Finite Group Databasebeta

← #34056310

Subgroups — #34056310

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
3#23750260C211yesmaximal, characteristic, complemented
4#23749875C212yesmaximal, complemented
5#23749822C211yesmaximal, characteristic, complemented
6#23749817C211yesmaximal, characteristic, complemented
7#23749767C212yesmaximal, complemented
8#23749757C212yesmaximal, complemented
9C4×#14558725C211yesmaximal, characteristic, complemented
10#23749704C211yesmaximal, characteristic, complemented
11#493×C149C211yesmaximal, characteristic, complemented
12C2×#14558735C2212yescomplemented
13C2×D4×D149—21no—
14C2×D4×D149C2212yes—
15#18011935—28no—
16C2×#14558725C2211yescharacteristic
17C2×#14558724C2212yescomplemented
18C2×#14558721C2212yes—
19C2×#14558719C2212yescomplemented
20#18011907—21no—
21#18011883—22no—
22#18011882—21no—
23#18011882C2212yes—
24#18011882C2212yescomplemented
25#18011879—22no—
26C4×Dic298C2211yescharacteristic, complemented
27C4×D596C2211yescharacteristic
28C4×D596C2211yescharacteristic
29D4×Dic149—21no—
30#18011843C2211yescharacteristic, complemented
31C4⋊#11924746C2211yescharacteristic, complemented
32#18011836C2211yescharacteristic, complemented
33#18011833C2211yescharacteristic
34#18011832C2212yescomplemented
35#18011832C2212yescomplemented
36#18011823C2212yescomplemented
37C2×D8×C149C2212yescomplemented
38C2×D8×C149C2212yescomplemented
39#127×C149C2211yescharacteristic
40#127×C149C2211yescharacteristic, complemented
41C4×C8×C149C2211yescharacteristic, complemented
42C23×D149—22no—
43#14558735—24no—
44#14558735—42no—
45C2×Dic298C2311yescharacteristic
46C2×D596D411yescharacteristic
47C2×D596C2311yescharacteristic
48C2×C22⋊D149—22no—
49C2×C22⋊D149—22no—
50C2×C22⋊D149—22no—
51C2×C22⋊D149—22no—
52D4×D149—24no—
53D4×D149—42no—
54D4×D149—44no—
55C22×Dic149—22no—
56C2×C4×D149—21no—
57C2×C4×D149C2312yes—
58#14558725D414yescomplemented
59#14558724—24no—
60#14558721—24no—
61#14558719—24no—
62C2×#11924746C2312yes—
63C4⋊Dic149D411yescharacteristic
64C4⋊Dic149C2311yescharacteristic
65C4⋊Dic149C2311yescharacteristic
66#14558715—21no—
67C22⋊Dic149—22no—
68C22⋊Dic149—22no—
69#14558713—21no—
70#14558713D412yes—
71C4×Dic149—21no—
72C2×D4×C149—21no—
73C2×D4×C149—21no—
74C2×D4×C149—21no—
75C2×D4×C149—21no—
76C2×D4×C149C2312yes—
77C2×D4×C149C2312yes—
78D8×C149—24no—
79D8×C149—42no—
80C2×C8×C149C2311yescharacteristic
81C2×C8×C149C2311yescharacteristic
82C42×C149C2311yescharacteristic
83C22×D149—21no—
84C22×D149—44no—
85C22×D149—44no—
86C22×D149—82no—
87C22×D149C2×D412yes—
88Dic298—21no—
89Dic298C2×D412yescomplemented
90D596—22no—
91D596—21no—
92D596C2×D412yes—
93C22⋊D149—44no—
94C22⋊D149—44no—
95C22⋊D149—44no—
96C22⋊D149—82no—
97C2×Dic149—21no—
98C2×Dic149—21no—
99C2×Dic149—21no—
100C2×Dic149—44no—
101C2×Dic149C2×D412yes—
102C4×D149—22no—
103C4×D149—22no—
104C4×D149—42no—
105#11924746—22no—
106C23×C149—22no—
107C23×C149—22no—
108D4×C149—22no—
109D4×C149—22no—
110D4×C149—22no—
111D4×C149—22no—
112D4×C149—42no—
113D4×C149—42no—
114D4×C149—81no—
115D4×C149—81no—
116C2×C4×C149C2×D411yescharacteristic
117C2×C4×C149C2×D411yescharacteristic
118C2×C4×C149C2411yescharacteristic
119C8×C149—21no—
120C8×C149C2×D412yescomplemented
121C22×C149—42no—
122C22×C149—42no—
123C22×C149—42no—
124C22×C149—42no—
125C22×C149—42no—
126C22×C149—42no—
127C22×C149C22×D411yescharacteristic
128C2×D149—22no—
129C2×D149—22no—
130C2×D149—22no—
131C2×D149—22no—
132C2×D149—41no—
133C2×D149—41no—
134C2×D149—88no—
135C4×C149—21no—
136C4×C149AGL(1,ℤ/8)12yescomplemented
137C4×C149C22×D411yescharacteristic
138C4×C149H(5,𝔽2)11yescharacteristic
139Dic149—22no—
140Dic149—22no—
141Dic149—41no—
142Dic149—41no—
143C2×C149—82no—
144C2×C149—82no—
145C2×C149D4211yescharacteristic
146C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C149#57611yescharacteristic
148D149—42no—
149D149—42no—
150C149#327411yescharacteristic, complemented, Sylow
151#3274—1491nomaximal, Sylow
152C2×AGL(1,ℤ/8)—1492no—
153#550—1491no—
154D42—1491no—
155#493D14911yescharacteristic, complemented
156#469—1492no—
157#460—1491no—
158#459—1491no—
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—1492no—
168C2×SD16—1491no—
169C2×D8—1492no—
170C2×D8C2×D14912yescomplemented
171C2×D8C2×D14912yescomplemented
172C2×M16—1492no—
173#127—2981no—
174#127C2×D14911yescharacteristic
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×D4—1491no—
185#106—1491no—
186C4⋊C8—1491no—
187#103—1491no—
188#102—1492no—
189#102—1492no—
190#102—1491no—
191C22⋊C8—1492no—
192C4×C8C2×D14911yescharacteristic, complemented
193C24—2982no—
194#41—2984no—
195#41—5962no—
196C2×Q8—1491no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—2984no—
202C2×D4—2982no—
203C2×D4—2982no—
204C2×D4—2982no—
205C2×D4—2982no—
206C2×D4—5962no—
207C2×D4—5964no—
208C2×D4—1491no—
209C2×D4—1491no—
210C2×D4C22×D14912yes—
211C2×D4C22×D14912yes—
212C22×C4—2982no—
213C22×C4—2981no—
214C22×C4—1492no—
215SD16—2984no—
216SD16—1494no—
217D8—24no—
218D8—42no—
219D8—2984no—
220M16—2984no—
221C2×C8—1492no—
222C2×C8C22×D14911yescharacteristic
223C2×C8C22×D14911yescharacteristic
224C4⋊C4—2981no—
225C4⋊C4—1491no—
226C4⋊C4—1491no—
227C4⋊C4—1491no—
228C22⋊C4—2982no—
229C22⋊C4—2982no—
230C22⋊C4—2981no—
231C22⋊C4—1492no—
232C42—2981no—
233C42C22×D14911yescharacteristic
234C23—22no—
235C23—22no—
236C23—2981no—
237C23—5964no—
238C23—5964no—
239C23—1492no—
240C23—11922no—
241Q8—2981no—
242Q8—1492no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—22no—
247D4—42no—
248D4—42no—
249D4—81no—
250D4—81no—
251D4—2982no—
252D4—2981no—
253D4—5964no—
254D4—5964no—
255D4—5964no—
256D4—1492no—
257D4—11922no—
258C2×C4—2981no—
259C2×C4—2982no—
260C2×C4—2982no—
261C2×C4—2981no—
262C2×C4—2981no—
263C2×C4—5964no—
264C2×C4—5962no—
265C2×C4—1492no—
266C2×C4D4×D14911yescharacteristic
267C2×C4D4×D14911yescharacteristic
268C2×C4C23×D14911yescharacteristic
269C8—21no—
270C8—2982no—
271C8D4×D14912yescomplemented
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—2982no—
279C22—2982no—
280C22—2982no—
281C22—2982no—
282C22—5961no—
283C22—5961no—
284C22—11928no—
285C22C2×D4×D14911yescharacteristic
286C4—21no—
287C4—2982no—
288C4—2982no—
289C4—5961no—
290C4—5961no—
291C4#1801193512yescomplemented
292C4C2×D4×D14911yescharacteristic
293C4#1801195611yescharacteristic
294C2—82no—
295C2—82no—
296C2—5962no—
297C2—5962no—
298C2#2375026011yescharacteristic
299C2C2×#1801193511yescharacteristic
300C2#2375042611yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#493×C149C2
socleC22×C149C22×D4
hypercenter#493D149
hypocenterC149#3274
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#18011935C2semidirect
2. #23750295C2semidirect
3. #23750260C2semidirect
4. #23749875C2semidirect
5. #23749822C2semidirect
6. #23749817C2semidirect
7. #23749767C2semidirect
8. #23749757C2semidirect
9. C4×#14558725C2semidirect
10. #23749704C2semidirect
11. #493×C149C2semidirect
12. C2×#14558735C22semidirect
17. C2×#14558724C22semidirect
19. C2×#14558719C22semidirect
24. #18011882C22semidirect
26. C4×Dic298C22semidirect
30. #18011843C22semidirect
31. C4⋊#11924746C22semidirect
32. #18011836C22semidirect
34. #18011832C22semidirect
35. #18011832C22semidirect
36. #18011823C22semidirect
37. C2×D8×C149C22semidirect
38. C2×D8×C149C22semidirect
40. #127×C149C22semidirect
41. C4×C8×C149C22semidirect
58. #14558725D4semidirect
89. Dic298C2×D4semidirect
120. C8×C149C2×D4semidirect
136. C4×C149AGL(1,ℤ/8)semidirect
150. C149#3274semidirect
155. #493D149semidirect
170. C2×D8C2×D149semidirect
171. C2×D8C2×D149semidirect
175. #127C2×D149semidirect
192. C4×C8C2×D149semidirect
271. C8D4×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
#237502602
#237498752
#237498222
#237498172
#237497672
#237497572
C4×#145587252
#237497042
#493×C1492
#3274149