Finite Group Databasebeta

← #21429845

Subgroups — #21429845

Order 7552.

728 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 143 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×#14273723C211yesmaximal, characteristic, complemented
2C2×#14273699C212yesmaximal, complemented
3C2×#14273683C211yesmaximal, characteristic, complemented
4#16837320C211yesmaximal, characteristic, complemented
5#16837317C212yesmaximal, complemented
6#16837316C211yesmaximal, characteristic, complemented
7#16837311C211yesmaximal, characteristic, complemented
8#16837310C212yesmaximal, complemented
9#16837302C212yesmaximal, complemented
10#16837266C211yesmaximal, characteristic, complemented
11C59×#392C211yesmaximal, characteristic, complemented
12C22×D236C2211yescharacteristic
13C22×C22⋊D59C2212yes—
14C22×C22⋊D59C2212yes—
15C22×C22⋊D59C2212yes—
16C23×Dic59C2211yescharacteristic
17#14273723—24no—
18#14273699—28no—
19#14273683—24no—
20C2×Dic59⋊C4C2212yes—
21C2×Dic59⋊C4C2211yescharacteristic
22C2×Dic59⋊C4C2211yescharacteristic
23C2×C22⋊Dic59C2212yes—
24C2×C22⋊Dic59C2211yescharacteristic
25C2×C22⋊Dic59C2212yes—
26C2×(C2×D59)⋊C4C2212yes—
27C2×(C2×D59)⋊C4C2212yes—
28C2×(C2×D59)⋊C4C2212yes—
29C2×C4×Dic59C2211yescharacteristic
30C2×C4×Dic59C2211yescharacteristic
31C59⋊#95C2212yes—
32C59⋊#95C2211yescharacteristic
33C22×D4×C59C2212yes—
34C22×D4×C59C2211yescharacteristic
35C2×C22⋊C4×C59C2211yescharacteristic
36C2×C22⋊C4×C59C2212yes—
37#95×C59C2211yescharacteristic
38C24×C59C2312yes—
39C24×C59C2311yescharacteristic
40C23×D59C2312yes—
41C2×D4×C59—24no—
42C2×D4×C59—22no—
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—24no—
51C2×C22⋊D59—24no—
52C2×C22⋊D59—24no—
53C2×C22⋊D59—24no—
54C2×C22⋊D59—42no—
55C2×C22⋊D59—42no—
56C2×C22⋊D59—42no—
57C2×C22⋊D59—42no—
58C22×Dic59—22no—
59C22×Dic59—22no—
60C22×Dic59D414yescomplemented
61C22×Dic59C2312yes—
62C22×Dic59C2311yescharacteristic
63C22×Dic59C2311yescharacteristic
64C22×Dic59C2311yescharacteristic
65C22×Dic59C2311yescharacteristic
66C2×D236—22no—
67C2×D236—42no—
68C22⋊C4×C59—22no—
69C22⋊C4×C59—42no—
70C22⋊Dic59—22no—
71C22⋊Dic59—24no—
72C22⋊Dic59—42no—
73(C2×D59)⋊C4—24no—
74(C2×D59)⋊C4—42no—
75(C2×D59)⋊C4—42no—
76Dic59⋊C4—22no—
77Dic59⋊C4—22no—
78Dic59⋊C4—24no—
79C4×Dic59—22no—
80C4×Dic59D414yescomplemented
81C23×C59—21no—
82C23×C59—21no—
83C23×C59—22no—
84C23×C59—22no—
85C23×C59—22no—
86C23×C59—22no—
87C23×C59—22no—
88C23×C59—22no—
89C23×C59—22no—
90C23×C59—22no—
91C23×C59—22no—
92C23×C59C2×D412yes—
93C23×C59C2411yescharacteristic
94C22×D59—22no—
95C22×D59—22no—
96C22×D59—22no—
97C22×D59—22no—
98C22×D59—22no—
99C22×D59—22no—
100C22×D59—22no—
101D4×C59—44no—
102D4×C59—48no—
103C2×C4×C59—22no—
104C2×C4×C59—21no—
105C2×C4×C59—22no—
106C2×C4×C59—22no—
107C2×C4×C59—22no—
108C2×C4×C59—22no—
109C2×C4×C59C2×D412yes—
110C22⋊D59—48no—
111C22⋊D59—48no—
112C22⋊D59—48no—
113C2×Dic59—21no—
114C2×Dic59—21no—
115C2×Dic59—22no—
116C2×Dic59—28no—
117C2×Dic59—21no—
118C2×Dic59—22no—
119C2×Dic59—22no—
120C2×Dic59—22no—
121C2×Dic59—22no—
122C2×Dic59C2×D412yes—
123C2×Dic59C2×D412yes—
124C2×Dic59C2×D412yes—
125C2×Dic59C2×D412yes—
126C2×Dic59#4112yes—
127D236—44no—
128C22×C59—22no—
129C22×C59—24no—
130C22×C59—41no—
131C22×C59—41no—
132C22×C59—42no—
133C22×C59—42no—
134C22×C59—42no—
135C22×C59—42no—
136C22×C59—42no—
137C22×C59—42no—
138C22×C59—42no—
139C22×C59—42no—
140C22×C59C22×D411yescharacteristic
141C22×C59C22×D411yescharacteristic
142C22×C59C2×#4111yescharacteristic
143C22×C59H(5,𝔽2)12yes—
144C22×C59H(5,𝔽2)12yes—
145C2×D59—42no—
146C2×D59—42no—
147C2×D59—42no—
148C2×D59—42no—
149C2×D59—42no—
150C2×D59—42no—
151C2×D59—42no—
152C4×C59—22no—
153C4×C59—41no—
154C4×C59—42no—
155Dic59—22no—
156Dic59—22no—
157Dic59—41no—
158Dic59—42no—
159Dic59#12114yes—
160#2619—591nomaximal, Sylow
161C2×C59—42no—
162C2×C59—82no—
163C2×C59C2×#12111yescharacteristic
164C2×C59#53511yescharacteristic
165C2×C59#53511yescharacteristic
166C2×C59#53512yes—
167C2×C59#54012yes—
168D59—82no—
169C2×#127—591no—
170C2×#121—592no—
171C2×#117—591no—
172#394—591no—
173#394—592no—
174#394—591no—
175#392—592no—
176#392—591no—
177#392D5911yescharacteristic, complemented
178#390—592no—
179#382—591no—
180C59#261911yescharacteristic, complemented, Sylow
181C22×D4—592no—
182C22×D4—592no—
183C22×D4—592no—
184C22×D4—591no—
185C22×D4C2×D5912yes—
186C22×D4C2×D5911yescharacteristic
187C23×C4—591no—
188#127—1184no—
189#121—1188no—
190#117—1184no—
191C2×C4⋊C4—591no—
192C2×C4⋊C4—591no—
193C2×C4⋊C4—592no—
194C2×C22⋊C4—591no—
195C2×C22⋊C4—592no—
196C2×C22⋊C4—592no—
197C2×C22⋊C4—592no—
198C2×C22⋊C4—592no—
199C2×C22⋊C4—592no—
200C2×C22⋊C4C2×D5911yescharacteristic
201C2×C22⋊C4C2×D5912yes—
202C2×C42—591no—
203C2×C42—591no—
204#95—592no—
205#95—591no—
206#95C2×D5911yescharacteristic
207C24—592no—
208C24C22×D5912yes—
209C24C22×D5911yescharacteristic
210C2×D4—24no—
211C2×D4—22no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—1184no—
216C2×D4—1184no—
217C2×D4—1184no—
218C2×D4—1184no—
219C2×D4—1184no—
220C2×D4—1182no—
221C2×D4—2362no—
222C2×D4—2362no—
223C2×D4—2362no—
224C2×D4—2362no—
225C2×D4—2362no—
226C22×C4—591no—
227C22×C4—594no—
228C22×C4—591no—
229C22×C4—591no—
230C22×C4—592no—
231C22×C4—591no—
232C22×C4—1182no—
233C22×C4—1182no—
234C22×C4C22×D5912yes—
235C22×C4C22×D5911yescharacteristic
236C22×C4C22×D5911yescharacteristic
237C4⋊C4—1182no—
238C4⋊C4—1184no—
239C4⋊C4—1182no—
240C22⋊C4—22no—
241C22⋊C4—42no—
242C22⋊C4—1184no—
243C22⋊C4—1182no—
244C22⋊C4—1184no—
245C22⋊C4—2362no—
246C22⋊C4—2362no—
247C22⋊C4—2362no—
248C42—594no—
249C42—1182no—
250C23—21no—
251C23—21no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—1182no—
262C23—1182no—
263C23—1182no—
264C23—1182no—
265C23—1182no—
266C23—1182no—
267C23—1182no—
268C23D4×D5912yes—
269C23C23×D5911yescharacteristic
270D4—44no—
271D4—48no—
272D4—2364no—
273D4—2368no—
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—592no—
284C2×C4—592no—
285C2×C4—592no—
286C2×C4—592no—
287C2×C4—1188no—
288C2×C4—1181no—
289C2×C4—1181no—
290C2×C4—1182no—
291C2×C4—1181no—
292C2×C4—1182no—
293C2×C4—1182no—
294C2×C4—1182no—
295C2×C4—1182no—
296C2×C4D4×D5912yes—
297C22—22no—
298C22—24no—
299C22—41no—
300C22—41no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—42no—
308C22—42no—
309C22—2362no—
310C22—2362no—
311C22—2362no—
312C22—2362no—
313C22—2362no—
314C22—2362no—
315C22—2362no—
316C22C2×D4×D5911yescharacteristic
317C22C2×D4×D5911yescharacteristic
318C22#41×D5911yescharacteristic
319C22#1427377212yes—
320C22#1427377212yes—
321C4—22no—
322C4—41no—
323C4—42no—
324C4—594no—
325C4—1182no—
326C4—1182no—
327C4—2362no—
328C4—2361no—
329C2—42no—
330C2—82no—
331C2—4722no—
332C2#121×D5911yescharacteristic
333C2#1683796511yescharacteristic
334C2#1683796711yescharacteristic
335C2#1683796712yes—
336C2#1683800212yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#392C2
hypercenter#392D59
hypocenterC59#2619
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#14273723C2semidirect
2. C2×#14273699C2semidirect
3. C2×#14273683C2semidirect
4. #16837320C2semidirect
5. #16837317C2semidirect
6. #16837316C2semidirect
7. #16837311C2semidirect
8. #16837310C2semidirect
9. #16837302C2semidirect
10. #16837266C2semidirect
11. C59×#392C2semidirect
60. C22×Dic59D4semidirect
80. C4×Dic59D4semidirect
177. #392D59semidirect
180. C59#2619semidirect