Finite Group Databasebeta

← #21435744

Subgroups — #21435744

Order 7552.

472 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14273760C212yesmaximal, complemented
2D4×Dic118C211yesmaximal, characteristic, complemented
3#16838057C211yesmaximal, characteristic
4#16837782C211yesmaximal, characteristic, complemented
5#16837660C212yesmaximal, complemented
6#16837613C211yesmaximal, characteristic, complemented
7#16837608C211yesmaximal, characteristic, complemented
8#16837566C212yesmaximal, complemented
9#16837560C212yesmaximal, complemented
10C4×#11745660C211yesmaximal, characteristic, complemented
11C59×#436C211yesmaximal, characteristic, complemented
12C2×#11745685C2212yes—
13C22×Dic118C2212yes—
14#14273760—28no—
15C2×Dic236C2212yes—
16C2×Dic236C2212yes—
17C2×Dic236C2212yes—
18C2×#11745660C2212yes—
19C2×#11745660C2211yescharacteristic
20#14273727C2211yescharacteristic
21#14273727C2211yescharacteristic
22#14273726—21no—
23#14273709—22no—
24#14273706C2212yes—
25#14273706C2212yes—
26#14273704—22no—
27#14273698—21no—
28C4×Dic118C2211yescharacteristic
29C4×D236C2211yescharacteristic
30D4×Dic59—21no—
31C4⋊C4×D59—21no—
32#14273659C2211yescharacteristic
33#14273652C2212yes—
34#14273652C2212yes—
35#14273652C2211yescharacteristic
36#14273649C2211yescharacteristic
37C2×M16×C59C2212yes—
38C4×D4×C59C2211yescharacteristic
39C4⋊C8×C59C2211yescharacteristic
40C22⋊C8×C59C2212yes—
41C4×C8×C59C2211yescharacteristic
42C2×D4×C59D411yescharacteristic
43C22×C4×C59C2312yes—
44C2×C22⋊D59—22no—
45C22×Dic59—22no—
46#11745685—24no—
47#11745685—42no—
48C2×D236C2311yescharacteristic
49C2×C4×D59—21no—
50C2×C4×D59—22no—
51C2×Dic118—24no—
52C2×Dic118—22no—
53C2×Dic118—42no—
54C2×Dic118C2312yes—
55C2×Dic118C2312yes—
56C2×Dic118C2311yescharacteristic
57M16×C59—24no—
58C2×C8×C59C2311yescharacteristic
59C2×C8×C59C2312yes—
60C2×C8×C59C2311yescharacteristic
61C4⋊C4×C59D411yescharacteristic
62C22⋊C4×C59D412yes—
63C42×C59C2311yescharacteristic
64C22⋊Dic59—22no—
65C22⋊Dic59—22no—
66(C2×D59)⋊C4—21no—
67C4⋊Dic59—21no—
68C4⋊Dic59—21no—
69C4⋊Dic59—21no—
70C4⋊Dic59—21no—
71C4⋊Dic59C2311yescharacteristic
72C4⋊Dic59C2311yescharacteristic
73Dic59⋊C4—22no—
74Dic59⋊C4—21no—
75Dic59⋊C4—22no—
76C4×Dic59—21no—
77Dic236—24no—
78Dic236—24no—
79Dic236—42no—
80#11745660—24no—
81#11745660D414yescomplemented
82C23×C59C2×D412yes—
83C22×D59—21no—
84D4×C59—22no—
85C2×C4×C59—22no—
86C2×C4×C59—22no—
87C2×C4×C59C2×D412yes—
88C2×C4×C59C2×D411yescharacteristic
89C2×C4×C59C2×D411yescharacteristic
90C2×C4×C59C2411yescharacteristic
91C22⋊D59—44no—
92C2×Dic59—21no—
93C2×Dic59—21no—
94C2×Dic59—21no—
95C2×Dic59—22no—
96C2×Dic59—22no—
97C2×Dic59—44no—
98D236—21no—
99D236C2×D412yes—
100C4×D59—44no—
101C4×D59—42no—
102Dic118—21no—
103Dic118—22no—
104Dic118—22no—
105Dic118—22no—
106Dic118—22no—
107Dic118—44no—
108Dic118C2×D412yes—
109C8×C59—22no—
110C8×C59—21no—
111C8×C59C2×D412yes—
112C22×C59—22no—
113C22×C59—22no—
114C22×C59—22no—
115C22×C59C22×D411yescharacteristic
116C2×D59—22no—
117C2×D59—41no—
118C2×D59—41no—
119C4×C59—22no—
120C4×C59—21no—
121C4×C59#13712yescomplemented
122C4×C59C22×D411yescharacteristic
123C4×C59H(5,𝔽2)11yescharacteristic
124Dic59—22no—
125Dic59—42no—
126Dic59—42no—
127Dic59—41no—
128Dic59—41no—
129#3276—591nomaximal, Sylow
130C2×C59—42no—
131C2×C59D4211yescharacteristic
132C2×C59C2×#13711yescharacteristic
133C2×C59#57811yescharacteristic
134D59—42no—
135C2×#137—592no—
136D4×Q8—591no—
137#547—591no—
138#494—591no—
139#470—592no—
140#462—591no—
141#461—591no—
142#452—592no—
143#451—592no—
144C4×SD16—591no—
145#436D5911yescharacteristic, complemented
146C59#327611yescharacteristic, complemented, Sylow
147C2×#41—592no—
148C22×Q8—592no—
149#137—1188no—
150C2×Q16—592no—
151C2×Q16—592no—
152C2×Q16—592no—
153C2×SD16—591no—
154C2×SD16—592no—
155C2×M16C2×D5912yes—
156C4⋊Q8—591no—
157C4⋊Q8—591no—
158C4⋊Q8—1181no—
159#123—1182no—
160C22⋊Q8—592no—
161C22⋊Q8—592no—
162C22⋊Q8—1182no—
163#121—1181no—
164C4×Q8—591no—
165C4×D4—591no—
166C4×D4—1181no—
167C4×D4C2×D5911yescharacteristic
168C2×C4⋊C4—1181no—
169#106—591no—
170C4⋊C8C2×D5911yescharacteristic
171#103—591no—
172#103—592no—
173#103—592no—
174#102—591no—
175C22⋊C8C2×D5912yes—
176C4×C8C2×D5911yescharacteristic
177#41—1184no—
178#41—2362no—
179C2×Q8—591no—
180C2×Q8—592no—
181C2×Q8—592no—
182C2×Q8—1184no—
183C2×Q8—1182no—
184C2×Q8—2362no—
185C2×D4—591no—
186C2×D4—1182no—
187C2×D4D23611yescharacteristic
188C22×C4—1182no—
189C22×C4—1181no—
190C22×C4—1182no—
191C22×C4C22×D5912yes—
192Q16—1184no—
193Q16—1184no—
194Q16—2362no—
195SD16—594no—
196SD16—1184no—
197M16—24no—
198C2×C8C22×D5911yescharacteristic
199C2×C8C22×D5912yes—
200C2×C8C22×D5911yescharacteristic
201C4⋊C4—591no—
202C4⋊C4—591no—
203C4⋊C4—1181no—
204C4⋊C4—1182no—
205C4⋊C4—1181no—
206C4⋊C4—1181no—
207C4⋊C4—1182no—
208C4⋊C4—1181no—
209C4⋊C4—1181no—
210C4⋊C4D23611yescharacteristic
211C22⋊C4—1182no—
212C22⋊C4—1182no—
213C22⋊C4—1181no—
214C22⋊C4D23612yes—
215C42—1181no—
216C42C22×D5911yescharacteristic
217C23—1181no—
218C23C2×D23612yes—
219Q8—592no—
220Q8—1181no—
221Q8—1182no—
222Q8—1182no—
223Q8—1182no—
224Q8—1182no—
225Q8—2364no—
226D4—22no—
227D4—592no—
228D4—1181no—
229D4—2364no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—1181no—
233C2×C4—1181no—
234C2×C4—1181no—
235C2×C4—1182no—
236C2×C4—1182no—
237C2×C4—2364no—
238C2×C4—2364no—
239C2×C4—2362no—
240C2×C4C2×D23611yescharacteristic
241C2×C4C2×D23612yes—
242C2×C4C2×D23611yescharacteristic
243C2×C4C23×D5911yescharacteristic
244C8—22no—
245C8—21no—
246C8D4×D5912yes—
247C22—22no—
248C22—22no—
249C22—22no—
250C22—1182no—
251C22—2361no—
252C22—2361no—
253C22C22×D23611yescharacteristic
254C4—21no—
255C4—22no—
256C4—1182no—
257C4—2361no—
258C4—2361no—
259C4—2362no—
260C4—2362no—
261C4#1427376012yescomplemented
262C4C2×D4×D5911yescharacteristic
263C4#1427377311yescharacteristic
264C2—42no—
265C2—2362no—
266C2D4×D23611yescharacteristic
267C2C2×#1427376011yescharacteristic
268C2#1683821911yescharacteristic
subgroupG/H
centerC22C22×D236
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#436C2
socleC22×C59C22×D4
hypercenter#436D59
hypocenterC59#3276
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#14273760C2semidirect
2. D4×Dic118C2semidirect
4. #16837782C2semidirect
5. #16837660C2semidirect
6. #16837613C2semidirect
7. #16837608C2semidirect
8. #16837566C2semidirect
9. #16837560C2semidirect
10. C4×#11745660C2semidirect
11. C59×#436C2semidirect
81. #11745660D4semidirect
121. C4×C59#137semidirect
145. #436D59semidirect
146. C59#3276semidirect
261. C4#14273760semidirect