Finite Group Databasebeta

← #34053957

Subgroups — #34053957

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×#18011957C211yesmaximal, characteristic, complemented
2C2×#18011937C211yesmaximal, characteristic, complemented
3C22×#14558721C211yesmaximal, characteristic, complemented
4#23749768C214yesmaximal, complemented
5#23749758C214yesmaximal, complemented
6#23749629C211yesmaximal, characteristic, complemented
7#23749619C211yesmaximal, characteristic, complemented
8#23749592C211yesmaximal, characteristic, complemented
9#535×C149C211yesmaximal, characteristic, complemented
10#18011957—24no—
11#18011957—24no—
12C2×#14558737—21no—
13C2×#14558736C2211yescharacteristic
14C2×#14558736C2212yescomplemented
15C22×D596C2211yescharacteristic
16C22×D596C2212yescomplemented
17C2×D4×D149—21no—
18C2×D4×D149—22no—
19#18011937—24no—
20C2×#14558724C2212yescomplemented
21C2×#14558721—24no—
22C2×#14558721C2212yescomplemented
23C2×#14558721C2214yescomplemented
24C2×#14558720C2211yescharacteristic, complemented
25C22×#11924746C2211yescharacteristic, complemented
26#18011834C2212yescomplemented
27#18011831C2212yescomplemented
28#18011831C2214yescomplemented
29C22⋊#11924746C2214yescomplemented
30C2×#41×C149C2211yescharacteristic
31C22×D4×C149C2211yescharacteristic, complemented
32#127×C149—21no—
33#124×C149—21no—
34#121×C149C2214yescomplemented
35#120×C149—22no—
36#117×C149C2211yescharacteristic, complemented
37C23×D149—21no—
38C23×D149—22no—
39#14558737—28no—
40#14558736—24no—
41#14558736—24no—
42#14558736—22no—
43#14558736D414yescomplemented
44C2×Dic298C2311yescharacteristic
45C2×D596—24no—
46C2×D596—24no—
47C2×D596—21no—
48C2×D596—22no—
49C2×D596—22no—
50C2×D596D414yescomplemented
51C2×D596C2312yes—
52C2×D596C2311yescharacteristic
53C2×C22⋊D149—21no—
54C2×C22⋊D149—22no—
55D4×D149—216no—
56D4×D149—28no—
57C2×C4×D149—21no—
58C2×C4×D149—22no—
59#14558724—24no—
60#14558721—28no—
61#14558721—24no—
62#14558720—22no—
63C2×#11924746—22no—
64C2×#11924746C2312yes—
65C2×#11924746C2312yes—
66C24×C149—21no—
67#41×C149—22no—
68#41×C149—22no—
69C2×Q8×C149D411yescharacteristic
70C2×D4×C149—24no—
71C2×D4×C149—22no—
72C2×D4×C149—22no—
73C2×D4×C149—21no—
74C2×D4×C149D412yes—
75C2×D4×C149D411yescharacteristic
76C2×D4×C149C2312yes—
77C22×C4×C149C2312yes—
78C22×C4×C149C2311yescharacteristic
79C4⋊C4×C149C2312yes—
80C22⋊C4×C149—21no—
81C22⋊C4×C149—24no—
82C42×C149—21no—
83C22×D149—21no—
84C22×D149—22no—
85C22×D149—28no—
86C22×D149—24no—
87C22×D149—82no—
88C22×D149—84no—
89Dic298—21no—
90Dic298C2×D412yes—
91D596—28no—
92D596—24no—
93D596—22no—
94D596—21no—
95D596C2×D412yes—
96D596C2×D414yes—
97C22⋊D149—28no—
98C22⋊D149—24no—
99C2×Dic149—21no—
100C4×D149—28no—
101C4×D149—24no—
102#11924746—22no—
103#11924746—22no—
104C23×C149—22no—
105C23×C149—42no—
106C23×C149—82no—
107C23×C149C2×D412yes—
108C23×C149C2×D411yescharacteristic
109Q8×C149—22no—
110D4×C149—24no—
111D4×C149—22no—
112D4×C149—22no—
113D4×C149—22no—
114D4×C149—42no—
115D4×C149—84no—
116C2×C4×C149—22no—
117C2×C4×C149—22no—
118C2×C4×C149—22no—
119C2×C4×C149C2×D412yes—
120C2×C4×C149C2×D412yes—
121C2×C4×C149C2×D412yes—
122C2×C4×C149C2×D411yescharacteristic
123C2×C4×C149C2411yescharacteristic
124C22×C149—22no—
125C22×C149—22no—
126C22×C149—24no—
127C22×C149—42no—
128C22×C149—44no—
129C22×C149—84no—
130C22×C149#12012yes—
131C22×C149C22×D411yescharacteristic
132C2×D149—24no—
133C2×D149—22no—
134C2×D149—44no—
135C2×D149—42no—
136C2×D149—84no—
137C2×D149—88no—
138C4×C149—22no—
139C4×C149—42no—
140C4×C149#12012yes—
141C4×C149C22×D411yescharacteristic
142C4×C149C22×D411yescharacteristic
143Dic149—22no—
144C2×C149—22no—
145C2×C149—42no—
146C2×C149—82no—
147C2×C149C2×#12011yescharacteristic
148C2×C149#57612yes—
149D149—44no—
150D149—42no—
151C149#300811yescharacteristic, complemented, Sylow
152#3008—1491nomaximal, Sylow
153C2×H(5,𝔽2)—1491no—
154C2×AGL(1,ℤ/8)—1491no—
155C22×D8—1491no—
156#535D14911yescharacteristic, complemented
157#449—1494no—
158#447—1494no—
159#418—1491no—
160#416—1491no—
161#408—1491no—
162H(5,𝔽2)—2984no—
163H(5,𝔽2)—2984no—
164C2×#41—2981no—
165C2×#41—1492no—
166C2×#41—1491no—
167C2×#41C2×D14911yescharacteristic
168C22×D4—2982no—
169C22×D4—2981no—
170C22×D4—1492no—
171C22×D4—1491no—
172C22×D4C2×D14911yescharacteristic, complemented
173AGL(1,ℤ/8)—2984no—
174C2×SD16—1492no—
175C2×D8—2984no—
176C2×D8—1494no—
177C2×D8—1492no—
178C2×M16—1491no—
179C22×C8—1491no—
180#127—21no—
181#124—21no—
182#121C2×D14914yescomplemented
183#120—22no—
184#117C2×D14911yescharacteristic, complemented
185#103—1492no—
186#102—1492no—
187#102—1494no—
188C22⋊C8—1494no—
189C24—21no—
190C24—2982no—
191C24—2981no—
192#41—22no—
193#41—22no—
194#41—2984no—
195#41—2984no—
196#41—2988no—
197#41—2982no—
198#41—1494no—
199C2×Q8—1491no—
200C2×Q8C22⋊D14911yescharacteristic
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—2984no—
206C2×D4—2984no—
207C2×D4—29816no—
208C2×D4—2988no—
209C2×D4—2982no—
210C2×D4—2981no—
211C2×D4—2982no—
212C2×D4—2982no—
213C2×D4—2981no—
214C2×D4—1494no—
215C2×D4—1491no—
216C2×D4—1492no—
217C2×D4C22⋊D14912yes—
218C2×D4C22⋊D14911yescharacteristic
219C2×D4C22×D14912yes—
220C22×C4—2982no—
221C22×C4—2981no—
222C22×C4C22×D14912yes—
223C22×C4C22×D14911yescharacteristic
224SD16—2984no—
225D8—2988no—
226D8—2984no—
227M16—2982no—
228C2×C8—2982no—
229C2×C8—1492no—
230C2×C8—1492no—
231C4⋊C4C22×D14912yes—
232C22⋊C4—21no—
233C22⋊C4—24no—
234C42—21no—
235C23—22no—
236C23—42no—
237C23—82no—
238C23—2988no—
239C23—2982no—
240C23—2981no—
241C23—2984no—
242C23—11922no—
243C23—11924no—
244C23C2×C22⋊D14912yes—
245C23C2×C22⋊D14911yescharacteristic
246Q8—22no—
247Q8—2981no—
248Q8—1492no—
249D4—24no—
250D4—22no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—84no—
255D4—2988no—
256D4—2988no—
257D4—2984no—
258D4—2982no—
259D4—2981no—
260D4—2984no—
261D4—1492no—
262D4—1494no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—2988no—
267C2×C4—2981no—
268C2×C4—2984no—
269C2×C4D4×D14912yes—
270C2×C4D4×D14912yes—
271C2×C4C2×C22⋊D14912yes—
272C2×C4C2×C22⋊D14911yescharacteristic
273C2×C4C23×D14911yescharacteristic
274C8—2982no—
275C8—2982no—
276C22—22no—
277C22—24no—
278C22—22no—
279C22—44no—
280C22—42no—
281C22—84no—
282C22—2984no—
283C22—2982no—
284C22—5964no—
285C22—5962no—
286C22—11924no—
287C22—11928no—
288C22#1801187912yes—
289C22C22×C22⋊D14911yescharacteristic
290C4—22no—
291C4—42no—
292C4—2982no—
293C4#1801187912yes—
294C4C2×D4×D14911yescharacteristic
295C4C2×D4×D14911yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—5964no—
300C2—5962no—
301C2C2×#1801187911yescharacteristic
302C2#2375042812yes—
subgroupG/H
centerC22C22×C22⋊D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#535×C149C2
socleC22×C149C22×D4
hypercenter#535D149
hypocenterC149#3008
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#18011957C2semidirect
2. C2×#18011937C2semidirect
3. C22×#14558721C2semidirect
4. #23749768C2semidirect
5. #23749758C2semidirect
6. #23749629C2semidirect
7. #23749619C2semidirect
8. #23749592C2semidirect
9. #535×C149C2semidirect
14. C2×#14558736C22semidirect
16. C22×D596C22semidirect
20. C2×#14558724C22semidirect
22. C2×#14558721C22semidirect
23. C2×#14558721C22semidirect
24. C2×#14558720C22semidirect
25. C22×#11924746C22semidirect
26. #18011834C22semidirect
27. #18011831C22semidirect
28. #18011831C22semidirect
29. C22⋊#11924746C22semidirect
31. C22×D4×C149C22semidirect
34. #121×C149C22semidirect
36. #117×C149C22semidirect
43. #14558736D4semidirect
50. C2×D596D4semidirect
151. C149#3008semidirect
156. #535D149semidirect
172. C22×D4C2×D149semidirect
182. #121C2×D149semidirect
184. #117C2×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119572
C2×#180119372
C22×#145587212
#237497682
#237497582
#237496292
#237496192
#237495922
#535×C1492
#3008149