Finite Group Databasebeta

← #21430225

Subgroups — #21430225

Order 7552.

598 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 137 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
2#16837333C211yesmaximal, characteristic, complemented
3#16837328C211yesmaximal, characteristic, complemented
4#16837316C211yesmaximal, characteristic, complemented
5#16837316C212yesmaximal, complemented
6#16837310C211yesmaximal, characteristic, complemented
7#16837310C212yesmaximal, complemented
8#16837302C212yesmaximal, complemented
9#16837296C212yesmaximal, complemented
10C4×C22⋊Dic59C211yesmaximal, characteristic, complemented
11C2×#121×C59C211yesmaximal, characteristic, complemented
12C22×C22⋊D59C2211yescharacteristic
13C22×C22⋊D59C2212yes—
14#14273710—24no—
15C2×Dic59⋊C4C2211yescharacteristic
16C2×Dic59⋊C4C2211yescharacteristic
17C2×Dic59⋊C4C2212yes—
18C2×C22⋊Dic59C2211yescharacteristic
19C2×C22⋊Dic59C2212yes—
20C2×C22⋊Dic59C2212yes—
21C2×C22⋊Dic59C2212yes—
22C2×C22⋊Dic59C2212yes—
23C2×(C2×D59)⋊C4C2211yescharacteristic
24C2×(C2×D59)⋊C4C2211yescharacteristic
25C2×(C2×D59)⋊C4C2212yes—
26C2×C4×Dic59C2212yes—
27C59⋊#95C2211yescharacteristic
28C59⋊#95C2211yescharacteristic
29C59⋊#95C2212yes—
30C59⋊#95C2212yes—
31C22×D4×C59C2212yes—
32C22×D4×C59C2211yescharacteristic
33C23×C4×C59C2211yescharacteristic
34#121×C59—24no—
35C2×C4⋊C4×C59C2211yescharacteristic
36C2×C22⋊C4×C59C2212yes—
37C24×C59C2312yes—
38C24×C59C2311yescharacteristic
39C23×D59C2311yescharacteristic
40C2×D4×C59—24no—
41C2×D4×C59—22no—
42C2×D4×C59—42no—
43C2×D4×C59—42no—
44C2×D4×C59—42no—
45C22×C4×C59—42no—
46C22×C4×C59D414yescomplemented
47C22×C4×C59C2312yes—
48C22×C4×C59C2311yescharacteristic
49C22×C4×C59C2311yescharacteristic
50C2×C22⋊D59—22no—
51C2×C22⋊D59—24no—
52C2×C22⋊D59—42no—
53C2×C22⋊D59—42no—
54C2×C22⋊D59—42no—
55C22×Dic59C2312yes—
56C22×Dic59C2311yescharacteristic
57C22×Dic59C2311yescharacteristic
58C22×Dic59C2311yescharacteristic
59C22×Dic59C2312yes—
60C4⋊C4×C59—22no—
61C22⋊C4×C59—24no—
62C22⋊Dic59—22no—
63C22⋊Dic59—24no—
64C22⋊Dic59—24no—
65C22⋊Dic59—42no—
66C22⋊Dic59—42no—
67(C2×D59)⋊C4—22no—
68(C2×D59)⋊C4—22no—
69(C2×D59)⋊C4—42no—
70Dic59⋊C4—22no—
71Dic59⋊C4—22no—
72Dic59⋊C4—42no—
73C4×Dic59—42no—
74C23×C59—22no—
75C23×C59—22no—
76C23×C59—21no—
77C23×C59—21no—
78C23×C59—22no—
79C23×C59—22no—
80C23×C59—22no—
81C23×C59—22no—
82C23×C59—22no—
83C23×C59—22no—
84C23×C59—22no—
85C23×C59C2×D412yes—
86C23×C59C2411yescharacteristic
87C22×D59—21no—
88C22×D59—22no—
89C22×D59—21no—
90C22×D59—22no—
91C22×D59—21no—
92D4×C59—44no—
93D4×C59—48no—
94C2×C4×C59—22no—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59—22no—
98C2×C4×C59—22no—
99C2×C4×C59—44no—
100C2×C4×C59C2×D412yes—
101C2×C4×C59C2×D412yes—
102C22⋊D59—48no—
103C22⋊D59—44no—
104C2×Dic59—22no—
105C2×Dic59—22no—
106C2×Dic59—21no—
107C2×Dic59—21no—
108C2×Dic59—21no—
109C2×Dic59—22no—
110C2×Dic59—22no—
111C2×Dic59—22no—
112C2×Dic59—21no—
113C2×Dic59—22no—
114C2×Dic59—21no—
115C2×Dic59—22no—
116C2×Dic59#4114yes—
117C22×C59—22no—
118C22×C59—22no—
119C22×C59—22no—
120C22×C59—42no—
121C22×C59—42no—
122C22×C59—42no—
123C22×C59—42no—
124C22×C59—42no—
125C22×C59—42no—
126C22×C59—42no—
127C22×C59—42no—
128C22×C59—42no—
129C22×C59C22×D411yescharacteristic
130C22×C59C2×#4112yes—
131C22×C59H(5,𝔽2)12yes—
132C22×C59H(5,𝔽2)11yescharacteristic
133C22×C59H(5,𝔽2)11yescharacteristic
134C2×D59—41no—
135C2×D59—41no—
136C2×D59—41no—
137C2×D59—42no—
138C2×D59—42no—
139C4×C59—22no—
140C4×C59—22no—
141C4×C59—42no—
142Dic59—24no—
143Dic59—41no—
144Dic59—41no—
145Dic59—41no—
146Dic59—42no—
147#2661—591nomaximal, Sylow
148C2×C59—42no—
149C2×C59—82no—
150C2×C59#53211yescharacteristic
151C2×C59#53411yescharacteristic
152C2×C59#53511yescharacteristic
153C2×C59#53812yes—
154C2×C59#54012yes—
155D59—81no—
156C2×#123—591no—
157C2×#121D5911yescharacteristic, complemented
158#398—591no—
159#397—591no—
160#394—591no—
161#394—592no—
162#392—591no—
163#392—592no—
164#390—592no—
165#388—592no—
166C4×C22⋊C4—591no—
167C59#266111yescharacteristic, complemented, Sylow
168C22×D4—591no—
169C22×D4—592no—
170C22×D4C2×D5912yes—
171C22×D4C2×D5911yescharacteristic
172C23×C4C2×D5911yescharacteristic
173#123—1184no—
174#121—24no—
175C2×C4⋊C4—591no—
176C2×C4⋊C4—591no—
177C2×C4⋊C4—592no—
178C2×C4⋊C4C2×D5911yescharacteristic
179C2×C22⋊C4—591no—
180C2×C22⋊C4—592no—
181C2×C22⋊C4—592no—
182C2×C22⋊C4—591no—
183C2×C22⋊C4—591no—
184C2×C22⋊C4—592no—
185C2×C22⋊C4—592no—
186C2×C22⋊C4—592no—
187C2×C22⋊C4C2×D5912yes—
188C2×C42—592no—
189#95—591no—
190#95—591no—
191#95—592no—
192#95—592no—
193C24—591no—
194C24C22×D5912yes—
195C24C22×D5911yescharacteristic
196C2×D4—24no—
197C2×D4—22no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—1184no—
202C2×D4—1182no—
203C2×D4—2362no—
204C2×D4—2362no—
205C2×D4—2362no—
206C22×C4—42no—
207C22×C4—592no—
208C22×C4—592no—
209C22×C4—591no—
210C22×C4—591no—
211C22×C4—591no—
212C22×C4C22⋊D5914yescomplemented
213C22×C4C22×D5912yes—
214C22×C4C22×D5911yescharacteristic
215C22×C4C22×D5911yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—1182no—
218C4⋊C4—1182no—
219C4⋊C4—2362no—
220C22⋊C4—24no—
221C22⋊C4—1184no—
222C22⋊C4—1184no—
223C22⋊C4—1182no—
224C22⋊C4—1182no—
225C22⋊C4—1182no—
226C22⋊C4—2362no—
227C22⋊C4—2362no—
228C22⋊C4—2362no—
229C42—2362no—
230C23—22no—
231C23—22no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—1181no—
242C23—1182no—
243C23—1181no—
244C23—1182no—
245C23—1181no—
246C23C2×C22⋊D5912yes—
247C23C23×D5911yescharacteristic
248D4—44no—
249D4—48no—
250D4—2368no—
251D4—2364no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—594no—
259C2×C4—1182no—
260C2×C4—1182no—
261C2×C4—1181no—
262C2×C4—1181no—
263C2×C4—1181no—
264C2×C4—1182no—
265C2×C4—1182no—
266C2×C4—1182no—
267C2×C4—1181no—
268C2×C4—1182no—
269C2×C4—1182no—
270C2×C4—1181no—
271C2×C4C2×C22⋊D5912yes—
272C2×C4C2×C22⋊D5912yes—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—2361no—
286C22—2361no—
287C22—2361no—
288C22—2362no—
289C22—2362no—
290C22C22×C22⋊D5911yescharacteristic
291C22#41×D5912yes—
292C22#1427377212yes—
293C22#1427377211yescharacteristic
294C22#1427377211yescharacteristic
295C4—22no—
296C4—22no—
297C4—42no—
298C4—1184no—
299C4—2361no—
300C4—2361no—
301C4—2361no—
302C4—2362no—
303C2—42no—
304C2—82no—
305C2—4721no—
306C2#1683794811yescharacteristic
307C2#1683796111yescharacteristic
308C2#1683796911yescharacteristic
309C2#1683798412yes—
310C2#1683800212yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC2×#121×C59C2
hypercenterC2×#121D59
hypocenterC59#2661
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#14273710C2semidirect
2. #16837333C2semidirect
3. #16837328C2semidirect
4. #16837316C2semidirect
5. #16837316C2semidirect
6. #16837310C2semidirect
7. #16837310C2semidirect
8. #16837302C2semidirect
9. #16837296C2semidirect
10. C4×C22⋊Dic59C2semidirect
11. C2×#121×C59C2semidirect
46. C22×C4×C59D4semidirect
157. C2×#121D59semidirect
167. C59#2661semidirect
212. C22×C4C22⋊D59semidirect