Finite Group Databasebeta

← #21432117

Subgroups — #21432117

Order 7552.

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

normal subgroups only

This group has 644 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×#14273700C211yesmaximal, characteristic, complemented
2C2×#14273698C211yesmaximal, characteristic, complemented
3#16837330C212yesmaximal, complemented
4#16837329C211yesmaximal, characteristic, complemented
5#16837327C211yesmaximal, characteristic, complemented
6#16837311C212yesmaximal, complemented
7#16837310C211yesmaximal, characteristic, complemented
8#16837310C212yesmaximal, complemented
9#16837310C212yesmaximal, complemented
10#16837264C211yesmaximal, characteristic, complemented
11C59×#392C211yesmaximal, characteristic, complemented
12C22×D236C2211yescharacteristic
13C22×C22⋊D59C2212yes—
14C22×C22⋊D59C2211yescharacteristic
15C22×C22⋊D59C2211yescharacteristic
16C22×C22⋊D59C2212yes—
17C22×C4×D59C2211yescharacteristic
18#14273700—24no—
19#14273698—24no—
20C2×C4⋊Dic59C2211yescharacteristic
21C2×Dic59⋊C4C2211yescharacteristic
22C2×C22⋊Dic59C2212yes—
23C2×C22⋊Dic59C2212yes—
24C2×C22⋊Dic59C2212yes—
25C2×(C2×D59)⋊C4C2211yescharacteristic
26C2×(C2×D59)⋊C4C2212yes—
27C2×(C2×D59)⋊C4C2212yes—
28C59⋊#95C2211yescharacteristic
29C59⋊#95C2212yes—
30C59⋊#95C2212yes—
31C59⋊#95C2212yes—
32C22×D4×C59C2211yescharacteristic
33C22×D4×C59C2212yes—
34C2×C22⋊C4×C59C2211yescharacteristic
35C2×C22⋊C4×C59C2212yes—
36#95×C59C2211yescharacteristic
37C24×C59C2312yes—
38C24×C59C2311yescharacteristic
39C23×D59C2311yescharacteristic
40C23×D59C2311yescharacteristic
41C2×D4×C59—22no—
42C2×D4×C59—42no—
43C2×D4×C59—42no—
44C2×D4×C59—42no—
45C2×D4×C59—42no—
46C22×C4×C59C2312yes—
47C22×C4×C59C2311yescharacteristic
48C22×C4×C59C2311yescharacteristic
49C2×C22⋊D59—24no—
50C2×C22⋊D59—22no—
51C2×C22⋊D59—22no—
52C2×C22⋊D59—42no—
53C2×C22⋊D59—42no—
54C2×C22⋊D59—42no—
55C2×C22⋊D59—42no—
56C2×C22⋊D59—42no—
57C2×C22⋊D59—42no—
58C2×C22⋊D59—42no—
59C22×Dic59C2312yes—
60C22×Dic59C2311yescharacteristic
61C22×Dic59C2312yes—
62C22×Dic59C2311yescharacteristic
63C2×D236—22no—
64C2×D236—42no—
65C2×C4×D59—42no—
66C2×C4×D59D414yescomplemented
67C22⋊C4×C59—22no—
68C22⋊C4×C59—42no—
69C22⋊Dic59—24no—
70C22⋊Dic59—42no—
71C22⋊Dic59—42no—
72(C2×D59)⋊C4—22no—
73(C2×D59)⋊C4—42no—
74(C2×D59)⋊C4—42no—
75C4⋊Dic59—22no—
76Dic59⋊C4—22no—
77C23×C59—22no—
78C23×C59—21no—
79C23×C59—21no—
80C23×C59—22no—
81C23×C59—22no—
82C23×C59—22no—
83C23×C59—22no—
84C23×C59—22no—
85C23×C59—22no—
86C23×C59—22no—
87C23×C59—22no—
88C23×C59—21no—
89C23×C59C2411yescharacteristic
90C22×D59—22no—
91C22×D59—22no—
92C22×D59—21no—
93C22×D59—21no—
94C22×D59—21no—
95C22×D59—21no—
96C22×D59—22no—
97C22×D59—22no—
98C22×D59—21no—
99C22×D59C2×D412yes—
100D4×C59—44no—
101D4×C59—48no—
102C2×C4×C59—22no—
103C2×C4×C59—21no—
104C2×C4×C59—22no—
105C2×C4×C59—22no—
106C2×C4×C59—22no—
107C2×C4×C59—22no—
108C2×C4×C59C2×D412yes—
109C22⋊D59—44no—
110C22⋊D59—44no—
111C22⋊D59—48no—
112C22⋊D59—48no—
113C2×Dic59—22no—
114C2×Dic59—21no—
115C2×Dic59—22no—
116C2×Dic59—22no—
117C2×Dic59—22no—
118C2×Dic59—22no—
119C2×Dic59—22no—
120C2×Dic59—22no—
121C2×Dic59—22no—
122C2×Dic59C2×D412yes—
123D236—44no—
124C4×D59—44no—
125C22×C59—41no—
126C22×C59—42no—
127C22×C59—41no—
128C22×C59—42no—
129C22×C59—42no—
130C22×C59—41no—
131C22×C59—42no—
132C22×C59—42no—
133C22×C59—42no—
134C22×C59—42no—
135C22×C59—42no—
136C22×C59—42no—
137C22×C59C22×D411yescharacteristic
138C22×C59H(5,𝔽2)12yes—
139C22×C59H(5,𝔽2)12yes—
140C22×C59H(5,𝔽2)11yescharacteristic
141C22×C59H(5,𝔽2)11yescharacteristic
142C2×D59—22no—
143C2×D59—22no—
144C2×D59—22no—
145C2×D59—42no—
146C2×D59—42no—
147C2×D59—41no—
148C2×D59—41no—
149C2×D59—41no—
150C2×D59—42no—
151C2×D59—42no—
152C4×C59—22no—
153C4×C59—41no—
154C4×C59—42no—
155Dic59—22no—
156Dic59—41no—
157Dic59—42no—
158Dic59—42no—
159#2815—591nomaximal, Sylow
160C2×C59—81no—
161C2×C59—82no—
162C2×C59#53411yescharacteristic
163C2×C59#53511yescharacteristic
164C2×C59#53511yescharacteristic
165C2×C59#56012yes—
166C2×C59#56012yes—
167D59—42no—
168D59—81no—
169C2×#121—591no—
170C2×#121—591no—
171#397—591no—
172#397—592no—
173#397—591no—
174#392—592no—
175#392—591no—
176#392—592no—
177#392—592no—
178#392D5911yescharacteristic, complemented
179#381—591no—
180C59#281511yescharacteristic, complemented, Sylow
181C22×D4—591no—
182C22×D4—592no—
183C22×D4—591no—
184C22×D4—592no—
185C22×D4—591no—
186C22×D4C2×D5911yescharacteristic
187C22×D4C2×D5912yes—
188C23×C4—591no—
189#121—1184no—
190#121—1184no—
191C2×C4⋊C4—591no—
192C2×C4⋊C4—591no—
193C2×C22⋊C4—592no—
194C2×C22⋊C4—591no—
195C2×C22⋊C4—592no—
196C2×C22⋊C4—592no—
197C2×C22⋊C4—592no—
198C2×C22⋊C4—592no—
199C2×C22⋊C4C2×D5911yescharacteristic
200C2×C22⋊C4C2×D5912yes—
201#95—592no—
202#95—591no—
203#95—592no—
204#95—592no—
205#95C2×D5911yescharacteristic
206C24—591no—
207C24—591no—
208C24C22×D5911yescharacteristic
209C24C22×D5912yes—
210C2×D4—22no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—1182no—
216C2×D4—1182no—
217C2×D4—1182no—
218C2×D4—1184no—
219C2×D4—2362no—
220C2×D4—2362no—
221C2×D4—2362no—
222C2×D4—2362no—
223C2×D4—2362no—
224C2×D4—2362no—
225C2×D4—2362no—
226C2×D4—2362no—
227C22×C4—591no—
228C22×C4—594no—
229C22×C4—592no—
230C22×C4—591no—
231C22×C4—592no—
232C22×C4—2362no—
233C22×C4C22×D5912yes—
234C22×C4C22×D5911yescharacteristic
235C22×C4C22×D5911yescharacteristic
236C4⋊C4—1182no—
237C4⋊C4—1182no—
238C22⋊C4—22no—
239C22⋊C4—42no—
240C22⋊C4—1184no—
241C22⋊C4—1182no—
242C22⋊C4—2362no—
243C22⋊C4—2362no—
244C22⋊C4—2362no—
245C22⋊C4—2362no—
246C23—21no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—21no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—21no—
257C23—22no—
258C23—592no—
259C23—1182no—
260C23—1182no—
261C23—1181no—
262C23—1181no—
263C23—1181no—
264C23—1181no—
265C23—1182no—
266C23—1182no—
267C23—1181no—
268C23C23×D5911yescharacteristic
269D4—44no—
270D4—48no—
271D4—2364no—
272D4—2364no—
273D4—2364no—
274D4—2368no—
275D4—2368no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—592no—
283C2×C4—1182no—
284C2×C4—1181no—
285C2×C4—1182no—
286C2×C4—1182no—
287C2×C4—1182no—
288C2×C4—1182no—
289C2×C4—1182no—
290C2×C4—1182no—
291C2×C4—1182no—
292C2×C4—2364no—
293C2×C4D4×D5912yes—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—1182no—
307C22—1182no—
308C22—1182no—
309C22—2361no—
310C22—2361no—
311C22—2361no—
312C22—2362no—
313C22—2362no—
314C22—2362no—
315C22—2362no—
316C22C2×D4×D5911yescharacteristic
317C22#1427377212yes—
318C22#1427377212yes—
319C22#1427377211yescharacteristic
320C22#1427377311yescharacteristic
321C4—22no—
322C4—41no—
323C4—42no—
324C4—1182no—
325C4—2361no—
326C4—2362no—
327C4—2362no—
328C2—82no—
329C2—81no—
330C2—4721no—
331C2—2362no—
332C2#1683796211yescharacteristic
333C2#1683796511yescharacteristic
334C2#1683796811yescharacteristic
335C2#1683814012yes—
336C2#1683814112yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#392C2
hypercenter#392D59
hypocenterC59#2815
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#14273700C2semidirect
2. C2×#14273698C2semidirect
3. #16837330C2semidirect
4. #16837329C2semidirect
5. #16837327C2semidirect
6. #16837311C2semidirect
7. #16837310C2semidirect
8. #16837310C2semidirect
9. #16837310C2semidirect
10. #16837264C2semidirect
11. C59×#392C2semidirect
66. C2×C4×D59D4semidirect
178. #392D59semidirect
180. C59#2815semidirect