Finite Group Databasebeta

← #21432138

Subgroups — #21432138

Order 7552.

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

normal subgroups only

This group has 630 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×#14273723C211yesmaximal, characteristic, complemented
2C2×#14273722C211yesmaximal, characteristic, complemented
3#16837319C211yesmaximal, characteristic, complemented
4#16837318C212yesmaximal, complemented
5#16837317C212yesmaximal, complemented
6#16837312C211yesmaximal, characteristic, complemented
7#16837311C212yesmaximal, complemented
8#16837311C212yesmaximal, complemented
9#16837310C211yesmaximal, characteristic, complemented
10#16837269C211yesmaximal, characteristic, complemented
11C59×#394C211yesmaximal, characteristic, complemented
12C22×D236C2211yescharacteristic
13C22×D236C2212yes—
14C22×C22⋊D59C2212yes—
15C22×C22⋊D59C2212yes—
16C22×C22⋊D59C2212yes—
17#14273723—24no—
18#14273722—24no—
19C2×C4⋊Dic59C2211yescharacteristic
20C2×Dic59⋊C4C2211yescharacteristic
21C2×C22⋊Dic59C2212yes—
22C2×(C2×D59)⋊C4C2211yescharacteristic
23C2×(C2×D59)⋊C4C2211yescharacteristic
24C2×(C2×D59)⋊C4C2211yescharacteristic
25C2×(C2×D59)⋊C4C2212yes—
26C2×(C2×D59)⋊C4C2212yes—
27C2×(C2×D59)⋊C4C2212yes—
28C2×C4×Dic59C2211yescharacteristic
29C59⋊#95C2211yescharacteristic
30C59⋊#95C2212yes—
31C59⋊#95C2212yes—
32C22×D4×C59C2211yescharacteristic
33C2×C4⋊C4×C59C2211yescharacteristic
34C2×C22⋊C4×C59C2212yes—
35C2×C22⋊C4×C59C2212yes—
36#95×C59C2211yescharacteristic
37C24×C59C2312yes—
38C23×D59C2311yescharacteristic
39C23×D59C2312yes—
40C2×D4×C59—22no—
41C2×D4×C59—42no—
42C22×C4×C59C2312yes—
43C22×C4×C59C2311yescharacteristic
44C22×C4×C59C2311yescharacteristic
45C22×C4×C59C2311yescharacteristic
46C2×C22⋊D59—24no—
47C2×C22⋊D59—24no—
48C2×C22⋊D59—42no—
49C2×C22⋊D59—42no—
50C2×C22⋊D59—42no—
51C2×C22⋊D59—42no—
52C2×C22⋊D59—42no—
53C2×C22⋊D59—42no—
54C2×C22⋊D59—42no—
55C22×Dic59C2312yes—
56C22×Dic59C2311yescharacteristic
57C22×Dic59C2311yescharacteristic
58C22×Dic59C2311yescharacteristic
59C2×D236—22no—
60C2×D236—42no—
61C2×D236—42no—
62C2×D236—42no—
63C2×D236—42no—
64C4⋊C4×C59—22no—
65C22⋊C4×C59—42no—
66C22⋊C4×C59—42no—
67C22⋊Dic59—42no—
68(C2×D59)⋊C4—22no—
69(C2×D59)⋊C4—22no—
70(C2×D59)⋊C4—22no—
71(C2×D59)⋊C4—42no—
72(C2×D59)⋊C4—42no—
73(C2×D59)⋊C4—42no—
74C4⋊Dic59—22no—
75Dic59⋊C4—22no—
76C4×Dic59D414yescomplemented
77C23×C59—22no—
78C23×C59—22no—
79C23×C59—22no—
80C23×C59—22no—
81C23×C59—22no—
82C23×C59—22no—
83C23×C59—22no—
84C23×C59C2411yescharacteristic
85C22×D59—21no—
86C22×D59—22no—
87C22×D59—22no—
88C22×D59—22no—
89C22×D59—22no—
90C22×D59—22no—
91C22×D59—21no—
92C22×D59—22no—
93C22×D59—22no—
94C22×D59—22no—
95C22×D59—22no—
96C22×D59—21no—
97D4×C59—44no—
98C2×C4×C59—22no—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—21no—
102C2×C4×C59—22no—
103C2×C4×C59—22no—
104C2×C4×C59—21no—
105C2×C4×C59—22no—
106C2×C4×C59—22no—
107C2×C4×C59C2×D412yes—
108C22⋊D59—48no—
109C22⋊D59—48no—
110C22⋊D59—48no—
111C2×Dic59—21no—
112C2×Dic59—21no—
113C2×Dic59—22no—
114C2×Dic59—21no—
115C2×Dic59—22no—
116C2×Dic59—22no—
117C2×Dic59—22no—
118C2×Dic59—22no—
119C2×Dic59C2×D412yes—
120C2×Dic59C2×D412yes—
121D236—44no—
122D236—48no—
123C22×C59—42no—
124C22×C59—42no—
125C22×C59—42no—
126C22×C59—42no—
127C22×C59—42no—
128C22×C59—42no—
129C22×C59—42no—
130C22×C59C22×D411yescharacteristic
131C22×C59H(5,𝔽2)12yes—
132C22×C59H(5,𝔽2)11yescharacteristic
133C22×C59H(5,𝔽2)11yescharacteristic
134C22×C59H(5,𝔽2)12yes—
135C2×D59—42no—
136C2×D59—42no—
137C2×D59—42no—
138C2×D59—41no—
139C2×D59—41no—
140C2×D59—41no—
141C2×D59—42no—
142C2×D59—42no—
143C2×D59—42no—
144C2×D59—42no—
145C2×D59—42no—
146C2×D59—42no—
147C4×C59—22no—
148C4×C59—41no—
149C4×C59—41no—
150C4×C59—42no—
151Dic59—22no—
152Dic59—22no—
153Dic59—41no—
154Dic59—42no—
155#2818—591nomaximal, Sylow
156C2×C59—82no—
157C2×C59#53511yescharacteristic
158C2×C59#53511yescharacteristic
159C2×C59#53511yescharacteristic
160C2×C59#56012yes—
161C2×C59#56012yes—
162D59—82no—
163D59—81no—
164C2×#127—591no—
165C2×#126—591no—
166#394—591no—
167#394—592no—
168#394—592no—
169#394D5911yescharacteristic, complemented
170#392—591no—
171#392—592no—
172#392—592no—
173#392—591no—
174#383—591no—
175C59#281811yescharacteristic, complemented, Sylow
176C22×D4—592no—
177C22×D4—592no—
178C22×D4—592no—
179C22×D4—591no—
180C22×D4—592no—
181C22×D4C2×D5911yescharacteristic
182#127—1184no—
183#126—1184no—
184C2×C4⋊C4—591no—
185C2×C4⋊C4—591no—
186C2×C4⋊C4C2×D5911yescharacteristic
187C2×C22⋊C4—591no—
188C2×C22⋊C4—591no—
189C2×C22⋊C4—591no—
190C2×C22⋊C4—592no—
191C2×C22⋊C4—592no—
192C2×C22⋊C4—592no—
193C2×C22⋊C4—592no—
194C2×C22⋊C4C2×D5912yes—
195C2×C22⋊C4C2×D5912yes—
196C2×C42—591no—
197#95—592no—
198#95—592no—
199#95—591no—
200#95C2×D5911yescharacteristic
201C24—591no—
202C24—592no—
203C24C22×D5912yes—
204C2×D4—22no—
205C2×D4—42no—
206C2×D4—1184no—
207C2×D4—1184no—
208C2×D4—1182no—
209C2×D4—2362no—
210C2×D4—2362no—
211C2×D4—2362no—
212C2×D4—2362no—
213C2×D4—2362no—
214C2×D4—2362no—
215C2×D4—2362no—
216C2×D4—2362no—
217C2×D4—2362no—
218C2×D4—2362no—
219C2×D4—2362no—
220C22×C4—591no—
221C22×C4—591no—
222C22×C4—592no—
223C22×C4—591no—
224C22×C4C22×D5912yes—
225C22×C4C22×D5911yescharacteristic
226C22×C4C22×D5911yescharacteristic
227C22×C4C22×D5911yescharacteristic
228C4⋊C4—22no—
229C4⋊C4—1182no—
230C4⋊C4—1182no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—1182no—
234C22⋊C4—1182no—
235C22⋊C4—1182no—
236C22⋊C4—2362no—
237C22⋊C4—2362no—
238C22⋊C4—2362no—
239C22⋊C4—2362no—
240C42—594no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—1181no—
249C23—1182no—
250C23—1182no—
251C23—1182no—
252C23—1182no—
253C23—1182no—
254C23—1181no—
255C23—1182no—
256C23—1182no—
257C23—1181no—
258C23—1182no—
259C23—1182no—
260C23C23×D5911yescharacteristic
261D4—44no—
262D4—2364no—
263D4—2368no—
264D4—2368no—
265D4—2368no—
266D4—2368no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—592no—
277C2×C4—592no—
278C2×C4—1181no—
279C2×C4—1181no—
280C2×C4—1182no—
281C2×C4—1181no—
282C2×C4—1182no—
283C2×C4—1182no—
284C2×C4—1182no—
285C2×C4—1182no—
286C2×C4D4×D5912yes—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—2362no—
295C22—2361no—
296C22—2362no—
297C22—2361no—
298C22—2362no—
299C22—2361no—
300C22—2362no—
301C22—2362no—
302C22—2362no—
303C22—2362no—
304C22—2362no—
305C22—2362no—
306C22C2×D4×D5911yescharacteristic
307C22#1427377212yes—
308C22#1427377211yescharacteristic
309C22#1427377311yescharacteristic
310C22#1427377312yes—
311C4—22no—
312C4—41no—
313C4—41no—
314C4—42no—
315C4—1182no—
316C4—1182no—
317C4—2361no—
318C4—2362no—
319C2—82no—
320C2—4722no—
321C2—4721no—
322C2#1683796711yescharacteristic
323C2#1683796811yescharacteristic
324C2#1683797011yescharacteristic
325C2#1683814012yes—
326C2#1683814212yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#394C2
hypercenter#394D59
hypocenterC59#2818
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#14273723C2semidirect
2. C2×#14273722C2semidirect
3. #16837319C2semidirect
4. #16837318C2semidirect
5. #16837317C2semidirect
6. #16837312C2semidirect
7. #16837311C2semidirect
8. #16837311C2semidirect
9. #16837310C2semidirect
10. #16837269C2semidirect
11. C59×#394C2semidirect
76. C4×Dic59D4semidirect
169. #394D59semidirect
175. C59#2818semidirect