Finite Group Databasebeta

← #21434046

Subgroups — #21434046

Order 7552.

410 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 127 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×#14273707C211yesmaximal, characteristic
2#117×D59C211yesmaximal, characteristic
3#16837746C212yesmaximal, complemented
4#16837745C212yesmaximal, complemented
5#16837739C212yesmaximal, complemented
6#16837734C212yesmaximal, complemented
7#16837442C211yesmaximal, characteristic, complemented
8#16837412C211yesmaximal, characteristic, complemented
9#16837410C211yesmaximal, characteristic, complemented
10#16837365C211yesmaximal, characteristic
11C59×#428C211yesmaximal, characteristic, complemented
12C22×Dic118C2211yescharacteristic
13C22×C4×D59C2211yescharacteristic
14C2×#11745664C2211yescharacteristic
15#14273707—21no—
16#14273707—21no—
17#14273707C2212yes—
18#14273707C2212yes—
19#14273686C2211yescharacteristic
20#14273685C2211yescharacteristic
21#14273685C2211yescharacteristic
22#14273684—21no—
23#14273683—21no—
24C2×C4⋊Dic59—21no—
25C4⋊C4×D59C2211yescharacteristic
26C4⋊C4×D59C2211yescharacteristic
27C2×(C2×D59)⋊C4—21no—
28C22⋊C4×D59—21no—
29C42×D59—21no—
30#14273660C2212yescomplemented
31#14273658C2212yescomplemented
32#14273652C2212yescomplemented
33#14273652C2212yescomplemented
34#14273650C2212yescomplemented
35#14273650C2212yescomplemented
36(C2×D59)⋊C8C2212yes—
37(C2×D59)⋊C8C2212yes—
38C2×M16×C59C2211yescharacteristic
39#117×C59C2211yescharacteristic
40C2×C4⋊C4×C59C2211yescharacteristic
41#107×C59C2212yescomplemented
42#106×C59C2212yescomplemented
43C23×D59D411yescharacteristic
44C22×C4×C59—21no—
45C22×C4×C59C2311yescharacteristic
46C22×Dic59—21no—
47C22×Dic59D411yescharacteristic
48C2×C4×D59—22no—
49C2×C4×D59—22no—
50C2×C4×D59—21no—
51C2×C4×D59—21no—
52C2×C4×D59D411yescharacteristic
53C2×C4×D59D411yescharacteristic
54C2×C4×D59C2311yescharacteristic
55C2×C4×D59C2311yescharacteristic
56C2×Dic118—22no—
57C2×Dic118—22no—
58C2×Dic118—21no—
59C2×Dic118C2312yes—
60M16×C59—22no—
61C2×C8×C59C2312yes—
62C4⋊C4×C59—21no—
63C4⋊C4×C59C2311yescharacteristic
64C4⋊C4×C59C2312yes—
65C4⋊C4×C59C2311yescharacteristic
66C22⋊C4×C59—21no—
67C42×C59—21no—
68C22⋊Dic59—21no—
69(C2×D59)⋊C4—21no—
70(C2×D59)⋊C4—21no—
71(C2×D59)⋊C4—22no—
72(C2×D59)⋊C4—22no—
73C4⋊Dic59—22no—
74C4⋊Dic59—22no—
75C4⋊Dic59C2311yescharacteristic
76C4⋊Dic59C2311yescharacteristic
77Dic59⋊C4—21no—
78Dic59⋊C4—21no—
79C4×Dic59—21no—
80C4×Dic59—21no—
81C4×Dic59—21no—
82#11745664—22no—
83C2×C59⋊C8C2312yes—
84C23×C59C2×D411yescharacteristic
85C22×D59—22no—
86C22×D59—22no—
87C22×D59—41no—
88C22×D59C2×D411yescharacteristic
89C22×D59C2×D411yescharacteristic
90C2×C4×C59—21no—
91C2×C4×C59—21no—
92C2×C4×C59—22no—
93C2×C4×C59—42no—
94C2×C4×C59C2×D411yescharacteristic
95C2×C4×C59#4112yes—
96C2×C4×C59#4112yes—
97C2×C4×C59C2411yescharacteristic
98C2×Dic59—22no—
99C2×Dic59—21no—
100C2×Dic59—21no—
101C2×Dic59—22no—
102C2×Dic59—42no—
103C2×Dic59C2×D411yescharacteristic
104C2×Dic59C2×D411yescharacteristic
105C4×D59—22no—
106C4×D59—22no—
107C4×D59—21no—
108C4×D59—21no—
109C4×D59—21no—
110C4×D59—21no—
111C4×D59—42no—
112C4×D59—42no—
113Dic118—22no—
114Dic118—22no—
115Dic118—42no—
116C8×C59—22no—
117C59⋊C8—22no—
118C22×C59—21no—
119C22×C59—21no—
120C22×C59#12312yes—
121C22×C59C22×D411yescharacteristic
122C2×D59—21no—
123C2×D59—21no—
124C2×D59—21no—
125C2×D59—21no—
126C2×D59—21no—
127C2×D59—44no—
128C2×D59#13712yes—
129C4×C59—41no—
130C4×C59—41no—
131C4×C59—42no—
132C4×C59#12312yes—
133C4×C59C2×#4111yescharacteristic
134C4×C59C2×#4111yescharacteristic
135Dic59—21no—
136Dic59—21no—
137Dic59—42no—
138Dic59—41no—
139Dic59—41no—
140#3095—591nomaximal, Sylow
141C2×C59—22no—
142C2×C59C2×#12311yescharacteristic
143C2×C59C2×#13711yescharacteristic
144C2×C59#57511yescharacteristic
145D59—22no—
146D59—41no—
147C2×C22⋊Q8—591no—
148C2×#117—591no—
149#485—592no—
150#485—592no—
151#484—592no—
152#483—592no—
153#428—591no—
154#428D5911yescharacteristic, complemented
155#419—591no—
156#419—591no—
157#407—591no—
158C59#309511yescharacteristic, complemented, Sylow
159C22×Q8—591no—
160C23×C4—591no—
161C2×M16—591no—
162C2×M16C2×D5911yescharacteristic
163C22⋊Q8—592no—
164C22⋊Q8—592no—
165C22⋊Q8—1181no—
166C22⋊Q8—1181no—
167#117—591no—
168#117—591no—
169#117—591no—
170#117—1181no—
171#117—1181no—
172#117C2×D5911yescharacteristic
173C2×C4⋊C4—591no—
174C2×C4⋊C4—591no—
175C2×C4⋊C4—1181no—
176C2×C4⋊C4C2×D5911yescharacteristic
177C2×C22⋊C4—1181no—
178C2×C22⋊C4—1181no—
179C2×C42—1181no—
180#107—592no—
181#107C2×D5912yescomplemented
182#106—592no—
183#106C2×D5912yescomplemented
184#103—592no—
185#103—592no—
186#103—592no—
187#103—592no—
188C22⋊C8—592no—
189C22⋊C8—592no—
190C24—591no—
191C2×Q8—592no—
192C2×Q8—1181no—
193C2×Q8—1182no—
194C2×Q8—1182no—
195C22×C4—21no—
196C22×C4—591no—
197C22×C4—591no—
198C22×C4—591no—
199C22×C4—591no—
200C22×C4—591no—
201C22×C4—1181no—
202C22×C4—1181no—
203C22×C4—1182no—
204C22×C4—1182no—
205C22×C4—1181no—
206C22×C4C22×D5911yescharacteristic
207M16—22no—
208M16—1182no—
209C2×C8—592no—
210C2×C8C22×D5912yes—
211C4⋊C4—21no—
212C4⋊C4—591no—
213C4⋊C4—591no—
214C4⋊C4—1181no—
215C4⋊C4—1181no—
216C4⋊C4—1182no—
217C4⋊C4—1182no—
218C4⋊C4C22×D5911yescharacteristic
219C4⋊C4C22×D5911yescharacteristic
220C4⋊C4C22×D5912yes—
221C22⋊C4—21no—
222C22⋊C4—1181no—
223C22⋊C4—1181no—
224C22⋊C4—1182no—
225C22⋊C4—1182no—
226C22⋊C4—1181no—
227C42—21no—
228C42—1181no—
229C42—1181no—
230C42—1181no—
231C23—591no—
232C23—591no—
233C23—1182no—
234C23—1182no—
235C23—2361no—
236C23D4×D5911yescharacteristic
237Q8—1182no—
238Q8—1182no—
239Q8—2362no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—591no—
245C2×C4—591no—
246C2×C4—1182no—
247C2×C4—1182no—
248C2×C4—1182no—
249C2×C4—1181no—
250C2×C4—1181no—
251C2×C4—1181no—
252C2×C4—1181no—
253C2×C4—1181no—
254C2×C4—1181no—
255C2×C4—1182no—
256C2×C4—2362no—
257C2×C4—2362no—
258C2×C4—2362no—
259C2×C4#1174568512yes—
260C2×C4D4×D5911yescharacteristic
261C2×C4#1174568912yes—
262C2×C4C23×D5911yescharacteristic
263C8—22no—
264C8—1182no—
265C22—21no—
266C22—21no—
267C22—592no—
268C22—1181no—
269C22—1181no—
270C22—1181no—
271C22—1181no—
272C22—1181no—
273C22—2364no—
274C22#1427371212yes—
275C22C2×D4×D5911yescharacteristic
276C4—41no—
277C4—42no—
278C4—41no—
279C4—1181no—
280C4—1181no—
281C4—2362no—
282C4—2361no—
283C4—2361no—
284C4#1427371212yes—
285C4C2×#1174568511yescharacteristic
286C4C2×#1174568911yescharacteristic
287C2—22no—
288C2—1182no—
289C2—2361no—
290C2C2×#1427371211yescharacteristic
291C2#137×D5911yescharacteristic
292C2#1683819811yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#428C2
socleC22×C59C22×D4
hypercenter#428D59
hypocenterC59#3095
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #16837746C2semidirect
4. #16837745C2semidirect
5. #16837739C2semidirect
6. #16837734C2semidirect
7. #16837442C2semidirect
8. #16837412C2semidirect
9. #16837410C2semidirect
11. C59×#428C2semidirect
30. #14273660C22semidirect
31. #14273658C22semidirect
32. #14273652C22semidirect
33. #14273652C22semidirect
34. #14273650C22semidirect
35. #14273650C22semidirect
41. #107×C59C22semidirect
42. #106×C59C22semidirect
154. #428D59semidirect
158. C59#3095semidirect
181. #107C2×D59semidirect
183. #106C2×D59semidirect