Finite Group Databasebeta

← #34112020

Subgroups — #34112020

Order 20096.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757961C211yesmaximal, characteristic, complemented
2#23757935C211yesmaximal, characteristic
3#23757630C211yesmaximal, characteristic, complemented
4#23757617C211yesmaximal, characteristic, complemented
5#23757588C211yesmaximal, characteristic, complemented
6#23757515C211yesmaximal, characteristic, complemented
7#23757503C211yesmaximal, characteristic, complemented
8#23757496C211yesmaximal, characteristic, complemented
9#23757438C211yesmaximal, characteristic, complemented
10#23757430C211yesmaximal, characteristic, complemented
11C4×#14578944C211yesmaximal, characteristic, complemented
12#23757338C211yesmaximal, characteristic, complemented
13#23757268C211yesmaximal, characteristic, complemented
14#103×D157C211yesmaximal, characteristic, complemented
15#479×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×Q8×D157C2211yescharacteristic
18C2×Dic628C2211yescharacteristic
19C2×#14578944C2211yescharacteristic
20C2×#14578943C2211yescharacteristic
21C2×C8×D157C2211yescharacteristic
22C2×C8×D157C2211yescharacteristic
23#19097766—21no—
24#19097762—21no—
25#19097759—21no—
26#19097759C2211yescharacteristic
27#19097756—21no—
28#19097751—21no—
29#19097751C2211yescharacteristic
30#19097751C2211yescharacteristic
31#19097751C2211yescharacteristic
32#19097746—21no—
33#19097746C2211yescharacteristic
34C4×Dic314C2211yescharacteristic
35C4×D628—21no—
36C4×D628C2211yescharacteristic
37#19097729—21no—
38#19097729C2211yescharacteristic
39C4⋊C4×D157C2211yescharacteristic
40#19097704C2211yescharacteristic
41#19097696C2211yescharacteristic
42#19097695C2211yescharacteristic
43#19097694C2211yescharacteristic
44#19097694C2211yescharacteristic
45#19097694C2211yescharacteristic
46#19097693C2211yescharacteristic
47#19097691C2211yescharacteristic
48#19097691C2211yescharacteristic
49#19097683C2211yescharacteristic
50#19097683C2211yescharacteristic
51C4×#11929434C2211yescharacteristic
52#125×C157C2211yescharacteristic
53C4×Q8×C157C2211yescharacteristic
54#107×C157C2211yescharacteristic
55#106×C157C2211yescharacteristic
56C4⋊C8×C157C2211yescharacteristic
57#103×C157C2211yescharacteristic
58#103×C157C2211yescharacteristic
59#14578955—22no—
60#14578955—41no—
61C2×Dic314—21no—
62C2×Dic314C2311yescharacteristic
63C2×Dic314C2311yescharacteristic
64Q8×D157—22no—
65Q8×D157—41no—
66C2×D628—21no—
67C2×D628—21no—
68C2×D628D411yescharacteristic
69C2×D628C2311yescharacteristic
70C2×C4×D157—21no—
71C2×C4×D157—21no—
72C2×C4×D157—21no—
73C2×C4×D157C2311yescharacteristic
74C2×C4×D157C2311yescharacteristic
75Dic628—41no—
76#14578944—22no—
77#14578943—41no—
78C2×#11929434C2311yescharacteristic
79C2×#11929434C2311yescharacteristic
80C8×D157—41no—
81C8×D157—41no—
82C4⋊Dic157—21no—
83C4⋊Dic157—21no—
84C4⋊Dic157—21no—
85C4⋊Dic157D411yescharacteristic
86C4⋊Dic157C2311yescharacteristic
87#14578933—21no—
88#14578933—21no—
89#14578931—21no—
90#14578931—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—
100C2×Q8×C157C2311yescharacteristic
101C2×C8×C157C2311yescharacteristic
102C2×C8×C157C2311yescharacteristic
103C4⋊C4×C157—21no—
104C4⋊C4×C157—21no—
105C4⋊C4×C157—21no—
106C4⋊C4×C157C2311yescharacteristic
107C4⋊C4×C157C2311yescharacteristic
108C4⋊C4×C157C2311yescharacteristic
109C42×C157—21no—
110C42×C157C2311yescharacteristic
111C22×D157—21no—
112C22×D157C2×D411yescharacteristic
113C22×D157C2×D411yescharacteristic
114Dic314—21no—
115Dic314—21no—
116Dic314—21no—
117Dic314—21no—
118Dic314—42no—
119D628—22no—
120D628—21no—
121D628—21no—
122D628—42no—
123D628—42no—
124C2×Dic157—21no—
125C2×Dic157—21no—
126C2×Dic157—21no—
127C2×Dic157C2×D411yescharacteristic
128C2×Dic157C2×D411yescharacteristic
129C4×D157—21no—
130C4×D157—21no—
131C4×D157—21no—
132C4×D157—21no—
133C4×D157—42no—
134C4×D157—42no—
135C4×D157—42no—
136#11929434—21no—
137#11929434—21no—
138Q8×C157—21no—
139Q8×C157#4112yes—
140C2×C4×C157—21no—
141C2×C4×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157C2×D411yescharacteristic
145C2×C4×C157C2×D411yescharacteristic
146C2×C4×C157C2411yescharacteristic
147C8×C157—21no—
148C8×C157—21no—
149C22×C157C22×D411yescharacteristic
150C2×D157—21no—
151C2×D157—21no—
152C2×D157—21no—
153C2×D157—21no—
154C2×D157—21no—
155C2×D157—41no—
156C2×D157—41no—
157C2×D157—41no—
158C2×D157#13512yes—
159C4×C157—22no—
160C4×C157—21no—
161C4×C157—21no—
162C4×C157—41no—
163C4×C157—41no—
164C4×C157—41no—
165C4×C157C2×#4111yescharacteristic
166C4×C157H(5,𝔽2)11yescharacteristic
167Dic157—21no—
168Dic157—21no—
169Dic157—41no—
170Dic157—41no—
171Dic157—41no—
172C2×C157#54611yescharacteristic
173C2×C157C2×#13511yescharacteristic
174C2×C157#57811yescharacteristic
175D157—22no—
176D157—41no—
177D157—81no—
178C157#330011yescharacteristic, complemented, Sylow
179#3300—1571nomaximal, Sylow
180#552—1571no—
181#548—1571no—
182#488—1571no—
183#485—1571no—
184#480—1571no—
185#479D15711yescharacteristic, complemented
186#467—1571no—
187#465—1571no—
188#464—1571no—
189#452—1571no—
190#451—1571no—
191C4×Q16—1571no—
192C8×D4—1571no—
193#416—1571no—
194C2×#103—1571no—
195C2×#41—1571no—
196C22×Q8—1571no—
197C2×Q16—1571no—
198C2×Q16—1571no—
199C2×SD16—1571no—
200C22×C8—1571no—
201C22×C8—1571no—
202#126—3141no—
203#125—3141no—
204#125C2×D15711yescharacteristic
205#124—3141no—
206#124—1571no—
207#123—3141no—
208C22⋊Q8—3141no—
209C22⋊Q8—1571no—
210C22⋊Q8—1571no—
211C22⋊Q8—1571no—
212#121—3141no—
213#121—1571no—
214C4×Q8—1571no—
215C4×Q8C2×D15711yescharacteristic
216C4×D4—3141no—
217C4×D4—1571no—
218#117—3141no—
219#117—1571no—
220C2×C4⋊C4—1571no—
221#107—1571no—
222#107C2×D15711yescharacteristic
223#106C2×D15711yescharacteristic
224C4⋊C8C2×D15711yescharacteristic
225#103—1571no—
226#103—1571no—
227#103—1571no—
228#103—1571no—
229#103—1571no—
230#103C2×D15711yescharacteristic
231#103C2×D15711yescharacteristic
232#102—1571no—
233#102—1571no—
234#102—1571no—
235C22⋊C8—1571no—
236C22⋊C8—1571no—
237C4×C8—1571no—
238#41—3142no—
239#41—6281no—
240C2×Q8—3141no—
241C2×Q8—3142no—
242C2×Q8—6281no—
243C2×Q8—1571no—
244C2×Q8—1571no—
245C2×Q8C22×D15711yescharacteristic
246C2×D4—3141no—
247C2×D4—3141no—
248C2×D4—1571no—
249C2×D4—1571no—
250C22×C4—3141no—
251C22×C4—3141no—
252C22×C4—3141no—
253C22×C4—1571no—
254C22×C4—1571no—
255Q16—3142no—
256Q16—6281no—
257SD16—6281no—
258C2×C8—6281no—
259C2×C8—6281no—
260C2×C8—1571no—
261C2×C8—1571no—
262C2×C8C22×D15711yescharacteristic
263C2×C8C22×D15711yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—3141no—
268C4⋊C4—3141no—
269C4⋊C4—3141no—
270C4⋊C4—3141no—
271C4⋊C4—3141no—
272C4⋊C4—1571no—
273C4⋊C4—1571no—
274C4⋊C4C22×D15711yescharacteristic
275C4⋊C4C22×D15711yescharacteristic
276C4⋊C4C22×D15711yescharacteristic
277C22⋊C4—3141no—
278C22⋊C4—3141no—
279C22⋊C4—3141no—
280C22⋊C4—3141no—
281C22⋊C4—3141no—
282C22⋊C4—3141no—
283C22⋊C4—3141no—
284C22⋊C4—1571no—
285C22⋊C4—1571no—
286C42—21no—
287C42—3141no—
288C42—3141no—
289C42C22×D15711yescharacteristic
290C23—3141no—
291C23—1571no—
292C23—1571no—
293Q8—21no—
294Q8—3141no—
295Q8—3141no—
296Q8—3141no—
297Q8—3141no—
298Q8—6282no—
299Q8#1457895512yes—
300D4—3142no—
301D4—3141no—
302D4—3141no—
303D4—6282no—
304D4—6282no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—3141no—
310C2×C4—3141no—
311C2×C4—3141no—
312C2×C4—3141no—
313C2×C4—3141no—
314C2×C4—3141no—
315C2×C4—3141no—
316C2×C4—6282no—
317C2×C4—6282no—
318C2×C4—6282no—
319C2×C4—1571no—
320C2×C4—1571no—
321C2×C4D4×D15711yescharacteristic
322C2×C4D4×D15711yescharacteristic
323C2×C4C23×D15711yescharacteristic
324C8—21no—
325C8—21no—
326C8—3141no—
327C8—3141no—
328C22—3141no—
329C22—3141no—
330C22—3141no—
331C22—3141no—
332C22—3141no—
333C22—6281no—
334C22—6281no—
335C22—6281no—
336C22—1572no—
337C22C2×D4×D15711yescharacteristic
338C4—21no—
339C4—22no—
340C4—21no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—3141no—
345C4—3141no—
346C4—6281no—
347C4—6281no—
348C4—6281no—
349C4C2×#1457895511yescharacteristic
350C4#1909781711yescharacteristic
351C2—3142no—
352C2—6281no—
353C2—12561no—
354C2#2375791411yescharacteristic
355C2#135×D15711yescharacteristic
356C2#2375808911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#479×C157C2
socleC22×C157C22×D4
prime radical#23757935C2
hypercenter#479D157
hypocenterC157#3300
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
1. #23757961C2semidirect
3. #23757630C2semidirect
4. #23757617C2semidirect
5. #23757588C2semidirect
6. #23757515C2semidirect
7. #23757503C2semidirect
8. #23757496C2semidirect
9. #23757438C2semidirect
10. #23757430C2semidirect
11. C4×#14578944C2semidirect
12. #23757338C2semidirect
13. #23757268C2semidirect
14. #103×D157C2semidirect
15. #479×C157C2semidirect
178. C157#3300semidirect
185. #479D157semidirect

Maximal subgroups

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

subgroupindex
#237579612
#237579352
#237576302
#237576172
#237575882
#237575152
#237575032
#237574962
#237574382
#237574302
C4×#145789442
#237573382
#237572682
#103×D1572
#479×C1572
#3300157