Finite Group Databasebeta

← #34103514

Subgroups — #34103514

Order 20096.

786 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×#19097753C211yesmaximal, characteristic, complemented
2C2×#19097739C211yesmaximal, characteristic
3C2×#19097729C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D157C211yesmaximal, characteristic, complemented
5#23757196C211yesmaximal, characteristic, complemented
6#23757195C211yesmaximal, characteristic, complemented
7#23757189C211yesmaximal, characteristic, complemented
8#23757176C211yesmaximal, characteristic, complemented
9#23757173C211yesmaximal, characteristic, complemented
10#23757166C211yesmaximal, characteristic, complemented
11#23757165C211yesmaximal, characteristic, complemented
12#23757149C211yesmaximal, characteristic, complemented
13#23757147C211yesmaximal, characteristic, complemented
14#23757128C211yesmaximal, characteristic, complemented
15#394×C157C211yesmaximal, characteristic, complemented
16C24×D157C2211yescharacteristic
17C22×C22⋊D157C2211yescharacteristic
18C22×C22⋊D157C2211yescharacteristic
19C22×C4×D157C2211yescharacteristic
20C22×C4×D157C2211yescharacteristic
21#19097753—24no—
22#19097739—24no—
23#19097729—24no—
24C2×C4⋊Dic157C2211yescharacteristic, complemented
25C2×C4⋊Dic157C2211yescharacteristic, complemented
26C2×C4⋊Dic157C2211yescharacteristic, complemented
27C2×#14578933C2211yescharacteristic, complemented
28C2×#14578933C2211yescharacteristic, complemented
29C2×C22⋊Dic157C2211yescharacteristic
30C2×C22⋊Dic157C2211yescharacteristic
31C2×C22⋊Dic157C2211yescharacteristic
32C2×C22⋊Dic157C2211yescharacteristic, complemented
33C2×C22⋊Dic157C2211yescharacteristic, complemented
34C2×#14578931C2211yescharacteristic
35C2×#14578931C2211yescharacteristic
36C2×#14578931C2211yescharacteristic
37C2×#14578931C2211yescharacteristic
38C2×#14578931C2211yescharacteristic
39C2×#14578931C2211yescharacteristic
40C2×#14578931C2211yescharacteristic
41C2×#14578931C2211yescharacteristic
42C22⋊C4×D157—24no—
43C2×C4×Dic157C2211yescharacteristic, complemented
44C2×C4×Dic157C2211yescharacteristic, complemented
45#19097678C2211yescharacteristic, complemented
46#19097678C2211yescharacteristic, complemented
47#19097678C2211yescharacteristic, complemented
48C22×D4×C157C2211yescharacteristic
49C2×C4⋊C4×C157C2211yescharacteristic, complemented
50C2×C22⋊C4×C157C2211yescharacteristic
51C2×C22⋊C4×C157C2211yescharacteristic
52C2×C22⋊C4×C157C2211yescharacteristic, complemented
53C2×C22⋊C4×C157C2211yescharacteristic, complemented
54#95×C157C2211yescharacteristic, complemented
55C23×D157—22no—
56C23×D157—22no—
57C23×D157—22no—
58C23×D157—22no—
59C23×D157—22no—
60C23×D157—41no—
61C23×D157—41no—
62C23×D157C2311yescharacteristic
63C23×D157C2311yescharacteristic
64C2×C22⋊D157—22no—
65C2×C22⋊D157—22no—
66C2×C22⋊D157—22no—
67C2×C22⋊D157—22no—
68C2×C22⋊D157—22no—
69C2×C22⋊D157—41no—
70C22×Dic157C2311yescharacteristic
71C22×Dic157C2311yescharacteristic
72C22×Dic157C2311yescharacteristic
73C22×Dic157C2311yescharacteristic
74C22×Dic157C2311yescharacteristic
75C22×Dic157C2311yescharacteristic
76C2×C4×D157—22no—
77C2×C4×D157—22no—
78C2×C4×D157—22no—
79C2×C4×D157—41no—
80C2×C4×D157—41no—
81C2×C4×D157D414yescomplemented
82C4⋊Dic157—22no—
83C4⋊Dic157—22no—
84C4⋊Dic157—41no—
85#14578933—22no—
86#14578933—41no—
87C22⋊Dic157—22no—
88C22⋊Dic157—22no—
89C22⋊Dic157—22no—
90C22⋊Dic157—41no—
91C22⋊Dic157—41no—
92#14578931—22no—
93#14578931—22no—
94#14578931—22no—
95#14578931—22no—
96#14578931—22no—
97#14578931—22no—
98#14578931—41no—
99#14578931D414yes—
100C4×Dic157—22no—
101C4×Dic157—22no—
102C24×C157C2311yescharacteristic
103C24×C157C2311yescharacteristic
104C2×D4×C157—22no—
105C2×D4×C157—22no—
106C2×D4×C157—41no—
107C22×C4×C157C2311yescharacteristic
108C22×C4×C157C2311yescharacteristic
109C22×C4×C157C2311yescharacteristic
110C22×C4×C157C2311yescharacteristic
111C22×C4×C157C2311yescharacteristic
112C4⋊C4×C157—22no—
113C22⋊C4×C157—22no—
114C22⋊C4×C157—22no—
115C22⋊C4×C157—22no—
116C22⋊C4×C157—41no—
117C22×D157—21no—
118C22×D157—21no—
119C22×D157—21no—
120C22×D157—21no—
121C22×D157—28no—
122C22×D157—21no—
123C22×D157—21no—
124C22×D157—21no—
125C22×D157—21no—
126C22×D157—21no—
127C22×D157—21no—
128C22×D157—44no—
129C22×D157—44no—
130C22×D157—44no—
131C22×D157—44no—
132C22×D157—44no—
133C22×D157—44no—
134C22×D157C2×D412yes—
135C22×D157C2×D412yes—
136C22×D157#4112yes—
137C22×D157#4112yes—
138C22⋊D157—44no—
139C22⋊D157—44no—
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—21no—
149C2×Dic157—21no—
150C2×Dic157—21no—
151C2×Dic157—21no—
152C2×Dic157—21no—
153C2×Dic157—21no—
154C2×Dic157—21no—
155C2×Dic157—21no—
156C2×Dic157C2×D412yes—
157C2×Dic157C2×D412yes—
158C4×D157—28no—
159C4×D157—44no—
160C23×C157—21no—
161C23×C157—21no—
162C23×C157—21no—
163C23×C157—21no—
164C23×C157—21no—
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#4112yes—
174C23×C157C2411yescharacteristic
175D4×C157—44no—
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×C157—21no—
189C2×C4×C157C2×D412yes—
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×C157—41no—
199C22×C157—41no—
200C22×C157—41no—
201C22×C157—41no—
202C22×C157—41no—
203C22×C157—41no—
204C22×C157—41no—
205C22×C157C22×D411yescharacteristic
206C22×C157C22×D411yescharacteristic
207C22×C157C2×#4111yescharacteristic
208C22×C157C2×#4111yescharacteristic
209C22×C157C2×#4111yescharacteristic
210C22×C157H(5,𝔽2)11yescharacteristic
211C22×C157H(5,𝔽2)11yescharacteristic
212C2×D157—22no—
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—416no—
222C2×D157—41no—
223C2×D157—41no—
224C2×D157—41no—
225C2×D157—41no—
226C2×D157#12114yes—
227C4×C157—22no—
228C4×C157—22no—
229C4×C157—41no—
230C4×C157—41no—
231C4×C157—41no—
232Dic157—22no—
233Dic157—22no—
234Dic157—41no—
235Dic157—41no—
236Dic157—41no—
237Dic157—41no—
238C2×C157—42no—
239C2×C157—81no—
240C2×C157C2×#12111yescharacteristic
241C2×C157#52511yescharacteristic
242C2×C157#53511yescharacteristic
243C2×C157#54611yescharacteristic
244C2×C157#54611yescharacteristic
245C2×C157#55111yescharacteristic
246C2×C157#55111yescharacteristic
247D157—24no—
248D157—42no—
249C157#241111yescharacteristic, complemented, Sylow
250#2411—1571nomaximal, Sylow
251C2×#123—1571no—
252C2×#120—1571no—
253C2×#117—1571no—
254C22×C22⋊C4—1571no—
255#396—1571no—
256#396—1571no—
257#394—1571no—
258#394D15711yescharacteristic, complemented
259#390—1571no—
260#390—1571no—
261#388—1571no—
262#388—1571no—
263#386—1571no—
264#385—1571no—
265#379—1571no—
266C25—1571no—
267C22×D4—1571no—
268C22×D4—1571no—
269C22×D4C2×D15711yescharacteristic
270C23×C4—1571no—
271C23×C4—1571no—
272#123—3144no—
273#120—3144no—
274#117—3144no—
275C2×C4⋊C4—1571no—
276C2×C4⋊C4—1571no—
277C2×C4⋊C4—1571no—
278C2×C4⋊C4—1571no—
279C2×C4⋊C4—1571no—
280C2×C4⋊C4C2×D15711yescharacteristic, complemented
281C2×C22⋊C4—3144no—
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⋊C4—1571no—
293C2×C22⋊C4—1571no—
294C2×C22⋊C4—1571no—
295C2×C22⋊C4C2×D15711yescharacteristic
296C2×C22⋊C4C2×D15711yescharacteristic
297C2×C22⋊C4C2×D15711yescharacteristic, complemented
298C2×C22⋊C4C2×D15711yescharacteristic, complemented
299C2×C42—1571no—
300C2×C42—1571no—
301#95—1571no—
302#95—1571no—
303#95—1571no—
304#95C2×D15711yescharacteristic, complemented
305C24—3142no—
306C24—3142no—
307C24—3142no—
308C24—3142no—
309C24—3142no—
310C24—6281no—
311C24—6281no—
312C24—1571no—
313C24—1571no—
314C24C22×D15711yescharacteristic
315C24C22×D15711yescharacteristic
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—41no—
319C2×D4—3142no—
320C2×D4—3142no—
321C2×D4—3142no—
322C2×D4—3142no—
323C2×D4—3142no—
324C2×D4—6281no—
325C22×C4—3142no—
326C22×C4—3142no—
327C22×C4—3142no—
328C22×C4—6281no—
329C22×C4—6281no—
330C22×C4—1571no—
331C22×C4—1571no—
332C22×C4—1571no—
333C22×C4—1571no—
334C22×C4—1571no—
335C22×C4—1571no—
336C22×C4—1574no—
337C22×C4C22×D15711yescharacteristic
338C22×C4C22×D15711yescharacteristic
339C22×C4C22×D15711yescharacteristic
340C22×C4C22×D15711yescharacteristic
341C22×C4C22×D15711yescharacteristic
342C4⋊C4—22no—
343C4⋊C4—3142no—
344C4⋊C4—3142no—
345C4⋊C4—3142no—
346C4⋊C4—6281no—
347C4⋊C4—6281no—
348C22⋊C4—22no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—41no—
352C22⋊C4—3142no—
353C22⋊C4—3142no—
354C22⋊C4—3142no—
355C22⋊C4—3142no—
356C22⋊C4—3142no—
357C22⋊C4—3142no—
358C22⋊C4—3142no—
359C22⋊C4—3142no—
360C22⋊C4—3142no—
361C22⋊C4—6281no—
362C22⋊C4—6281no—
363C22⋊C4—6281no—
364C22⋊C4—1574no—
365C42—3142no—
366C42—3142no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—3148no—
381C23—3141no—
382C23—3141no—
383C23—3141no—
384C23—3141no—
385C23—3141no—
386C23—3141no—
387C23—3141no—
388C23—3141no—
389C23—3141no—
390C23—3141no—
391C23—6284no—
392C23—6284no—
393C23—6284no—
394C23—6284no—
395C23—6284no—
396C23—6284no—
397C23—1572no—
398C23—1572no—
399C23—1572no—
400C23—1572no—
401C23#1457895312yes—
402C23C23×D15711yescharacteristic
403D4—44no—
404D4—6284no—
405D4—6284no—
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—3148no—
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—3141no—
434C2×C4—3141no—
435C2×C4—3141no—
436C2×C4—6284no—
437C2×C4—1572no—
438C2×C4—1572no—
439C2×C4D4×D15712yes—
440C2×C4D4×D15712yes—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—3142no—
456C22—3142no—
457C22—3142no—
458C22—3142no—
459C22—3142no—
460C22—3142no—
461C22—3142no—
462C22—3142no—
463C22—3142no—
464C22—62816no—
465C22—6281no—
466C22—6281no—
467C22—6281no—
468C22—6281no—
469C22—1574no—
470C22C2×D4×D15711yescharacteristic
471C22C2×D4×D15711yescharacteristic
472C22#41×D15711yescharacteristic
473C22#41×D15711yescharacteristic
474C22C2×#1457895311yescharacteristic
475C22#1909781611yescharacteristic
476C22#1909781711yescharacteristic
477C4—22no—
478C4—22no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—3142no—
483C4—3142no—
484C4—6281no—
485C4—6281no—
486C4—6281no—
487C4—6281no—
488C2—42no—
489C2—81no—
490C2—3144no—
491C2—6282no—
492C2#121×D15711yescharacteristic
493C2#2375778311yescharacteristic
494C2#2375783911yescharacteristic
495C2#2375791011yescharacteristic
496C2#2375791611yescharacteristic
497C2#2375794611yescharacteristic
498C2#2375795111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D157
commutator, socle, cosocleC23×C157C24
Fitting#394×C157C2
prime radicalC2×#19097739C2
hypercenter#394D157
hypocenterC157#2411
LCSG ▷ C23×C157 ▷ C157
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#19097753C2semidirect
3. C2×#19097729C2semidirect
4. C2×C22⋊C4×D157C2semidirect
5. #23757196C2semidirect
6. #23757195C2semidirect
7. #23757189C2semidirect
8. #23757176C2semidirect
9. #23757173C2semidirect
10. #23757166C2semidirect
11. #23757165C2semidirect
12. #23757149C2semidirect
13. #23757147C2semidirect
14. #23757128C2semidirect
15. #394×C157C2semidirect
24. C2×C4⋊Dic157C22semidirect
25. C2×C4⋊Dic157C22semidirect
26. C2×C4⋊Dic157C22semidirect
27. C2×#14578933C22semidirect
28. C2×#14578933C22semidirect
32. C2×C22⋊Dic157C22semidirect
33. C2×C22⋊Dic157C22semidirect
43. C2×C4×Dic157C22semidirect
44. C2×C4×Dic157C22semidirect
45. #19097678C22semidirect
46. #19097678C22semidirect
47. #19097678C22semidirect
49. C2×C4⋊C4×C157C22semidirect
52. C2×C22⋊C4×C157C22semidirect
53. C2×C22⋊C4×C157C22semidirect
54. #95×C157C22semidirect
81. C2×C4×D157D4semidirect
249. C157#2411semidirect
258. #394D157semidirect
280. C2×C4⋊C4C2×D157semidirect
297. C2×C22⋊C4C2×D157semidirect
298. C2×C22⋊C4C2×D157semidirect
304. #95C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977532
C2×#190977392
C2×#190977292
C2×C22⋊C4×D1572
#237571962
#237571952
#237571892
#237571762
#237571732
#237571662
#237571652
#237571492
#237571472
#237571282
#394×C1572
#2411157