Finite Group Databasebeta

← #34047667

Subgroups — #34047667

Order 19072.

928 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18011881C211yesmaximal, characteristic
2C2×#18011873C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D149C211yesmaximal, characteristic, complemented
4C22×#14558713C211yesmaximal, characteristic, complemented
5C2×C22⋊C4×D149C211yesmaximal, characteristic, complemented
6#23749526C211yesmaximal, characteristic, complemented
7#23749515C211yesmaximal, characteristic, complemented
8#23749514C211yesmaximal, characteristic, complemented
9#23749503C211yesmaximal, characteristic, complemented
10#23749503C211yesmaximal, characteristic, complemented
11#23749502C211yesmaximal, characteristic, complemented
12#23749496C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic149C211yesmaximal, characteristic, complemented
14#23749472C211yesmaximal, characteristic, complemented
15#385×C149C211yesmaximal, characteristic, complemented
16C24×D149C2211yescharacteristic
17C22×D596C2211yescharacteristic
18C22×D596C2211yescharacteristic
19C22×C22⋊D149C2211yescharacteristic
20C23×Dic149C2211yescharacteristic
21C22×C4×D149C2211yescharacteristic
22C22×C4×D149C2211yescharacteristic
23C22×C4×D149C2211yescharacteristic, complemented
24#18011881C418yescomplemented
25#18011873—24no—
26C4×C22⋊D149—24no—
27C2×#14558715C2211yescharacteristic, complemented
28C2×C22⋊Dic149C2211yescharacteristic
29C2×#14558713—24no—
30C2×#14558713C2211yescharacteristic
31C2×#14558713C2211yescharacteristic
32C2×#14558713C2211yescharacteristic
33C2×#14558713C2211yescharacteristic
34C2×#14558713C2211yescharacteristic
35C2×#14558713C2211yescharacteristic
36C2×#14558713C2211yescharacteristic
37C2×#14558713C2211yescharacteristic
38C2×#14558713C2211yescharacteristic, complemented
39C2×#14558713C2211yescharacteristic, complemented
40C2×#14558713C2211yescharacteristic, complemented
41C22⋊C4×D149—24no—
42C2×C4×Dic149C2211yescharacteristic, complemented
43C2×C4×Dic149C2211yescharacteristic, complemented
44C2×C4×Dic149C2211yescharacteristic, complemented
45#18011818C2211yescharacteristic, complemented
46#18011818C2211yescharacteristic, complemented
47#18011818C2211yescharacteristic, complemented
48#18011818C2211yescharacteristic, complemented
49C23×C4×C149C2211yescharacteristic
50C2×C4⋊C4×C149C2211yescharacteristic, complemented
51C2×C4⋊C4×C149C2211yescharacteristic, complemented
52C2×C22⋊C4×C149C2211yescharacteristic
53C2×C22⋊C4×C149C2211yescharacteristic
54#95×C149C2211yescharacteristic, complemented
55#95×C149C2211yescharacteristic, complemented
56C23×D149—22no—
57C23×D149—22no—
58C23×D149—22no—
59C23×D149—22no—
60C23×D149—41no—
61C23×D149—41no—
62C23×D149C2×C414yescomplemented
63C23×D149C2311yescharacteristic
64C23×D149C2311yescharacteristic
65C23×D149C2311yescharacteristic
66C2×D596—22no—
67C2×D596—22no—
68C2×D596—41no—
69C2×D596—41no—
70C2×D596C2×C414yescomplemented
71C2×D596C2×C414yescomplemented
72C2×C22⋊D149—22no—
73C2×C22⋊D149—22no—
74C2×C22⋊D149C2×C414yescomplemented
75C22×Dic149—22no—
76C22×Dic149—22no—
77C22×Dic149—41no—
78C22×Dic149C2311yescharacteristic
79C22×Dic149C2311yescharacteristic
80C22×Dic149C2311yescharacteristic
81C22×Dic149C2311yescharacteristic
82C22×Dic149C2311yescharacteristic
83C2×C4×D149—22no—
84C2×C4×D149—22no—
85C2×C4×D149—22no—
86C2×C4×D149—22no—
87C2×C4×D149—22no—
88C2×C4×D149—22no—
89C2×C4×D149—41no—
90C2×C4×D149—41no—
91C2×C4×D149—41no—
92#14558715—22no—
93C22⋊Dic149D414yescomplemented
94#14558713—22no—
95#14558713—22no—
96#14558713—22no—
97#14558713—22no—
98#14558713—22no—
99#14558713—22no—
100#14558713—22no—
101#14558713—41no—
102#14558713C2×C414yescomplemented
103#14558713C2×C414yescomplemented
104#14558713D414yescomplemented
105C4×Dic149—22no—
106C4×Dic149—22no—
107C4×Dic149—41no—
108C24×C149C2311yescharacteristic
109C22×C4×C149—22no—
110C22×C4×C149—22no—
111C22×C4×C149—41no—
112C22×C4×C149C2311yescharacteristic
113C22×C4×C149C2311yescharacteristic
114C22×C4×C149C2311yescharacteristic
115C22×C4×C149C2311yescharacteristic
116C22×C4×C149C2311yescharacteristic
117C22×C4×C149C2311yescharacteristic
118C4⋊C4×C149—22no—
119C4⋊C4×C149—41no—
120C22⋊C4×C149—22no—
121C22⋊C4×C149C2×C414yescomplemented
122C22×D149—21no—
123C22×D149—21no—
124C22×D149—21no—
125C22×D149—21no—
126C22×D149—28no—
127C22×D149—21no—
128C22×D149—21no—
129C22×D149—21no—
130C22×D149—21no—
131C22×D149—21no—
132C22×D149—21no—
133C22×D149—21no—
134C22×D149—21no—
135C22×D149—21no—
136C22×D149—21no—
137C22×D149—21no—
138C22×D149—21no—
139C22×D149—28no—
140C22×D149—44no—
141C22×D149—44no—
142C22×D149—44no—
143C22×D149—44no—
144C22×D149—44no—
145C22×D149C22×C412yes—
146C22×D149C22×C412yes—
147C22×D149C22×C412yes—
148C22×D149C2×D412yes—
149C22×D149#4112yes—
150D596—28no—
151D596—44no—
152C22⋊D149—28no—
153C2×Dic149—21no—
154C2×Dic149—21no—
155C2×Dic149—21no—
156C2×Dic149—21no—
157C2×Dic149—21no—
158C2×Dic149—21no—
159C2×Dic149—21no—
160C2×Dic149—21no—
161C2×Dic149—21no—
162C2×Dic149—21no—
163C2×Dic149—21no—
164C2×Dic149—44no—
165C2×Dic149C22×C412yes—
166C2×Dic149C2×D412yes—
167C2×Dic149C2×D412yes—
168C2×Dic149C2×D412yes—
169C4×D149—44no—
170C4×D149—44no—
171C4×D149—44no—
172C23×C149—21no—
173C23×C149—21no—
174C23×C149—21no—
175C23×C149—21no—
176C23×C149C22×C412yes—
177C23×C149C2×D412yes—
178C23×C149#4112yes—
179C23×C149C2411yescharacteristic
180C2×C4×C149—21no—
181C2×C4×C149—21no—
182C2×C4×C149—21no—
183C2×C4×C149—21no—
184C2×C4×C149—21no—
185C2×C4×C149—21no—
186C2×C4×C149—21no—
187C2×C4×C149—21no—
188C2×C4×C149—21no—
189C2×C4×C149—21no—
190C2×C4×C149—21no—
191C2×C4×C149—21no—
192C2×C4×C149—21no—
193C2×C4×C149—21no—
194C2×C4×C149—21no—
195C2×C4×C149—44no—
196C2×C4×C149C22×C412yes—
197C2×C4×C149C22×C412yes—
198C2×C4×C149C2×D412yes—
199C22×C149—22no—
200C22×C149—22no—
201C22×C149—22no—
202C22×C149—22no—
203C22×C149—22no—
204C22×C149—22no—
205C22×C149C4×D414yescomplemented
206C22×C149C23×C411yescharacteristic
207C22×C149C22×D411yescharacteristic
208C22×C149C22×D411yescharacteristic
209C22×C149C2×#4111yescharacteristic
210C22×C149C2×#4111yescharacteristic
211C22×C149H(5,𝔽2)11yescharacteristic
212C22×C149H(5,𝔽2)11yescharacteristic
213C2×D149—22no—
214C2×D149—22no—
215C2×D149—22no—
216C2×D149—22no—
217C2×D149—22no—
218C2×D149—22no—
219C2×D149—22no—
220C2×D149—22no—
221C2×D149—22no—
222C2×D149—22no—
223C2×D149—22no—
224C2×D149—22no—
225C2×D149—416no—
226C2×D149—41no—
227C2×D149—41no—
228C2×D149—41no—
229C2×D149—41no—
230C2×D149—41no—
231C2×D149—41no—
232C2×D149—41no—
233C2×D149—41no—
234C2×D149C4×D414yescomplemented
235C4×C149—22no—
236C4×C149—22no—
237C4×C149—22no—
238C4×C149—41no—
239C4×C149—41no—
240C4×C149—41no—
241Dic149—22no—
242Dic149—22no—
243Dic149—22no—
244Dic149—22no—
245Dic149—41no—
246C2×C149—24no—
247C2×C149C2×C4×D411yescharacteristic
248C2×C149C2×C4×D411yescharacteristic
249C2×C149#51811yescharacteristic
250C2×C149D4211yescharacteristic
251C2×C149#54611yescharacteristic
252C2×C149#54611yescharacteristic
253C2×C149#55111yescharacteristic
254D149—24no—
255D149—42no—
256D149—42no—
257C149#235811yescharacteristic, complemented, Sylow
258#2358—1491nomaximal, Sylow
259C2×#120—1491no—
260C2×C4×D4—1491no—
261C2×C4×D4—1491no—
262C22×C22⋊C4—1491no—
263C22×C22⋊C4—1491no—
264#390—1491no—
265#388—1491no—
266#388—1491no—
267#386—1491no—
268#386—1491no—
269#386—1491no—
270#385—1491no—
271#385D14911yescharacteristic, complemented
272C4×C22⋊C4—1491no—
273C4×C22⋊C4—1491no—
274C25—1491no—
275C22×D4—1491no—
276C22×D4—1491no—
277C22×D4—1491no—
278C23×C4—1491no—
279C23×C4—1491no—
280C23×C4—1491no—
281C23×C4—1491no—
282C23×C4C2×D14911yescharacteristic
283#120—1498no—
284C4×D4—2984no—
285C4×D4—2984no—
286C2×C4⋊C4—1491no—
287C2×C4⋊C4C2×D14911yescharacteristic, complemented
288C2×C4⋊C4C2×D14911yescharacteristic, complemented
289C2×C22⋊C4—2984no—
290C2×C22⋊C4—2984no—
291C2×C22⋊C4—1491no—
292C2×C22⋊C4—1491no—
293C2×C22⋊C4—1491no—
294C2×C22⋊C4—1491no—
295C2×C22⋊C4—1491no—
296C2×C22⋊C4—1491no—
297C2×C22⋊C4—1491no—
298C2×C22⋊C4—1491no—
299C2×C22⋊C4—1491no—
300C2×C22⋊C4—1491no—
301C2×C22⋊C4—1491no—
302C2×C22⋊C4—1491no—
303C2×C22⋊C4C2×D14911yescharacteristic
304C2×C22⋊C4C2×D14911yescharacteristic
305C2×C42—1491no—
306C2×C42—1491no—
307C2×C42—1491no—
308#95—1491no—
309#95—1491no—
310#95—1491no—
311#95—1491no—
312#95C2×D14911yescharacteristic, complemented
313#95C2×D14911yescharacteristic, complemented
314C24—2982no—
315C24—2982no—
316C24—2982no—
317C24—2982no—
318C24—5961no—
319C24—5961no—
320C24—1491no—
321C24—1491no—
322C24—1494no—
323C24—1491no—
324C24C22×D14911yescharacteristic
325C2×D4—2982no—
326C2×D4—2982no—
327C2×D4—2982no—
328C2×D4—2982no—
329C2×D4—5961no—
330C2×D4—5961no—
331C2×D4—1494no—
332C2×D4—1494no—
333C2×D4—1494no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—41no—
337C22×C4—2982no—
338C22×C4—2982no—
339C22×C4—2982no—
340C22×C4—2982no—
341C22×C4—2982no—
342C22×C4—2982no—
343C22×C4—2982no—
344C22×C4—2982no—
345C22×C4—5961no—
346C22×C4—5961no—
347C22×C4—5961no—
348C22×C4—5961no—
349C22×C4—1491no—
350C22×C4—1491no—
351C22×C4—1491no—
352C22×C4—1491no—
353C22×C4—1491no—
354C22×C4C22×D14911yescharacteristic
355C22×C4C22×D14911yescharacteristic
356C22×C4C22×D14911yescharacteristic
357C22×C4C22×D14911yescharacteristic
358C22×C4C22×D14911yescharacteristic
359C22×C4C22×D14911yescharacteristic
360C4⋊C4—22no—
361C4⋊C4—41no—
362C4⋊C4—2982no—
363C22⋊C4—22no—
364C22⋊C4—2982no—
365C22⋊C4—2982no—
366C22⋊C4—2982no—
367C22⋊C4—2982no—
368C22⋊C4—2982no—
369C22⋊C4—2982no—
370C22⋊C4—2982no—
371C22⋊C4—5961no—
372C22⋊C4—1494no—
373C22⋊C4—1494no—
374C22⋊C4—1494no—
375C22⋊C4—1494no—
376C22⋊C4C4×D14914yescomplemented
377C42—2982no—
378C42—2982no—
379C42—5961no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—2988no—
385C23—2981no—
386C23—2981no—
387C23—2981no—
388C23—2981no—
389C23—2981no—
390C23—2981no—
391C23—2981no—
392C23—2981no—
393C23—2981no—
394C23—2981no—
395C23—2981no—
396C23—2981no—
397C23—2981no—
398C23—2981no—
399C23—2981no—
400C23—2981no—
401C23—2988no—
402C23—5964no—
403C23—5964no—
404C23—5964no—
405C23—5964no—
406C23—5964no—
407C23—1492no—
408C23—1492no—
409C23—1492no—
410C23—1492no—
411C23—1492no—
412C23C2×C4×D14912yes—
413C23D4×D14912yes—
414C23#1455873712yes—
415C23C23×D14911yescharacteristic
416D4—2988no—
417D4—2988no—
418D4—5964no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—44no—
435C2×C4—2981no—
436C2×C4—2981no—
437C2×C4—2981no—
438C2×C4—2981no—
439C2×C4—2981no—
440C2×C4—2981no—
441C2×C4—2981no—
442C2×C4—2981no—
443C2×C4—2981no—
444C2×C4—2981no—
445C2×C4—2981no—
446C2×C4—5964no—
447C2×C4—5964no—
448C2×C4—5964no—
449C2×C4—5964no—
450C2×C4—1492no—
451C2×C4—1492no—
452C2×C4—1492no—
453C2×C4—1492no—
454C2×C4C2×C4×D14912yes—
455C2×C4C2×C4×D14912yes—
456C2×C4D4×D14912yes—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—2982no—
464C22—2982no—
465C22—2982no—
466C22—2982no—
467C22—2982no—
468C22—2982no—
469C22—2982no—
470C22—2982no—
471C22—2982no—
472C22—2982no—
473C22—2982no—
474C22—2982no—
475C22—59616no—
476C22—5961no—
477C22—5961no—
478C22—5961no—
479C22—5961no—
480C22—5961no—
481C22—5961no—
482C22—5961no—
483C22—5961no—
484C22—1494no—
485C22#1801187314yescomplemented
486C22C22×C4×D14911yescharacteristic
487C22C2×D4×D14911yescharacteristic
488C22C2×D4×D14911yescharacteristic
489C22#41×D14911yescharacteristic
490C22C2×#1455873711yescharacteristic
491C22#1801195611yescharacteristic
492C22#1801195711yescharacteristic
493C4—22no—
494C4—22no—
495C4—22no—
496C4—41no—
497C4—41no—
498C4—41no—
499C4—2982no—
500C4—2982no—
501C4—2982no—
502C4—2982no—
503C4—5961no—
504C2—24no—
505C2—2984no—
506C2—5962no—
507C2—5962no—
508C2C4×D4×D14911yescharacteristic
509C2C2×#1801187311yescharacteristic
510C2#2375009711yescharacteristic
511C2#2375025711yescharacteristic
512C2#2375026611yescharacteristic
513C2#2375026711yescharacteristic
514C2#2375030211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D149
commutatorC22×C149C23×C4
Fitting#385×C149C2
socle, cosocleC23×C149C24
prime radicalC2×#18011881C2
hypercenter#385D149
hypocenterC149#2358
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#18011873C2semidirect
3. C2×C4×C22⋊D149C2semidirect
4. C22×#14558713C2semidirect
5. C2×C22⋊C4×D149C2semidirect
6. #23749526C2semidirect
7. #23749515C2semidirect
8. #23749514C2semidirect
9. #23749503C2semidirect
10. #23749503C2semidirect
11. #23749502C2semidirect
12. #23749496C2semidirect
13. C22⋊C4×Dic149C2semidirect
14. #23749472C2semidirect
15. #385×C149C2semidirect
23. C22×C4×D149C22semidirect
24. #18011881C4semidirect
27. C2×#14558715C22semidirect
38. C2×#14558713C22semidirect
39. C2×#14558713C22semidirect
40. C2×#14558713C22semidirect
42. C2×C4×Dic149C22semidirect
43. C2×C4×Dic149C22semidirect
44. C2×C4×Dic149C22semidirect
45. #18011818C22semidirect
46. #18011818C22semidirect
47. #18011818C22semidirect
48. #18011818C22semidirect
50. C2×C4⋊C4×C149C22semidirect
51. C2×C4⋊C4×C149C22semidirect
54. #95×C149C22semidirect
55. #95×C149C22semidirect
62. C23×D149C2×C4semidirect
70. C2×D596C2×C4semidirect
71. C2×D596C2×C4semidirect
74. C2×C22⋊D149C2×C4semidirect
93. C22⋊Dic149D4semidirect
102. #14558713C2×C4semidirect
103. #14558713C2×C4semidirect
104. #14558713D4semidirect
121. C22⋊C4×C149C2×C4semidirect
205. C22×C149C4×D4semidirect
234. C2×D149C4×D4semidirect
257. C149#2358semidirect
271. #385D149semidirect
287. C2×C4⋊C4C2×D149semidirect
288. C2×C4⋊C4C2×D149semidirect
312. #95C2×D149semidirect
313. #95C2×D149semidirect
376. C22⋊C4C4×D149semidirect
485. C22#18011873semidirect

Maximal subgroups

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

subgroupindex
C2×#180118812
C2×#180118732
C2×C4×C22⋊D1492
C22×#145587132
C2×C22⋊C4×D1492
#237495262
#237495152
#237495142
#237495032
#237495022
#237494962
C22⋊C4×Dic1492
#237494722
#385×C1492
#2358149