Finite Group Databasebeta

← #21429573

Subgroups — #21429573

Order 7552.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14273699C211yesmaximal, characteristic
2C22×C4⋊Dic59C211yesmaximal, characteristic, complemented
3#16837344C212yesmaximal, complemented
4#16837318C212yesmaximal, complemented
5#16837295C212yesmaximal, complemented
6#16837294C212yesmaximal, complemented
7#16837282C211yesmaximal, characteristic, complemented
8#16837281C211yesmaximal, characteristic, complemented
9#16837279C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic59C211yesmaximal, characteristic, complemented
11C59×#386C211yesmaximal, characteristic, complemented
12C22×C22⋊D59C2211yescharacteristic
13C22×C22⋊D59C2211yescharacteristic
14C23×Dic59C2211yescharacteristic
15C23×Dic59C2211yescharacteristic
16#14273699—24no—
17C2×C4⋊Dic59—24no—
18C2×C4⋊Dic59C2212yes—
19C2×C4⋊Dic59C2212yes—
20C2×Dic59⋊C4C2211yescharacteristic
21C2×C22⋊Dic59C2211yescharacteristic
22C2×C22⋊Dic59C2211yescharacteristic
23C2×C22⋊Dic59C2212yes—
24C2×(C2×D59)⋊C4C2211yescharacteristic
25C2×(C2×D59)⋊C4C2212yes—
26C2×(C2×D59)⋊C4C2212yes—
27C2×C4×Dic59C2212yes—
28C59⋊#95C2212yes—
29C59⋊#95C2212yes—
30C59⋊#95C2212yes—
31C59⋊#95C2212yes—
32C22×D4×C59C2211yescharacteristic
33C23×C4×C59C2211yescharacteristic
34C2×C22⋊C4×C59C2211yescharacteristic
35C2×C22⋊C4×C59C2212yes—
36#95×C59C2212yes—
37C24×C59C2311yescharacteristic
38C24×C59C2311yescharacteristic
39C23×D59C2311yescharacteristic
40C2×D4×C59—22no—
41C2×D4×C59—22no—
42C2×D4×C59—22no—
43C22×C4×C59—22no—
44C22×C4×C59—24no—
45C22×C4×C59C2311yescharacteristic
46C22×C4×C59C2312yes—
47C22×C4×C59C2312yes—
48C2×C22⋊D59—22no—
49C2×C22⋊D59—22no—
50C2×C22⋊D59—22no—
51C2×C22⋊D59—22no—
52C2×C22⋊D59—42no—
53C22×Dic59—22no—
54C22×Dic59—42no—
55C22×Dic59—42no—
56C22×Dic59D414yes—
57C22×Dic59C2311yescharacteristic
58C22×Dic59C2312yes—
59C22×Dic59C2312yes—
60C22×Dic59C2311yescharacteristic
61C22×Dic59C2311yescharacteristic
62C22⋊C4×C59—22no—
63C22⋊C4×C59—24no—
64C22⋊Dic59—22no—
65C22⋊Dic59—22no—
66C22⋊Dic59—42no—
67(C2×D59)⋊C4—24no—
68(C2×D59)⋊C4—22no—
69(C2×D59)⋊C4—42no—
70C4⋊Dic59—24no—
71C4⋊Dic59—24no—
72Dic59⋊C4—22no—
73C4×Dic59—42no—
74C23×C59—22no—
75C23×C59—22no—
76C23×C59—21no—
77C23×C59—22no—
78C23×C59—21no—
79C23×C59—21no—
80C23×C59—22no—
81C23×C59C2×D412yes—
82C23×C59#4112yes—
83C23×C59#4112yes—
84C23×C59C2411yescharacteristic
85C22×D59—21no—
86C22×D59—22no—
87C22×D59—21no—
88C22×D59—21no—
89C22×D59—22no—
90D4×C59—44no—
91C2×C4×C59—28no—
92C2×C4×C59—22no—
93C2×C4×C59—22no—
94C2×C4×C59—21no—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59—21no—
98C2×C4×C59—21no—
99C2×C4×C59—22no—
100C2×C4×C59#4114yes—
101C22⋊D59—44no—
102C22⋊D59—44no—
103C2×Dic59—22no—
104C2×Dic59—22no—
105C2×Dic59—21no—
106C2×Dic59—22no—
107C2×Dic59—22no—
108C2×Dic59—22no—
109C2×Dic59—21no—
110C2×Dic59—21no—
111C2×Dic59—22no—
112C2×Dic59—22no—
113C2×Dic59—22no—
114C2×Dic59—44no—
115C2×Dic59—44no—
116C2×Dic59C2×D412yes—
117C2×Dic59C2×D412yes—
118C22×C59—22no—
119C22×C59—22no—
120C22×C59—24no—
121C22×C59—24no—
122C22×C59—41no—
123C22×C59—41no—
124C22×C59—41no—
125C22×C59—42no—
126C22×C59—42no—
127C22×C59#12414yes—
128C22×C59C22×D411yescharacteristic
129C22×C59C2×#4112yes—
130C22×C59C2×#4111yescharacteristic
131C22×C59C2×#4111yescharacteristic
132C22×C59H(5,𝔽2)12yes—
133C2×D59—41no—
134C2×D59—41no—
135C2×D59—41no—
136C2×D59—42no—
137C2×D59—42no—
138C4×C59—24no—
139C4×C59—41no—
140C4×C59—42no—
141Dic59—22no—
142Dic59—22no—
143Dic59—41no—
144Dic59—42no—
145Dic59—42no—
146#2591—591nomaximal, Sylow
147C2×C59—24no—
148C2×C59—81no—
149C2×C59#52511yescharacteristic
150C2×C59C2×#12411yescharacteristic
151C2×C59#53411yescharacteristic
152C2×C59#55312yes—
153C2×C59#55312yes—
154D59—81no—
155C2×#121—591no—
156C22×C4⋊C4—591no—
157#400—592no—
158#394—592no—
159#388—592no—
160#388—592no—
161#386—591no—
162#386—591no—
163#386—591no—
164#386D5911yescharacteristic, complemented
165C4×C22⋊C4—591no—
166C59#259111yescharacteristic, complemented, Sylow
167C22×D4—591no—
168C22×D4—591no—
169C22×D4C2×D5911yescharacteristic
170C23×C4—591no—
171C23×C4—591no—
172C23×C4C2×D5911yescharacteristic
173#121—1184no—
174C2×C4⋊C4—592no—
175C2×C4⋊C4—592no—
176C2×C4⋊C4—591no—
177C2×C4⋊C4—1184no—
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⋊C4C2×D5911yescharacteristic
185C2×C22⋊C4C2×D5912yes—
186C2×C42—592no—
187#95—592no—
188#95—592no—
189#95—592no—
190#95—592no—
191#95C2×D5912yes—
192C24—591no—
193C24C22×D5911yescharacteristic
194C24C22×D5911yescharacteristic
195C2×D4—22no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—1182no—
199C2×D4—1182no—
200C2×D4—1182no—
201C2×D4—1182no—
202C2×D4—2362no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—591no—
206C22×C4—591no—
207C22×C4—594no—
208C22×C4—591no—
209C22×C4—592no—
210C22×C4—592no—
211C22×C4—1182no—
212C22×C4—2362no—
213C22×C4—2362no—
214C22×C4C22×D5911yescharacteristic
215C22×C4C22×D5912yes—
216C22×C4C22×D5912yes—
217C4⋊C4—1184no—
218C4⋊C4—1184no—
219C4⋊C4—1182no—
220C22⋊C4—22no—
221C22⋊C4—24no—
222C22⋊C4—1182no—
223C22⋊C4—1182no—
224C22⋊C4—1182no—
225C22⋊C4—1184no—
226C22⋊C4—2362no—
227C22⋊C4—2362no—
228C42—2362no—
229C23—22no—
230C23—22no—
231C23—21no—
232C23—22no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—1181no—
237C23—1182no—
238C23—1181no—
239C23—1181no—
240C23—1182no—
241C23#1174568512yes—
242C23D4×D5912yes—
243C23#1174568712yes—
244C23C23×D5911yescharacteristic
245D4—44no—
246D4—2364no—
247D4—2364no—
248C2×C4—28no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—592no—
258C2×C4—592no—
259C2×C4—1182no—
260C2×C4—1182no—
261C2×C4—1181no—
262C2×C4—1182no—
263C2×C4—1182no—
264C2×C4—1181no—
265C2×C4—1182no—
266C2×C4—1181no—
267C2×C4—1182no—
268C2×C4—1182no—
269C2×C4—1182no—
270C2×C4—2364no—
271C2×C4—2364no—
272C2×C4#1174568714yes—
273C22—22no—
274C22—22no—
275C22—24no—
276C22—24no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—42no—
281C22—42no—
282C22—2361no—
283C22—2361no—
284C22—2361no—
285C22—2362no—
286C22—2362no—
287C22#1427371314yes—
288C22C2×D4×D5911yescharacteristic
289C22C2×#1174568712yes—
290C22C2×#1174568711yescharacteristic
291C22C2×#1174568511yescharacteristic
292C22#1427377312yes—
293C4—24no—
294C4—41no—
295C4—42no—
296C4—1182no—
297C4—1182no—
298C4—2361no—
299C4—2362no—
300C4—2362no—
301C2—24no—
302C2—81no—
303C2—4721no—
304C2#1683790811yescharacteristic
305C2C2×#1427371311yescharacteristic
306C2#1683796311yescharacteristic
307C2#1683809812yes—
308C2#1683810412yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#386C2
hypercenter#386D59
hypocenterC59#2591
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊Dic59C2semidirect
3. #16837344C2semidirect
4. #16837318C2semidirect
5. #16837295C2semidirect
6. #16837294C2semidirect
7. #16837282C2semidirect
8. #16837281C2semidirect
9. #16837279C2semidirect
10. C22⋊C4×Dic59C2semidirect
11. C59×#386C2semidirect
164. #386D59semidirect
166. C59#2591semidirect