Finite Group Databasebeta

← #34058344

Subgroups — #34058344

Order 19072.

1536 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18011958C211yesmaximal, characteristic, complemented
2C2×#41×D149C211yesmaximal, characteristic, complemented
3#23750295C213yesmaximal, complemented
4#23750294C211yesmaximal, characteristic, complemented
5#23750293C213yesmaximal, complemented
6#23750287C211yesmaximal, characteristic, complemented
7#23750284C213yesmaximal, complemented
8#23750282C211yesmaximal, characteristic, complemented
9#23750281C213yesmaximal, complemented
10#23750191C213yesmaximal, complemented
11#23750188C213yesmaximal, complemented
12#23750135C213yesmaximal, complemented
13#23750134C213yesmaximal, complemented
14#23750101C211yesmaximal, characteristic, complemented
15#550×C149C211yesmaximal, characteristic, complemented
16#18011958—28no—
17C2×#14558737C2211yescharacteristic
18C2×#14558736C2213yes—
19C2×#14558736C2213yes—
20C2×#14558735C2213yes—
21C2×#14558735C2213yes—
22C2×#14558735C2213yescomplemented
23#41×D149—28no—
24C22×Dic298C2213yescomplemented
25C2×Q8×D149C2211yescharacteristic
26C2×Q8×D149C2211yescharacteristic, complemented
27C2×D4×D149C2213yes—
28C22×C4×D149C2213yes—
29#18011907C2213yescomplemented
30#18011907C2216yescomplemented
31#18011899C2213yescomplemented
32#18011898C2213yescomplemented
33#18011897C2216yescomplemented
34#18011896C2211yescharacteristic
35#18011893C2213yes—
36#18011891C2213yescomplemented
37#18011890C2213yescomplemented
38#18011889C2213yescomplemented
39#18011888C2213yescomplemented
40#18011886C2213yescomplemented
41#18011885C2213yescomplemented
42#18011884C2213yescomplemented
43#18011883C2216yescomplemented
44#18011883C2216yescomplemented
45#18011883C2213yescomplemented
46#18011882C2213yescomplemented
47#18011882C2213yescomplemented
48#18011881C2213yes—
49#18011879C2213yes—
50#18011879C2216yes—
51Q8×Dic149C2211yescharacteristic, complemented
52C4×Dic298C2213yescomplemented
53#18011876C2213yescomplemented
54C4×D596C2213yescomplemented
55D4×Dic149C2216yescomplemented
56C4×C22⋊D149C2213yescomplemented
57C4×C22⋊D149C2213yescomplemented
58#18011871C2216yescomplemented
59#18011869C2213yes—
60#18011868C2213yes—
61C2×#41×C149C2211yescharacteristic
62C2×#41×C149C2211yescharacteristic, complemented
63#127×C149C2213yescomplemented
64#121×C149C2213yescomplemented
65#121×C149C2213yescomplemented
66C4×Q8×C149C2211yescharacteristic, complemented
67C4×D4×C149C2213yescomplemented
68C23×D149C2313yes—
69#14558737—24no—
70#14558736—212no—
71#14558736—212no—
72#14558735—212no—
73#14558735—212no—
74#14558735—212no—
75C2×Dic298—212no—
76C2×Dic298C2313yes—
77C2×Dic298C2313yes—
78C2×Dic298C2313yes—
79Q8×D149—24no—
80Q8×D149D418yescomplemented
81C2×D596C2313yes—
82C2×D596C2311yescharacteristic
83C2×D596C2313yes—
84C2×C22⋊D149C2313yes—
85C2×C22⋊D149C2316yes—
86C2×C22⋊D149C2313yes—
87C2×C22⋊D149C2316yes—
88C2×C22⋊D149C2313yes—
89C2×C22⋊D149C2313yes—
90D4×D149—212no—
91C22×Dic149C2313yes—
92C22×Dic149C2316yes—
93C2×C4×D149—212no—
94C2×C4×D149C2311yescharacteristic
95C2×C4×D149C2313yes—
96C2×C4×D149C2313yes—
97C2×C4×D149C2313yes—
98C2×C4×D149C2313yes—
99C4⋊Dic149C2313yes—
100C4⋊Dic149C2313yes—
101#14558715C2313yes—
102#14558715C2313yes—
103#14558715C2313yes—
104#14558715C2311yescharacteristic
105C22⋊Dic149C2313yes—
106C22⋊Dic149C2313yes—
107C22⋊Dic149C2316yes—
108#14558713C2311yescharacteristic
109#14558713C2313yes—
110#14558713C2316yes—
111#14558713C2311yescharacteristic
112#14558713C2313yes—
113#14558713C2311yescharacteristic
114#14558713C2313yes—
115C4×Dic149C2313yes—
116C4×Dic149C2316yes—
117#41×C149—24no—
118#41×C149—24no—
119C2×Q8×C149C2311yescharacteristic
120C2×D4×C149C2313yes—
121C2×D4×C149C2313yes—
122C2×D4×C149C2316yes—
123C2×D4×C149C2313yes—
124C22×C4×C149C2313yes—
125C22×C4×C149C2313yes—
126C4⋊C4×C149C2311yescharacteristic
127C4⋊C4×C149C2313yes—
128C22⋊C4×C149C2313yes—
129C22⋊C4×C149C2313yes—
130C42×C149C2313yes—
131C22×D149—26no—
132C22×D149—43no—
133C22×D149—43no—
134C22×D149C2411yescharacteristic
135C22×D149C2413yes—
136C22×D149C2411yescharacteristic
137Dic298—26no—
138Dic298—26no—
139Dic298C2×D4112yes—
140D596—22no—
141D596—26no—
142D596—43no—
143C22⋊D149—26no—
144C22⋊D149—212no—
145C22⋊D149—26no—
146C22⋊D149—43no—
147C22⋊D149—46no—
148C22⋊D149—43no—
149C2×Dic149—26no—
150C2×Dic149—212no—
151C2×Dic149C2411yescharacteristic
152C2×Dic149C2413yes—
153C2×Dic149C2411yescharacteristic
154C2×Dic149C2413yes—
155C2×Dic149C2413yes—
156C4×D149—22no—
157C4×D149—26no—
158C4×D149—26no—
159C4×D149—26no—
160C4×D149C2×D4112yes—
161C23×C149C2413yes—
162C23×C149C2413yes—
163Q8×C149C2×D414yes—
164D4×C149—26no—
165D4×C149—26no—
166D4×C149—43no—
167D4×C149—46no—
168C2×C4×C149—26no—
169C2×C4×C149—26no—
170C2×C4×C149C2411yescharacteristic
171C2×C4×C149C2411yescharacteristic
172C2×C4×C149C2413yes—
173C2×C4×C149C2411yescharacteristic
174C2×C4×C149C2413yes—
175C22×C149—23no—
176C22×C149—23no—
177C22×C149—23no—
178C22×C149—23no—
179C22×C149—23no—
180C22×C149—23no—
181C22×C149C2511yescharacteristic
182C2×D149—21no—
183C2×D149—21no—
184C2×D149—21no—
185C2×D149—23no—
186C2×D149—21no—
187C2×D149—23no—
188C2×D149—23no—
189C2×D149—21no—
190C2×D149—412no—
191C2×D149C22×D412yes—
192C4×C149—21no—
193C4×C149—21no—
194C4×C149—21no—
195C4×C149—23no—
196C4×C149C22×D416yes—
197Dic149—21no—
198Dic149—23no—
199Dic149—21no—
200Dic149—23no—
201Dic149C22×D416yes—
202C2×C149—43no—
203C2×C149—43no—
204C2×C149C23×D411yescharacteristic
205C2×C149#58511yescharacteristic
206C2×C149#58511yescharacteristic
207D149—22no—
208D149—41no—
209D149—43no—
210C149#348711yescharacteristic, complemented, Sylow
211#3487—1491nomaximal, Sylow
212C2×2−1+4—1491no—
213C22×#41—1491no—
214#550—1491no—
215#550—1493no—
216#550—1493no—
217#550D14911yescharacteristic, complemented
218#548—1491no—
219#548—1493no—
220#548—1491no—
221#548—1493no—
222#535—1493no—
223#535—1493no—
224#525—1493no—
225#525—1493no—
226#519—1491no—
2272−1+4—2988no—
228C2×#41—2988no—
229C2×#41—1491no—
230C2×#41—1493no—
231C2×#41—1493no—
232C2×#41—1493no—
233C2×#41—1493no—
234C2×#41—1493no—
235C2×#41C2×D14911yescharacteristic
236C2×#41C2×D14911yescharacteristic, complemented
237C22×Q8—1491no—
238C22×Q8—1491no—
239C22×Q8—1493no—
240C22×D4—1493no—
241C23×C4—1493no—
242#127—1496no—
243#127—1493no—
244#127C2×D14913yescomplemented
245#124—1493no—
246#124—1496no—
247#124—1493no—
248#123—1491no—
249#123—1493no—
250C22⋊Q8—1493no—
251C22⋊Q8—1493no—
252C22⋊Q8—1493no—
253C22⋊Q8—1493no—
254#121—1493no—
255#121—1493no—
256#121—1493no—
257#121—1496no—
258#121—1496no—
259#121—1493no—
260#121—1493no—
261#121—1493no—
262#121C2×D14913yescomplemented
263#121C2×D14913yescomplemented
264#120—1496no—
265#120—1493no—
266#120—1493no—
267C4×Q8—1491no—
268C4×Q8—1493no—
269C4×Q8—1493no—
270C4×Q8C2×D14911yescharacteristic, complemented
271C4×D4—1496no—
272C4×D4—1493no—
273C4×D4—1493no—
274C4×D4—1496no—
275C4×D4—1493no—
276C4×D4C2×D14913yescomplemented
277#117—1493no—
278#117—1493no—
279C24—1493no—
280#41—24no—
281#41—24no—
282#41—29812no—
283#41—29812no—
284#41—29812no—
285#41—29812no—
286#41—2984no—
287#41—29812no—
288C2×Q8—29812no—
289C2×Q8—2984no—
290C2×Q8—1498no—
291C2×Q8—1493no—
292C2×Q8—1493no—
293C2×Q8—1493no—
294C2×Q8C22×D14911yescharacteristic
295C2×D4—29812no—
296C2×D4—1493no—
297C2×D4—1496no—
298C2×D4—1493no—
299C2×D4—1496no—
300C2×D4—1493no—
301C2×D4—1493no—
302C2×D4—1491no—
303C2×D4—1493no—
304C2×D4—1493no—
305C2×D4C22×D14913yes—
306C2×D4C22×D14913yes—
307C2×D4C22×D14916yes—
308C2×D4C22×D14913yes—
309C22×C4—29812no—
310C22×C4—1496no—
311C22×C4—1493no—
312C22×C4—1493no—
313C22×C4—1493no—
314C22×C4—1493no—
315C22×C4—1491no—
316C22×C4—1493no—
317C22×C4C22×D14913yes—
318C22×C4C22×D14913yes—
319C4⋊C4—1493no—
320C4⋊C4—1493no—
321C4⋊C4—1493no—
322C4⋊C4—1493no—
323C4⋊C4—1491no—
324C4⋊C4—1493no—
325C4⋊C4C22×D14913yes—
326C4⋊C4C22×D14911yescharacteristic
327C22⋊C4—1493no—
328C22⋊C4—1496no—
329C22⋊C4—1493no—
330C22⋊C4—1493no—
331C22⋊C4—1491no—
332C22⋊C4—1491no—
333C22⋊C4—1491no—
334C22⋊C4—1493no—
335C22⋊C4—1496no—
336C22⋊C4—1493no—
337C22⋊C4C22×D14913yes—
338C22⋊C4C22×D14913yes—
339C42—1493no—
340C42—1496no—
341C42C22×D14913yes—
342C23—2986no—
343C23—5963no—
344C23—5963no—
345C23—1493no—
346C23—1491no—
347C23—1491no—
348C23C23×D14913yes—
349C23C23×D14913yes—
350Q8—2986no—
351Q8—2986no—
352Q8—14912no—
353Q8D4×D14914yes—
354D4—26no—
355D4—26no—
356D4—46no—
357D4—43no—
358D4—2982no—
359D4—2986no—
360D4—29812no—
361D4—2986no—
362D4—2986no—
363D4—5963no—
364D4—5963no—
365D4—5966no—
366D4—5963no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—2982no—
370C2×C4—2986no—
371C2×C4—2986no—
372C2×C4—29812no—
373C2×C4—2986no—
374C2×C4—2986no—
375C2×C4—1491no—
376C2×C4—1493no—
377C2×C4—1493no—
378C2×C4—1491no—
379C2×C4—1493no—
380C2×C4—14912no—
381C2×C4C23×D14913yes—
382C2×C4C23×D14911yescharacteristic
383C2×C4C23×D14911yescharacteristic
384C2×C4C23×D14911yescharacteristic
385C2×C4C23×D14913yes—
386C22—23no—
387C22—23no—
388C22—23no—
389C22—23no—
390C22—23no—
391C22—23no—
392C22—2981no—
393C22—2981no—
394C22—2981no—
395C22—2983no—
396C22—2981no—
397C22—2983no—
398C22—2981no—
399C22—2983no—
400C22—59612no—
401C22—1492no—
402C22C24×D14911yescharacteristic
403C4—21no—
404C4—21no—
405C4—23no—
406C4—21no—
407C4—2981no—
408C4—2983no—
409C4—2981no—
410C4—2983no—
411C4—1496no—
412C4C2×D4×D14916yes—
413C2—43no—
414C2—43no—
415C2—2982no—
416C2—5963no—
417C2—5961no—
418C2C22×D4×D14911yescharacteristic
419C2#2375046011yescharacteristic
420C2#2375046111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D149
commutator, socle, cosocleC22×C149C25
Fitting#550×C149C2
hypercenter#550D149
hypocenterC149#3487
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#18011958C2semidirect
2. C2×#41×D149C2semidirect
3. #23750295C2semidirect
4. #23750294C2semidirect
5. #23750293C2semidirect
6. #23750287C2semidirect
7. #23750284C2semidirect
8. #23750282C2semidirect
9. #23750281C2semidirect
10. #23750191C2semidirect
11. #23750188C2semidirect
12. #23750135C2semidirect
13. #23750134C2semidirect
14. #23750101C2semidirect
15. #550×C149C2semidirect
22. C2×#14558735C22semidirect
24. C22×Dic298C22semidirect
26. C2×Q8×D149C22semidirect
29. #18011907C22semidirect
30. #18011907C22semidirect
31. #18011899C22semidirect
32. #18011898C22semidirect
33. #18011897C22semidirect
36. #18011891C22semidirect
37. #18011890C22semidirect
38. #18011889C22semidirect
39. #18011888C22semidirect
40. #18011886C22semidirect
41. #18011885C22semidirect
42. #18011884C22semidirect
43. #18011883C22semidirect
44. #18011883C22semidirect
45. #18011883C22semidirect
46. #18011882C22semidirect
47. #18011882C22semidirect
51. Q8×Dic149C22semidirect
52. C4×Dic298C22semidirect
53. #18011876C22semidirect
54. C4×D596C22semidirect
55. D4×Dic149C22semidirect
56. C4×C22⋊D149C22semidirect
57. C4×C22⋊D149C22semidirect
58. #18011871C22semidirect
62. C2×#41×C149C22semidirect
63. #127×C149C22semidirect
64. #121×C149C22semidirect
65. #121×C149C22semidirect
66. C4×Q8×C149C22semidirect
67. C4×D4×C149C22semidirect
80. Q8×D149D4semidirect
210. C149#3487semidirect
217. #550D149semidirect
236. C2×#41C2×D149semidirect
244. #127C2×D149semidirect
262. #121C2×D149semidirect
263. #121C2×D149semidirect
270. C4×Q8C2×D149semidirect
276. C4×D4C2×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119582
C2×#41×D1492
#237502952
#237502942
#237502932
#237502872
#237502842
#237502822
#237502812
#237501912
#237501882
#237501352
#237501342
#237501012
#550×C1492
#3487149