Finite Group Databasebeta

← #34056311

Subgroups — #34056311

Order 19072.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 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×#18011936C212yesmaximal, complemented
2D4×D596C211yesmaximal, characteristic, complemented
3#23750001C211yesmaximal, characteristic, complemented
4#23749876C212yesmaximal, complemented
5#23749825C211yesmaximal, characteristic, complemented
6#23749817C211yesmaximal, characteristic, complemented
7#23749770C212yesmaximal, complemented
8#23749758C212yesmaximal, complemented
9C4×#14558723C211yesmaximal, characteristic, complemented
10#23749703C211yesmaximal, characteristic, complemented
11#550×C149C211yesmaximal, characteristic, complemented
12C22×D596C2212yes—
13C2×D4×D149—21no—
14#18011936—28no—
15C2×#14558723C2211yescharacteristic
16C2×#14558723C2212yescomplemented
17C2×#14558721C2212yes—
18C2×#14558721C2212yescomplemented
19C2×#14558721C2212yescomplemented
20C2×#14558720C2212yescomplemented
21#18011908C2211yescharacteristic
22#18011886—21no—
23#18011885C2212yes—
24#18011881—22no—
25C4×D596C2211yescharacteristic
26#18011842C2211yescharacteristic, complemented
27C4⋊#11924746C2211yescharacteristic, complemented
28#18011835C2211yescharacteristic, complemented
29#18011832C2212yescomplemented
30#18011831C2211yescharacteristic
31#18011831C2212yescomplemented
32C22⋊#11924746C2212yescomplemented
33C4×#11924746C2211yescharacteristic, complemented
34C2×#41×C149C2212yescomplemented
35#127×C149—21no—
36#127×C149C2211yescharacteristic, complemented
37#121×C149—22no—
38#121×C149C2212yescomplemented
39C4×Q8×C149C2211yescharacteristic, complemented
40C4×D4×C149—21no—
41C4×D4×C149C2211yescharacteristic
42C23×D149—22no—
43C2×D596—24no—
44C2×D596—21no—
45C2×D596—22no—
46C2×D596—21no—
47C2×D596—42no—
48C2×D596C2312yes—
49C2×D596C2311yescharacteristic
50C2×C22⋊D149—22no—
51D4×D149—44no—
52C2×C4×D149—21no—
53#14558723—24no—
54#14558723D414yescomplemented
55#14558721—24no—
56#14558721—24no—
57#14558721—42no—
58#14558720—24no—
59C2×#11924746C2312yes—
60C2×#11924746C2311yescharacteristic
61C2×#11924746C2311yescharacteristic
62C4⋊Dic149C2311yescharacteristic
63#14558713—21no—
64#14558713—22no—
65#41×C149—24no—
66#41×C149—42no—
67C2×Q8×C149C2311yescharacteristic
68C2×D4×C149—22no—
69C2×D4×C149—22no—
70C2×D4×C149—21no—
71C2×D4×C149—21no—
72C2×D4×C149D411yescharacteristic
73C2×D4×C149C2312yes—
74C22×C4×C149—22no—
75C22×C4×C149C2312yes—
76C4⋊C4×C149—21no—
77C4⋊C4×C149D411yescharacteristic
78C4⋊C4×C149C2311yescharacteristic
79C22⋊C4×C149—22no—
80C22⋊C4×C149D412yes—
81C42×C149—21no—
82C42×C149C2311yescharacteristic
83C22×D149—21no—
84C22×D149—22no—
85C22×D149—44no—
86C22×D149—44no—
87C22×D149—82no—
88D596—22no—
89D596—21no—
90D596—22no—
91D596—44no—
92D596—42no—
93D596—81no—
94D596C2×D412yes—
95C22⋊D149—44no—
96C2×Dic149—21no—
97C4×D149—42no—
98#11924746—22no—
99#11924746—21no—
100#11924746C2×D412yescomplemented
101C23×C149—22no—
102C23×C149C2×D412yes—
103Q8×C149—21no—
104Q8×C149C2×D412yescomplemented
105D4×C149—22no—
106D4×C149—22no—
107D4×C149—22no—
108D4×C149—44no—
109D4×C149—42no—
110D4×C149—81no—
111D4×C149—82no—
112C2×C4×C149—21no—
113C2×C4×C149—22no—
114C2×C4×C149—22no—
115C2×C4×C149—21no—
116C2×C4×C149—44no—
117C2×C4×C149C2×D411yescharacteristic
118C2×C4×C149C2×D412yes—
119C2×C4×C149C2×D411yescharacteristic
120C2×C4×C149C2411yescharacteristic
121C22×C149—22no—
122C22×C149—22no—
123C22×C149—22no—
124C22×C149—42no—
125C22×C149—42no—
126C22×C149—42no—
127C22×C149C22×D411yescharacteristic
128C2×D149—22no—
129C2×D149—41no—
130C2×D149—41no—
131C2×D149—42no—
132C2×D149—42no—
133C2×D149—42no—
134C2×D149—88no—
135C4×C149—22no—
136C4×C149—22no—
137C4×C149—21no—
138C4×C149—41no—
139C4×C149AGL(1,ℤ/8)12yescomplemented
140C4×C149C22×D411yescharacteristic
141C4×C149H(5,𝔽2)11yescharacteristic
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#550D14911yescharacteristic, complemented
154D42—1491no—
155#493—1491no—
156#469—1492no—
157#460—1491no—
158#459—1491no—
159#449—1492no—
160#447—1492no—
161C4×SD16—1491no—
162#436—1491no—
163C2×#41C2×D14912yescomplemented
164C22×D4—2981no—
165C22×D4—1492no—
166AGL(1,ℤ/8)—2988no—
167C2×SD16—1492no—
168C2×SD16—1491no—
169C2×D8—1492no—
170C2×D8—1492no—
171C2×D8—1492no—
172C2×M16—1492no—
173#127—21no—
174#127—1491no—
175#127C2×D14911yescharacteristic, complemented
176#121—22no—
177#121—2981no—
178#121—1492no—
179#121C2×D14912yescomplemented
180#120—2982no—
181C4×Q8C2×D14911yescharacteristic, complemented
182C4×D4—21no—
183C4×D4—1491no—
184C4×D4C2×D14911yescharacteristic
185#106—1491no—
186C4⋊C8—1491no—
187#103—1491no—
188#102—1492no—
189#102—1491no—
190#102—1492no—
191C22⋊C8—1492no—
192C4×C8—1491no—
193C24—2982no—
194#41—24no—
195#41—42no—
196C2×Q8C22×D14911yescharacteristic
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—2984no—
202C2×D4—2982no—
203C2×D4—2982no—
204C2×D4—2981no—
205C2×D4—2981no—
206C2×D4—5962no—
207C2×D4—5964no—
208C2×D4—1492no—
209C2×D4—1491no—
210C2×D4C22⋊D14911yescharacteristic
211C2×D4C22×D14912yes—
212C22×C4—22no—
213C22×C4—2981no—
214C22×C4C22×D14912yes—
215SD16—2984no—
216SD16—1494no—
217D8—2984no—
218D8—2984no—
219D8—5962no—
220M16—2984no—
221C2×C8—1491no—
222C2×C8—1492no—
223C2×C8—1491no—
224C4⋊C4—21no—
225C4⋊C4—1491no—
226C4⋊C4C22⋊D14911yescharacteristic
227C4⋊C4C22×D14911yescharacteristic
228C22⋊C4—22no—
229C22⋊C4—2981no—
230C22⋊C4—2982no—
231C22⋊C4C22⋊D14912yes—
232C42—21no—
233C42C22×D14911yescharacteristic
234C23—22no—
235C23—2981no—
236C23—2982no—
237C23—5964no—
238C23—5964no—
239C23—11922no—
240C23C2×C22⋊D14912yes—
241Q8—21no—
242Q8D4×D14912yescomplemented
243D4—22no—
244D4—22no—
245D4—22no—
246D4—44no—
247D4—42no—
248D4—81no—
249D4—82no—
250D4—2982no—
251D4—2981no—
252D4—2982no—
253D4—5964no—
254D4—5964no—
255D4—5962no—
256D4—1492no—
257D4—11921no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—44no—
263C2×C4—2981no—
264C2×C4—5962no—
265C2×C4C2×C22⋊D14911yescharacteristic
266C2×C4C2×C22⋊D14912yes—
267C2×C4C2×C22⋊D14911yescharacteristic
268C2×C4C23×D14911yescharacteristic
269C8—2982no—
270C8—2981no—
271C8—1492no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—2982no—
279C22—5961no—
280C22—5961no—
281C22—5962no—
282C22—5962no—
283C22—5962no—
284C22—11928no—
285C22C22×C22⋊D14911yescharacteristic
286C4—21no—
287C4—22no—
288C4—22no—
289C4—41no—
290C4—5961no—
291C4#1801193612yescomplemented
292C4C2×D4×D14911yescharacteristic
293C4#1801195611yescharacteristic
294C2—42no—
295C2—82no—
296C2—5962no—
297C2—11922no—
298C2D4×C22⋊D14911yescharacteristic
299C2C2×#1801193611yescharacteristic
300C2#2375042811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#550×C149C2
socleC22×C149C22×D4
hypercenter#550D149
hypocenterC149#3274
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#18011936C2semidirect
2. D4×D596C2semidirect
3. #23750001C2semidirect
4. #23749876C2semidirect
5. #23749825C2semidirect
6. #23749817C2semidirect
7. #23749770C2semidirect
8. #23749758C2semidirect
9. C4×#14558723C2semidirect
10. #23749703C2semidirect
11. #550×C149C2semidirect
16. C2×#14558723C22semidirect
18. C2×#14558721C22semidirect
19. C2×#14558721C22semidirect
20. C2×#14558720C22semidirect
26. #18011842C22semidirect
27. C4⋊#11924746C22semidirect
28. #18011835C22semidirect
29. #18011832C22semidirect
31. #18011831C22semidirect
32. C22⋊#11924746C22semidirect
33. C4×#11924746C22semidirect
34. C2×#41×C149C22semidirect
36. #127×C149C22semidirect
38. #121×C149C22semidirect
39. C4×Q8×C149C22semidirect
54. #14558723D4semidirect
100. #11924746C2×D4semidirect
104. Q8×C149C2×D4semidirect
139. C4×C149AGL(1,ℤ/8)semidirect
150. C149#3274semidirect
153. #550D149semidirect
163. C2×#41C2×D149semidirect
175. #127C2×D149semidirect
179. #121C2×D149semidirect
181. C4×Q8C2×D149semidirect
242. Q8D4×D149semidirect
291. C4#18011936semidirect

Maximal subgroups

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

subgroupindex
C2×#180119362
D4×D5962
#237500012
#237498762
#237498252
#237498172
#237497702
#237497582
C4×#145587232
#237497032
#550×C1492
#3274149