Finite Group Databasebeta

← #21427503

Subgroups — #21427503

Order 7552.

610 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14273710C211yesmaximal, characteristic
2C2×#14273710C211yesmaximal, characteristic
3C2×#14273686C211yesmaximal, characteristic, complemented
4#16837335C212yesmaximal, complemented
5#16837324C212yesmaximal, complemented
6#16837297C212yesmaximal, complemented
7#16837292C212yesmaximal, complemented
8#16837281C211yesmaximal, characteristic
9#16837275C211yesmaximal, characteristic, complemented
10#16837265C211yesmaximal, characteristic, complemented
11C22×C4⋊C4×C59C211yesmaximal, characteristic, complemented
12C22×C22⋊D59C2211yescharacteristic
13#14273710—24no—
14#14273710—24no—
15#14273686—24no—
16C2×C4⋊Dic59C2212yes—
17C2×Dic59⋊C4C2211yescharacteristic
18C2×Dic59⋊C4C2212yes—
19C2×Dic59⋊C4C2211yescharacteristic
20C2×Dic59⋊C4C2212yes—
21C2×C22⋊Dic59C2211yescharacteristic
22C2×C22⋊Dic59C2211yescharacteristic
23C2×C22⋊Dic59C2211yescharacteristic
24C2×(C2×D59)⋊C4C2211yescharacteristic
25C2×(C2×D59)⋊C4C2212yes—
26C2×(C2×D59)⋊C4C2211yescharacteristic
27C2×(C2×D59)⋊C4C2212yes—
28C2×C4×Dic59C2212yes—
29C59⋊#95C2212yes—
30C59⋊#95C2212yes—
31C59⋊#95C2212yes—
32C59⋊#95C2212yes—
33C23×C4×C59C2211yescharacteristic
34C23×C4×C59C2211yescharacteristic
35C23×C4×C59C2211yescharacteristic
36C2×C4⋊C4×C59—24no—
37C2×C4⋊C4×C59C2212yes—
38C2×C4⋊C4×C59C2212yes—
39C24×C59C2311yescharacteristic
40C23×D59C2311yescharacteristic
41C22×C4×C59—22no—
42C22×C4×C59—22no—
43C22×C4×C59—22no—
44C22×C4×C59—24no—
45C22×C4×C59—22no—
46C22×C4×C59—22no—
47C22×C4×C59—22no—
48C22×C4×C59D414yes—
49C22×C4×C59C2312yes—
50C22×C4×C59C2311yescharacteristic
51C22×C4×C59C2311yescharacteristic
52C22×C4×C59C2312yes—
53C2×C22⋊D59—22no—
54C2×C22⋊D59—22no—
55C2×C22⋊D59—22no—
56C22×Dic59C2311yescharacteristic
57C22×Dic59C2312yes—
58C22×Dic59C2311yescharacteristic
59C22×Dic59C2311yescharacteristic
60C22×Dic59C2312yes—
61C4⋊C4×C59—24no—
62C4⋊C4×C59—24no—
63C22⋊Dic59—22no—
64C22⋊Dic59—22no—
65C22⋊Dic59D414yes—
66(C2×D59)⋊C4—22no—
67(C2×D59)⋊C4—22no—
68(C2×D59)⋊C4—24no—
69(C2×D59)⋊C4—24no—
70C4⋊Dic59—24no—
71Dic59⋊C4—22no—
72Dic59⋊C4—22no—
73Dic59⋊C4—24no—
74Dic59⋊C4—24no—
75C4×Dic59—24no—
76C23×C59—22no—
77C23×C59—22no—
78C23×C59C2×D412yes—
79C23×C59C2×D412yes—
80C23×C59#4112yes—
81C23×C59C2411yescharacteristic
82C22×D59—21no—
83C22×D59—21no—
84C22×D59—22no—
85C22×D59—21no—
86C22×D59—22no—
87C2×C4×C59—28no—
88C2×C4×C59—28no—
89C2×C4×C59—21no—
90C2×C4×C59—21no—
91C2×C4×C59—22no—
92C2×C4×C59—21no—
93C2×C4×C59—21no—
94C2×C4×C59—22no—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59—22no—
98C2×C4×C59—44no—
99C2×C4×C59C2×D412yes—
100C2×C4×C59C2×D412yes—
101C2×C4×C59#4114yes—
102C22⋊D59—44no—
103C2×Dic59—22no—
104C2×Dic59—22no—
105C2×Dic59—22no—
106C2×Dic59—21no—
107C2×Dic59—21no—
108C2×Dic59—21no—
109C2×Dic59—21no—
110C2×Dic59—22no—
111C2×Dic59—22no—
112C2×Dic59—21no—
113C2×Dic59—22no—
114C2×Dic59—22no—
115C2×Dic59C2×D414yes—
116C22×C59—22no—
117C22×C59—22no—
118C22×C59—24no—
119C22×C59—24no—
120C22×C59#12114yes—
121C22×C59C22×D411yescharacteristic
122C22×C59C22×D411yescharacteristic
123C22×C59C2×#4111yescharacteristic
124C22×C59C2×#4112yes—
125C22×C592−1+412yes—
126C2×D59—41no—
127C2×D59—41no—
128C2×D59—41no—
129C2×D59—42no—
130C2×D59—42no—
131C4×C59—22no—
132C4×C59—22no—
133C4×C59—24no—
134C4×C59—42no—
135Dic59—24no—
136Dic59—41no—
137Dic59—41no—
138Dic59—41no—
139Dic59—42no—
140#2413—591nomaximal, Sylow
141C2×C59—24no—
142C2×C59C2×#12111yescharacteristic
143C2×C59#52511yescharacteristic
144C2×C59#53611yescharacteristic
145C2×C59#54712yes—
146C2×C59#55212yes—
147D59—81no—
148C2×#123—591no—
149C2×#123—591no—
150C2×#117—591no—
151C22×C4⋊C4D5911yescharacteristic, complemented
152#398—592no—
153#396—592no—
154#389—592no—
155#388—592no—
156#386—591no—
157#385—591no—
158#381—591no—
159C59#241311yescharacteristic, complemented, Sylow
160C22×D4—591no—
161C23×C4C2×D5911yescharacteristic
162C23×C4C2×D5911yescharacteristic
163C23×C4C2×D5911yescharacteristic
164#123—1184no—
165#123—1184no—
166#117—1184no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—592no—
169C2×C4⋊C4—591no—
170C2×C4⋊C4—592no—
171C2×C4⋊C4—591no—
172C2×C4⋊C4—592no—
173C2×C4⋊C4C2×D5912yes—
174C2×C4⋊C4C2×D5912yes—
175C2×C22⋊C4—591no—
176C2×C22⋊C4—591no—
177C2×C22⋊C4—591no—
178C2×C22⋊C4—591no—
179C2×C22⋊C4—592no—
180C2×C22⋊C4—591no—
181C2×C22⋊C4—592no—
182C2×C42—592no—
183#95—592no—
184#95—592no—
185#95—592no—
186#95—592no—
187C24—591no—
188C24C22×D5911yescharacteristic
189C2×D4—1182no—
190C2×D4—1182no—
191C2×D4—1182no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—592no—
200C22×C4—591no—
201C22×C4—592no—
202C22×C4—591no—
203C22×C4—591no—
204C22×C4C22⋊D5914yes—
205C22×C4C22×D5912yes—
206C22×C4C22×D5912yes—
207C22×C4C22×D5911yescharacteristic
208C22×C4C22×D5911yescharacteristic
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—1184no—
212C4⋊C4—1184no—
213C4⋊C4—1184no—
214C4⋊C4—1182no—
215C4⋊C4—1182no—
216C22⋊C4—594no—
217C22⋊C4—1184no—
218C22⋊C4—1182no—
219C22⋊C4—1182no—
220C22⋊C4—1184no—
221C22⋊C4—1182no—
222C22⋊C4—1182no—
223C42—1184no—
224C23—22no—
225C23—22no—
226C23—1181no—
227C23—1181no—
228C23—1181no—
229C23—1182no—
230C23—1182no—
231C23D4×D5912yes—
232C23#1174568712yes—
233C23C2×C22⋊D5912yes—
234C23C23×D5911yescharacteristic
235D4—2364no—
236C2×C4—28no—
237C2×C4—28no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—44no—
248C2×C4—594no—
249C2×C4—1182no—
250C2×C4—1182no—
251C2×C4—1182no—
252C2×C4—1181no—
253C2×C4—1181no—
254C2×C4—1181no—
255C2×C4—1181no—
256C2×C4—1181no—
257C2×C4—1182no—
258C2×C4—1182no—
259C2×C4—1182no—
260C2×C4—1182no—
261C2×C4#1174568514yes—
262C2×C4C2×C22⋊D5912yes—
263C2×C4C2×C22⋊D5912yes—
264C22—22no—
265C22—22no—
266C22—24no—
267C22—24no—
268C22—2361no—
269C22—2361no—
270C22—2361no—
271C22—2362no—
272C22—2362no—
273C22#1427369914yes—
274C22C2×D4×D5911yescharacteristic
275C22C22×C22⋊D5911yescharacteristic
276C22C2×#1174568711yescharacteristic
277C22C2×#1174568512yes—
278C22#1427377512yes—
279C4—22no—
280C4—22no—
281C4—24no—
282C4—42no—
283C4—1184no—
284C4—2361no—
285C4—2361no—
286C4—2361no—
287C4—2362no—
288C2—24no—
289C2—4721no—
290C2C2×#1427369911yescharacteristic
291C2#1683791311yescharacteristic
292C2#1683797611yescharacteristic
293C2#1683805312yes—
294C2#1683808712yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC22×C4⋊C4×C59C2
hypercenterC22×C4⋊C4D59
hypocenterC59#2413
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#14273686C2semidirect
4. #16837335C2semidirect
5. #16837324C2semidirect
6. #16837297C2semidirect
7. #16837292C2semidirect
9. #16837275C2semidirect
10. #16837265C2semidirect
11. C22×C4⋊C4×C59C2semidirect
151. C22×C4⋊C4D59semidirect
159. C59#2413semidirect