Finite Group Databasebeta

← #18005358

Subgroups — #18005358

Order 4736.

472 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 115 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#14558043C211yesmaximal, characteristic, complemented
2C2×#11924507C211yesmaximal, characteristic, complemented
3#14557869C211yesmaximal, characteristic, complemented
4#14557855C212yesmaximal, complemented
5#14557849C211yesmaximal, characteristic, complemented
6#14557766C212yesmaximal, complemented
7#14557761C212yesmaximal, complemented
8#14557684C211yesmaximal, characteristic, complemented
9#14557675C212yesmaximal, complemented
10C22⋊C8×D37C211yesmaximal, characteristic, complemented
11C37×#447C211yesmaximal, characteristic, complemented
12C2×#10596964C2211yescharacteristic, complemented
13C22×C22⋊D37—21no—
14C22×C4×D37C2211yescharacteristic
15C2×#10596940C2212yescomplemented
16C2×C8×D37C2212yescomplemented
17#11924518—21no—
18#11924516C2211yescharacteristic, complemented
19#11924515C2211yescharacteristic
20#11924507—21no—
21#11924507—21no—
22#11924507C2212yes—
23#11924507C2212yes—
24#11924507C2211yescharacteristic, complemented
25#11924504—21no—
26D4×Dic37—21no—
27#11924496—21no—
28#11924494C2211yescharacteristic, complemented
29C2×C4⋊Dic37C2211yescharacteristic
30C2×C22⋊Dic37—21no—
31#11924470C2212yescomplemented
32#11924467C2212yescomplemented
33#11924461C2212yescomplemented
34#11924457C2212yescomplemented
35#11924457C2212yescomplemented
36C22⋊#64436C2211yescharacteristic
37#11924448C2212yescomplemented
38C22×D4×C37C2211yescharacteristic
39C2×D8×C37C2212yescomplemented
40#121×C37C2211yescharacteristic
41#102×C37C2212yescomplemented
42C22⋊C8×C37C2211yescharacteristic
43C24×C37—21no—
44C23×D37D411yescharacteristic
45C2×D4×C37—21no—
46C2×D4×C37—21no—
47C2×D4×C37—42no—
48C2×D4×C37C2312yes—
49C2×D4×C37C2311yescharacteristic
50C22×C4×C37C2311yescharacteristic
51C2×C22⋊D37—21no—
52C2×C22⋊D37—21no—
53C2×C22⋊D37—22no—
54C2×C22⋊D37—22no—
55C2×C22⋊D37—44no—
56C22×Dic37—21no—
57C22×Dic37—21no—
58C22×Dic37D411yescharacteristic
59#10596964—42no—
60C2×C4×D37—42no—
61C2×C4×D37D411yescharacteristic
62C2×C4×D37D411yescharacteristic
63C2×C4×D37C2311yescharacteristic
64C2×C4×D37C2311yescharacteristic
65C2×Dic74C2311yescharacteristic
66D8×C37—42no—
67C2×C8×C37C2312yes—
68C4⋊C4×C37C2311yescharacteristic
69C22⋊C4×C37—21no—
70C22⋊Dic37—21no—
71C22⋊Dic37—22no—
72C22⋊Dic37—22no—
73C22⋊Dic37—21no—
74#10596940—42no—
75#10596938—21no—
76#10596938—21no—
77C4⋊Dic37—21no—
78C4⋊Dic37—21no—
79C4⋊Dic37C2312yes—
80C4⋊Dic37C2311yescharacteristic
81#10596936—21no—
82C4×Dic37—21no—
83C2×#64436C2312yes—
84C8×D37—42no—
85C23×C37—22no—
86C23×C37—21no—
87C23×C37—42no—
88C23×C37—42no—
89C23×C37—42no—
90C23×C37C2×D411yescharacteristic
91C22×D37—22no—
92C22×D37—22no—
93C22×D37—41no—
94C22×D37C2×D411yescharacteristic
95C22×D37C2×D411yescharacteristic
96D4×C37—22no—
97D4×C37—22no—
98D4×C37—22no—
99D4×C37—42no—
100D4×C37—42no—
101C2×C4×C37—22no—
102C2×C4×C37—21no—
103C2×C4×C37C2×D411yescharacteristic
104C2×C4×C37C2411yescharacteristic
105C22⋊D37—42no—
106C22⋊D37—44no—
107C22⋊D37—42no—
108C22⋊D37—44no—
109C2×Dic37—21no—
110C2×Dic37—21no—
111C2×Dic37—22no—
112C2×Dic37—41no—
113C2×Dic37—42no—
114C2×Dic37—42no—
115C2×Dic37C2×D411yescharacteristic
116C2×Dic37C2×D411yescharacteristic
117C4×D37—22no—
118C4×D37—22no—
119C4×D37—41no—
120C4×D37—41no—
121Dic74—22no—
122C8×C37—22no—
123#64436—22no—
124C22×C37—21no—
125C22×C37—21no—
126C22×C37—41no—
127C22×C37—42no—
128C22×C37—41no—
129C22×C37—42no—
130C22×C37—42no—
131C22×C37—41no—
132C22×C37—84no—
133C22×C37#13512yes—
134C22×C37C22×D411yescharacteristic
135C2×D37—21no—
136C2×D37—21no—
137C2×D37—21no—
138C2×D37—21no—
139C2×D37—21no—
140C2×D37—44no—
141C2×D37AGL(1,ℤ/8)12yes—
142C4×C37—21no—
143C4×C37—41no—
144C4×C37H(5,𝔽2)12yes—
145Dic37—21no—
146Dic37—21no—
147Dic37—41no—
148Dic37—42no—
149Dic37—41no—
150#3188—371nomaximal, Sylow
151C2×C37—22no—
152C2×C37—81no—
153C2×C37—82no—
154C2×C37#53411yescharacteristic
155C2×C37C2×#13511yescharacteristic
156C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
157D37—22no—
158D37—41no—
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#447D3711yescharacteristic, complemented
169C2×C22⋊C8—371no—
170C37#318811yescharacteristic, complemented, Sylow
171C2×#41—371no—
172C22×D4—741no—
173C22×D4C2×D3711yescharacteristic
174C23×C4—371no—
175C2×SD16—372no—
176C2×D8C2×D3712yescomplemented
177C22×C8—372no—
178#123—741no—
179C22⋊Q8—371no—
180C22⋊Q8—371no—
181#121—372no—
182#121—372no—
183#121—371no—
184#121—741no—
185#121—741no—
186#121C2×D3711yescharacteristic
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#102C2×D3712yescomplemented
199C22⋊C8—371no—
200C22⋊C8—372no—
201C22⋊C8C2×D3711yescharacteristic
202C24—21no—
203C24—371no—
204#41—1482no—
205C2×Q8—371no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—42no—
209C2×D4—741no—
210C2×D4—741no—
211C2×D4—742no—
212C2×D4—742no—
213C2×D4—1484no—
214C2×D4C22×D3711yescharacteristic
215C2×D4C22×D3712yes—
216C22×C4—371no—
217C22×C4—371no—
218C22×C4—371no—
219C22×C4—371no—
220C22×C4—371no—
221C22×C4—741no—
222C22×C4—741no—
223C22×C4—1482no—
224C22×C4C22×D3711yescharacteristic
225SD16—1482no—
226D8—42no—
227C2×C8—372no—
228C2×C8—1482no—
229C2×C8C22×D3712yes—
230C4⋊C4—371no—
231C4⋊C4—372no—
232C4⋊C4—741no—
233C4⋊C4—741no—
234C4⋊C4—741no—
235C4⋊C4C22×D3711yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—741no—
238C22⋊C4—741no—
239C22⋊C4—742no—
240C22⋊C4—742no—
241C22⋊C4—741no—
242C22⋊C4—741no—
243C42—741no—
244C23—21no—
245C23—22no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—371no—
250C23—371no—
251C23—742no—
252C23—742no—
253C23—1481no—
254C23D4×D3711yescharacteristic
255Q8—742no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—1484no—
262D4—1482no—
263D4—1484no—
264D4—1482no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—371no—
268C2×C4—371no—
269C2×C4—742no—
270C2×C4—742no—
271C2×C4—741no—
272C2×C4—742no—
273C2×C4—741no—
274C2×C4—1481no—
275C2×C4—1481no—
276C2×C4—1481no—
277C2×C4—1482no—
278C2×C4—1482no—
279C2×C4D4×D3711yescharacteristic
280C2×C4C23×D3711yescharacteristic
281C8—22no—
282C8—742no—
283C22—21no—
284C22—21no—
285C22—41no—
286C22—42no—
287C22—42no—
288C22—41no—
289C22—41no—
290C22—42no—
291C22—84no—
292C22—372no—
293C22—741no—
294C22—741no—
295C22—741no—
296C22—741no—
297C22—741no—
298C22—1484no—
299C22#1192455512yes—
300C22C2×D4×D3711yescharacteristic
301C4—21no—
302C4—41no—
303C4—741no—
304C4—741no—
305C4—1482no—
306C4—1481no—
307C4—1481no—
308C4#1192458112yes—
309C2—22no—
310C2—82no—
311C2—81no—
312C2—742no—
313C2—1481no—
314C2#1455809311yescharacteristic
315C2C2×#1192455511yescharacteristic
316C2AGL(1,ℤ/8)×D3711yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#447C2
socleC22×C37C22×D4
hypercenter#447D37
hypocenterC37#3188
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #14558043C2semidirect
2. C2×#11924507C2semidirect
3. #14557869C2semidirect
4. #14557855C2semidirect
5. #14557849C2semidirect
6. #14557766C2semidirect
7. #14557761C2semidirect
8. #14557684C2semidirect
9. #14557675C2semidirect
10. C22⋊C8×D37C2semidirect
11. C37×#447C2semidirect
12. C2×#10596964C22semidirect
15. C2×#10596940C22semidirect
16. C2×C8×D37C22semidirect
18. #11924516C22semidirect
24. #11924507C22semidirect
28. #11924494C22semidirect
31. #11924470C22semidirect
32. #11924467C22semidirect
33. #11924461C22semidirect
34. #11924457C22semidirect
35. #11924457C22semidirect
37. #11924448C22semidirect
39. C2×D8×C37C22semidirect
41. #102×C37C22semidirect
168. #447D37semidirect
170. C37#3188semidirect
176. C2×D8C2×D37semidirect
198. #102C2×D37semidirect

Maximal subgroups

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

subgroupindex
#145580432
C2×#119245072
#145578692
#145578552
#145578492
#145577662
#145577612
#145576842
#145576752
C22⋊C8×D372
C37×#4472
#318837