Finite Group Databasebeta

← #34056292

Subgroups — #34056292

Order 19072.

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

normal subgroups only

This group has 546 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×#18011935C212yesmaximal, complemented
2#23750279C211yesmaximal, characteristic, complemented
3D4×D596C211yesmaximal, characteristic, complemented
4#23749998C211yesmaximal, characteristic, complemented
5#23749872C212yesmaximal, complemented
6#23749830C211yesmaximal, characteristic, complemented
7#23749820C211yesmaximal, characteristic, complemented
8#23749777C212yesmaximal, complemented
9#23749771C212yesmaximal, complemented
10#23749704C211yesmaximal, characteristic, complemented
11C4×D8×C149C211yesmaximal, characteristic, complemented
12C2×#14558735C2212yescomplemented
13C22×D596—21no—
14C2×D4×D149C2212yes—
15#18011935—28no—
16C2×#14558725C2212yescomplemented
17C2×#14558725C2212yescomplemented
18C2×#14558724C2212yes—
19C2×#14558721C2212yescomplemented
20C2×#14558719C2212yescomplemented
21#18011911C2211yescharacteristic, complemented
22#18011908C2211yescharacteristic
23#18011895—22no—
24#18011891C2212yescomplemented
25#18011886C2212yes—
26#18011885—21no—
27#18011885—21no—
28#18011881—22no—
29C4×D596C2211yescharacteristic
30C2×C4⋊Dic149—21no—
31C4⋊#11924746C2211yescharacteristic, complemented
32#18011834C2212yescomplemented
33#18011831C2212yescomplemented
34#18011823C2212yescomplemented
35C2×D8×C149C2211yescharacteristic
36C4×D4×C149C2211yescharacteristic
37C4×D4×C149C2211yescharacteristic, complemented
38#107×C149C2211yescharacteristic, complemented
39#102×C149C2211yescharacteristic
40#102×C149C2211yescharacteristic, complemented
41C4×C8×C149C2211yescharacteristic, complemented
42C23×D149—22no—
43#14558735—24no—
44#14558735—42no—
45C2×Dic298C2312yes—
46C2×D596—22no—
47C2×D596—21no—
48C2×D596—21no—
49C2×D596—44no—
50C2×D596D411yescharacteristic
51C2×D596C2312yes—
52C2×C22⋊D149—22no—
53C2×C22⋊D149—22no—
54D4×D149—24no—
55D4×D149—42no—
56C22×Dic149—22no—
57C2×C4×D149C2312yes—
58#14558725—24no—
59#14558725—42no—
60#14558724—24no—
61#14558721—24no—
62#14558719—24no—
63C2×#11924746C2312yes—
64C4⋊Dic149—22no—
65C4⋊Dic149—21no—
66C4⋊Dic149—21no—
67C4⋊Dic149D411yescharacteristic
68#14558713—22no—
69#14558713—22no—
70#14558713D412yes—
71C2×D4×C149C2311yescharacteristic
72C2×D4×C149C2311yescharacteristic
73C22×C4×C149—21no—
74C22×C4×C149—21no—
75D8×C149D414yescomplemented
76C2×C8×C149C2311yescharacteristic
77C2×C8×C149C2311yescharacteristic
78C4⋊C4×C149C2311yescharacteristic
79C4⋊C4×C149C2311yescharacteristic
80C22⋊C4×C149—21no—
81C22⋊C4×C149—21no—
82C42×C149C2311yescharacteristic
83C22×D149—22no—
84C22×D149—44no—
85C22×D149—44no—
86C22×D149—82no—
87C22×D149C2×D412yes—
88Dic298—22no—
89Dic298—22no—
90D596—22no—
91D596—22no—
92D596—22no—
93D596—44no—
94D596—42no—
95D596—81no—
96C22⋊D149—44no—
97C22⋊D149—44no—
98C2×Dic149—22no—
99C2×Dic149—44no—
100C2×Dic149C2×D412yes—
101C4×D149—22no—
102C4×D149—22no—
103#11924746—22no—
104C23×C149—21no—
105C23×C149—21no—
106D4×C149—21no—
107D4×C149—21no—
108D4×C149C2×D412yes—
109D4×C149C2×D412yescomplemented
110C2×C4×C149—21no—
111C2×C4×C149—21no—
112C2×C4×C149—42no—
113C2×C4×C149—42no—
114C2×C4×C149C2×D411yescharacteristic
115C2×C4×C149C2×D411yescharacteristic
116C2×C4×C149C2411yescharacteristic
117C8×C149—21no—
118C8×C149C2×D412yescomplemented
119C22×C149—22no—
120C22×C149—22no—
121C22×C149—41no—
122C22×C149—41no—
123C22×C149—41no—
124C22×C149—41no—
125C22×C149C22×D411yescharacteristic
126C2×D149—22no—
127C2×D149—22no—
128C2×D149—22no—
129C2×D149—42no—
130C2×D149—42no—
131C2×D149—42no—
132C2×D149—88no—
133C4×C149—21no—
134C4×C149—41no—
135C4×C149—41no—
136C4×C149AGL(1,ℤ/8)12yescomplemented
137C4×C149C22×D411yescharacteristic
138C4×C149H(5,𝔽2)11yescharacteristic
139Dic149—22no—
140Dic149—42no—
141C2×C149—42no—
142C2×C149—42no—
143C2×C149D4211yescharacteristic
144C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C149#57711yescharacteristic
146D149—42no—
147D149—82no—
148C149#327211yescharacteristic, complemented, Sylow
149#3272—1491nomaximal, Sylow
150C2×AGL(1,ℤ/8)—1492no—
151#547—1491no—
152D42—1491no—
153#492—1491no—
154#468—1492no—
155#461—1491no—
156#459—1491no—
157#450—1492no—
158#449—1492no—
159C4×D8D14911yescharacteristic, complemented
160#436—1491no—
161C2×#41—1492no—
162C22×D4—2981no—
163C22×D4—1492no—
164AGL(1,ℤ/8)—2988no—
165C2×SD16—1492no—
166C2×SD16—1492no—
167C2×SD16—1492no—
168C2×D8—1492no—
169C2×D8C2×D14911yescharacteristic
170C2×M16—1492no—
171C4⋊Q8—1491no—
172#127—1491no—
173#123—2982no—
174C22⋊Q8—1492no—
175#121—2981no—
176#121—2981no—
177#121—1492no—
178#120—2982no—
179C4×D4—1491no—
180C4×D4C2×D14911yescharacteristic
181C4×D4C2×D14911yescharacteristic, complemented
182C2×C4⋊C4—2981no—
183#107C2×D14911yescharacteristic, complemented
184C4⋊C8—1491no—
185#103—1492no—
186#102—1492no—
187#102C2×D14911yescharacteristic
188#102C2×D14911yescharacteristic, complemented
189C22⋊C8—1492no—
190C4×C8C2×D14911yescharacteristic, complemented
191C24—2982no—
192#41—2984no—
193#41—5962no—
194C2×Q8—1492no—
195C2×D4—2982no—
196C2×D4—2984no—
197C2×D4—2982no—
198C2×D4—2982no—
199C2×D4—2981no—
200C2×D4—2981no—
201C2×D4—5964no—
202C2×D4—5962no—
203C2×D4—1491no—
204C2×D4—1492no—
205C2×D4C22×D14911yescharacteristic
206C2×D4C22×D14911yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—2982no—
210C22×C4—1492no—
211SD16—2984no—
212SD16—2984no—
213SD16—5962no—
214D8—2984no—
215D8D59614yescomplemented
216M16—2984no—
217C2×C8—1492no—
218C2×C8C22×D14911yescharacteristic
219C2×C8C22×D14911yescharacteristic
220C4⋊C4—2982no—
221C4⋊C4—2981no—
222C4⋊C4—2981no—
223C4⋊C4—1491no—
224C4⋊C4C22×D14911yescharacteristic
225C4⋊C4C22×D14911yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—21no—
228C22⋊C4—2982no—
229C22⋊C4—2982no—
230C22⋊C4—1492no—
231C42C22×D14911yescharacteristic
232C23—21no—
233C23—21no—
234C23—2982no—
235C23—5964no—
236C23—5964no—
237C23—1492no—
238C23—11922no—
239Q8—2982no—
240Q8—2982no—
241D4—21no—
242D4—21no—
243D4—2982no—
244D4—2982no—
245D4—2982no—
246D4—5964no—
247D4—5964no—
248D4—5964no—
249D4—5962no—
250D4—11921no—
251D4C2×D59612yes—
252D4C2×D59612yescomplemented
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—2982no—
258C2×C4—2982no—
259C2×C4—2982no—
260C2×C4—5964no—
261C2×C4—1492no—
262C2×C4D4×D14911yescharacteristic
263C2×C4D4×D14911yescharacteristic
264C2×C4C23×D14911yescharacteristic
265C8—21no—
266C8—2982no—
267C8C2×D59612yescomplemented
268C22—22no—
269C22—22no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—2982no—
275C22—2982no—
276C22—2982no—
277C22—5962no—
278C22—5962no—
279C22—5962no—
280C22—11928no—
281C22C2×D4×D14911yescharacteristic
282C4—21no—
283C4—41no—
284C4—41no—
285C4—2982no—
286C4—5962no—
287C4#1801193512yescomplemented
288C4C22×D59611yescharacteristic
289C4#1801195711yescharacteristic
290C2—42no—
291C2—42no—
292C2—5962no—
293C2—11922no—
294C2D4×D59611yescharacteristic
295C2C2×#1801193511yescharacteristic
296C2#2375043211yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
FittingC4×D8×C149C2
socleC22×C149C22×D4
hypercenterC4×D8D149
hypocenterC149#3272
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#18011935C2semidirect
2. #23750279C2semidirect
3. D4×D596C2semidirect
4. #23749998C2semidirect
5. #23749872C2semidirect
6. #23749830C2semidirect
7. #23749820C2semidirect
8. #23749777C2semidirect
9. #23749771C2semidirect
10. #23749704C2semidirect
11. C4×D8×C149C2semidirect
12. C2×#14558735C22semidirect
16. C2×#14558725C22semidirect
17. C2×#14558725C22semidirect
19. C2×#14558721C22semidirect
20. C2×#14558719C22semidirect
21. #18011911C22semidirect
24. #18011891C22semidirect
31. C4⋊#11924746C22semidirect
32. #18011834C22semidirect
33. #18011831C22semidirect
34. #18011823C22semidirect
37. C4×D4×C149C22semidirect
38. #107×C149C22semidirect
40. #102×C149C22semidirect
41. C4×C8×C149C22semidirect
75. D8×C149D4semidirect
109. D4×C149C2×D4semidirect
118. C8×C149C2×D4semidirect
136. C4×C149AGL(1,ℤ/8)semidirect
148. C149#3272semidirect
159. C4×D8D149semidirect
181. C4×D4C2×D149semidirect
183. #107C2×D149semidirect
188. #102C2×D149semidirect
190. C4×C8C2×D149semidirect
215. D8D596semidirect
252. D4C2×D596semidirect
267. C8C2×D596semidirect
287. C4#18011935semidirect

Maximal subgroups

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

subgroupindex
C2×#180119352
#237502792
D4×D5962
#237499982
#237498722
#237498302
#237498202
#237497772
#237497712
#237497042
C4×D8×C1492
#3272149