Finite Group Databasebeta

← #21429880

Subgroups — #21429880

Order 7552.

526 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14273708C211yesmaximal, characteristic
2C22×Dic59⋊C4C211yesmaximal, characteristic, complemented
3#16837343C212yesmaximal, complemented
4#16837338C212yesmaximal, complemented
5#16837336C212yesmaximal, complemented
6#16837330C212yesmaximal, complemented
7#16837282C211yesmaximal, characteristic
8#16837277C211yesmaximal, characteristic, complemented
9#16837275C211yesmaximal, characteristic, complemented
10#16837263C211yesmaximal, characteristic, complemented
11C59×#386C211yesmaximal, characteristic
12C23×Dic59C2211yescharacteristic
13C23×Dic59C2211yescharacteristic
14#14273708—24no—
15C2×C4⋊Dic59C2212yes—
16C2×Dic59⋊C4—24no—
17C2×Dic59⋊C4C2211yescharacteristic
18C2×Dic59⋊C4C2211yescharacteristic
19C2×Dic59⋊C4C2212yes—
20C2×Dic59⋊C4C2212yes—
21C2×Dic59⋊C4C2212yes—
22C2×C22⋊Dic59C2211yescharacteristic
23C2×C22⋊Dic59C2211yescharacteristic
24C2×C22⋊Dic59C2211yescharacteristic
25C2×C22⋊Dic59C2211yescharacteristic
26C2×C22⋊Dic59C2212yes—
27C59⋊#95C2212yes—
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
39C2×D4×C59—22no—
40C2×D4×C59—22no—
41C2×D4×C59—22no—
42C22×C4×C59—22no—
43C22×C4×C59—42no—
44C22×C4×C59C2311yescharacteristic
45C22×C4×C59C2312yes—
46C22×C4×C59C2312yes—
47C22×Dic59—22no—
48C22×Dic59—22no—
49C22×Dic59—22no—
50C22×Dic59—42no—
51C22×Dic59D414yes—
52C22×Dic59C2311yescharacteristic
53C22×Dic59C2312yes—
54C22×Dic59C2312yes—
55C22×Dic59C2311yescharacteristic
56C22×Dic59C2311yescharacteristic
57C22×Dic59C2311yescharacteristic
58C22⋊C4×C59—22no—
59C22⋊C4×C59—42no—
60C22⋊Dic59—22no—
61C22⋊Dic59—22no—
62C22⋊Dic59—22no—
63C22⋊Dic59—22no—
64C22⋊Dic59—42no—
65C4⋊Dic59—42no—
66Dic59⋊C4—24no—
67Dic59⋊C4—24no—
68Dic59⋊C4—22no—
69Dic59⋊C4—22no—
70Dic59⋊C4—42no—
71C23×C59—22no—
72C23×C59—22no—
73C23×C59—21no—
74C23×C59—21no—
75C23×C59—21no—
76C23×C59—22no—
77C23×C59—22no—
78C23×C59C2×D412yes—
79C23×C59#4112yes—
80C23×C59#4112yes—
81C23×C59C2411yescharacteristic
82D4×C59—44no—
83C2×C4×C59—22no—
84C2×C4×C59—21no—
85C2×C4×C59—22no—
86C2×C4×C59—21no—
87C2×C4×C59—21no—
88C2×C4×C59—22no—
89C2×C4×C59—22no—
90C2×C4×C59—22no—
91C2×C4×C59—22no—
92C2×C4×C59—44no—
93C2×Dic59—28no—
94C2×Dic59—21no—
95C2×Dic59—21no—
96C2×Dic59—21no—
97C2×Dic59—21no—
98C2×Dic59—22no—
99C2×Dic59—21no—
100C2×Dic59—21no—
101C2×Dic59—22no—
102C2×Dic59—22no—
103C2×Dic59—22no—
104C2×Dic59—22no—
105C2×Dic59—22no—
106C2×Dic59—22no—
107C2×Dic59—22no—
108C2×Dic59—44no—
109C2×Dic59C2×D412yes—
110C2×Dic59C2×D412yes—
111C22×C59—22no—
112C22×C59—22no—
113C22×C59—24no—
114C22×C59—24no—
115C22×C59—41no—
116C22×C59—41no—
117C22×C59—41no—
118C22×C59—42no—
119C22×C59—42no—
120C22×C59#12314yes—
121C22×C59C22×D411yescharacteristic
122C22×C59C2×#4111yescharacteristic
123C22×C59C2×#4111yescharacteristic
124C22×C59H(5,𝔽2)12yes—
125C22×C592−1+412yes—
126C4×C59—41no—
127C4×C59—42no—
128C4×C59—42no—
129Dic59—22no—
130Dic59—22no—
131Dic59—41no—
132Dic59—42no—
133Dic59—41no—
134Dic59—42no—
135#2623—591nomaximal, Sylow
136C2×C59—24no—
137C2×C59—81no—
138C2×C59C2×#12311yescharacteristic
139C2×C59#53511yescharacteristic
140C2×C59#53611yescharacteristic
141C2×C59#53912yes—
142C2×C59#53912yes—
143C2×#123—591no—
144C22×C4⋊C4—591no—
145#400—592no—
146#399—592no—
147#398—592no—
148#397—592no—
149#386—591no—
150#386D5911yescharacteristic
151#385—591no—
152#385—591no—
153#381—591no—
154C59#262311yescharacteristic, complemented, Sylow
155C22×D4C2×D5911yescharacteristic
156C23×C4—591no—
157C23×C4—591no—
158C23×C4C2×D5911yescharacteristic
159#123—1184no—
160C2×C4⋊C4—592no—
161C2×C4⋊C4—592no—
162C2×C4⋊C4—591no—
163C2×C4⋊C4—591no—
164C2×C4⋊C4—592no—
165C2×C4⋊C4—592no—
166C2×C4⋊C4—1184no—
167C2×C22⋊C4—591no—
168C2×C22⋊C4—591no—
169C2×C22⋊C4—591no—
170C2×C22⋊C4—591no—
171C2×C22⋊C4—592no—
172C2×C22⋊C4C2×D5911yescharacteristic
173C2×C22⋊C4C2×D5912yes—
174#95—592no—
175#95—592no—
176#95—592no—
177#95—592no—
178#95—592no—
179#95C2×D5912yes—
180C24C22×D5911yescharacteristic
181C24C22×D5911yescharacteristic
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—22no—
185C22×C4—22no—
186C22×C4—42no—
187C22×C4—594no—
188C22×C4—591no—
189C22×C4—591no—
190C22×C4—591no—
191C22×C4—592no—
192C22×C4—592no—
193C22×C4—591no—
194C22×C4—1182no—
195C22×C4—1182no—
196C22×C4—1182no—
197C22×C4—2362no—
198C22×C4C22×D5912yes—
199C22×C4C22×D5911yescharacteristic
200C22×C4C22×D5912yes—
201C4⋊C4—1184no—
202C4⋊C4—1184no—
203C4⋊C4—1182no—
204C4⋊C4—1182no—
205C4⋊C4—2362no—
206C4⋊C4—2362no—
207C22⋊C4—22no—
208C22⋊C4—42no—
209C22⋊C4—1182no—
210C22⋊C4—1182no—
211C22⋊C4—1182no—
212C22⋊C4—1182no—
213C22⋊C4—2362no—
214C23—22no—
215C23—22no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—22no—
220C23—22no—
221C23#1174568512yes—
222C23D4×D5912yes—
223C23#1174568712yes—
224C23C23×D5911yescharacteristic
225D4—44no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—44no—
236C2×C4—592no—
237C2×C4—592no—
238C2×C4—1188no—
239C2×C4—1181no—
240C2×C4—1181no—
241C2×C4—1181no—
242C2×C4—1181no—
243C2×C4—1181no—
244C2×C4—1181no—
245C2×C4—1182no—
246C2×C4—1182no—
247C2×C4—1182no—
248C2×C4—1182no—
249C2×C4—1182no—
250C2×C4—1182no—
251C2×C4—1182no—
252C2×C4—1182no—
253C2×C4—2364no—
254C22—22no—
255C22—22no—
256C22—24no—
257C22—24no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—42no—
263C22#1427370914yes—
264C22C2×D4×D5911yescharacteristic
265C22C2×#1174568711yescharacteristic
266C22C2×#1174568511yescharacteristic
267C22#1427377212yes—
268C22#1427377412yes—
269C4—41no—
270C4—42no—
271C4—42no—
272C4—1182no—
273C4—1182no—
274C4—2362no—
275C4—2361no—
276C4—2361no—
277C4—2362no—
278C2—24no—
279C2—81no—
280C2C2×#1427370911yescharacteristic
281C2#1683796711yescharacteristic
282C2#1683797411yescharacteristic
283C2#1683799412yes—
284C2#1683799712yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#386C2
hypercenter#386D59
hypocenterC59#2623
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×Dic59⋊C4C2semidirect
3. #16837343C2semidirect
4. #16837338C2semidirect
5. #16837336C2semidirect
6. #16837330C2semidirect
8. #16837277C2semidirect
9. #16837275C2semidirect
10. #16837263C2semidirect
154. C59#2623semidirect