Finite Group Databasebeta

← #21427419

Subgroups — #21427419

Order 7552.

1116 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 197 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×#14273708C212yesmaximal, complemented
2C22×C22⋊Dic59C211yesmaximal, characteristic, complemented
3#16837316C214yesmaximal, complemented
4#16837282C212yesmaximal, complemented
5#16837275C212yesmaximal, complemented
6#16837258C212yesmaximal, complemented
7#16837258C211yesmaximal, characteristic, complemented
8C23×D4×C59C211yesmaximal, characteristic
9C23×Dic59C2211yescharacteristic, complemented
10C23×Dic59C2211yescharacteristic, complemented
11#14273708—28no—
12C2×Dic59⋊C4C2214yescomplemented
13C2×C22⋊Dic59—24no—
14C2×C22⋊Dic59C2212yes—
15C2×C22⋊Dic59C2212yes—
16C2×C22⋊Dic59C2212yescomplemented
17C2×C22⋊Dic59C2214yescomplemented
18C2×C22⋊Dic59C2212yescomplemented
19C2×C22⋊Dic59C2214yescomplemented
20C2×C22⋊Dic59C2212yescomplemented
21C59⋊#95C2214yescomplemented
22C25×C59C2211yescharacteristic
23C25×C59C2211yescharacteristic
24C22×D4×C59—24no—
25C22×D4×C59—28no—
26C22×D4×C59C2212yes—
27C22×D4×C59C2212yes—
28C23×C4×C59C2211yescharacteristic
29C24×C59—24no—
30C24×C59—24no—
31C24×C59—22no—
32C24×C59—24no—
33C24×C59—24no—
34C24×C59—42no—
35C24×C59—42no—
36C24×C59D414yes—
37C24×C59C2311yescharacteristic
38C24×C59C2312yes—
39C24×C59C2311yescharacteristic
40C24×C59C2311yescharacteristic
41C2×D4×C59—232no—
42C2×D4×C59—24no—
43C2×D4×C59—24no—
44C2×D4×C59—24no—
45C2×D4×C59—24no—
46C2×D4×C59—24no—
47C2×D4×C59D418yes—
48C22×C4×C59—22no—
49C22×C4×C59—24no—
50C22×C4×C59C2312yes—
51C22×Dic59—22no—
52C22×Dic59—22no—
53C22×Dic59—24no—
54C22×Dic59—42no—
55C22×Dic59C2312yes—
56C22×Dic59C2314yes—
57C22×Dic59C2312yes—
58C22⋊Dic59—24no—
59C22⋊Dic59—24no—
60C22⋊Dic59—24no—
61C22⋊Dic59—28no—
62C22⋊Dic59—24no—
63C22⋊Dic59—42no—
64C22⋊Dic59—44no—
65Dic59⋊C4—28no—
66C23×C59—22no—
67C23×C59—22no—
68C23×C59—22no—
69C23×C59—22no—
70C23×C59—28no—
71C23×C59—28no—
72C23×C59—28no—
73C23×C59—22no—
74C23×C59—22no—
75C23×C59—22no—
76C23×C59—22no—
77C23×C59—24no—
78C23×C59—22no—
79C23×C59—24no—
80C23×C59—44no—
81C23×C59—416no—
82C23×C59—416no—
83C23×C59—44no—
84C23×C59—44no—
85C23×C59C2×D412yes—
86C23×C59C2×D414yes—
87C23×C59C2×D414yes—
88C23×C59C2×D412yes—
89C23×C59C2×D412yes—
90C23×C59#4114yes—
91C23×C59C2411yescharacteristic
92D4×C59—216no—
93D4×C59—216no—
94C2×C4×C59—28no—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59C2×D414yes—
98C2×Dic59—24no—
99C2×Dic59—24no—
100C2×Dic59—22no—
101C2×Dic59—22no—
102C2×Dic59—24no—
103C2×Dic59—24no—
104C2×Dic59—24no—
105C2×Dic59—44no—
106C2×Dic59—44no—
107C22×C59—22no—
108C22×C59—22no—
109C22×C59—22no—
110C22×C59—22no—
111C22×C59—28no—
112C22×C59—28no—
113C22×C59—22no—
114C22×C59—24no—
115C22×C59—22no—
116C22×C59—22no—
117C22×C59—28no—
118C22×C59—416no—
119C22×C59—416no—
120C22×C59—42no—
121C22×C59—42no—
122C22×C59—44no—
123C22×C59—44no—
124C22×C59#12014yes—
125C22×C59#12314yescomplemented
126C22×C59C22×D411yescharacteristic
127C22×C59C22×D412yes—
128C22×C59C2×#4112yes—
129C22×C59H(5,𝔽2)12yes—
130C4×C59—24no—
131Dic59—42no—
132Dic59—42no—
133Dic59—44no—
134#2405—591nomaximal, Sylow
135C2×C59—24no—
136C2×C59—24no—
137C2×C59—42no—
138C2×C59—44no—
139C2×C59C2×#12011yescharacteristic
140C2×C59C2×#12311yescharacteristic
141C2×C59#53411yescharacteristic
142C2×C59#54614yes—
143C23×D4D5911yescharacteristic
144C2×#123—592no—
145C22×C22⋊C4—591no—
146#394—594no—
147#386—592no—
148#385—592no—
149#379—592no—
150#379—591no—
151C59#240511yescharacteristic, complemented, Sylow
152C25C2×D5911yescharacteristic
153C25C2×D5911yescharacteristic
154C22×D4—24no—
155C22×D4—28no—
156C22×D4C2×D5912yes—
157C22×D4C2×D5912yes—
158C23×C4—591no—
159C23×C4—591no—
160C23×C4C2×D5911yescharacteristic
161#123—1188no—
162C2×C4⋊C4—594no—
163C2×C22⋊C4—592no—
164C2×C22⋊C4—592no—
165C2×C22⋊C4—592no—
166C2×C22⋊C4—594no—
167C2×C22⋊C4—592no—
168C2×C22⋊C4—594no—
169C2×C22⋊C4—592no—
170C2×C22⋊C4—1184no—
171#95—594no—
172C24—24no—
173C24—24no—
174C24—24no—
175C24—22no—
176C24—24no—
177C24—42no—
178C24—42no—
179C24C22⋊D5914yes—
180C24C22×D5911yescharacteristic
181C24C22×D5912yes—
182C24C22×D5911yescharacteristic
183C24C22×D5911yescharacteristic
184C2×D4—232no—
185C2×D4—24no—
186C2×D4—24no—
187C2×D4—24no—
188C2×D4—24no—
189C2×D4—24no—
190C2×D4C22⋊D5918yes—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—592no—
194C22×C4—594no—
195C22×C4—592no—
196C22×C4—1182no—
197C22×C4—1182no—
198C22×C4—1184no—
199C22×C4—2362no—
200C22×C4C22×D5912yes—
201C4⋊C4—1188no—
202C22⋊C4—1184no—
203C22⋊C4—1184no—
204C22⋊C4—1184no—
205C22⋊C4—1184no—
206C22⋊C4—1188no—
207C22⋊C4—2362no—
208C22⋊C4—2364no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—28no—
214C23—28no—
215C23—28no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—24no—
221C23—22no—
222C23—24no—
223C23—44no—
224C23—416no—
225C23—416no—
226C23—44no—
227C23—44no—
228C23#1174568714yes—
229C23C2×C22⋊D5912yes—
230C23C2×C22⋊D5914yes—
231C23C2×C22⋊D5914yes—
232C23C2×C22⋊D5912yes—
233C23C2×C22⋊D5912yes—
234C23C23×D5911yescharacteristic
235D4—216no—
236D4—216no—
237C2×C4—28no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—1184no—
241C2×C4—1182no—
242C2×C4—1182no—
243C2×C4—1184no—
244C2×C4—1184no—
245C2×C4—1184no—
246C2×C4—1184no—
247C2×C4—2364no—
248C2×C4—2364no—
249C2×C4C2×C22⋊D5914yes—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—28no—
255C22—28no—
256C22—22no—
257C22—24no—
258C22—22no—
259C22—22no—
260C22—28no—
261C22—416no—
262C22—416no—
263C22—42no—
264C22—42no—
265C22—44no—
266C22—44no—
267C22#1427369614yes—
268C22#1427370814yescomplemented
269C22C22×C22⋊D5911yescharacteristic
270C22C22×C22⋊D5912yes—
271C22C2×#1174568712yes—
272C22#1427377212yes—
273C4—24no—
274C4—2362no—
275C4—2362no—
276C4—2364no—
277C2—24no—
278C2—24no—
279C2—42no—
280C2—44no—
281C2C2×#1427369611yescharacteristic
282C2C2×#1427370811yescharacteristic
283C2#1683796111yescharacteristic
284C2#1683804614yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC23×D4×C59C2
hypercenterC23×D4D59
hypocenterC59#2405
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#14273708C2semidirect
2. C22×C22⋊Dic59C2semidirect
3. #16837316C2semidirect
4. #16837282C2semidirect
5. #16837275C2semidirect
6. #16837258C2semidirect
7. #16837258C2semidirect
9. C23×Dic59C22semidirect
10. C23×Dic59C22semidirect
12. C2×Dic59⋊C4C22semidirect
16. C2×C22⋊Dic59C22semidirect
17. C2×C22⋊Dic59C22semidirect
18. C2×C22⋊Dic59C22semidirect
19. C2×C22⋊Dic59C22semidirect
20. C2×C22⋊Dic59C22semidirect
21. C59⋊#95C22semidirect
125. C22×C59#123semidirect
151. C59#2405semidirect
268. C22#14273708semidirect