
40C23×D59C2311yescharacteristic
41C23×D59C2311yescharacteristic
42C22×C4×C59C2312yes—
43C22×C4×C59C2311yescharacteristic
44C22×C4×C59C2311yescharacteristic
45C22×C4×C59C2312yes—
46C2×C22⋊D59—22no—
47C2×C22⋊D59—24no—
48C2×C22⋊D59—22no—
49C2×C22⋊D59—24no—
50C22×Dic59C2312yes—
51C22×Dic59C2311yescharacteristic
52C22×Dic59C2312yes—
53C22×Dic59C2311yescharacteristic
54C2×D236—22no—
55C2×D236—24no—
56C2×C4×D59—24no—
57C2×C4×D59—24no—
58C2×C4×D59—24no—
59C2×C4×D59—24no—
60C2×C4×D59D414yes—
61C4⋊C4×C59—22no—
62C4⋊C4×C59—24no—
63C22⋊C4×C59—22no—
64C22⋊C4×C59—24no—
65C22⋊Dic59—24no—
66(C2×D59)⋊C4—22no—
67(C2×D59)⋊C4—24no—
68(C2×D59)⋊C4—24no—
69(C2×D59)⋊C4D418yescomplemented
70C4⋊Dic59—22no—
71C4⋊Dic59—24no—
72Dic59⋊C4—24no—
73Dic59⋊C4—22no—
74Dic59⋊C4—24no—
75Dic59⋊C4—24no—
76C4×Dic59—24no—
77C23×C59—22no—
78C23×C59—21no—
79C23×C59—22no—
80C23×C59—21no—
81C23×C59—21no—
82C23×C59C2411yescharacteristic
83C22×D59—22no—
84C22×D59—21no—
85C22×D59—21no—
86C22×D59—22no—
87C22×D59—21no—
88C22×D59—22no—
89C22×D59—21no—
90C22×D59—21no—
91C22×D59C2×D414yes—
92C22×D59C2×D412yes—
93C2×C4×C59—22no—
94C2×C4×C59—22no—
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×C59C2×D414yes—
102C2×C4×C59C2×D412yes—
103C22⋊D59—44no—
104C22⋊D59—44no—
105C2×Dic59—22no—
106C2×Dic59—22no—
107C2×Dic59—22no—
108C2×Dic59—22no—
109C2×Dic59—22no—
110C2×Dic59—21no—
111C2×Dic59—22no—
112C2×Dic59C2×D414yes—
113C2×Dic59C2×D412yes—
114C2×Dic59#4114yes—
115D236—44no—
116C4×D59—28no—
117C4×D59—48no—
118C22×C59—41no—
119C22×C59—41no—
120C22×C59—41no—
121C22×C59—42no—
122C22×C59—42no—
123C22×C59C22×D411yescharacteristic
124C22×C59C22×D412yes—
125C22×C59C2×#4112yes—
126C22×C592−1+411yescharacteristic
127C22×C592−1+411yescharacteristic
128C2×D59—22no—
129C2×D59—22no—
130C2×D59—24no—
131C2×D59—24no—
132C2×D59—41no—
133C2×D59—41no—
134C2×D59—41no—
135C2×D59—42no—
136C2×D59—42no—
137C2×D59#12714yes—
138C4×C59—22no—
139C4×C59—24no—
140C4×C59—42no—
141C4×C59—41no—
142Dic59—24no—
143Dic59—42no—
144Dic59—41no—
145Dic59#12314yescomplemented
146#2501—591nomaximal, Sylow
147C2×C59—81no—
148C2×C59C2×#12311yescharacteristic
149C2×C59C2×#12711yescharacteristic
150C2×C59#53611yescharacteristic
151C2×C59#54712yes—
152C2×C59#54712yes—
153D59—24no—
154D59—81no—
155C2×#123—592no—
156C2×#123—592no—
157C2×#121—591no—
158C22×C4⋊C4—591no—
159#399—591no—
160#399—591no—
161#399—591no—
162#399D5911yescharacteristic, complemented
163#390—592no—
164#384—592no—
165C4×C22⋊C4—591no—
166C59#250111yescharacteristic, complemented, Sylow
167C22×D4—591no—
168C22×D4—591no—
169C22×D4—591no—
170C23×C4—591no—
171C23×C4—592no—
172#123—1188no—
173#123—1188no—
174#121—1184no—
175C2×C4⋊C4—592no—
176C2×C4⋊C4—591no—
177C2×C4⋊C4—591no—
178C2×C4⋊C4—592no—
179C2×C4⋊C4—592no—
180C2×C4⋊C4—592no—
181C2×C4⋊C4—1184no—
182C2×C4⋊C4C2×D5911yescharacteristic
183C2×C4⋊C4C2×D5912yes—
184C2×C22⋊C4—592no—
185C2×C22⋊C4—592no—
186C2×C22⋊C4—591no—
187C2×C22⋊C4—592no—
188C2×C22⋊C4—592no—
189C2×C22⋊C4C2×D5911yescharacteristic
190C2×C22⋊C4C2×D5912yes—
191C2×C42—592no—
192#95—591no—
193#95C2×D5911yescharacteristic
194C24—591no—
195C24—591no—
196C24C22×D5911yescharacteristic
197C2×D4—1182no—
198C2×D4—1184no—
199C2×D4—1182no—
200C2×D4—1184no—
201C2×D4—1182no—
202C2×D4—1184no—
203C22×C4—591no—
204C22×C4—592no—
205C22×C4—594no—
206C22×C4—592no—
207C22×C4—591no—
208C22×C4—1184no—
209C22×C4—1184no—
210C22×C4—1184no—
211C22×C4—1184no—
212C22×C4C22×D5912yes—
213C22×C4C22×D5911yescharacteristic
214C22×C4C22×D5911yescharacteristic
215C22×C4C22×D5912yes—
216C4⋊C4—22no—
217C4⋊C4—24no—
218C4⋊C4—1184no—
219C4⋊C4—1182no—
220C4⋊C4—1184no—
221C4⋊C4—1184no—
222C4⋊C4—1182no—
223C4⋊C4—1184no—
224C22⋊C4—22no—
225C22⋊C4—24no—
226C22⋊C4—598no—
227C22⋊C4—1184no—
228C22⋊C4—1184no—
229C22⋊C4—1182no—
230C22⋊C4—1184no—
231C42—1184no—
232C23—22no—
233C23—21no—
234C23—22no—
235C23—21no—
236C23—21no—
237C23—594no—
238C23—592no—
239C23—1182no—
240C23—1181no—
241C23—1181no—
242C23—1182no—
243C23—1181no—
244C23—1182no—
245C23—1181no—
246C23—1181no—
247C23C23×D5911yescharacteristic
248D4—2364no—
249D4—2364no—
250D4—2364no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—592no—
260C2×C4—594no—
261C2×C4—594no—
262C2×C4—1188no—
263C2×C4—1182no—
264C2×C4—1182no—
265C2×C4—1182no—
266C2×C4—1182no—
267C2×C4—1181no—
268C2×C4—1182no—
269C2×C4—1182no—
270C2×C4—2368no—
271C2×C4D4×D5914yes—
272C2×C4D4×D5912yes—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—42no—
277C22—42no—
278C22—594no—
279C22—1182no—
280C22—1182no—
281C22—1184no—
282C22—1184no—
283C22—2361no—
284C22—2361no—
285C22—2361no—
286C22—2362no—
287C22—2362no—
288C22C2×D4×D5911yescharacteristic
289C22C2×D4×D5912yes—
290C22#41×D5912yes—
291C22#1427377411yescharacteristic
292C22#1427377511yescharacteristic
293C4—22no—
294C4—24no—
295C4—42no—
296C4—41no—
297C4—594no—
298C4—1184no—
299C4—2362no—
300C4—2361no—
301C2—81no—
302C2—1184no—
303C2—4721no—
304C2#123×D5911yescharacteristic
305C2#127×D5911yescharacteristic
306C2#1683797511yescharacteristic
307C2#1683805012yes—
308C2#1683805412yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#399C2
hypercenter#399D59
hypocenterC59#2501
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#14273712C2semidirect
2. C2×#14273709C2semidirect
4. C2×C4⋊C4×D59C2semidirect
5. #16837339C2semidirect
6. #16837338C2semidirect
7. #16837337C2semidirect
8. #16837302C2semidirect
9. #16837271C2semidirect
10. #16837249C2semidirect
11. C59×#399C2semidirect
69. (C2×D59)⋊C4D4semidirect
145. Dic59#123semidirect
162. #399D59semidirect
166. C59#2501semidirect