Finite Group Databasebeta

← #18005363

Subgroups — #18005363

Order 4736.

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

normal subgroups only

This group has 472 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
1#14558044C211yesmaximal, characteristic, complemented
2C2×#11924510C211yesmaximal, characteristic, complemented
3#14557869C211yesmaximal, characteristic, complemented
4#14557858C212yesmaximal, complemented
5#14557852C211yesmaximal, characteristic, complemented
6#14557769C212yesmaximal, complemented
7#14557762C212yesmaximal, complemented
8#14557686C211yesmaximal, characteristic, complemented
9#14557680C212yesmaximal, complemented
10#14557669C211yesmaximal, characteristic, complemented
11C2×C22⋊C8×C37C211yesmaximal, characteristic, complemented
12C2×#10596962C2211yescharacteristic, complemented
13C22×D148C2211yescharacteristic
14C22×C22⋊D37—21no—
15C2×#10596930C2212yescomplemented
16C2×D296C2212yescomplemented
17#11924519—21no—
18#11924515C2211yescharacteristic
19#11924515C2211yescharacteristic, complemented
20#11924510—21no—
21#11924510—21no—
22#11924510C2212yes—
23#11924510C2212yes—
24#11924510C2211yescharacteristic
25#11924510C2211yescharacteristic, complemented
26#11924505—21no—
27C4×C22⋊D37—21no—
28C4×C22⋊D37—21no—
29#11924495C2211yescharacteristic, complemented
30C2×C4⋊Dic37C2211yescharacteristic
31C2×#10596938—21no—
32#11924470C2212yescomplemented
33#11924468C2212yescomplemented
34#11924461C2212yescomplemented
35#11924458C2212yescomplemented
36#11924458C2212yescomplemented
37#11924458C2212yescomplemented
38C23×C4×C37C2211yescharacteristic
39C22×C8×C37C2212yescomplemented
40C22⋊C8×C37C2211yescharacteristic
41C22⋊C8×C37C2211yescharacteristic
42C22⋊C8×C37C2212yescomplemented
43C24×C37D411yescharacteristic
44C23×D37—21no—
45C22×C4×C37—42no—
46C22×C4×C37D411yescharacteristic
47C22×C4×C37D411yescharacteristic
48C22×C4×C37D411yescharacteristic
49C22×C4×C37C2311yescharacteristic
50C22×C4×C37C2311yescharacteristic
51C22×C4×C37C2311yescharacteristic
52C2×C22⋊D37—21no—
53C2×C22⋊D37—21no—
54C2×C22⋊D37—21no—
55C2×C22⋊D37—22no—
56C2×C22⋊D37—22no—
57C2×C22⋊D37—44no—
58C22×Dic37—21no—
59#10596962—42no—
60C2×D148—21no—
61C2×D148—42no—
62C2×D148C2312yes—
63C2×D148C2311yescharacteristic
64C2×C4×D37—21no—
65C2×Dic74C2311yescharacteristic
66C2×C8×C37—42no—
67C2×C8×C37C2312yes—
68C2×C8×C37C2312yes—
69C22⋊Dic37—21no—
70C22⋊Dic37—21no—
71C22⋊Dic37—21no—
72#10596938—22no—
73#10596938—22no—
74#10596938—21no—
75#10596938—21no—
76C4⋊Dic37—21no—
77C4⋊Dic37C2312yes—
78C4⋊Dic37C2311yescharacteristic
79C4⋊Dic37C2311yescharacteristic
80#10596936—21no—
81#10596936—21no—
82C4×Dic37—21no—
83D296—42no—
84#10596930—42no—
85C23×C37—22no—
86C23×C37—22no—
87C23×C37—41no—
88C23×C37C2×D411yescharacteristic
89C23×C37C2×D411yescharacteristic
90C23×C37C2×D411yescharacteristic
91C22×D37—21no—
92C22×D37—22no—
93C22×D37—42no—
94C22×D37—42no—
95C22×D37—42no—
96C2×C4×C37—22no—
97C2×C4×C37—22no—
98C2×C4×C37—22no—
99C2×C4×C37—41no—
100C2×C4×C37—41no—
101C2×C4×C37—41no—
102C2×C4×C37C2×D411yescharacteristic
103C2×C4×C37C2×D411yescharacteristic
104C2×C4×C37C2×D411yescharacteristic
105C2×C4×C37C2411yescharacteristic
106C22⋊D37—42no—
107C22⋊D37—44no—
108C22⋊D37—42no—
109C22⋊D37—42no—
110C22⋊D37—44no—
111C2×Dic37—21no—
112C2×Dic37—21no—
113C2×Dic37—21no—
114C2×Dic37—22no—
115C2×Dic37—42no—
116D148—22no—
117D148—22no—
118D148—22no—
119D148—42no—
120C4×D37—42no—
121Dic74—22no—
122C8×C37—22no—
123C8×C37—22no—
124C22×C37—21no—
125C22×C37—21no—
126C22×C37—21no—
127C22×C37—21no—
128C22×C37—21no—
129C22×C37—21no—
130C22×C37—21no—
131C22×C37—44no—
132C22×C37#13512yes—
133C22×C37AGL(1,ℤ/8)12yes—
134C22×C37C22×D411yescharacteristic
135C2×D37—41no—
136C2×D37—42no—
137C2×D37—41no—
138C2×D37—42no—
139C2×D37—42no—
140C2×D37—41no—
141C2×D37—84no—
142C4×C37—21no—
143C4×C37—21no—
144C4×C37—21no—
145C4×C37H(5,𝔽2)12yes—
146Dic37—41no—
147Dic37—41no—
148Dic37—42no—
149Dic37—41no—
150#3188—371nomaximal, Sylow
151C2×C37—22no—
152C2×C37—22no—
153C2×C37—41no—
154C2×C37#53411yescharacteristic
155C2×C37C2×#13511yescharacteristic
156C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
157D37—81no—
158D37—82no—
159#525—371no—
160C2×#121—371no—
161#484—371no—
162#482—372no—
163#481—371no—
164#466—372no—
165#465—372no—
166#450—371no—
167#449—372no—
168#447—371no—
169C2×C22⋊C8D3711yescharacteristic, complemented
170C37#318811yescharacteristic, complemented, Sylow
171C2×#41—371no—
172C22×D4—371no—
173C22×D4—741no—
174C23×C4C2×D3711yescharacteristic
175C2×SD16—372no—
176C2×D8—372no—
177C22×C8C2×D3712yescomplemented
178#123—741no—
179C22⋊Q8—371no—
180C22⋊Q8—371no—
181#121—372no—
182#121—372no—
183#121—371no—
184#121—371no—
185#121—741no—
186#121—741no—
187#120—741no—
188C4×D4—741no—
189C4×D4—741no—
190#117—371no—
191C2×C4⋊C4—371no—
192C2×C22⋊C4—741no—
193#107—372no—
194#106—372no—
195#103—372no—
196#102—372no—
197#102—372no—
198#102—372no—
199C22⋊C8C2×D3711yescharacteristic
200C22⋊C8C2×D3711yescharacteristic
201C22⋊C8C2×D3712yescomplemented
202C24—741no—
203C24D14811yescharacteristic
204#41—1482no—
205C2×Q8—371no—
206C2×D4—371no—
207C2×D4—372no—
208C2×D4—741no—
209C2×D4—741no—
210C2×D4—741no—
211C2×D4—742no—
212C2×D4—742no—
213C2×D4—741no—
214C2×D4—1482no—
215C2×D4—1484no—
216C22×C4—42no—
217C22×C4—741no—
218C22×C4—741no—
219C22×C4D14811yescharacteristic
220C22×C4D14811yescharacteristic
221C22×C4D14811yescharacteristic
222C22×C4C22×D3711yescharacteristic
223C22×C4C22×D3711yescharacteristic
224C22×C4C22×D3711yescharacteristic
225SD16—1482no—
226D8—1482no—
227C2×C8—42no—
228C2×C8C22×D3712yes—
229C2×C8C22×D3712yes—
230C4⋊C4—371no—
231C4⋊C4—371no—
232C4⋊C4—372no—
233C4⋊C4—741no—
234C4⋊C4—741no—
235C4⋊C4—741no—
236C22⋊C4—741no—
237C22⋊C4—741no—
238C22⋊C4—741no—
239C22⋊C4—742no—
240C22⋊C4—742no—
241C22⋊C4—741no—
242C22⋊C4—741no—
243C42—741no—
244C23—22no—
245C23—22no—
246C23—41no—
247C23—741no—
248C23—742no—
249C23—1482no—
250C23—1482no—
251C23—1482no—
252C23C2×D14811yescharacteristic
253C23C2×D14811yescharacteristic
254C23C2×D14811yescharacteristic
255Q8—742no—
256D4—742no—
257D4—742no—
258D4—742no—
259D4—1482no—
260D4—1484no—
261D4—1482no—
262D4—1482no—
263D4—1484no—
264D4—1482no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—741no—
272C2×C4—741no—
273C2×C4—742no—
274C2×C4—741no—
275C2×C4—1482no—
276C2×C4—1482no—
277C2×C4C2×D14811yescharacteristic
278C2×C4C2×D14811yescharacteristic
279C2×C4C2×D14811yescharacteristic
280C2×C4C23×D3711yescharacteristic
281C8—22no—
282C8—22no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—44no—
291C22—2964no—
292C22—1481no—
293C22—1482no—
294C22—1482no—
295C22—1481no—
296C22—1481no—
297C22—1482no—
298C22#1192455812yes—
299C22#1192456412yes—
300C22C22×D14811yescharacteristic
301C4—21no—
302C4—21no—
303C4—21no—
304C4—1481no—
305C4—1482no—
306C4—1481no—
307C4—1481no—
308C4#1192458112yes—
309C2—22no—
310C2—22no—
311C2—41no—
312C2—2962no—
313C2—2961no—
314C2#1455809511yescharacteristic
315C2C2×#1192455811yescharacteristic
316C2C2×#1192456411yescharacteristic
subgroupG/H
centerC22C22×D148
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC2×C22⋊C8×C37C2
socleC22×C37C22×D4
hypercenterC2×C22⋊C8D37
hypocenterC37#3188
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. #14558044C2semidirect
2. C2×#11924510C2semidirect
3. #14557869C2semidirect
4. #14557858C2semidirect
5. #14557852C2semidirect
6. #14557769C2semidirect
7. #14557762C2semidirect
8. #14557686C2semidirect
9. #14557680C2semidirect
10. #14557669C2semidirect
11. C2×C22⋊C8×C37C2semidirect
12. C2×#10596962C22semidirect
15. C2×#10596930C22semidirect
16. C2×D296C22semidirect
19. #11924515C22semidirect
25. #11924510C22semidirect
29. #11924495C22semidirect
32. #11924470C22semidirect
33. #11924468C22semidirect
34. #11924461C22semidirect
35. #11924458C22semidirect
36. #11924458C22semidirect
37. #11924458C22semidirect
39. C22×C8×C37C22semidirect
42. C22⋊C8×C37C22semidirect
169. C2×C22⋊C8D37semidirect
170. C37#3188semidirect
177. C22×C8C2×D37semidirect
201. C22⋊C8C2×D37semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#145580442
C2×#119245102
#145578692
#145578582
#145578522
#145577692
#145577622
#145576862
#145576802
#145576692
C2×C22⋊C8×C372
#318837