Finite Group Databasebeta

← #21435141

Subgroups — #21435141

Order 7552.

420 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 109 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
1#16838006C211yesmaximal, characteristic, complemented
2#16837970C211yesmaximal, characteristic, complemented
3#16837778C211yesmaximal, characteristic, complemented
4#16837767C211yesmaximal, characteristic, complemented
5#16837756C211yesmaximal, characteristic, complemented
6#16837619C212yesmaximal, complemented
7#16837595C212yesmaximal, complemented
8#16837546C212yesmaximal, complemented
9#16837534C212yesmaximal, complemented
10#16837464C211yesmaximal, characteristic, complemented
11C59×#496C211yesmaximal, characteristic, complemented
12C2×#11745689C2211yescharacteristic
13C2×D4×D59C2211yescharacteristic
14C2×#11745669C2212yes—
15C2×#11745669C2212yescomplemented
16#14273724C2211yescharacteristic
17#14273723C2211yescharacteristic
18#14273719C2211yescharacteristic, complemented
19#14273716C2211yescharacteristic
20#14273708—22no—
21#14273700—22no—
22#14273699—22no—
23#14273698C2212yes—
24#14273697—22no—
25D4×Dic59C2212yescomplemented
26#14273684C2211yescharacteristic
27Dic59⋊C8C2212yescomplemented
28#14273651C2212yescomplemented
29#14273649C2212yes—
30#14273649C2212yescomplemented
31#14273648C2212yes—
32(C2×D59)⋊C8C2212yes—
33C4×C59⋊C8C2211yescharacteristic, complemented
34C2×SD16×C59C2212yescomplemented
35C2×D8×C59C2212yes—
36#127×C59C2211yescharacteristic, complemented
37#124×C59C2211yescharacteristic
38#97×C59C2211yescharacteristic, complemented
39C23×D59—21no—
40C2×Q8×C59C2311yescharacteristic
41C2×D4×C59—22no—
42C2×D4×C59C2311yescharacteristic
43C2×D4×C59C2312yes—
44C2×C22⋊D59—22no—
45C2×C22⋊D59—21no—
46C2×C22⋊D59—22no—
47C22×Dic59—22no—
48#11745689—42no—
49D4×D59—42no—
50C2×D236—21no—
51C2×D236—22no—
52C2×D236C2311yescharacteristic
53C2×D236C2311yescharacteristic
54C2×C4×D59—21no—
55C2×C4×D59C2311yescharacteristic
56SD16×C59—42no—
57D8×C59—42no—
58C2×C8×C59C2312yes—
59C22⋊C4×C59—22no—
60C42×C59C2311yescharacteristic
61C22⋊Dic59—22no—
62C22⋊Dic59—22no—
63#11745669—42no—
64#11745669—42no—
65(C2×D59)⋊C4—22no—
66(C2×D59)⋊C4D412yes—
67C4⋊Dic59C2312yes—
68Dic59⋊C4—22no—
69Dic59⋊C4D412yes—
70C4×Dic59C2311yescharacteristic
71C2×C59⋊C8C2312yes—
72C23×C59—22no—
73C23×C59—21no—
74C22×D59—21no—
75C22×D59—21no—
76C22×D59—42no—
77C22×D59—82no—
78C22×D59C2×D411yescharacteristic
79Q8×C59—22no—
80D4×C59—22no—
81D4×C59—22no—
82D4×C59—22no—
83D4×C59—82no—
84C2×C4×C59—21no—
85C2×C4×C59C2×D412yes—
86C2×C4×C59C2411yescharacteristic
87C22⋊D59—42no—
88C22⋊D59—82no—
89C22⋊D59—82no—
90C2×Dic59—22no—
91C2×Dic59—82no—
92C2×Dic59C2×D412yes—
93C2×Dic59C2×D411yescharacteristic
94D236—22no—
95D236—22no—
96D236—42no—
97D236—82no—
98C4×D59—22no—
99C4×D59—42no—
100C8×C59—22no—
101C59⋊C8—22no—
102C22×C59—41no—
103C22×C59—42no—
104C22×C59—42no—
105C22×C59—42no—
106C22×C59—42no—
107C22×C59C22×D411yescharacteristic
108C2×D59—21no—
109C2×D59—22no—
110C2×D59—41no—
111C2×D59—41no—
112C2×D59—42no—
113C2×D59—42no—
114C2×D59—84no—
115C4×C59—22no—
116C4×C59—41no—
117C4×C59H(5,𝔽2)12yes—
118Dic59—21no—
119Dic59—22no—
120Dic59—42no—
121#3220—591nomaximal, Sylow
122C2×C59—81no—
123C2×C59—82no—
124C2×C59#53511yescharacteristic
125C2×C59#57612yes—
126D59—41no—
127D59—81no—
128D59—81no—
129#540—591no—
130#535—591no—
131#496D5911yescharacteristic, complemented
132#493—591no—
133#490—591no—
134#488—591no—
135#463—592no—
136#459—592no—
137#449—592no—
138#447—592no—
139#433—591no—
140C59#322011yescharacteristic, complemented, Sylow
141C2×#41—591no—
142C22×D4—591no—
143C2×SD16C2×D5912yescomplemented
144C2×D8—592no—
145C2×D8—592no—
146C2×D8C2×D5912yes—
147#127—591no—
148#127—591no—
149#127C2×D5911yescharacteristic, complemented
150#125—591no—
151#124—591no—
152#124C2×D5911yescharacteristic
153#123—1182no—
154#121—592no—
155#121—1182no—
156#121—1182no—
157#120—1182no—
158C4×D4—592no—
159#117—591no—
160C4⋊C8—592no—
161#103—592no—
162#102—592no—
163#102—592no—
164#102—592no—
165C22⋊C8—592no—
166#97C2×D5911yescharacteristic, complemented
167C4×C8—591no—
168C24—1181no—
169#41—2362no—
170C2×Q8C22×D5911yescharacteristic
171C2×D4—22no—
172C2×D4—591no—
173C2×D4—591no—
174C2×D4—1181no—
175C2×D4—1182no—
176C2×D4—1181no—
177C2×D4—1182no—
178C2×D4—1182no—
179C2×D4—2362no—
180C2×D4C22×D5911yescharacteristic
181C2×D4C22×D5912yes—
182C22×C4—591no—
183C22×C4—1181no—
184C22×C4—1182no—
185SD16—42no—
186D8—42no—
187D8—2362no—
188D8—2362no—
189C2×C8—592no—
190C2×C8C22×D5912yes—
191C4⋊C4—592no—
192C4⋊C4—592no—
193C4⋊C4—1182no—
194C22⋊C4—22no—
195C22⋊C4—592no—
196C22⋊C4—1182no—
197C22⋊C4—1182no—
198C22⋊C4—1182no—
199C42—591no—
200C42C22×D5911yescharacteristic
201C23—21no—
202C23—22no—
203C23—591no—
204C23—1181no—
205C23—1181no—
206C23—4722no—
207C23—2362no—
208Q8—22no—
209D4—22no—
210D4—22no—
211D4—22no—
212D4—82no—
213D4—1182no—
214D4—1182no—
215D4—4722no—
216D4—4722no—
217D4—4722no—
218D4—2362no—
219D4—2362no—
220C2×C4—21no—
221C2×C4—592no—
222C2×C4—591no—
223C2×C4—1182no—
224C2×C4—1182no—
225C2×C4—4722no—
226C2×C4—2362no—
227C2×C4D4×D5912yes—
228C2×C4C23×D5911yescharacteristic
229C8—22no—
230C8—1182no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—41no—
235C22—42no—
236C22—1182no—
237C22—1181no—
238C22—4724no—
239C22—2362no—
240C22—2362no—
241C22—2361no—
242C22—2361no—
243C22C2×D4×D5911yescharacteristic
244C4—22no—
245C4—41no—
246C4—1182no—
247C4—1181no—
248C4—2362no—
249C4#1427377212yes—
250C2—81no—
251C2—82no—
252C2—4721no—
253C2—4721no—
254C2—2361no—
255C2#1683796511yescharacteristic
256C2#1683820612yes—
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#496C2
socleC22×C59C22×D4
hypercenter#496D59
hypocenterC59#3220
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #16838006C2semidirect
2. #16837970C2semidirect
3. #16837778C2semidirect
4. #16837767C2semidirect
5. #16837756C2semidirect
6. #16837619C2semidirect
7. #16837595C2semidirect
8. #16837546C2semidirect
9. #16837534C2semidirect
10. #16837464C2semidirect
11. C59×#496C2semidirect
15. C2×#11745669C22semidirect
18. #14273719C22semidirect
25. D4×Dic59C22semidirect
27. Dic59⋊C8C22semidirect
28. #14273651C22semidirect
30. #14273649C22semidirect
33. C4×C59⋊C8C22semidirect
34. C2×SD16×C59C22semidirect
36. #127×C59C22semidirect
38. #97×C59C22semidirect
131. #496D59semidirect
140. C59#3220semidirect
143. C2×SD16C2×D59semidirect
149. #127C2×D59semidirect
166. #97C2×D59semidirect