Finite Group Databasebeta

← #34058387

Subgroups — #34058387

Order 19072.

1052 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750312C214yesmaximal, complemented
2#23750310C214yesmaximal, complemented
3#23750301C214yesmaximal, complemented
4#23750298C214yesmaximal, complemented
5#23750251C211yesmaximal, characteristic, complemented
6#23750219C214yesmaximal, complemented
7#23750184C211yesmaximal, characteristic
8#23750152C214yesmaximal, complemented
9C2×#18011894C211yesmaximal, characteristic, complemented
10#23750100C211yesmaximal, characteristic, complemented
11#23750098C211yesmaximal, characteristic, complemented
12C2×#18011870C211yesmaximal, characteristic, complemented
13C2×#123×C149C211yesmaximal, characteristic, complemented
14C22×C22⋊D149C2211yescharacteristic
15#18011906C2214yescomplemented
16#18011906C2214yescomplemented
17#18011904C2214yescomplemented
18#18011904C2214yescomplemented
19#18011902C2214yescomplemented
20#18011901C2214yescomplemented
21#18011901C2214yescomplemented
22#18011897C2214yescomplemented
23#18011895C2214yescomplemented
24#18011894C2212yes—
25#18011894C2214yes—
26#18011894C2212yescomplemented
27#18011894C2212yescomplemented
28#18011893C2214yescomplemented
29#18011892C2212yes—
30#18011891C2214yescomplemented
31#18011890C2214yescomplemented
32#18011888C2214yescomplemented
33#18011888C2214yescomplemented
34#18011888C2214yescomplemented
35#18011883C2214yes—
36#18011880C2212yes—
37#18011879C2212yes—
38#18011876C2214yescomplemented
39D4×Dic149C2214yes—
40D4×Dic149C2214yescomplemented
41C4×C22⋊D149C2214yes—
42#18011871C2214yescomplemented
43#18011871C2214yescomplemented
44#18011870C2214yescomplemented
45#18011870C2214yescomplemented
46#18011870C2212yescomplemented
47#18011869C2214yescomplemented
48#18011867C2214yescomplemented
49#18011867C2212yescomplemented
50C2×C4⋊Dic149C2212yescomplemented
51C2×#14558715C2212yescomplemented
52C2×C22⋊Dic149C2211yescharacteristic
53C2×C22⋊Dic149C2211yescharacteristic
54C2×C22⋊Dic149C2211yescharacteristic
55C2×C22⋊Dic149C2212yescomplemented
56C2×#14558713C2212yescomplemented
57C22⋊C4×D149C2212yescomplemented
58C2×C4×Dic149C2212yescomplemented
59C22×D4×C149C2211yescharacteristic
60C23×C4×C149C2211yescharacteristic
61#123×C149C2214yes—
62#123×C149C2212yescomplemented
63#123×C149C2212yescomplemented
64C2×C4⋊C4×C149C2212yescomplemented
65C2×C22⋊C4×C149C2211yescharacteristic
66C2×C22⋊C4×C149C2212yescomplemented
67C23×D149C2311yescharacteristic
68C2×Dic298C2314yes—
69C2×C22⋊D149—24no—
70C2×C22⋊D149C2312yes—
71C2×C22⋊D149C2312yes—
72C2×C22⋊D149C2314yes—
73C22×Dic149C2311yescharacteristic
74C22×Dic149C2312yes—
75C22×Dic149C2312yes—
76C22×Dic149C2312yes—
77C22×Dic149C2314yes—
78C2×C4×D149C2314yes—
79C4⋊Dic149C2314yes—
80C4⋊Dic149C2314yes—
81C4⋊Dic149C2314yes—
82C4⋊Dic149C2314yes—
83#14558715C2314yes—
84#14558715C2314yes—
85#14558715C2314yes—
86#14558715C2314yes—
87#14558715C2314yes—
88#14558715C2314yes—
89C22⋊Dic149C2314yes—
90C22⋊Dic149C2314yes—
91C22⋊Dic149C2312yes—
92C22⋊Dic149C2312yes—
93C22⋊Dic149C2312yes—
94C22⋊Dic149C2312yes—
95C22⋊Dic149C2312yes—
96C22⋊Dic149C2312yes—
97#14558713C2314yes—
98#14558713C2314yes—
99#14558713C2314yes—
100#14558713C2314yes—
101C4×Dic149C2314yes—
102C4×Dic149C2314yes—
103C4×Dic149C2314yes—
104C4×Dic149C2314yes—
105C24×C149C2311yescharacteristic
106C24×C149C2311yescharacteristic
107C2×D4×C149—24no—
108C2×D4×C149C2312yes—
109C2×D4×C149C2312yes—
110C22×C4×C149—24no—
111C22×C4×C149C2312yes—
112C22×C4×C149C2311yescharacteristic
113C22×C4×C149C2312yes—
114C22×C4×C149C2311yescharacteristic
115C22×C4×C149C2311yescharacteristic
116C22×C4×C149C2312yes—
117C4⋊C4×C149C2314yes—
118C4⋊C4×C149C2314yes—
119C22⋊C4×C149C2314yes—
120C22⋊C4×C149C2314yes—
121C22⋊C4×C149C2312yes—
122C22⋊C4×C149C2312yes—
123C22×D149—22no—
124C22×D149—42no—
125C22×D149C2412yes—
126Dic298—44no—
127C22⋊D149—24no—
128C22⋊D149—24no—
129C22⋊D149—44no—
130C2×Dic149—22no—
131C2×Dic149—42no—
132C2×Dic149—44no—
133C2×Dic149—42no—
134C2×Dic149—42no—
135C2×Dic149C2414yes—
136C2×Dic149C2414yes—
137C2×Dic149C2412yes—
138C2×Dic149C2414yes—
139C4×D149—44no—
140C23×C149—22no—
141C23×C149—42no—
142C23×C149—42no—
143C23×C149#4114yes—
144C23×C149C2412yes—
145C23×C149C2411yescharacteristic
146C23×C149C2411yescharacteristic
147C23×C149C2411yescharacteristic
148D4×C149—24no—
149D4×C149—24no—
150C2×C4×C149—24no—
151C2×C4×C149—24no—
152C2×C4×C149—22no—
153C2×C4×C149—22no—
154C2×C4×C149—22no—
155C2×C4×C149—42no—
156C2×C4×C149C2414yes—
157C2×C4×C149C2412yes—
158C2×C4×C149C2412yes—
159C2×C4×C149C2412yes—
160C22×C149—22no—
161C22×C149—22no—
162C22×C149—22no—
163C22×C149—22no—
164C22×C149—24no—
165C22×C149—44no—
166C22×C149—82no—
167C22×C149C2×#4112yes—
168C22×C149C2×#4112yes—
169C22×C149C2×#4112yes—
170C22×C149C2511yescharacteristic
171C2×D149—22no—
172C2×D149—24no—
173C2×D149—82no—
174C4×C149—22no—
175C4×C149—22no—
176C4×C149—22no—
177C4×C149—24no—
178Dic149—22no—
179Dic149—24no—
180Dic149—24no—
181Dic149—24no—
182C2×C149—22no—
183C2×C149—22no—
184C2×C149—22no—
185C2×C149—42no—
186C2×C149C22×#4111yescharacteristic
187C2×C149#58512yes—
188D149—42no—
189C149#349111yescharacteristic, complemented, Sylow
190#3491—1491nomaximal, Sylow
191#552—1494no—
192#552—1494no—
193#551—1494no—
194#551—1494no—
195#543—1491no—
196#539—1494no—
197#534—1491no—
198#529—1494no—
199C2×#123—1491no—
200C2×#123D14911yescharacteristic, complemented
201#518—1491no—
202#518—1491no—
203C2×#117—1491no—
204C22×D4—1491no—
205C22×D4C2×D14911yescharacteristic
206C23×C4C2×D14911yescharacteristic
207#126—1494no—
208#126—1494no—
209#126—1494no—
210#126—1494no—
211#125—1494no—
212#125—1494no—
213#125—1494no—
214#124—1494no—
215#123—1492no—
216#123—1492no—
217#123—1492no—
218#123—1494no—
219#123—1492no—
220#123—1494no—
221#123—1494no—
222#123C2×D14914yes—
223#123C2×D14912yescomplemented
224#123C2×D14912yescomplemented
225C22⋊Q8—1494no—
226C22⋊Q8—1494no—
227C22⋊Q8—1494no—
228C22⋊Q8—1494no—
229C22⋊Q8—1494no—
230#121—1494no—
231#120—1492no—
232#120—1492no—
233C4×Q8—1494no—
234C4×D4—1494no—
235C4×D4—1494no—
236C4×D4—1494no—
237C4×D4—1494no—
238C4×D4—1494no—
239#117—1494no—
240#117—1494no—
241#117—1492no—
242#117—1494no—
243#117—1492no—
244#117—1494no—
245C2×C4⋊C4—1492no—
246C2×C4⋊C4—1492no—
247C2×C4⋊C4C2×D14912yescomplemented
248C2×C22⋊C4—1491no—
249C2×C22⋊C4—1491no—
250C2×C22⋊C4—1491no—
251C2×C22⋊C4—1492no—
252C2×C22⋊C4—1492no—
253C2×C22⋊C4—1492no—
254C2×C22⋊C4C2×D14911yescharacteristic
255C2×C22⋊C4C2×D14912yescomplemented
256C2×C42—1492no—
257C24—1491no—
258C24C22×D14911yescharacteristic
259C24C22×D14911yescharacteristic
260C2×Q8—1494no—
261C2×D4—24no—
262C2×D4—2984no—
263C2×D4—1492no—
264C2×D4—1492no—
265C2×D4—1494no—
266C2×D4C22×D14912yes—
267C2×D4C22×D14912yes—
268C22×C4—24no—
269C22×C4—1491no—
270C22×C4—1492no—
271C22×C4—1492no—
272C22×C4—1492no—
273C22×C4—1494no—
274C22×C4—1494no—
275C22×C4C22×D14911yescharacteristic
276C22×C4C22×D14912yes—
277C22×C4C22×D14912yes—
278C22×C4C22×D14911yescharacteristic
279C22×C4C22×D14911yescharacteristic
280C22×C4C22×D14912yes—
281C4⋊C4—1494no—
282C4⋊C4—1494no—
283C4⋊C4—1494no—
284C4⋊C4—1494no—
285C4⋊C4—1494no—
286C4⋊C4—1494no—
287C4⋊C4—1494no—
288C4⋊C4—1494no—
289C4⋊C4—1494no—
290C4⋊C4—1494no—
291C4⋊C4C22×D14914yes—
292C4⋊C4C22×D14914yes—
293C22⋊C4—1492no—
294C22⋊C4—1492no—
295C22⋊C4—1492no—
296C22⋊C4—1492no—
297C22⋊C4—1492no—
298C22⋊C4—1492no—
299C22⋊C4—1494no—
300C22⋊C4—1494no—
301C22⋊C4—1494no—
302C22⋊C4—1494no—
303C22⋊C4—1494no—
304C22⋊C4—1494no—
305C22⋊C4C22×D14912yes—
306C22⋊C4C22×D14912yes—
307C22⋊C4C22×D14914yes—
308C22⋊C4C22×D14914yes—
309C42—1494no—
310C42—1494no—
311C42—1494no—
312C42—1494no—
313C23—22no—
314C23—42no—
315C23—42no—
316C23—2982no—
317C23—5962no—
318C23—1492no—
319C23#1455873514yes—
320C23C23×D14912yes—
321C23C23×D14911yescharacteristic
322C23C23×D14911yescharacteristic
323C23C23×D14911yescharacteristic
324Q8—5964no—
325D4—24no—
326D4—24no—
327D4—2984no—
328D4—2984no—
329D4—5964no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—42no—
336C2×C4—2982no—
337C2×C4—5964no—
338C2×C4—5962no—
339C2×C4—5964no—
340C2×C4—5962no—
341C2×C4—5962no—
342C2×C4—1494no—
343C2×C4—1494no—
344C2×C4—1492no—
345C2×C4—1494no—
346C2×C4C23×D14912yes—
347C2×C4C23×D14912yes—
348C2×C4C23×D14914yes—
349C2×C4C23×D14912yes—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—24no—
355C22—44no—
356C22—82no—
357C22—2982no—
358C22—2984no—
359C22—11922no—
360C22C2×#1455873512yes—
361C22C2×#1455873512yes—
362C22C2×#1455873512yes—
363C22C24×D14911yescharacteristic
364C4—22no—
365C4—22no—
366C4—22no—
367C4—24no—
368C4—2982no—
369C4—2984no—
370C4—2984no—
371C4—2984no—
372C2—22no—
373C2—22no—
374C2—22no—
375C2—42no—
376C2—5962no—
377C2C22×#1455873511yescharacteristic
378C2#2375046112yes—
subgroupG/H
center, FrattiniC22C24×D149
commutator, socle, cosocleC22×C149C25
FittingC2×#123×C149C2
prime radical#23750184C2
hypercenterC2×#123D149
hypocenterC149#3491
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. #23750312C2semidirect
2. #23750310C2semidirect
3. #23750301C2semidirect
4. #23750298C2semidirect
5. #23750251C2semidirect
6. #23750219C2semidirect
8. #23750152C2semidirect
9. C2×#18011894C2semidirect
10. #23750100C2semidirect
11. #23750098C2semidirect
12. C2×#18011870C2semidirect
13. C2×#123×C149C2semidirect
15. #18011906C22semidirect
16. #18011906C22semidirect
17. #18011904C22semidirect
18. #18011904C22semidirect
19. #18011902C22semidirect
20. #18011901C22semidirect
21. #18011901C22semidirect
22. #18011897C22semidirect
23. #18011895C22semidirect
26. #18011894C22semidirect
27. #18011894C22semidirect
28. #18011893C22semidirect
30. #18011891C22semidirect
31. #18011890C22semidirect
32. #18011888C22semidirect
33. #18011888C22semidirect
34. #18011888C22semidirect
38. #18011876C22semidirect
40. D4×Dic149C22semidirect
42. #18011871C22semidirect
43. #18011871C22semidirect
44. #18011870C22semidirect
45. #18011870C22semidirect
46. #18011870C22semidirect
47. #18011869C22semidirect
48. #18011867C22semidirect
49. #18011867C22semidirect
50. C2×C4⋊Dic149C22semidirect
51. C2×#14558715C22semidirect
55. C2×C22⋊Dic149C22semidirect
56. C2×#14558713C22semidirect
57. C22⋊C4×D149C22semidirect
58. C2×C4×Dic149C22semidirect
62. #123×C149C22semidirect
63. #123×C149C22semidirect
64. C2×C4⋊C4×C149C22semidirect
66. C2×C22⋊C4×C149C22semidirect
189. C149#3491semidirect
200. C2×#123D149semidirect
223. #123C2×D149semidirect
224. #123C2×D149semidirect
247. C2×C4⋊C4C2×D149semidirect
255. C2×C22⋊C4C2×D149semidirect

Maximal subgroups

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

subgroupindex
#237503122
#237503102
#237503012
#237502982
#237502512
#237502192
#237501842
#237501522
C2×#180118942
#237501002
#237500982
C2×#180118702
C2×#123×C1492
#3491149