Finite Group Databasebeta

← #34114248

Subgroups — #34114248

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#23758008C213yesmaximal, complemented
2#23757989C213yesmaximal, complemented
3#23757976C213yesmaximal, complemented
4#23757961C213yesmaximal, complemented
5#23757945C211yesmaximal, characteristic, complemented
6#23757935C211yesmaximal, characteristic, complemented
7#23757935C211yesmaximal, characteristic, complemented
8#23757897C213yesmaximal, complemented
9#23757879C213yesmaximal, complemented
10#23757810C213yesmaximal, complemented
11#23757808C213yesmaximal, complemented
12#23757755C211yesmaximal, characteristic, complemented
13#23757749C211yesmaximal, characteristic, complemented
14C4×#14578955C211yesmaximal, characteristic, complemented
15#557×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×#14578955C2211yescharacteristic
18C2×Q8×D157C2211yescharacteristic
19#19097768C2213yes—
20#19097766C2216yes—
21#19097766C2213yes—
22#19097766C2211yescharacteristic
23#19097762C2213yes—
24#19097760C2216yes—
25#19097760C2213yes—
26#19097759C2213yes—
27#19097759C2213yes—
28#19097756C2216yes—
29#19097756C2216yes—
30#19097756C2213yes—
31#19097756C2213yes—
32#19097751C2213yes—
33#19097751C2213yes—
34#19097747C2213yes—
35#19097746C2213yes—
36#19097746C2213yes—
37#19097746C2213yes—
38#19097746C2216yes—
39#19097746C2213yes—
40#19097746C2213yes—
41Q8×Dic157C2211yescharacteristic
42C4×Dic314C2213yes—
43#19097736C2213yes—
44C4×D628C2213yes—
45C4×D628C2213yes—
46C4×D628C2213yes—
47C4×D628C2213yes—
48C4×D628C2213yes—
49#19097733C2213yes—
50#19097733C2213yes—
51#19097733C2216yes—
52#19097729C2213yes—
53#19097729C2216yes—
54#19097729C2213yes—
55#19097728C2213yes—
56#19097728C2213yes—
57C4⋊C4×D157C2213yes—
58C42×D157C2213yes—
59C4⋊Q8×C157C2213yes—
60#125×C157C2213yes—
61#125×C157C2213yes—
62C4×Q8×C157C2211yescharacteristic
63C4×Q8×C157C2211yescharacteristic
64C4×Q8×C157C2211yescharacteristic
65C4×Q8×C157C2213yes—
66#14578955—24no—
67#14578955—24no—
68C2×Dic314C2313yes—
69Q8×D157—24no—
70C2×D628C2313yes—
71C2×D628C2313yes—
72C2×D628C2316yes—
73C2×D628C2313yes—
74C2×D628C2313yes—
75C2×D628C2313yes—
76C2×C4×D157C2313yes—
77C2×C4×D157C2313yes—
78C2×C4×D157C2313yes—
79C2×C4×D157C2313yes—
80C2×C4×D157C2313yes—
81C2×C4×D157C2316yes—
82C4⋊Dic157C2313yes—
83C4⋊Dic157C2313yes—
84C4⋊Dic157C2311yescharacteristic
85C4⋊Dic157C2313yes—
86#14578933C2313yes—
87#14578933C2311yescharacteristic
88#14578933C2313yes—
89#14578933C2313yes—
90#14578931C2311yescharacteristic
91#14578931C2313yes—
92#14578931C2313yes—
93#14578931C2313yes—
94#14578931C2313yes—
95#14578931C2316yes—
96#14578931C2316yes—
97#14578931C2313yes—
98#14578931C2316yes—
99#14578931C2311yescharacteristic
100#14578931C2311yescharacteristic
101#14578931C2313yes—
102#14578931C2313yes—
103C4×Dic157C2311yescharacteristic
104C4×Dic157C2313yes—
105C4×Dic157C2313yes—
106C4×Dic157C2316yes—
107C2×Q8×C157C2313yes—
108C2×Q8×C157C2311yescharacteristic
109C4⋊C4×C157C2311yescharacteristic
110C4⋊C4×C157C2313yes—
111C4⋊C4×C157C2313yes—
112C4⋊C4×C157C2313yes—
113C4⋊C4×C157C2313yes—
114C4⋊C4×C157C2313yes—
115C4⋊C4×C157C2316yes—
116C42×C157C2313yes—
117C42×C157C2313yes—
118C42×C157C2313yes—
119C22×D157C2411yescharacteristic
120C22×D157C2413yes—
121C22×D157C2413yes—
122Dic314—26no—
123D628—26no—
124D628—26no—
125D628—43no—
126D628—46no—
127D628—43no—
128D628—43no—
129C2×Dic157C2411yescharacteristic
130C2×Dic157C2413yes—
131C2×Dic157C2411yescharacteristic
132C2×Dic157C2411yescharacteristic
133C2×Dic157C2413yes—
134C4×D157—26no—
135C4×D157—26no—
136C4×D157—26no—
137C4×D157—43no—
138C4×D157—43no—
139C4×D157—46no—
140Q8×C157—43no—
141Q8×C157#4114yes—
142C2×C4×C157C2411yescharacteristic
143C2×C4×C157C2413yes—
144C2×C4×C157C2413yes—
145C2×C4×C157C2411yescharacteristic
146C2×C4×C157C2413yes—
147C2×C4×C157C2411yescharacteristic
148C2×C4×C157C2413yes—
149C22×C157C2511yescharacteristic
150C2×D157—21no—
151C2×D157—23no—
152C2×D157—23no—
153C2×D157—23no—
154C2×D157—23no—
155C2×D157—23no—
156C2×D157—21no—
157C2×D157—23no—
158C2×D157—21no—
159C4×C157—21no—
160C4×C157—23no—
161C4×C157—23no—
162C4×C157—21no—
163C4×C157—23no—
164C4×C157—21no—
165C4×C157C2×#4116yes—
166Dic157—21no—
167Dic157—21no—
168Dic157—23no—
169Dic157—21no—
170Dic157—23no—
171C2×C157C22×#4111yescharacteristic
172C2×C157#58511yescharacteristic
173C2×C157#58511yescharacteristic
174D157—41no—
175D157—43no—
176D157—43no—
177C157#352411yescharacteristic, complemented, Sylow
178#3524—1571nomaximal, Sylow
179#559—1573no—
180#557D15711yescharacteristic, complemented
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#520—1571no—
192#519—1571no—
193C4×#41—1571no—
194C2×#41—1571no—
195C2×#41—1571no—
196C22×Q8—1571no—
197C4⋊Q8C2×D15713yes—
198#127—1573no—
199#126—1571no—
200#126—1573no—
201#126—1576no—
202#125—1573no—
203#125C2×D15713yes—
204#125C2×D15713yes—
205#124—1573no—
206#124—1576no—
207#124—1573no—
208#124—1573no—
209#123—1576no—
210#123—1573no—
211#123—1573no—
212#123—1576no—
213C22⋊Q8—1573no—
214C22⋊Q8—1573no—
215C22⋊Q8—1573no—
216#121—1573no—
217#121—1573no—
218#121—1573no—
219#121—1573no—
220#121—1576no—
221#121—1573no—
222C4×Q8—1571no—
223C4×Q8—1573no—
224C4×Q8—1573no—
225C4×Q8C2×D15711yescharacteristic
226C4×Q8C2×D15711yescharacteristic
227C4×Q8C2×D15711yescharacteristic
228C4×Q8C2×D15713yes—
229C4×D4—1573no—
230C4×D4—1573no—
231C4×D4—1576no—
232C4×D4—1573no—
233C4×D4—1573no—
234C4×D4—1573no—
235C4×D4—1573no—
236C4×D4—1573no—
237#117—1573no—
238#117—1573no—
239#117—1576no—
240#117—1573no—
241#117—1573no—
242C2×C4⋊C4—1573no—
243C2×C42—1573no—
244#41—3144no—
245#41—3144no—
246C2×Q8—3144no—
247C2×Q8—1573no—
248C2×Q8C22×D15713yes—
249C2×Q8C22×D15711yescharacteristic
250C2×D4—1573no—
251C2×D4—1573no—
252C2×D4—1573no—
253C2×D4—1573no—
254C2×D4—1576no—
255C2×D4—1573no—
256C22×C4—1573no—
257C22×C4—1573no—
258C22×C4—1573no—
259C22×C4—1573no—
260C22×C4—1573no—
261C22×C4—1576no—
262C4⋊C4—1573no—
263C4⋊C4—1573no—
264C4⋊C4—1571no—
265C4⋊C4—1573no—
266C4⋊C4—1573no—
267C4⋊C4—1573no—
268C4⋊C4—1571no—
269C4⋊C4—1573no—
270C4⋊C4C22×D15713yes—
271C4⋊C4C22×D15713yes—
272C4⋊C4C22×D15713yes—
273C4⋊C4C22×D15716yes—
274C4⋊C4C22×D15713yes—
275C4⋊C4C22×D15711yescharacteristic
276C4⋊C4C22×D15713yes—
277C22⋊C4—1571no—
278C22⋊C4—1573no—
279C22⋊C4—1573no—
280C22⋊C4—1573no—
281C22⋊C4—1576no—
282C22⋊C4—1573no—
283C22⋊C4—1576no—
284C22⋊C4—1571no—
285C22⋊C4—1573no—
286C22⋊C4—1571no—
287C22⋊C4—1573no—
288C22⋊C4—1576no—
289C22⋊C4—1573no—
290C42—1571no—
291C42—1573no—
292C42—1573no—
293C42—1576no—
294C42C22×D15713yes—
295C42C22×D15713yes—
296C42C22×D15713yes—
297C23—1571no—
298C23—1573no—
299C23—1573no—
300Q8—43no—
301Q8—3146no—
302Q8#1457895514yes—
303D4—3146no—
304D4—3146no—
305D4—6283no—
306D4—6286no—
307D4—6283no—
308D4—6283no—
309C2×C4—3146no—
310C2×C4—3146no—
311C2×C4—3146no—
312C2×C4—6283no—
313C2×C4—6283no—
314C2×C4—6286no—
315C2×C4—1571no—
316C2×C4—1571no—
317C2×C4—1573no—
318C2×C4—1573no—
319C2×C4—1571no—
320C2×C4C23×D15713yes—
321C2×C4C23×D15711yescharacteristic
322C2×C4C23×D15713yes—
323C2×C4C23×D15711yescharacteristic
324C2×C4C23×D15711yescharacteristic
325C2×C4C23×D15713yes—
326C2×C4C23×D15713yes—
327C22—3141no—
328C22—3143no—
329C22—3143no—
330C22—3143no—
331C22—3141no—
332C22—3143no—
333C22—3143no—
334C22—3143no—
335C22—3141no—
336C22C24×D15711yescharacteristic
337C4—23no—
338C4—23no—
339C4—21no—
340C4—21no—
341C4—23no—
342C4—21no—
343C4—3141no—
344C4—3141no—
345C4—3143no—
346C4—3141no—
347C4—3143no—
348C4C2×#1457895516yes—
349C2—6281no—
350C2—6283no—
351C2—6283no—
352C2C22×#1457895511yescharacteristic
353C2#2375810911yescharacteristic
354C2#2375811011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D157
commutator, socle, cosocleC22×C157C25
Fitting#557×C157C2
hypercenter#557D157
hypocenterC157#3524
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. #23758008C2semidirect
2. #23757989C2semidirect
3. #23757976C2semidirect
4. #23757961C2semidirect
5. #23757945C2semidirect
6. #23757935C2semidirect
7. #23757935C2semidirect
8. #23757897C2semidirect
9. #23757879C2semidirect
10. #23757810C2semidirect
11. #23757808C2semidirect
12. #23757755C2semidirect
13. #23757749C2semidirect
14. C4×#14578955C2semidirect
15. #557×C157C2semidirect
177. C157#3524semidirect
180. #557D157semidirect

Maximal subgroups

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

subgroupindex
#237580082
#237579892
#237579762
#237579612
#237579452
#237579352
#237578972
#237578792
#237578102
#237578082
#237577552
#237577492
C4×#145789552
#557×C1572
#3524157