Finite Group Databasebeta

← #34102980

Subgroups — #34102980

Order 20096.

792 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097728C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D157C211yesmaximal, characteristic
3C2×C22⋊C4×D157C211yesmaximal, characteristic
4#23757167C211yesmaximal, characteristic, complemented
5#23757166C211yesmaximal, characteristic, complemented
6#23757161C211yesmaximal, characteristic, complemented
7#23757147C211yesmaximal, characteristic, complemented
8#23757144C211yesmaximal, characteristic, complemented
9#23757135C211yesmaximal, characteristic, complemented
10#23757130C211yesmaximal, characteristic, complemented
11#23757128C211yesmaximal, characteristic
12C4×#14578931C211yesmaximal, characteristic, complemented
13#23757120C211yesmaximal, characteristic, complemented
14#23757120C211yesmaximal, characteristic, complemented
15#388×C157C211yesmaximal, characteristic, complemented
16C24×D157C2211yescharacteristic
17C22×C4×D157C2211yescharacteristic
18C22×C4×D157C2211yescharacteristic
19C22×C4×D157C2211yescharacteristic
20C22×C4×D157C2211yescharacteristic
21#19097728—24no—
22C2×C4⋊Dic157C2211yescharacteristic, complemented
23C2×#14578933C2211yescharacteristic, complemented
24C2×#14578933C2211yescharacteristic, complemented
25C2×C22⋊Dic157C2211yescharacteristic
26C2×C22⋊Dic157C2211yescharacteristic
27C2×C22⋊Dic157C2211yescharacteristic
28C2×#14578931C2211yescharacteristic
29C2×#14578931C2211yescharacteristic
30C2×#14578931C2211yescharacteristic
31C2×#14578931C2211yescharacteristic
32C2×#14578931C2211yescharacteristic
33C2×#14578931C2211yescharacteristic
34C2×#14578931C2211yescharacteristic
35C2×#14578931C2211yescharacteristic
36C22⋊C4×D157—24no—
37C22⋊C4×D157C418yescomplemented
38C2×C4×Dic157C2211yescharacteristic, complemented
39C2×C4×Dic157C2211yescharacteristic, complemented
40C2×C4×Dic157C2211yescharacteristic, complemented
41#19097678C2211yescharacteristic, complemented
42#19097678C2211yescharacteristic, 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—22no—
57C23×D157—22no—
58C23×D157—41no—
59C23×D157—41no—
60C23×D157C2×C414yes—
61C23×D157C2311yescharacteristic
62C23×D157C2311yescharacteristic
63C22×Dic157C2311yescharacteristic
64C22×Dic157C2311yescharacteristic
65C22×Dic157C2311yescharacteristic
66C22×Dic157C2311yescharacteristic
67C22×Dic157C2311yescharacteristic
68C22×Dic157C2311yescharacteristic
69C2×C4×D157—22no—
70C2×C4×D157—22no—
71C2×C4×D157—22no—
72C2×C4×D157—22no—
73C2×C4×D157—22no—
74C2×C4×D157—22no—
75C2×C4×D157—41no—
76C2×C4×D157—41no—
77C2×C4×D157—41no—
78C2×C4×D157—41no—
79C2×C4×D157C2×C414yescomplemented
80C2×C4×D157C2×C414yescomplemented
81C4⋊Dic157—41no—
82#14578933—22no—
83#14578933—22no—
84C22⋊Dic157—22no—
85C22⋊Dic157—41no—
86C22⋊Dic157C2×C414yescomplemented
87#14578931—22no—
88#14578931—22no—
89#14578931—22no—
90#14578931—22no—
91#14578931—22no—
92#14578931C2×C414yes—
93#14578931C2×C414yes—
94#14578931D414yes—
95C4×Dic157—22no—
96C4×Dic157—41no—
97C4×Dic157—41no—
98C24×C157C2311yescharacteristic
99C22×C4×C157C2311yescharacteristic
100C22×C4×C157C2311yescharacteristic
101C22×C4×C157C2311yescharacteristic
102C22×C4×C157C2311yescharacteristic
103C22×C4×C157C2311yescharacteristic
104C22×C4×C157C2311yescharacteristic
105C4⋊C4×C157—41no—
106C22⋊C4×C157—22no—
107C22⋊C4×C157—41no—
108C22⋊C4×C157C2×C414yescomplemented
109C42×C157—22no—
110C22×D157—21no—
111C22×D157—21no—
112C22×D157—21no—
113C22×D157—21no—
114C22×D157—21no—
115C22×D157—21no—
116C22×D157—21no—
117C22×D157—21no—
118C22×D157—28no—
119C22×D157—28no—
120C22×D157—44no—
121C22×D157—44no—
122C22×D157—44no—
123C22×D157—44no—
124C22×D157—44no—
125C22×D157C22×C412yes—
126C22×D157C22×C412yes—
127C22×D157C2×D412yes—
128C22×D157#4112yes—
129C22×D157#4112yes—
130C22×D157#4112yes—
131C2×Dic157—21no—
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×Dic157C22×C412yes—
147C2×Dic157C22×C412yes—
148C2×Dic157C2×D412yes—
149C4×D157—28no—
150C4×D157—44no—
151C4×D157—44no—
152C4×D157—44no—
153C23×C157—21no—
154C23×C157—21no—
155C23×C157—21no—
156C23×C157—21no—
157C23×C157—21no—
158C23×C157—21no—
159C23×C157C22×C412yes—
160C23×C157C2411yescharacteristic
161C2×C4×C157—21no—
162C2×C4×C157—21no—
163C2×C4×C157—21no—
164C2×C4×C157—21no—
165C2×C4×C157—21no—
166C2×C4×C157—21no—
167C2×C4×C157—21no—
168C2×C4×C157—21no—
169C2×C4×C157—21no—
170C2×C4×C157—21no—
171C2×C4×C157—21no—
172C2×C4×C157—21no—
173C2×C4×C157—21no—
174C2×C4×C157—21no—
175C2×C4×C157—21no—
176C2×C4×C157C22×C412yes—
177C2×C4×C157C22×C412yes—
178C2×C4×C157C2×D412yes—
179C22×C157—22no—
180C22×C157—22no—
181C22×C157—22no—
182C22×C157—41no—
183C22×C157—41no—
184C22×C157—41no—
185C22×C157—41no—
186C22×C157C23×C411yescharacteristic
187C22×C157C22×D411yescharacteristic
188C22×C157C2×#4111yescharacteristic
189C22×C157C2×#4111yescharacteristic
190C22×C157C2×#4111yescharacteristic
191C22×C157H(5,𝔽2)11yescharacteristic
192C22×C157H(5,𝔽2)11yescharacteristic
193C2×D157—22no—
194C2×D157—22no—
195C2×D157—22no—
196C2×D157—22no—
197C2×D157—22no—
198C2×D157—22no—
199C2×D157—22no—
200C2×D157—22no—
201C2×D157—22no—
202C2×D157—22no—
203C2×D157—22no—
204C2×D157—22no—
205C2×D157—416no—
206C2×D157#11714yescomplemented
207C2×D157C4×D414yes—
208C4×C157—22no—
209C4×C157—22no—
210C4×C157—22no—
211C4×C157—41no—
212C4×C157—41no—
213C4×C157—41no—
214Dic157—22no—
215Dic157—22no—
216Dic157—22no—
217Dic157—41no—
218Dic157—41no—
219Dic157—41no—
220C2×C157—42no—
221C2×C157C2×#11711yescharacteristic
222C2×C157C2×C4×D411yescharacteristic
223C2×C157#51811yescharacteristic
224C2×C157#54611yescharacteristic
225C2×C157#54611yescharacteristic
226C2×C157#55111yescharacteristic
227C2×C157#55111yescharacteristic
228D157—24no—
229D157—24no—
230C157#234211yescharacteristic, complemented, Sylow
231#2342—1571nomaximal, Sylow
232C2×#117—1571no—
233C22×C22⋊C4—1571no—
234C22×C22⋊C4—1571no—
235#388—1571no—
236#388—1571no—
237#388—1571no—
238#388D15711yescharacteristic, complemented
239#385—1571no—
240#385—1571no—
241#381—1571no—
242#380—1571no—
243#379—1571no—
244C4×C22⋊C4—1571no—
245C4×C22⋊C4—1571no—
246C4×C22⋊C4—1571no—
247C25—1571no—
248C23×C4—1571no—
249C23×C4—1571no—
250C23×C4—1571no—
251C23×C4—1571no—
252#117—3144no—
253C2×C4⋊C4—1571no—
254C2×C4⋊C4—1571no—
255C2×C4⋊C4—1571no—
256C2×C4⋊C4C2×D15711yescharacteristic, complemented
257C2×C22⋊C4—3144no—
258C2×C22⋊C4—1571no—
259C2×C22⋊C4—1571no—
260C2×C22⋊C4—1571no—
261C2×C22⋊C4—1571no—
262C2×C22⋊C4—1571no—
263C2×C22⋊C4—1571no—
264C2×C22⋊C4—1571no—
265C2×C22⋊C4—1571no—
266C2×C22⋊C4—1571no—
267C2×C22⋊C4—1571no—
268C2×C22⋊C4—1571no—
269C2×C22⋊C4—1578no—
270C2×C22⋊C4C2×D15711yescharacteristic
271C2×C22⋊C4C2×D15711yescharacteristic
272C2×C22⋊C4C2×D15711yescharacteristic
273C2×C42—1571no—
274C2×C42—1571no—
275C2×C42—1571no—
276C2×C42C2×D15711yescharacteristic, complemented
277#95—1571no—
278#95—1571no—
279#95—1571no—
280#95—1571no—
281#95—1571no—
282#95—1571no—
283#95C2×D15711yescharacteristic, complemented
284#95C2×D15711yescharacteristic, complemented
285C24—3142no—
286C24—3142no—
287C24—3142no—
288C24—3142no—
289C24—6281no—
290C24—6281no—
291C24—1574no—
292C24—1571no—
293C24—1571no—
294C24C22×D15711yescharacteristic
295C22×C4—3142no—
296C22×C4—3142no—
297C22×C4—3142no—
298C22×C4—3142no—
299C22×C4—3142no—
300C22×C4—3142no—
301C22×C4—6281no—
302C22×C4—6281no—
303C22×C4—6281no—
304C22×C4—6281no—
305C22×C4—1571no—
306C22×C4—1571no—
307C22×C4—1571no—
308C22×C4—1571no—
309C22×C4—1571no—
310C22×C4—1571no—
311C22×C4—1574no—
312C22×C4—1574no—
313C22×C4C22×D15711yescharacteristic
314C22×C4C22×D15711yescharacteristic
315C22×C4C22×D15711yescharacteristic
316C22×C4C22×D15711yescharacteristic
317C22×C4C22×D15711yescharacteristic
318C22×C4C22×D15711yescharacteristic
319C4⋊C4—41no—
320C4⋊C4—3142no—
321C4⋊C4—3142no—
322C4⋊C4—6281no—
323C22⋊C4—22no—
324C22⋊C4—41no—
325C22⋊C4—3142no—
326C22⋊C4—3142no—
327C22⋊C4—3142no—
328C22⋊C4—3142no—
329C22⋊C4—3142no—
330C22⋊C4—3142no—
331C22⋊C4—6281no—
332C22⋊C4—1574no—
333C22⋊C4—1574no—
334C22⋊C4—1574no—
335C22⋊C4—1574no—
336C22⋊C4C4×D15714yescomplemented
337C42—22no—
338C42—3142no—
339C42—6281no—
340C42—6281no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—3148no—
348C23—3148no—
349C23—3141no—
350C23—3141no—
351C23—3141no—
352C23—3141no—
353C23—3141no—
354C23—3141no—
355C23—3141no—
356C23—3141no—
357C23—6284no—
358C23—6284no—
359C23—6284no—
360C23—6284no—
361C23—6284no—
362C23—1572no—
363C23—1572no—
364C23—1572no—
365C23—1572no—
366C23—1572no—
367C23—1572no—
368C23C2×C4×D15712yes—
369C23C23×D15711yescharacteristic
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—3148no—
386C2×C4—3141no—
387C2×C4—3141no—
388C2×C4—3141no—
389C2×C4—3141no—
390C2×C4—3141no—
391C2×C4—3141no—
392C2×C4—3141no—
393C2×C4—3141no—
394C2×C4—3141no—
395C2×C4—3141no—
396C2×C4—3141no—
397C2×C4—3141no—
398C2×C4—3141no—
399C2×C4—3141no—
400C2×C4—3141no—
401C2×C4—6284no—
402C2×C4—6284no—
403C2×C4—6284no—
404C2×C4—1572no—
405C2×C4—1572no—
406C2×C4—1572no—
407C2×C4C2×C4×D15712yes—
408C2×C4C2×C4×D15712yes—
409C2×C4D4×D15712yes—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—3142no—
418C22—3142no—
419C22—3142no—
420C22—3142no—
421C22—3142no—
422C22—3142no—
423C22—3142no—
424C22—3142no—
425C22—3142no—
426C22—3142no—
427C22—3142no—
428C22—3142no—
429C22—62816no—
430C22—1574no—
431C22—1574no—
432C22C22×C4×D15711yescharacteristic
433C22C2×D4×D15711yescharacteristic
434C22#41×D15711yescharacteristic
435C22#41×D15711yescharacteristic
436C22#41×D15711yescharacteristic
437C22#1909781611yescharacteristic
438C22#1909781711yescharacteristic
439C4—22no—
440C4—22no—
441C4—22no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—3142no—
446C4—3142no—
447C4—3142no—
448C4—6281no—
449C4—6281no—
450C4—6281no—
451C2—42no—
452C2—3144no—
453C2—3144no—
454C2#117×D15711yescharacteristic
455C2C4×D4×D15711yescharacteristic
456C2#2375774511yescharacteristic
457C2#2375791511yescharacteristic
458C2#2375792011yescharacteristic
459C2#2375794811yescharacteristic
460C2#2375795311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D157
commutatorC22×C157C23×C4
Fitting#388×C157C2
socle, cosocleC23×C157C24
prime radicalC24×D157C22
hypercenter#388D157
hypocenterC157#2342
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#19097728C2semidirect
4. #23757167C2semidirect
5. #23757166C2semidirect
6. #23757161C2semidirect
7. #23757147C2semidirect
8. #23757144C2semidirect
9. #23757135C2semidirect
10. #23757130C2semidirect
12. C4×#14578931C2semidirect
13. #23757120C2semidirect
14. #23757120C2semidirect
15. #388×C157C2semidirect
22. C2×C4⋊Dic157C22semidirect
23. C2×#14578933C22semidirect
24. C2×#14578933C22semidirect
37. C22⋊C4×D157C4semidirect
38. C2×C4×Dic157C22semidirect
39. C2×C4×Dic157C22semidirect
40. C2×C4×Dic157C22semidirect
41. #19097678C22semidirect
42. #19097678C22semidirect
43. #19097678C22semidirect
44. #19097678C22semidirect
45. #19097678C22semidirect
46. #19097678C22semidirect
47. C2×C4⋊C4×C157C22semidirect
51. C2×C42×C157C22semidirect
52. #95×C157C22semidirect
53. #95×C157C22semidirect
79. C2×C4×D157C2×C4semidirect
80. C2×C4×D157C2×C4semidirect
86. C22⋊Dic157C2×C4semidirect
108. C22⋊C4×C157C2×C4semidirect
206. C2×D157#117semidirect
230. C157#2342semidirect
238. #388D157semidirect
256. C2×C4⋊C4C2×D157semidirect
276. C2×C42C2×D157semidirect
283. #95C2×D157semidirect
284. #95C2×D157semidirect
336. C22⋊C4C4×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977282
C2×C22⋊C4×D1572
#237571672
#237571662
#237571612
#237571472
#237571442
#237571352
#237571302
#237571282
C4×#145789312
#237571202
#388×C1572
#2342157