Finite Group Databasebeta

← #34056314

Subgroups — #34056314

Order 19072.

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

normal subgroups only

This group has 556 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
1C2×#18011935C212yesmaximal, complemented
2#23750295C211yesmaximal, characteristic, complemented
3D4×D596C211yesmaximal, characteristic, complemented
4#23750001C211yesmaximal, characteristic, complemented
5#23749877C212yesmaximal, complemented
6#23749826C211yesmaximal, characteristic, complemented
7#23749771C212yesmaximal, complemented
8#23749757C212yesmaximal, complemented
9C4×#14558724C211yesmaximal, characteristic, complemented
10#23749702C211yesmaximal, characteristic, complemented
11#459×C149C211yesmaximal, characteristic, complemented
12C2×#14558735C2212yescomplemented
13C22×D596—21no—
14C2×D4×D149C2212yes—
15#18011935—28no—
16C2×#14558725C2212yescomplemented
17C2×#14558724C2211yescharacteristic
18C2×#14558721C2212yescomplemented
19C2×#14558721C2212yescomplemented
20C2×#14558719C2212yescomplemented
21#18011908C2211yescharacteristic
22#18011907—21no—
23#18011886C2212yes—
24#18011885—21no—
25#18011883—22no—
26#18011882C2212yescomplemented
27#18011881—22no—
28C4×Dic298C2211yescharacteristic, complemented
29C4×D596C2211yescharacteristic
30D4×Dic149—21no—
31#18011842C2211yescharacteristic, complemented
32#18011834C2211yescharacteristic, complemented
33#18011833C2212yescomplemented
34#18011832C2211yescharacteristic
35#18011823C2212yescomplemented
36C4×#11924746C2211yescharacteristic, complemented
37C2×D8×C149C2212yes—
38#127×C149C2211yescharacteristic, complemented
39C4×D4×C149C2211yescharacteristic
40C4⋊C8×C149C2211yescharacteristic, complemented
41#102×C149C2212yescomplemented
42C23×D149—22no—
43#14558735—24no—
44#14558735—42no—
45C2×Dic298C2311yescharacteristic
46C2×D596—22no—
47C2×D596—21no—
48C2×D596—21no—
49C2×D596—44no—
50C2×D596D411yescharacteristic
51C2×D596C2312yes—
52C2×C22⋊D149—22no—
53C2×C22⋊D149—22no—
54C2×C22⋊D149—22no—
55D4×D149—24no—
56D4×D149—42no—
57C22×Dic149—22no—
58C2×C4×D149C2312yes—
59#14558725—24no—
60#14558724D414yescomplemented
61#14558721—24no—
62#14558721—42no—
63#14558719—24no—
64C2×#11924746C2311yescharacteristic
65C2×#11924746C2311yescharacteristic
66C4⋊Dic149D411yescharacteristic
67C4⋊Dic149C2311yescharacteristic
68#14558715—21no—
69C22⋊Dic149—22no—
70#14558713—22no—
71#14558713D412yes—
72C4×Dic149—21no—
73C2×D4×C149—21no—
74C2×D4×C149—21no—
75C2×D4×C149C2312yes—
76C2×D4×C149C2311yescharacteristic
77C22×C4×C149—21no—
78D8×C149—24no—
79C2×C8×C149C2312yes—
80C4⋊C4×C149C2311yescharacteristic
81C22⋊C4×C149—21no—
82C42×C149C2311yescharacteristic
83C22×D149—22no—
84C22×D149—44no—
85C22×D149—44no—
86C22×D149—82no—
87C22×D149C2×D412yes—
88Dic298—21no—
89Dic298C2×D412yescomplemented
90D596—22no—
91D596—22no—
92D596—22no—
93D596—44no—
94D596—42no—
95D596—81no—
96C22⋊D149—44no—
97C22⋊D149—44no—
98C22⋊D149—82no—
99C2×Dic149—21no—
100C2×Dic149—21no—
101C2×Dic149—44no—
102C2×Dic149C2×D412yes—
103C4×D149—22no—
104C4×D149—22no—
105#11924746—21no—
106#11924746C2×D412yescomplemented
107C23×C149—21no—
108C23×C149—22no—
109D4×C149—22no—
110D4×C149—21no—
111D4×C149—22no—
112D4×C149—42no—
113D4×C149—81no—
114D4×C149C2×D412yes—
115C2×C4×C149—21no—
116C2×C4×C149—42no—
117C2×C4×C149C2×D411yescharacteristic
118C2×C4×C149C2×D411yescharacteristic
119C2×C4×C149C2411yescharacteristic
120C8×C149—22no—
121C22×C149—22no—
122C22×C149—41no—
123C22×C149—41no—
124C22×C149—42no—
125C22×C149—42no—
126C22×C149—42no—
127C22×C149C22×D411yescharacteristic
128C2×D149—22no—
129C2×D149—22no—
130C2×D149—22no—
131C2×D149—42no—
132C2×D149—42no—
133C2×D149—42no—
134C2×D149—88no—
135C4×C149—21no—
136C4×C149—41no—
137C4×C149AGL(1,ℤ/8)12yescomplemented
138C4×C149C22×D411yescharacteristic
139C4×C149H(5,𝔽2)11yescharacteristic
140Dic149—22no—
141Dic149—22no—
142Dic149—41no—
143C2×C149—42no—
144C2×C149—82no—
145C2×C149D4211yescharacteristic
146C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C149#57611yescharacteristic
148D149—42no—
149D149—82no—
150C149#327411yescharacteristic, complemented, Sylow
151#3274—1491nomaximal, Sylow
152C2×AGL(1,ℤ/8)—1492no—
153#550—1491no—
154D42—1491no—
155#493—1491no—
156#469—1492no—
157#460—1491no—
158#459D14911yescharacteristic, complemented
159#449—1492no—
160#447—1492no—
161C4×SD16—1491no—
162#436—1491no—
163C2×#41—1492no—
164C22×D4—2981no—
165C22×D4—1492no—
166AGL(1,ℤ/8)—2988no—
167C2×SD16—1491no—
168C2×SD16—1492no—
169C2×D8—1492no—
170C2×D8—1492no—
171C2×D8C2×D14912yes—
172C2×M16—1492no—
173#127—2981no—
174#127—1491no—
175#127C2×D14911yescharacteristic, complemented
176#121—2981no—
177#121—2982no—
178#121—1492no—
179#121—1492no—
180#120—2982no—
181C4×Q8—1491no—
182C4×D4—2981no—
183C4×D4—1491no—
184C4×D4C2×D14911yescharacteristic
185#106—1491no—
186C4⋊C8C2×D14911yescharacteristic, complemented
187#103—1491no—
188#102—1491no—
189#102—1492no—
190#102C2×D14912yescomplemented
191C22⋊C8—1492no—
192C4×C8—1491no—
193C24—2982no—
194#41—2984no—
195#41—5962no—
196C2×Q8—1491no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—2982no—
200C2×D4—2984no—
201C2×D4—2982no—
202C2×D4—2982no—
203C2×D4—2982no—
204C2×D4—2981no—
205C2×D4—2981no—
206C2×D4—5964no—
207C2×D4—5962no—
208C2×D4—1491no—
209C2×D4—1492no—
210C2×D4C22×D14912yes—
211C2×D4C22×D14911yescharacteristic
212C22×C4—21no—
213C22×C4—2982no—
214C22×C4—1492no—
215SD16—2984no—
216SD16—1494no—
217D8—24no—
218D8—2984no—
219D8—5962no—
220M16—2984no—
221C2×C8—1491no—
222C2×C8—1491no—
223C2×C8C22×D14912yes—
224C4⋊C4—2981no—
225C4⋊C4—1491no—
226C4⋊C4—1491no—
227C4⋊C4C22×D14911yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—2982no—
230C22⋊C4—2982no—
231C22⋊C4—1492no—
232C42—2981no—
233C42C22×D14911yescharacteristic
234C23—21no—
235C23—22no—
236C23—2982no—
237C23—5964no—
238C23—5964no—
239C23—1492no—
240C23—11922no—
241Q8—2981no—
242Q8—1492no—
243D4—22no—
244D4—21no—
245D4—22no—
246D4—42no—
247D4—81no—
248D4—2982no—
249D4—2982no—
250D4—2982no—
251D4—5964no—
252D4—5964no—
253D4—5964no—
254D4—5962no—
255D4—11921no—
256D4—11922no—
257D4D4×D14912yes—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—2981no—
261C2×C4—2982no—
262C2×C4—2982no—
263C2×C4—2981no—
264C2×C4—5964no—
265C2×C4—1492no—
266C2×C4D4×D14911yescharacteristic
267C2×C4D4×D14911yescharacteristic
268C2×C4C23×D14911yescharacteristic
269C8—22no—
270C8—2981no—
271C8—1492no—
272C22—22no—
273C22—41no—
274C22—41no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—2982no—
279C22—2982no—
280C22—2982no—
281C22—5962no—
282C22—5962no—
283C22—5962no—
284C22—11928no—
285C22C2×D4×D14911yescharacteristic
286C4—21no—
287C4—41no—
288C4—2982no—
289C4—2982no—
290C4—5961no—
291C4#1801193512yescomplemented
292C4C2×D4×D14911yescharacteristic
293C4#1801195711yescharacteristic
294C2—42no—
295C2—82no—
296C2—5962no—
297C2—11922no—
298C2#2375025911yescharacteristic
299C2C2×#1801193511yescharacteristic
300C2#2375042911yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#459×C149C2
socleC22×C149C22×D4
hypercenter#459D149
hypocenterC149#3274
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#18011935C2semidirect
2. #23750295C2semidirect
3. D4×D596C2semidirect
4. #23750001C2semidirect
5. #23749877C2semidirect
6. #23749826C2semidirect
7. #23749771C2semidirect
8. #23749757C2semidirect
9. C4×#14558724C2semidirect
10. #23749702C2semidirect
11. #459×C149C2semidirect
12. C2×#14558735C22semidirect
16. C2×#14558725C22semidirect
18. C2×#14558721C22semidirect
19. C2×#14558721C22semidirect
20. C2×#14558719C22semidirect
26. #18011882C22semidirect
28. C4×Dic298C22semidirect
31. #18011842C22semidirect
32. #18011834C22semidirect
33. #18011833C22semidirect
35. #18011823C22semidirect
36. C4×#11924746C22semidirect
38. #127×C149C22semidirect
40. C4⋊C8×C149C22semidirect
41. #102×C149C22semidirect
60. #14558724D4semidirect
89. Dic298C2×D4semidirect
106. #11924746C2×D4semidirect
137. C4×C149AGL(1,ℤ/8)semidirect
150. C149#3274semidirect
158. #459D149semidirect
175. #127C2×D149semidirect
186. C4⋊C8C2×D149semidirect
190. #102C2×D149semidirect
291. C4#18011935semidirect

Maximal subgroups

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

subgroupindex
C2×#180119352
#237502952
D4×D5962
#237500012
#237498772
#237498262
#237497712
#237497572
C4×#145587242
#237497022
#459×C1492
#3274149