Finite Group Databasebeta

← #34113179

Subgroups — #34113179

Order 20096.

418 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23758006C211yesmaximal, characteristic, complemented
2#23757944C211yesmaximal, characteristic, complemented
3#23757662C211yesmaximal, characteristic, complemented
4#23757661C211yesmaximal, characteristic, complemented
5#23757648C211yesmaximal, characteristic, complemented
6#23757488C211yesmaximal, characteristic, complemented
7#23757487C211yesmaximal, characteristic, complemented
8#23757472C211yesmaximal, characteristic, complemented
9#23757468C211yesmaximal, characteristic, complemented
10#23757467C211yesmaximal, characteristic, complemented
11#23757467C211yesmaximal, characteristic, complemented
12#23757401C211yesmaximal, characteristic, complemented
13#23757384C211yesmaximal, characteristic, complemented
14#23757360C211yesmaximal, characteristic, complemented
15#442×C157C211yesmaximal, characteristic, complemented
16C2×#14578954—21no—
17C2×#14578943C2211yescharacteristic
18C2×#14578943C2211yescharacteristic
19C2×#14578942C2211yescharacteristic
20C2×D1256C2211yescharacteristic
21C2×D1256C2211yescharacteristic
22C2×#14578939C2211yescharacteristic
23C2×#14578939C2211yescharacteristic
24C2×#14578939C2211yescharacteristic
25#19097768C2211yescharacteristic
26#19097767C2211yescharacteristic
27#19097767C2211yescharacteristic
28#19097767C2211yescharacteristic
29#19097759C2211yescharacteristic
30#19097758C2211yescharacteristic
31#19097757—21no—
32#19097754—21no—
33#19097753—21no—
34#19097745—21no—
35#19097744—21no—
36#19097744—21no—
37C4×Dic314C2211yescharacteristic
38#19097736C2211yescharacteristic
39#19097733C2211yescharacteristic
40#19097733C2211yescharacteristic
41#19097733C2211yescharacteristic
42C4×C22⋊D157—21no—
43C4⋊#11929434C2211yescharacteristic
44#19097700C2211yescharacteristic
45#19097700C2211yescharacteristic
46#19097694C2211yescharacteristic
47#19097691C2211yescharacteristic
48#19097691C2211yescharacteristic
49#19097691C2211yescharacteristic
50#19097681C2211yescharacteristic
51C8×Dic157C2211yescharacteristic
52C2×D8×C157C2211yescharacteristic
53C4×D4×C157C2211yescharacteristic
54C4×D4×C157C2211yescharacteristic
55#106×C157C2211yescharacteristic
56#102×C157C2211yescharacteristic
57#102×C157C2211yescharacteristic
58#97×C157C2211yescharacteristic
59#14578954—44no—
60C2×Dic314D411yescharacteristic
61C2×Dic314C2311yescharacteristic
62C2×D628—21no—
63C2×D628—21no—
64C2×D628C2311yescharacteristic
65C2×D628C2311yescharacteristic
66C2×D628C2311yescharacteristic
67C2×C22⋊D157—21no—
68C2×C22⋊D157—21no—
69C2×C22⋊D157—21no—
70C2×C22⋊D157—21no—
71C2×C22⋊D157—21no—
72C2×C22⋊D157—21no—
73C2×C4×D157—21no—
74C2×C4×D157—21no—
75C2×C4×D157—21no—
76#14578943—22no—
77#14578943—41no—
78#14578942—41no—
79D1256—22no—
80D1256—41no—
81#14578939—41no—
82#14578939—41no—
83#14578939—41no—
84C2×#11929434C2311yescharacteristic
85C2×#11929434C2311yescharacteristic
86C4⋊Dic157D411yescharacteristic
87#14578933—21no—
88#14578933D411yescharacteristic
89#14578933D411yescharacteristic
90C22⋊Dic157—21no—
91C22⋊Dic157—21no—
92#14578931—21no—
93#14578931—21no—
94#14578931—21no—
95#14578931—21no—
96#14578931—21no—
97C4×Dic157—21no—
98C4×Dic157C2311yescharacteristic
99C4×Dic157C2311yescharacteristic
100C2×D4×C157C2311yescharacteristic
101C2×D4×C157C2311yescharacteristic
102C22×C4×C157—21no—
103C22×C4×C157—21no—
104D8×C157—22no—
105C2×C8×C157C2311yescharacteristic
106C2×C8×C157C2311yescharacteristic
107C4⋊C4×C157C2311yescharacteristic
108C4⋊C4×C157C2311yescharacteristic
109C22⋊C4×C157—21no—
110C22⋊C4×C157—21no—
111C42×C157C2311yescharacteristic
112C22×D157—21no—
113C22×D157—21no—
114C22×D157—21no—
115Dic314—22no—
116Dic314—21no—
117Dic314—21no—
118D628—21no—
119D628—21no—
120D628—21no—
121D628—21no—
122D628—21no—
123D628—21no—
124D628—42no—
125D628—81no—
126C22⋊D157—42no—
127C22⋊D157—42no—
128C22⋊D157—42no—
129C22⋊D157—81no—
130C22⋊D157—81no—
131C22⋊D157—81no—
132C2×Dic157—21no—
133C2×Dic157C2×D411yescharacteristic
134C2×Dic157C2×D411yescharacteristic
135C2×Dic157C2×D411yescharacteristic
136C2×Dic157C2×D411yescharacteristic
137C4×D157—42no—
138C4×D157—42no—
139C4×D157—81no—
140#11929434—21no—
141#11929434—21no—
142C23×C157—21no—
143C23×C157—21no—
144D4×C157—21no—
145D4×C157—21no—
146D4×C157—21no—
147D4×C157—21no—
148C2×C4×C157—21no—
149C2×C4×C157—21no—
150C2×C4×C157—42no—
151C2×C4×C157—81no—
152C2×C4×C157C2×D411yescharacteristic
153C2×C4×C157C2×D411yescharacteristic
154C2×C4×C157C2411yescharacteristic
155C8×C157—21no—
156C8×C157#4112yescomplemented
157C22×C157—41no—
158C22×C157—41no—
159C22×C157—41no—
160C22×C157—41no—
161C22×C157—41no—
162C22×C157—41no—
163C22×C157C22×D411yescharacteristic
164C2×D157—41no—
165C2×D157—41no—
166C2×D157—41no—
167C2×D157—41no—
168C2×D157—41no—
169C2×D157—41no—
170C2×D157—41no—
171C2×D157—41no—
172C2×D157—41no—
173C4×C157—21no—
174C4×C157—21no—
175C4×C157—41no—
176C4×C157—41no—
177C4×C157C2×#4111yescharacteristic
178C4×C157H(5,𝔽2)11yescharacteristic
179Dic157—21no—
180Dic157—21no—
181Dic157—21no—
182Dic157—41no—
183Dic157AGL(1,ℤ/8)12yes—
184C2×C157—81no—
185C2×C157—81no—
186C2×C157#55011yescharacteristic
187C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
188C2×C157#57611yescharacteristic
189D157—81no—
190D157—81no—
191D157—81no—
192C157#339911yescharacteristic, complemented, Sylow
193#3399—1571nomaximal, Sylow
194#559—1571no—
195#550—1571no—
196#496—1571no—
197#496—1571no—
198#493—1571no—
199#463—1571no—
200#463—1571no—
201#460—1571no—
202#459—1571no—
203#459—1571no—
204#459—1571no—
205#446—1571no—
206#442—1571no—
207#442D15711yescharacteristic, complemented
208C4×SD16—1571no—
209C2×#41—3141no—
210C2×SD16—1571no—
211C2×SD16—1571no—
212C2×SD16—1571no—
213C2×D8—1571no—
214C2×D8—1571no—
215C2×D8—1571no—
216C2×D8—1571no—
217C2×D8—1571no—
218C2×D8C2×D15711yescharacteristic
219#127—1571no—
220#127—1571no—
221#127—1571no—
222#127—1571no—
223#124—3141no—
224#124—1571no—
225#124—1571no—
226#123—3141no—
227#123—3141no—
228#121—3141no—
229#121—3141no—
230#121—3141no—
231C4×Q8—1571no—
232C4×Q8—1571no—
233C4×D4—3141no—
234C4×D4—1571no—
235C4×D4—1571no—
236C4×D4—1571no—
237C4×D4C2×D15711yescharacteristic
238C4×D4C2×D15711yescharacteristic
239#106C2×D15711yescharacteristic
240C4⋊C8—1571no—
241C4⋊C8—1571no—
242C4⋊C8—1571no—
243#103—1571no—
244#102—1571no—
245#102—1571no—
246#102—1571no—
247#102C2×D15711yescharacteristic
248#102C2×D15711yescharacteristic
249#97—1571no—
250#97C2×D15711yescharacteristic
251C4×C8—1571no—
252#41—6284no—
253C2×Q8—1571no—
254C2×Q8—1571no—
255C2×D4—3141no—
256C2×D4—3141no—
257C2×D4—3141no—
258C2×D4—3141no—
259C2×D4—3141no—
260C2×D4—3141no—
261C2×D4—3141no—
262C2×D4—3141no—
263C2×D4—1571no—
264C2×D4—1571no—
265C2×D4—1571no—
266C2×D4C22×D15711yescharacteristic
267C2×D4C22×D15711yescharacteristic
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—3141no—
271C22×C4—3141no—
272C22×C4—3141no—
273SD16—3142no—
274SD16—6281no—
275SD16—6281no—
276D8—22no—
277D8—3142no—
278D8—6281no—
279D8—6281no—
280D8—6281no—
281D8—6281no—
282C2×C8—1571no—
283C2×C8—1571no—
284C2×C8C22×D15711yescharacteristic
285C2×C8C22×D15711yescharacteristic
286C4⋊C4—3141no—
287C4⋊C4—1571no—
288C4⋊C4—1571no—
289C4⋊C4—1571no—
290C4⋊C4C22×D15711yescharacteristic
291C4⋊C4C22×D15711yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—3141no—
295C22⋊C4—3141no—
296C22⋊C4—3141no—
297C22⋊C4—3141no—
298C22⋊C4—3141no—
299C22⋊C4—3141no—
300C22⋊C4—3141no—
301C42—3141no—
302C42—1571no—
303C42—1571no—
304C42C22×D15711yescharacteristic
305C23—21no—
306C23—21no—
307C23—3141no—
308C23—3141no—
309C23—3141no—
310Q8—3142no—
311Q8—3141no—
312Q8—3141no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—3141no—
318D4—3141no—
319D4—3141no—
320D4—3141no—
321D4—3141no—
322D4—3141no—
323D4—6282no—
324D4—6282no—
325D4—6282no—
326D4—6282no—
327D4—12561no—
328D4—12561no—
329D4—12561no—
330D4—12561no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—42no—
334C2×C4—81no—
335C2×C4—3141no—
336C2×C4—6282no—
337C2×C4—6282no—
338C2×C4—1571no—
339C2×C4—1571no—
340C2×C4—1571no—
341C2×C4—1571no—
342C2×C4—12561no—
343C2×C4D4×D15711yescharacteristic
344C2×C4D4×D15711yescharacteristic
345C2×C4C23×D15711yescharacteristic
346C8—21no—
347C8—3141no—
348C8—3141no—
349C8#1457895412yescomplemented
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—6281no—
357C22—6281no—
358C22—6281no—
359C22—6281no—
360C22—6281no—
361C22—6281no—
362C22—6281no—
363C22—6281no—
364C22—6281no—
365C22C2×D4×D15711yescharacteristic
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—3141no—
371C4—3141no—
372C4—3141no—
373C4—6281no—
374C4—1572no—
375C4C2×#1457895411yescharacteristic
376C4#1909781711yescharacteristic
377C2—81no—
378C2—81no—
379C2—12561no—
380C2—12561no—
381C2—12561no—
382C2#2375794411yescharacteristic
383C2AGL(1,ℤ/8)×D15711yescharacteristic
384C2#2375807711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#442×C157C2
socleC22×C157C22×D4
hypercenter#442D157
hypocenterC157#3399
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #23758006C2semidirect
2. #23757944C2semidirect
3. #23757662C2semidirect
4. #23757661C2semidirect
5. #23757648C2semidirect
6. #23757488C2semidirect
7. #23757487C2semidirect
8. #23757472C2semidirect
9. #23757468C2semidirect
10. #23757467C2semidirect
11. #23757467C2semidirect
12. #23757401C2semidirect
13. #23757384C2semidirect
14. #23757360C2semidirect
15. #442×C157C2semidirect
156. C8×C157#41semidirect
192. C157#3399semidirect
207. #442D157semidirect
349. C8#14578954semidirect

Maximal subgroups

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

subgroupindex
#237580062
#237579442
#237576622
#237576612
#237576482
#237574882
#237574872
#237574722
#237574682
#237574672
#237574012
#237573842
#237573602
#442×C1572
#3399157