Finite Group Databasebeta

← #21430034

Subgroups — #21430034

Order 7552.

500 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14273725C211yesmaximal, characteristic, complemented
2C2×#14273713C211yesmaximal, characteristic
3#16837331C212yesmaximal, complemented
4#16837330C212yesmaximal, complemented
5#16837315C211yesmaximal, characteristic, complemented
6#16837314C211yesmaximal, characteristic, complemented
7#16837313C211yesmaximal, characteristic, complemented
8#16837306C212yesmaximal, complemented
9#16837302C212yesmaximal
10#16837247C211yesmaximal, characteristic, complemented
11C59×#393C211yesmaximal, characteristic, complemented
12C22×Dic118C2211yescharacteristic
13C22×Dic118C2211yescharacteristic
14C22×C22⋊D59C2211yescharacteristic
15#14273725—24no—
16#14273713—24no—
17C2×Dic59⋊C4C2212yes—
18C2×Dic59⋊C4C2212yes—
19C2×C22⋊Dic59C2211yescharacteristic
20C2×C22⋊Dic59C2212yes—
21C2×(C2×D59)⋊C4C2211yescharacteristic
22C2×(C2×D59)⋊C4C2211yescharacteristic
23C2×(C2×D59)⋊C4C2212yes—
24C2×(C2×D59)⋊C4C2212yes—
25C2×C4×Dic59C2212yes—
26C2×C4×Dic59C2211yescharacteristic
27C59⋊#95C2211yescharacteristic
28C59⋊#95C2212yes—
29C59⋊#95C2212yes—
30C59⋊#95C2212yes—
31C59⋊#95C2212yes—
32C22×Q8×C59C2211yescharacteristic
33C2×C22⋊C4×C59C2211yescharacteristic
34C2×C22⋊C4×C59C2212yes—
35#95×C59C2211yescharacteristic
36#95×C59C2212yes—
37C24×C59C2311yescharacteristic
38C23×D59C2311yescharacteristic
39C2×Q8×C59—22no—
40C2×Q8×C59—42no—
41C22×C4×C59C2312yes—
42C22×C4×C59C2311yescharacteristic
43C22×C4×C59C2312yes—
44C22×C4×C59C2311yescharacteristic
45C2×C22⋊D59—22no—
46C2×C22⋊D59—24no—
47C22×Dic59C2312yes—
48C22×Dic59C2311yescharacteristic
49C22×Dic59C2312yes—
50C22×Dic59C2311yescharacteristic
51C22×Dic59C2311yescharacteristic
52C2×Dic118—22no—
53C2×Dic118—22no—
54C2×Dic118—24no—
55C2×Dic118—42no—
56C22⋊C4×C59—22no—
57C22⋊C4×C59—42no—
58C22⋊Dic59—22no—
59C22⋊Dic59—42no—
60(C2×D59)⋊C4—22no—
61(C2×D59)⋊C4—22no—
62(C2×D59)⋊C4—42no—
63(C2×D59)⋊C4—42no—
64Dic59⋊C4—24no—
65Dic59⋊C4—24no—
66C4×Dic59—24no—
67C4×Dic59D414yes—
68C23×C59—21no—
69C23×C59—22no—
70C23×C59—22no—
71C23×C59—21no—
72C23×C59—21no—
73C23×C59C2411yescharacteristic
74C22×D59—21no—
75C22×D59—22no—
76C22×D59—22no—
77C22×D59—21no—
78C22×D59—21no—
79Q8×C59—44no—
80C2×C4×C59—22no—
81C2×C4×C59—22no—
82C2×C4×C59—21no—
83C2×C4×C59—22no—
84C2×C4×C59—22no—
85C2×C4×C59—22no—
86C2×C4×C59—22no—
87C2×C4×C59—22no—
88C2×C4×C59—22no—
89C2×C4×C59C2×D412yes—
90C22⋊D59—44no—
91C2×Dic59—21no—
92C2×Dic59—21no—
93C2×Dic59—22no—
94C2×Dic59—21no—
95C2×Dic59—22no—
96C2×Dic59—22no—
97C2×Dic59—22no—
98C2×Dic59—22no—
99C2×Dic59—22no—
100C2×Dic59—22no—
101C2×Dic59C2×D412yes—
102C2×Dic59C2×D412yes—
103C2×Dic59#4114yes—
104Dic118—44no—
105Dic118—44no—
106C22×C59—41no—
107C22×C59—41no—
108C22×C59—41no—
109C22×C59—42no—
110C22×C59—42no—
111C22×C59C22×D411yescharacteristic
112C22×C59C2×#4112yes—
113C22×C59H(5,𝔽2)12yes—
114C22×C592−1+411yescharacteristic
115C22×C592−1+411yescharacteristic
116C2×D59—41no—
117C2×D59—41no—
118C2×D59—41no—
119C2×D59—42no—
120C2×D59—42no—
121C4×C59—22no—
122C4×C59—41no—
123C4×C59—42no—
124C4×C59—42no—
125Dic59—22no—
126Dic59—41no—
127Dic59—42no—
128Dic59—42no—
129Dic59#12414yes—
130#2640—591nomaximal, Sylow
131C2×C59—81no—
132C2×C59C2×#12411yescharacteristic
133C2×C59#53511yescharacteristic
134C2×C59#53611yescharacteristic
135C2×C59#54212yes—
136C2×C59#54212yes—
137D59—81no—
138C2×C4⋊Q8—591no—
139C2×#124—591no—
140#397—592no—
141#397—592no—
142#393—591no—
143#393—591no—
144#393—591no—
145#393D5911yescharacteristic, complemented
146#391—592no—
147#390—592no—
148#376—591no—
149C59#264011yescharacteristic, complemented, Sylow
150C22×Q8—591no—
151C22×Q8—591no—
152C22×Q8C2×D5911yescharacteristic
153C22×D4—591no—
154C4⋊Q8—1184no—
155#124—1184no—
156C2×C4⋊C4—592no—
157C2×C4⋊C4—592no—
158C2×C22⋊C4—591no—
159C2×C22⋊C4—591no—
160C2×C22⋊C4—591no—
161C2×C22⋊C4—592no—
162C2×C22⋊C4—592no—
163C2×C22⋊C4—592no—
164C2×C22⋊C4C2×D5911yescharacteristic
165C2×C22⋊C4C2×D5912yes—
166C2×C42—591no—
167C2×C42—592no—
168#95—591no—
169#95—592no—
170#95—592no—
171#95—592no—
172#95—592no—
173#95C2×D5911yescharacteristic
174#95C2×D5912yes—
175C24—591no—
176C24C22×D5911yescharacteristic
177C2×Q8—22no—
178C2×Q8—42no—
179C2×Q8—1182no—
180C2×Q8—1184no—
181C2×Q8—1182no—
182C2×Q8—2362no—
183C2×D4—1182no—
184C2×D4—1184no—
185C22×C4—591no—
186C22×C4—591no—
187C22×C4—592no—
188C22×C4—591no—
189C22×C4—592no—
190C22×C4C22×D5912yes—
191C22×C4C22×D5911yescharacteristic
192C22×C4C22×D5912yes—
193C22×C4C22×D5911yescharacteristic
194C4⋊C4—1184no—
195C4⋊C4—1184no—
196C22⋊C4—22no—
197C22⋊C4—42no—
198C22⋊C4—1182no—
199C22⋊C4—1182no—
200C22⋊C4—1182no—
201C22⋊C4—2362no—
202C22⋊C4—2362no—
203C22⋊C4—2362no—
204C42—594no—
205C42—1184no—
206C23—21no—
207C23—22no—
208C23—22no—
209C23—21no—
210C23—21no—
211C23—1181no—
212C23—1182no—
213C23—1182no—
214C23—1181no—
215C23—1181no—
216C23C23×D5911yescharacteristic
217Q8—44no—
218Q8—2364no—
219Q8—2364no—
220D4—2364no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—592no—
231C2×C4—594no—
232C2×C4—592no—
233C2×C4—1181no—
234C2×C4—1181no—
235C2×C4—1181no—
236C2×C4—1182no—
237C2×C4—1182no—
238C2×C4—1182no—
239C2×C4—1182no—
240C2×C4—1182no—
241C2×C4—1182no—
242C2×C4—1182no—
243C2×C4D4×D5912yes—
244C22—41no—
245C22—41no—
246C22—41no—
247C22—42no—
248C22—42no—
249C22—2361no—
250C22—2361no—
251C22—2361no—
252C22—2362no—
253C22—2362no—
254C22C2×D4×D5911yescharacteristic
255C22#41×D5912yes—
256C22#1427377212yes—
257C22#1427377411yescharacteristic
258C22#1427377511yescharacteristic
259C4—22no—
260C4—41no—
261C4—42no—
262C4—42no—
263C4—594no—
264C4—1182no—
265C4—2361no—
266C4—2362no—
267C4—2362no—
268C2—81no—
269C2—4721no—
270C2#124×D5911yescharacteristic
271C2#1683796711yescharacteristic
272C2#1683797511yescharacteristic
273C2#1683801612yes—
274C2#1683802212yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#393C2
hypercenter#393D59
hypocenterC59#2640
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#14273725C2semidirect
3. #16837331C2semidirect
4. #16837330C2semidirect
5. #16837315C2semidirect
6. #16837314C2semidirect
7. #16837313C2semidirect
8. #16837306C2semidirect
10. #16837247C2semidirect
11. C59×#393C2semidirect
145. #393D59semidirect
149. C59#2640semidirect