Finite Group Databasebeta

← #34111971

Subgroups — #34111971

Order 20096.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757963C211yesmaximal, characteristic, complemented
2#23757627C211yesmaximal, characteristic, complemented
3#23757601C211yesmaximal, characteristic, complemented
4#23757594C211yesmaximal, characteristic, complemented
5#23757573C211yesmaximal, characteristic, complemented
6#23757512C211yesmaximal, characteristic, complemented
7#23757500C211yesmaximal, characteristic, complemented
8#23757492C211yesmaximal, characteristic, complemented
9#23757435C211yesmaximal, characteristic, complemented
10#23757424C211yesmaximal, characteristic, complemented
11C4×#14578942C211yesmaximal, characteristic, complemented
12D4×#11929434C211yesmaximal, characteristic, complemented
13#23757269C211yesmaximal, characteristic, complemented
14C2×#19097692C211yesmaximal, characteristic, complemented
15#546×C157C211yesmaximal, characteristic
16C2×#14578944C2211yescharacteristic, complemented
17C2×#14578942C2211yescharacteristic
18C2×#14578942C2211yescharacteristic
19C22×#11929434C2211yescharacteristic
20C22×#11929434C2211yescharacteristic, complemented
21#19097764—21no—
22#19097763C2211yescharacteristic, complemented
23#19097761—21no—
24#19097752—21no—
25#19097750C2211yescharacteristic
26#19097750C2211yescharacteristic, complemented
27C4×Dic314C2211yescharacteristic, complemented
28#19097730C2211yescharacteristic, complemented
29#19097727—21no—
30C2×C4⋊Dic157C2211yescharacteristic
31#19097704C2211yescharacteristic, complemented
32#19097702C2211yescharacteristic, complemented
33#19097702C2211yescharacteristic, complemented
34C4⋊#11929434C2211yescharacteristic, complemented
35#19097695C2211yescharacteristic, complemented
36#19097695C2211yescharacteristic, complemented
37#19097695C2211yescharacteristic, complemented
38#19097694C2211yescharacteristic, complemented
39#19097692C2211yescharacteristic
40#19097692C2211yescharacteristic
41#19097692C2211yescharacteristic
42#19097692C2211yescharacteristic
43#19097692C2211yescharacteristic
44#19097692C2211yescharacteristic
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×Dic314C2311yescharacteristic
60C22×Dic157—21no—
61#14578944—41no—
62#14578942—22no—
63#14578942—41no—
64C2×#11929434—41no—
65C2×#11929434—41no—
66C2×#11929434C2311yescharacteristic
67C2×#11929434C2311yescharacteristic
68C2×#11929434C2311yescharacteristic
69C2×#11929434C2311yescharacteristic
70C4⋊Dic157—21no—
71C4⋊Dic157C2311yescharacteristic
72C4⋊Dic157C2311yescharacteristic
73C4⋊Dic157C2311yescharacteristic
74#14578933—21no—
75#14578933—21no—
76#14578933—21no—
77#14578933—21no—
78#14578933—21no—
79C22⋊Dic157—21no—
80C22⋊Dic157—21no—
81C22⋊Dic157—21no—
82C4×Dic157—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
111Dic314—21no—
112Dic314—21no—
113C2×Dic157—21no—
114C2×Dic157—21no—
115C2×Dic157—21no—
116C2×Dic157—21no—
117C2×Dic157—42no—
118#11929434—21no—
119#11929434—21no—
120#11929434—21no—
121#11929434—21no—
122C23×C157—21no—
123C23×C157—21no—
124C23×C157—42no—
125C23×C157—42no—
126C23×C157—81no—
127C23×C157C2×D411yescharacteristic
128C23×C157C2×D411yescharacteristic
129Q8×C157—21no—
130Q8×C157—21no—
131D4×C157—22no—
132D4×C157—21no—
133D4×C157—21no—
134D4×C157—21no—
135D4×C157—42no—
136D4×C157—42no—
137D4×C157—81no—
138D4×C157#4112yes—
139C2×C4×C157—21no—
140C2×C4×C157—21no—
141C2×C4×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157—21no—
145C2×C4×C157—42no—
146C2×C4×C157—81no—
147C2×C4×C157C2×D411yescharacteristic
148C2×C4×C157C2×D411yescharacteristic
149C2×C4×C157C2×D411yescharacteristic
150C2×C4×C157C2×D411yescharacteristic
151C2×C4×C157C2411yescharacteristic
152C22×C157—21no—
153C22×C157—21no—
154C22×C157—21no—
155C22×C157—22no—
156C22×C157—21no—
157C22×C157—21no—
158C22×C157—41no—
159C22×C157—41no—
160C22×C157—41no—
161C22×C157—41no—
162C22×C157—41no—
163C22×C157—84no—
164C22×C157#13512yes—
165C22×C157C22×D411yescharacteristic
166C4×C157—21no—
167C4×C157—21no—
168C4×C157—21no—
169C4×C157—21no—
170C4×C157—41no—
171C4×C157—41no—
172C4×C157C2×#4111yescharacteristic
173C4×C157H(5,𝔽2)11yescharacteristic
174Dic157—41no—
175Dic157—41no—
176Dic157—41no—
177Dic157—41no—
178C2×C157—22no—
179C2×C157—41no—
180C2×C157—42no—
181C2×C157—81no—
182C2×C157#54611yescharacteristic
183C2×C157C2×#13511yescharacteristic
184C2×C157#57711yescharacteristic
185C157#329611yescharacteristic, complemented, Sylow
186#3296—1571nomaximal, Sylow
187#552—1571no—
188#546D15711yescharacteristic
189#488—1571no—
190#483—1571no—
191#482—1571no—
192#478—1571no—
193#467—1571no—
194#465—1571no—
195#464—1571no—
196#452—1571no—
197#450—1571no—
198C4×SD16—1571no—
199C8×D4—1571no—
200#416—1571no—
201C2×#102—1571no—
202C2×#41C2×D15711yescharacteristic
203C22×D4C2×D15711yescharacteristic
204C2×Q16—1571no—
205C2×SD16—1571no—
206C2×SD16—1571no—
207C22×C8—1571no—
208C22×C8—1571no—
209#126—3141no—
210#125—3141no—
211#125—1571no—
212#124C2×D15711yescharacteristic
213#123—21no—
214#123—3141no—
215C22⋊Q8—1571no—
216C22⋊Q8—1571no—
217C22⋊Q8C2×D15711yescharacteristic
218#121—21no—
219#121C2×D15711yescharacteristic
220#120—21no—
221C4×Q8—1571no—
222C4×D4C2×D15711yescharacteristic
223C4×D4C2×D15711yescharacteristic
224#117—3141no—
225#117—1571no—
226C2×C4⋊C4—1571no—
227C2×C22⋊C4—21no—
228#107—1571no—
229#106—1571no—
230#106—1571no—
231C4⋊C8—1571no—
232#103—1571no—
233#103—1571no—
234#103—1571no—
235#103—1571no—
236#102—1571no—
237#102—1571no—
238#102—1571no—
239#102—1571no—
240#102—1571no—
241#102—1571no—
242C22⋊C8—1571no—
243C22⋊C8—1571no—
244C4×C8—1571no—
245C24—21no—
246#41—22no—
247#41—41no—
248C2×Q8—1571no—
249C2×Q8C22×D15711yescharacteristic
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—41no—
255C2×D4C22⋊D15711yescharacteristic
256C2×D4C22×D15711yescharacteristic
257C2×D4C22×D15711yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—3141no—
261C22×C4C22×D15711yescharacteristic
262C22×C4C22×D15711yescharacteristic
263Q16—6281no—
264SD16—3142no—
265SD16—6281no—
266C2×C8—6281no—
267C2×C8—6281no—
268C2×C8—1571no—
269C2×C8—1571no—
270C2×C8—1571no—
271C2×C8—1571no—
272C4⋊C4—21no—
273C4⋊C4—3141no—
274C4⋊C4—3141no—
275C4⋊C4—3141no—
276C4⋊C4—3141no—
277C4⋊C4—3141no—
278C4⋊C4—3141no—
279C4⋊C4—1571no—
280C4⋊C4—1571no—
281C4⋊C4—1571no—
282C4⋊C4C22⋊D15711yescharacteristic
283C4⋊C4C22×D15711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—3141no—
290C22⋊C4—3141no—
291C22⋊C4—3141no—
292C22⋊C4C22⋊D15711yescharacteristic
293C22⋊C4C22⋊D15711yescharacteristic
294C42—3141no—
295C42—3141no—
296C42C22×D15711yescharacteristic
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23C2×C22⋊D15711yescharacteristic
303C23C2×C22⋊D15711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—3141no—
307Q8—3141no—
308D4—22no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—42no—
313D4—42no—
314D4—81no—
315D4#1457895312yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—81no—
324C2×C4—3141no—
325C2×C4—3141no—
326C2×C4—3141no—
327C2×C4—3141no—
328C2×C4—6282no—
329C2×C4C2×C22⋊D15711yescharacteristic
330C2×C4C2×C22⋊D15711yescharacteristic
331C2×C4C2×C22⋊D15711yescharacteristic
332C2×C4C2×C22⋊D15711yescharacteristic
333C2×C4C23×D15711yescharacteristic
334C8—3141no—
335C8—3141no—
336C8—3141no—
337C8—3141no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—22no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—84no—
350C22#1909779112yes—
351C22C22×C22⋊D15711yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—6281no—
359C4—6281no—
360C4—6281no—
361C4—6281no—
362C4C2×#1457895311yescharacteristic
363C4#1909781611yescharacteristic
364C2—22no—
365C2—41no—
366C2—42no—
367C2—81no—
368C2#2375791711yescharacteristic
369C2C2×#1909779111yescharacteristic
370C2#2375808411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting, prime radical#546×C157C2
socleC22×C157C22×D4
hypercenter#546D157
hypocenterC157#3296
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. #23757963C2semidirect
2. #23757627C2semidirect
3. #23757601C2semidirect
4. #23757594C2semidirect
5. #23757573C2semidirect
6. #23757512C2semidirect
7. #23757500C2semidirect
8. #23757492C2semidirect
9. #23757435C2semidirect
10. #23757424C2semidirect
11. C4×#14578942C2semidirect
12. D4×#11929434C2semidirect
13. #23757269C2semidirect
14. C2×#19097692C2semidirect
16. C2×#14578944C22semidirect
20. C22×#11929434C22semidirect
22. #19097763C22semidirect
26. #19097750C22semidirect
27. C4×Dic314C22semidirect
28. #19097730C22semidirect
31. #19097704C22semidirect
32. #19097702C22semidirect
33. #19097702C22semidirect
34. C4⋊#11929434C22semidirect
35. #19097695C22semidirect
36. #19097695C22semidirect
37. #19097695C22semidirect
38. #19097694C22semidirect
46. C22⋊#11929434C22semidirect
47. C4×#11929434C22semidirect
185. C157#3296semidirect

Maximal subgroups

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

subgroupindex
#237579632
#237576272
#237576012
#237575942
#237575732
#237575122
#237575002
#237574922
#237574352
#237574242
C4×#145789422
D4×#119294342
#237572692
C2×#190976922
#546×C1572
#3296157