Finite Group Databasebeta

← #34053960

Subgroups — #34053960

Order 19072.

784 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 149 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×#18011956C211yesmaximal, characteristic, complemented
2C2×#18011939C211yesmaximal, characteristic, complemented
3C22×#14558721C211yesmaximal, characteristic, complemented
4#23750194C211yesmaximal, characteristic, complemented
5#23749771C214yesmaximal, complemented
6#23749760C214yesmaximal, complemented
7#23749619C211yesmaximal, characteristic, complemented
8#23749593C211yesmaximal, characteristic, complemented
9#418×C149C211yesmaximal, characteristic, complemented
10#18011956—24no—
11#18011956—24no—
12C2×#14558736C2211yescharacteristic
13C2×#14558736C2211yescharacteristic
14C2×#14558735—21no—
15C2×#14558735C2212yescomplemented
16C22×D596C2211yescharacteristic, complemented
17C22×C22⋊D149—22no—
18C2×D4×D149—21no—
19C2×D4×D149C2212yescomplemented
20#18011939—24no—
21C2×#14558725C2212yescomplemented
22C2×D1192C2212yescomplemented
23C2×#14558721—24no—
24C2×#14558721C2214yescomplemented
25C22×#11924746C2211yescharacteristic, complemented
26#18011908—21no—
27#18011899—21no—
28#18011886C2214yescomplemented
29#18011881—22no—
30#18011834C2212yescomplemented
31#18011831C2212yescomplemented
32#18011823C2214yescomplemented
33C22×D4×C149C2211yescharacteristic
34C2×M16×C149C2211yescharacteristic, complemented
35#117×C149C2211yescharacteristic, complemented
36#102×C149C2214yescomplemented
37C23×D149—22no—
38C23×D149—21no—
39#14558736—22no—
40#14558736—22no—
41#14558736—22no—
42#14558736D414yescomplemented
43#14558735—28no—
44#14558735—24no—
45#14558735—24no—
46C2×Dic298D411yescharacteristic
47C2×Dic298C2311yescharacteristic
48C2×D596—24no—
49C2×D596—22no—
50C2×D596—22no—
51C2×D596—21no—
52C2×D596D411yescharacteristic
53C2×D596C2312yes—
54C2×D596C2311yescharacteristic
55C2×C22⋊D149—216no—
56C2×C22⋊D149—22no—
57C2×C22⋊D149—22no—
58C2×C22⋊D149—21no—
59C2×C22⋊D149D412yes—
60D4×D149—28no—
61D4×D149—24no—
62D4×D149—24no—
63C22×Dic149—22no—
64C2×C4×D149—21no—
65C2×C4×D149C2312yes—
66#14558725—24no—
67D1192—24no—
68#14558721—28no—
69C2×#11924746—22no—
70C2×#11924746C2312yes—
71#14558713—24no—
72C24×C149—21no—
73C2×D4×C149—21no—
74C2×D4×C149—22no—
75C2×D4×C149D414yescomplemented
76C2×D4×C149C2312yes—
77C22×C4×C149C2311yescharacteristic
78M16×C149—22no—
79C2×C8×C149C2312yes—
80C4⋊C4×C149C2312yes—
81C22⋊C4×C149—21no—
82C42×C149—21no—
83C22×D149—21no—
84C22×D149—28no—
85C22×D149—22no—
86C22×D149—42no—
87C22×D149—84no—
88C22×D149—82no—
89C22×D149C2×D412yes—
90Dic298—22no—
91Dic298—21no—
92Dic298C2×D412yes—
93D596—22no—
94D596—21no—
95D596—22no—
96D596—22no—
97D596—42no—
98D596—84no—
99D596C2×D412yes—
100C22⋊D149—28no—
101C22⋊D149—28no—
102C22⋊D149—24no—
103C22⋊D149—24no—
104C2×Dic149—28no—
105C2×Dic149—21no—
106C2×Dic149C2×D412yes—
107C4×D149—24no—
108C4×D149—22no—
109C4×D149—22no—
110#11924746—22no—
111C23×C149—22no—
112C23×C149—24no—
113C23×C149—82no—
114C23×C149C2×D411yescharacteristic
115D4×C149—24no—
116D4×C149—22no—
117D4×C149C2×D414yes—
118C2×C4×C149—22no—
119C2×C4×C149C2×D411yescharacteristic
120C2×C4×C149C2×D412yes—
121C2×C4×C149C2×D412yes—
122C2×C4×C149C2411yescharacteristic
123C8×C149—22no—
124C22×C149—22no—
125C22×C149—24no—
126C22×C149—44no—
127C22×C149—84no—
128C22×C149#12012yes—
129C22×C149C22×D411yescharacteristic
130C2×D149—22no—
131C2×D149—22no—
132C2×D149—24no—
133C2×D149—42no—
134C2×D149—42no—
135C2×D149—44no—
136C2×D149—88no—
137C2×D149—84no—
138C4×C149—42no—
139C4×C149#12012yes—
140C4×C149C22×D411yescharacteristic
141C4×C149C22×D411yescharacteristic
142Dic149—22no—
143Dic149—22no—
144C2×C149—22no—
145C2×C149—44no—
146C2×C149C2×#12011yescharacteristic
147C2×C149#57612yes—
148D149—42no—
149D149—42no—
150D149—82no—
151C149#300811yescharacteristic, complemented, Sylow
152#3008—1491nomaximal, Sylow
153C2×H(5,𝔽2)—1491no—
154C2×AGL(1,ℤ/8)—1491no—
155C22×D8—1491no—
156#535—1491no—
157#449—1494no—
158#447—1494no—
159#418D14911yescharacteristic, complemented
160#416—1491no—
161#408—1491no—
162H(5,𝔽2)—2984no—
163H(5,𝔽2)—2984no—
164C2×#41—2981no—
165C2×#41—1492no—
166C2×#41—1491no—
167C2×#41—1491no—
168C22×D4—2981no—
169C22×D4—2982no—
170C22×D4—1492no—
171C22×D4—1491no—
172C22×D4C2×D14911yescharacteristic
173AGL(1,ℤ/8)—2984no—
174C2×SD16—1492no—
175C2×D8—2984no—
176C2×D8—1494no—
177C2×D8—1492no—
178C2×M16C2×D14911yescharacteristic, complemented
179C22×C8—1491no—
180#127—2981no—
181#124—2981no—
182#121—1494no—
183#120—2982no—
184#117C2×D14911yescharacteristic, complemented
185#103—1492no—
186#102—1492no—
187#102C2×D14914yescomplemented
188C22⋊C8—1494no—
189C24—21no—
190C24—2982no—
191C24—2981no—
192#41—2988no—
193#41—2984no—
194#41—2984no—
195#41—2982no—
196#41—2982no—
197#41—2982no—
198#41—1494no—
199C2×Q8—1491no—
200C2×Q8—1491no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—2984no—
204C2×D4—2988no—
205C2×D4—2984no—
206C2×D4—29816no—
207C2×D4—2982no—
208C2×D4—2981no—
209C2×D4—2984no—
210C2×D4—2982no—
211C2×D4—2982no—
212C2×D4—2981no—
213C2×D4—2982no—
214C2×D4—1492no—
215C2×D4—1491no—
216C2×D4—1492no—
217C2×D4—1491no—
218C2×D4D59614yescomplemented
219C2×D4C22×D14912yes—
220C22×C4—2982no—
221C22×C4—2981no—
222C22×C4—1492no—
223C22×C4C22×D14911yescharacteristic
224SD16—2984no—
225D8—2984no—
226D8—2988no—
227M16—22no—
228C2×C8—2982no—
229C2×C8—1492no—
230C2×C8C22×D14912yes—
231C4⋊C4C22×D14912yes—
232C22⋊C4—21no—
233C22⋊C4—2984no—
234C42—21no—
235C23—22no—
236C23—24no—
237C23—82no—
238C23—2988no—
239C23—2981no—
240C23—2982no—
241C23—5962no—
242C23—1492no—
243C23—11922no—
244C23—11924no—
245C23D4×D14911yescharacteristic
246Q8—2981no—
247Q8—2982no—
248Q8—1492no—
249D4—22no—
250D4—24no—
251D4—2988no—
252D4—2988no—
253D4—2984no—
254D4—2984no—
255D4—2982no—
256D4—2981no—
257D4—2982no—
258D4—2982no—
259D4—5962no—
260D4—1492no—
261D4—11924no—
262D4C2×D59614yes—
263C2×C4—22no—
264C2×C4—2988no—
265C2×C4—2981no—
266C2×C4—2982no—
267C2×C4—2982no—
268C2×C4—2984no—
269C2×C4—1492no—
270C2×C4D4×D14912yes—
271C2×C4D4×D14911yescharacteristic
272C2×C4C2×D59612yes—
273C2×C4C23×D14911yescharacteristic
274C8—22no—
275C8—2982no—
276C22—22no—
277C22—24no—
278C22—44no—
279C22—84no—
280C22—2984no—
281C22—2982no—
282C22—2982no—
283C22—5962no—
284C22—5964no—
285C22—5962no—
286C22—11924no—
287C22—11928no—
288C22#1801188112yes—
289C22C2×D4×D14911yescharacteristic
290C4—42no—
291C4—2982no—
292C4—2982no—
293C4#1801188112yes—
294C4C2×D4×D14911yescharacteristic
295C4C22×D59611yescharacteristic
296C2—22no—
297C2—44no—
298C2—5962no—
299C2—5962no—
300C2—11922no—
301C2C2×#1801188111yescharacteristic
302C2#2375042912yes—
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#418×C149C2
socleC22×C149C22×D4
hypercenter#418D149
hypocenterC149#3008
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#18011956C2semidirect
2. C2×#18011939C2semidirect
3. C22×#14558721C2semidirect
4. #23750194C2semidirect
5. #23749771C2semidirect
6. #23749760C2semidirect
7. #23749619C2semidirect
8. #23749593C2semidirect
9. #418×C149C2semidirect
15. C2×#14558735C22semidirect
16. C22×D596C22semidirect
19. C2×D4×D149C22semidirect
21. C2×#14558725C22semidirect
22. C2×D1192C22semidirect
24. C2×#14558721C22semidirect
25. C22×#11924746C22semidirect
28. #18011886C22semidirect
30. #18011834C22semidirect
31. #18011831C22semidirect
32. #18011823C22semidirect
34. C2×M16×C149C22semidirect
35. #117×C149C22semidirect
36. #102×C149C22semidirect
42. #14558736D4semidirect
75. C2×D4×C149D4semidirect
151. C149#3008semidirect
159. #418D149semidirect
178. C2×M16C2×D149semidirect
184. #117C2×D149semidirect
187. #102C2×D149semidirect
218. C2×D4D596semidirect

Maximal subgroups

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

subgroupindex
C2×#180119562
C2×#180119392
C22×#145587212
#237501942
#237497712
#237497602
#237496192
#237495932
#418×C1492
#3008149