Finite Group Databasebeta

← #34056313

Subgroups — #34056313

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×#18011939C212yesmaximal, complemented
2#23750296C211yesmaximal, characteristic, complemented
3D4×D596C211yesmaximal, characteristic, complemented
4#23750002C211yesmaximal, characteristic, complemented
5#23749878C212yesmaximal, complemented
6#23749826C211yesmaximal, characteristic, complemented
7#23749821C211yesmaximal, characteristic, complemented
8#23749772C212yesmaximal, complemented
9#23749761C212yesmaximal, complemented
10C4×#14558725C211yesmaximal, characteristic, complemented
11#436×C149C211yesmaximal, characteristic, complemented
12C2×#14558736C2212yescomplemented
13C22×D596C2212yes—
14C2×D4×D149—21no—
15#18011939—28no—
16C2×#14558725C2211yescharacteristic
17C2×#14558725C2212yescomplemented
18C2×D1192C2212yes—
19C2×D1192C2212yescomplemented
20C2×D1192C2212yescomplemented
21#18011908C2211yescharacteristic
22#18011908C2211yescharacteristic, complemented
23#18011907—21no—
24#18011886—21no—
25#18011885C2212yes—
26#18011885C2212yescomplemented
27#18011884—22no—
28#18011881—22no—
29C4×Dic298C2211yescharacteristic, complemented
30C4×D596C2211yescharacteristic
31#18011873—21no—
32#18011843C2211yescharacteristic, complemented
33#18011836C2211yescharacteristic, complemented
34#18011833C2211yescharacteristic
35#18011833C2212yescomplemented
36#18011833C2212yescomplemented
37C2×M16×C149C2212yescomplemented
38C4×D4×C149C2211yescharacteristic
39C4⋊C8×C149C2211yescharacteristic, complemented
40C22⋊C8×C149C2212yescomplemented
41C4×C8×C149C2211yescharacteristic, complemented
42C23×D149—22no—
43#14558736—24no—
44#14558736—42no—
45C2×Dic298C2311yescharacteristic
46C2×D596—24no—
47C2×D596—21no—
48C2×D596—21no—
49C2×D596—22no—
50C2×D596—21no—
51C2×D596—21no—
52C2×D596—42no—
53C2×D596C2312yes—
54C2×D596C2312yes—
55C2×D596C2311yescharacteristic
56C2×C22⋊D149—22no—
57C2×C22⋊D149—22no—
58C2×C22⋊D149—22no—
59D4×D149—44no—
60C2×C4×D149—21no—
61C2×C4×D149—22no—
62#14558725—24no—
63#14558725D414yescomplemented
64D1192—24no—
65D1192—24no—
66D1192—42no—
67C4⋊Dic149C2311yescharacteristic
68C4⋊Dic149C2311yescharacteristic
69#14558715—21no—
70#14558713—22no—
71#14558713—21no—
72#14558713—22no—
73C4×Dic149—21no—
74C2×D4×C149D411yescharacteristic
75C22×C4×C149C2312yes—
76M16×C149—24no—
77C2×C8×C149C2312yes—
78C2×C8×C149C2311yescharacteristic
79C2×C8×C149C2311yescharacteristic
80C4⋊C4×C149D411yescharacteristic
81C22⋊C4×C149D412yes—
82C42×C149C2311yescharacteristic
83C22×D149—21no—
84C22×D149—22no—
85C22×D149—22no—
86C22×D149—44no—
87C22×D149—44no—
88C22×D149—82no—
89Dic298—21no—
90Dic298C2×D412yescomplemented
91D596—22no—
92D596—22no—
93D596—21no—
94D596—22no—
95D596—22no—
96D596—44no—
97D596—42no—
98D596—42no—
99D596—81no—
100D596—81no—
101D596C2×D412yes—
102C22⋊D149—44no—
103C22⋊D149—44no—
104C22⋊D149—82no—
105C2×Dic149—21no—
106C2×Dic149—21no—
107C2×Dic149—21no—
108C4×D149—44no—
109C4×D149—42no—
110C23×C149C2×D412yes—
111D4×C149—22no—
112C2×C4×C149—22no—
113C2×C4×C149—22no—
114C2×C4×C149C2×D412yes—
115C2×C4×C149C2×D411yescharacteristic
116C2×C4×C149C2×D411yescharacteristic
117C2×C4×C149C2411yescharacteristic
118C8×C149—22no—
119C8×C149—21no—
120C8×C149C2×D412yescomplemented
121C22×C149—22no—
122C22×C149—22no—
123C22×C149—22no—
124C22×C149C22×D411yescharacteristic
125C2×D149—22no—
126C2×D149—41no—
127C2×D149—41no—
128C2×D149—42no—
129C2×D149—42no—
130C2×D149—42no—
131C2×D149—42no—
132C2×D149—42no—
133C2×D149—42no—
134C2×D149—88no—
135C4×C149—22no—
136C4×C149—21no—
137C4×C149AGL(1,ℤ/8)12yescomplemented
138C4×C149C22×D411yescharacteristic
139C4×C149H(5,𝔽2)11yescharacteristic
140Dic149—22no—
141Dic149—41no—
142Dic149—41no—
143C2×C149—42no—
144C2×C149D4211yescharacteristic
145C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C149#57611yescharacteristic
147D149—42no—
148D149—82no—
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#459—1491no—
159#449—1492no—
160#447—1492no—
161C4×SD16—1491no—
162#436D14911yescharacteristic, complemented
163C2×#41—1492no—
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×M16C2×D14912yescomplemented
173#127—2981no—
174#127—1491no—
175#127—1491no—
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#102—1492no—
191C22⋊C8C2×D14912yescomplemented
192C4×C8C2×D14911yescharacteristic, complemented
193C24—2982no—
194#41—2984no—
195#41—5962no—
196C2×Q8—1491no—
197C2×D4—2984no—
198C2×D4—2982no—
199C2×D4—2982no—
200C2×D4—2982no—
201C2×D4—2981no—
202C2×D4—2982no—
203C2×D4—2981no—
204C2×D4—2981no—
205C2×D4—2981no—
206C2×D4—5962no—
207C2×D4—5964no—
208C2×D4—1492no—
209C2×D4—1492no—
210C2×D4—1491no—
211C2×D4D59611yescharacteristic
212C22×C4—2982no—
213C22×C4—2981no—
214C22×C4C22×D14912yes—
215SD16—2984no—
216SD16—1494no—
217D8—2984no—
218D8—2984no—
219D8—5962no—
220M16—24no—
221C2×C8C22×D14911yescharacteristic
222C2×C8C22×D14912yes—
223C2×C8C22×D14911yescharacteristic
224C4⋊C4—2981no—
225C4⋊C4—1491no—
226C4⋊C4—1491no—
227C4⋊C4D59611yescharacteristic
228C22⋊C4—2982no—
229C22⋊C4—2981no—
230C22⋊C4—2982no—
231C22⋊C4D59612yes—
232C42—2981no—
233C42C22×D14911yescharacteristic
234C23—2981no—
235C23—2982no—
236C23—2982no—
237C23—5964no—
238C23—5964no—
239C23—11922no—
240C23C2×D59612yes—
241Q8—2981no—
242Q8—1492no—
243D4—22no—
244D4—2982no—
245D4—2982no—
246D4—2981no—
247D4—2982no—
248D4—2982no—
249D4—5964no—
250D4—5964no—
251D4—5964no—
252D4—5962no—
253D4—5962no—
254D4—1492no—
255D4—11921no—
256D4—11921no—
257D4—11922no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—2981no—
261C2×C4—2981no—
262C2×C4—2981no—
263C2×C4—5964no—
264C2×C4—5962no—
265C2×C4C2×D59611yescharacteristic
266C2×C4C2×D59612yes—
267C2×C4C2×D59611yescharacteristic
268C2×C4C23×D14911yescharacteristic
269C8—22no—
270C8—21no—
271C8D4×D14912yescomplemented
272C22—22no—
273C22—22no—
274C22—22no—
275C22—2982no—
276C22—5961no—
277C22—5961no—
278C22—5962no—
279C22—5962no—
280C22—5962no—
281C22—5962no—
282C22—5962no—
283C22—5962no—
284C22—11928no—
285C22C22×D59611yescharacteristic
286C4—21no—
287C4—22no—
288C4—2982no—
289C4—5961no—
290C4—5961no—
291C4#1801193912yescomplemented
292C4C2×D4×D14911yescharacteristic
293C4#1801195711yescharacteristic
294C2—42no—
295C2—5962no—
296C2—11922no—
297C2—11922no—
298C2D4×D59611yescharacteristic
299C2C2×#1801193911yescharacteristic
300C2#2375043011yescharacteristic
subgroupG/H
centerC22C22×D596
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#436×C149C2
socleC22×C149C22×D4
hypercenter#436D149
hypocenterC149#3274
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#18011939C2semidirect
2. #23750296C2semidirect
3. D4×D596C2semidirect
4. #23750002C2semidirect
5. #23749878C2semidirect
6. #23749826C2semidirect
7. #23749821C2semidirect
8. #23749772C2semidirect
9. #23749761C2semidirect
10. C4×#14558725C2semidirect
11. #436×C149C2semidirect
12. C2×#14558736C22semidirect
17. C2×#14558725C22semidirect
19. C2×D1192C22semidirect
20. C2×D1192C22semidirect
22. #18011908C22semidirect
26. #18011885C22semidirect
29. C4×Dic298C22semidirect
32. #18011843C22semidirect
33. #18011836C22semidirect
35. #18011833C22semidirect
36. #18011833C22semidirect
37. C2×M16×C149C22semidirect
39. C4⋊C8×C149C22semidirect
40. C22⋊C8×C149C22semidirect
41. C4×C8×C149C22semidirect
63. #14558725D4semidirect
90. Dic298C2×D4semidirect
120. C8×C149C2×D4semidirect
137. C4×C149AGL(1,ℤ/8)semidirect
150. C149#3274semidirect
162. #436D149semidirect
172. C2×M16C2×D149semidirect
186. C4⋊C8C2×D149semidirect
191. C22⋊C8C2×D149semidirect
192. C4×C8C2×D149semidirect
271. C8D4×D149semidirect
291. C4#18011939semidirect

Maximal subgroups

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

subgroupindex
C2×#180119392
#237502962
D4×D5962
#237500022
#237498782
#237498262
#237498212
#237497722
#237497612
C4×#145587252
#436×C1492
#3274149