Finite Group Databasebeta

← #21429783

Subgroups — #21429783

Order 7552.

576 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 145 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×#14273715C211yesmaximal, characteristic, complemented
2#16837322C211yesmaximal, characteristic, complemented
3#16837319C211yesmaximal, characteristic, complemented
4#16837314C211yesmaximal, characteristic, complemented
5#16837311C211yesmaximal, characteristic
6#16837302C212yesmaximal, complemented
7#16837293C212yesmaximal, complemented
8#16837292C212yesmaximal, complemented
9#16837281C212yesmaximal, complemented
10#16837248C211yesmaximal, characteristic, complemented
11C2×#117×C59C211yesmaximal, characteristic, complemented
12C22×Dic118C2211yescharacteristic
13C22×D236C2211yescharacteristic
14C22×C22⋊D59C2212yes—
15#14273715—24no—
16C2×Dic59⋊C4C2212yes—
17C2×C22⋊Dic59C2212yes—
18C2×(C2×D59)⋊C4C2212yes—
19C2×(C2×D59)⋊C4C2212yes—
20C2×(C2×D59)⋊C4C2212yes—
21C2×(C2×D59)⋊C4C2212yes—
22C2×(C2×D59)⋊C4C2212yes—
23C2×C4×Dic59C2212yes—
24C59⋊#95C2212yes—
25C59⋊#95C2211yescharacteristic
26C59⋊#95C2211yescharacteristic
27C59⋊#95C2212yes—
28C59⋊#95C2212yes—
29C23×C4×C59C2211yescharacteristic
30#117×C59—24no—
31C2×C4⋊C4×C59C2211yescharacteristic
32C2×C4⋊C4×C59C2211yescharacteristic
33C2×C22⋊C4×C59C2211yescharacteristic
34C2×C22⋊C4×C59C2211yescharacteristic
35C2×C42×C59C2211yescharacteristic
36C2×C42×C59C2211yescharacteristic
37C24×C59C2311yescharacteristic
38C23×D59C2312yes—
39C22×C4×C59—22no—
40C22×C4×C59—22no—
41C22×C4×C59—22no—
42C22×C4×C59C2311yescharacteristic
43C22×C4×C59C2311yescharacteristic
44C22×C4×C59C2311yescharacteristic
45C22×C4×C59C2311yescharacteristic
46C22×C4×C59C2311yescharacteristic
47C22×C4×C59C2311yescharacteristic
48C2×C22⋊D59—24no—
49C2×C22⋊D59—24no—
50C2×C22⋊D59—42no—
51C22×Dic59C2312yes—
52C22×Dic59C2312yes—
53C22×Dic59C2312yes—
54C2×D236—22no—
55C2×D236—42no—
56C2×Dic118—22no—
57C2×Dic118—42no—
58C4⋊C4×C59—22no—
59C4⋊C4×C59—22no—
60C22⋊C4×C59—22no—
61C22⋊C4×C59—22no—
62C42×C59—22no—
63C42×C59D414yescomplemented
64C22⋊Dic59—42no—
65(C2×D59)⋊C4—24no—
66(C2×D59)⋊C4—24no—
67(C2×D59)⋊C4—24no—
68(C2×D59)⋊C4—24no—
69(C2×D59)⋊C4—42no—
70Dic59⋊C4—42no—
71C4×Dic59—24no—
72C23×C59—21no—
73C23×C59—22no—
74C23×C59—21no—
75C23×C59—22no—
76C23×C59#4112yes—
77C23×C59C2411yescharacteristic
78C22×D59—22no—
79C22×D59—22no—
80C22×D59—22no—
81C22×D59—22no—
82C22×D59—22no—
83C22×D59—22no—
84C22×D59—22no—
85C2×C4×C59—28no—
86C2×C4×C59—21no—
87C2×C4×C59—22no—
88C2×C4×C59—21no—
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×C59C2×D412yes—
97C2×C4×C59C2×D412yes—
98C2×C4×C59C2×D412yes—
99C2×C4×C59#4112yes—
100C22⋊D59—48no—
101C2×Dic59—22no—
102C2×Dic59—22no—
103C2×Dic59—22no—
104C2×Dic59—22no—
105C2×Dic59—22no—
106C2×Dic59—22no—
107C2×Dic59—22no—
108C2×Dic59—22no—
109C2×Dic59#4114yes—
110D236—44no—
111Dic118—44no—
112C22×C59—22no—
113C22×C59—24no—
114C22×C59—41no—
115C22×C59—41no—
116C22×C59—42no—
117C22×C59C22×D411yescharacteristic
118C22×C59C2×#4112yes—
119C22×C59C2×#4111yescharacteristic
120C22×C59C2×#4111yescharacteristic
121C22×C59H(5,𝔽2)12yes—
122C2×D59—42no—
123C2×D59—42no—
124C2×D59—42no—
125C2×D59—42no—
126C2×D59—42no—
127C2×D59—42no—
128C2×D59—42no—
129C4×C59—22no—
130C4×C59—22no—
131C4×C59—22no—
132C4×C59—22no—
133C4×C59—41no—
134C4×C59—41no—
135Dic59—24no—
136Dic59—42no—
137Dic59—42no—
138#2610—591nomaximal, Sylow
139C2×C59—42no—
140C2×C59#52511yescharacteristic
141C2×C59#53211yescharacteristic
142C2×C59#53511yescharacteristic
143C2×C59#55112yes—
144C2×C59#55512yes—
145D59—82no—
146C2×#124—591no—
147C2×#117D5911yescharacteristic, complemented
148#395—591no—
149#394—591no—
150#393—591no—
151#392—591no—
152#390—592no—
153#388—592no—
154#388—592no—
155#386—592no—
156#376—591no—
157C59#261011yescharacteristic, complemented, Sylow
158C22×Q8—591no—
159C22×D4—592no—
160C22×D4—591no—
161C23×C4C2×D5911yescharacteristic
162#124—1184no—
163#117—24no—
164C2×C4⋊C4—592no—
165C2×C4⋊C4C2×D5911yescharacteristic
166C2×C4⋊C4C2×D5911yescharacteristic
167C2×C22⋊C4—592no—
168C2×C22⋊C4—592no—
169C2×C22⋊C4—592no—
170C2×C22⋊C4—592no—
171C2×C22⋊C4—592no—
172C2×C22⋊C4—592no—
173C2×C22⋊C4C2×D5911yescharacteristic
174C2×C22⋊C4C2×D5911yescharacteristic
175C2×C42—592no—
176C2×C42C2×D5911yescharacteristic
177C2×C42C2×D5911yescharacteristic
178#95—592no—
179#95—591no—
180#95—591no—
181#95—592no—
182#95—592no—
183C24—592no—
184C24C22×D5911yescharacteristic
185C2×Q8—1182no—
186C2×Q8—2362no—
187C2×D4—1184no—
188C2×D4—1184no—
189C2×D4—1182no—
190C2×D4—2362no—
191C2×D4—2362no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—592no—
196C22×C4—592no—
197C22×C4—592no—
198C22×C4C22×D5911yescharacteristic
199C22×C4C22×D5911yescharacteristic
200C22×C4C22×D5911yescharacteristic
201C22×C4C22×D5911yescharacteristic
202C22×C4C22×D5911yescharacteristic
203C22×C4C22×D5911yescharacteristic
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—2362no—
207C22⋊C4—22no—
208C22⋊C4—22no—
209C22⋊C4—1184no—
210C22⋊C4—1184no—
211C22⋊C4—1184no—
212C22⋊C4—1184no—
213C22⋊C4—2362no—
214C22⋊C4—2362no—
215C42—22no—
216C42—1184no—
217C42C22⋊D5914yescomplemented
218C23—21no—
219C23—22no—
220C23—21no—
221C23—22no—
222C23—1182no—
223C23—1182no—
224C23—1182no—
225C23—1182no—
226C23—1182no—
227C23—1182no—
228C23—1182no—
229C23#1174568512yes—
230C23C23×D5911yescharacteristic
231Q8—2364no—
232D4—2364no—
233D4—2368no—
234C2×C4—28no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—594no—
246C2×C4—1182no—
247C2×C4—1182no—
248C2×C4—1182no—
249C2×C4—1182no—
250C2×C4—1182no—
251C2×C4—1182no—
252C2×C4—1182no—
253C2×C4—1182no—
254C2×C4#1174568512yes—
255C2×C4C2×C22⋊D5912yes—
256C2×C4C2×C22⋊D5912yes—
257C2×C4C2×C22⋊D5912yes—
258C22—22no—
259C22—24no—
260C22—41no—
261C22—41no—
262C22—42no—
263C22—2362no—
264C22—2362no—
265C22—2362no—
266C22—2362no—
267C22—2362no—
268C22—2362no—
269C22—2362no—
270C22C22×C22⋊D5911yescharacteristic
271C22#41×D5912yes—
272C22C2×#1174568511yescharacteristic
273C22C2×#1174568511yescharacteristic
274C22#1427377312yes—
275C4—22no—
276C4—22no—
277C4—22no—
278C4—22no—
279C4—41no—
280C4—41no—
281C4—1184no—
282C4—2362no—
283C4—2362no—
284C2—42no—
285C2—4722no—
286C2#1683791311yescharacteristic
287C2#1683794811yescharacteristic
288C2#1683796611yescharacteristic
289C2#1683807612yes—
290C2#1683811512yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC2×#117×C59C2
hypercenterC2×#117D59
hypocenterC59#2610
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×#14273715C2semidirect
2. #16837322C2semidirect
3. #16837319C2semidirect
4. #16837314C2semidirect
6. #16837302C2semidirect
7. #16837293C2semidirect
8. #16837292C2semidirect
9. #16837281C2semidirect
10. #16837248C2semidirect
11. C2×#117×C59C2semidirect
63. C42×C59D4semidirect
147. C2×#117D59semidirect
157. C59#2610semidirect
217. C42C22⋊D59semidirect