Finite Group Databasebeta

← #21433609

Subgroups — #21433609

Order 7552.

494 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14273755C211yesmaximal, characteristic
2C22×#11745660C211yesmaximal, characteristic, complemented
3#16837978C211yesmaximal, characteristic, complemented
4#16837876C211yesmaximal, characteristic, complemented
5#16837618C214yesmaximal, complemented
6#16837608C214yesmaximal, complemented
7#16837412C211yesmaximal, characteristic, complemented
8#16837399C211yesmaximal, characteristic, complemented
9C59×#424C211yesmaximal, characteristic, complemented
10C2×#11745685C2211yescharacteristic
11C22×Dic118C2211yescharacteristic
12C22×D236C2211yescharacteristic
13#14273755—24no—
14C2×#11745660—24no—
15C2×#11745660C2214yes—
16C2×#11745660C2212yes—
17C2×D472C2212yes—
18#14273727C2212yescomplemented
19#14273715C2212yescomplemented
20#14273711—22no—
21#14273707—22no—
22C4×D236C2214yes—
23#14273689—22no—
24#14273686C2211yescharacteristic
25C22⋊C4×D59—22no—
26#14273652C2212yescomplemented
27#14273652C2214yescomplemented
28#14273649C2212yescomplemented
29C2×M16×C59C2211yescharacteristic
30C22×C8×C59C2211yescharacteristic
31#117×C59C2211yescharacteristic
32C4⋊C8×C59C2214yescomplemented
33C23×D59—21no—
34C22×C4×C59C2311yescharacteristic
35C2×C22⋊D59—21no—
36C22×Dic59—21no—
37#11745685—22no—
38#11745685—22no—
39C2×D236—21no—
40C2×D236—22no—
41C2×D236D414yes—
42C2×D236C2311yescharacteristic
43C2×D236C2312yes—
44C2×C4×D59—21no—
45C2×C4×D59—24no—
46C2×Dic118—22no—
47C2×Dic118—22no—
48C2×Dic118—21no—
49C2×Dic118C2312yes—
50C2×Dic118C2311yescharacteristic
51M16×C59—22no—
52C2×C8×C59—22no—
53C2×C8×C59C2312yes—
54C2×C8×C59C2312yes—
55C4⋊C4×C59D412yes—
56C22⋊C4×C59D412yes—
57C42×C59C2312yes—
58C22⋊Dic59—21no—
59(C2×D59)⋊C4—24no—
60(C2×D59)⋊C4—24no—
61C4⋊Dic59—24no—
62C4⋊Dic59C2312yes—
63C4×Dic59—21no—
64D472—24no—
65#11745660—28no—
66#11745660—24no—
67C23×C59C2×D411yescharacteristic
68C22×D59—22no—
69C22×D59—24no—
70C22×D59—21no—
71C22×D59—82no—
72C2×C4×C59C2×D414yes—
73C2×C4×C59C2×D411yescharacteristic
74C2×C4×C59C2×D412yes—
75C2×C4×C59#4112yes—
76C2×C4×C59C2411yescharacteristic
77C22⋊D59—42no—
78C2×Dic59—22no—
79C2×Dic59—21no—
80C2×Dic59—22no—
81C2×Dic59—42no—
82D236—24no—
83D236—21no—
84D236—22no—
85D236—21no—
86D236C2×D414yes—
87C4×D59—42no—
88C4×D59—84no—
89Dic118—22no—
90Dic118—21no—
91Dic118—22no—
92Dic118—21no—
93Dic118—42no—
94C8×C59—22no—
95C8×C59—22no—
96C22×C59—22no—
97C22×C59#12112yes—
98C22×C59C22×D411yescharacteristic
99C2×D59—24no—
100C2×D59—44no—
101C2×D59—41no—
102C2×D59—42no—
103C2×D59—84no—
104C4×C59—24no—
105C4×C59#12112yes—
106C4×C59C22×D411yescharacteristic
107C4×C59C2×#4111yescharacteristic
108Dic59—42no—
109Dic59—41no—
110Dic59—42no—
111#3042—591nomaximal, Sylow
112C2×C59—22no—
113C2×C59C2×#12111yescharacteristic
114C2×C59#57712yes—
115D59—44no—
116D59—81no—
117C2×AGL(1,ℤ/8)—591no—
118C22×SD16—591no—
119#536—591no—
120#518—591no—
121#463—594no—
122#461—594no—
123#424D5911yescharacteristic, complemented
124#419—591no—
125#416—591no—
126C59#304211yescharacteristic, complemented, Sylow
127C2×#41—591no—
128C22×Q8—591no—
129C22×D4—591no—
130AGL(1,ℤ/8)—1184no—
131C2×SD16—594no—
132C2×SD16—592no—
133C2×SD16—1184no—
134C2×D8—592no—
135C2×M16C2×D5911yescharacteristic
136C22×C8C2×D5911yescharacteristic
137C4⋊Q8—592no—
138#124—592no—
139#123—1182no—
140C22⋊Q8—1182no—
141C4×D4—594no—
142C4×D4—1182no—
143#117—591no—
144#117C2×D5911yescharacteristic
145C2×C22⋊C4—1182no—
146C4⋊C8C2×D5914yescomplemented
147#103—592no—
148#103—594no—
149#102—592no—
150C24—1181no—
151#41—1182no—
152#41—1182no—
153C2×Q8—592no—
154C2×Q8—591no—
155C2×Q8—1181no—
156C2×Q8—1182no—
157C2×Q8—1182no—
158C2×D4—594no—
159C2×D4—592no—
160C2×D4—591no—
161C2×D4—1181no—
162C2×D4—1181no—
163C2×D4—1182no—
164C22×C4—1181no—
165C22×C4—1184no—
166C22×C4—1181no—
167C22×C4C22×D5911yescharacteristic
168SD16—1188no—
169SD16—1184no—
170D8—1184no—
171M16—22no—
172C2×C8—22no—
173C2×C8C22×D5912yes—
174C2×C8C22×D5912yes—
175C4⋊C4—592no—
176C4⋊C4—1184no—
177C4⋊C4D23612yes—
178C22⋊C4—1181no—
179C22⋊C4—1184no—
180C22⋊C4—1184no—
181C22⋊C4D23612yes—
182C42—1181no—
183C42C22×D5912yes—
184C23—1184no—
185C23—1182no—
186C23—1181no—
187C23—4722no—
188C23C2×D23611yescharacteristic
189Q8—1182no—
190Q8—1181no—
191Q8—1182no—
192Q8—1181no—
193Q8—2362no—
194D4—594no—
195D4—1184no—
196D4—1181no—
197D4—1182no—
198D4—1181no—
199D4—2362no—
200C2×C4—1182no—
201C2×C4—1182no—
202C2×C4—1181no—
203C2×C4—4724no—
204C2×C4—2362no—
205C2×C4—2362no—
206C2×C4C2×D23614yes—
207C2×C4C2×D23611yescharacteristic
208C2×C4D4×D5912yes—
209C2×C4#1174568912yes—
210C2×C4C23×D5911yescharacteristic
211C8—22no—
212C8—22no—
213C22—22no—
214C22—1184no—
215C22—4724no—
216C22—2361no—
217C22—2362no—
218C22—2364no—
219C22#1427370212yes—
220C22C22×D23611yescharacteristic
221C4—24no—
222C4—2361no—
223C4—2362no—
224C4—2362no—
225C4#1427370212yes—
226C4C2×D4×D5911yescharacteristic
227C4C2×#1174568911yescharacteristic
228C2—22no—
229C2—4721no—
230C2—2364no—
231C2C2×#1427370211yescharacteristic
232C2#1683821412yes—
subgroupG/H
centerC22C22×D236
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#424C2
socleC22×C59C22×D4
hypercenter#424D59
hypocenterC59#3042
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
2. C22×#11745660C2semidirect
3. #16837978C2semidirect
4. #16837876C2semidirect
5. #16837618C2semidirect
6. #16837608C2semidirect
7. #16837412C2semidirect
8. #16837399C2semidirect
9. C59×#424C2semidirect
18. #14273727C22semidirect
19. #14273715C22semidirect
26. #14273652C22semidirect
27. #14273652C22semidirect
28. #14273649C22semidirect
32. C4⋊C8×C59C22semidirect
123. #424D59semidirect
126. C59#3042semidirect
146. C4⋊C8C2×D59semidirect