Finite Group Databasebeta

← #34114237

Subgroups — #34114237

Order 20096.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23758006C213yesmaximal, complemented
2#23758002C211yesmaximal, characteristic, complemented
3#23757988C213yesmaximal, complemented
4#23757971C213yesmaximal, complemented
5#23757958C213yesmaximal, complemented
6#23757942C211yesmaximal, characteristic, complemented
7#23757935C211yesmaximal, characteristic, complemented
8#23757930C211yesmaximal, characteristic, complemented
9#23757894C213yesmaximal, complemented
10#23757874C213yesmaximal, complemented
11#23757805C213yesmaximal, complemented
12#23757801C213yesmaximal, complemented
13#23757749C211yesmaximal, characteristic, complemented
14C4×#14578955C211yesmaximal, characteristic, complemented
15#520×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×Q8×D157C2211yescharacteristic
18#19097771C2213yes—
19#19097767C2213yes—
20#19097766C2216yes—
21#19097766C2211yescharacteristic
22#19097764C2213yes—
23#19097762C2213yes—
24#19097762C2213yes—
25#19097761C2213yes—
26#19097759C2213yes—
27#19097758C2213yes—
28#19097757C2216yes—
29#19097757C2213yes—
30#19097756C2213yes—
31#19097754C2216yes—
32#19097753C2216yes—
33#19097753C2213yes—
34#19097751C2213yes—
35#19097750C2213yes—
36#19097749C2213yes—
37#19097746C2213yes—
38#19097745C2216yes—
39#19097745C2213yes—
40#19097744C2213yes—
41#19097742C2213yes—
42#19097742C2213yes—
43Q8×Dic157C2211yescharacteristic
44Q8×Dic157C2211yescharacteristic
45C4×Dic314C2213yes—
46C4×Dic314C2213yes—
47#19097736C2213yes—
48C4×D628C2213yes—
49C4×D628C2213yes—
50#19097733C2213yes—
51C4×C22⋊D157C2213yes—
52C4×C22⋊D157C2213yes—
53C4×C22⋊D157C2216yes—
54#19097730C2216yes—
55#19097729C2213yes—
56#19097729C2213yes—
57#19097728C2213yes—
58C42×D157C2213yes—
59C2×#41×C157C2211yescharacteristic
60C4×Q8×C157C2211yescharacteristic
61C4×Q8×C157C2211yescharacteristic
62C4×D4×C157C2213yes—
63C4×D4×C157C2213yes—
64#117×C157C2213yes—
65C2×C4⋊C4×C157C2213yes—
66#14578955—24no—
67C2×Dic314C2313yes—
68C2×Dic314C2313yes—
69Q8×D157—24no—
70C2×D628C2313yes—
71C2×D628C2313yes—
72C2×C22⋊D157C2313yes—
73C2×C22⋊D157C2316yes—
74C2×C22⋊D157C2313yes—
75C2×C4×D157C2313yes—
76C2×C4×D157C2313yes—
77C2×C4×D157C2313yes—
78C2×C4×D157C2313yes—
79C4⋊Dic157C2313yes—
80C4⋊Dic157C2313yes—
81C4⋊Dic157C2313yes—
82C4⋊Dic157C2316yes—
83C4⋊Dic157C2311yescharacteristic
84#14578933C2313yes—
85#14578933C2313yes—
86#14578933C2311yescharacteristic
87#14578933C2313yes—
88#14578933C2313yes—
89#14578933C2313yes—
90C22⋊Dic157C2313yes—
91C22⋊Dic157C2316yes—
92C22⋊Dic157C2313yes—
93#14578931C2311yescharacteristic
94#14578931C2313yes—
95#14578931C2316yes—
96#14578931C2311yescharacteristic
97#14578931C2313yes—
98#14578931C2316yes—
99#14578931C2311yescharacteristic
100#14578931C2313yes—
101C4×Dic157C2311yescharacteristic
102C4×Dic157C2313yes—
103C4×Dic157C2313yes—
104C4×Dic157C2313yes—
105C4×Dic157C2316yes—
106#41×C157—24no—
107C2×Q8×C157C2311yescharacteristic
108C2×D4×C157C2313yes—
109C22×C4×C157C2313yes—
110C22×C4×C157C2316yes—
111C4⋊C4×C157C2313yes—
112C4⋊C4×C157C2313yes—
113C4⋊C4×C157C2313yes—
114C4⋊C4×C157C2311yescharacteristic
115C22⋊C4×C157C2313yes—
116C22⋊C4×C157C2313yes—
117C42×C157C2313yes—
118C42×C157C2313yes—
119C22×D157C2411yescharacteristic
120C22×D157C2413yes—
121Dic314—26no—
122Dic314—43no—
123D628—26no—
124D628—43no—
125C22⋊D157—43no—
126C22⋊D157—43no—
127C22⋊D157—46no—
128C2×Dic157C2411yescharacteristic
129C2×Dic157C2411yescharacteristic
130C2×Dic157C2413yes—
131C2×Dic157C2413yes—
132C2×Dic157C2411yescharacteristic
133C2×Dic157C2413yes—
134C4×D157—26no—
135C4×D157—26no—
136C4×D157—43no—
137C4×D157—43no—
138C23×C157C2413yes—
139Q8×C157#4114yes—
140D4×C157—26no—
141C2×C4×C157—26no—
142C2×C4×C157—46no—
143C2×C4×C157C2411yescharacteristic
144C2×C4×C157C2413yes—
145C2×C4×C157C2413yes—
146C2×C4×C157C2411yescharacteristic
147C2×C4×C157C2411yescharacteristic
148C2×C4×C157C2413yes—
149C22×C157—23no—
150C22×C157—23no—
151C22×C157—23no—
152C22×C157C2511yescharacteristic
153C2×D157—21no—
154C2×D157—23no—
155C2×D157—23no—
156C2×D157—23no—
157C2×D157—21no—
158C2×D157—21no—
159C4×C157—21no—
160C4×C157—23no—
161C4×C157—23no—
162C4×C157—21no—
163C4×C157—21no—
164C4×C157C2×#4116yes—
165Dic157—21no—
166Dic157—23no—
167Dic157—21no—
168Dic157—23no—
169Dic157—21no—
170Dic157—23no—
171C2×C157—43no—
172C2×C157C22×#4111yescharacteristic
173C2×C157#58511yescharacteristic
174C2×C157#58511yescharacteristic
175D157—41no—
176D157—43no—
177C157#352411yescharacteristic, complemented, Sylow
178#3524—1571nomaximal, Sylow
179#559—1573no—
180#557—1571no—
181#555—1573no—
182#553—1573no—
183#552—1573no—
184#550—1571no—
185#548—1571no—
186#548—1571no—
187#542—1573no—
188#540—1573no—
189#529—1573no—
190#529—1573no—
191#520D15711yescharacteristic, complemented
192#519—1571no—
193C4×#41—1571no—
194C2×#41—1571no—
195C2×#41C2×D15711yescharacteristic
196C22×Q8—1571no—
197C4⋊Q8—1573no—
198#127—1573no—
199#126—1571no—
200#126—1576no—
201#126—1573no—
202#125—1573no—
203#125—1573no—
204#125—1573no—
205#124—1573no—
206#124—1573no—
207#124—1576no—
208#124—1573no—
209#123—1576no—
210#123—1573no—
211#123—1576no—
212#123—1573no—
213C22⋊Q8—1573no—
214C22⋊Q8—1573no—
215C22⋊Q8—1573no—
216#121—1573no—
217#121—1573no—
218#121—1576no—
219#121—1573no—
220#121—1573no—
221#121—1573no—
222C4×Q8—1571no—
223C4×Q8—1571no—
224C4×Q8—1573no—
225C4×Q8—1573no—
226C4×Q8—1573no—
227C4×Q8C2×D15711yescharacteristic
228C4×Q8C2×D15711yescharacteristic
229C4×D4—1573no—
230C4×D4—1573no—
231C4×D4—1576no—
232C4×D4—1573no—
233C4×D4—1573no—
234C4×D4—1573no—
235C4×D4C2×D15713yes—
236C4×D4C2×D15713yes—
237#117—1576no—
238#117—1573no—
239#117—1573no—
240#117—1573no—
241#117C2×D15713yes—
242C2×C4⋊C4C2×D15713yes—
243C2×C42—1573no—
244#41—24no—
245#41—3144no—
246C2×Q8—3144no—
247C2×Q8—1573no—
248C2×Q8—1573no—
249C2×Q8C22×D15711yescharacteristic
250C2×D4—1573no—
251C2×D4—1576no—
252C2×D4—1573no—
253C2×D4—1573no—
254C2×D4—1573no—
255C2×D4C22×D15713yes—
256C22×C4—1573no—
257C22×C4—1573no—
258C22×C4—1573no—
259C22×C4—1573no—
260C22×C4C22×D15713yes—
261C22×C4C22×D15716yes—
262C4⋊C4—1573no—
263C4⋊C4—1573no—
264C4⋊C4—1573no—
265C4⋊C4—1573no—
266C4⋊C4—1573no—
267C4⋊C4—1576no—
268C4⋊C4—1571no—
269C4⋊C4—1573no—
270C4⋊C4—1573no—
271C4⋊C4—1571no—
272C4⋊C4—1573no—
273C4⋊C4C22×D15713yes—
274C4⋊C4C22×D15713yes—
275C4⋊C4C22×D15713yes—
276C4⋊C4C22×D15711yescharacteristic
277C22⋊C4—1573no—
278C22⋊C4—1576no—
279C22⋊C4—1573no—
280C22⋊C4—1571no—
281C22⋊C4—1573no—
282C22⋊C4—1576no—
283C22⋊C4—1571no—
284C22⋊C4—1571no—
285C22⋊C4—1573no—
286C22⋊C4—1576no—
287C22⋊C4—1573no—
288C22⋊C4C22×D15713yes—
289C22⋊C4C22×D15713yes—
290C42—1571no—
291C42—1573no—
292C42—1573no—
293C42—1576no—
294C42—1573no—
295C42C22×D15713yes—
296C42C22×D15713yes—
297C23—1571no—
298C23—1573no—
299C23C23×D15713yes—
300Q8—3146no—
301Q8—6283no—
302Q8#1457895414yes—
303D4—26no—
304D4—3146no—
305D4—6283no—
306D4—6283no—
307D4—6283no—
308D4—6286no—
309C2×C4—26no—
310C2×C4—46no—
311C2×C4—3146no—
312C2×C4—3146no—
313C2×C4—6283no—
314C2×C4—6283no—
315C2×C4—1571no—
316C2×C4—1571no—
317C2×C4—1573no—
318C2×C4—1571no—
319C2×C4—1573no—
320C2×C4—1573no—
321C2×C4C23×D15713yes—
322C2×C4C23×D15711yescharacteristic
323C2×C4C23×D15711yescharacteristic
324C2×C4C23×D15713yes—
325C2×C4C23×D15711yescharacteristic
326C2×C4C23×D15713yes—
327C22—23no—
328C22—23no—
329C22—23no—
330C22—3141no—
331C22—3143no—
332C22—3143no—
333C22—3141no—
334C22—3143no—
335C22—3141no—
336C22C24×D15711yescharacteristic
337C4—21no—
338C4—23no—
339C4—23no—
340C4—21no—
341C4—21no—
342C4—3141no—
343C4—3143no—
344C4—3141no—
345C4—3141no—
346C4—3143no—
347C4—3143no—
348C4C2×#1457895416yes—
349C2—43no—
350C2—6281no—
351C2—6283no—
352C2C22×#1457895411yescharacteristic
353C2#2375810811yescharacteristic
354C2#2375811011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D157
commutator, socle, cosocleC22×C157C25
Fitting#520×C157C2
hypercenter#520D157
hypocenterC157#3524
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #23758006C2semidirect
2. #23758002C2semidirect
3. #23757988C2semidirect
4. #23757971C2semidirect
5. #23757958C2semidirect
6. #23757942C2semidirect
7. #23757935C2semidirect
8. #23757930C2semidirect
9. #23757894C2semidirect
10. #23757874C2semidirect
11. #23757805C2semidirect
12. #23757801C2semidirect
13. #23757749C2semidirect
14. C4×#14578955C2semidirect
15. #520×C157C2semidirect
177. C157#3524semidirect
191. #520D157semidirect

Maximal subgroups

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

subgroupindex
#237580062
#237580022
#237579882
#237579712
#237579582
#237579422
#237579352
#237579302
#237578942
#237578742
#237578052
#237578012
#237577492
C4×#145789552
#520×C1572
#3524157