Finite Group Databasebeta

← #21429883

Subgroups — #21429883

Order 7552.

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

normal subgroups only

This group has 526 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×#14273712C211yesmaximal, characteristic
2C2×C4⋊C4×D59C211yesmaximal, characteristic, complemented
3#16837341C211yesmaximal, characteristic, complemented
4#16837339C211yesmaximal, characteristic, complemented
5#16837337C211yesmaximal, characteristic, complemented
6#16837336C212yesmaximal, complemented
7#16837331C212yesmaximal, complemented
8#16837280C212yesmaximal
9#16837276C212yesmaximal, complemented
10#16837264C211yesmaximal, characteristic, complemented
11C59×#400C211yesmaximal, characteristic, complemented
12C22×D236C2211yescharacteristic
13C22×C4×D59C2211yescharacteristic
14C22×C4×D59C2212yes—
15#14273712—24no—
16C2×C4⋊Dic59C2211yescharacteristic
17C2×C4⋊Dic59C2212yes—
18C2×Dic59⋊C4C2211yescharacteristic
19C2×Dic59⋊C4C2212yes—
20C2×Dic59⋊C4C2212yes—
21C4⋊C4×D59—24no—
22C2×(C2×D59)⋊C4C2211yescharacteristic
23C2×(C2×D59)⋊C4C2211yescharacteristic
24C2×(C2×D59)⋊C4C2211yescharacteristic
25C2×(C2×D59)⋊C4C2212yes—
26C2×(C2×D59)⋊C4C2212yes—
27C2×(C2×D59)⋊C4C2212yes—
28C59⋊#95C2212yes—
29C59⋊#95C2211yescharacteristic
30C59⋊#95C2212yes—
31C59⋊#95C2212yes—
32C2×C4⋊C4×C59C2211yescharacteristic
33C2×C4⋊C4×C59C2211yescharacteristic
34#95×C59C2211yescharacteristic
35#95×C59C2212yes—
36#95×C59C2212yes—
37C23×D59C2311yescharacteristic
38C23×D59C2311yescharacteristic
39C22×C4×C59C2312yes—
40C22×C4×C59C2311yescharacteristic
41C22×C4×C59C2311yescharacteristic
42C22×C4×C59C2312yes—
43C22×C4×C59C2311yescharacteristic
44C22×Dic59C2312yes—
45C22×Dic59C2312yes—
46C22×Dic59C2311yescharacteristic
47C22×Dic59C2311yescharacteristic
48C2×D236—22no—
49C2×D236—24no—
50C2×C4×D59—24no—
51C2×C4×D59—24no—
52C2×C4×D59—42no—
53C2×C4×D59—42no—
54C2×C4×D59D414yes—
55C4⋊C4×C59—22no—
56C4⋊C4×C59—22no—
57(C2×D59)⋊C4—22no—
58(C2×D59)⋊C4—24no—
59(C2×D59)⋊C4—22no—
60(C2×D59)⋊C4—22no—
61(C2×D59)⋊C4—42no—
62(C2×D59)⋊C4—42no—
63C4⋊Dic59—22no—
64C4⋊Dic59—42no—
65Dic59⋊C4—24no—
66Dic59⋊C4—22no—
67Dic59⋊C4—42no—
68C23×C59C2411yescharacteristic
69C22×D59—22no—
70C22×D59—21no—
71C22×D59—21no—
72C22×D59—21no—
73C22×D59—22no—
74C22×D59—22no—
75C22×D59—21no—
76C22×D59—21no—
77C22×D59C2×D412yes—
78C22×D59#4114yes—
79C2×C4×C59—22no—
80C2×C4×C59—22no—
81C2×C4×C59—21no—
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—21no—
89C2×C4×C59—22no—
90C2×C4×C59—21no—
91C2×C4×C59C2×D412yes—
92C2×Dic59—22no—
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—
102D236—44no—
103C4×D59—28no—
104C4×D59—48no—
105C22×C59C22×D411yescharacteristic
106C22×C59C2×#4112yes—
107C22×C59H(5,𝔽2)12yes—
108C22×C592−1+411yescharacteristic
109C22×C592−1+411yescharacteristic
110C2×D59—22no—
111C2×D59—22no—
112C2×D59—24no—
113C2×D59—24no—
114C2×D59—41no—
115C2×D59—41no—
116C2×D59—41no—
117C2×D59—42no—
118C2×D59—42no—
119C2×D59#12314yes—
120C4×C59—22no—
121C4×C59—41no—
122C4×C59—42no—
123C4×C59—41no—
124C4×C59—42no—
125Dic59—22no—
126Dic59—42no—
127Dic59—41no—
128Dic59—42no—
129#2623—591nomaximal, Sylow
130C2×C59C2×#12311yescharacteristic
131C2×C59#53511yescharacteristic
132C2×C59#53611yescharacteristic
133C2×C59#53912yes—
134C2×C59#53912yes—
135D59—24no—
136D59—81no—
137C2×#123—591no—
138C22×C4⋊C4—591no—
139#400—591no—
140#400D5911yescharacteristic, complemented
141#399—591no—
142#399—591no—
143#398—592no—
144#397—592no—
145#386—592no—
146#385—592no—
147#381—591no—
148C59#262311yescharacteristic, complemented, Sylow
149C22×D4—591no—
150C23×C4—591no—
151C23×C4—592no—
152#123—1184no—
153C2×C4⋊C4—592no—
154C2×C4⋊C4—591no—
155C2×C4⋊C4—591no—
156C2×C4⋊C4—592no—
157C2×C4⋊C4—592no—
158C2×C4⋊C4—1184no—
159C2×C4⋊C4C2×D5911yescharacteristic
160C2×C4⋊C4C2×D5911yescharacteristic
161C2×C22⋊C4—591no—
162C2×C22⋊C4—591no—
163C2×C22⋊C4—591no—
164C2×C22⋊C4—592no—
165C2×C22⋊C4—592no—
166C2×C22⋊C4—592no—
167#95—592no—
168#95—591no—
169#95—592no—
170#95—592no—
171#95C2×D5911yescharacteristic
172#95C2×D5912yes—
173#95C2×D5912yes—
174C24—591no—
175C24—591no—
176C2×D4—1182no—
177C2×D4—1184no—
178C22×C4—591no—
179C22×C4—594no—
180C22×C4—592no—
181C22×C4—592no—
182C22×C4—591no—
183C22×C4—1184no—
184C22×C4—1184no—
185C22×C4—2362no—
186C22×C4—2362no—
187C22×C4C22×D5911yescharacteristic
188C22×C4C22×D5911yescharacteristic
189C22×C4C22×D5912yes—
190C22×C4C22×D5912yes—
191C22×C4C22×D5911yescharacteristic
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—1184no—
195C4⋊C4—1182no—
196C4⋊C4—1182no—
197C4⋊C4—2362no—
198C4⋊C4—2362no—
199C22⋊C4—1182no—
200C22⋊C4—1182no—
201C22⋊C4—1184no—
202C22⋊C4—1182no—
203C22⋊C4—2362no—
204C22⋊C4—2362no—
205C23—592no—
206C23—594no—
207C23—1182no—
208C23—1181no—
209C23—1181no—
210C23—1181no—
211C23—1182no—
212C23—1182no—
213C23—1181no—
214C23—1181no—
215C23C23×D5911yescharacteristic
216D4—2364no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—592no—
230C2×C4—1188no—
231C2×C4—1182no—
232C2×C4—1181no—
233C2×C4—1182no—
234C2×C4—1182no—
235C2×C4—1182no—
236C2×C4—1182no—
237C2×C4—1182no—
238C2×C4—1182no—
239C2×C4—1182no—
240C2×C4—2368no—
241C2×C4D4×D5912yes—
242C22—594no—
243C22—1182no—
244C22—1182no—
245C22—1184no—
246C22—1184no—
247C22—2361no—
248C22—2361no—
249C22—2361no—
250C22—2362no—
251C22—2362no—
252C22C2×D4×D5911yescharacteristic
253C22#41×D5912yes—
254C22#1427377312yes—
255C22#1427377411yescharacteristic
256C22#1427377511yescharacteristic
257C4—22no—
258C4—41no—
259C4—42no—
260C4—41no—
261C4—42no—
262C4—1182no—
263C4—2362no—
264C4—2361no—
265C4—2362no—
266C2—1184no—
267C2—4721no—
268C2#123×D5911yescharacteristic
269C2#1683797011yescharacteristic
270C2#1683797511yescharacteristic
271C2#1683799312yes—
272C2#1683800012yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#400C2
hypercenter#400D59
hypocenterC59#2623
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D59C2semidirect
3. #16837341C2semidirect
4. #16837339C2semidirect
5. #16837337C2semidirect
6. #16837336C2semidirect
7. #16837331C2semidirect
9. #16837276C2semidirect
10. #16837264C2semidirect
11. C59×#400C2semidirect
140. #400D59semidirect
148. C59#2623semidirect