Finite Group Databasebeta

← #34111986

Subgroups — #34111986

Order 20096.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757969C211yesmaximal, characteristic, complemented
2#23757627C211yesmaximal, characteristic, complemented
3#23757606C211yesmaximal, characteristic, complemented
4#23757594C211yesmaximal, characteristic, complemented
5#23757573C211yesmaximal, characteristic, complemented
6#23757507C211yesmaximal, characteristic, complemented
7#23757504C211yesmaximal, characteristic, complemented
8#23757485C211yesmaximal, characteristic, complemented
9#23757418C211yesmaximal, characteristic, complemented
10#23757407C211yesmaximal, characteristic, complemented
11C4×#14578939C211yesmaximal, characteristic, complemented
12D4×#11929434C211yesmaximal, characteristic, complemented
13#23757269C211yesmaximal, characteristic, complemented
14C2×#19097692C211yesmaximal, characteristic, complemented
15#546×C157C211yesmaximal, characteristic, complemented
16C2×#14578941C2211yescharacteristic, complemented
17C2×#14578939C2211yescharacteristic
18C2×#14578939C2211yescharacteristic
19C22×#11929434C2211yescharacteristic
20C22×#11929434C2211yescharacteristic, complemented
21#19097766—21no—
22#19097763C2211yescharacteristic, complemented
23#19097753—21no—
24#19097745C2211yescharacteristic
25#19097745C2211yescharacteristic, complemented
26#19097743—21no—
27C4×D628C2211yescharacteristic, complemented
28#19097731—21no—
29#19097730C2211yescharacteristic, complemented
30C2×C4⋊Dic157C2211yescharacteristic
31#19097704C2211yescharacteristic, complemented
32#19097704C2211yescharacteristic, complemented
33#19097702C2211yescharacteristic, complemented
34C4⋊#11929434C2211yescharacteristic, complemented
35#19097695C2211yescharacteristic, complemented
36#19097695C2211yescharacteristic, complemented
37#19097695C2211yescharacteristic, complemented
38#19097692C2211yescharacteristic
39#19097692C2211yescharacteristic
40#19097692C2211yescharacteristic
41#19097692C2211yescharacteristic
42#19097692C2211yescharacteristic
43#19097692C2211yescharacteristic
44#19097691C2211yescharacteristic, complemented
45C22⋊#11929434C2211yescharacteristic
46C22⋊#11929434C2211yescharacteristic, complemented
47C4×#11929434C2211yescharacteristic, complemented
48C2×#41×C157C2211yescharacteristic
49C22×D4×C157C2211yescharacteristic
50#124×C157C2211yescharacteristic
51#123×C157—21no—
52C22⋊Q8×C157C2211yescharacteristic
53#121×C157—21no—
54#121×C157C2211yescharacteristic
55#120×C157—21no—
56C4×D4×C157C2211yescharacteristic
57C4×D4×C157C2211yescharacteristic
58C2×C22⋊C4×C157—21no—
59C2×D628C2311yescharacteristic
60C2×C22⋊D157—21no—
61C2×C22⋊D157—21no—
62C22×Dic157—21no—
63C2×C4×D157—21no—
64#14578941—41no—
65#14578939—22no—
66#14578939—41no—
67C2×#11929434—41no—
68C2×#11929434—41no—
69C2×#11929434C2311yescharacteristic
70C2×#11929434C2311yescharacteristic
71C2×#11929434C2311yescharacteristic
72C2×#11929434C2311yescharacteristic
73C4⋊Dic157—21no—
74C4⋊Dic157C2311yescharacteristic
75C4⋊Dic157C2311yescharacteristic
76C4⋊Dic157C2311yescharacteristic
77#14578933—21no—
78#14578933—21no—
79C22⋊Dic157—21no—
80#14578931—21no—
81#14578931—21no—
82#14578931—21no—
83C4×Dic157—21no—
84C24×C157—21no—
85#41×C157—22no—
86#41×C157—41no—
87C2×Q8×C157C2311yescharacteristic
88C2×D4×C157—22no—
89C2×D4×C157—21no—
90C2×D4×C157—21no—
91C2×D4×C157—21no—
92C2×D4×C157—41no—
93C2×D4×C157D411yescharacteristic
94C2×D4×C157C2311yescharacteristic
95C2×D4×C157C2311yescharacteristic
96C22×C4×C157—21no—
97C22×C4×C157—21no—
98C22×C4×C157C2311yescharacteristic
99C22×C4×C157C2311yescharacteristic
100C4⋊C4×C157—21no—
101C4⋊C4×C157D411yescharacteristic
102C4⋊C4×C157C2311yescharacteristic
103C22⋊C4×C157—21no—
104C22⋊C4×C157—21no—
105C22⋊C4×C157—21no—
106C22⋊C4×C157—21no—
107C22⋊C4×C157—21no—
108C22⋊C4×C157D411yescharacteristic
109C22⋊C4×C157D411yescharacteristic
110C42×C157C2311yescharacteristic
111C22×D157—21no—
112D628—21no—
113D628—21no—
114C22⋊D157—42no—
115C22⋊D157—81no—
116C2×Dic157—21no—
117C2×Dic157—21no—
118C2×Dic157—21no—
119C2×Dic157—81no—
120C4×D157—81no—
121#11929434—21no—
122#11929434—21no—
123#11929434—21no—
124#11929434—21no—
125C23×C157—21no—
126C23×C157—21no—
127C23×C157—42no—
128C23×C157—42no—
129C23×C157—81no—
130C23×C157C2×D411yescharacteristic
131C23×C157C2×D411yescharacteristic
132Q8×C157—21no—
133Q8×C157—21no—
134D4×C157—22no—
135D4×C157—21no—
136D4×C157—21no—
137D4×C157—21no—
138D4×C157—42no—
139D4×C157—42no—
140D4×C157—81no—
141D4×C157#4112yes—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157—21no—
145C2×C4×C157—21no—
146C2×C4×C157—21no—
147C2×C4×C157—21no—
148C2×C4×C157—42no—
149C2×C4×C157—81no—
150C2×C4×C157C2×D411yescharacteristic
151C2×C4×C157C2×D411yescharacteristic
152C2×C4×C157C2×D411yescharacteristic
153C2×C4×C157C2×D411yescharacteristic
154C2×C4×C157C2411yescharacteristic
155C22×C157—21no—
156C22×C157—21no—
157C22×C157—21no—
158C22×C157—22no—
159C22×C157—21no—
160C22×C157—21no—
161C22×C157—41no—
162C22×C157—41no—
163C22×C157—41no—
164C22×C157—41no—
165C22×C157—41no—
166C22×C157—84no—
167C22×C157#13512yes—
168C22×C157C22×D411yescharacteristic
169C2×D157—41no—
170C2×D157—41no—
171C2×D157—41no—
172C4×C157—21no—
173C4×C157—21no—
174C4×C157—21no—
175C4×C157—21no—
176C4×C157—41no—
177C4×C157—41no—
178C4×C157C2×#4111yescharacteristic
179C4×C157H(5,𝔽2)11yescharacteristic
180Dic157—41no—
181Dic157—41no—
182Dic157—41no—
183C2×C157—22no—
184C2×C157—41no—
185C2×C157—42no—
186C2×C157—81no—
187C2×C157#54611yescharacteristic
188C2×C157C2×#13511yescharacteristic
189C2×C157#57611yescharacteristic
190D157—81no—
191C157#329711yescharacteristic, complemented, Sylow
192#3297—1571nomaximal, Sylow
193#553—1571no—
194#546D15711yescharacteristic, complemented
195#488—1571no—
196#484—1571no—
197#482—1571no—
198#478—1571no—
199#466—1571no—
200#465—1571no—
201#463—1571no—
202#449—1571no—
203#447—1571no—
204C4×D8—1571no—
205C8×D4—1571no—
206#416—1571no—
207C2×#102—1571no—
208C2×#41C2×D15711yescharacteristic
209C22×D4C2×D15711yescharacteristic
210C2×SD16—1571no—
211C2×D8—1571no—
212C2×D8—1571no—
213C22×C8—1571no—
214C22×C8—1571no—
215#126—3141no—
216#125—1571no—
217#124C2×D15711yescharacteristic
218#123—21no—
219#123—3141no—
220C22⋊Q8C2×D15711yescharacteristic
221#121—21no—
222#121—3141no—
223#121—1571no—
224#121—1571no—
225#121C2×D15711yescharacteristic
226#120—21no—
227C4×D4—3141no—
228C4×D4—1571no—
229C4×D4C2×D15711yescharacteristic
230C4×D4C2×D15711yescharacteristic
231#117—1571no—
232C2×C4⋊C4—1571no—
233C2×C22⋊C4—21no—
234#107—1571no—
235#107—1571no—
236#106—1571no—
237C4⋊C8—1571no—
238#103—1571no—
239#103—1571no—
240#103—1571no—
241#102—1571no—
242#102—1571no—
243#102—1571no—
244#102—1571no—
245#102—1571no—
246#102—1571no—
247#102—1571no—
248C22⋊C8—1571no—
249C22⋊C8—1571no—
250C4×C8—1571no—
251C24—21no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D15711yescharacteristic
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—41no—
260C2×D4—3141no—
261C2×D4—3141no—
262C2×D4—1571no—
263C2×D4C22⋊D15711yescharacteristic
264C2×D4C22×D15711yescharacteristic
265C2×D4C22×D15711yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—3141no—
269C22×C4—3141no—
270C22×C4C22×D15711yescharacteristic
271C22×C4C22×D15711yescharacteristic
272SD16—6281no—
273D8—3142no—
274D8—6281no—
275C2×C8—6281no—
276C2×C8—6281no—
277C2×C8—1571no—
278C2×C8—1571no—
279C2×C8—1571no—
280C2×C8—1571no—
281C4⋊C4—21no—
282C4⋊C4—3141no—
283C4⋊C4—3141no—
284C4⋊C4—3141no—
285C4⋊C4—1571no—
286C4⋊C4—1571no—
287C4⋊C4—1571no—
288C4⋊C4C22⋊D15711yescharacteristic
289C4⋊C4C22×D15711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—3141no—
296C22⋊C4—3141no—
297C22⋊C4—3141no—
298C22⋊C4—3141no—
299C22⋊C4C22⋊D15711yescharacteristic
300C22⋊C4C22⋊D15711yescharacteristic
301C42—3141no—
302C42C22×D15711yescharacteristic
303C23—21no—
304C23—21no—
305C23—42no—
306C23—42no—
307C23—81no—
308C23—3141no—
309C23C2×C22⋊D15711yescharacteristic
310C23C2×C22⋊D15711yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—42no—
318D4—42no—
319D4—81no—
320D4—3141no—
321D4—3141no—
322D4—6282no—
323D4—12561no—
324D4#1457895312yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—81no—
333C2×C4—3141no—
334C2×C4—3141no—
335C2×C4—3141no—
336C2×C4—12561no—
337C2×C4—12561no—
338C2×C4C2×C22⋊D15711yescharacteristic
339C2×C4C2×C22⋊D15711yescharacteristic
340C2×C4C2×C22⋊D15711yescharacteristic
341C2×C4C2×C22⋊D15711yescharacteristic
342C2×C4C23×D15711yescharacteristic
343C8—3141no—
344C8—3141no—
345C8—3141no—
346C8—3141no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—22no—
351C22—21no—
352C22—21no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—84no—
359C22—6281no—
360C22—6281no—
361C22—6281no—
362C22#1909779112yes—
363C22C22×C22⋊D15711yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—6281no—
371C4—6281no—
372C4—6281no—
373C4C2×#1457895311yescharacteristic
374C4#1909781611yescharacteristic
375C2—22no—
376C2—41no—
377C2—42no—
378C2—81no—
379C2—12561no—
380C2#2375791711yescharacteristic
381C2C2×#1909779111yescharacteristic
382C2#2375807611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#546×C157C2
socleC22×C157C22×D4
hypercenter#546D157
hypocenterC157#3297
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #23757969C2semidirect
2. #23757627C2semidirect
3. #23757606C2semidirect
4. #23757594C2semidirect
5. #23757573C2semidirect
6. #23757507C2semidirect
7. #23757504C2semidirect
8. #23757485C2semidirect
9. #23757418C2semidirect
10. #23757407C2semidirect
11. C4×#14578939C2semidirect
12. D4×#11929434C2semidirect
13. #23757269C2semidirect
14. C2×#19097692C2semidirect
15. #546×C157C2semidirect
16. C2×#14578941C22semidirect
20. C22×#11929434C22semidirect
22. #19097763C22semidirect
25. #19097745C22semidirect
27. C4×D628C22semidirect
29. #19097730C22semidirect
31. #19097704C22semidirect
32. #19097704C22semidirect
33. #19097702C22semidirect
34. C4⋊#11929434C22semidirect
35. #19097695C22semidirect
36. #19097695C22semidirect
37. #19097695C22semidirect
44. #19097691C22semidirect
46. C22⋊#11929434C22semidirect
47. C4×#11929434C22semidirect
191. C157#3297semidirect
194. #546D157semidirect

Maximal subgroups

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

subgroupindex
#237579692
#237576272
#237576062
#237575942
#237575732
#237575072
#237575042
#237574852
#237574182
#237574072
C4×#145789392
D4×#119294342
#237572692
C2×#190976922
#546×C1572
#3297157