Finite Group Databasebeta

← #21432099

Subgroups — #21432099

Order 7552.

578 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 125 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×#14273709C211yesmaximal, characteristic, complemented
2C2×#14273702C211yesmaximal, characteristic, complemented
3#16837340C211yesmaximal, characteristic, complemented
4#16837324C212yesmaximal, complemented
5#16837320C211yesmaximal, characteristic, complemented
6#16837319C212yesmaximal, complemented
7#16837317C212yesmaximal, complemented
8#16837311C211yesmaximal, characteristic, complemented
9#16837311C212yesmaximal, complemented
10#16837261C211yesmaximal, characteristic, complemented
11C59×#399C211yesmaximal, characteristic, complemented
12C22×D236C2212yes—
13C22×D236C2211yescharacteristic
14C22×C22⋊D59C2211yescharacteristic
15C22×C22⋊D59C2212yes—
16C22×C4×D59C2211yescharacteristic
17#14273709—24no—
18#14273702—24no—
19C2×C4⋊Dic59C2211yescharacteristic
20C2×Dic59⋊C4C2211yescharacteristic
21C2×Dic59⋊C4C2212yes—
22C2×C22⋊Dic59C2211yescharacteristic
23C2×(C2×D59)⋊C4C2212yes—
24C2×(C2×D59)⋊C4C2211yescharacteristic
25C2×(C2×D59)⋊C4C2212yes—
26C2×(C2×D59)⋊C4C2212yes—
27C2×(C2×D59)⋊C4C2212yes—
28C2×(C2×D59)⋊C4C2212yes—
29C59⋊#95C2211yescharacteristic
30C59⋊#95C2212yes—
31C59⋊#95C2212yes—
32C2×C4⋊C4×C59C2211yescharacteristic
33C2×C4⋊C4×C59C2212yes—
34C2×C22⋊C4×C59C2211yescharacteristic
35C2×C22⋊C4×C59C2212yes—
36#95×C59C2211yescharacteristic
37C24×C59C2311yescharacteristic
38C23×D59C2312yes—
39C23×D59C2311yescharacteristic
40C22×C4×C59C2312yes—
41C22×C4×C59C2312yes—
42C22×C4×C59C2311yescharacteristic
43C22×C4×C59C2311yescharacteristic
44C2×C22⋊D59—22no—
45C2×C22⋊D59—42no—
46C2×C22⋊D59—42no—
47C2×C22⋊D59—42no—
48C2×C22⋊D59—42no—
49C22×Dic59C2312yes—
50C22×Dic59C2311yescharacteristic
51C22×Dic59C2311yescharacteristic
52C22×Dic59C2311yescharacteristic
53C2×D236—24no—
54C2×D236—22no—
55C2×D236—42no—
56C2×D236—42no—
57C2×D236—42no—
58C2×C4×D59—42no—
59C2×C4×D59D414yescomplemented
60C4⋊C4×C59—22no—
61C4⋊C4×C59—42no—
62C22⋊C4×C59—22no—
63C22⋊C4×C59—42no—
64C22⋊Dic59—22no—
65(C2×D59)⋊C4—22no—
66(C2×D59)⋊C4—24no—
67(C2×D59)⋊C4—42no—
68(C2×D59)⋊C4—42no—
69(C2×D59)⋊C4—42no—
70(C2×D59)⋊C4—42no—
71C4⋊Dic59—22no—
72Dic59⋊C4—22no—
73Dic59⋊C4—42no—
74C23×C59—21no—
75C23×C59—22no—
76C23×C59—22no—
77C23×C59—21no—
78C23×C59—21no—
79C23×C59C2411yescharacteristic
80C22×D59—21no—
81C22×D59—22no—
82C22×D59—22no—
83C22×D59—21no—
84C22×D59—22no—
85C22×D59—22no—
86C22×D59—22no—
87C22×D59—22no—
88C22×D59—22no—
89C22×D59—22no—
90C22×D59—22no—
91C22×D59C2×D412yes—
92C2×C4×C59—22no—
93C2×C4×C59—21no—
94C2×C4×C59—22no—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59—22no—
98C2×C4×C59—22no—
99C2×C4×C59—22no—
100C2×C4×C59—22no—
101C2×C4×C59C2×D412yes—
102C22⋊D59—44no—
103C22⋊D59—48no—
104C2×Dic59—22no—
105C2×Dic59—21no—
106C2×Dic59—21no—
107C2×Dic59—22no—
108C2×Dic59—21no—
109C2×Dic59—22no—
110C2×Dic59—22no—
111C2×Dic59—22no—
112C2×Dic59—21no—
113C2×Dic59C2×D412yes—
114D236—44no—
115D236—48no—
116C4×D59—44no—
117C22×C59—41no—
118C22×C59—41no—
119C22×C59—41no—
120C22×C59—42no—
121C22×C59—42no—
122C22×C59C22×D411yescharacteristic
123C22×C59H(5,𝔽2)12yes—
124C22×C59H(5,𝔽2)12yes—
125C22×C59H(5,𝔽2)11yescharacteristic
126C22×C592−1+411yescharacteristic
127C2×D59—22no—
128C2×D59—22no—
129C2×D59—22no—
130C2×D59—42no—
131C2×D59—42no—
132C2×D59—42no—
133C2×D59—42no—
134C2×D59—42no—
135C2×D59—42no—
136C2×D59—42no—
137C2×D59—42no—
138C2×D59—42no—
139C4×C59—22no—
140C4×C59—42no—
141C4×C59—41no—
142C4×C59—42no—
143Dic59—22no—
144Dic59—41no—
145Dic59—41no—
146Dic59—42no—
147#2813—591nomaximal, Sylow
148C2×C59—81no—
149C2×C59#53411yescharacteristic
150C2×C59#53511yescharacteristic
151C2×C59#53711yescharacteristic
152C2×C59#56012yes—
153C2×C59#56212yes—
154D59—42no—
155D59—82no—
156C2×#123—591no—
157C2×#121—591no—
158#400—591no—
159#399D5911yescharacteristic, complemented
160#396—592no—
161#394—592no—
162#394—592no—
163#394—591no—
164#392—591no—
165#392—592no—
166#380—591no—
167C59#281311yescharacteristic, complemented, Sylow
168C22×D4—591no—
169C22×D4—592no—
170C22×D4—592no—
171C22×D4—591no—
172C23×C4—591no—
173#123—1184no—
174#121—1184no—
175C2×C4⋊C4—591no—
176C2×C4⋊C4—591no—
177C2×C4⋊C4—592no—
178C2×C4⋊C4C2×D5911yescharacteristic
179C2×C4⋊C4C2×D5912yes—
180C2×C22⋊C4—591no—
181C2×C22⋊C4—592no—
182C2×C22⋊C4—591no—
183C2×C22⋊C4—592no—
184C2×C22⋊C4—592no—
185C2×C22⋊C4—592no—
186C2×C22⋊C4—592no—
187C2×C22⋊C4C2×D5911yescharacteristic
188C2×C22⋊C4C2×D5912yes—
189#95—591no—
190#95—592no—
191#95—592no—
192#95C2×D5911yescharacteristic
193C24—591no—
194C24—592no—
195C24C22×D5911yescharacteristic
196C2×D4—1184no—
197C2×D4—1182no—
198C2×D4—1182no—
199C2×D4—2362no—
200C2×D4—2362no—
201C2×D4—2362no—
202C2×D4—2362no—
203C2×D4—2362no—
204C2×D4—2362no—
205C2×D4—2362no—
206C22×C4—591no—
207C22×C4—594no—
208C22×C4—592no—
209C22×C4—591no—
210C22×C4—591no—
211C22×C4—2362no—
212C22×C4C22×D5912yes—
213C22×C4C22×D5912yes—
214C22×C4C22×D5911yescharacteristic
215C22×C4C22×D5911yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—42no—
218C4⋊C4—1182no—
219C4⋊C4—1182no—
220C4⋊C4—2362no—
221C22⋊C4—22no—
222C22⋊C4—42no—
223C22⋊C4—1184no—
224C22⋊C4—1182no—
225C22⋊C4—1182no—
226C22⋊C4—2362no—
227C22⋊C4—2362no—
228C22⋊C4—2362no—
229C22⋊C4—2362no—
230C23—21no—
231C23—22no—
232C23—22no—
233C23—21no—
234C23—21no—
235C23—592no—
236C23—1182no—
237C23—1181no—
238C23—1182no—
239C23—1181no—
240C23—1182no—
241C23—1182no—
242C23—1182no—
243C23—1182no—
244C23—1182no—
245C23—1182no—
246C23—1182no—
247C23C23×D5911yescharacteristic
248D4—2364no—
249D4—2368no—
250D4—2364no—
251D4—2368no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—592no—
262C2×C4—1182no—
263C2×C4—1181no—
264C2×C4—1181no—
265C2×C4—1182no—
266C2×C4—1181no—
267C2×C4—1182no—
268C2×C4—1182no—
269C2×C4—1182no—
270C2×C4—1181no—
271C2×C4—2364no—
272C2×C4D4×D5912yes—
273C22—42no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—42no—
278C22—1182no—
279C22—1182no—
280C22—1182no—
281C22—2362no—
282C22—2362no—
283C22—2362no—
284C22—2362no—
285C22—2362no—
286C22—2362no—
287C22—2362no—
288C22—2362no—
289C22—2362no—
290C22C2×D4×D5911yescharacteristic
291C22#1427377212yes—
292C22#1427377312yes—
293C22#1427377311yescharacteristic
294C22#1427377511yescharacteristic
295C4—22no—
296C4—42no—
297C4—41no—
298C4—42no—
299C4—1182no—
300C4—2361no—
301C4—2361no—
302C4—2362no—
303C2—81no—
304C2—4722no—
305C2—2362no—
306C2#1683796311yescharacteristic
307C2#1683796511yescharacteristic
308C2#1683798211yescharacteristic
309C2#1683814212yes—
310C2#1683814812yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#399C2
hypercenter#399D59
hypocenterC59#2813
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#14273709C2semidirect
2. C2×#14273702C2semidirect
3. #16837340C2semidirect
4. #16837324C2semidirect
5. #16837320C2semidirect
6. #16837319C2semidirect
7. #16837317C2semidirect
8. #16837311C2semidirect
9. #16837311C2semidirect
10. #16837261C2semidirect
11. C59×#399C2semidirect
59. C2×C4×D59D4semidirect
159. #399D59semidirect
167. C59#2813semidirect