Finite Group Databasebeta

← #509×D37 #18007716

Subgroups — #509×D37 #18007716

Order 4736.

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

normal subgroups only

This group has 400 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#135×D37C211yesmaximal, characteristic, complemented
2C2×D8×D37C211yesmaximal, characteristic, complemented
3#14557971C211yesmaximal, characteristic, complemented
4#14557970C212yesmaximal, complemented
5#14557969C211yesmaximal, characteristic, complemented
6#14557968C211yesmaximal, characteristic, complemented
7#14557967C212yesmaximal, complemented
8SD32×D37C212yesmaximal, complemented
9D16×D37C212yesmaximal, complemented
10M32×D37C211yesmaximal, characteristic, complemented
11C37×#509C211yesmaximal, characteristic, complemented
12#41×D37—21no—
13C2×D4×D37—21no—
14#11924559C2211yescharacteristic, complemented
15#11924558C2211yescharacteristic, complemented
16#11924557—21no—
17#11924556—21no—
18#11924555C2211yescharacteristic, complemented
19Q16×D37C2211yescharacteristic, complemented
20SD16×D37—21no—
21C2×D296C2211yescharacteristic, complemented
22C2×#10596939—21no—
23D8×D37—21no—
24D8×D37—21no—
25D8×D37C2212yes—
26D8×D37C2212yescomplemented
27D8×D37C2211yescharacteristic, complemented
28C2×C8×D37C2211yescharacteristic
29#11924481C2212yescomplemented
30#11924480C2212yescomplemented
31#11924479C2212yescomplemented
32D592C2212yescomplemented
33#11924477C2212yescomplemented
34#11924477C2212yescomplemented
35#11924476C2211yescharacteristic, complemented
36#11924475C2212yescomplemented
37C16×D37C2212yescomplemented
38#135×C37C2211yescharacteristic, complemented
39C2×D8×C37C2211yescharacteristic, complemented
40SD32×C37C2212yescomplemented
41D16×C37C2212yescomplemented
42M32×C37C2211yescharacteristic, complemented
43C23×D37—41no—
44#41×C37—21no—
45C2×D4×C37—21no—
46C2×C22⋊D37—41no—
47#10596966—21no—
48Q8×D37—21no—
49#10596964—21no—
50#10596964—41no—
51D4×D37—22no—
52D4×D37—22no—
53D4×D37—21no—
54D4×D37—41no—
55D4×D37—41no—
56D4×D37—41no—
57#10596962—21no—
58#10596962—41no—
59C2×D148—21no—
60C2×C4×D37—41no—
61C2×C4×D37D411yescharacteristic
62Q16×C37C2311yescharacteristic, complemented
63SD16×C37—21no—
64D8×C37—21no—
65D8×C37C2312yes—
66D8×C37C2311yescharacteristic, complemented
67C2×C8×C37C2311yescharacteristic
68#10596942—21no—
69#10596941—21no—
70#10596940—21no—
71#10596939—22no—
72#10596939—22no—
73#10596939—21no—
74C2×#64436D411yescharacteristic
75Dic148C2311yescharacteristic, complemented
76D296—21no—
77D296C2312yes—
78D296C2311yescharacteristic, complemented
79#10596930—21no—
80C8×D37D411yescharacteristic
81C8×D37D411yescharacteristic
82C8×D37C2311yescharacteristic
83C8×D37C2311yescharacteristic
84C16×C37C2312yescomplemented
85#10596925C2312yescomplemented
86C23×C37—41no—
87C22×D37—41no—
88C22×D37—41no—
89C22×D37—42no—
90C22×D37—42no—
91C22×D37—84no—
92C22×D37D811yescharacteristic
93Q8×C37—21no—
94D4×C37—22no—
95D4×C37—21no—
96D4×C37—41no—
97D4×C37—41no—
98C2×C4×C37—41no—
99C2×C4×C37C2×D411yescharacteristic
100C22⋊D37—41no—
101C22⋊D37—41no—
102C22⋊D37—41no—
103C22⋊D37—42no—
104C22⋊D37—42no—
105C2×Dic37—41no—
106C2×Dic37D811yescharacteristic
107D148—21no—
108D148—22no—
109D148—41no—
110D148—41no—
111C4×D37—41no—
112C4×D37—41no—
113C4×D37—41no—
114C4×D37D811yescharacteristic
115C4×D37D811yescharacteristic
116C4×D37C2×D411yescharacteristic
117C4×D37C2×D411yescharacteristic
118Dic74—21no—
119Dic74—41no—
120C8×C37C2×D411yescharacteristic
121C8×C37C2411yescharacteristic
122#64436C2×D411yescharacteristic
123#64436C2×D411yescharacteristic
124C22×C37—41no—
125C22×C37—42no—
126C22×C37—82no—
127C22×C37C2×D811yescharacteristic
128C2×D37—22no—
129C2×D37—41no—
130C2×D37—42no—
131C2×D37—84no—
132C2×D37—82no—
133C2×D37—82no—
134C2×D37—84no—
135C2×D37C2×D811yescharacteristic
136C2×D37C2×D811yescharacteristic
137C4×C37—41no—
138C4×C37C2×D811yescharacteristic
139C4×C37C22×D411yescharacteristic
140Dic37—41no—
141Dic37C2×D811yescharacteristic
142Dic37C2×D811yescharacteristic
143C2×#509—371nomaximal, Sylow
144C2×C37—21no—
145C2×C37—82no—
146C2×C37—81no—
147C2×C37C22×D811yescharacteristic
148D37—21no—
149D37—81no—
150D37—82no—
151D37#50912yesdirect product factor
152C2×#135—371no—
153C22×D8—371no—
154#509—371no—
155#509—371no—
156#509—372no—
157#509—372no—
158#509—371no—
159#509D3711yescharacteristic, direct product factor
160C2×SD32—372no—
161C2×D16—372no—
162C2×M32—371no—
163C37C2×#50911yescharacteristic, complemented, Sylow
164C2×#41—741no—
165C22×D4—741no—
166#135—371no—
167#135—371no—
168#135—371no—
169#135—741no—
170#135—741no—
171#135C2×D3711yescharacteristic, complemented
172C2×Q16—371no—
173C2×SD16—741no—
174C2×D8—372no—
175C2×D8—372no—
176C2×D8—371no—
177C2×D8—371no—
178C2×D8—741no—
179C2×D8—741no—
180C2×D8—741no—
181C2×D8C2×D3711yescharacteristic, complemented
182C22×C8—371no—
183SD32—372no—
184SD32—372no—
185SD32—372no—
186SD32C2×D3712yescomplemented
187D16—372no—
188D16—372no—
189D16—372no—
190D16C2×D3712yescomplemented
191M32—371no—
192M32—372no—
193M32C2×D3711yescharacteristic, complemented
194C2×C16—372no—
195C24—1481no—
196#41—21no—
197#41—741no—
198#41—741no—
199#41—741no—
200#41—1481no—
201#41—1481no—
202C2×Q8—741no—
203C2×D4—21no—
204C2×D4—742no—
205C2×D4—742no—
206C2×D4—741no—
207C2×D4—741no—
208C2×D4—1481no—
209C2×D4—1481no—
210C2×D4—1481no—
211C2×D4—1481no—
212C22×C4—371no—
213C22×C4—1481no—
214Q16—371no—
215Q16—741no—
216Q16C22×D3711yescharacteristic, complemented
217SD16—21no—
218SD16—741no—
219SD16—741no—
220SD16—741no—
221D8—21no—
222D8—372no—
223D8—371no—
224D8—742no—
225D8—742no—
226D8—741no—
227D8—741no—
228D8C22×D3712yes—
229D8C22×D3711yescharacteristic, complemented
230C2×C8—371no—
231C2×C8—371no—
232C2×C8—371no—
233C2×C8—371no—
234C2×C8—371no—
235C2×C8C22×D3711yescharacteristic
236C16—372no—
237C16C22×D3712yescomplemented
238C23—41no—
239C23—371no—
240C23—2964no—
241C23—1481no—
242C23—1482no—
243C23—1482no—
244C23—1481no—
245Q8—21no—
246Q8—741no—
247Q8—1481no—
248D4—22no—
249D4—21no—
250D4—41no—
251D4—41no—
252D4—742no—
253D4—741no—
254D4—1481no—
255D4—1482no—
256D4—1482no—
257D4—1481no—
258D4—1481no—
259D4—1481no—
260D4—1481no—
261C2×C4—41no—
262C2×C4—371no—
263C2×C4—371no—
264C2×C4—371no—
265C2×C4—371no—
266C2×C4—371no—
267C2×C4—1481no—
268C2×C4—1481no—
269C2×C4—1481no—
270C2×C4—1481no—
271C2×C4D4×D3711yescharacteristic
272C8—371no—
273C8—371no—
274C8D4×D3711yescharacteristic
275C8C23×D3711yescharacteristic
276C22—41no—
277C22—42no—
278C22—82no—
279C22—371no—
280C22—371no—
281C22—2962no—
282C22—2964no—
283C22—2962no—
284C22—2964no—
285C22—742no—
286C22—1481no—
287C22—1482no—
288C22D8×D3711yescharacteristic
289C4—41no—
290C4—371no—
291C4—371no—
292C4—1481no—
293C4D8×D3711yescharacteristic
294C4C2×D4×D3711yescharacteristic
295C2—21no—
296C2—82no—
297C2—81no—
298C2—372no—
299C2—2962no—
300C2—2961no—
301C2—741no—
302C2C2×D8×D3711yescharacteristic
subgroupG/H
centerC2C2×D8×D37
commutator, cosocleC8×C37C24
FrattiniC8C23×D37
FittingC37×#509C2
socleC2×C37C22×D8
hypercenter#509D37
hypocenterC37C2×#509
LCSG ▷ C8×C37 ▷ C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #135×D37C2semidirect
2. C2×D8×D37C2semidirect
3. #14557971C2semidirect
4. #14557970C2semidirect
5. #14557969C2semidirect
6. #14557968C2semidirect
7. #14557967C2semidirect
8. SD32×D37C2semidirect
9. D16×D37C2semidirect
10. M32×D37C2semidirect
11. C37×#509C2semidirect
14. #11924559C22semidirect
15. #11924558C22semidirect
18. #11924555C22semidirect
19. Q16×D37C22semidirect
21. C2×D296C22semidirect
26. D8×D37C22semidirect
27. D8×D37C22semidirect
29. #11924481C22semidirect
30. #11924480C22semidirect
31. #11924479C22semidirect
32. D592C22semidirect
33. #11924477C22semidirect
34. #11924477C22semidirect
35. #11924476C22semidirect
36. #11924475C22semidirect
37. C16×D37C22semidirect
38. #135×C37C22semidirect
39. C2×D8×C37C22semidirect
40. SD32×C37C22semidirect
41. D16×C37C22semidirect
42. M32×C37C22semidirect
62. Q16×C37C23semidirect
66. D8×C37C23semidirect
75. Dic148C23semidirect
78. D296C23semidirect
84. C16×C37C23semidirect
85. #10596925C23semidirect
151. D37#509direct (N × H)
159. #509D37direct (N × H)
163. C37C2×#509semidirect
171. #135C2×D37semidirect
181. C2×D8C2×D37semidirect
186. SD32C2×D37semidirect
190. D16C2×D37semidirect
193. M32C2×D37semidirect
216. Q16C22×D37semidirect
229. D8C22×D37semidirect
237. C16C22×D37semidirect

Maximal subgroups

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

subgroupindex
#135×D372
C2×D8×D372
#145579712
#145579702
#145579692
#145579682
#145579672
SD32×D372
D16×D372
M32×D372
C37×#5092
C2×#50937