Finite Group Databasebeta

← #21435741

Subgroups — #21435741

Order 7552.

472 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 125 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×#14273759C212yesmaximal, complemented
2#16838067C211yesmaximal, characteristic, complemented
3#16838055C211yesmaximal, characteristic
4#16837657C212yesmaximal, complemented
5#16837609C211yesmaximal, characteristic, complemented
6#16837604C211yesmaximal, characteristic, complemented
7#16837563C212yesmaximal, complemented
8#16837556C212yesmaximal, complemented
9C4×#11745660C211yesmaximal, characteristic, complemented
10#16837481C211yesmaximal, characteristic, complemented
11C59×#494C211yesmaximal, characteristic, complemented
12C2×#11745689C2212yes—
13C2×Q8×D59C2212yes—
14#14273759—28no—
15C2×#11745672C2212yes—
16C2×#11745660C2211yescharacteristic
17C2×#11745671C2212yes—
18C2×#11745659C2212yes—
19#14273726—21no—
20#14273712—22no—
21#14273705—22no—
22#14273703C2212yes—
23#14273703C2212yes—
24#14273702—21no—
25C4×Dic118C2211yescharacteristic
26C4×D236C2211yescharacteristic
27C4×D236C2211yescharacteristic
28#14273689—21no—
29C4⋊C4×D59—21no—
30#14273659C2211yescharacteristic
31C4⋊(C59⋊C8)C2211yescharacteristic
32#14273652C2211yescharacteristic
33#14273651C2212yes—
34#14273651C2212yes—
35#14273649C2211yescharacteristic
36(C2×D59)⋊C8C2212yes—
37C2×Q16×C59C2212yes—
38C2×Q16×C59C2212yes—
39C4⋊Q8×C59C2211yescharacteristic
40C4⋊Q8×C59C2211yescharacteristic
41C4×C8×C59C2211yescharacteristic
42C2×Q8×C59C2312yes—
43C2×Q8×C59C2312yes—
44#11745689—24no—
45#11745689—42no—
46Q8×D59—24no—
47Q8×D59—42no—
48C2×D236—22no—
49C2×D236D411yescharacteristic
50C2×D236C2311yescharacteristic
51C2×C4×D59—22no—
52C2×C4×D59—21no—
53C2×C4×D59—22no—
54C2×C4×D59C2312yes—
55C2×Dic118—22no—
56C2×Dic118C2311yescharacteristic
57Q16×C59—24no—
58Q16×C59—42no—
59C2×C8×C59C2311yescharacteristic
60C2×C8×C59C2311yescharacteristic
61C4⋊C4×C59—21no—
62C4⋊C4×C59—21no—
63C4⋊C4×C59—21no—
64C4⋊C4×C59—21no—
65C42×C59C2311yescharacteristic
66#11745672—24no—
67#11745671—24no—
68(C2×D59)⋊C4—22no—
69(C2×D59)⋊C4—22no—
70(C2×D59)⋊C4—21no—
71(C2×D59)⋊C4D412yes—
72C4⋊Dic59D411yescharacteristic
73C4⋊Dic59C2311yescharacteristic
74C4⋊Dic59C2311yescharacteristic
75Dic59⋊C4—21no—
76Dic59⋊C4—22no—
77Dic59⋊C4—22no—
78C4×Dic59—21no—
79C2×C59⋊C8C2312yes—
80#11745660D414yescomplemented
81#11745659—24no—
82C22×D59—21no—
83C22×D59C2×D412yes—
84Q8×C59—22no—
85Q8×C59—22no—
86Q8×C59—22no—
87Q8×C59—22no—
88C2×C4×C59—22no—
89C2×C4×C59—22no—
90C2×C4×C59C2×D411yescharacteristic
91C2×C4×C59C2×D411yescharacteristic
92C2×C4×C59C2411yescharacteristic
93C2×Dic59—21no—
94C2×Dic59—21no—
95C2×Dic59—21no—
96C2×Dic59C2×D412yes—
97D236—22no—
98D236—21no—
99D236—44no—
100D236C2×D412yes—
101C4×D59—22no—
102C4×D59—22no—
103C4×D59—44no—
104C4×D59—44no—
105C4×D59—42no—
106Dic118—21no—
107Dic118—44no—
108Dic118C2×D412yes—
109C8×C59—21no—
110C8×C59C2×D412yes—
111C59⋊C8—22no—
112C22×C59C22×D411yescharacteristic
113C2×D59—22no—
114C2×D59—22no—
115C2×D59—22no—
116C2×D59—22no—
117C2×D59—41no—
118C2×D59—41no—
119C4×C59—21no—
120C4×C59—42no—
121C4×C59—42no—
122C4×C59#13712yescomplemented
123C4×C59C22×D411yescharacteristic
124C4×C59H(5,𝔽2)11yescharacteristic
125Dic59—22no—
126Dic59—22no—
127Dic59—41no—
128Dic59—41no—
129#3276—591nomaximal, Sylow
130C2×C59D4211yescharacteristic
131C2×C59C2×#13711yescharacteristic
132C2×C59#57811yescharacteristic
133D59—42no—
134D59—42no—
135C2×#137—592no—
136D4×Q8—591no—
137#547—591no—
138#494D5911yescharacteristic, complemented
139#470—592no—
140#462—591no—
141#461—591no—
142#452—592no—
143#451—592no—
144C4×SD16—591no—
145#436—591no—
146C59#327611yescharacteristic, complemented, Sylow
147C2×#41—592no—
148C22×Q8—592no—
149#137—1188no—
150C2×Q16—592no—
151C2×Q16C2×D5912yes—
152C2×Q16C2×D5912yes—
153C2×SD16—592no—
154C2×SD16—591no—
155C2×M16—592no—
156C4⋊Q8—1181no—
157C4⋊Q8C2×D5911yescharacteristic
158C4⋊Q8C2×D5911yescharacteristic
159#123—1182no—
160C22⋊Q8—592no—
161C22⋊Q8—592no—
162C22⋊Q8—1182no—
163#121—1181no—
164C4×Q8—591no—
165C4×D4—591no—
166C4×D4—591no—
167C4×D4—1181no—
168C2×C4⋊C4—1181no—
169#106—591no—
170C4⋊C8—591no—
171#103—592no—
172#103—592no—
173#103—591no—
174#102—591no—
175C22⋊C8—592no—
176C4×C8C2×D5911yescharacteristic
177#41—1184no—
178#41—2362no—
179C2×Q8—591no—
180C2×Q8—1182no—
181C2×Q8—1184no—
182C2×Q8—2362no—
183C2×Q8C22×D5912yes—
184C2×Q8C22×D5912yes—
185C2×D4—591no—
186C2×D4—591no—
187C2×D4—1182no—
188C22×C4—592no—
189C22×C4—1182no—
190C22×C4—1182no—
191C22×C4—1181no—
192Q16—24no—
193Q16—42no—
194Q16—1184no—
195SD16—594no—
196SD16—1184no—
197M16—1184no—
198C2×C8—592no—
199C2×C8C22×D5911yescharacteristic
200C2×C8C22×D5911yescharacteristic
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—591no—
206C4⋊C4—591no—
207C4⋊C4—591no—
208C4⋊C4—1182no—
209C4⋊C4—1182no—
210C4⋊C4—1181no—
211C22⋊C4—592no—
212C22⋊C4—1182no—
213C22⋊C4—1182no—
214C22⋊C4—1181no—
215C42—1181no—
216C42C22×D5911yescharacteristic
217C23—592no—
218C23—1181no—
219Q8—22no—
220Q8—22no—
221Q8—22no—
222Q8—22no—
223Q8—592no—
224Q8—1181no—
225Q8—2364no—
226D4—592no—
227D4—1182no—
228D4—1181no—
229D4—2364no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—592no—
233C2×C4—1181no—
234C2×C4—1182no—
235C2×C4—1182no—
236C2×C4—1181no—
237C2×C4—1181no—
238C2×C4—2364no—
239C2×C4—2364no—
240C2×C4—2362no—
241C2×C4D4×D5911yescharacteristic
242C2×C4D4×D5911yescharacteristic
243C2×C4C23×D5911yescharacteristic
244C8—21no—
245C8—1182no—
246C8D4×D5912yes—
247C22—1182no—
248C22—1182no—
249C22—1182no—
250C22—1182no—
251C22—2361no—
252C22—2361no—
253C22C2×D4×D5911yescharacteristic
254C4—21no—
255C4—42no—
256C4—42no—
257C4—1182no—
258C4—1182no—
259C4—2361no—
260C4—2361no—
261C4#1427375912yescomplemented
262C4C2×D4×D5911yescharacteristic
263C4#1427377211yescharacteristic
264C2—2362no—
265C2—2362no—
266C2#1683803711yescharacteristic
267C2C2×#1427375911yescharacteristic
268C2#1683821511yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#494C2
socleC22×C59C22×D4
hypercenter#494D59
hypocenterC59#3276
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
1. C2×#14273759C2semidirect
2. #16838067C2semidirect
4. #16837657C2semidirect
5. #16837609C2semidirect
6. #16837604C2semidirect
7. #16837563C2semidirect
8. #16837556C2semidirect
9. C4×#11745660C2semidirect
10. #16837481C2semidirect
11. C59×#494C2semidirect
80. #11745660D4semidirect
122. C4×C59#137semidirect
138. #494D59semidirect
146. C59#3276semidirect
261. C4#14273759semidirect