Finite Group Databasebeta

← #21434022

Subgroups — #21434022

Order 7552.

458 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 129 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#16837914C211yesmaximal, characteristic
2C2×C4⋊C4×D59C211yesmaximal, characteristic, complemented
3#16837739C212yesmaximal, complemented
4#16837734C212yesmaximal, complemented
5#16837722C212yesmaximal, complemented
6#16837717C212yesmaximal, complemented
7#16837445C211yesmaximal, characteristic, complemented
8#16837404C211yesmaximal, characteristic, complemented
9#16837401C211yesmaximal, characteristic, complemented
10#16837365C211yesmaximal, characteristic, complemented
11C59×#428C211yesmaximal, characteristic, complemented
12C2×#11745685C2211yescharacteristic
13C22×C4×D59—21no—
14C22×C4×D59C2211yescharacteristic
15C2×#11745664C2211yescharacteristic
16#14273711—21no—
17#14273707C2211yescharacteristic
18#14273707C2211yescharacteristic
19#14273702C2211yescharacteristic
20#14273702C2211yescharacteristic
21#14273697—21no—
22C4×D236—21no—
23C4×D236—21no—
24C2×C4⋊Dic59C2211yescharacteristic
25C2×Dic59⋊C4—21no—
26C4⋊C4×D59—21no—
27C4⋊C4×D59—21no—
28C4⋊C4×D59C2212yes—
29C4⋊C4×D59C2212yes—
30#14273660C2212yescomplemented
31#14273658C2212yescomplemented
32#14273652C2212yescomplemented
33#14273650C2212yescomplemented
34#14273649C2212yescomplemented
35#14273647C2212yescomplemented
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
46C2×C22⋊D59—21no—
47C22×Dic59—21no—
48C22×Dic59D411yescharacteristic
49#11745685—22no—
50#11745685—22no—
51C2×D236—21no—
52C2×D236—21no—
53C2×D236C2311yescharacteristic
54C2×C4×D59—22no—
55C2×C4×D59—22no—
56C2×C4×D59—22no—
57C2×C4×D59—22no—
58C2×C4×D59—21no—
59C2×C4×D59—44no—
60C2×C4×D59D411yescharacteristic
61C2×C4×D59D411yescharacteristic
62C2×C4×D59C2311yescharacteristic
63C2×C4×D59C2311yescharacteristic
64C2×Dic118C2311yescharacteristic
65M16×C59—22no—
66C2×C8×C59C2312yes—
67C4⋊C4×C59—21no—
68C4⋊C4×C59C2311yescharacteristic
69C4⋊C4×C59C2312yes—
70C4⋊C4×C59C2311yescharacteristic
71C22⋊C4×C59—21no—
72C42×C59—21no—
73(C2×D59)⋊C4—21no—
74(C2×D59)⋊C4—21no—
75(C2×D59)⋊C4—21no—
76(C2×D59)⋊C4—21no—
77C4⋊Dic59—21no—
78C4⋊Dic59—21no—
79C4⋊Dic59—21no—
80C4⋊Dic59C2312yes—
81Dic59⋊C4—22no—
82Dic59⋊C4—22no—
83#11745664—22no—
84C2×C59⋊C8C2312yes—
85C23×C59C2×D411yescharacteristic
86C22×D59—22no—
87C22×D59—22no—
88C22×D59—22no—
89C22×D59—21no—
90C22×D59C2×D411yescharacteristic
91C22×D59C2×D411yescharacteristic
92C2×C4×C59—22no—
93C2×C4×C59—21no—
94C2×C4×C59—21no—
95C2×C4×C59—42no—
96C2×C4×C59C2×D411yescharacteristic
97C2×C4×C59#4112yes—
98C2×C4×C59#4112yes—
99C2×C4×C59C2411yescharacteristic
100C22⋊D59—42no—
101C2×Dic59—22no—
102C2×Dic59—22no—
103C2×Dic59—21no—
104C2×Dic59—42no—
105C2×Dic59C2×D411yescharacteristic
106C2×Dic59C2×D411yescharacteristic
107D236—21no—
108D236—21no—
109D236—42no—
110D236—42no—
111C4×D59—22no—
112C4×D59—22no—
113C4×D59—21no—
114C4×D59—21no—
115C4×D59—21no—
116C4×D59—21no—
117C4×D59—44no—
118C4×D59—44no—
119C4×D59—42no—
120Dic118—21no—
121Dic118—21no—
122C8×C59—22no—
123C59⋊C8—22no—
124C22×C59—21no—
125C22×C59—21no—
126C22×C59#12312yes—
127C22×C59C22×D411yescharacteristic
128C2×D59—28no—
129C2×D59—21no—
130C2×D59—21no—
131C2×D59—21no—
132C2×D59—21no—
133C2×D59—41no—
134C2×D59—41no—
135C2×D59—41no—
136C2×D59AGL(1,ℤ/8)12yes—
137C2×D59#13712yes—
138C4×C59—42no—
139C4×C59—41no—
140C4×C59—41no—
141C4×C59#12312yes—
142C4×C59C2×#4111yescharacteristic
143C4×C59C2×#4111yescharacteristic
144Dic59—21no—
145Dic59—21no—
146Dic59—41no—
147Dic59—42no—
148#3092—591nomaximal, Sylow
149C2×C59—22no—
150C2×C59C2×#12311yescharacteristic
151C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C59C2×#13711yescharacteristic
153D59—22no—
154D59—22no—
155D59—81no—
156#525—591no—
157C22×C4⋊C4—591no—
158#484—592no—
159#483—592no—
160#481—592no—
161#480—592no—
162#428—591no—
163#428D5911yescharacteristic, complemented
164#417—591no—
165#417—591no—
166#407—591no—
167C59#309211yescharacteristic, complemented, Sylow
168C2×#41—591no—
169C23×C4—591no—
170C23×C4—1181no—
171C2×M16—591no—
172C2×M16C2×D5911yescharacteristic
173#123—1181no—
174C22⋊Q8—591no—
175C22⋊Q8—591no—
176#121—591no—
177#121—591no—
178#120—1181no—
179C4×D4—1181no—
180C4×D4—1181no—
181#117C2×D5911yescharacteristic
182C2×C4⋊C4—591no—
183C2×C4⋊C4—592no—
184C2×C4⋊C4—592no—
185C2×C4⋊C4—1181no—
186C2×C4⋊C4—1181no—
187C2×C4⋊C4—1181no—
188C2×C4⋊C4C2×D5911yescharacteristic
189#107—592no—
190#107C2×D5912yescomplemented
191#106—592no—
192#106C2×D5912yescomplemented
193#103—592no—
194#103—592no—
195#102—592no—
196#102—592no—
197C22⋊C8—592no—
198C22⋊C8—592no—
199C24—591no—
200#41—1182no—
201#41—1182no—
202C2×Q8—591no—
203C2×D4—591no—
204C2×D4—1181no—
205C2×D4—1181no—
206C2×D4—1181no—
207C22×C4—21no—
208C22×C4—591no—
209C22×C4—591no—
210C22×C4—591no—
211C22×C4—591no—
212C22×C4—591no—
213C22×C4—1182no—
214C22×C4—1182no—
215C22×C4—1182no—
216C22×C4—1182no—
217C22×C4—1181no—
218C22×C4—1181no—
219C22×C4—2364no—
220C22×C4C22×D5911yescharacteristic
221M16—22no—
222M16—1182no—
223C2×C8—592no—
224C2×C8C22×D5912yes—
225C4⋊C4—21no—
226C4⋊C4—592no—
227C4⋊C4—1182no—
228C4⋊C4—1182no—
229C4⋊C4—1181no—
230C4⋊C4—1181no—
231C4⋊C4—1181no—
232C4⋊C4C22×D5912yes—
233C4⋊C4C22×D5911yescharacteristic
234C4⋊C4C22×D5911yescharacteristic
235C22⋊C4—21no—
236C22⋊C4—1181no—
237C22⋊C4—1181no—
238C22⋊C4—1181no—
239C22⋊C4—1181no—
240C42—21no—
241C23—591no—
242C23—591no—
243C23—1182no—
244C23—1182no—
245C23—1182no—
246C23—1181no—
247C23D4×D5911yescharacteristic
248Q8—1181no—
249Q8—1181no—
250D4—1181no—
251D4—1181no—
252D4—2362no—
253D4—2362no—
254D4—2362no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—591no—
260C2×C4—591no—
261C2×C4—1182no—
262C2×C4—1182no—
263C2×C4—1182no—
264C2×C4—1181no—
265C2×C4—1181no—
266C2×C4—1181no—
267C2×C4—1181no—
268C2×C4—1181no—
269C2×C4—1182no—
270C2×C4—2364no—
271C2×C4—2364no—
272C2×C4—2362no—
273C2×C4—2362no—
274C2×C4#1174568512yes—
275C2×C4D4×D5911yescharacteristic
276C2×C4#1174568912yes—
277C2×C4C23×D5911yescharacteristic
278C8—22no—
279C8—1182no—
280C22—21no—
281C22—21no—
282C22—592no—
283C22—592no—
284C22—1188no—
285C22—1181no—
286C22—1181no—
287C22—1181no—
288C22—1181no—
289C22—2361no—
290C22—2361no—
291C22—2361no—
292C22#1427371212yes—
293C22C2×D4×D5911yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—1181no—
298C4—1181no—
299C4—2361no—
300C4—2362no—
301C4#1427371212yes—
302C4C2×#1174568511yescharacteristic
303C4C2×#1174568911yescharacteristic
304C2—22no—
305C2—1182no—
306C2—1182no—
307C2—4721no—
308C2C2×#1427371211yescharacteristic
309C2AGL(1,ℤ/8)×D5911yescharacteristic
310C2#137×D5911yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#428C2
socleC22×C59C22×D4
hypercenter#428D59
hypocenterC59#3092
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
2. C2×C4⋊C4×D59C2semidirect
3. #16837739C2semidirect
4. #16837734C2semidirect
5. #16837722C2semidirect
6. #16837717C2semidirect
7. #16837445C2semidirect
8. #16837404C2semidirect
9. #16837401C2semidirect
10. #16837365C2semidirect
11. C59×#428C2semidirect
30. #14273660C22semidirect
31. #14273658C22semidirect
32. #14273652C22semidirect
33. #14273650C22semidirect
34. #14273649C22semidirect
35. #14273647C22semidirect
41. #107×C59C22semidirect
42. #106×C59C22semidirect
163. #428D59semidirect
167. C59#3092semidirect
190. #107C2×D59semidirect
192. #106C2×D59semidirect