Finite Group Databasebeta

← #21430242

Subgroups — #21430242

Order 7552.

534 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 129 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×#14273720C211yesmaximal, characteristic, complemented
2C2×#14273716C211yesmaximal, characteristic, complemented
3#16837326C212yesmaximal, complemented
4#16837321C211yesmaximal, characteristic, complemented
5#16837320C211yesmaximal, characteristic, complemented
6#16837317C212yesmaximal, complemented
7#16837311C211yesmaximal, characteristic
8#16837302C212yesmaximal, complemented
9#16837294C212yesmaximal, complemented
10#16837253C211yesmaximal, characteristic, complemented
11C59×#393C211yesmaximal, characteristic, complemented
12C22×D236C2211yescharacteristic
13C22×C22⋊D59C2212yes—
14#14273720—24no—
15#14273716—24no—
16C2×C4⋊Dic59C2211yescharacteristic
17C2×Dic59⋊C4C2211yescharacteristic
18C2×Dic59⋊C4C2211yescharacteristic
19C2×Dic59⋊C4C2212yes—
20C2×Dic59⋊C4C2212yes—
21C2×C22⋊Dic59C2211yescharacteristic
22C2×C22⋊Dic59C2211yescharacteristic
23C2×(C2×D59)⋊C4C2212yes—
24C2×(C2×D59)⋊C4C2212yes—
25C2×(C2×D59)⋊C4C2212yes—
26C2×(C2×D59)⋊C4C2212yes—
27C2×(C2×D59)⋊C4C2212yes—
28C2×C4×Dic59C2212yes—
29C2×C4×Dic59C2211yescharacteristic
30C59⋊#95C2211yescharacteristic
31C59⋊#95C2212yes—
32C22×Q8×C59C2211yescharacteristic
33C2×C22⋊C4×C59C2211yescharacteristic
34C2×C22⋊C4×C59C2212yes—
35#95×C59C2211yescharacteristic
36#95×C59C2212yes—
37C24×C59C2311yescharacteristic
38C23×D59C2312yes—
39C2×Q8×C59—22no—
40C2×Q8×C59—42no—
41C22×C4×C59C2312yes—
42C22×C4×C59C2311yescharacteristic
43C22×C4×C59C2312yes—
44C22×C4×C59C2311yescharacteristic
45C2×C22⋊D59—24no—
46C2×C22⋊D59—42no—
47C2×C22⋊D59—42no—
48C22×Dic59C2311yescharacteristic
49C22×Dic59C2311yescharacteristic
50C22×Dic59C2311yescharacteristic
51C22×Dic59C2311yescharacteristic
52C22×Dic59C2312yes—
53C2×D236—22no—
54C2×D236—42no—
55C22⋊C4×C59—22no—
56C22⋊C4×C59—42no—
57C22⋊Dic59—22no—
58C22⋊Dic59—22no—
59(C2×D59)⋊C4—24no—
60(C2×D59)⋊C4—24no—
61(C2×D59)⋊C4—24no—
62(C2×D59)⋊C4—42no—
63(C2×D59)⋊C4—42no—
64C4⋊Dic59—22no—
65Dic59⋊C4—22no—
66Dic59⋊C4—22no—
67Dic59⋊C4—24no—
68Dic59⋊C4—42no—
69C4×Dic59—42no—
70C4×Dic59D414yescomplemented
71C23×C59—21no—
72C23×C59—22no—
73C23×C59—22no—
74C23×C59—21no—
75C23×C59—21no—
76C23×C59C2411yescharacteristic
77C22×D59—22no—
78C22×D59—22no—
79C22×D59—22no—
80C22×D59—22no—
81C22×D59—22no—
82C22×D59—22no—
83C22×D59—22no—
84Q8×C59—44no—
85C2×C4×C59—22no—
86C2×C4×C59—21no—
87C2×C4×C59—22no—
88C2×C4×C59—22no—
89C2×C4×C59—22no—
90C2×C4×C59—22no—
91C2×C4×C59—22no—
92C2×C4×C59—22no—
93C2×C4×C59—22no—
94C2×C4×C59C2×D412yes—
95C22⋊D59—48no—
96C2×Dic59—22no—
97C2×Dic59—21no—
98C2×Dic59—22no—
99C2×Dic59—22no—
100C2×Dic59—21no—
101C2×Dic59—21no—
102C2×Dic59—21no—
103C2×Dic59—22no—
104C2×Dic59—21no—
105C2×Dic59—21no—
106C2×Dic59C2×D412yes—
107C2×Dic59C2×D412yes—
108C2×Dic59#4114yes—
109D236—44no—
110C22×C59—41no—
111C22×C59—41no—
112C22×C59—41no—
113C22×C59—42no—
114C22×C59—42no—
115C22×C59C22×D411yescharacteristic
116C22×C59C2×#4112yes—
117C22×C59H(5,𝔽2)12yes—
118C22×C59H(5,𝔽2)11yescharacteristic
119C22×C592−1+411yescharacteristic
120C2×D59—42no—
121C2×D59—42no—
122C2×D59—42no—
123C2×D59—42no—
124C2×D59—42no—
125C2×D59—42no—
126C2×D59—42no—
127C4×C59—22no—
128C4×C59—41no—
129C4×C59—42no—
130C4×C59—42no—
131Dic59—22no—
132Dic59—22no—
133Dic59—24no—
134Dic59—41no—
135Dic59—41no—
136#2662—591nomaximal, Sylow
137C2×C59—81no—
138C2×C59#53211yescharacteristic
139C2×C59#53511yescharacteristic
140C2×C59#53711yescharacteristic
141C2×C59#53812yes—
142C2×C59#54212yes—
143D59—82no—
144C2×#126—591no—
145C2×#124—591no—
146#396—592no—
147#395—591no—
148#394—592no—
149#394—591no—
150#393D5911yescharacteristic, complemented
151#392—591no—
152#390—592no—
153#388—592no—
154C4×C4⋊C4—591no—
155C59#266211yescharacteristic, complemented, Sylow
156C22×Q8C2×D5911yescharacteristic
157C22×D4—592no—
158C22×D4—591no—
159#126—1184no—
160#124—1184no—
161C2×C4⋊C4—591no—
162C2×C4⋊C4—591no—
163C2×C4⋊C4—591no—
164C2×C4⋊C4—592no—
165C2×C4⋊C4—592no—
166C2×C22⋊C4—591no—
167C2×C22⋊C4—591no—
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×D5912yes—
175C2×C42—591no—
176C2×C42—592no—
177#95—591no—
178#95—592no—
179#95C2×D5911yescharacteristic
180#95C2×D5912yes—
181C24—592no—
182C24C22×D5911yescharacteristic
183C2×Q8—22no—
184C2×Q8—42no—
185C2×D4—1184no—
186C2×D4—1182no—
187C2×D4—2362no—
188C2×D4—2362no—
189C2×D4—2362no—
190C22×C4—592no—
191C22×C4—591no—
192C22×C4—591no—
193C22×C4—591no—
194C22×C4—591no—
195C22×C4C22×D5912yes—
196C22×C4C22×D5912yes—
197C22×C4C22×D5911yescharacteristic
198C22×C4C22×D5911yescharacteristic
199C4⋊C4—1182no—
200C4⋊C4—1182no—
201C4⋊C4—1184no—
202C4⋊C4—1182no—
203C4⋊C4—2362no—
204C22⋊C4—22no—
205C22⋊C4—42no—
206C22⋊C4—1184no—
207C22⋊C4—1184no—
208C22⋊C4—1184no—
209C22⋊C4—1182no—
210C22⋊C4—1182no—
211C22⋊C4—2362no—
212C22⋊C4—2362no—
213C42—594no—
214C42—2362no—
215C23—21no—
216C23—22no—
217C23—22no—
218C23—21no—
219C23—21no—
220C23—1182no—
221C23—1182no—
222C23—1182no—
223C23—1182no—
224C23—1182no—
225C23—1182no—
226C23—1182no—
227C23C23×D5911yescharacteristic
228Q8—44no—
229D4—2364no—
230D4—2368no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—592no—
241C2×C4—592no—
242C2×C4—594no—
243C2×C4—1182no—
244C2×C4—1182no—
245C2×C4—1181no—
246C2×C4—1182no—
247C2×C4—1181no—
248C2×C4—1181no—
249C2×C4—1181no—
250C2×C4—1182no—
251C2×C4—1181no—
252C2×C4—1181no—
253C2×C4D4×D5912yes—
254C22—41no—
255C22—41no—
256C22—41no—
257C22—42no—
258C22—42no—
259C22—2362no—
260C22—2362no—
261C22—2362no—
262C22—2362no—
263C22—2362no—
264C22—2362no—
265C22—2362no—
266C22C2×D4×D5911yescharacteristic
267C22#41×D5912yes—
268C22#1427377212yes—
269C22#1427377311yescharacteristic
270C22#1427377511yescharacteristic
271C4—22no—
272C4—41no—
273C4—42no—
274C4—42no—
275C4—1182no—
276C4—1182no—
277C4—1184no—
278C4—2361no—
279C4—2361no—
280C2—81no—
281C2—4722no—
282C2#1683795111yescharacteristic
283C2#1683796711yescharacteristic
284C2#1683798211yescharacteristic
285C2#1683798612yes—
286C2#1683802212yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#393C2
hypercenter#393D59
hypocenterC59#2662
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#14273720C2semidirect
2. C2×#14273716C2semidirect
3. #16837326C2semidirect
4. #16837321C2semidirect
5. #16837320C2semidirect
6. #16837317C2semidirect
8. #16837302C2semidirect
9. #16837294C2semidirect
10. #16837253C2semidirect
11. C59×#393C2semidirect
70. C4×Dic59D4semidirect
150. #393D59semidirect
155. C59#2662semidirect