Finite Group Databasebeta

← #34108967

Subgroups — #34108967

Order 20096.

506 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097798C211yesmaximal, characteristic
2#23757755C211yesmaximal, characteristic, complemented
3#23757745C211yesmaximal, characteristic, complemented
4#23757386C211yesmaximal, characteristic, complemented
5#23757382C211yesmaximal, characteristic, complemented
6#23757376C211yesmaximal, characteristic, complemented
7SD16×Dic157C211yesmaximal, characteristic, complemented
8C4×#14578941C211yesmaximal, characteristic, complemented
9#23757355C211yesmaximal, characteristic, complemented
10C4×#14578939C211yesmaximal, characteristic, complemented
11#23757314C211yesmaximal, characteristic, complemented
12#23757268C211yesmaximal, characteristic, complemented
13#102×D157C211yesmaximal, characteristic, complemented
14#23757230C211yesmaximal, characteristic, complemented
15#440×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×D4×D157C2211yescharacteristic
18#19097798C418yescomplemented
19C2×#14578941C2211yescharacteristic
20C2×D1256C2211yescharacteristic
21C2×#14578939C2211yescharacteristic
22C2×#14578937C2211yescharacteristic
23C2×C8×D157C2211yescharacteristic
24Q8×Dic157C2211yescharacteristic, complemented
25C4×D628—21no—
26C4×D628C2211yescharacteristic
27C4×D628C2211yescharacteristic, complemented
28D4×Dic157C2211yescharacteristic
29#19097733—21no—
30#19097733C2211yescharacteristic
31#19097733C2211yescharacteristic, complemented
32C4×C22⋊D157—21no—
33#19097731—21no—
34#19097729C2211yescharacteristic
35#19097728—21no—
36#19097728C2211yescharacteristic
37C4⋊C4×D157—21no—
38C4⋊C4×D157C2211yescharacteristic
39C2×#14578931—21no—
40C22⋊C4×D157—21no—
41#19097704C2211yescharacteristic, complemented
42#19097703C2211yescharacteristic, complemented
43#19097702C2211yescharacteristic, complemented
44#19097695C2211yescharacteristic, complemented
45#19097693C2211yescharacteristic
46#19097693C2211yescharacteristic, complemented
47#19097692C2211yescharacteristic
48#19097691C2211yescharacteristic
49#19097691C2211yescharacteristic, complemented
50#19097681C2211yescharacteristic, complemented
51C4×#11929434C2211yescharacteristic, complemented
52C8×Dic157C2211yescharacteristic, complemented
53C2×SD16×C157C2211yescharacteristic
54C4×Q8×C157C2211yescharacteristic, complemented
55C4×D4×C157C2211yescharacteristic
56#107×C157C2211yescharacteristic, complemented
57#103×C157C2211yescharacteristic, complemented
58#102×C157C2211yescharacteristic
59#97×C157C2211yescharacteristic, complemented
60C23×D157—21no—
61#14578955—22no—
62#14578955C2×C414yescomplemented
63C2×D628—21no—
64C2×D628C2311yescharacteristic
65C2×D628C2311yescharacteristic
66C2×C22⋊D157—21no—
67D4×D157—22no—
68D4×D157C2×C414yescomplemented
69C22×Dic157—21no—
70C2×C4×D157—21no—
71C2×C4×D157—21no—
72C2×C4×D157—21no—
73C2×C4×D157—21no—
74C2×C4×D157—21no—
75C2×C4×D157—21no—
76C2×C4×D157C2311yescharacteristic
77#14578941C2×C414yescomplemented
78D1256C2×C414yescomplemented
79#14578939C2×C414yescomplemented
80#14578937C2×C414yescomplemented
81C2×#11929434C2311yescharacteristic
82C2×#11929434C2311yescharacteristic
83C8×D157—22no—
84C4⋊Dic157—21no—
85C4⋊Dic157C2311yescharacteristic
86C4⋊Dic157C2311yescharacteristic
87#14578933—21no—
88#14578933D411yescharacteristic
89#14578933D411yescharacteristic
90C22⋊Dic157—21no—
91#14578931—21no—
92#14578931—21no—
93#14578931—21no—
94#14578931—21no—
95#14578931—21no—
96#14578931D411yescharacteristic
97#14578931D411yescharacteristic
98C4×Dic157—21no—
99C4×Dic157—21no—
100C4×Dic157C2311yescharacteristic
101C2×Q8×C157C2311yescharacteristic
102C2×D4×C157C2311yescharacteristic
103C22×C4×C157—21no—
104SD16×C157C2×C414yescomplemented
105C2×C8×C157C2311yescharacteristic
106C2×C8×C157C2311yescharacteristic
107C4⋊C4×C157—21no—
108C4⋊C4×C157C2311yescharacteristic
109C4⋊C4×C157C2311yescharacteristic
110C22⋊C4×C157—21no—
111C42×C157—21no—
112C42×C157C2311yescharacteristic
113C22×D157—21no—
114C22×D157—21no—
115C22×D157—24no—
116C22×D157—81no—
117C22×D157—81no—
118C22×D157C2×D411yescharacteristic
119D628—24no—
120D628—21no—
121D628—21no—
122D628C22×C412yes—
123D628C22×C412yes—
124C22⋊D157—24no—
125C2×Dic157—21no—
126C2×Dic157—21no—
127C2×Dic157—81no—
128C2×Dic157C2×D411yescharacteristic
129C2×Dic157C2×D411yescharacteristic
130C2×Dic157C2×D411yescharacteristic
131C4×D157—24no—
132C4×D157—81no—
133C4×D157—81no—
134C4×D157—81no—
135C4×D157—81no—
136C4×D157—81no—
137C4×D157C22×C412yes—
138C4×D157#4112yes—
139#11929434—21no—
140#11929434C22×C412yes—
141C23×C157—21no—
142Q8×C157—21no—
143Q8×C157C22×C412yes—
144D4×C157—21no—
145D4×C157C22×C412yes—
146C2×C4×C157—21no—
147C2×C4×C157—21no—
148C2×C4×C157—21no—
149C2×C4×C157—81no—
150C2×C4×C157C2×D411yescharacteristic
151C2×C4×C157C2×D411yescharacteristic
152C2×C4×C157C2411yescharacteristic
153C8×C157—21no—
154C8×C157C22×C412yes—
155C22×C157—22no—
156C22×C157—41no—
157C22×C157—41no—
158C22×C157C22×D411yescharacteristic
159C2×D157—21no—
160C2×D157—21no—
161C2×D157—22no—
162C2×D157—22no—
163C2×D157—41no—
164C2×D157—41no—
165C2×D157—41no—
166C2×D157—41no—
167C2×D157—84no—
168C2×D157C4×D412yes—
169C4×C157—22no—
170C4×C157—21no—
171C4×C157—21no—
172C4×C157—41no—
173C4×C157—41no—
174C4×C157C23×C411yescharacteristic
175C4×C157C2×#4111yescharacteristic
176Dic157—21no—
177Dic157—21no—
178Dic157—41no—
179Dic157—41no—
180Dic157C4×D412yes—
181C2×C157—42no—
182C2×C157C2×C4×D411yescharacteristic
183C2×C157#57611yescharacteristic
184C2×C157#57711yescharacteristic
185D157—22no—
186D157—42no—
187D157—42no—
188C157#294611yescharacteristic, complemented, Sylow
189#2946—1571nomaximal, Sylow
190C2×AGL(1,ℤ/8)—1571no—
191#520—1571no—
192#518—1571no—
193#442—1571no—
194#442—1571no—
195#440—1571no—
196#440D15711yescharacteristic, complemented
197C4×SD16—1571no—
198C4×SD16—1571no—
199C4×D8—1571no—
200C4×D8—1571no—
201#428—1571no—
202#416—1571no—
203C2×#102—1571no—
204#405—1571no—
205C2×#41—1571no—
206C22×D4—1571no—
207AGL(1,ℤ/8)—1578no—
208C2×SD16—1571no—
209C2×SD16C2×D15711yescharacteristic
210C2×D8—1571no—
211C2×D8—1571no—
212C2×M16—1571no—
213C22×C8—1571no—
214C4×Q8—1571no—
215C4×Q8C2×D15711yescharacteristic, complemented
216C4×D4—3141no—
217C4×D4—3141no—
218C4×D4—3141no—
219C4×D4—3141no—
220C4×D4—1571no—
221C4×D4—1571no—
222C4×D4—1571no—
223C4×D4—1571no—
224C4×D4—1571no—
225C4×D4C2×D15711yescharacteristic
226#117—3141no—
227#117—1571no—
228#117—1571no—
229C2×C4⋊C4—3141no—
230C2×C4⋊C4—1571no—
231C2×C22⋊C4—3141no—
232C2×C22⋊C4—3141no—
233#107—1571no—
234#107C2×D15711yescharacteristic, complemented
235#106—1571no—
236#106—1571no—
237#103—1571no—
238#103C2×D15711yescharacteristic, complemented
239#102—1571no—
240#102—1571no—
241#102—1571no—
242#102—1571no—
243#102—1571no—
244#102C2×D15711yescharacteristic
245#97—1571no—
246#97C2×D15711yescharacteristic, complemented
247C4×C8—1571no—
248C4×C8—1571no—
249C24—3141no—
250#41—3142no—
251#41—1574no—
252C2×Q8C22×D15711yescharacteristic
253C2×D4—3141no—
254C2×D4—3142no—
255C2×D4—3141no—
256C2×D4—1574no—
257C2×D4—1571no—
258C2×D4—1571no—
259C2×D4C22×D15711yescharacteristic
260C22×C4—21no—
261C22×C4—3141no—
262C22×C4—3141no—
263C22×C4—3141no—
264C22×C4—3141no—
265C22×C4—3141no—
266C22×C4—3141no—
267C22×C4—3141no—
268C22×C4—1571no—
269SD16—1574no—
270SD16C4×D15714yescomplemented
271D8—1574no—
272D8—1574no—
273M16—1574no—
274C2×C8—3142no—
275C2×C8—1571no—
276C2×C8—1571no—
277C2×C8C22×D15711yescharacteristic
278C2×C8C22×D15711yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—3141no—
281C4⋊C4—3141no—
282C4⋊C4—1571no—
283C4⋊C4—1571no—
284C4⋊C4—1571no—
285C4⋊C4—1571no—
286C4⋊C4C22×D15711yescharacteristic
287C4⋊C4C22×D15711yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—3141no—
290C22⋊C4—3141no—
291C22⋊C4—3141no—
292C22⋊C4—3141no—
293C22⋊C4—3141no—
294C22⋊C4—3141no—
295C22⋊C4—1571no—
296C22⋊C4—1571no—
297C42—21no—
298C42—3141no—
299C42—3141no—
300C42—1571no—
301C42C22×D15711yescharacteristic
302C23—21no—
303C23—3144no—
304C23—3141no—
305C23—3141no—
306C23—1571no—
307C23—12561no—
308C23—12561no—
309Q8—21no—
310Q8C2×C4×D15712yes—
311D4—21no—
312D4—3144no—
313D4—3144no—
314D4—3141no—
315D4—3141no—
316D4—1572no—
317D4—1572no—
318D4C2×C4×D15712yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—81no—
323C2×C4—3144no—
324C2×C4—3141no—
325C2×C4—3141no—
326C2×C4—1571no—
327C2×C4—1571no—
328C2×C4—1571no—
329C2×C4—1572no—
330C2×C4—1572no—
331C2×C4—12561no—
332C2×C4—12561no—
333C2×C4—12561no—
334C2×C4—12561no—
335C2×C4—12561no—
336C2×C4—12561no—
337C2×C4D4×D15711yescharacteristic
338C2×C4D4×D15711yescharacteristic
339C2×C4C23×D15711yescharacteristic
340C8—21no—
341C8—3141no—
342C8—1572no—
343C8C2×C4×D15712yes—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—3141no—
348C22—3141no—
349C22—3142no—
350C22—3142no—
351C22—6281no—
352C22—6281no—
353C22—6281no—
354C22—6281no—
355C22—1572no—
356C22—12564no—
357C22C2×D4×D15711yescharacteristic
358C4—21no—
359C4—22no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—3141no—
364C4—3141no—
365C4—6281no—
366C4—6281no—
367C4—1572no—
368C4C22×C4×D15711yescharacteristic
369C4#41×D15711yescharacteristic
370C2—42no—
371C2—3142no—
372C2—6282no—
373C2—6282no—
374C2C4×D4×D15711yescharacteristic
375C2#2375807711yescharacteristic
376C2#2375808211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutatorC4×C157C23×C4
FrattiniC2×C4C23×D157
Fitting#440×C157C2
socleC22×C157C22×D4
prime radicalC2×#19097798C2
cosocleC2×C4×C157C24
hypercenter#440D157
hypocenterC157#2946
LCSG ▷ C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. #23757755C2semidirect
3. #23757745C2semidirect
4. #23757386C2semidirect
5. #23757382C2semidirect
6. #23757376C2semidirect
7. SD16×Dic157C2semidirect
8. C4×#14578941C2semidirect
9. #23757355C2semidirect
10. C4×#14578939C2semidirect
11. #23757314C2semidirect
12. #23757268C2semidirect
13. #102×D157C2semidirect
14. #23757230C2semidirect
15. #440×C157C2semidirect
18. #19097798C4semidirect
24. Q8×Dic157C22semidirect
27. C4×D628C22semidirect
31. #19097733C22semidirect
41. #19097704C22semidirect
42. #19097703C22semidirect
43. #19097702C22semidirect
44. #19097695C22semidirect
46. #19097693C22semidirect
49. #19097691C22semidirect
50. #19097681C22semidirect
51. C4×#11929434C22semidirect
52. C8×Dic157C22semidirect
54. C4×Q8×C157C22semidirect
56. #107×C157C22semidirect
57. #103×C157C22semidirect
59. #97×C157C22semidirect
62. #14578955C2×C4semidirect
68. D4×D157C2×C4semidirect
77. #14578941C2×C4semidirect
78. D1256C2×C4semidirect
79. #14578939C2×C4semidirect
80. #14578937C2×C4semidirect
104. SD16×C157C2×C4semidirect
188. C157#2946semidirect
196. #440D157semidirect
215. C4×Q8C2×D157semidirect
234. #107C2×D157semidirect
238. #103C2×D157semidirect
246. #97C2×D157semidirect
270. SD16C4×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977982
#237577552
#237577452
#237573862
#237573822
#237573762
SD16×Dic1572
C4×#145789412
#237573552
C4×#145789392
#237573142
#237572682
#102×D1572
#237572302
#440×C1572
#2946157