Finite Group Databasebeta

← #34103478

Subgroups — #34103478

Order 20096.

908 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14578954C211yesmaximal, characteristic, complemented
2C2×#19097748C211yesmaximal, characteristic, complemented
3C2×#19097743C211yesmaximal, characteristic, complemented
4C2×#19097731C211yesmaximal, characteristic, complemented
5#23757191C211yesmaximal, characteristic, complemented
6#23757189C211yesmaximal, characteristic, complemented
7#23757182C211yesmaximal, characteristic, complemented
8#23757181C211yesmaximal, characteristic, complemented
9#23757177C211yesmaximal, characteristic, complemented
10#23757173C211yesmaximal, characteristic, complemented
11#23757149C211yesmaximal, characteristic, complemented
12#23757147C211yesmaximal, characteristic, complemented
13#23757146C211yesmaximal, characteristic, complemented
14#23757132C211yesmaximal, characteristic, complemented
15#386×C157C211yesmaximal, characteristic, complemented
16C2×#14578954—24no—
17C2×#14578954—24no—
18C2×#14578954—24no—
19C22×Dic314C2211yescharacteristic
20C22×D628C2211yescharacteristic
21C22×C22⋊D157C2211yescharacteristic
22C22×C22⋊D157C2211yescharacteristic
23C22×C22⋊D157C2211yescharacteristic
24C22×C22⋊D157C2211yescharacteristic
25C23×Dic157C2211yescharacteristic
26C22×C4×D157C2211yescharacteristic
27C22×C4×D157C2211yescharacteristic
28#19097748—24no—
29#19097743—24no—
30#19097731—24no—
31C2×C4⋊Dic157C2211yescharacteristic
32C2×#14578933C2211yescharacteristic
33C2×#14578933C2211yescharacteristic
34C2×#14578933C2211yescharacteristic
35C2×#14578933C2211yescharacteristic
36C2×C22⋊Dic157C2211yescharacteristic
37C2×C22⋊Dic157C2211yescharacteristic
38C2×C22⋊Dic157C2211yescharacteristic
39C2×#14578931C2211yescharacteristic
40C2×#14578931C2211yescharacteristic
41C2×#14578931C2211yescharacteristic
42C2×#14578931C2211yescharacteristic
43C2×#14578931C2211yescharacteristic
44C2×#14578931C2211yescharacteristic
45C2×C4×Dic157C2211yescharacteristic
46#19097678C2211yescharacteristic
47#19097678C2211yescharacteristic
48#19097678C2211yescharacteristic
49#19097678C2211yescharacteristic
50C22×D4×C157C2211yescharacteristic
51C23×C4×C157C2211yescharacteristic
52C2×C22⋊C4×C157C2211yescharacteristic
53C2×C22⋊C4×C157C2211yescharacteristic
54C2×C22⋊C4×C157C2211yescharacteristic
55#95×C157C2211yescharacteristic
56#95×C157C2211yescharacteristic
57C23×D157C2311yescharacteristic
58C23×D157C2311yescharacteristic
59#14578954—232no—
60C2×Dic314—22no—
61C2×Dic314—22no—
62C2×Dic314D414yescomplemented
63C2×D628—22no—
64C2×D628—22no—
65C2×D628—22no—
66C2×C22⋊D157—22no—
67C2×C22⋊D157—22no—
68C2×C22⋊D157—22no—
69C2×C22⋊D157—22no—
70C2×C22⋊D157—22no—
71C2×C22⋊D157—22no—
72C2×C22⋊D157—22no—
73C2×C22⋊D157—22no—
74C2×C22⋊D157—22no—
75C2×C22⋊D157—22no—
76C2×C22⋊D157—41no—
77C2×C22⋊D157D414yescomplemented
78C22×Dic157—22no—
79C22×Dic157—22no—
80C22×Dic157—22no—
81C22×Dic157C2311yescharacteristic
82C22×Dic157C2311yescharacteristic
83C22×Dic157C2311yescharacteristic
84C22×Dic157C2311yescharacteristic
85C22×Dic157C2311yescharacteristic
86C22×Dic157C2311yescharacteristic
87C2×C4×D157—22no—
88C2×C4×D157—22no—
89C2×C4×D157—22no—
90C2×C4×D157—22no—
91C2×C4×D157—22no—
92C2×C4×D157D414yescomplemented
93C4⋊Dic157—22no—
94#14578933—22no—
95#14578933—22no—
96#14578933—22no—
97#14578933—22no—
98C22⋊Dic157—22no—
99C22⋊Dic157—22no—
100C22⋊Dic157—22no—
101#14578931—22no—
102#14578931—22no—
103#14578931—22no—
104#14578931—41no—
105#14578931—41no—
106#14578931—41no—
107C4×Dic157—22no—
108C24×C157C2311yescharacteristic
109C24×C157C2311yescharacteristic
110C2×D4×C157—22no—
111C2×D4×C157—22no—
112C2×D4×C157—41no—
113C22×C4×C157—22no—
114C22×C4×C157—22no—
115C22×C4×C157—22no—
116C22×C4×C157C2311yescharacteristic
117C22×C4×C157C2311yescharacteristic
118C22×C4×C157C2311yescharacteristic
119C22×C4×C157C2311yescharacteristic
120C22×C4×C157C2311yescharacteristic
121C22⋊C4×C157—22no—
122C22⋊C4×C157—22no—
123C22⋊C4×C157—41no—
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—21no—
133C22×D157—21no—
134C22×D157—21no—
135C22×D157—21no—
136C22×D157C2×D412yes—
137C22×D157C2×D412yes—
138Dic314—28no—
139D628—28no—
140C22⋊D157—28no—
141C22⋊D157—28no—
142C22⋊D157—28no—
143C22⋊D157—44no—
144C2×Dic157—21no—
145C2×Dic157—21no—
146C2×Dic157—21no—
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—44no—
158C2×Dic157C2×D412yes—
159C2×Dic157C2×D412yes—
160C2×Dic157C2×D412yes—
161C2×Dic157C2×D412yes—
162C2×Dic157#4112yescomplemented
163C4×D157—28no—
164C4×D157—28no—
165C23×C157—21no—
166C23×C157—21no—
167C23×C157—21no—
168C23×C157—21no—
169C23×C157—21no—
170C23×C157—21no—
171C23×C157—21no—
172C23×C157—21no—
173C23×C157—21no—
174C23×C157—21no—
175C23×C157—21no—
176C23×C157—21no—
177C23×C157C2×D412yes—
178C23×C157#4112yescomplemented
179C23×C157C2411yescharacteristic
180D4×C157—44no—
181C2×C4×C157—28no—
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—21no—
194C2×C4×C157—21no—
195C2×C4×C157C2×D412yes—
196C2×C4×C157C2×D412yes—
197C22×C157—22no—
198C22×C157—22no—
199C22×C157—22no—
200C22×C157—22no—
201C22×C157—22no—
202C22×C157—22no—
203C22×C157—41no—
204C22×C157—41no—
205C22×C157—41no—
206C22×C157—41no—
207C22×C157—41no—
208C22×C157—41no—
209C22×C157—41no—
210C22×C157—41no—
211C22×C157C22×D411yescharacteristic
212C22×C157C22×D411yescharacteristic
213C22×C157C22×D411yescharacteristic
214C22×C157C2×#4111yescharacteristic
215C22×C157C2×#4111yescharacteristic
216C22×C157H(5,𝔽2)11yescharacteristic
217C22×C1572−1+411yescharacteristic
218C2×D157—22no—
219C2×D157—22no—
220C2×D157—22no—
221C2×D157—22no—
222C2×D157—22no—
223C2×D157—22no—
224C2×D157—41no—
225C2×D157—41no—
226C2×D157—41no—
227C2×D157—41no—
228C2×D157—41no—
229C2×D157—41no—
230C2×D157—41no—
231C2×D157—41no—
232C4×C157—22no—
233C4×C157—22no—
234C4×C157—41no—
235C4×C157—41no—
236C4×C157—41no—
237Dic157—22no—
238Dic157—22no—
239Dic157—41no—
240Dic157—41no—
241Dic157—41no—
242Dic157#12014yes—
243C2×C157—42no—
244C2×C157—42no—
245C2×C157C2×#12011yescharacteristic
246C2×C157#52511yescharacteristic
247C2×C157#53711yescharacteristic
248C2×C157#54611yescharacteristic
249C2×C157#54711yescharacteristic
250C2×C157#54811yescharacteristic
251C2×C157#55011yescharacteristic
252D157—42no—
253D157—42no—
254C157#240711yescharacteristic, complemented, Sylow
255#2407—1571nomaximal, Sylow
256C22×#41—1571no—
257C2×C22⋊Q8—1571no—
258C2×#121—1571no—
259C2×C4×D4—1571no—
260#394—1571no—
261#394—1571no—
262#392—1571no—
263#392—1571no—
264#391—1571no—
265#390—1571no—
266#386—1571no—
267#386D15711yescharacteristic, complemented
268#385—1571no—
269#385—1571no—
270#380—1571no—
271C2×#41—3144no—
272C2×#41—3144no—
273C2×#41—3144no—
274C22×Q8—1571no—
275C22×D4—1571no—
276C22×D4—1571no—
277C22×D4—1571no—
278C22×D4—1571no—
279C22×D4—1571no—
280C22×D4C2×D15711yescharacteristic
281C23×C4—1571no—
282C23×C4—1571no—
283C23×C4—1571no—
284C23×C4C2×D15711yescharacteristic
285C22⋊Q8—3144no—
286#121—3144no—
287C4×D4—3144no—
288C2×C4⋊C4—1571no—
289C2×C4⋊C4—1571no—
290C2×C4⋊C4—1571no—
291C2×C4⋊C4—1571no—
292C2×C4⋊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⋊C4C2×D15711yescharacteristic
303C2×C22⋊C4C2×D15711yescharacteristic
304C2×C22⋊C4C2×D15711yescharacteristic
305C2×C42—1571no—
306#95—1571no—
307#95—1571no—
308#95—1571no—
309#95—1571no—
310#95C2×D15711yescharacteristic
311#95C2×D15711yescharacteristic
312C24—1571no—
313C24—1571no—
314C24C22×D15711yescharacteristic
315C24C22×D15711yescharacteristic
316#41—31432no—
317C2×Q8—3142no—
318C2×Q8—3142no—
319C2×Q8—1574no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—41no—
323C2×D4—3142no—
324C2×D4—3142no—
325C2×D4—3142no—
326C2×D4—3142no—
327C2×D4—3142no—
328C2×D4—3142no—
329C2×D4—3142no—
330C2×D4—3142no—
331C2×D4—3142no—
332C2×D4—3142no—
333C2×D4—3142no—
334C2×D4—3142no—
335C2×D4—3142no—
336C2×D4—6281no—
337C2×D4—1574no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—3142no—
342C22×C4—3142no—
343C22×C4—3142no—
344C22×C4—3142no—
345C22×C4—3142no—
346C22×C4—3142no—
347C22×C4—3142no—
348C22×C4—3142no—
349C22×C4—1571no—
350C22×C4—1571no—
351C22×C4—1571no—
352C22×C4—1571no—
353C22×C4—1571no—
354C22×C4—1574no—
355C22×C4—1571no—
356C22×C4C22×D15711yescharacteristic
357C22×C4C22×D15711yescharacteristic
358C22×C4C22×D15711yescharacteristic
359C22×C4C22×D15711yescharacteristic
360C22×C4C22×D15711yescharacteristic
361C4⋊C4—3142no—
362C4⋊C4—3142no—
363C4⋊C4—3142no—
364C4⋊C4—3142no—
365C4⋊C4—3142no—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—41no—
369C22⋊C4—3142no—
370C22⋊C4—3142no—
371C22⋊C4—3142no—
372C22⋊C4—3142no—
373C22⋊C4—3142no—
374C22⋊C4—3142no—
375C22⋊C4—6281no—
376C22⋊C4—6281no—
377C22⋊C4—6281no—
378C42—3142no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—3141no—
392C23—3141no—
393C23—3141no—
394C23—3141no—
395C23—3141no—
396C23—3141no—
397C23—3141no—
398C23—3141no—
399C23—3141no—
400C23—3141no—
401C23—3141no—
402C23—3141no—
403C23—1572no—
404C23—1572no—
405C23D4×D15712yes—
406C23#1457895412yescomplemented
407C23C23×D15711yescharacteristic
408Q8—3148no—
409D4—44no—
410D4—3148no—
411D4—3148no—
412D4—3148no—
413D4—3148no—
414D4—6284no—
415C2×C4—28no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
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—3148no—
430C2×C4—3148no—
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—6284no—
445C2×C4—1572no—
446C2×C4—1572no—
447C2×C4—1572no—
448C2×C4—1572no—
449C2×C4—1572no—
450C2×C4D4×D15712yes—
451C2×C4D4×D15712yes—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—3142no—
467C22—3142no—
468C22—3142no—
469C22—3142no—
470C22—3142no—
471C22—3142no—
472C22—6281no—
473C22—6281no—
474C22—6281no—
475C22—6281no—
476C22—6281no—
477C22—6281no—
478C22—6281no—
479C22—6281no—
480C22C2×D4×D15711yescharacteristic
481C22C2×D4×D15711yescharacteristic
482C22C2×D4×D15711yescharacteristic
483C22#41×D15711yescharacteristic
484C22C2×#1457895411yescharacteristic
485C22#1909781611yescharacteristic
486C22#1909781811yescharacteristic
487C4—22no—
488C4—22no—
489C4—41no—
490C4—41no—
491C4—41no—
492C4—3142no—
493C4—3142no—
494C4—6281no—
495C4—6281no—
496C4—6281no—
497C4—1574no—
498C2—42no—
499C2—42no—
500C2—6282no—
501C2—6282no—
502C2#120×D15711yescharacteristic
503C2#2375778211yescharacteristic
504C2#2375785211yescharacteristic
505C2#2375791311yescharacteristic
506C2#2375792111yescharacteristic
507C2#2375793211yescharacteristic
508C2#2375794111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D157
commutator, socle, cosocleC23×C157C24
Fitting#386×C157C2
hypercenter#386D157
hypocenterC157#2407
LCSG ▷ C23×C157 ▷ C157
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C22×#14578954C2semidirect
2. C2×#19097748C2semidirect
3. C2×#19097743C2semidirect
4. C2×#19097731C2semidirect
5. #23757191C2semidirect
6. #23757189C2semidirect
7. #23757182C2semidirect
8. #23757181C2semidirect
9. #23757177C2semidirect
10. #23757173C2semidirect
11. #23757149C2semidirect
12. #23757147C2semidirect
13. #23757146C2semidirect
14. #23757132C2semidirect
15. #386×C157C2semidirect
62. C2×Dic314D4semidirect
77. C2×C22⋊D157D4semidirect
92. C2×C4×D157D4semidirect
162. C2×Dic157#41semidirect
178. C23×C157#41semidirect
254. C157#2407semidirect
267. #386D157semidirect
406. C23#14578954semidirect

Maximal subgroups

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

subgroupindex
C22×#145789542
C2×#190977482
C2×#190977432
C2×#190977312
#237571912
#237571892
#237571822
#237571812
#237571772
#237571732
#237571492
#237571472
#237571462
#237571322
#386×C1572
#2407157