Finite Group Databasebeta

← #34103118

Subgroups — #34103118

Order 20096.

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×#19097741C211yesmaximal, characteristic
2C2×C4×D628C211yesmaximal, characteristic, complemented
3C2×#19097731C211yesmaximal, characteristic, complemented
4C22×#14578931C211yesmaximal, characteristic, complemented
5C2×C22⋊C4×D157C211yesmaximal, characteristic, complemented
6#23757175C211yesmaximal, characteristic, complemented
7#23757165C211yesmaximal, characteristic, complemented
8#23757164C211yesmaximal, characteristic, complemented
9#23757152C211yesmaximal, characteristic, complemented
10#23757151C211yesmaximal, characteristic, complemented
11#23757151C211yesmaximal, characteristic, complemented
12#23757148C211yesmaximal, characteristic, complemented
13#23757147C211yesmaximal, characteristic, complemented
14C4×#14578931C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C157C211yesmaximal, characteristic, complemented
16C24×D157C2211yescharacteristic
17C22×D628C2211yescharacteristic
18C22×D628C2211yescharacteristic
19C22×C22⋊D157C2211yescharacteristic
20C23×Dic157C2211yescharacteristic
21C22×C4×D157C2211yescharacteristic
22C22×C4×D157C2211yescharacteristic
23C22×C4×D157C2211yescharacteristic, complemented
24#19097741C418yescomplemented
25C4×D628—24no—
26#19097731—24no—
27C2×C4⋊Dic157C2211yescharacteristic, complemented
28C2×C4⋊Dic157C2211yescharacteristic, complemented
29C2×#14578933C2211yescharacteristic, complemented
30C2×C22⋊Dic157C2211yescharacteristic
31C2×#14578931—24no—
32C2×#14578931C2211yescharacteristic
33C2×#14578931C2211yescharacteristic
34C2×#14578931C2211yescharacteristic
35C2×#14578931C2211yescharacteristic
36C2×#14578931C2211yescharacteristic
37C2×#14578931C2211yescharacteristic
38C2×#14578931C2211yescharacteristic
39C2×#14578931C2211yescharacteristic
40C2×#14578931C2211yescharacteristic, complemented
41C2×#14578931C2211yescharacteristic, complemented
42C2×#14578931C2211yescharacteristic, complemented
43C22⋊C4×D157—24no—
44C2×C4×Dic157C2211yescharacteristic, complemented
45#19097678C2211yescharacteristic, complemented
46#19097678C2211yescharacteristic, complemented
47#19097678C2211yescharacteristic, complemented
48#19097678C2211yescharacteristic, complemented
49C23×C4×C157C2211yescharacteristic
50C2×C22⋊C4×C157C2211yescharacteristic
51C2×C22⋊C4×C157C2211yescharacteristic
52C2×C42×C157C2211yescharacteristic, complemented
53C2×C42×C157C2211yescharacteristic, complemented
54#95×C157C2211yescharacteristic, complemented
55#95×C157C2211yescharacteristic, complemented
56C23×D157—22no—
57C23×D157—22no—
58C23×D157—22no—
59C23×D157—22no—
60C23×D157—41no—
61C23×D157—41no—
62C23×D157C2×C414yescomplemented
63C23×D157C2311yescharacteristic
64C23×D157C2311yescharacteristic
65C23×D157C2311yescharacteristic
66C2×D628—22no—
67C2×D628—22no—
68C2×D628—41no—
69C2×D628—41no—
70C2×D628C2×C414yescomplemented
71C2×D628C2×C414yescomplemented
72C2×C22⋊D157—22no—
73C2×C22⋊D157—22no—
74C2×C22⋊D157C2×C414yescomplemented
75C22×Dic157—22no—
76C22×Dic157—22no—
77C22×Dic157—41no—
78C22×Dic157C2311yescharacteristic
79C22×Dic157C2311yescharacteristic
80C22×Dic157C2311yescharacteristic
81C22×Dic157C2311yescharacteristic
82C22×Dic157C2311yescharacteristic
83C2×C4×D157—22no—
84C2×C4×D157—22no—
85C2×C4×D157—22no—
86C2×C4×D157—22no—
87C2×C4×D157—22no—
88C2×C4×D157—22no—
89C2×C4×D157—41no—
90C2×C4×D157—41no—
91C2×C4×D157—41no—
92C4⋊Dic157—22no—
93C4⋊Dic157—41no—
94#14578933—22no—
95C22⋊Dic157—22no—
96#14578931—22no—
97#14578931—22no—
98#14578931—22no—
99#14578931—22no—
100#14578931—22no—
101#14578931—22no—
102#14578931—22no—
103#14578931—41no—
104#14578931C2×C414yescomplemented
105#14578931C2×C414yescomplemented
106#14578931D414yescomplemented
107C4×Dic157—22no—
108C24×C157C2311yescharacteristic
109C22×C4×C157—22no—
110C22×C4×C157—22no—
111C22×C4×C157—41no—
112C22×C4×C157C2311yescharacteristic
113C22×C4×C157C2311yescharacteristic
114C22×C4×C157C2311yescharacteristic
115C22×C4×C157C2311yescharacteristic
116C22×C4×C157C2311yescharacteristic
117C22×C4×C157C2311yescharacteristic
118C22⋊C4×C157C2×C414yescomplemented
119C22⋊C4×C157D414yescomplemented
120C42×C157—22no—
121C42×C157—41no—
122C22×D157—21no—
123C22×D157—21no—
124C22×D157—21no—
125C22×D157—21no—
126C22×D157—28no—
127C22×D157—21no—
128C22×D157—21no—
129C22×D157—21no—
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—21no—
139C22×D157—28no—
140C22×D157—44no—
141C22×D157—44no—
142C22×D157—44no—
143C22×D157—44no—
144C22×D157—44no—
145C22×D157C22×C412yes—
146C22×D157C22×C412yes—
147C22×D157C22×C412yes—
148C22×D157C2×D412yes—
149C22×D157#4112yes—
150D628—28no—
151D628—44no—
152C22⋊D157—28no—
153C2×Dic157—21no—
154C2×Dic157—21no—
155C2×Dic157—21no—
156C2×Dic157—21no—
157C2×Dic157—21no—
158C2×Dic157—21no—
159C2×Dic157—21no—
160C2×Dic157—21no—
161C2×Dic157—21no—
162C2×Dic157—21no—
163C2×Dic157—21no—
164C2×Dic157—21no—
165C2×Dic157—21no—
166C2×Dic157—44no—
167C2×Dic157C22×C412yes—
168C2×Dic157C2×D412yes—
169C4×D157—44no—
170C4×D157—44no—
171C4×D157—44no—
172C23×C157—21no—
173C23×C157—21no—
174C23×C157—21no—
175C23×C157—21no—
176C23×C157C22×C412yes—
177C23×C157C2×D412yes—
178C23×C157#4112yes—
179C23×C157C2411yescharacteristic
180C2×C4×C157—21no—
181C2×C4×C157—21no—
182C2×C4×C157—21no—
183C2×C4×C157—21no—
184C2×C4×C157—21no—
185C2×C4×C157—21no—
186C2×C4×C157—21no—
187C2×C4×C157—21no—
188C2×C4×C157—21no—
189C2×C4×C157—21no—
190C2×C4×C157—21no—
191C2×C4×C157—21no—
192C2×C4×C157—21no—
193C2×C4×C157—44no—
194C2×C4×C157C22×C412yes—
195C2×C4×C157C22×C412yes—
196C2×C4×C157C2×D412yes—
197C2×C4×C157C2×D412yes—
198C2×C4×C157C2×D412yes—
199C22×C157—22no—
200C22×C157—22no—
201C22×C157—22no—
202C22×C157—22no—
203C22×C157—22no—
204C22×C157—22no—
205C22×C157C4×D414yescomplemented
206C22×C157C23×C411yescharacteristic
207C22×C157C22×D411yescharacteristic
208C22×C157C22×D411yescharacteristic
209C22×C157C2×#4111yescharacteristic
210C22×C157C2×#4111yescharacteristic
211C22×C157H(5,𝔽2)11yescharacteristic
212C22×C157H(5,𝔽2)11yescharacteristic
213C2×D157—22no—
214C2×D157—22no—
215C2×D157—22no—
216C2×D157—22no—
217C2×D157—22no—
218C2×D157—22no—
219C2×D157—22no—
220C2×D157—22no—
221C2×D157—22no—
222C2×D157—22no—
223C2×D157—22no—
224C2×D157—22no—
225C2×D157—416no—
226C2×D157—41no—
227C2×D157—41no—
228C2×D157—41no—
229C2×D157—41no—
230C2×D157—41no—
231C2×D157—41no—
232C2×D157—41no—
233C2×D157—41no—
234C2×D157C4×D414yescomplemented
235C4×C157—22no—
236C4×C157—22no—
237C4×C157—22no—
238C4×C157—22no—
239C4×C157—22no—
240C4×C157—41no—
241Dic157—22no—
242Dic157—22no—
243Dic157—41no—
244Dic157—41no—
245Dic157—41no—
246C2×C157—24no—
247C2×C157C2×C4×D411yescharacteristic
248C2×C157C2×C4×D411yescharacteristic
249C2×C157#51811yescharacteristic
250C2×C157D4211yescharacteristic
251C2×C157#54611yescharacteristic
252C2×C157#54611yescharacteristic
253C2×C157#55111yescharacteristic
254D157—24no—
255D157—42no—
256D157—42no—
257C157#235811yescharacteristic, complemented, Sylow
258#2358—1571nomaximal, Sylow
259C2×#120—1571no—
260C2×C4×D4—1571no—
261C2×C4×D4—1571no—
262C22×C22⋊C4—1571no—
263C22×C22⋊C4—1571no—
264#390—1571no—
265#388—1571no—
266#388—1571no—
267#386—1571no—
268#386—1571no—
269#386—1571no—
270#385—1571no—
271#385—1571no—
272C4×C22⋊C4—1571no—
273C4×C22⋊C4D15711yescharacteristic, complemented
274C25—1571no—
275C22×D4—1571no—
276C22×D4—1571no—
277C22×D4—1571no—
278C23×C4—1571no—
279C23×C4—1571no—
280C23×C4—1571no—
281C23×C4—1571no—
282C23×C4C2×D15711yescharacteristic
283#120—1578no—
284C4×D4—3144no—
285C4×D4—3144no—
286C2×C4⋊C4—1571no—
287C2×C4⋊C4—1571no—
288C2×C4⋊C4—1571no—
289C2×C22⋊C4—3144no—
290C2×C22⋊C4—3144no—
291C2×C22⋊C4—1571no—
292C2×C22⋊C4—1571no—
293C2×C22⋊C4—1571no—
294C2×C22⋊C4—1571no—
295C2×C22⋊C4—1571no—
296C2×C22⋊C4—1571no—
297C2×C22⋊C4—1571no—
298C2×C22⋊C4—1571no—
299C2×C22⋊C4—1571no—
300C2×C22⋊C4—1571no—
301C2×C22⋊C4—1571no—
302C2×C22⋊C4—1571no—
303C2×C22⋊C4C2×D15711yescharacteristic
304C2×C22⋊C4C2×D15711yescharacteristic
305C2×C42—1571no—
306C2×C42C2×D15711yescharacteristic, complemented
307C2×C42C2×D15711yescharacteristic, complemented
308#95—1571no—
309#95—1571no—
310#95—1571no—
311#95—1571no—
312#95C2×D15711yescharacteristic, complemented
313#95C2×D15711yescharacteristic, complemented
314C24—3142no—
315C24—3142no—
316C24—3142no—
317C24—3142no—
318C24—6281no—
319C24—6281no—
320C24—1571no—
321C24—1571no—
322C24—1574no—
323C24—1571no—
324C24C22×D15711yescharacteristic
325C2×D4—3142no—
326C2×D4—3142no—
327C2×D4—3142no—
328C2×D4—3142no—
329C2×D4—6281no—
330C2×D4—6281no—
331C2×D4—1574no—
332C2×D4—1574no—
333C2×D4—1574no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—41no—
337C22×C4—3142no—
338C22×C4—3142no—
339C22×C4—3142no—
340C22×C4—3142no—
341C22×C4—3142no—
342C22×C4—3142no—
343C22×C4—3142no—
344C22×C4—3142no—
345C22×C4—6281no—
346C22×C4—6281no—
347C22×C4—6281no—
348C22×C4—6281no—
349C22×C4—1571no—
350C22×C4—1571no—
351C22×C4—1571no—
352C22×C4—1571no—
353C22×C4—1571no—
354C22×C4C22×D15711yescharacteristic
355C22×C4C22×D15711yescharacteristic
356C22×C4C22×D15711yescharacteristic
357C22×C4C22×D15711yescharacteristic
358C22×C4C22×D15711yescharacteristic
359C22×C4C22×D15711yescharacteristic
360C4⋊C4—3142no—
361C4⋊C4—3142no—
362C4⋊C4—6281no—
363C22⋊C4—3142no—
364C22⋊C4—3142no—
365C22⋊C4—3142no—
366C22⋊C4—3142no—
367C22⋊C4—3142no—
368C22⋊C4—3142no—
369C22⋊C4—3142no—
370C22⋊C4—3142no—
371C22⋊C4—6281no—
372C22⋊C4—1574no—
373C22⋊C4—1574no—
374C22⋊C4—1574no—
375C22⋊C4C4×D15714yescomplemented
376C22⋊C4D62814yescomplemented
377C42—22no—
378C42—41no—
379C42—3142no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—3148no—
385C23—3141no—
386C23—3141no—
387C23—3141no—
388C23—3141no—
389C23—3148no—
390C23—3141no—
391C23—3141no—
392C23—3141no—
393C23—3141no—
394C23—3141no—
395C23—3141no—
396C23—3141no—
397C23—3141no—
398C23—3141no—
399C23—3141no—
400C23—3141no—
401C23—3141no—
402C23—6284no—
403C23—6284no—
404C23—6284no—
405C23—6284no—
406C23—6284no—
407C23—1572no—
408C23—1572no—
409C23—1572no—
410C23—1572no—
411C23—1572no—
412C23C2×C4×D15712yes—
413C23C2×D62812yes—
414C23#1457895412yes—
415C23C23×D15711yescharacteristic
416D4—3148no—
417D4—3148no—
418D4—6284no—
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—44no—
433C2×C4—3141no—
434C2×C4—3141no—
435C2×C4—3141no—
436C2×C4—3141no—
437C2×C4—3141no—
438C2×C4—3141no—
439C2×C4—3141no—
440C2×C4—3141no—
441C2×C4—3141no—
442C2×C4—3141no—
443C2×C4—3141no—
444C2×C4—3141no—
445C2×C4—3141no—
446C2×C4—6284no—
447C2×C4—6284no—
448C2×C4—6284no—
449C2×C4—6284no—
450C2×C4—1572no—
451C2×C4—1572no—
452C2×C4C2×C4×D15712yes—
453C2×C4C2×C4×D15712yes—
454C2×C4D4×D15712yes—
455C2×C4C2×D62812yes—
456C2×C4C2×D62812yes—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—3142no—
464C22—3142no—
465C22—3142no—
466C22—3142no—
467C22—3142no—
468C22—3142no—
469C22—3142no—
470C22—3142no—
471C22—3142no—
472C22—3142no—
473C22—3142no—
474C22—3142no—
475C22—62816no—
476C22—6281no—
477C22—6281no—
478C22—6281no—
479C22—6281no—
480C22—6281no—
481C22—6281no—
482C22—6281no—
483C22—6281no—
484C22—1574no—
485C22C4×D62814yescomplemented
486C22C22×C4×D15711yescharacteristic
487C22C2×D4×D15711yescharacteristic
488C22C22×D62811yescharacteristic
489C22#41×D15711yescharacteristic
490C22C2×#1457895411yescharacteristic
491C22#1909781611yescharacteristic
492C22#1909781711yescharacteristic
493C4—22no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—22no—
498C4—41no—
499C4—3142no—
500C4—3142no—
501C4—6281no—
502C4—6281no—
503C4—6281no—
504C2—24no—
505C2—3144no—
506C2—6282no—
507C2—6282no—
508C2C4×D4×D15711yescharacteristic
509C2C2×C4×D62811yescharacteristic
510C2#2375774511yescharacteristic
511C2D4×D62811yescharacteristic
512C2#2375791311yescharacteristic
513C2#2375791811yescharacteristic
514C2#2375794711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D157
commutatorC22×C157C23×C4
FittingC4×C22⋊C4×C157C2
socle, cosocleC23×C157C24
prime radicalC2×#19097741C2
hypercenterC4×C22⋊C4D157
hypocenterC157#2358
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×D628C2semidirect
3. C2×#19097731C2semidirect
4. C22×#14578931C2semidirect
5. C2×C22⋊C4×D157C2semidirect
6. #23757175C2semidirect
7. #23757165C2semidirect
8. #23757164C2semidirect
9. #23757152C2semidirect
10. #23757151C2semidirect
11. #23757151C2semidirect
12. #23757148C2semidirect
13. #23757147C2semidirect
14. C4×#14578931C2semidirect
15. C4×C22⋊C4×C157C2semidirect
23. C22×C4×D157C22semidirect
24. #19097741C4semidirect
27. C2×C4⋊Dic157C22semidirect
28. C2×C4⋊Dic157C22semidirect
29. C2×#14578933C22semidirect
40. C2×#14578931C22semidirect
41. C2×#14578931C22semidirect
42. C2×#14578931C22semidirect
44. C2×C4×Dic157C22semidirect
45. #19097678C22semidirect
46. #19097678C22semidirect
47. #19097678C22semidirect
48. #19097678C22semidirect
52. C2×C42×C157C22semidirect
53. C2×C42×C157C22semidirect
54. #95×C157C22semidirect
55. #95×C157C22semidirect
62. C23×D157C2×C4semidirect
70. C2×D628C2×C4semidirect
71. C2×D628C2×C4semidirect
74. C2×C22⋊D157C2×C4semidirect
104. #14578931C2×C4semidirect
105. #14578931C2×C4semidirect
106. #14578931D4semidirect
118. C22⋊C4×C157C2×C4semidirect
119. C22⋊C4×C157D4semidirect
205. C22×C157C4×D4semidirect
234. C2×D157C4×D4semidirect
257. C157#2358semidirect
273. C4×C22⋊C4D157semidirect
306. C2×C42C2×D157semidirect
307. C2×C42C2×D157semidirect
312. #95C2×D157semidirect
313. #95C2×D157semidirect
375. C22⋊C4C4×D157semidirect
376. C22⋊C4D628semidirect
485. C22C4×D628semidirect

Maximal subgroups

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

subgroupindex
C2×#190977412
C2×C4×D6282
C2×#190977312
C22×#145789312
C2×C22⋊C4×D1572
#237571752
#237571652
#237571642
#237571522
#237571512
#237571482
#237571472
C4×#145789312
C4×C22⋊C4×C1572
#2358157