Finite Group Databasebeta

← #34103458

Subgroups — #34103458

Order 20096.

1116 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×D157C211yesmaximal, characteristic
2C2×#19097756C211yesmaximal, characteristic, complemented
3C2×#19097753C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D157C211yesmaximal, characteristic, complemented
5#23757189C211yesmaximal, characteristic, complemented
6#23757188C211yesmaximal, characteristic, complemented
7#23757187C211yesmaximal, characteristic, complemented
8#23757151C211yesmaximal, characteristic, complemented
9#23757149C211yesmaximal, characteristic, complemented
10#23757147C211yesmaximal, characteristic, complemented
11#23757144C211yesmaximal, characteristic, complemented
12#23757128C211yesmaximal, characteristic, complemented
13#23757128C211yesmaximal, characteristic, complemented
14#23757128C211yesmaximal, characteristic, complemented
15#394×C157C211yesmaximal, characteristic, complemented
16C24×D157C2211yescharacteristic
17C24×D157C2211yescharacteristic
18C22×D628C2211yescharacteristic
19C22×C22⋊D157C2211yescharacteristic
20C22×C22⋊D157C2211yescharacteristic
21C2×D4×D157—24no—
22C2×D4×D157—24no—
23C2×D4×D157—24no—
24C22×C4×D157C2211yescharacteristic
25C22×C4×D157C2211yescharacteristic, complemented
26C22×C4×D157C2211yescharacteristic, complemented
27#19097756—24no—
28#19097753—24no—
29C2×C4⋊Dic157C2211yescharacteristic, complemented
30C2×#14578933C2211yescharacteristic, complemented
31C2×#14578933C2211yescharacteristic, complemented
32C2×C22⋊Dic157C2211yescharacteristic, complemented
33C2×C22⋊Dic157C2211yescharacteristic, complemented
34C2×C22⋊Dic157C2211yescharacteristic, complemented
35C2×C22⋊Dic157C2211yescharacteristic, complemented
36C2×#14578931C2211yescharacteristic
37C2×#14578931C2211yescharacteristic
38C2×#14578931C2211yescharacteristic
39C2×#14578931C2211yescharacteristic
40C2×#14578931C2211yescharacteristic, complemented
41C2×#14578931C2211yescharacteristic, complemented
42C2×#14578931C2211yescharacteristic, complemented
43C2×#14578931C2211yescharacteristic, complemented
44C2×#14578931C2211yescharacteristic, complemented
45C2×#14578931C2211yescharacteristic, complemented
46C22⋊C4×D157—24no—
47#19097678C2211yescharacteristic, complemented
48#19097678C2211yescharacteristic, complemented
49#19097678C2211yescharacteristic, complemented
50C22×D4×C157C2211yescharacteristic
51C2×C4⋊C4×C157C2211yescharacteristic, complemented
52C2×C22⋊C4×C157C2211yescharacteristic, complemented
53C2×C22⋊C4×C157C2211yescharacteristic, complemented
54C2×C22⋊C4×C157C2211yescharacteristic, complemented
55C2×C22⋊C4×C157C2211yescharacteristic, complemented
56#95×C157C2211yescharacteristic, complemented
57C23×D157—22no—
58C23×D157—22no—
59C23×D157—22no—
60C23×D157—22no—
61C23×D157—22no—
62C23×D157—22no—
63C23×D157—22no—
64C23×D157—22no—
65C23×D157—22no—
66C23×D157—41no—
67C23×D157—41no—
68C23×D157—41no—
69C23×D157—41no—
70C23×D157D414yes—
71C23×D157C2311yescharacteristic
72C23×D157C2311yescharacteristic
73C23×D157C2311yescharacteristic
74C2×D628—22no—
75C2×D628—22no—
76C2×D628D414yes—
77C2×C22⋊D157—22no—
78C2×C22⋊D157—22no—
79C2×C22⋊D157—22no—
80C2×C22⋊D157—22no—
81C2×C22⋊D157—22no—
82C2×C22⋊D157D414yes—
83D4×D157—232no—
84C22×Dic157C2311yescharacteristic
85C22×Dic157C2311yescharacteristic
86C22×Dic157C2311yescharacteristic
87C22×Dic157C2311yescharacteristic
88C22×Dic157C2311yescharacteristic
89C2×C4×D157—22no—
90C2×C4×D157—22no—
91C2×C4×D157—22no—
92C2×C4×D157—22no—
93C2×C4×D157—22no—
94C2×C4×D157—22no—
95C2×C4×D157—22no—
96C2×C4×D157—41no—
97C2×C4×D157—41no—
98C4⋊Dic157—22no—
99#14578933—22no—
100#14578933—22no—
101C22⋊Dic157—22no—
102C22⋊Dic157—22no—
103C22⋊Dic157—41no—
104C22⋊Dic157—41no—
105#14578931—22no—
106#14578931—22no—
107#14578931—22no—
108#14578931—22no—
109#14578931—22no—
110#14578931—22no—
111#14578931—22no—
112#14578931—22no—
113#14578931—41no—
114#14578931—41no—
115C24×C157C2311yescharacteristic
116C24×C157C2311yescharacteristic
117C2×D4×C157—22no—
118C2×D4×C157—22no—
119C2×D4×C157—22no—
120C22×C4×C157C2311yescharacteristic
121C22×C4×C157C2311yescharacteristic
122C22×C4×C157C2311yescharacteristic
123C22×C4×C157C2311yescharacteristic
124C22×C4×C157C2311yescharacteristic
125C4⋊C4×C157—22no—
126C22⋊C4×C157—22no—
127C22⋊C4×C157—22no—
128C22⋊C4×C157—41no—
129C22⋊C4×C157—41no—
130C22×D157—21no—
131C22×D157—21no—
132C22×D157—21no—
133C22×D157—21no—
134C22×D157—21no—
135C22×D157—21no—
136C22×D157—21no—
137C22×D157—21no—
138C22×D157—28no—
139C22×D157—21no—
140C22×D157—21no—
141C22×D157—21no—
142C22×D157—21no—
143C22×D157—21no—
144C22×D157—21no—
145C22×D157—28no—
146C22×D157—28no—
147C22×D157—44no—
148C22×D157—44no—
149C22×D157—44no—
150C22×D157—44no—
151C22×D157—44no—
152C22×D157—44no—
153C22×D157—44no—
154C22×D157—44no—
155C22×D157—44no—
156C22×D157—44no—
157C22×D157—44no—
158C22×D157C2×D412yes—
159C22×D157C2×D412yes—
160C22×D157C2×D412yes—
161C22×D157C2×D412yes—
162C22×D157C2×D412yes—
163C22×D157#4112yes—
164C22×D157#4112yes—
165D628—28no—
166C22⋊D157—28no—
167C22⋊D157—28no—
168C2×Dic157—21no—
169C2×Dic157—21no—
170C2×Dic157—21no—
171C2×Dic157—21no—
172C2×Dic157—21no—
173C2×Dic157—21no—
174C2×Dic157—21no—
175C2×Dic157—21no—
176C2×Dic157—21no—
177C2×Dic157—21no—
178C2×Dic157—21no—
179C2×Dic157—21no—
180C2×Dic157—21no—
181C2×Dic157—21no—
182C2×Dic157C2×D412yes—
183C4×D157—28no—
184C4×D157—44no—
185C4×D157—44no—
186C23×C157—21no—
187C23×C157—21no—
188C23×C157—21no—
189C23×C157—21no—
190C23×C157—21no—
191C23×C157—21no—
192C23×C157—21no—
193C23×C157—21no—
194C23×C157—21no—
195C23×C157—21no—
196C23×C157—21no—
197C23×C157—21no—
198C23×C157C2×D412yes—
199C23×C157C2×D412yes—
200C23×C157C2411yescharacteristic
201D4×C157—28no—
202C2×C4×C157—21no—
203C2×C4×C157—21no—
204C2×C4×C157—21no—
205C2×C4×C157—21no—
206C2×C4×C157—21no—
207C2×C4×C157—21no—
208C2×C4×C157—21no—
209C2×C4×C157—21no—
210C2×C4×C157—21no—
211C2×C4×C157—21no—
212C2×C4×C157—21no—
213C2×C4×C157—21no—
214C2×C4×C157—21no—
215C2×C4×C157—21no—
216C2×C4×C157C2×D412yes—
217C22×C157—22no—
218C22×C157—22no—
219C22×C157—22no—
220C22×C157—22no—
221C22×C157—22no—
222C22×C157—22no—
223C22×C157—41no—
224C22×C157—41no—
225C22×C157—41no—
226C22×C157—41no—
227C22×C157—41no—
228C22×C157—41no—
229C22×C157—41no—
230C22×C157—41no—
231C22×C157C22×D411yescharacteristic
232C22×C157C22×D411yescharacteristic
233C22×C157C22×D411yescharacteristic
234C22×C157C2×#4111yescharacteristic
235C22×C157C2×#4111yescharacteristic
236C22×C157H(5,𝔽2)11yescharacteristic
237C22×C157H(5,𝔽2)11yescharacteristic
238C2×D157—22no—
239C2×D157—22no—
240C2×D157—22no—
241C2×D157—22no—
242C2×D157—22no—
243C2×D157—22no—
244C2×D157—22no—
245C2×D157—22no—
246C2×D157—22no—
247C2×D157—22no—
248C2×D157—22no—
249C2×D157—22no—
250C2×D157—22no—
251C2×D157—22no—
252C2×D157—22no—
253C2×D157—416no—
254C2×D157—416no—
255C2×D157—41no—
256C2×D157—41no—
257C2×D157—41no—
258C2×D157—41no—
259C2×D157#12014yes—
260C2×D157#12314yescomplemented
261C4×C157—22no—
262C4×C157—41no—
263C4×C157—41no—
264C4×C157—41no—
265C4×C157—41no—
266Dic157—22no—
267Dic157—41no—
268Dic157—41no—
269Dic157—41no—
270Dic157—41no—
271C2×C157—42no—
272C2×C157—42no—
273C2×C157C2×#12011yescharacteristic
274C2×C157C2×#12311yescharacteristic
275C2×C157#53411yescharacteristic
276C2×C157#54611yescharacteristic
277C2×C157#54611yescharacteristic
278C2×C157#54611yescharacteristic
279C2×C157#54611yescharacteristic
280D157—24no—
281D157—24no—
282D157—42no—
283C157#240511yescharacteristic, complemented, Sylow
284#2405—1571nomaximal, Sylow
285C23×D4—1571no—
286C2×#123—1571no—
287C2×#123—1571no—
288C22×C22⋊C4—1571no—
289#394—1571no—
290#394—1571no—
291#394—1571no—
292#394D15711yescharacteristic, complemented
293#386—1571no—
294#386—1571no—
295#385—1571no—
296#385—1571no—
297#379—1571no—
298#379—1571no—
299#379—1571no—
300C25—1571no—
301C25—1571no—
302C22×D4—3144no—
303C22×D4—3144no—
304C22×D4—3144no—
305C22×D4—1571no—
306C22×D4—1571no—
307C22×D4—1571no—
308C22×D4C2×D15711yescharacteristic
309C23×C4—1571no—
310C23×C4—1571no—
311C23×C4—1571no—
312#123—3144no—
313#123—3144no—
314C2×C4⋊C4—1571no—
315C2×C4⋊C4—1571no—
316C2×C4⋊C4—1571no—
317C2×C4⋊C4C2×D15711yescharacteristic, complemented
318C2×C22⋊C4—3144no—
319C2×C22⋊C4—1571no—
320C2×C22⋊C4—1571no—
321C2×C22⋊C4—1571no—
322C2×C22⋊C4—1571no—
323C2×C22⋊C4—1571no—
324C2×C22⋊C4—1571no—
325C2×C22⋊C4—1571no—
326C2×C22⋊C4—1571no—
327C2×C22⋊C4—1571no—
328C2×C22⋊C4—1571no—
329C2×C22⋊C4—1571no—
330C2×C22⋊C4—1571no—
331C2×C22⋊C4—1571no—
332C2×C22⋊C4—1571no—
333C2×C22⋊C4C2×D15711yescharacteristic, complemented
334C2×C22⋊C4C2×D15711yescharacteristic, complemented
335C2×C22⋊C4C2×D15711yescharacteristic, complemented
336C2×C22⋊C4C2×D15711yescharacteristic, complemented
337#95—1571no—
338#95—1571no—
339#95—1571no—
340#95C2×D15711yescharacteristic, complemented
341C24—3142no—
342C24—3142no—
343C24—3142no—
344C24—3142no—
345C24—3142no—
346C24—3142no—
347C24—3142no—
348C24—3142no—
349C24—3142no—
350C24—6281no—
351C24—6281no—
352C24—6281no—
353C24—6281no—
354C24—1571no—
355C24—1574no—
356C24—1571no—
357C24—1571no—
358C24C22×D15711yescharacteristic
359C24C22×D15711yescharacteristic
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—3142no—
364C2×D4—3142no—
365C2×D4—31432no—
366C2×D4—3142no—
367C2×D4—3142no—
368C2×D4—3142no—
369C2×D4—3142no—
370C2×D4—3142no—
371C2×D4—1574no—
372C2×D4—1574no—
373C22×C4—3142no—
374C22×C4—3142no—
375C22×C4—3142no—
376C22×C4—3142no—
377C22×C4—3142no—
378C22×C4—3142no—
379C22×C4—3142no—
380C22×C4—6281no—
381C22×C4—6281no—
382C22×C4—1571no—
383C22×C4—1571no—
384C22×C4—1571no—
385C22×C4—1571no—
386C22×C4—1571no—
387C22×C4C22×D15711yescharacteristic
388C22×C4C22×D15711yescharacteristic
389C22×C4C22×D15711yescharacteristic
390C22×C4C22×D15711yescharacteristic
391C22×C4C22×D15711yescharacteristic
392C4⋊C4—22no—
393C4⋊C4—3142no—
394C4⋊C4—3142no—
395C4⋊C4—3142no—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—3142no—
401C22⋊C4—3142no—
402C22⋊C4—3142no—
403C22⋊C4—3142no—
404C22⋊C4—3142no—
405C22⋊C4—3142no—
406C22⋊C4—3142no—
407C22⋊C4—3142no—
408C22⋊C4—3142no—
409C22⋊C4—3142no—
410C22⋊C4—6281no—
411C22⋊C4—6281no—
412C22⋊C4—6281no—
413C22⋊C4—6281no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—3148no—
427C23—3148no—
428C23—3148no—
429C23—3141no—
430C23—3141no—
431C23—3141no—
432C23—3141no—
433C23—3141no—
434C23—3141no—
435C23—3141no—
436C23—3141no—
437C23—3141no—
438C23—3141no—
439C23—3141no—
440C23—3141no—
441C23—3141no—
442C23—3141no—
443C23—6284no—
444C23—6284no—
445C23—6284no—
446C23—6284no—
447C23—6284no—
448C23—6284no—
449C23—6284no—
450C23—6284no—
451C23—6284no—
452C23—6284no—
453C23—6284no—
454C23—1572no—
455C23—1572no—
456C23—1572no—
457C23—1572no—
458C23—1572no—
459C23—1572no—
460C23—1572no—
461C23D4×D15712yes—
462C23D4×D15712yes—
463C23C23×D15711yescharacteristic
464D4—28no—
465D4—3148no—
466D4—3148no—
467D4—3148no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—3148no—
483C2×C4—3141no—
484C2×C4—3141no—
485C2×C4—3141no—
486C2×C4—3141no—
487C2×C4—3141no—
488C2×C4—3141no—
489C2×C4—3141no—
490C2×C4—3141no—
491C2×C4—3141no—
492C2×C4—3141no—
493C2×C4—3141no—
494C2×C4—3141no—
495C2×C4—3141no—
496C2×C4—3141no—
497C2×C4—6284no—
498C2×C4—6284no—
499C2×C4—1572no—
500C2×C4D4×D15712yes—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—3142no—
516C22—3142no—
517C22—3142no—
518C22—3142no—
519C22—3142no—
520C22—3142no—
521C22—3142no—
522C22—3142no—
523C22—3142no—
524C22—3142no—
525C22—3142no—
526C22—3142no—
527C22—3142no—
528C22—3142no—
529C22—3142no—
530C22—62816no—
531C22—62816no—
532C22—6281no—
533C22—6281no—
534C22—6281no—
535C22—6281no—
536C22—1574no—
537C22—1574no—
538C22C2×D4×D15711yescharacteristic
539C22C2×D4×D15711yescharacteristic
540C22C2×D4×D15711yescharacteristic
541C22#41×D15711yescharacteristic
542C22#41×D15711yescharacteristic
543C22#1909781611yescharacteristic
544C22#1909781711yescharacteristic
545C4—22no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—3142no—
551C4—6281no—
552C4—6281no—
553C4—6281no—
554C4—6281no—
555C2—42no—
556C2—42no—
557C2—3144no—
558C2—3144no—
559C2—6282no—
560C2#120×D15711yescharacteristic
561C2#123×D15711yescharacteristic
562C2#2375783111yescharacteristic
563C2#2375791211yescharacteristic
564C2#2375791511yescharacteristic
565C2#2375791611yescharacteristic
566C2#2375792011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D157
commutator, socle, cosocleC23×C157C24
Fitting#394×C157C2
prime radicalC22×D4×D157C2
hypercenter#394D157
hypocenterC157#2405
LCSG ▷ C23×C157 ▷ C157
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#19097756C2semidirect
3. C2×#19097753C2semidirect
4. C2×C22⋊C4×D157C2semidirect
5. #23757189C2semidirect
6. #23757188C2semidirect
7. #23757187C2semidirect
8. #23757151C2semidirect
9. #23757149C2semidirect
10. #23757147C2semidirect
11. #23757144C2semidirect
12. #23757128C2semidirect
13. #23757128C2semidirect
14. #23757128C2semidirect
15. #394×C157C2semidirect
25. C22×C4×D157C22semidirect
26. C22×C4×D157C22semidirect
29. C2×C4⋊Dic157C22semidirect
30. C2×#14578933C22semidirect
31. C2×#14578933C22semidirect
32. C2×C22⋊Dic157C22semidirect
33. C2×C22⋊Dic157C22semidirect
34. C2×C22⋊Dic157C22semidirect
35. C2×C22⋊Dic157C22semidirect
40. C2×#14578931C22semidirect
41. C2×#14578931C22semidirect
42. C2×#14578931C22semidirect
43. C2×#14578931C22semidirect
44. C2×#14578931C22semidirect
45. C2×#14578931C22semidirect
47. #19097678C22semidirect
48. #19097678C22semidirect
49. #19097678C22semidirect
51. C2×C4⋊C4×C157C22semidirect
52. C2×C22⋊C4×C157C22semidirect
53. C2×C22⋊C4×C157C22semidirect
54. C2×C22⋊C4×C157C22semidirect
55. C2×C22⋊C4×C157C22semidirect
56. #95×C157C22semidirect
260. C2×D157#123semidirect
283. C157#2405semidirect
292. #394D157semidirect
317. C2×C4⋊C4C2×D157semidirect
333. C2×C22⋊C4C2×D157semidirect
334. C2×C22⋊C4C2×D157semidirect
335. C2×C22⋊C4C2×D157semidirect
336. C2×C22⋊C4C2×D157semidirect
340. #95C2×D157semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1572
C2×#190977562
C2×#190977532
C2×C22⋊C4×D1572
#237571892
#237571882
#237571872
#237571512
#237571492
#237571472
#237571442
#237571282
#394×C1572
#2405157