Finite Group Databasebeta

← #34058097

Subgroups — #34058097

Order 19072.

1160 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750251C211yesmaximal, characteristic, complemented
2#23750220C212yesmaximal, complemented
3#23750219C214yesmaximal, complemented
4#23750212C212yesmaximal, complemented
5#23750210C214yesmaximal, complemented
6#23750184C212yesmaximal, complemented
7#23750156C212yesmaximal, complemented
8#23750152C214yesmaximal, complemented
9#23750135C212yesmaximal, complemented
10#23750131C214yesmaximal, complemented
11C2×D4×Dic149C212yesmaximal, complemented
12C2×#18011871C211yesmaximal, characteristic, complemented
13#534×C149C211yesmaximal, characteristic, complemented
14C2×#14558735C2212yescomplemented
15C22×C22⋊D149C2211yescharacteristic
16C23×Dic149C2211yescharacteristic
17C23×Dic149C2212yescomplemented
18C22×C4×D149C2211yescharacteristic, complemented
19#18011906C2214yescomplemented
20#18011904C2214yescomplemented
21#18011904C2214yescomplemented
22#18011902C2212yescomplemented
23#18011901C2214yescomplemented
24#18011898C2212yescomplemented
25#18011897C2214yescomplemented
26#18011895C2214yescomplemented
27#18011894C2214yescomplemented
28#18011893C2214yescomplemented
29#18011892C2214yescomplemented
30#18011892C2214yescomplemented
31#18011891C2212yescomplemented
32#18011890C2212yescomplemented
33#18011890C2212yescomplemented
34#18011888C2214yescomplemented
35#18011888C2214yescomplemented
36#18011888C2214yescomplemented
37#18011883C2214yes—
38#18011883C2214yes—
39#18011883C2214yescomplemented
40#18011882C2212yescomplemented
41#18011880C2214yescomplemented
42#18011879C2214yescomplemented
43#18011876C2212yescomplemented
44D4×Dic149C2214yescomplemented
45D4×Dic149C2214yescomplemented
46D4×Dic149C2212yescomplemented
47D4×Dic149C2214yescomplemented
48D4×Dic149C2214yescomplemented
49#18011871C2214yescomplemented
50#18011871C2214yescomplemented
51#18011871C2214yescomplemented
52#18011869C2212yescomplemented
53#18011867C2214yescomplemented
54C2×C4⋊Dic149C2212yescomplemented
55C2×#14558715C2211yescharacteristic, complemented
56C2×C22⋊Dic149C2212yes—
57C2×C22⋊Dic149C2212yescomplemented
58C2×#14558713C2211yescharacteristic, complemented
59C2×C4×Dic149C2212yescomplemented
60C2×C4×Dic149C2211yescharacteristic, complemented
61C22×D4×C149C2212yes—
62#123×C149C2214yescomplemented
63#121×C149C2212yes—
64#121×C149C2212yes—
65#120×C149C2214yescomplemented
66C2×C22⋊C4×C149C2211yescharacteristic
67C23×D149C2311yescharacteristic
68#14558735—44no—
69C2×Dic298C2312yes—
70C2×Dic298C2312yes—
71C2×C22⋊D149—24no—
72C2×C22⋊D149C2314yes—
73C2×C22⋊D149C2314yes—
74C2×C22⋊D149C2314yes—
75C22×Dic149—42no—
76C22×Dic149—44no—
77C22×Dic149C2311yescharacteristic
78C22×Dic149C2312yes—
79C22×Dic149C2314yes—
80C22×Dic149C2311yescharacteristic
81C22×Dic149C2311yescharacteristic
82C22×Dic149C2312yes—
83C22×Dic149C2312yes—
84C22×Dic149C2314yes—
85C22×Dic149C2312yes—
86C2×C4×D149—42no—
87C2×C4×D149C2312yes—
88C2×C4×D149C2312yes—
89C4⋊Dic149C2312yes—
90C4⋊Dic149C2314yes—
91C4⋊Dic149C2314yes—
92C4⋊Dic149C2312yes—
93#14558715C2314yes—
94#14558715C2314yes—
95#14558715C2314yes—
96#14558715C2314yes—
97#14558715C2314yes—
98C22⋊Dic149C2314yes—
99C22⋊Dic149C2314yes—
100C22⋊Dic149C2314yes—
101C22⋊Dic149C2314yes—
102C22⋊Dic149C2314yes—
103C22⋊Dic149C2314yes—
104#14558713C2314yes—
105#14558713C2314yes—
106#14558713C2314yes—
107C4×Dic149C2312yes—
108C4×Dic149C2314yes—
109C4×Dic149C2312yes—
110C4×Dic149C2314yes—
111C24×C149C2312yes—
112C24×C149C2311yescharacteristic
113C2×D4×C149—28no—
114C2×D4×C149C2314yes—
115C2×D4×C149C2314yes—
116C2×D4×C149C2312yes—
117C2×D4×C149C2312yes—
118C22×C4×C149C2312yes—
119C22×C4×C149C2311yescharacteristic
120C22×C4×C149C2311yescharacteristic
121C4⋊C4×C149C2312yes—
122C4⋊C4×C149C2312yes—
123C22⋊C4×C149C2314yes—
124C22⋊C4×C149C2314yes—
125C22⋊C4×C149C2314yes—
126C22×D149—22no—
127C22×D149—41no—
128C22×D149—41no—
129C22×D149C2412yes—
130Dic298—42no—
131Dic298—42no—
132C22⋊D149—28no—
133C22⋊D149—44no—
134C22⋊D149—44no—
135C2×Dic149—22no—
136C2×Dic149—22no—
137C2×Dic149—24no—
138C2×Dic149—28no—
139C2×Dic149—42no—
140C2×Dic149—41no—
141C2×Dic149—44no—
142C2×Dic149—42no—
143C2×Dic149—42no—
144C2×Dic149C2411yescharacteristic
145C2×Dic149C2412yes—
146C2×Dic149C2414yes—
147C2×Dic149C2412yes—
148C2×Dic149C2414yes—
149C2×Dic149C2411yescharacteristic
150C4×D149—24no—
151C4×D149—42no—
152C23×C149—24no—
153C23×C149—41no—
154C23×C149—42no—
155C23×C149—41no—
156C23×C149—42no—
157C23×C149#4114yescomplemented
158C23×C149C2414yes—
159C23×C149C2412yes—
160C23×C149C2411yescharacteristic
161D4×C149—28no—
162D4×C149—28no—
163D4×C149—42no—
164D4×C149—42no—
165C2×C4×C149—24no—
166C2×C4×C149—41no—
167C2×C4×C149—41no—
168C2×C4×C149C2412yes—
169C2×C4×C149C2412yes—
170C2×C4×C149C2414yes—
171C22×C149—21no—
172C22×C149—22no—
173C22×C149—21no—
174C22×C149—22no—
175C22×C149—24no—
176C22×C149—24no—
177C22×C149—24no—
178C22×C149—44no—
179C22×C149—48no—
180C22×C149C2×#4114yes—
181C22×C149C2×#4112yes—
182C22×C149C2511yescharacteristic
183C2×D149—22no—
184C2×D149—22no—
185C2×D149—22no—
186C2×D149—44no—
187C4×C149—24no—
188C4×C149—22no—
189C4×C149—22no—
190Dic149—22no—
191Dic149—21no—
192Dic149—22no—
193Dic149—24no—
194Dic149—24no—
195Dic149H(5,𝔽2)12yes—
196C2×C149—22no—
197C2×C149—24no—
198C2×C149—44no—
199C2×C149C22×#4111yescharacteristic
200C2×C149C2×H(5,𝔽2)11yescharacteristic
201C2×C149#58511yescharacteristic
202D149—42no—
203C149#345911yescharacteristic, complemented, Sylow
204#3459—1491nomaximal, Sylow
205#543—1491no—
206#539—1494no—
207#539—1492no—
208#538—1492no—
209#538—1494no—
210#534—1492no—
211#534D14911yescharacteristic, complemented
212#529—1494no—
213#529—1492no—
214#525—1492no—
215#525—1494no—
216C2×C4×D4—1492no—
217C2×C4×D4—1491no—
218C2×#41—1492no—
219C22×D4—1491no—
220C22×D4C2×D14912yes—
221C23×C4—1491no—
222C23×C4—1492no—
223C23×C4—1491no—
224#126—1494no—
225#126—1494no—
226#126—1494no—
227#125—1494no—
228#125—1492no—
229#124—1492no—
230#124—1494no—
231#123—1494no—
232#123—1494no—
233#123—1494no—
234#123—1494no—
235#123—1494no—
236#123C2×D14914yescomplemented
237C22⋊Q8—1492no—
238C22⋊Q8—1492no—
239C22⋊Q8—1492no—
240C22⋊Q8—1494no—
241C22⋊Q8—1494no—
242C22⋊Q8—1494no—
243#121—1494no—
244#121—1492no—
245#121—1494no—
246#121—1494no—
247#121C2×D14912yes—
248#121C2×D14912yes—
249#120—1494no—
250#120—1494no—
251#120C2×D14914yescomplemented
252C4×Q8—1492no—
253C4×D4—1494no—
254C4×D4—1494no—
255C4×D4—1494no—
256C4×D4—1494no—
257C4×D4—1492no—
258C4×D4—1494no—
259C4×D4—1494no—
260C4×D4—1494no—
261#117—1492no—
262#117—1494no—
263C2×C4⋊C4—1491no—
264C2×C4⋊C4—1492no—
265C2×C22⋊C4—1492no—
266C2×C22⋊C4—1492no—
267C2×C22⋊C4—1491no—
268C2×C22⋊C4C2×D14911yescharacteristic
269C2×C42—1491no—
270C2×C42—1492no—
271C24—1491no—
272C24C22×D14912yes—
273C24C22×D14911yescharacteristic
274#41—5964no—
275C2×Q8—1492no—
276C2×Q8—1492no—
277C2×D4—28no—
278C2×D4—2984no—
279C2×D4—1494no—
280C2×D4—1494no—
281C2×D4—1494no—
282C2×D4C22×D14914yes—
283C2×D4C22×D14914yes—
284C2×D4C22×D14912yes—
285C2×D4C22×D14912yes—
286C22×C4—5962no—
287C22×C4—5962no—
288C22×C4—5964no—
289C22×C4—1491no—
290C22×C4—1491no—
291C22×C4—1492no—
292C22×C4—1494no—
293C22×C4—1492no—
294C22×C4—1492no—
295C22×C4—1491no—
296C22×C4—1492no—
297C22×C4—1492no—
298C22×C4—1494no—
299C22×C4—1492no—
300C22×C4C22×D14912yes—
301C22×C4C22×D14911yescharacteristic
302C22×C4C22×D14911yescharacteristic
303C4⋊C4—1494no—
304C4⋊C4—1494no—
305C4⋊C4—1494no—
306C4⋊C4—1494no—
307C4⋊C4—1492no—
308C4⋊C4—1494no—
309C4⋊C4—1494no—
310C4⋊C4—1492no—
311C4⋊C4—1494no—
312C4⋊C4C22×D14912yes—
313C4⋊C4C22×D14912yes—
314C22⋊C4—1494no—
315C22⋊C4—1494no—
316C22⋊C4—1494no—
317C22⋊C4—1494no—
318C22⋊C4—1494no—
319C22⋊C4—1494no—
320C22⋊C4—1494no—
321C22⋊C4—1494no—
322C22⋊C4—1494no—
323C22⋊C4C22×D14914yes—
324C22⋊C4C22×D14914yes—
325C22⋊C4C22×D14914yes—
326C42—1494no—
327C42—1492no—
328C42—1494no—
329C42—1492no—
330C23—24no—
331C23—41no—
332C23—42no—
333C23—41no—
334C23—42no—
335C23—2982no—
336C23—5961no—
337C23—5961no—
338C23—1492no—
339C23#1455873514yescomplemented
340C23C23×D14914yes—
341C23C23×D14911yescharacteristic
342C23C23×D14912yes—
343Q8—5962no—
344Q8—5962no—
345D4—28no—
346D4—28no—
347D4—42no—
348D4—42no—
349D4—2988no—
350D4—5964no—
351D4—5964no—
352C2×C4—24no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—2984no—
356C2×C4—2982no—
357C2×C4—2982no—
358C2×C4—2984no—
359C2×C4—2988no—
360C2×C4—5962no—
361C2×C4—5961no—
362C2×C4—5962no—
363C2×C4—5964no—
364C2×C4—5962no—
365C2×C4—5962no—
366C2×C4—1494no—
367C2×C4—1492no—
368C2×C4—1491no—
369C2×C4—1492no—
370C2×C4—1494no—
371C2×C4—1491no—
372C2×C4C23×D14912yes—
373C2×C4C23×D14914yes—
374C2×C4C23×D14912yes—
375C22—21no—
376C22—22no—
377C22—21no—
378C22—22no—
379C22—24no—
380C22—24no—
381C22—24no—
382C22—44no—
383C22—48no—
384C22—2982no—
385C22—2982no—
386C22—2982no—
387C22—5964no—
388C22C2×#1455873514yes—
389C22C2×#1455873512yes—
390C22C24×D14911yescharacteristic
391C4—24no—
392C4—22no—
393C4—22no—
394C4—2982no—
395C4—2981no—
396C4—2982no—
397C4—2984no—
398C4—2984no—
399C4—1492no—
400C2—24no—
401C2—22no—
402C2—44no—
403C2—5962no—
404C2C22×#1455873511yescharacteristic
405C2H(5,𝔽2)×D14911yescharacteristic
406C2#2375046011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D149
commutator, socle, cosocleC22×C149C25
Fitting#534×C149C2
hypercenter#534D149
hypocenterC149#3459
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ #534

Complements

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

NH ≅ G/Ntype
1. #23750251C2semidirect
2. #23750220C2semidirect
3. #23750219C2semidirect
4. #23750212C2semidirect
5. #23750210C2semidirect
6. #23750184C2semidirect
7. #23750156C2semidirect
8. #23750152C2semidirect
9. #23750135C2semidirect
10. #23750131C2semidirect
11. C2×D4×Dic149C2semidirect
12. C2×#18011871C2semidirect
13. #534×C149C2semidirect
14. C2×#14558735C22semidirect
17. C23×Dic149C22semidirect
18. C22×C4×D149C22semidirect
19. #18011906C22semidirect
20. #18011904C22semidirect
21. #18011904C22semidirect
22. #18011902C22semidirect
23. #18011901C22semidirect
24. #18011898C22semidirect
25. #18011897C22semidirect
26. #18011895C22semidirect
27. #18011894C22semidirect
28. #18011893C22semidirect
29. #18011892C22semidirect
30. #18011892C22semidirect
31. #18011891C22semidirect
32. #18011890C22semidirect
33. #18011890C22semidirect
34. #18011888C22semidirect
35. #18011888C22semidirect
36. #18011888C22semidirect
39. #18011883C22semidirect
40. #18011882C22semidirect
41. #18011880C22semidirect
42. #18011879C22semidirect
43. #18011876C22semidirect
44. D4×Dic149C22semidirect
45. D4×Dic149C22semidirect
46. D4×Dic149C22semidirect
47. D4×Dic149C22semidirect
48. D4×Dic149C22semidirect
49. #18011871C22semidirect
50. #18011871C22semidirect
51. #18011871C22semidirect
52. #18011869C22semidirect
53. #18011867C22semidirect
54. C2×C4⋊Dic149C22semidirect
55. C2×#14558715C22semidirect
57. C2×C22⋊Dic149C22semidirect
58. C2×#14558713C22semidirect
59. C2×C4×Dic149C22semidirect
60. C2×C4×Dic149C22semidirect
62. #123×C149C22semidirect
65. #120×C149C22semidirect
157. C23×C149#41semidirect
203. C149#3459semidirect
211. #534D149semidirect
236. #123C2×D149semidirect
251. #120C2×D149semidirect
339. C23#14558735semidirect

Maximal subgroups

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

subgroupindex
#237502512
#237502202
#237502192
#237502122
#237502102
#237501842
#237501562
#237501522
#237501352
#237501312
C2×D4×Dic1492
C2×#180118712
#534×C1492
#3459149