Finite Group Databasebeta

← #509×D109 #28376152

Subgroups — #509×D109 #28376152

Order 13952.

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×D109C211yesmaximal, characteristic, complemented
2C2×D8×D109C211yesmaximal, characteristic, complemented
3#20295464C211yesmaximal, characteristic, complemented
4#20295463C212yesmaximal, complemented
5#20295462C211yesmaximal, characteristic, complemented
6#20295461C211yesmaximal, characteristic, complemented
7#20295460C212yesmaximal, complemented
8SD32×D109C212yesmaximal, complemented
9D16×D109C212yesmaximal, complemented
10M32×D109C211yesmaximal, characteristic, complemented
11#509×C109C211yesmaximal, characteristic, complemented
12#41×D109—21no—
13C2×D4×D109—21no—
14#16812722C2211yescharacteristic, complemented
15#16812721C2211yescharacteristic, complemented
16#16812720—21no—
17#16812719—21no—
18#16812718C2211yescharacteristic, complemented
19Q16×D109C2211yescharacteristic, complemented
20SD16×D109—21no—
21C2×D872C2211yescharacteristic, complemented
22C2×#13182014—21no—
23D8×D109—21no—
24D8×D109—21no—
25D8×D109C2212yes—
26D8×D109C2212yescomplemented
27D8×D109C2211yescharacteristic, complemented
28C2×C8×D109C2211yescharacteristic
29#16812644C2212yescomplemented
30#16812643C2212yescomplemented
31#16812642C2212yescomplemented
32D1744C2212yescomplemented
33#16812640C2212yescomplemented
34#16812640C2212yescomplemented
35#16812639C2211yescharacteristic, complemented
36#16812638C2212yescomplemented
37C16×D109C2212yescomplemented
38#135×C109C2211yescharacteristic, complemented
39C2×D8×C109C2211yescharacteristic, complemented
40SD32×C109C2212yescomplemented
41D16×C109C2212yescomplemented
42M32×C109C2211yescharacteristic, complemented
43C23×D109—41no—
44#13182030—21no—
45#13182029—21no—
46#13182029—41no—
47#13182028—21no—
48#13182028—41no—
49Q8×D109—21no—
50C2×D436—21no—
51C2×C22⋊D109—41no—
52D4×D109—22no—
53D4×D109—22no—
54D4×D109—21no—
55D4×D109—41no—
56D4×D109—41no—
57D4×D109—41no—
58C2×C4×D109—41no—
59C2×C4×D109D411yescharacteristic
60Dic436C2311yescharacteristic, complemented
61#13182019—21no—
62#13182018—21no—
63#13182017—21no—
64#13182016—21no—
65D872—21no—
66D872C2312yes—
67D872C2311yescharacteristic, complemented
68#13182014—22no—
69#13182014—22no—
70#13182014—21no—
71C2×#11724784D411yescharacteristic
72C8×D109D411yescharacteristic
73C8×D109D411yescharacteristic
74C8×D109C2311yescharacteristic
75C8×D109C2311yescharacteristic
76#13182004C2312yescomplemented
77#41×C109—21no—
78C2×D4×C109—21no—
79Q16×C109C2311yescharacteristic, complemented
80SD16×C109—21no—
81D8×C109—21no—
82D8×C109C2312yes—
83D8×C109C2311yescharacteristic, complemented
84C2×C8×C109C2311yescharacteristic
85C16×C109C2312yescomplemented
86C23×C109—41no—
87C22×D109—41no—
88C22×D109—41no—
89C22×D109—42no—
90C22×D109—42no—
91C22×D109—84no—
92C22×D109D811yescharacteristic
93Q8×C109—21no—
94D4×C109—22no—
95D4×C109—21no—
96D4×C109—41no—
97D4×C109—41no—
98C2×C4×C109—41no—
99C2×C4×C109C2×D411yescharacteristic
100C22⋊D109—41no—
101C22⋊D109—41no—
102C22⋊D109—42no—
103C22⋊D109—42no—
104C22⋊D109—41no—
105C2×Dic109—41no—
106C2×Dic109D811yescharacteristic
107D436—22no—
108D436—21no—
109D436—41no—
110D436—41no—
111C4×D109—41no—
112C4×D109—41no—
113C4×D109—41no—
114C4×D109D811yescharacteristic
115C4×D109D811yescharacteristic
116C4×D109C2×D411yescharacteristic
117C4×D109C2×D411yescharacteristic
118Dic218—21no—
119Dic218—41no—
120C8×C109C2×D411yescharacteristic
121C8×C109C2411yescharacteristic
122#11724784C2×D411yescharacteristic
123#11724784C2×D411yescharacteristic
124C22×C109—41no—
125C22×C109—42no—
126C22×C109—82no—
127C22×C109C2×D811yescharacteristic
128C2×D109—22no—
129C2×D109—41no—
130C2×D109—42no—
131C2×D109—84no—
132C2×D109—82no—
133C2×D109—82no—
134C2×D109—84no—
135C2×D109C2×D811yescharacteristic
136C2×D109C2×D811yescharacteristic
137C4×C109—41no—
138C4×C109C2×D811yescharacteristic
139C4×C109C22×D411yescharacteristic
140Dic109—41no—
141Dic109C2×D811yescharacteristic
142Dic109C2×D811yescharacteristic
143C2×C109—21no—
144C2×C109—82no—
145C2×C109—81no—
146C2×C109C22×D811yescharacteristic
147D109—21no—
148D109—81no—
149D109—82no—
150D109#50912yesdirect product factor
151C2×#509—1091nomaximal, Sylow
152C109C2×#50911yescharacteristic, complemented, Sylow
153C2×#135—1091no—
154C22×D8—1091no—
155#509—1091no—
156#509—1091no—
157#509—1092no—
158#509—1092no—
159#509—1091no—
160#509D10911yescharacteristic, direct product factor
161C2×SD32—1092no—
162C2×D16—1092no—
163C2×M32—1091no—
164C2×#41—2181no—
165C22×D4—2181no—
166#135—1091no—
167#135—1091no—
168#135—1091no—
169#135—2181no—
170#135—2181no—
171#135C2×D10911yescharacteristic, complemented
172C2×Q16—1091no—
173C2×SD16—2181no—
174C2×D8—1092no—
175C2×D8—1092no—
176C2×D8—1091no—
177C2×D8—1091no—
178C2×D8—2181no—
179C2×D8—2181no—
180C2×D8—2181no—
181C2×D8C2×D10911yescharacteristic, complemented
182C22×C8—1091no—
183SD32—1092no—
184SD32—1092no—
185SD32—1092no—
186SD32C2×D10912yescomplemented
187D16—1092no—
188D16—1092no—
189D16—1092no—
190D16C2×D10912yescomplemented
191M32—1091no—
192M32—1092no—
193M32C2×D10911yescharacteristic, complemented
194C2×C16—1092no—
195C24—4361no—
196#41—21no—
197#41—4361no—
198#41—4361no—
199#41—2181no—
200#41—2181no—
201#41—2181no—
202C2×Q8—2181no—
203C2×D4—21no—
204C2×D4—4361no—
205C2×D4—4361no—
206C2×D4—4361no—
207C2×D4—4361no—
208C2×D4—2182no—
209C2×D4—2182no—
210C2×D4—2181no—
211C2×D4—2181no—
212C22×C4—1091no—
213C22×C4—4361no—
214Q16—1091no—
215Q16—2181no—
216Q16C22×D10911yescharacteristic, complemented
217SD16—21no—
218SD16—2181no—
219SD16—2181no—
220SD16—2181no—
221D8—21no—
222D8—1092no—
223D8—1091no—
224D8—2182no—
225D8—2182no—
226D8—2181no—
227D8—2181no—
228D8C22×D10912yes—
229D8C22×D10911yescharacteristic, complemented
230C2×C8—1091no—
231C2×C8—1091no—
232C2×C8—1091no—
233C2×C8—1091no—
234C2×C8—1091no—
235C2×C8C22×D10911yescharacteristic
236C16—1092no—
237C16C22×D10912yescomplemented
238C23—41no—
239C23—8724no—
240C23—1091no—
241C23—4361no—
242C23—4362no—
243C23—4362no—
244C23—4361no—
245Q8—21no—
246Q8—4361no—
247Q8—2181no—
248D4—22no—
249D4—21no—
250D4—41no—
251D4—41no—
252D4—4361no—
253D4—4362no—
254D4—4362no—
255D4—4361no—
256D4—4361no—
257D4—4361no—
258D4—4361no—
259D4—2182no—
260D4—2181no—
261C2×C4—41no—
262C2×C4—1091no—
263C2×C4—1091no—
264C2×C4—1091no—
265C2×C4—1091no—
266C2×C4—1091no—
267C2×C4—4361no—
268C2×C4—4361no—
269C2×C4—4361no—
270C2×C4—4361no—
271C2×C4D4×D10911yescharacteristic
272C8—1091no—
273C8—1091no—
274C8D4×D10911yescharacteristic
275C8C23×D10911yescharacteristic
276C22—41no—
277C22—42no—
278C22—82no—
279C22—8722no—
280C22—8724no—
281C22—8722no—
282C22—8724no—
283C22—1091no—
284C22—1091no—
285C22—4361no—
286C22—4362no—
287C22—2182no—
288C22D8×D10911yescharacteristic
289C4—41no—
290C4—1091no—
291C4—1091no—
292C4—4361no—
293C4D8×D10911yescharacteristic
294C4C2×D4×D10911yescharacteristic
295C2—21no—
296C2—82no—
297C2—81no—
298C2—8722no—
299C2—8721no—
300C2—1092no—
301C2—2181no—
302C2C2×D8×D10911yescharacteristic
subgroupG/H
centerC2C2×D8×D109
commutator, cosocleC8×C109C24
FrattiniC8C23×D109
Fitting#509×C109C2
socleC2×C109C22×D8
hypercenter#509D109
hypocenterC109C2×#509
LCSG ▷ C8×C109 ▷ C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #135×D109C2semidirect
2. C2×D8×D109C2semidirect
3. #20295464C2semidirect
4. #20295463C2semidirect
5. #20295462C2semidirect
6. #20295461C2semidirect
7. #20295460C2semidirect
8. SD32×D109C2semidirect
9. D16×D109C2semidirect
10. M32×D109C2semidirect
11. #509×C109C2semidirect
14. #16812722C22semidirect
15. #16812721C22semidirect
18. #16812718C22semidirect
19. Q16×D109C22semidirect
21. C2×D872C22semidirect
26. D8×D109C22semidirect
27. D8×D109C22semidirect
29. #16812644C22semidirect
30. #16812643C22semidirect
31. #16812642C22semidirect
32. D1744C22semidirect
33. #16812640C22semidirect
34. #16812640C22semidirect
35. #16812639C22semidirect
36. #16812638C22semidirect
37. C16×D109C22semidirect
38. #135×C109C22semidirect
39. C2×D8×C109C22semidirect
40. SD32×C109C22semidirect
41. D16×C109C22semidirect
42. M32×C109C22semidirect
60. Dic436C23semidirect
67. D872C23semidirect
76. #13182004C23semidirect
79. Q16×C109C23semidirect
83. D8×C109C23semidirect
85. C16×C109C23semidirect
150. D109#509direct (N × H)
152. C109C2×#509semidirect
160. #509D109direct (N × H)
171. #135C2×D109semidirect
181. C2×D8C2×D109semidirect
186. SD32C2×D109semidirect
190. D16C2×D109semidirect
193. M32C2×D109semidirect
216. Q16C22×D109semidirect
229. D8C22×D109semidirect
237. C16C22×D109semidirect

Maximal subgroups

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

subgroupindex
#135×D1092
C2×D8×D1092
#202954642
#202954632
#202954622
#202954612
#202954602
SD32×D1092
D16×D1092
M32×D1092
#509×C1092
C2×#509109