Finite Group Databasebeta

← #21429044

Subgroups — #21429044

Order 7552.

676 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 171 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×#14273726C211yesmaximal, characteristic, complemented
2C2×#14273705C212yesmaximal, complemented
3C2×#14273704C212yesmaximal, complemented
4C2×(C22⋊D59)⋊C4C211yesmaximal, characteristic
5#16837339C211yesmaximal, characteristic, complemented
6#16837338C211yesmaximal, characteristic, complemented
7#16837337C211yesmaximal, characteristic, complemented
8#16837302C212yesmaximal
9#16837271C212yesmaximal, complemented
10#16837253C211yesmaximal, characteristic, complemented
11C59×#399C211yesmaximal, characteristic, complemented
12C22×Dic118C2212yes—
13C22×C22⋊D59C2211yescharacteristic
14C23×Dic59C2211yescharacteristic
15C22×C4×D59C2211yescharacteristic
16#14273726—24no—
17#14273705—28no—
18#14273704—28no—
19(C22⋊D59)⋊C4—24no—
20C2×C4⋊Dic59C2212yes—
21C2×C4⋊Dic59C2211yescharacteristic
22C2×Dic59⋊C4C2212yes—
23C2×Dic59⋊C4C2212yes—
24C2×Dic59⋊C4C2212yes—
25C2×Dic59⋊C4C2211yescharacteristic
26C2×Dic59⋊C4C2212yes—
27C2×C22⋊Dic59C2212yes—
28C2×(C2×D59)⋊C4C2212yes—
29C2×(C2×D59)⋊C4C2212yes—
30C2×(C2×D59)⋊C4C2211yescharacteristic
31C2×C4×Dic59C2212yes—
32C2×C4×Dic59C2211yescharacteristic
33C59⋊#95C2211yescharacteristic
34C2×C4⋊C4×C59C2212yes—
35C2×C4⋊C4×C59C2211yescharacteristic
36C2×C22⋊C4×C59C2212yes—
37C2×C22⋊C4×C59C2211yescharacteristic
38#95×C59C2211yescharacteristic
39C24×C59C2311yescharacteristic
40C23×D59C2311yescharacteristic
41C22×C4×C59C2312yes—
42C22×C4×C59C2311yescharacteristic
43C22×C4×C59C2311yescharacteristic
44C22×C4×C59C2312yes—
45C2×C22⋊D59—22no—
46C2×C22⋊D59—22no—
47C2×C22⋊D59Q814yescomplemented
48C22×Dic59—22no—
49C22×Dic59—24no—
50C22×Dic59C2312yes—
51C22×Dic59C2311yescharacteristic
52C22×Dic59C2312yes—
53C22×Dic59C2311yescharacteristic
54C22×Dic59C2311yescharacteristic
55C2×C4×D59—22no—
56C2×C4×D59—24no—
57C2×Dic118—24no—
58C2×Dic118—24no—
59C2×Dic118—24no—
60C4⋊C4×C59—24no—
61C4⋊C4×C59—22no—
62C22⋊C4×C59—22no—
63C22⋊C4×C59—24no—
64C22⋊Dic59—24no—
65(C2×D59)⋊C4—22no—
66(C2×D59)⋊C4—24no—
67(C2×D59)⋊C4—24no—
68C4⋊Dic59—24no—
69C4⋊Dic59—22no—
70Dic59⋊C4—22no—
71Dic59⋊C4—24no—
72Dic59⋊C4—24no—
73Dic59⋊C4—24no—
74Dic59⋊C4D418yescomplemented
75C4×Dic59—24no—
76C4×Dic59D414yes—
77C23×C59—22no—
78C23×C59—21no—
79C23×C59—21no—
80C23×C59—22no—
81C23×C59C2×Q812yes—
82C23×C59C2411yescharacteristic
83C22×D59—22no—
84C22×D59—21no—
85C22×D59—21no—
86C22×D59—22no—
87C22×D59C2×Q812yes—
88C2×C4×C59—22no—
89C2×C4×C59—22no—
90C2×C4×C59—22no—
91C2×C4×C59—22no—
92C2×C4×C59—21no—
93C2×C4×C59—22no—
94C2×C4×C59—22no—
95C2×C4×C59—22no—
96C2×C4×C59C2×D412yes—
97C2×C4×C59C2×D414yes—
98C22⋊D59—28no—
99C2×Dic59—22no—
100C2×Dic59—22no—
101C2×Dic59—22no—
102C2×Dic59—22no—
103C2×Dic59—22no—
104C2×Dic59—21no—
105C2×Dic59—22no—
106C2×Dic59—44no—
107C2×Dic59C2×D414yes—
108C2×Dic59C2×D412yes—
109C2×Dic59C2×D412yes—
110C2×Dic59C2×D414yes—
111C2×Dic59C2×Q812yes—
112C2×Dic59#4112yes—
113C4×D59—44no—
114Dic118—48no—
115C22×C59—22no—
116C22×C59—24no—
117C22×C59—41no—
118C22×C59—41no—
119C22×C59—42no—
120C22×C59C22×D411yescharacteristic
121C22×C59C22×D412yes—
122C22×C59C22×Q811yescharacteristic
123C22×C59C2×#4111yescharacteristic
124C22×C592−1+412yes—
125C2×D59—22no—
126C2×D59—24no—
127C2×D59—41no—
128C2×D59—41no—
129C2×D59—42no—
130C4×C59—22no—
131C4×C59—24no—
132C4×C59—42no—
133C4×C59—41no—
134Dic59—24no—
135Dic59—42no—
136Dic59—41no—
137Dic59C22⋊Q814yescomplemented
138Dic59#12714yes—
139#2542—591nomaximal, Sylow
140C2×C59—42no—
141C2×C59C2×C22⋊Q811yescharacteristic
142C2×C59C2×#12711yescharacteristic
143C2×C59#53611yescharacteristic
144C2×C59#54712yes—
145C2×C59D4×Q812yes—
146D59—42no—
147C2×C4⋊Q8—591no—
148C2×C22⋊Q8—592no—
149C2×C22⋊Q8—592no—
150C2×C4×D4—591no—
151#399—591no—
152#399—591no—
153#399—591no—
154#399D5911yescharacteristic, complemented
155#390—592no—
156#384—592no—
157C4×C4⋊C4—591no—
158C59#254211yescharacteristic, complemented, Sylow
159C22×Q8—592no—
160C22×D4—591no—
161C23×C4—591no—
162C23×C4—591no—
163C4⋊Q8—1184no—
164C22⋊Q8—1188no—
165C22⋊Q8—1188no—
166C4×D4—1184no—
167C2×C4⋊C4—592no—
168C2×C4⋊C4—591no—
169C2×C4⋊C4—592no—
170C2×C4⋊C4—592no—
171C2×C4⋊C4—592no—
172C2×C4⋊C4—592no—
173C2×C4⋊C4—591no—
174C2×C4⋊C4C2×D5912yes—
175C2×C4⋊C4C2×D5911yescharacteristic
176C2×C22⋊C4—592no—
177C2×C22⋊C4—591no—
178C2×C22⋊C4—592no—
179C2×C22⋊C4—592no—
180C2×C22⋊C4C2×D5911yescharacteristic
181C2×C22⋊C4C2×D5912yes—
182C2×C42—591no—
183C2×C42—592no—
184#95—591no—
185#95C2×D5911yescharacteristic
186C24—591no—
187C24C22×D5911yescharacteristic
188C2×Q8—1184no—
189C2×Q8—1184no—
190C2×Q8—1184no—
191C2×D4—594no—
192C2×D4—1182no—
193C2×D4—1182no—
194C22×C4—591no—
195C22×C4—591no—
196C22×C4—592no—
197C22×C4—592no—
198C22×C4—591no—
199C22×C4—1182no—
200C22×C4—1184no—
201C22×C4—1182no—
202C22×C4—1184no—
203C22×C4C22×D5912yes—
204C22×C4C22×D5911yescharacteristic
205C22×C4C22×D5911yescharacteristic
206C22×C4C22×D5912yes—
207C4⋊C4—24no—
208C4⋊C4—22no—
209C4⋊C4—598no—
210C4⋊C4—1182no—
211C4⋊C4—1184no—
212C4⋊C4—1184no—
213C4⋊C4—1184no—
214C4⋊C4—1184no—
215C4⋊C4—1182no—
216C22⋊C4—22no—
217C22⋊C4—24no—
218C22⋊C4—1184no—
219C22⋊C4—1184no—
220C22⋊C4—1182no—
221C22⋊C4—1184no—
222C42—594no—
223C42—1184no—
224C23—22no—
225C23—21no—
226C23—21no—
227C23—22no—
228C23—592no—
229C23—1182no—
230C23—1181no—
231C23—1181no—
232C23—1182no—
233C23Q8×D5912yes—
234C23C23×D5911yescharacteristic
235Q8—2368no—
236D4—1188no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—592no—
246C2×C4—592no—
247C2×C4—594no—
248C2×C4—592no—
249C2×C4—592no—
250C2×C4—594no—
251C2×C4—1182no—
252C2×C4—1182no—
253C2×C4—1181no—
254C2×C4—1182no—
255C2×C4—1182no—
256C2×C4—1182no—
257C2×C4—1182no—
258C2×C4—2364no—
259C2×C4—2364no—
260C2×C4D4×D5912yes—
261C2×C4D4×D5914yes—
262C22—22no—
263C22—24no—
264C22—41no—
265C22—41no—
266C22—42no—
267C22—1182no—
268C22—1184no—
269C22—2361no—
270C22—2361no—
271C22—2362no—
272C22C2×D4×D5911yescharacteristic
273C22C2×D4×D5912yes—
274C22C2×Q8×D5911yescharacteristic
275C22#41×D5911yescharacteristic
276C22#1427377412yes—
277C4—22no—
278C4—24no—
279C4—41no—
280C4—42no—
281C4—594no—
282C4—594no—
283C4—1184no—
284C4—2361no—
285C4—2362no—
286C2—42no—
287C2—2362no—
288C2C22⋊Q8×D5911yescharacteristic
289C2#127×D5911yescharacteristic
290C2#1683797411yescharacteristic
291C2#1683805012yes—
292C2#1683806512yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#399C2
hypercenter#399D59
hypocenterC59#2542
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#14273726C2semidirect
2. C2×#14273705C2semidirect
3. C2×#14273704C2semidirect
5. #16837339C2semidirect
6. #16837338C2semidirect
7. #16837337C2semidirect
9. #16837271C2semidirect
10. #16837253C2semidirect
11. C59×#399C2semidirect
47. C2×C22⋊D59Q8semidirect
74. Dic59⋊C4D4semidirect
137. Dic59C22⋊Q8semidirect
154. #399D59semidirect
158. C59#2542semidirect