Finite Group Databasebeta

← #22621554

Subgroups — #22621554

Order 9344.

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

normal subgroups only

This group has 556 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×#14556615C212yesmaximal, complemented
2#17989257C211yesmaximal, characteristic, complemented
3D4×D292C211yesmaximal, characteristic, complemented
4#17988963C211yesmaximal, characteristic, complemented
5#17988839C212yesmaximal, complemented
6#17988788C211yesmaximal, characteristic, complemented
7#17988733C212yesmaximal, complemented
8#17988719C212yesmaximal, complemented
9C4×#11924016C211yesmaximal, characteristic, complemented
10#17988664C211yesmaximal, characteristic, complemented
11#459×C73C211yesmaximal, characteristic, complemented
12C2×#11924027C2212yescomplemented
13C22×D292—21no—
14C2×D4×D73C2212yes—
15#14556615—28no—
16C2×#11924017C2212yescomplemented
17C2×#11924016C2211yescharacteristic
18C2×#11924013C2212yescomplemented
19C2×#11924013C2212yescomplemented
20C2×#11924011C2212yescomplemented
21#14556588C2211yescharacteristic
22#14556587—21no—
23#14556566C2212yes—
24#14556565—21no—
25#14556563—22no—
26#14556562C2212yescomplemented
27#14556561—22no—
28C4×Dic146C2211yescharacteristic, complemented
29C4×D292C2211yescharacteristic
30D4×Dic73—21no—
31#14556522C2211yescharacteristic, complemented
32#14556514C2211yescharacteristic, complemented
33#14556513C2212yescomplemented
34#14556512C2211yescharacteristic
35#14556503C2212yescomplemented
36C4×#10596831C2211yescharacteristic, complemented
37C2×D8×C73C2212yes—
38#127×C73C2211yescharacteristic, complemented
39C4×D4×C73C2211yescharacteristic
40C4⋊C8×C73C2211yescharacteristic, complemented
41#102×C73C2212yescomplemented
42C23×D73—22no—
43#11924027—24no—
44#11924027—42no—
45C2×Dic146C2311yescharacteristic
46C2×D292—22no—
47C2×D292—21no—
48C2×D292—21no—
49C2×D292—44no—
50C2×D292D411yescharacteristic
51C2×D292C2312yes—
52C2×C22⋊D73—22no—
53C2×C22⋊D73—22no—
54C2×C22⋊D73—22no—
55D4×D73—24no—
56D4×D73—42no—
57C22×Dic73—22no—
58C2×C4×D73C2312yes—
59#11924017—24no—
60#11924016D414yescomplemented
61#11924013—24no—
62#11924013—42no—
63#11924011—24no—
64C2×#10596831C2311yescharacteristic
65C2×#10596831C2311yescharacteristic
66C4⋊Dic73D411yescharacteristic
67C4⋊Dic73C2311yescharacteristic
68#11924007—21no—
69C22⋊Dic73—22no—
70#11924005—22no—
71#11924005D412yes—
72C4×Dic73—21no—
73C2×D4×C73—21no—
74C2×D4×C73—21no—
75C2×D4×C73C2312yes—
76C2×D4×C73C2311yescharacteristic
77C22×C4×C73—21no—
78D8×C73—24no—
79C2×C8×C73C2312yes—
80C4⋊C4×C73C2311yescharacteristic
81C22⋊C4×C73—21no—
82C42×C73C2311yescharacteristic
83C23×C73—21no—
84C23×C73—22no—
85C22×D73—22no—
86C22×D73—44no—
87C22×D73—44no—
88C22×D73—82no—
89C22×D73C2×D412yes—
90D4×C73—22no—
91D4×C73—21no—
92D4×C73—22no—
93D4×C73—42no—
94D4×C73—81no—
95D4×C73C2×D412yes—
96C2×C4×C73—21no—
97C2×C4×C73—42no—
98C2×C4×C73C2×D411yescharacteristic
99C2×C4×C73C2×D411yescharacteristic
100C2×C4×C73C2411yescharacteristic
101C22⋊D73—44no—
102C22⋊D73—44no—
103C22⋊D73—82no—
104C2×Dic73—21no—
105C2×Dic73—21no—
106C2×Dic73—44no—
107C2×Dic73C2×D412yes—
108D292—22no—
109D292—22no—
110D292—22no—
111D292—44no—
112D292—42no—
113D292—81no—
114C4×D73—22no—
115C4×D73—22no—
116Dic146—21no—
117Dic146C2×D412yescomplemented
118C8×C73—22no—
119#10596831—21no—
120#10596831C2×D412yescomplemented
121C22×C73—22no—
122C22×C73—41no—
123C22×C73—41no—
124C22×C73—42no—
125C22×C73—42no—
126C22×C73—42no—
127C22×C73C22×D411yescharacteristic
128C2×D73—22no—
129C2×D73—22no—
130C2×D73—22no—
131C2×D73—42no—
132C2×D73—42no—
133C2×D73—42no—
134C2×D73—88no—
135C4×C73—21no—
136C4×C73—41no—
137C4×C73AGL(1,ℤ/8)12yescomplemented
138C4×C73C22×D411yescharacteristic
139C4×C73H(5,𝔽2)11yescharacteristic
140Dic73—22no—
141Dic73—22no—
142Dic73—41no—
143C2×C73—42no—
144C2×C73—82no—
145C2×C73D4211yescharacteristic
146C2×C73C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C73#57611yescharacteristic
148D73—42no—
149D73—82no—
150#3274—731nomaximal, Sylow
151C73#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—732no—
153#550—731no—
154D42—731no—
155#493—731no—
156#469—732no—
157#460—731no—
158#459D7311yescharacteristic, complemented
159#449—732no—
160#447—732no—
161C4×SD16—731no—
162#436—731no—
163C2×#41—732no—
164C22×D4—732no—
165C22×D4—1461no—
166AGL(1,ℤ/8)—1468no—
167C2×SD16—731no—
168C2×SD16—732no—
169C2×D8—732no—
170C2×D8—732no—
171C2×D8C2×D7312yes—
172C2×M16—732no—
173#127—731no—
174#127—1461no—
175#127C2×D7311yescharacteristic, complemented
176#121—732no—
177#121—732no—
178#121—1462no—
179#121—1461no—
180#120—1462no—
181C4×Q8—731no—
182C4×D4—731no—
183C4×D4—1461no—
184C4×D4C2×D7311yescharacteristic
185#106—731no—
186C4⋊C8C2×D7311yescharacteristic, complemented
187#103—731no—
188#102—731no—
189#102—732no—
190#102C2×D7312yescomplemented
191C22⋊C8—732no—
192C4×C8—731no—
193C24—1462no—
194#41—2922no—
195#41—1464no—
196C2×Q8—731no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—2924no—
200C2×D4—2922no—
201C2×D4—731no—
202C2×D4—732no—
203C2×D4—1462no—
204C2×D4—1462no—
205C2×D4—1462no—
206C2×D4—1464no—
207C2×D4—1462no—
208C2×D4—1461no—
209C2×D4—1461no—
210C2×D4C22×D7312yes—
211C2×D4C22×D7311yescharacteristic
212C22×C4—21no—
213C22×C4—732no—
214C22×C4—1462no—
215SD16—734no—
216SD16—1464no—
217D8—24no—
218D8—2922no—
219D8—1464no—
220M16—1464no—
221C2×C8—731no—
222C2×C8—731no—
223C2×C8C22×D7312yes—
224C4⋊C4—731no—
225C4⋊C4—731no—
226C4⋊C4—1461no—
227C4⋊C4C22×D7311yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—732no—
230C22⋊C4—1462no—
231C22⋊C4—1462no—
232C42—1461no—
233C42C22×D7311yescharacteristic
234C23—21no—
235C23—22no—
236C23—2924no—
237C23—2924no—
238C23—5842no—
239C23—732no—
240C23—1462no—
241Q8—732no—
242Q8—1461no—
243D4—22no—
244D4—21no—
245D4—22no—
246D4—42no—
247D4—81no—
248D4—2924no—
249D4—2924no—
250D4—2924no—
251D4—2922no—
252D4—5841no—
253D4—5842no—
254D4—1462no—
255D4—1462no—
256D4—1462no—
257D4D4×D7312yes—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—2924no—
261C2×C4—732no—
262C2×C4—1461no—
263C2×C4—1461no—
264C2×C4—1462no—
265C2×C4—1462no—
266C2×C4D4×D7311yescharacteristic
267C2×C4D4×D7311yescharacteristic
268C2×C4C23×D7311yescharacteristic
269C8—22no—
270C8—732no—
271C8—1461no—
272C22—22no—
273C22—41no—
274C22—41no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—2922no—
279C22—2922no—
280C22—2922no—
281C22—5848no—
282C22—1462no—
283C22—1462no—
284C22—1462no—
285C22C2×D4×D7311yescharacteristic
286C4—21no—
287C4—41no—
288C4—2921no—
289C4—1462no—
290C4—1462no—
291C4#1455661512yescomplemented
292C4C2×D4×D7311yescharacteristic
293C4#1455663711yescharacteristic
294C2—42no—
295C2—82no—
296C2—2922no—
297C2—5842no—
298C2#1798922111yescharacteristic
299C2C2×#1455661511yescharacteristic
300C2#1798939111yescharacteristic
subgroupG/H
centerC22C2×D4×D73
commutatorC2×C4×C73C24
FrattiniC2×C4C23×D73
Fitting#459×C73C2
socleC22×C73C22×D4
hypercenter#459D73
hypocenterC73#3274
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#14556615C2semidirect
2. #17989257C2semidirect
3. D4×D292C2semidirect
4. #17988963C2semidirect
5. #17988839C2semidirect
6. #17988788C2semidirect
7. #17988733C2semidirect
8. #17988719C2semidirect
9. C4×#11924016C2semidirect
10. #17988664C2semidirect
11. #459×C73C2semidirect
12. C2×#11924027C22semidirect
16. C2×#11924017C22semidirect
18. C2×#11924013C22semidirect
19. C2×#11924013C22semidirect
20. C2×#11924011C22semidirect
26. #14556562C22semidirect
28. C4×Dic146C22semidirect
31. #14556522C22semidirect
32. #14556514C22semidirect
33. #14556513C22semidirect
35. #14556503C22semidirect
36. C4×#10596831C22semidirect
38. #127×C73C22semidirect
40. C4⋊C8×C73C22semidirect
41. #102×C73C22semidirect
60. #11924016D4semidirect
117. Dic146C2×D4semidirect
120. #10596831C2×D4semidirect
137. C4×C73AGL(1,ℤ/8)semidirect
151. C73#3274semidirect
158. #459D73semidirect
175. #127C2×D73semidirect
186. C4⋊C8C2×D73semidirect
190. #102C2×D73semidirect
291. C4#14556615semidirect