Finite Group Databasebeta

← #509×D149 #34057811

Subgroups — #509×D149 #34057811

Order 19072.

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×D149C211yesmaximal, characteristic, complemented
2C2×D8×D149C211yesmaximal, characteristic, complemented
3#23750063C211yesmaximal, characteristic, complemented
4#23750062C212yesmaximal, complemented
5#23750061C211yesmaximal, characteristic, complemented
6#23750060C211yesmaximal, characteristic, complemented
7#23750059C212yesmaximal, complemented
8SD32×D149C212yesmaximal, complemented
9D16×D149C212yesmaximal, complemented
10M32×D149C211yesmaximal, characteristic, complemented
11#509×C149C211yesmaximal, characteristic, complemented
12#41×D149—21no—
13C2×D4×D149—21no—
14#18011934C2211yescharacteristic, complemented
15#18011933C2211yescharacteristic, complemented
16#18011932—21no—
17#18011931—21no—
18#18011930C2211yescharacteristic, complemented
19Q16×D149C2211yescharacteristic, complemented
20SD16×D149—21no—
21C2×D1192C2211yescharacteristic, complemented
22C2×#14558721—21no—
23D8×D149—21no—
24D8×D149—21no—
25D8×D149C2212yes—
26D8×D149C2212yescomplemented
27D8×D149C2211yescharacteristic, complemented
28C2×C8×D149C2211yescharacteristic
29#18011856C2212yescomplemented
30#18011855C2212yescomplemented
31#18011854C2212yescomplemented
32D2384C2212yescomplemented
33#18011852C2212yescomplemented
34#18011852C2212yescomplemented
35#18011851C2211yescharacteristic, complemented
36#18011850C2212yescomplemented
37C16×D149C2212yescomplemented
38#135×C149C2211yescharacteristic, complemented
39C2×D8×C149C2211yescharacteristic, complemented
40SD32×C149C2212yescomplemented
41D16×C149C2212yescomplemented
42M32×C149C2211yescharacteristic, complemented
43C23×D149—41no—
44#14558737—21no—
45#14558736—21no—
46#14558736—41no—
47#14558735—21no—
48#14558735—41no—
49Q8×D149—21no—
50C2×D596—21no—
51C2×C22⋊D149—41no—
52D4×D149—22no—
53D4×D149—22no—
54D4×D149—21no—
55D4×D149—41no—
56D4×D149—41no—
57D4×D149—41no—
58C2×C4×D149—41no—
59C2×C4×D149D411yescharacteristic
60Dic596C2311yescharacteristic, complemented
61#14558726—21no—
62#14558725—21no—
63#14558724—21no—
64#14558723—21no—
65D1192—21no—
66D1192C2312yes—
67D1192C2311yescharacteristic, complemented
68#14558721—22no—
69#14558721—22no—
70#14558721—21no—
71C2×#11924746D411yescharacteristic
72C8×D149D411yescharacteristic
73C8×D149D411yescharacteristic
74C8×D149C2311yescharacteristic
75C8×D149C2311yescharacteristic
76#14558711C2312yescomplemented
77#41×C149—21no—
78C2×D4×C149—21no—
79Q16×C149C2311yescharacteristic, complemented
80SD16×C149—21no—
81D8×C149—21no—
82D8×C149C2312yes—
83D8×C149C2311yescharacteristic, complemented
84C2×C8×C149C2311yescharacteristic
85C16×C149C2312yescomplemented
86C22×D149—41no—
87C22×D149—41no—
88C22×D149—42no—
89C22×D149—42no—
90C22×D149—84no—
91C22×D149D811yescharacteristic
92Dic298—21no—
93Dic298—41no—
94D596—22no—
95D596—21no—
96D596—41no—
97D596—41no—
98C22⋊D149—41no—
99C22⋊D149—41no—
100C22⋊D149—42no—
101C22⋊D149—42no—
102C22⋊D149—41no—
103C2×Dic149—41no—
104C2×Dic149D811yescharacteristic
105C4×D149—41no—
106C4×D149—41no—
107C4×D149—41no—
108C4×D149D811yescharacteristic
109C4×D149D811yescharacteristic
110C4×D149C2×D411yescharacteristic
111C4×D149C2×D411yescharacteristic
112#11924746C2×D411yescharacteristic
113#11924746C2×D411yescharacteristic
114C23×C149—41no—
115Q8×C149—21no—
116D4×C149—22no—
117D4×C149—21no—
118D4×C149—41no—
119D4×C149—41no—
120C2×C4×C149—41no—
121C2×C4×C149C2×D411yescharacteristic
122C8×C149C2×D411yescharacteristic
123C8×C149C2411yescharacteristic
124C22×C149—41no—
125C22×C149—42no—
126C22×C149—82no—
127C22×C149C2×D811yescharacteristic
128C2×D149—22no—
129C2×D149—41no—
130C2×D149—42no—
131C2×D149—84no—
132C2×D149—82no—
133C2×D149—82no—
134C2×D149—84no—
135C2×D149C2×D811yescharacteristic
136C2×D149C2×D811yescharacteristic
137C4×C149—41no—
138C4×C149C2×D811yescharacteristic
139C4×C149C22×D411yescharacteristic
140Dic149—41no—
141Dic149C2×D811yescharacteristic
142Dic149C2×D811yescharacteristic
143C2×C149—21no—
144C2×C149—82no—
145C2×C149—81no—
146C2×C149C22×D811yescharacteristic
147D149—21no—
148D149—81no—
149D149—82no—
150D149#50912yesdirect product factor
151C149C2×#50911yescharacteristic, complemented, Sylow
152C2×#509—1491nomaximal, Sylow
153C2×#135—1491no—
154C22×D8—1491no—
155#509—1491no—
156#509—1491no—
157#509—1492no—
158#509—1492no—
159#509—1491no—
160#509D14911yescharacteristic, direct product factor
161C2×SD32—1492no—
162C2×D16—1492no—
163C2×M32—1491no—
164C2×#41—2981no—
165C22×D4—2981no—
166#135—2981no—
167#135—2981no—
168#135—1491no—
169#135—1491no—
170#135—1491no—
171#135C2×D14911yescharacteristic, complemented
172C2×Q16—1491no—
173C2×SD16—2981no—
174C2×D8—2981no—
175C2×D8—2981no—
176C2×D8—2981no—
177C2×D8—1492no—
178C2×D8—1492no—
179C2×D8—1491no—
180C2×D8—1491no—
181C2×D8C2×D14911yescharacteristic, complemented
182C22×C8—1491no—
183SD32—1492no—
184SD32—1492no—
185SD32—1492no—
186SD32C2×D14912yescomplemented
187D16—1492no—
188D16—1492no—
189D16—1492no—
190D16C2×D14912yescomplemented
191M32—1491no—
192M32—1492no—
193M32C2×D14911yescharacteristic, complemented
194C2×C16—1492no—
195C24—5961no—
196#41—21no—
197#41—2981no—
198#41—2981no—
199#41—2981no—
200#41—5961no—
201#41—5961no—
202C2×Q8—2981no—
203C2×D4—21no—
204C2×D4—2982no—
205C2×D4—2982no—
206C2×D4—2981no—
207C2×D4—2981no—
208C2×D4—5961no—
209C2×D4—5961no—
210C2×D4—5961no—
211C2×D4—5961no—
212C22×C4—5961no—
213C22×C4—1491no—
214Q16—2981no—
215Q16—1491no—
216Q16C22×D14911yescharacteristic, complemented
217SD16—21no—
218SD16—2981no—
219SD16—2981no—
220SD16—2981no—
221D8—21no—
222D8—2982no—
223D8—2982no—
224D8—2981no—
225D8—2981no—
226D8—1492no—
227D8—1491no—
228D8C22×D14912yes—
229D8C22×D14911yescharacteristic, complemented
230C2×C8—1491no—
231C2×C8—1491no—
232C2×C8—1491no—
233C2×C8—1491no—
234C2×C8—1491no—
235C2×C8C22×D14911yescharacteristic
236C16—1492no—
237C16C22×D14912yescomplemented
238C23—41no—
239C23—5961no—
240C23—5962no—
241C23—5962no—
242C23—5961no—
243C23—1491no—
244C23—11924no—
245Q8—21no—
246Q8—2981no—
247Q8—5961no—
248D4—22no—
249D4—21no—
250D4—41no—
251D4—41no—
252D4—2982no—
253D4—2981no—
254D4—5961no—
255D4—5961no—
256D4—5961no—
257D4—5962no—
258D4—5962no—
259D4—5961no—
260D4—5961no—
261C2×C4—41no—
262C2×C4—5961no—
263C2×C4—5961no—
264C2×C4—5961no—
265C2×C4—5961no—
266C2×C4—1491no—
267C2×C4—1491no—
268C2×C4—1491no—
269C2×C4—1491no—
270C2×C4—1491no—
271C2×C4D4×D14911yescharacteristic
272C8—1491no—
273C8—1491no—
274C8D4×D14911yescharacteristic
275C8C23×D14911yescharacteristic
276C22—41no—
277C22—42no—
278C22—82no—
279C22—2982no—
280C22—5961no—
281C22—5962no—
282C22—1491no—
283C22—1491no—
284C22—11922no—
285C22—11924no—
286C22—11924no—
287C22—11922no—
288C22D8×D14911yescharacteristic
289C4—41no—
290C4—5961no—
291C4—1491no—
292C4—1491no—
293C4D8×D14911yescharacteristic
294C4C2×D4×D14911yescharacteristic
295C2—21no—
296C2—82no—
297C2—81no—
298C2—2981no—
299C2—1492no—
300C2—11922no—
301C2—11921no—
302C2C2×D8×D14911yescharacteristic
subgroupG/H
centerC2C2×D8×D149
commutator, cosocleC8×C149C24
FrattiniC8C23×D149
Fitting#509×C149C2
socleC2×C149C22×D8
hypercenter#509D149
hypocenterC149C2×#509
LCSG ▷ C8×C149 ▷ C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #135×D149C2semidirect
2. C2×D8×D149C2semidirect
3. #23750063C2semidirect
4. #23750062C2semidirect
5. #23750061C2semidirect
6. #23750060C2semidirect
7. #23750059C2semidirect
8. SD32×D149C2semidirect
9. D16×D149C2semidirect
10. M32×D149C2semidirect
11. #509×C149C2semidirect
14. #18011934C22semidirect
15. #18011933C22semidirect
18. #18011930C22semidirect
19. Q16×D149C22semidirect
21. C2×D1192C22semidirect
26. D8×D149C22semidirect
27. D8×D149C22semidirect
29. #18011856C22semidirect
30. #18011855C22semidirect
31. #18011854C22semidirect
32. D2384C22semidirect
33. #18011852C22semidirect
34. #18011852C22semidirect
35. #18011851C22semidirect
36. #18011850C22semidirect
37. C16×D149C22semidirect
38. #135×C149C22semidirect
39. C2×D8×C149C22semidirect
40. SD32×C149C22semidirect
41. D16×C149C22semidirect
42. M32×C149C22semidirect
60. Dic596C23semidirect
67. D1192C23semidirect
76. #14558711C23semidirect
79. Q16×C149C23semidirect
83. D8×C149C23semidirect
85. C16×C149C23semidirect
150. D149#509direct (N × H)
151. C149C2×#509semidirect
160. #509D149direct (N × H)
171. #135C2×D149semidirect
181. C2×D8C2×D149semidirect
186. SD32C2×D149semidirect
190. D16C2×D149semidirect
193. M32C2×D149semidirect
216. Q16C22×D149semidirect
229. D8C22×D149semidirect
237. C16C22×D149semidirect

Maximal subgroups

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

subgroupindex
#135×D1492
C2×D8×D1492
#237500632
#237500622
#237500612
#237500602
#237500592
SD32×D1492
D16×D1492
M32×D1492
#509×C1492
C2×#509149