Finite Group Databasebeta

← #34109419

Subgroups — #34109419

Order 20096.

732 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097818C211yesmaximal, characteristic, complemented
2C2×#19097795C211yesmaximal, characteristic, complemented
3C2×SD16×D157C211yesmaximal, characteristic, complemented
4#23757839C211yesmaximal, characteristic, complemented
5#23757419C211yesmaximal, characteristic, complemented
6#23757418C211yesmaximal, characteristic, complemented
7#23757415C211yesmaximal, characteristic, complemented
8#23757414C211yesmaximal, characteristic, complemented
9#23757413C211yesmaximal, characteristic, complemented
10#23757411C211yesmaximal, characteristic, complemented
11#23757410C211yesmaximal, characteristic, complemented
12#23757280C211yesmaximal, characteristic, complemented
13#23757266C211yesmaximal, characteristic, complemented
14#23757238C211yesmaximal, characteristic, complemented
15#449×C157C211yesmaximal, characteristic, complemented
16#19097818—24no—
17#19097818—24no—
18C2×#14578954—21no—
19C2×#14578954C2211yescharacteristic
20C2×#14578953—21no—
21C2×#14578953—21no—
22C2×#14578953C2211yescharacteristic
23C2×#14578953C2211yescharacteristic, complemented
24C22×Dic314—21no—
25C22×Dic314C2211yescharacteristic, complemented
26C2×Q8×D157C2211yescharacteristic
27C2×Q8×D157C2211yescharacteristic, complemented
28C2×D4×D157C2211yescharacteristic
29#19097795—24no—
30C2×#14578943C2211yescharacteristic
31C2×#14578943C2211yescharacteristic
32C2×#14578942C2211yescharacteristic
33C2×#14578942C2211yescharacteristic
34C2×#14578941C2211yescharacteristic
35SD16×D157—24no—
36C2×#14578939C2211yescharacteristic
37C2×#14578937C2211yescharacteristic
38C2×C8×D157C2211yescharacteristic
39#19097767—21no—
40#19097757—21no—
41#19097746C2211yescharacteristic
42#19097745C2211yescharacteristic
43#19097742C2211yescharacteristic
44#19097741—21no—
45#19097739—21no—
46#19097729C2211yescharacteristic
47#19097696C2211yescharacteristic, complemented
48#19097696C2211yescharacteristic, complemented
49#19097695C2211yescharacteristic, complemented
50#19097694C2211yescharacteristic, complemented
51#19097694C2211yescharacteristic, complemented
52#19097692C2211yescharacteristic, complemented
53C22⋊#11929434C2211yescharacteristic, complemented
54#19097683C2211yescharacteristic, complemented
55#19097683C2211yescharacteristic, complemented
56C2×#41×C157C2211yescharacteristic, complemented
57C2×SD16×C157C2211yescharacteristic
58C2×D8×C157C2211yescharacteristic
59#121×C157C2211yescharacteristic
60#103×C157C2211yescharacteristic, complemented
61#102×C157C2211yescharacteristic, complemented
62C22⋊C8×C157C2211yescharacteristic, complemented
63C23×D157—21no—
64#14578954—28no—
65#14578954—22no—
66#14578954—22no—
67#14578953—28no—
68#14578953—28no—
69#14578953—22no—
70#14578953—22no—
71#14578953—22no—
72#14578953D414yescomplemented
73C2×Dic314—28no—
74C2×Dic314—22no—
75C2×Dic314—22no—
76C2×Dic314—21no—
77C2×Dic314—21no—
78C2×Dic314—21no—
79C2×Dic314D411yescharacteristic
80C2×Dic314C2311yescharacteristic
81C2×Dic314C2311yescharacteristic
82Q8×D157—22no—
83Q8×D157—22no—
84Q8×D157—22no—
85Q8×D157D414yescomplemented
86C2×D628—21no—
87C2×D628D411yescharacteristic
88C2×D628C2311yescharacteristic
89C2×C22⋊D157—21no—
90C2×C22⋊D157—21no—
91C2×C22⋊D157—21no—
92C2×C22⋊D157—21no—
93C2×C22⋊D157—21no—
94C2×C22⋊D157D411yescharacteristic
95C2×C22⋊D157D411yescharacteristic
96D4×D157—22no—
97D4×D157—22no—
98C22×Dic157—21no—
99C22×Dic157—21no—
100C22×Dic157—21no—
101C2×C4×D157—21no—
102C2×C4×D157—21no—
103C2×C4×D157C2311yescharacteristic
104C2×C4×D157C2311yescharacteristic
105#14578943—22no—
106#14578943—22no—
107#14578942—22no—
108#14578942—22no—
109#14578941—22no—
110#14578939—22no—
111#14578937—22no—
112C2×#11929434C2311yescharacteristic
113C2×#11929434C2311yescharacteristic
114C8×D157—22no—
115C4⋊Dic157C2311yescharacteristic
116C22⋊Dic157—21no—
117#14578931—21no—
118#14578931—21no—
119#14578931—21no—
120C4×Dic157—21no—
121#41×C157—22no—
122#41×C157—22no—
123C2×Q8×C157C2311yescharacteristic
124C2×D4×C157—21no—
125C2×D4×C157—21no—
126C2×D4×C157C2311yescharacteristic
127C2×D4×C157C2311yescharacteristic
128C22×C4×C157—21no—
129C22×C4×C157C2311yescharacteristic
130SD16×C157—22no—
131D8×C157—22no—
132C2×C8×C157C2311yescharacteristic
133C2×C8×C157C2311yescharacteristic
134C4⋊C4×C157C2311yescharacteristic
135C22⋊C4×C157—21no—
136C22×D157—21no—
137C22×D157—42no—
138C22×D157—81no—
139C22×D157—81no—
140C22×D157C2×D411yescharacteristic
141C22×D157C2×D411yescharacteristic
142Dic314—24no—
143Dic314—24no—
144Dic314—21no—
145Dic314—21no—
146Dic314—24no—
147Dic314—22no—
148Dic314C2×D412yes—
149Dic314C2×D412yes—
150D628—22no—
151D628—21no—
152D628—21no—
153D628—81no—
154C22⋊D157—24no—
155C22⋊D157—24no—
156C22⋊D157—22no—
157C22⋊D157—22no—
158C22⋊D157—42no—
159C22⋊D157—81no—
160C22⋊D157—81no—
161C2×Dic157—24no—
162C2×Dic157—24no—
163C2×Dic157—21no—
164C2×Dic157—21no—
165C2×Dic157—24no—
166C2×Dic157—21no—
167C2×Dic157C2×D411yescharacteristic
168C2×Dic157C2×D411yescharacteristic
169C4×D157—24no—
170C4×D157—24no—
171C4×D157—21no—
172C4×D157—21no—
173C4×D157C2×D412yes—
174C4×D157C2×D412yes—
175#11929434—21no—
176#11929434—21no—
177C23×C157—21no—
178C23×C157—21no—
179C23×C157C2×D411yescharacteristic
180Q8×C157—21no—
181Q8×C157C2×D412yes—
182D4×C157—24no—
183D4×C157—21no—
184D4×C157—21no—
185D4×C157—21no—
186D4×C157—81no—
187D4×C157C2×D412yes—
188C2×C4×C157—24no—
189C2×C4×C157—21no—
190C2×C4×C157—21no—
191C2×C4×C157—21no—
192C2×C4×C157—21no—
193C2×C4×C157C2×D411yescharacteristic
194C2×C4×C157C2411yescharacteristic
195C8×C157—21no—
196C8×C157—21no—
197C22×C157—21no—
198C22×C157—22no—
199C22×C157—21no—
200C22×C157—21no—
201C22×C157—41no—
202C22×C157—41no—
203C22×C157—41no—
204C22×C157—41no—
205C22×C157—41no—
206C22×C157C22×D411yescharacteristic
207C2×D157—21no—
208C2×D157—21no—
209C2×D157—21no—
210C2×D157—21no—
211C2×D157—21no—
212C2×D157—41no—
213C2×D157—41no—
214C2×D157—41no—
215C2×D157—84no—
216C2×D157#12012yes—
217C4×C157—21no—
218C4×C157—22no—
219C4×C157—41no—
220C4×C157C22×D411yescharacteristic
221C4×C157C22×D411yescharacteristic
222Dic157—22no—
223Dic157—22no—
224Dic157—21no—
225Dic157—41no—
226Dic157#12012yes—
227C2×C157—41no—
228C2×C157—42no—
229C2×C157—81no—
230C2×C157C2×#12011yescharacteristic
231C2×C157#57711yescharacteristic
232C2×C157#57711yescharacteristic
233D157—22no—
234D157—41no—
235D157—81no—
236C157#301011yescharacteristic, complemented, Sylow
237#3010—1571nomaximal, Sylow
238C2×2−1+4—1571no—
239C2×AGL(1,ℤ/8)—1571no—
240C22×SD16—1571no—
241#535—1571no—
242#449—1571no—
243#449—1571no—
244#449—1571no—
245#449D15711yescharacteristic, complemented
246#448—1571no—
247#448—1571no—
248#448—1571no—
249#448—1571no—
250#419—1571no—
251#416—1571no—
252#408—1571no—
2532−1+4—3144no—
2542−1+4—3144no—
255C2×#41—3141no—
256C2×#41—3141no—
257C2×#41—3141no—
258C2×#41—1571no—
259C2×#41—1571no—
260C2×#41—1571no—
261C2×#41C2×D15711yescharacteristic, complemented
262C22×Q8—3141no—
263C22×Q8—1571no—
264C22×Q8—1571no—
265C22×Q8—1571no—
266C22×D4—1571no—
267AGL(1,ℤ/8)—3144no—
268C2×SD16—3144no—
269C2×SD16—1571no—
270C2×SD16—1571no—
271C2×SD16—1571no—
272C2×SD16—1571no—
273C2×SD16—1571no—
274C2×SD16C2×D15711yescharacteristic
275C2×D8—1571no—
276C2×D8C2×D15711yescharacteristic
277C2×M16—1571no—
278C22×C8—1571no—
279#127—3141no—
280#124—3141no—
281#121—1571no—
282#121—1571no—
283#121—1571no—
284#121C2×D15711yescharacteristic
285#120—3141no—
286#120—3141no—
287#117—1571no—
288#103—1571no—
289#103—1571no—
290#103—1571no—
291#103—1571no—
292#103—1571no—
293#103C2×D15711yescharacteristic, complemented
294#102—1571no—
295#102C2×D15711yescharacteristic, complemented
296C22⋊C8—1571no—
297C22⋊C8—1571no—
298C22⋊C8—1571no—
299C22⋊C8C2×D15711yescharacteristic, complemented
300C24—3141no—
301#41—22no—
302#41—22no—
303#41—3148no—
304#41—3142no—
305#41—3142no—
306#41—3148no—
307#41—3148no—
308#41—3142no—
309#41—3142no—
310#41—3142no—
311#41—1574no—
312C2×Q8—3148no—
313C2×Q8—3142no—
314C2×Q8—3142no—
315C2×Q8—3141no—
316C2×Q8—3141no—
317C2×Q8—3142no—
318C2×Q8—3142no—
319C2×Q8—3142no—
320C2×Q8—3141no—
321C2×Q8—1574no—
322C2×Q8—1571no—
323C2×Q8—1571no—
324C2×Q8—1571no—
325C2×Q8C22×D15711yescharacteristic
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—3141no—
329C2×D4—3141no—
330C2×D4—3141no—
331C2×D4—3141no—
332C2×D4—3141no—
333C2×D4—3142no—
334C2×D4—3142no—
335C2×D4—3141no—
336C2×D4—1571no—
337C2×D4—1571no—
338C2×D4—1571no—
339C2×D4—1571no—
340C2×D4C22×D15711yescharacteristic
341C2×D4C22×D15711yescharacteristic
342C22×C4—21no—
343C22×C4—3141no—
344C22×C4—3141no—
345C22×C4—3141no—
346C22×C4—3141no—
347C22×C4—3141no—
348C22×C4—1571no—
349C22×C4—1571no—
350C22×C4C22×D15711yescharacteristic
351SD16—22no—
352SD16—3142no—
353SD16—3142no—
354SD16—3142no—
355SD16—3142no—
356SD16—3142no—
357D8—22no—
358D8—3142no—
359M16—3142no—
360C2×C8—3142no—
361C2×C8—1571no—
362C2×C8—1571no—
363C2×C8C22×D15711yescharacteristic
364C2×C8C22×D15711yescharacteristic
365C4⋊C4—1571no—
366C4⋊C4C22×D15711yescharacteristic
367C22⋊C4—21no—
368C22⋊C4—3141no—
369C22⋊C4—3141no—
370C22⋊C4—3141no—
371C22⋊C4—3141no—
372C42—3141no—
373C23—21no—
374C23—21no—
375C23—3141no—
376C23—6282no—
377C23—1571no—
378C23—1571no—
379C23—12561no—
380C23—12561no—
381C23D4×D15711yescharacteristic
382Q8—21no—
383Q8—3144no—
384Q8—3144no—
385Q8—3142no—
386Q8—3144no—
387Q8—3141no—
388Q8—3141no—
389Q8—1572no—
390Q8—1572no—
391Q8D4×D15712yes—
392D4—24no—
393D4—21no—
394D4—21no—
395D4—21no—
396D4—81no—
397D4—3142no—
398D4—3144no—
399D4—3144no—
400D4—3142no—
401D4—3142no—
402D4—3141no—
403D4—3141no—
404D4—6282no—
405D4—12561no—
406D4—12561no—
407D4—12561no—
408D4D4×D15712yes—
409C2×C4—24no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—3144no—
415C2×C4—3144no—
416C2×C4—3144no—
417C2×C4—3144no—
418C2×C4—3141no—
419C2×C4—3141no—
420C2×C4—3144no—
421C2×C4—3141no—
422C2×C4—3141no—
423C2×C4—3141no—
424C2×C4—1571no—
425C2×C4—1571no—
426C2×C4—1572no—
427C2×C4—1572no—
428C2×C4D4×D15711yescharacteristic
429C2×C4C23×D15711yescharacteristic
430C8—21no—
431C8—21no—
432C8—3141no—
433C8—3141no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—3141no—
444C22—3141no—
445C22—3141no—
446C22—3141no—
447C22—3141no—
448C22—6281no—
449C22—6281no—
450C22—6281no—
451C22—1572no—
452C22—12564no—
453C22C2×D4×D15711yescharacteristic
454C4—21no—
455C4—22no—
456C4—41no—
457C4—3142no—
458C4—3142no—
459C4—3141no—
460C4—6281no—
461C4—1572no—
462C4C2×D4×D15711yescharacteristic
463C4C2×D4×D15711yescharacteristic
464C2—41no—
465C2—42no—
466C2—81no—
467C2—3142no—
468C2—6281no—
469C2—12561no—
470C2#120×D15711yescharacteristic
471C2#2375808011yescharacteristic
472C2#2375808311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#449×C157C2
socleC22×C157C22×D4
hypercenter#449D157
hypocenterC157#3010
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#19097818C2semidirect
2. C2×#19097795C2semidirect
3. C2×SD16×D157C2semidirect
4. #23757839C2semidirect
5. #23757419C2semidirect
6. #23757418C2semidirect
7. #23757415C2semidirect
8. #23757414C2semidirect
9. #23757413C2semidirect
10. #23757411C2semidirect
11. #23757410C2semidirect
12. #23757280C2semidirect
13. #23757266C2semidirect
14. #23757238C2semidirect
15. #449×C157C2semidirect
23. C2×#14578953C22semidirect
25. C22×Dic314C22semidirect
27. C2×Q8×D157C22semidirect
47. #19097696C22semidirect
48. #19097696C22semidirect
49. #19097695C22semidirect
50. #19097694C22semidirect
51. #19097694C22semidirect
52. #19097692C22semidirect
53. C22⋊#11929434C22semidirect
54. #19097683C22semidirect
55. #19097683C22semidirect
56. C2×#41×C157C22semidirect
60. #103×C157C22semidirect
61. #102×C157C22semidirect
62. C22⋊C8×C157C22semidirect
72. #14578953D4semidirect
85. Q8×D157D4semidirect
236. C157#3010semidirect
245. #449D157semidirect
261. C2×#41C2×D157semidirect
293. #103C2×D157semidirect
295. #102C2×D157semidirect
299. C22⋊C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978182
C2×#190977952
C2×SD16×D1572
#237578392
#237574192
#237574182
#237574152
#237574142
#237574132
#237574112
#237574102
#237572802
#237572662
#237572382
#449×C1572
#3010157