Finite Group Databasebeta

← #21434018

Subgroups — #21434018

Order 7552.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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#16837912C211yesmaximal, characteristic
2C2×C4⋊C4×D59C211yesmaximal, characteristic, complemented
3#16837738C211yesmaximal, characteristic, complemented
4#16837737C211yesmaximal, characteristic, complemented
5#16837734C212yesmaximal, complemented
6#16837718C211yesmaximal, characteristic, complemented
7#16837714C212yesmaximal, complemented
8#16837441C212yesmaximal, complemented
9#16837400C212yesmaximal, complemented
10#16837364C211yesmaximal, characteristic, complemented
11C59×#481C211yesmaximal, characteristic, complemented
12C2×#11745687C2211yescharacteristic
13C22×C4×D59—21no—
14C22×C4×D59C2211yescharacteristic
15C2×#11745659C2212yes—
16#14273709—21no—
17#14273707C2211yescharacteristic
18#14273706C2211yescharacteristic
19#14273698C2211yescharacteristic
20#14273695—21no—
21D4×Dic59—21no—
22(C22⋊D59)⋊C4—21no—
23#14273685C2211yescharacteristic
24C2×C4⋊Dic59C2211yescharacteristic
25C2×Dic59⋊C4—21no—
26C4⋊C4×D59—21no—
27C4⋊C4×D59—21no—
28C4⋊C4×D59C2212yes—
29C4⋊C4×D59C2212yes—
30#14273661C2212yescomplemented
31#14273660C2212yescomplemented
32#14273658C2212yescomplemented
33#14273652C2212yescomplemented
34#14273650C2212yescomplemented
35#14273648C2212yescomplemented
36C22⋊(C59⋊C8)C2211yescharacteristic
37(C2×D59)⋊C8C2212yes—
38#121×C59C2211yescharacteristic
39C2×C4⋊C4×C59C2211yescharacteristic
40#106×C59C2212yescomplemented
41#102×C59C2212yescomplemented
42C22⋊C8×C59C2211yescharacteristic
43C23×D59D411yescharacteristic
44C2×D4×C59—21no—
45C2×D4×C59C2311yescharacteristic
46C22×C4×C59—21no—
47C22×C4×C59C2311yescharacteristic
48C2×C22⋊D59—21no—
49C2×C22⋊D59—21no—
50C22×Dic59—21no—
51C22×Dic59—21no—
52C22×Dic59D411yescharacteristic
53#11745687—24no—
54C2×C4×D59—24no—
55C2×C4×D59—22no—
56C2×C4×D59—22no—
57C2×C4×D59—44no—
58C2×C4×D59D411yescharacteristic
59C2×C4×D59D411yescharacteristic
60C2×C4×D59C2311yescharacteristic
61C2×C4×D59C2311yescharacteristic
62C2×Dic118C2311yescharacteristic
63C2×C8×C59C2312yes—
64C4⋊C4×C59—21no—
65C4⋊C4×C59C2312yes—
66C4⋊C4×C59C2311yescharacteristic
67C22⋊C4×C59—21no—
68C22⋊Dic59—21no—
69C22⋊Dic59—21no—
70(C2×D59)⋊C4—21no—
71(C2×D59)⋊C4—21no—
72C4⋊Dic59—21no—
73C4⋊Dic59—21no—
74C4⋊Dic59C2312yes—
75C4⋊Dic59C2311yescharacteristic
76Dic59⋊C4—21no—
77Dic59⋊C4—22no—
78Dic59⋊C4—22no—
79C4×Dic59—21no—
80C2×C59⋊C8C2312yes—
81#11745659—24no—
82C23×C59—21no—
83C23×C59C2×D411yescharacteristic
84C22×D59—22no—
85C22×D59—22no—
86C22×D59—22no—
87C22×D59C2×D411yescharacteristic
88C22×D59C2×D411yescharacteristic
89D4×C59—22no—
90D4×C59—42no—
91C2×C4×C59—22no—
92C2×C4×C59—21no—
93C2×C4×C59—22no—
94C2×C4×C59—42no—
95C2×C4×C59C2×D411yescharacteristic
96C2×C4×C59C2411yescharacteristic
97C22⋊D59—42no—
98C22⋊D59—42no—
99C2×Dic59—22no—
100C2×Dic59—21no—
101C2×Dic59—22no—
102C2×Dic59—21no—
103C2×Dic59—42no—
104C2×Dic59—42no—
105C2×Dic59C2×D411yescharacteristic
106C2×Dic59C2×D411yescharacteristic
107C4×D59—22no—
108C4×D59—22no—
109C4×D59—22no—
110C4×D59—44no—
111C4×D59—44no—
112C4×D59#4114yes—
113Dic118—22no—
114C8×C59—22no—
115C59⋊C8—22no—
116C22×C59—21no—
117C22×C59—21no—
118C22×C59—41no—
119C22×C59—41no—
120C22×C59—41no—
121C22×C59#13712yes—
122C22×C59C22×D411yescharacteristic
123C2×D59—28no—
124C2×D59—21no—
125C2×D59—21no—
126C2×D59—21no—
127C2×D59—21no—
128C2×D59#12312yes—
129C2×D59AGL(1,ℤ/8)12yes—
130C4×C59—21no—
131C4×C59—42no—
132C4×C59—41no—
133C4×C59C2×#4112yes—
134Dic59—21no—
135Dic59—41no—
136Dic59—42no—
137Dic59—41no—
138Dic59#12312yes—
139#3092—591nomaximal, Sylow
140C2×C59—22no—
141C2×C59—81no—
142C2×C59C2×#12311yescharacteristic
143C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C59C2×#13711yescharacteristic
145D59—22no—
146D59—22no—
147#525—591no—
148C22×C4⋊C4—591no—
149#484—591no—
150#484—591no—
151#483—592no—
152#481—591no—
153#481D5911yescharacteristic, complemented
154#480—592no—
155#428—592no—
156#417—592no—
157#407—591no—
158C59#309211yescharacteristic, complemented, Sylow
159C2×#41—591no—
160C23×C4—591no—
161C23×C4—1181no—
162C2×M16—592no—
163#123—1181no—
164C22⋊Q8—591no—
165C22⋊Q8—591no—
166#121—591no—
167#121C2×D5911yescharacteristic
168#120—1181no—
169C4×D4—1181no—
170C4×D4—1181no—
171#117—591no—
172C2×C4⋊C4—591no—
173C2×C4⋊C4—592no—
174C2×C4⋊C4—592no—
175C2×C4⋊C4—1181no—
176C2×C4⋊C4—1181no—
177C2×C4⋊C4—1181no—
178C2×C4⋊C4C2×D5911yescharacteristic
179#107—592no—
180#107—592no—
181#106—592no—
182#106C2×D5912yescomplemented
183#103—592no—
184#103—592no—
185#102—592no—
186#102C2×D5912yescomplemented
187C22⋊C8—591no—
188C22⋊C8—592no—
189C22⋊C8C2×D5911yescharacteristic
190C24—591no—
191#41—1184no—
192C2×Q8—591no—
193C2×D4—21no—
194C2×D4—1181no—
195C2×D4—1181no—
196C2×D4C22×D5911yescharacteristic
197C22×C4—21no—
198C22×C4—591no—
199C22×C4—591no—
200C22×C4—591no—
201C22×C4—591no—
202C22×C4—591no—
203C22×C4—1184no—
204C22×C4—1182no—
205C22×C4—1182no—
206C22×C4—1181no—
207C22×C4—1181no—
208C22×C4—2364no—
209C22×C4C22×D5911yescharacteristic
210M16—1184no—
211C2×C8—592no—
212C2×C8C22×D5912yes—
213C4⋊C4—21no—
214C4⋊C4—592no—
215C4⋊C4—591no—
216C4⋊C4—1182no—
217C4⋊C4—1182no—
218C4⋊C4—1181no—
219C4⋊C4—1181no—
220C4⋊C4—1181no—
221C4⋊C4C22×D5912yes—
222C4⋊C4C22×D5911yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—1181no—
225C22⋊C4—1181no—
226C22⋊C4—1181no—
227C22⋊C4—1181no—
228C42—1181no—
229C23—21no—
230C23—591no—
231C23—591no—
232C23—1182no—
233C23—1182no—
234C23—1182no—
235C23D4×D5911yescharacteristic
236Q8—1182no—
237D4—22no—
238D4—42no—
239D4—2362no—
240D4—2362no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—591no—
246C2×C4—594no—
247C2×C4—591no—
248C2×C4—1182no—
249C2×C4—1182no—
250C2×C4—1182no—
251C2×C4—1182no—
252C2×C4—1181no—
253C2×C4—1181no—
254C2×C4—1182no—
255C2×C4—2364no—
256C2×C4—2364no—
257C2×C4—2362no—
258C2×C4—2362no—
259C2×C4D4×D5911yescharacteristic
260C2×C4C23×D5911yescharacteristic
261C8—22no—
262C8—1182no—
263C22—21no—
264C22—21no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—592no—
269C22—592no—
270C22—1188no—
271C22—1181no—
272C22—1181no—
273C22—1181no—
274C22—1181no—
275C22#1427375612yes—
276C22C2×D4×D5911yescharacteristic
277C4—21no—
278C4—42no—
279C4—41no—
280C4—592no—
281C4—1181no—
282C4—2361no—
283C4—2362no—
284C4—2361no—
285C4#41×D5912yes—
286C2—22no—
287C2—81no—
288C2—1182no—
289C2—1182no—
290C2#123×D5911yescharacteristic
291C2AGL(1,ℤ/8)×D5911yescharacteristic
292C2C2×#1427375611yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#481C2
socleC22×C59C22×D4
hypercenter#481D59
hypocenterC59#3092
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D59C2semidirect
3. #16837738C2semidirect
4. #16837737C2semidirect
5. #16837734C2semidirect
6. #16837718C2semidirect
7. #16837714C2semidirect
8. #16837441C2semidirect
9. #16837400C2semidirect
10. #16837364C2semidirect
11. C59×#481C2semidirect
30. #14273661C22semidirect
31. #14273660C22semidirect
32. #14273658C22semidirect
33. #14273652C22semidirect
34. #14273650C22semidirect
35. #14273648C22semidirect
40. #106×C59C22semidirect
41. #102×C59C22semidirect
153. #481D59semidirect
158. C59#3092semidirect
182. #106C2×D59semidirect
186. #102C2×D59semidirect