Finite Group Databasebeta

← #34103115

Subgroups — #34103115

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×#19097740C211yesmaximal, characteristic
2C2×C4×C22⋊D157C211yesmaximal, characteristic, complemented
3C2×#19097731C211yesmaximal, characteristic, complemented
4C22×C22⋊Dic157C211yesmaximal, characteristic, complemented
5#23757173C211yesmaximal, characteristic, complemented
6#23757167C211yesmaximal, characteristic, complemented
7#23757161C211yesmaximal, characteristic, complemented
8#23757152C211yesmaximal, characteristic, complemented
9#23757150C211yesmaximal, characteristic, complemented
10#23757149C211yesmaximal, characteristic, complemented
11#23757146C211yesmaximal, characteristic, complemented
12#23757145C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic157C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic157C211yesmaximal, characteristic, complemented
15C22×C22⋊C4×C157C211yesmaximal, characteristic, complemented
16C22×C22⋊D157C2211yescharacteristic
17C22×C22⋊D157C2211yescharacteristic
18C22×C22⋊D157C2211yescharacteristic
19C23×Dic157C2211yescharacteristic
20C23×Dic157C2211yescharacteristic
21C22×C4×D157C2211yescharacteristic, complemented
22#19097740C418yescomplemented
23C4×C22⋊D157—24no—
24#19097731—24no—
25C2×#14578933C2211yescharacteristic, complemented
26C2×#14578933C2211yescharacteristic, complemented
27C2×#14578933C2211yescharacteristic, complemented
28C2×C22⋊Dic157—24no—
29C2×C22⋊Dic157C2211yescharacteristic
30C2×C22⋊Dic157C2211yescharacteristic
31C2×C22⋊Dic157C2211yescharacteristic
32C2×C22⋊Dic157C2211yescharacteristic
33C2×C22⋊Dic157C2211yescharacteristic
34C2×C22⋊Dic157C2211yescharacteristic
35C2×C22⋊Dic157C2211yescharacteristic
36C2×#14578931C2211yescharacteristic, complemented
37C2×#14578931C2211yescharacteristic, complemented
38C2×#14578931C2211yescharacteristic, complemented
39C2×C4×Dic157C2211yescharacteristic, complemented
40C2×C4×Dic157C2211yescharacteristic, complemented
41C2×C4×Dic157C2211yescharacteristic, complemented
42#19097678C2211yescharacteristic, complemented
43#19097678C2211yescharacteristic, complemented
44#19097678C2211yescharacteristic, complemented
45#19097678C2211yescharacteristic, complemented
46#19097678C2211yescharacteristic, complemented
47#19097678C2211yescharacteristic, complemented
48C25×C157C2211yescharacteristic
49C23×C4×C157C2211yescharacteristic
50C23×C4×C157C2211yescharacteristic
51C2×C22⋊C4×C157—24no—
52C2×C22⋊C4×C157C2211yescharacteristic
53C2×C22⋊C4×C157C2211yescharacteristic
54C2×C22⋊C4×C157C2211yescharacteristic
55C2×C22⋊C4×C157C2211yescharacteristic
56C23×D157C2311yescharacteristic
57C2×C22⋊D157—22no—
58C2×C22⋊D157—22no—
59C2×C22⋊D157—22no—
60C2×C22⋊D157—22no—
61C2×C22⋊D157—41no—
62C2×C22⋊D157—41no—
63C2×C22⋊D157C2×C414yescomplemented
64C2×C22⋊D157C2×C414yescomplemented
65C2×C22⋊D157C2×C414yescomplemented
66C22×Dic157—22no—
67C22×Dic157—22no—
68C22×Dic157—22no—
69C22×Dic157—22no—
70C22×Dic157—41no—
71C22×Dic157—41no—
72C22×Dic157C2311yescharacteristic
73C22×Dic157C2311yescharacteristic
74C22×Dic157C2311yescharacteristic
75C22×Dic157C2311yescharacteristic
76C22×Dic157C2311yescharacteristic
77C22×Dic157C2311yescharacteristic
78C22×Dic157C2311yescharacteristic
79C2×C4×D157—22no—
80C2×C4×D157—22no—
81C2×C4×D157—41no—
82#14578933—22no—
83#14578933—22no—
84#14578933—41no—
85C22⋊Dic157—22no—
86C22⋊Dic157—22no—
87C22⋊Dic157—22no—
88C22⋊Dic157C2×C414yescomplemented
89C22⋊Dic157C2×C414yescomplemented
90C22⋊Dic157C2×C414yescomplemented
91C22⋊Dic157D414yescomplemented
92#14578931—22no—
93#14578931—22no—
94#14578931—41no—
95C4×Dic157—22no—
96C4×Dic157—22no—
97C4×Dic157—41no—
98C24×C157—22no—
99C24×C157—22no—
100C24×C157—22no—
101C24×C157—22no—
102C24×C157—41no—
103C24×C157—41no—
104C24×C157C2×C414yescomplemented
105C24×C157C2311yescharacteristic
106C24×C157C2311yescharacteristic
107C24×C157C2311yescharacteristic
108C22×C4×C157—22no—
109C22×C4×C157—22no—
110C22×C4×C157—22no—
111C22×C4×C157—22no—
112C22×C4×C157—41no—
113C22×C4×C157—41no—
114C22×C4×C157C2311yescharacteristic
115C22×C4×C157C2311yescharacteristic
116C22×C4×C157C2311yescharacteristic
117C22×C4×C157C2311yescharacteristic
118C22⋊C4×C157—22no—
119C22⋊C4×C157—22no—
120C22⋊C4×C157—22no—
121C22⋊C4×C157D414yescomplemented
122C22×D157—21no—
123C22×D157—21no—
124C22×D157—21no—
125C22×D157—21no—
126C22×D157—21no—
127C22×D157—21no—
128C22×D157C22×C412yes—
129C22⋊D157—28no—
130C22⋊D157—28no—
131C22⋊D157—44no—
132C2×Dic157—21no—
133C2×Dic157—21no—
134C2×Dic157—21no—
135C2×Dic157—21no—
136C2×Dic157—21no—
137C2×Dic157—21no—
138C2×Dic157—21no—
139C2×Dic157—21no—
140C2×Dic157—21no—
141C2×Dic157—21no—
142C2×Dic157—21no—
143C2×Dic157—21no—
144C2×Dic157—21no—
145C2×Dic157—21no—
146C2×Dic157—21no—
147C2×Dic157—21no—
148C2×Dic157—44no—
149C2×Dic157—44no—
150C2×Dic157C22×C412yes—
151C2×Dic157C22×C412yes—
152C2×Dic157C22×C412yes—
153C2×Dic157C2×D412yes—
154C2×Dic157C2×D412yes—
155C4×D157—44no—
156C23×C157—21no—
157C23×C157—21no—
158C23×C157—21no—
159C23×C157—21no—
160C23×C157—21no—
161C23×C157—21no—
162C23×C157—21no—
163C23×C157—21no—
164C23×C157—28no—
165C23×C157—28no—
166C23×C157—21no—
167C23×C157—21no—
168C23×C157—21no—
169C23×C157—21no—
170C23×C157—21no—
171C23×C157—21no—
172C23×C157—44no—
173C23×C157—44no—
174C23×C157—44no—
175C23×C157—44no—
176C23×C157—44no—
177C23×C157C22×C412yes—
178C23×C157C22×C412yes—
179C23×C157C22×C412yes—
180C23×C157C2×D412yes—
181C23×C157C2×D412yes—
182C23×C157#4112yes—
183C23×C157#4112yes—
184C23×C157C2411yescharacteristic
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—21no—
194C2×C4×C157—21no—
195C2×C4×C157—44no—
196C2×C4×C157—44no—
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×C157—22no—
206C22×C157—22no—
207C22×C157—22no—
208C22×C157—22no—
209C22×C157—22no—
210C22×C157—22no—
211C22×C157—22no—
212C22×C157—22no—
213C22×C157—22no—
214C22×C157—416no—
215C22×C157—41no—
216C22×C157—41no—
217C22×C157—41no—
218C22×C157—41no—
219C22×C157C4×D414yescomplemented
220C22×C157C4×D414yescomplemented
221C22×C157C23×C411yescharacteristic
222C22×C157C22×D411yescharacteristic
223C22×C157C22×D411yescharacteristic
224C22×C157C2×#4111yescharacteristic
225C22×C157C2×#4111yescharacteristic
226C22×C157H(5,𝔽2)11yescharacteristic
227C22×C157H(5,𝔽2)11yescharacteristic
228C2×D157—22no—
229C2×D157—22no—
230C2×D157—22no—
231C2×D157—41no—
232C2×D157—41no—
233C2×D157—41no—
234C2×D157—41no—
235C4×C157—22no—
236C4×C157—22no—
237C4×C157—41no—
238C4×C157—41no—
239Dic157—22no—
240Dic157—22no—
241Dic157—22no—
242Dic157—22no—
243Dic157—22no—
244Dic157—41no—
245Dic157—41no—
246C2×C157—24no—
247C2×C157—24no—
248C2×C157—42no—
249C2×C157C2×C4×D411yescharacteristic
250C2×C157C2×C4×D411yescharacteristic
251C2×C157#51811yescharacteristic
252C2×C157D4211yescharacteristic
253C2×C157#54611yescharacteristic
254C2×C157#54611yescharacteristic
255C2×C157#55111yescharacteristic
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⋊C4D15711yescharacteristic, complemented
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⋊C4—1571no—
274C25C2×D15711yescharacteristic
275C22×D4—1571no—
276C22×D4—1571no—
277C22×D4—1571no—
278C23×C4—1571no—
279C23×C4—1571no—
280C23×C4—1571no—
281C23×C4C2×D15711yescharacteristic
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—24no—
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⋊C4C2×D15711yescharacteristic
302C2×C22⋊C4C2×D15711yescharacteristic
303C2×C22⋊C4C2×D15711yescharacteristic
304C2×C22⋊C4C2×D15711yescharacteristic
305C2×C42—1571no—
306C2×C42—1571no—
307C2×C42—1571no—
308#95—1571no—
309#95—1571no—
310#95—1571no—
311#95—1571no—
312#95—1571no—
313#95—1571no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—41no—
319C24—41no—
320C24—1571no—
321C24C4×D15714yescomplemented
322C24C22×D15711yescharacteristic
323C24C22×D15711yescharacteristic
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—22no—
337C22×C4—22no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—3142no—
341C22×C4—3142no—
342C22×C4—3142no—
343C22×C4—3142no—
344C22×C4—3142no—
345C22×C4—3142no—
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×C4—1571no—
355C22×C4—1571no—
356C22×C4C22×D15711yescharacteristic
357C22×C4C22×D15711yescharacteristic
358C22×C4C22×D15711yescharacteristic
359C22×C4C22×D15711yescharacteristic
360C4⋊C4—3142no—
361C4⋊C4—3142no—
362C4⋊C4—6281no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
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⋊C4—1574no—
376C22⋊C4C22⋊D15714yescomplemented
377C42—3142no—
378C42—3142no—
379C42—6281no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—28no—
389C23—28no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23—44no—
400C23—44no—
401C23—3141no—
402C23—3141no—
403C23—3141no—
404C23—3141no—
405C23—3141no—
406C23—3141no—
407C23—1572no—
408C23C2×C4×D15712yes—
409C23C2×C4×D15712yes—
410C23C2×C4×D15712yes—
411C23D4×D15712yes—
412C23C2×C22⋊D15712yes—
413C23#1457895312yes—
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—44no—
430C2×C4—44no—
431C2×C4—3141no—
432C2×C4—3141no—
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—3141no—
447C2×C4—6284no—
448C2×C4—6284no—
449C2×C4—6284no—
450C2×C4—1572no—
451C2×C4—1572no—
452C2×C4—1572no—
453C2×C4—1572no—
454C2×C4—1572no—
455C2×C4C2×C22⋊D15712yes—
456C2×C4C2×C22⋊D15712yes—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—416no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—3142no—
478C22—3142no—
479C22—3142no—
480C22—6281no—
481C22—6281no—
482C22—6281no—
483C22—6281no—
484C22#1909773114yescomplemented
485C22C4×C22⋊D15714yescomplemented
486C22C22×C4×D15711yescharacteristic
487C22C2×D4×D15711yescharacteristic
488C22C22×C22⋊D15711yescharacteristic
489C22C2×#1457895311yescharacteristic
490C22C2×#1457895411yescharacteristic
491C22#1909781611yescharacteristic
492C22#1909781611yescharacteristic
493C4—22no—
494C4—22no—
495C4—41no—
496C4—41no—
497C4—3142no—
498C4—3142no—
499C4—3142no—
500C4—3142no—
501C4—3142no—
502C4—6281no—
503C4—6281no—
504C2—24no—
505C2—24no—
506C2—42no—
507C2—6282no—
508C2C2×#1909773111yescharacteristic
509C2C2×C4×C22⋊D15711yescharacteristic
510C2#2375774611yescharacteristic
511C2D4×C22⋊D15711yescharacteristic
512C2#2375791311yescharacteristic
513C2#2375791711yescharacteristic
514C2#2375794911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D157
commutatorC22×C157C23×C4
FittingC22×C22⋊C4×C157C2
socle, cosocleC23×C157C24
prime radicalC2×#19097740C2
hypercenterC22×C22⋊C4D157
hypocenterC157#2358
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D157C2semidirect
3. C2×#19097731C2semidirect
4. C22×C22⋊Dic157C2semidirect
5. #23757173C2semidirect
6. #23757167C2semidirect
7. #23757161C2semidirect
8. #23757152C2semidirect
9. #23757150C2semidirect
10. #23757149C2semidirect
11. #23757146C2semidirect
12. #23757145C2semidirect
13. C22⋊C4×Dic157C2semidirect
14. C4×C22⋊Dic157C2semidirect
15. C22×C22⋊C4×C157C2semidirect
21. C22×C4×D157C22semidirect
22. #19097740C4semidirect
25. C2×#14578933C22semidirect
26. C2×#14578933C22semidirect
27. C2×#14578933C22semidirect
36. C2×#14578931C22semidirect
37. C2×#14578931C22semidirect
38. C2×#14578931C22semidirect
39. C2×C4×Dic157C22semidirect
40. C2×C4×Dic157C22semidirect
41. C2×C4×Dic157C22semidirect
42. #19097678C22semidirect
43. #19097678C22semidirect
44. #19097678C22semidirect
45. #19097678C22semidirect
46. #19097678C22semidirect
47. #19097678C22semidirect
63. C2×C22⋊D157C2×C4semidirect
64. C2×C22⋊D157C2×C4semidirect
65. C2×C22⋊D157C2×C4semidirect
88. C22⋊Dic157C2×C4semidirect
89. C22⋊Dic157C2×C4semidirect
90. C22⋊Dic157C2×C4semidirect
91. C22⋊Dic157D4semidirect
104. C24×C157C2×C4semidirect
121. C22⋊C4×C157D4semidirect
219. C22×C157C4×D4semidirect
220. C22×C157C4×D4semidirect
257. C157#2358semidirect
263. C22×C22⋊C4D157semidirect
321. C24C4×D157semidirect
376. C22⋊C4C22⋊D157semidirect
484. C22#19097731semidirect
485. C22C4×C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977402
C2×C4×C22⋊D1572
C2×#190977312
C22×C22⋊Dic1572
#237571732
#237571672
#237571612
#237571522
#237571502
#237571492
#237571462
#237571452
C22⋊C4×Dic1572
C4×C22⋊Dic1572
C22×C22⋊C4×C1572
#2358157