Finite Group Databasebeta

← #21434939

Subgroups — #21434939

Order 7552.

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

normal subgroups only

This group has 418 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#16837915C212yesmaximal, complemented
2#16837741C212yesmaximal, complemented
3#16837728C212yesmaximal, complemented
4#16837655C211yesmaximal, characteristic, complemented
5#16837650C212yesmaximal, complemented
6#16837563C212yesmaximal, complemented
7#16837550C212yesmaximal, complemented
8#16837365C211yesmaximal, characteristic, complemented
9C59×#470C211yesmaximal, characteristic, complemented
10C2×#11745689C2212yes—
11C22×C4×D59C2211yescharacteristic
12C2×#11745671C2212yescomplemented
13C2×D472C2211yescharacteristic, complemented
14C2×#11745664C2211yescharacteristic
15#14273712—22no—
16#14273703C2212yes—
17#14273702C2212yes—
18#14273701C2212yes—
19#14273697—22no—
20#14273689—22no—
21(C22⋊D59)⋊C4—22no—
22#14273685C2212yes—
23#14273660C2212yescomplemented
24#14273659C2211yescharacteristic, complemented
25#14273651C2212yescomplemented
26#14273649C2212yescomplemented
27#14273647C2212yescomplemented
28(C2×D59)⋊C8C2212yes—
29(C2×D59)⋊C8C2212yes—
30C2×Q16×C59C2211yescharacteristic, complemented
31C2×M16×C59C2211yescharacteristic
32C22⋊Q8×C59C2212yes—
33#106×C59C2211yescharacteristic, complemented
34#103×C59C2212yescomplemented
35C23×D59D411yescharacteristic
36C2×Q8×C59C2312yes—
37C22×C4×C59C2311yescharacteristic
38C2×C22⋊D59—22no—
39C22×Dic59D411yescharacteristic
40#11745689—42no—
41#11745689—42no—
42C2×D236—22no—
43C2×D236—22no—
44C2×D236C2312yes—
45C2×C4×D59—22no—
46C2×C4×D59—41no—
47C2×C4×D59—41no—
48C2×C4×D59D412yes—
49C2×C4×D59C2312yes—
50Q16×C59—41no—
51M16×C59—41no—
52C2×C8×C59C2311yescharacteristic
53C2×C8×C59C2311yescharacteristic
54C4⋊C4×C59—22no—
55C4⋊C4×C59C2312yes—
56C22⋊C4×C59—22no—
57#11745671—42no—
58(C2×D59)⋊C4—22no—
59(C2×D59)⋊C4—22no—
60(C2×D59)⋊C4—22no—
61(C2×D59)⋊C4—22no—
62C4⋊Dic59C2312yes—
63Dic59⋊C4—22no—
64C4×Dic59—22no—
65#11745664—41no—
66C2×C59⋊C8C2312yes—
67D472—41no—
68C23×C59C2×D411yescharacteristic
69C22×D59—22no—
70C22×D59—41no—
71C22×D59—42no—
72C22×D59C2×D412yes—
73Q8×C59—22no—
74Q8×C59—22no—
75C2×C4×C59—21no—
76C2×C4×C59—21no—
77C2×C4×C59—22no—
78C2×C4×C59—22no—
79C2×C4×C59C2×D411yescharacteristic
80C2×C4×C59C2411yescharacteristic
81C22⋊D59—44no—
82C2×Dic59—22no—
83C2×Dic59—41no—
84C2×Dic59C2×D412yes—
85D236—22no—
86D236—22no—
87D236—44no—
88D236—44no—
89C4×D59—22no—
90C4×D59—22no—
91C4×D59—42no—
92C4×D59—44no—
93C8×C59—21no—
94C8×C59—21no—
95C59⋊C8—22no—
96C22×C59—21no—
97C22×C59—22no—
98C22×C59C22×D411yescharacteristic
99C2×D59—22no—
100C2×D59—22no—
101C2×D59—22no—
102C2×D59—44no—
103C2×D59—42no—
104C2×D59—42no—
105C2×D59—42no—
106C4×C59—21no—
107C4×C59—42no—
108C4×C59—42no—
109C4×C59H(5,𝔽2)11yescharacteristic
110C4×C59H(5,𝔽2)11yescharacteristic
111Dic59—22no—
112Dic59—42no—
113#3197—591nomaximal, Sylow
114C2×C59—41no—
115C2×C59#53411yescharacteristic
116C2×C59#57512yes—
117D59—42no—
118D59—82no—
119#525—592no—
120#485—592no—
121#482—592no—
122#470D5911yescharacteristic, complemented
123#469—591no—
124#468—592no—
125#452—592no—
126#449—592no—
127#407—591no—
128C59#319711yescharacteristic, complemented, Sylow
129C2×#41—592no—
130C23×C4—591no—
131C2×Q16C2×D5911yescharacteristic, complemented
132C2×SD16—592no—
133C2×D8—591no—
134C2×M16—591no—
135C2×M16C2×D5911yescharacteristic
136#123—1182no—
137C22⋊Q8—592no—
138C22⋊Q8C2×D5912yes—
139#121—592no—
140#121—592no—
141#120—1182no—
142C4×D4—1182no—
143C4×D4—1182no—
144#117—592no—
145#107—592no—
146#106—591no—
147#106C2×D5911yescharacteristic, complemented
148#103—592no—
149#103C2×D5912yescomplemented
150#102—592no—
151#102—592no—
152C22⋊C8—592no—
153C22⋊C8—592no—
154C24—591no—
155#41—2362no—
156#41—2362no—
157C2×Q8C22×D5912yes—
158C2×D4—592no—
159C2×D4—1182no—
160C2×D4—1182no—
161C2×D4—1182no—
162C22×C4—591no—
163C22×C4—592no—
164C22×C4—592no—
165C22×C4—1182no—
166C22×C4—2361no—
167C22×C4—2361no—
168C22×C4C22×D5911yescharacteristic
169Q16—41no—
170SD16—2362no—
171D8—2361no—
172M16—41no—
173M16—2361no—
174C2×C8—592no—
175C2×C8C22×D5911yescharacteristic
176C2×C8C22×D5911yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—592no—
179C4⋊C4—1182no—
180C4⋊C4C22×D5912yes—
181C22⋊C4—22no—
182C22⋊C4—1182no—
183C22⋊C4—1182no—
184C22⋊C4—1182no—
185C22⋊C4—1182no—
186C42—1182no—
187C23—592no—
188C23—1182no—
189C23—2361no—
190C23—2362no—
191C23D4×D5911yescharacteristic
192Q8—22no—
193Q8—22no—
194D4—1182no—
195D4—1182no—
196D4—2364no—
197D4—2364no—
198D4—2364no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—592no—
204C2×C4—1182no—
205C2×C4—1182no—
206C2×C4—1182no—
207C2×C4—2362no—
208C2×C4—2364no—
209C2×C4—2361no—
210C2×C4D4×D5911yescharacteristic
211C2×C4C23×D5911yescharacteristic
212C8—21no—
213C8—21no—
214C8—1182no—
215C22—22no—
216C22—21no—
217C22—1182no—
218C22—1182no—
219C22—1182no—
220C22—2364no—
221C22—2362no—
222C22—2362no—
223C22—2362no—
224C22C2×D4×D5911yescharacteristic
225C4—21no—
226C4—42no—
227C4—42no—
228C4—1182no—
229C4—2362no—
230C4#1427377211yescharacteristic
231C4#1427377311yescharacteristic
232C2—41no—
233C2—4722no—
234C2—2362no—
235C2#1683796011yescharacteristic
236C2#1683820112yes—
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#470C2
socleC22×C59C22×D4
hypercenter#470D59
hypocenterC59#3197
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #16837915C2semidirect
2. #16837741C2semidirect
3. #16837728C2semidirect
4. #16837655C2semidirect
5. #16837650C2semidirect
6. #16837563C2semidirect
7. #16837550C2semidirect
8. #16837365C2semidirect
9. C59×#470C2semidirect
12. C2×#11745671C22semidirect
13. C2×D472C22semidirect
23. #14273660C22semidirect
24. #14273659C22semidirect
25. #14273651C22semidirect
26. #14273649C22semidirect
27. #14273647C22semidirect
30. C2×Q16×C59C22semidirect
33. #106×C59C22semidirect
34. #103×C59C22semidirect
122. #470D59semidirect
128. C59#3197semidirect
131. C2×Q16C2×D59semidirect
147. #106C2×D59semidirect
149. #103C2×D59semidirect