Finite Group Databasebeta

← #34111906

Subgroups — #34111906

Order 20096.

448 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23758089C211yesmaximal, characteristic, complemented
2#23758088C211yesmaximal, characteristic, complemented
3#23758082C211yesmaximal, characteristic, complemented
4#23758081C211yesmaximal, characteristic, complemented
5#23757665C211yesmaximal, characteristic, complemented
6#23757547C211yesmaximal, characteristic, complemented
7#23757546C211yesmaximal, characteristic, complemented
8#23757539C211yesmaximal, characteristic, complemented
9#23757538C211yesmaximal, characteristic, complemented
10#23757455C211yesmaximal, characteristic, complemented
11#23757452C211yesmaximal, characteristic, complemented
12#23757447C211yesmaximal, characteristic, complemented
13#23757392C211yesmaximal, characteristic, complemented
14#23757392C211yesmaximal, characteristic, complemented
15#456×C157C211yesmaximal, characteristic, complemented
16#19097819C2211yescharacteristic
17#19097818C2211yescharacteristic, complemented
18#19097817C2211yescharacteristic
19#19097804C2211yescharacteristic
20#19097803—21no—
21#19097803—21no—
22#19097802—21no—
23#19097802—21no—
24#19097802C2211yescharacteristic
25#19097802C2211yescharacteristic, complemented
26#19097801—21no—
27#19097801C2211yescharacteristic
28#19097800—21no—
29#19097799C2211yescharacteristic
30#19097798—21no—
31#19097796—21no—
32#19097796C2211yescharacteristic
33#19097794—21no—
34#19097792—21no—
35#19097791—21no—
36#19097791—21no—
37#19097791C2211yescharacteristic
38#19097791C2211yescharacteristic
39C2×#14578944—21no—
40C2×#14578944C2211yescharacteristic, complemented
41C2×#14578944C2211yescharacteristic, complemented
42Q16×D157—21no—
43C2×#14578941—21no—
44C2×#14578941C2211yescharacteristic
45C2×#14578941C2211yescharacteristic
46SD16×D157—21no—
47#19097779C2211yescharacteristic
48#19097779C2211yescharacteristic, complemented
49#19097778C2211yescharacteristic
50#19097778C2211yescharacteristic, complemented
51#19097759C2211yescharacteristic
52#19097706C2211yescharacteristic, complemented
53#19097706C2211yescharacteristic, complemented
54#19097699C2211yescharacteristic
55#19097699C2211yescharacteristic, complemented
56#19097690C2211yescharacteristic, complemented
57#19097689C2211yescharacteristic, complemented
58#19097687C2211yescharacteristic
59#19097682C2211yescharacteristic, complemented
602−1+4×C157C2211yescharacteristic
61#137×C157C2211yescharacteristic
62#137×C157C2211yescharacteristic, complemented
63C4⋊Q8×C157C2211yescharacteristic, complemented
64#104×C157C2211yescharacteristic
65#104×C157C2211yescharacteristic, complemented
66#101×C157C2211yescharacteristic, complemented
67#14578955—21no—
68#14578955—21no—
69#14578955—21no—
70#14578954—21no—
71#14578954—21no—
72#14578954—41no—
73#14578954C2311yescharacteristic
74#14578954C2311yescharacteristic
75#14578953—21no—
76#14578953—41no—
77C2×Dic314—21no—
78C2×Dic314C2311yescharacteristic
79Q8×D157—21no—
80Q8×D157—21no—
81Q8×D157—21no—
82C2×D628—21no—
83C2×D628C2311yescharacteristic
84D4×D157—21no—
85D4×D157—41no—
86Dic628—21no—
87#14578944—21no—
88#14578944—21no—
89#14578944—21no—
90#14578944—21no—
91#14578944—21no—
92#14578944—21no—
93#14578944—21no—
94#14578944D411yescharacteristic
95#14578944D411yescharacteristic
96#14578943—21no—
97#14578943—21no—
98#14578942—21no—
99#14578942D411yescharacteristic
100#14578942D411yescharacteristic
101#14578941—21no—
102#14578941—21no—
103#14578941—21no—
104#14578941—21no—
105#14578941—21no—
106#14578941—21no—
107#14578941—21no—
108#14578941D411yescharacteristic
109#14578941D411yescharacteristic
110D1256—21no—
111#14578939—21no—
112#14578939D411yescharacteristic
113#14578939D411yescharacteristic
114#14578938—21no—
115#14578938—21no—
116#14578938C2311yescharacteristic
117#14578938C2311yescharacteristic
118#14578937—21no—
119#14578937—21no—
120C2×#11929434—21no—
121C2×#11929434—21no—
122C2×#11929434C2311yescharacteristic
123C2×#11929434C2311yescharacteristic
124C8×D157—21no—
125C8×D157—21no—
126#14578931—41no—
127#41×C157—21no—
128#41×C157—21no—
129#41×C157—41no—
130#41×C157C2311yescharacteristic
131#41×C157C2311yescharacteristic
132C2×Q8×C157—21no—
133C2×Q8×C157—21no—
134C2×Q8×C157C2311yescharacteristic
135C2×Q8×C157C2311yescharacteristic
136Q16×C157—21no—
137Q16×C157—21no—
138SD16×C157—21no—
139SD16×C157—21no—
140M16×C157C2311yescharacteristic
141M16×C157C2311yescharacteristic
142C4⋊C4×C157—41no—
143C42×C157C2311yescharacteristic
144C22×D157—21no—
145C22×D157—41no—
146Dic314—21no—
147Dic314—21no—
148Dic314—41no—
149Dic314—41no—
150Dic314C2×D411yescharacteristic
151Dic314C2×D411yescharacteristic
152D628—21no—
153D628—21no—
154D628—41no—
155D628—41no—
156D628C2×D411yescharacteristic
157D628C2×D411yescharacteristic
158C22⋊D157—21no—
159C22⋊D157—21no—
160C22⋊D157—41no—
161C22⋊D157—41no—
162C2×Dic157—21no—
163C2×Dic157—41no—
164C4×D157—21no—
165C4×D157—21no—
166C4×D157—41no—
167C4×D157—41no—
168C4×D157—41no—
169C4×D157—41no—
170#11929434—21no—
171#11929434—21no—
172#11929434C2×D411yescharacteristic
173#11929434C2×D411yescharacteristic
174#11929434C2×D411yescharacteristic
175#11929434C2×D411yescharacteristic
176Q8×C157—21no—
177Q8×C157—21no—
178Q8×C157—21no—
179Q8×C157—21no—
180Q8×C157—41no—
181Q8×C157C2×D411yescharacteristic
182Q8×C157C2×D411yescharacteristic
183D4×C157—21no—
184D4×C157—21no—
185D4×C157—41no—
186D4×C157C2×D411yescharacteristic
187D4×C157C2×D411yescharacteristic
188C2×C4×C157—21no—
189C2×C4×C157—21no—
190C2×C4×C157—21no—
191C2×C4×C157—21no—
192C2×C4×C157—21no—
193C2×C4×C157—41no—
194C2×C4×C157—41no—
195C2×C4×C157C2411yescharacteristic
196C8×C157—21no—
197C8×C157—21no—
198C22×C157—21no—
199C22×C157—21no—
200C22×C157H(5,𝔽2)11yescharacteristic
201C2×D157—21no—
202C2×D157—21no—
203C2×D157—41no—
204C2×D157—81no—
205C2×D157—161no—
206C4×C157—21no—
207C4×C157—21no—
208C4×C157—41no—
209C4×C157—41no—
210C4×C157—41no—
211C4×C157C22×D411yescharacteristic
212C4×C157C22×D411yescharacteristic
213Dic157—21no—
214Dic157—21no—
215Dic157—41no—
216C2×C157—21no—
217C2×C157—41no—
218C2×C157—41no—
219C2×C157D4211yescharacteristic
220D157—41no—
221D157—41no—
222D157—81no—
223C157#328911yescharacteristic, complemented, Sylow
224#3289—1571nomaximal, Sylow
225#578—1571no—
226#578—1571no—
227#577—1571no—
228#577—1571no—
229#497—1571no—
230#473—1571no—
231#473—1571no—
232#471—1571no—
233#471—1571no—
234#456—1571no—
235#456D15711yescharacteristic, complemented
236#455—1571no—
237#454—1571no—
238#444—1571no—
239#444—1571no—
2402−1+4—1571no—
2412−1+4—1571no—
2422−1+4C2×D15711yescharacteristic
243H(5,𝔽2)—1571no—
244#137—3141no—
245#137—3141no—
246#137—3141no—
247#137—3141no—
248#137—3141no—
249#137—3141no—
250#137—1571no—
251#137—1571no—
252#137—1571no—
253#137—1571no—
254#137C2×D15711yescharacteristic
255#137C2×D15711yescharacteristic, complemented
256AGL(1,ℤ/8)—3141no—
257AGL(1,ℤ/8)—3141no—
258AGL(1,ℤ/8)—1571no—
259AGL(1,ℤ/8)—1571no—
260#135—3141no—
261#135—3141no—
262#135—3141no—
263#135—3141no—
264#135—1571no—
265#135—1571no—
266C2×Q16—3141no—
267C2×Q16—3141no—
268C2×Q16—1571no—
269C2×Q16—1571no—
270C2×SD16—3141no—
271C2×SD16—3141no—
272C2×SD16—1571no—
273C2×SD16—1571no—
274#131—1571no—
275#131—1571no—
276#131—1571no—
277#131—1571no—
278C4⋊Q8C2×D15711yescharacteristic, complemented
279#124—1571no—
280#108—1571no—
281#108—1571no—
282#104—1571no—
283#104—1571no—
284#104C2×D15711yescharacteristic
285#104C2×D15711yescharacteristic, complemented
286#101—1571no—
287#101—1571no—
288#101C2×D15711yescharacteristic, complemented
289#100—1571no—
290#97—1571no—
291#41—21no—
292#41—21no—
293#41—41no—
294#41—3141no—
295#41—3141no—
296#41—3141no—
297#41—3141no—
298#41—3141no—
299#41—3141no—
300#41—6281no—
301#41—6281no—
302#41—1571no—
303#41—1571no—
304#41C22×D15711yescharacteristic
305#41C22×D15711yescharacteristic
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—3141no—
309C2×Q8—3141no—
310C2×Q8—3141no—
311C2×Q8—3141no—
312C2×Q8—1571no—
313C2×Q8C22×D15711yescharacteristic
314C2×Q8C22×D15711yescharacteristic
315C2×D4—3141no—
316C2×D4—3141no—
317C2×D4—6281no—
318C2×D4—1571no—
319Q16—21no—
320Q16—21no—
321Q16—3141no—
322Q16—3141no—
323Q16—3141no—
324Q16—3141no—
325Q16—3141no—
326Q16—3141no—
327Q16—3141no—
328Q16—3141no—
329Q16—1571no—
330Q16—1571no—
331SD16—21no—
332SD16—21no—
333SD16—3141no—
334SD16—3141no—
335SD16—3141no—
336SD16—3141no—
337SD16—3141no—
338SD16—3141no—
339SD16—3141no—
340SD16—3141no—
341SD16—3141no—
342SD16—3141no—
343SD16—1571no—
344SD16—1571no—
345SD16—1571no—
346SD16—1571no—
347D8—3141no—
348D8—3141no—
349D8—1571no—
350D8—1571no—
351M16—3141no—
352M16—3141no—
353M16—3141no—
354M16—3141no—
355M16—1571no—
356M16—1571no—
357M16C22×D15711yescharacteristic
358M16C22×D15711yescharacteristic
359C2×C8—3141no—
360C2×C8—3141no—
361C2×C8—3141no—
362C2×C8—3141no—
363C2×C8—1571no—
364C2×C8—1571no—
365C4⋊C4—41no—
366C22⋊C4—6281no—
367C42C22×D15711yescharacteristic
368C23—3141no—
369C23—6281no—
370Q8—21no—
371Q8—21no—
372Q8—21no—
373Q8—21no—
374Q8—41no—
375Q8—3141no—
376Q8—3141no—
377Q8—6281no—
378Q8—6281no—
379Q8—1571no—
380Q8—1571no—
381Q8D4×D15711yescharacteristic
382Q8D4×D15711yescharacteristic
383D4—21no—
384D4—21no—
385D4—41no—
386D4—3141no—
387D4—3141no—
388D4—3141no—
389D4—3141no—
390D4—6281no—
391D4—6281no—
392D4—6281no—
393D4—6281no—
394D4—1571no—
395D4—1571no—
396D4D4×D15711yescharacteristic
397D4D4×D15711yescharacteristic
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—3141no—
406C2×C4—3141no—
407C2×C4—3141no—
408C2×C4—6281no—
409C2×C4—6281no—
410C2×C4—6281no—
411C2×C4—6281no—
412C2×C4—6281no—
413C2×C4C23×D15711yescharacteristic
414C8—21no—
415C8—21no—
416C8—3141no—
417C8—3141no—
418C8—1571no—
419C8—1571no—
420C8—1571no—
421C8—1571no—
422C22—21no—
423C22—21no—
424C22—3141no—
425C22—3141no—
426C22—6281no—
427C22—25121no—
428C22—12561no—
429C22#1909781611yescharacteristic
430C4—21no—
431C4—21no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—3141no—
436C4—3141no—
437C4—6281no—
438C4C2×D4×D15711yescharacteristic
439C4C2×D4×D15711yescharacteristic
440C2—21no—
441C2—41no—
442C2—41no—
443C2—6281no—
444C2—6281no—
445C2—12561no—
446C2#2375790511yescharacteristic
subgroupG/H
centerC2#23757905
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#456×C157C2
socleC2×C157D42
hypercenter#456D157
hypocenterC157#3289
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #23758089C2semidirect
2. #23758088C2semidirect
3. #23758082C2semidirect
4. #23758081C2semidirect
5. #23757665C2semidirect
6. #23757547C2semidirect
7. #23757546C2semidirect
8. #23757539C2semidirect
9. #23757538C2semidirect
10. #23757455C2semidirect
11. #23757452C2semidirect
12. #23757447C2semidirect
13. #23757392C2semidirect
14. #23757392C2semidirect
15. #456×C157C2semidirect
17. #19097818C22semidirect
25. #19097802C22semidirect
40. C2×#14578944C22semidirect
41. C2×#14578944C22semidirect
48. #19097779C22semidirect
50. #19097778C22semidirect
52. #19097706C22semidirect
53. #19097706C22semidirect
55. #19097699C22semidirect
56. #19097690C22semidirect
57. #19097689C22semidirect
59. #19097682C22semidirect
62. #137×C157C22semidirect
63. C4⋊Q8×C157C22semidirect
65. #104×C157C22semidirect
66. #101×C157C22semidirect
223. C157#3289semidirect
235. #456D157semidirect
255. #137C2×D157semidirect
278. C4⋊Q8C2×D157semidirect
285. #104C2×D157semidirect
288. #101C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580892
#237580882
#237580822
#237580812
#237576652
#237575472
#237575462
#237575392
#237575382
#237574552
#237574522
#237574472
#237573922
#456×C1572
#3289157