Finite Group Databasebeta

← #34102789

Subgroups — #34102789

Order 20096.

808 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097741C211yesmaximal, characteristic
2C2×#19097733C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D157C211yesmaximal, characteristic, complemented
4#23757175C211yesmaximal, characteristic, complemented
5#23757173C211yesmaximal, characteristic, complemented
6#23757166C211yesmaximal, characteristic, complemented
7#23757163C211yesmaximal, characteristic, complemented
8#23757162C211yesmaximal, characteristic, complemented
9#23757151C211yesmaximal, characteristic, complemented
10#23757151C211yesmaximal, characteristic, complemented
11#23757149C211yesmaximal, characteristic, complemented
12#23757147C211yesmaximal, characteristic, complemented
13#23757128C211yesmaximal, characteristic, complemented
14C4×#14578931C211yesmaximal, characteristic, complemented
15#388×C157C211yesmaximal, characteristic, complemented
16C24×D157C2211yescharacteristic
17C22×D628C2211yescharacteristic
18C22×D628C2211yescharacteristic
19C22×C22⋊D157C2211yescharacteristic
20C22×C4×D157C2211yescharacteristic
21C22×C4×D157C2211yescharacteristic
22C22×C4×D157C2211yescharacteristic, complemented
23#19097741C418yescomplemented
24#19097733—24no—
25C2×C4⋊Dic157C2211yescharacteristic, complemented
26C2×#14578933C2211yescharacteristic, complemented
27C2×C22⋊Dic157C2211yescharacteristic
28C2×C22⋊Dic157C2211yescharacteristic
29C2×#14578931C2211yescharacteristic
30C2×#14578931C2211yescharacteristic
31C2×#14578931C2211yescharacteristic
32C2×#14578931C2211yescharacteristic
33C2×#14578931C2211yescharacteristic
34C2×#14578931C2211yescharacteristic
35C2×#14578931C2211yescharacteristic
36C2×#14578931C2211yescharacteristic
37C2×#14578931C2211yescharacteristic, complemented
38C2×#14578931C2211yescharacteristic, complemented
39C2×#14578931C2211yescharacteristic, complemented
40C22⋊C4×D157—24no—
41C2×C4×Dic157C2211yescharacteristic, complemented
42C2×C4×Dic157C2211yescharacteristic, complemented
43#19097678C2211yescharacteristic, complemented
44#19097678C2211yescharacteristic, complemented
45#19097678C2211yescharacteristic, complemented
46#19097678C2211yescharacteristic, complemented
47C2×C4⋊C4×C157C2211yescharacteristic, complemented
48C2×C22⋊C4×C157C2211yescharacteristic
49C2×C22⋊C4×C157C2211yescharacteristic
50C2×C22⋊C4×C157C2211yescharacteristic
51C2×C42×C157C2211yescharacteristic, complemented
52#95×C157C2211yescharacteristic, complemented
53#95×C157C2211yescharacteristic, complemented
54C23×D157—22no—
55C23×D157—22no—
56C23×D157—41no—
57C23×D157—41no—
58C23×D157—41no—
59C23×D157—41no—
60C23×D157C2×C414yescomplemented
61C23×D157C2311yescharacteristic
62C23×D157C2311yescharacteristic
63C23×D157C2311yescharacteristic
64C2×D628—22no—
65C2×D628—22no—
66C2×D628—41no—
67C2×D628—41no—
68C2×D628C2×C414yes—
69C2×D628C2×C414yescomplemented
70C2×C22⋊D157—41no—
71C2×C22⋊D157—41no—
72C2×C22⋊D157C2×C414yescomplemented
73C22×Dic157C2311yescharacteristic
74C22×Dic157C2311yescharacteristic
75C22×Dic157C2311yescharacteristic
76C22×Dic157C2311yescharacteristic
77C22×Dic157C2311yescharacteristic
78C2×C4×D157—22no—
79C2×C4×D157—22no—
80C2×C4×D157—41no—
81C2×C4×D157—41no—
82C2×C4×D157—41no—
83C2×C4×D157—41no—
84C2×C4×D157—41no—
85C2×C4×D157—41no—
86C2×C4×D157D414yescomplemented
87C4⋊Dic157—41no—
88#14578933—41no—
89C22⋊Dic157—22no—
90C22⋊Dic157—41no—
91#14578931—22no—
92#14578931—22no—
93#14578931—22no—
94#14578931—22no—
95#14578931—22no—
96#14578931—41no—
97#14578931—41no—
98#14578931—41no—
99#14578931—41no—
100#14578931C2×C414yescomplemented
101#14578931C2×C414yescomplemented
102C4×Dic157—22no—
103C4×Dic157—41no—
104C24×C157C2311yescharacteristic
105C22×C4×C157C2311yescharacteristic
106C22×C4×C157C2311yescharacteristic
107C22×C4×C157C2311yescharacteristic
108C22×C4×C157C2311yescharacteristic
109C22×C4×C157C2311yescharacteristic
110C22×C4×C157C2311yescharacteristic
111C4⋊C4×C157—22no—
112C22⋊C4×C157—22no—
113C22⋊C4×C157—41no—
114C22⋊C4×C157C2×C414yescomplemented
115C42×C157—41no—
116C22×D157—21no—
117C22×D157—21no—
118C22×D157—21no—
119C22×D157—21no—
120C22×D157—21no—
121C22×D157—21no—
122C22×D157—21no—
123C22×D157—21no—
124C22×D157—21no—
125C22×D157—21no—
126C22×D157—21no—
127C22×D157—21no—
128C22×D157—21no—
129C22×D157—21no—
130C22×D157—21no—
131C22×D157—21no—
132C22×D157—28no—
133C22×D157—44no—
134C22×D157—44no—
135C22×D157—44no—
136C22×D157—44no—
137C22×D157—44no—
138C22×D157—44no—
139C22×D157C22×C412yes—
140C22×D157C22×C412yes—
141C22×D157C22×C412yes—
142C22×D157C2×D412yes—
143C22×D157#4112yes—
144D628—28no—
145D628—44no—
146C22⋊D157—44no—
147C2×Dic157—21no—
148C2×Dic157—21no—
149C2×Dic157—21no—
150C2×Dic157—21no—
151C2×Dic157—21no—
152C2×Dic157—21no—
153C2×Dic157—21no—
154C2×Dic157—21no—
155C2×Dic157—21no—
156C2×Dic157—21no—
157C2×Dic157—21no—
158C2×Dic157—21no—
159C2×Dic157—21no—
160C2×Dic157C22×C412yes—
161C2×Dic157C2×D412yes—
162C4×D157—44no—
163C4×D157—44no—
164C4×D157—44no—
165C23×C157—21no—
166C23×C157—21no—
167C23×C157—21no—
168C23×C157—21no—
169C23×C157—21no—
170C23×C157—21no—
171C23×C157C22×C412yes—
172C23×C157C2411yescharacteristic
173C2×C4×C157—21no—
174C2×C4×C157—21no—
175C2×C4×C157—21no—
176C2×C4×C157—21no—
177C2×C4×C157—21no—
178C2×C4×C157—21no—
179C2×C4×C157—21no—
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×C157C22×C412yes—
189C2×C4×C157C22×C412yes—
190C2×C4×C157C2×D412yes—
191C22×C157—22no—
192C22×C157—22no—
193C22×C157—22no—
194C22×C157—41no—
195C22×C157—41no—
196C22×C157—41no—
197C22×C157—41no—
198C22×C157C23×C411yescharacteristic
199C22×C157C22×D411yescharacteristic
200C22×C157C2×#4111yescharacteristic
201C22×C157H(5,𝔽2)11yescharacteristic
202C22×C157H(5,𝔽2)11yescharacteristic
203C22×C157H(5,𝔽2)11yescharacteristic
204C22×C157H(5,𝔽2)11yescharacteristic
205C2×D157—22no—
206C2×D157—22no—
207C2×D157—22no—
208C2×D157—22no—
209C2×D157—22no—
210C2×D157—22no—
211C2×D157—22no—
212C2×D157—22no—
213C2×D157—22no—
214C2×D157—22no—
215C2×D157—22no—
216C2×D157—22no—
217C2×D157—416no—
218C2×D157—41no—
219C2×D157—41no—
220C2×D157—41no—
221C2×D157—41no—
222C2×D157—41no—
223C2×D157—41no—
224C2×D157—41no—
225C2×D157—41no—
226C2×D157C4×D414yes—
227C4×C157—22no—
228C4×C157—22no—
229C4×C157—22no—
230C4×C157—41no—
231C4×C157—41no—
232C4×C157—41no—
233Dic157—22no—
234Dic157—22no—
235Dic157—41no—
236Dic157—41no—
237Dic157—41no—
238C2×C157—42no—
239C2×C157C2×C4×D411yescharacteristic
240C2×C157#51811yescharacteristic
241C2×C157#51811yescharacteristic
242C2×C157#53411yescharacteristic
243C2×C157#53511yescharacteristic
244C2×C157#53811yescharacteristic
245C2×C157#53811yescharacteristic
246D157—24no—
247D157—42no—
248D157—42no—
249C157#232111yescharacteristic, complemented, Sylow
250#2321—1571nomaximal, Sylow
251C2×#120—1571no—
252C2×C4×D4—1571no—
253C22×C22⋊C4—1571no—
254#390—1571no—
255#390—1571no—
256#388—1571no—
257#388—1571no—
258#388—1571no—
259#388D15711yescharacteristic, complemented
260#386—1571no—
261#386—1571no—
262#386—1571no—
263#385—1571no—
264#379—1571no—
265C4×C22⋊C4—1571no—
266C25—1571no—
267C22×D4—1571no—
268C22×D4—1571no—
269C22×D4—1571no—
270C23×C4—1571no—
271C23×C4—1571no—
272C23×C4—1571no—
273#120—1578no—
274C4×D4—3144no—
275C2×C4⋊C4—1571no—
276C2×C4⋊C4—1571no—
277C2×C4⋊C4C2×D15711yescharacteristic, complemented
278C2×C22⋊C4—3144no—
279C2×C22⋊C4—1571no—
280C2×C22⋊C4—1571no—
281C2×C22⋊C4—1571no—
282C2×C22⋊C4—1571no—
283C2×C22⋊C4—1571no—
284C2×C22⋊C4—1571no—
285C2×C22⋊C4—1571no—
286C2×C22⋊C4—1571no—
287C2×C22⋊C4—1571no—
288C2×C22⋊C4—1571no—
289C2×C22⋊C4—1571no—
290C2×C22⋊C4—1571no—
291C2×C22⋊C4—1571no—
292C2×C22⋊C4C2×D15711yescharacteristic
293C2×C22⋊C4C2×D15711yescharacteristic
294C2×C22⋊C4C2×D15711yescharacteristic
295C2×C42—1571no—
296C2×C42—1571no—
297C2×C42C2×D15711yescharacteristic, complemented
298#95—1571no—
299#95—1571no—
300#95—1571no—
301#95—1571no—
302#95C2×D15711yescharacteristic, complemented
303#95C2×D15711yescharacteristic, complemented
304C24—3142no—
305C24—3142no—
306C24—6281no—
307C24—6281no—
308C24—6281no—
309C24—6281no—
310C24—1571no—
311C24—1571no—
312C24—1571no—
313C24—1574no—
314C24C22×D15711yescharacteristic
315C2×D4—3142no—
316C2×D4—3142no—
317C2×D4—6281no—
318C2×D4—6281no—
319C2×D4—6281no—
320C2×D4—6281no—
321C2×D4—1574no—
322C2×D4—1574no—
323C2×D4—1574no—
324C22×C4—3142no—
325C22×C4—3142no—
326C22×C4—6281no—
327C22×C4—6281no—
328C22×C4—6281no—
329C22×C4—6281no—
330C22×C4—6281no—
331C22×C4—6281no—
332C22×C4—1571no—
333C22×C4—1571no—
334C22×C4—1571no—
335C22×C4—1571no—
336C22×C4—1571no—
337C22×C4—1574no—
338C22×C4C22×D15711yescharacteristic
339C22×C4C22×D15711yescharacteristic
340C22×C4C22×D15711yescharacteristic
341C22×C4C22×D15711yescharacteristic
342C22×C4C22×D15711yescharacteristic
343C22×C4C22×D15711yescharacteristic
344C4⋊C4—22no—
345C4⋊C4—6281no—
346C4⋊C4—6281no—
347C22⋊C4—22no—
348C22⋊C4—41no—
349C22⋊C4—3142no—
350C22⋊C4—3142no—
351C22⋊C4—3142no—
352C22⋊C4—3142no—
353C22⋊C4—3142no—
354C22⋊C4—3142no—
355C22⋊C4—6281no—
356C22⋊C4—6281no—
357C22⋊C4—6281no—
358C22⋊C4—6281no—
359C22⋊C4—6281no—
360C22⋊C4—1574no—
361C22⋊C4—1574no—
362C22⋊C4C4×D15714yescomplemented
363C42—41no—
364C42—3142no—
365C42—6281no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—3148no—
373C23—3141no—
374C23—3141no—
375C23—3141no—
376C23—3141no—
377C23—3141no—
378C23—3141no—
379C23—3141no—
380C23—3141no—
381C23—3141no—
382C23—3141no—
383C23—3141no—
384C23—3141no—
385C23—3141no—
386C23—3141no—
387C23—3141no—
388C23—3141no—
389C23—6284no—
390C23—6284no—
391C23—6284no—
392C23—6284no—
393C23—6284no—
394C23—6284no—
395C23—1572no—
396C23—1572no—
397C23—1572no—
398C23—1572no—
399C23—1572no—
400C23C2×C4×D15712yes—
401C23C23×D15711yescharacteristic
402D4—3148no—
403D4—6284no—
404D4—6284no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—3141no—
421C2×C4—3141no—
422C2×C4—3141no—
423C2×C4—3141no—
424C2×C4—3141no—
425C2×C4—3141no—
426C2×C4—3141no—
427C2×C4—3141no—
428C2×C4—3141no—
429C2×C4—3141no—
430C2×C4—3141no—
431C2×C4—3141no—
432C2×C4—3141no—
433C2×C4—6284no—
434C2×C4—6284no—
435C2×C4—6284no—
436C2×C4—1572no—
437C2×C4—1572no—
438C2×C4C2×C4×D15712yes—
439C2×C4C2×C4×D15712yes—
440C2×C4D4×D15712yes—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—3142no—
449C22—3142no—
450C22—3142no—
451C22—3142no—
452C22—3142no—
453C22—3142no—
454C22—3142no—
455C22—3142no—
456C22—3142no—
457C22—3142no—
458C22—3142no—
459C22—3142no—
460C22—62816no—
461C22—6281no—
462C22—6281no—
463C22—6281no—
464C22—6281no—
465C22—6281no—
466C22—6281no—
467C22—6281no—
468C22—6281no—
469C22—1574no—
470C22C22×C4×D15711yescharacteristic
471C22C2×D4×D15711yescharacteristic
472C22#41×D15711yescharacteristic
473C22#1909781611yescharacteristic
474C22#1909781711yescharacteristic
475C22#1909781711yescharacteristic
476C22#1909781711yescharacteristic
477C4—22no—
478C4—22no—
479C4—22no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—3142no—
484C4—3142no—
485C4—6281no—
486C4—6281no—
487C4—6281no—
488C2—42no—
489C2—3144no—
490C2—6282no—
491C2—6282no—
492C2C4×D4×D15711yescharacteristic
493C2#2375774511yescharacteristic
494C2#2375774711yescharacteristic
495C2#2375783111yescharacteristic
496C2#2375784111yescharacteristic
497C2#2375785911yescharacteristic
498C2#2375786211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D157
commutatorC22×C157C23×C4
Fitting#388×C157C2
socle, cosocleC23×C157C24
prime radicalC2×#19097741C2
hypercenter#388D157
hypocenterC157#2321
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#19097733C2semidirect
3. C2×C22⋊C4×D157C2semidirect
4. #23757175C2semidirect
5. #23757173C2semidirect
6. #23757166C2semidirect
7. #23757163C2semidirect
8. #23757162C2semidirect
9. #23757151C2semidirect
10. #23757151C2semidirect
11. #23757149C2semidirect
12. #23757147C2semidirect
13. #23757128C2semidirect
14. C4×#14578931C2semidirect
15. #388×C157C2semidirect
22. C22×C4×D157C22semidirect
23. #19097741C4semidirect
25. C2×C4⋊Dic157C22semidirect
26. C2×#14578933C22semidirect
37. C2×#14578931C22semidirect
38. C2×#14578931C22semidirect
39. C2×#14578931C22semidirect
41. C2×C4×Dic157C22semidirect
42. C2×C4×Dic157C22semidirect
43. #19097678C22semidirect
44. #19097678C22semidirect
45. #19097678C22semidirect
46. #19097678C22semidirect
47. C2×C4⋊C4×C157C22semidirect
51. C2×C42×C157C22semidirect
52. #95×C157C22semidirect
53. #95×C157C22semidirect
60. C23×D157C2×C4semidirect
69. C2×D628C2×C4semidirect
72. C2×C22⋊D157C2×C4semidirect
86. C2×C4×D157D4semidirect
100. #14578931C2×C4semidirect
101. #14578931C2×C4semidirect
114. C22⋊C4×C157C2×C4semidirect
249. C157#2321semidirect
259. #388D157semidirect
277. C2×C4⋊C4C2×D157semidirect
297. C2×C42C2×D157semidirect
302. #95C2×D157semidirect
303. #95C2×D157semidirect
362. C22⋊C4C4×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977412
C2×#190977332
C2×C22⋊C4×D1572
#237571752
#237571732
#237571662
#237571632
#237571622
#237571512
#237571492
#237571472
#237571282
C4×#145789312
#388×C1572
#2321157