Finite Group Databasebeta

← #34058359

Subgroups — #34058359

Order 19072.

1658 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18011957C211yesmaximal, characteristic, complemented
2C2×#41×D149C211yesmaximal, characteristic, complemented
3#23750294C211yesmaximal, characteristic, complemented
4#23750287C211yesmaximal, characteristic, complemented
5#23750282C211yesmaximal, characteristic, complemented
6#23750272C213yesmaximal, complemented
7#23750271C213yesmaximal, complemented
8#23750260C213yesmaximal, complemented
9#23750259C213yesmaximal, complemented
10#23750204C213yesmaximal, complemented
11#23750188C213yesmaximal, complemented
12#23750134C213yesmaximal, complemented
13#121×D149C213yesmaximal, complemented
14#23750107C211yesmaximal, characteristic, complemented
15#550×C149C211yesmaximal, characteristic, complemented
16#18011957—28no—
17C2×#14558737C2211yescharacteristic
18C2×#14558737C2211yescharacteristic, complemented
19C2×#14558736C2213yescomplemented
20C2×#14558736C2213yescomplemented
21C2×#14558735C2213yescomplemented
22#41×D149—28no—
23C2×Q8×D149C2211yescharacteristic, complemented
24C22×D596C2213yescomplemented
25C2×D4×D149C2213yescomplemented
26C2×D4×D149C2213yescomplemented
27C2×D4×D149C2213yescomplemented
28C2×D4×D149C2216yescomplemented
29C22×C4×D149C2213yescomplemented
30#18011907C2213yescomplemented
31#18011899C2213yescomplemented
32#18011898C2213yescomplemented
33#18011896C2213yescomplemented
34#18011893C2213yescomplemented
35#18011891C2213yescomplemented
36#18011890C2213yescomplemented
37#18011889C2211yescharacteristic, complemented
38#18011888C2213yescomplemented
39#18011886C2213yescomplemented
40#18011885C2213yescomplemented
41#18011884C2216yescomplemented
42#18011884C2213yescomplemented
43#18011883C2213yescomplemented
44#18011883C2216yescomplemented
45#18011882C2216yescomplemented
46#18011882C2213yescomplemented
47#18011882C2213yescomplemented
48#18011881C2213yescomplemented
49#18011879C2216yescomplemented
50#18011879C2216yescomplemented
51#18011879C2213yescomplemented
52Q8×Dic149C2211yescharacteristic, complemented
53C4×D596C2213yescomplemented
54C4×D596C2213yescomplemented
55#18011873C2213yescomplemented
56C4×C22⋊D149C2213yescomplemented
57C4×C22⋊D149C2213yescomplemented
58#18011868C2213yescomplemented
59C4⋊C4×D149C2213yescomplemented
60C22⋊C4×D149C2216yescomplemented
61C2×#41×C149C2211yescharacteristic, complemented
62C2×#41×C149C2211yescharacteristic, complemented
63#127×C149C2213yescomplemented
64#121×C149C2213yescomplemented
65#121×C149C2213yescomplemented
66C4×Q8×C149C2211yescharacteristic, complemented
67C4×D4×C149C2213yescomplemented
68C23×D149C2313yes—
69C23×D149C2316yes—
70#14558737—24no—
71#14558737D418yescomplemented
72#14558736—212no—
73#14558736—212no—
74#14558735—212no—
75C2×Dic298C2313yes—
76C2×Dic298C2311yescharacteristic
77Q8×D149—24no—
78C2×D596—212no—
79C2×D596C2313yes—
80C2×D596C2313yes—
81C2×D596C2313yes—
82C2×D596C2313yes—
83C2×C22⋊D149C2313yes—
84C2×C22⋊D149C2313yes—
85C2×C22⋊D149C2313yes—
86C2×C22⋊D149C2316yes—
87C2×C22⋊D149C2313yes—
88C2×C22⋊D149C2316yes—
89D4×D149—212no—
90D4×D149—212no—
91D4×D149—212no—
92D4×D149—412no—
93C22×Dic149C2313yes—
94C2×C4×D149—212no—
95C2×C4×D149C2311yescharacteristic
96C2×C4×D149C2313yes—
97C2×C4×D149C2313yes—
98C2×C4×D149C2313yes—
99C2×C4×D149C2316yes—
100C2×C4×D149C2313yes—
101C4⋊Dic149C2313yes—
102C4⋊Dic149C2313yes—
103#14558715C2313yes—
104#14558715C2311yescharacteristic
105#14558715C2311yescharacteristic
106#14558715C2313yes—
107C22⋊Dic149C2316yes—
108C22⋊Dic149C2313yes—
109C22⋊Dic149C2313yes—
110#14558713C2313yes—
111#14558713C2316yes—
112#14558713C2313yes—
113#14558713C2313yes—
114#14558713C2311yescharacteristic
115#14558713C2311yescharacteristic
116#14558713C2313yes—
117C4×Dic149C2313yes—
118#41×C149—24no—
119#41×C149—24no—
120C2×Q8×C149C2311yescharacteristic
121C2×D4×C149C2313yes—
122C2×D4×C149C2313yes—
123C2×D4×C149C2316yes—
124C2×D4×C149C2313yes—
125C22×C4×C149C2313yes—
126C22×C4×C149C2313yes—
127C4⋊C4×C149C2311yescharacteristic
128C4⋊C4×C149C2313yes—
129C22⋊C4×C149C2313yes—
130C22⋊C4×C149C2313yes—
131C42×C149C2313yes—
132C22×D149—26no—
133C22×D149—26no—
134C22×D149—212no—
135C22×D149—43no—
136C22×D149—46no—
137C22×D149—46no—
138C22×D149C2413yes—
139C22×D149C2413yes—
140C22×D149C2411yescharacteristic
141Dic298—26no—
142Dic298—22no—
143D596—26no—
144D596—26no—
145D596—26no—
146D596C2×D4112yescomplemented
147C22⋊D149—26no—
148C22⋊D149—212no—
149C22⋊D149—26no—
150C22⋊D149—26no—
151C22⋊D149—46no—
152C22⋊D149—43no—
153C2×Dic149—26no—
154C2×Dic149C2411yescharacteristic
155C2×Dic149C2413yes—
156C2×Dic149C2413yes—
157C2×Dic149C2411yescharacteristic
158C2×Dic149C2411yescharacteristic
159C4×D149—26no—
160C4×D149—26no—
161C4×D149—26no—
162C4×D149—22no—
163C4×D149—46no—
164C4×D149C2×D4112yescomplemented
165C23×C149C2413yes—
166C23×C149C2413yes—
167Q8×C149C2×D414yescomplemented
168D4×C149—26no—
169D4×C149—26no—
170D4×C149—46no—
171D4×C149—43no—
172C2×C4×C149—26no—
173C2×C4×C149—26no—
174C2×C4×C149C2411yescharacteristic
175C2×C4×C149C2413yes—
176C2×C4×C149C2411yescharacteristic
177C2×C4×C149C2411yescharacteristic
178C2×C4×C149C2413yes—
179C22×C149—23no—
180C22×C149—23no—
181C22×C149—23no—
182C22×C149—23no—
183C22×C149—23no—
184C22×C149—23no—
185C22×C149C2511yescharacteristic
186C2×D149—21no—
187C2×D149—23no—
188C2×D149—21no—
189C2×D149—23no—
190C2×D149—23no—
191C2×D149—23no—
192C2×D149—23no—
193C2×D149—412no—
194C2×D149—812no—
195C2×D149C22×D416yes—
196C2×D149H(5,𝔽2)12yescomplemented
197C4×C149—21no—
198C4×C149—21no—
199C4×C149—23no—
200C4×C149—21no—
201C4×C149C22×D416yes—
202Dic149—21no—
203Dic149—23no—
204Dic149—23no—
205Dic149—21no—
206Dic149C22×D412yes—
207C2×C149—43no—
208C2×C149—43no—
209C2×C149C23×D411yescharacteristic
210C2×C149C2×H(5,𝔽2)11yescharacteristic
211C2×C149#58511yescharacteristic
212D149—22no—
213D149—26no—
214D149—43no—
215C149#348811yescharacteristic, complemented, Sylow
216#3488—1491nomaximal, Sylow
217C2×H(5,𝔽2)—1491no—
218C22×#41—1491no—
219#550—1491no—
220#550D14911yescharacteristic, complemented
221#548—1491no—
222#548—1491no—
223#546—1493no—
224#546—1493no—
225D42—1493no—
226D42—1493no—
227#537—1493no—
228#535—1493no—
229#525—1493no—
230C2×#121—1493no—
231#520—1491no—
232H(5,𝔽2)—2988no—
233C2×#41—2988no—
234C2×#41—1491no—
235C2×#41—1491no—
236C2×#41—1493no—
237C2×#41—1493no—
238C2×#41—1493no—
239C2×#41C2×D14911yescharacteristic, complemented
240C2×#41C2×D14911yescharacteristic, complemented
241C22×Q8—1491no—
242C22×D4—1493no—
243C22×D4—1493no—
244C22×D4—1493no—
245C22×D4—1496no—
246C22×D4—1493no—
247C23×C4—1493no—
248#127—1493no—
249#127C2×D14913yescomplemented
250#124—1493no—
251#124—1493no—
252#123—1493no—
253#123—1493no—
254C22⋊Q8—1493no—
255C22⋊Q8—1491no—
256C22⋊Q8—1493no—
257C22⋊Q8—1493no—
258#121—1496no—
259#121—1493no—
260#121—1493no—
261#121—1496no—
262#121—1493no—
263#121—1493no—
264#121—1493no—
265#121—1493no—
266#121—1496no—
267#121C2×D14913yescomplemented
268#121C2×D14913yescomplemented
269#120—1493no—
270#120—1496no—
271#120—1496no—
272#120—1493no—
273C4×Q8—1491no—
274C4×Q8C2×D14911yescharacteristic, complemented
275C4×D4—1493no—
276C4×D4—1493no—
277C4×D4—1493no—
278C4×D4—1493no—
279C4×D4—1493no—
280C4×D4C2×D14913yescomplemented
281#117—1493no—
282C2×C4⋊C4—1493no—
283C2×C22⋊C4—1496no—
284C24—1493no—
285C24—1496no—
286#41—24no—
287#41—24no—
288#41—29812no—
289#41—29812no—
290#41—2984no—
291#41—29812no—
292#41—1498no—
293C2×Q8—2984no—
294C2×Q8—1491no—
295C2×Q8—1493no—
296C2×Q8C22×D14911yescharacteristic
297C2×D4—29812no—
298C2×D4—29812no—
299C2×D4—29812no—
300C2×D4—29812no—
301C2×D4—59612no—
302C2×D4—1493no—
303C2×D4—1496no—
304C2×D4—1493no—
305C2×D4—1493no—
306C2×D4—1493no—
307C2×D4—1496no—
308C2×D4—1493no—
309C2×D4—1493no—
310C2×D4—1493no—
311C2×D4—1493no—
312C2×D4C22×D14913yes—
313C2×D4C22×D14913yes—
314C2×D4C22×D14913yes—
315C2×D4C22×D14916yes—
316C22×C4—29812no—
317C22×C4—1493no—
318C22×C4—1493no—
319C22×C4—1493no—
320C22×C4—1493no—
321C22×C4—1491no—
322C22×C4—1496no—
323C22×C4—1493no—
324C22×C4C22×D14913yes—
325C22×C4C22×D14913yes—
326C4⋊C4—1493no—
327C4⋊C4—1493no—
328C4⋊C4—1491no—
329C4⋊C4—1491no—
330C4⋊C4—1493no—
331C4⋊C4—1493no—
332C4⋊C4C22×D14913yes—
333C4⋊C4C22×D14911yescharacteristic
334C22⋊C4—1496no—
335C22⋊C4—1493no—
336C22⋊C4—1493no—
337C22⋊C4—1493no—
338C22⋊C4—1496no—
339C22⋊C4—1493no—
340C22⋊C4—1493no—
341C22⋊C4—1491no—
342C22⋊C4—1491no—
343C22⋊C4—1493no—
344C22⋊C4C22×D14913yes—
345C22⋊C4C22×D14913yes—
346C42—1493no—
347C42C22×D14913yes—
348C23—2986no—
349C23—2986no—
350C23—29812no—
351C23—5963no—
352C23—5966no—
353C23—5966no—
354C23—1493no—
355C23—1491no—
356C23—1493no—
357C23C23×D14913yes—
358C23C23×D14913yes—
359Q8—2982no—
360Q8—2986no—
361Q8D4×D14914yescomplemented
362D4—26no—
363D4—26no—
364D4—46no—
365D4—43no—
366D4—2986no—
367D4—2986no—
368D4—2986no—
369D4—29812no—
370D4—2986no—
371D4—2986no—
372D4—2986no—
373D4—5966no—
374D4—5963no—
375D4—14912no—
376C2×C4—26no—
377C2×C4—26no—
378C2×C4—2986no—
379C2×C4—2982no—
380C2×C4—2986no—
381C2×C4—2986no—
382C2×C4—2986no—
383C2×C4—5966no—
384C2×C4—1491no—
385C2×C4—1493no—
386C2×C4—1493no—
387C2×C4—1491no—
388C2×C4—1491no—
389C2×C4—14912no—
390C2×C4C23×D14911yescharacteristic
391C2×C4C23×D14913yes—
392C2×C4C23×D14911yescharacteristic
393C2×C4C23×D14911yescharacteristic
394C2×C4C23×D14913yes—
395C22—23no—
396C22—23no—
397C22—23no—
398C22—23no—
399C22—23no—
400C22—23no—
401C22—2981no—
402C22—2983no—
403C22—2981no—
404C22—2983no—
405C22—2983no—
406C22—2983no—
407C22—2983no—
408C22—59612no—
409C22—1496no—
410C22—1492no—
411C22—119212no—
412C22C24×D14911yescharacteristic
413C4—21no—
414C4—21no—
415C4—23no—
416C4—21no—
417C4—2981no—
418C4—2983no—
419C4—2983no—
420C4—2981no—
421C4—1492no—
422C4C2×D4×D14916yes—
423C2—43no—
424C2—43no—
425C2—2982no—
426C2—2986no—
427C2—5963no—
428C2C22×D4×D14911yescharacteristic
429C2H(5,𝔽2)×D14911yescharacteristic
430C2#2375046111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D149
commutator, socle, cosocleC22×C149C25
Fitting#550×C149C2
hypercenter#550D149
hypocenterC149#3488
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#18011957C2semidirect
2. C2×#41×D149C2semidirect
3. #23750294C2semidirect
4. #23750287C2semidirect
5. #23750282C2semidirect
6. #23750272C2semidirect
7. #23750271C2semidirect
8. #23750260C2semidirect
9. #23750259C2semidirect
10. #23750204C2semidirect
11. #23750188C2semidirect
12. #23750134C2semidirect
13. #121×D149C2semidirect
14. #23750107C2semidirect
15. #550×C149C2semidirect
18. C2×#14558737C22semidirect
19. C2×#14558736C22semidirect
20. C2×#14558736C22semidirect
21. C2×#14558735C22semidirect
23. C2×Q8×D149C22semidirect
24. C22×D596C22semidirect
25. C2×D4×D149C22semidirect
26. C2×D4×D149C22semidirect
27. C2×D4×D149C22semidirect
28. C2×D4×D149C22semidirect
29. C22×C4×D149C22semidirect
30. #18011907C22semidirect
31. #18011899C22semidirect
32. #18011898C22semidirect
33. #18011896C22semidirect
34. #18011893C22semidirect
35. #18011891C22semidirect
36. #18011890C22semidirect
37. #18011889C22semidirect
38. #18011888C22semidirect
39. #18011886C22semidirect
40. #18011885C22semidirect
41. #18011884C22semidirect
42. #18011884C22semidirect
43. #18011883C22semidirect
44. #18011883C22semidirect
45. #18011882C22semidirect
46. #18011882C22semidirect
47. #18011882C22semidirect
48. #18011881C22semidirect
49. #18011879C22semidirect
50. #18011879C22semidirect
51. #18011879C22semidirect
52. Q8×Dic149C22semidirect
53. C4×D596C22semidirect
54. C4×D596C22semidirect
55. #18011873C22semidirect
56. C4×C22⋊D149C22semidirect
57. C4×C22⋊D149C22semidirect
58. #18011868C22semidirect
59. C4⋊C4×D149C22semidirect
60. C22⋊C4×D149C22semidirect
61. C2×#41×C149C22semidirect
62. C2×#41×C149C22semidirect
63. #127×C149C22semidirect
64. #121×C149C22semidirect
65. #121×C149C22semidirect
66. C4×Q8×C149C22semidirect
67. C4×D4×C149C22semidirect
71. #14558737D4semidirect
146. D596C2×D4semidirect
164. C4×D149C2×D4semidirect
167. Q8×C149C2×D4semidirect
196. C2×D149H(5,𝔽2)semidirect
215. C149#3488semidirect
220. #550D149semidirect
239. C2×#41C2×D149semidirect
240. C2×#41C2×D149semidirect
249. #127C2×D149semidirect
267. #121C2×D149semidirect
268. #121C2×D149semidirect
274. C4×Q8C2×D149semidirect
280. C4×D4C2×D149semidirect
361. Q8D4×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119572
C2×#41×D1492
#237502942
#237502872
#237502822
#237502722
#237502712
#237502602
#237502592
#237502042
#237501882
#237501342
#121×D1492
#237501072
#550×C1492
#3488149