Finite Group Databasebeta

← #34114215

Subgroups — #34114215

Order 20096.

1104 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23758006C213yesmaximal, complemented
2Q8×Dic314C211yesmaximal, characteristic, complemented
3#23757978C213yesmaximal, complemented
4#23757942C211yesmaximal, characteristic, complemented
5#23757935C211yesmaximal, characteristic, complemented
6#23757930C211yesmaximal, characteristic, complemented
7#23757928C213yesmaximal, complemented
8#23757924C213yesmaximal, complemented
9#23757894C213yesmaximal, complemented
10#23757888C213yesmaximal, complemented
11#23757820C213yesmaximal, complemented
12#23757801C213yesmaximal, complemented
13#23757755C211yesmaximal, characteristic, complemented
14C4×Q8×D157C211yesmaximal, characteristic, complemented
15#520×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×#14578954C2216yes—
18C2×Q8×D157C2211yescharacteristic
19#19097771C2213yes—
20#19097770C2213yes—
21#19097770C2213yes—
22#19097767C2213yes—
23#19097766C2213yes—
24#19097764C2213yes—
25#19097761C2213yes—
26#19097759C2213yes—
27#19097758C2213yes—
28#19097757C2216yes—
29#19097757C2213yes—
30#19097756C2213yes—
31#19097754C2216yes—
32#19097753C2216yes—
33#19097753C2213yes—
34#19097751C2213yes—
35#19097750C2216yes—
36#19097750C2213yes—
37#19097749C2216yes—
38#19097749C2211yescharacteristic
39#19097746C2213yes—
40#19097745C2213yes—
41#19097744C2213yes—
42#19097742C2213yes—
43#19097742C2213yes—
44Q8×Dic157C2211yescharacteristic
45Q8×Dic157C2211yescharacteristic
46C4×Dic314C2213yes—
47C4×Dic314C2213yes—
48#19097736C2213yes—
49C4×D628C2213yes—
50C4×D628C2213yes—
51#19097733C2213yes—
52C4×C22⋊D157C2213yes—
53C4×C22⋊D157C2213yes—
54C4×C22⋊D157C2216yes—
55#19097728C2213yes—
56C4⋊C4×D157C2213yes—
57C4⋊C4×D157C2213yes—
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—
67#14578954—412no—
68C2×Dic314C2313yes—
69C2×Dic314C2311yescharacteristic
70C2×Dic314C2316yes—
71Q8×D157—24no—
72C2×D628C2313yes—
73C2×D628C2313yes—
74C2×C22⋊D157C2313yes—
75C2×C22⋊D157C2313yes—
76C2×C22⋊D157C2316yes—
77C2×C4×D157C2311yescharacteristic
78C2×C4×D157C2313yes—
79C2×C4×D157C2313yes—
80C2×C4×D157C2313yes—
81C2×C4×D157C2316yes—
82C4⋊Dic157C2313yes—
83C4⋊Dic157C2313yes—
84C4⋊Dic157C2313yes—
85C4⋊Dic157C2313yes—
86#14578933C2313yes—
87#14578933C2316yes—
88#14578933C2311yescharacteristic
89#14578933C2311yescharacteristic
90#14578933C2313yes—
91#14578933C2313yes—
92#14578933C2313yes—
93C22⋊Dic157C2313yes—
94C22⋊Dic157C2316yes—
95C22⋊Dic157C2313yes—
96#14578931C2313yes—
97#14578931C2313yes—
98#14578931C2311yescharacteristic
99#14578931C2313yes—
100#14578931C2316yes—
101#14578931C2311yescharacteristic
102#14578931C2313yes—
103C4×Dic157C2313yes—
104C4×Dic157C2313yes—
105C4×Dic157C2313yes—
106C4×Dic157C2313yes—
107#41×C157—24no—
108C2×Q8×C157C2311yescharacteristic
109C2×D4×C157C2313yes—
110C22×C4×C157C2313yes—
111C22×C4×C157C2316yes—
112C4⋊C4×C157C2313yes—
113C4⋊C4×C157C2313yes—
114C4⋊C4×C157C2313yes—
115C4⋊C4×C157C2311yescharacteristic
116C22⋊C4×C157C2313yes—
117C22⋊C4×C157C2313yes—
118C42×C157C2313yes—
119C42×C157C2313yes—
120C22×D157C2413yes—
121C22×D157C2411yescharacteristic
122Dic314—26no—
123Dic314—22no—
124Dic314—46no—
125D628—26no—
126D628—26no—
127C22⋊D157—26no—
128C22⋊D157—26no—
129C22⋊D157—46no—
130C2×Dic157C2411yescharacteristic
131C2×Dic157C2413yes—
132C2×Dic157C2413yes—
133C2×Dic157C2413yes—
134C2×Dic157C2411yescharacteristic
135C2×Dic157C2411yescharacteristic
136C4×D157—26no—
137C4×D157—26no—
138C4×D157—26no—
139C4×D157—22no—
140C4×D157—46no—
141C23×C157C2413yes—
142Q8×C157#4114yescomplemented
143D4×C157—26no—
144C2×C4×C157—26no—
145C2×C4×C157—46no—
146C2×C4×C157C2413yes—
147C2×C4×C157C2411yescharacteristic
148C2×C4×C157C2413yes—
149C2×C4×C157C2411yescharacteristic
150C2×C4×C157C2411yescharacteristic
151C2×C4×C157C2413yes—
152C22×C157—23no—
153C22×C157—23no—
154C22×C157—23no—
155C22×C157C2511yescharacteristic
156C2×D157—21no—
157C2×D157—21no—
158C2×D157—23no—
159C2×D157—23no—
160C2×D157—23no—
161C2×D157H(5,𝔽2)12yes—
162C4×C157—21no—
163C4×C157—23no—
164C4×C157—23no—
165C4×C157—21no—
166C4×C157—21no—
167C4×C157C2×#4116yes—
168Dic157—21no—
169Dic157—23no—
170Dic157—23no—
171Dic157—23no—
172Dic157—21no—
173Dic1572−1+412yes—
174C2×C157—43no—
175C2×C157C22×#4111yescharacteristic
176C2×C157C2×H(5,𝔽2)11yescharacteristic
177C2×C157C2×2−1+411yescharacteristic
178D157—22no—
179D157—43no—
180C157#352211yescharacteristic, complemented, Sylow
181#3522—1571nomaximal, Sylow
182#559—1573no—
183Q82—1571no—
184#554—1573no—
185#550—1571no—
186#548—1571no—
187#548—1571no—
188#547—1573no—
189#547—1573no—
190#542—1573no—
191#542—1573no—
192#532—1573no—
193#529—1573no—
194#520—1571no—
195#520D15711yescharacteristic, complemented
196C2×C4×Q8—1571no—
197C2×#41—1571no—
198C2×#41—1576no—
199C2×#41C2×D15711yescharacteristic
200C22×Q8—1571no—
201C4⋊Q8—1573no—
202C4⋊Q8—1573no—
203C4⋊Q8—1573no—
204#127—1573no—
205#126—1573no—
206#126—1573no—
207#125—1573no—
208#124—1573no—
209#124—1573no—
210#124—1576no—
211#124—1573no—
212#123—1576no—
213#123—1573no—
214#123—1573no—
215#123—1576no—
216C22⋊Q8—1576no—
217C22⋊Q8—1573no—
218C22⋊Q8—1571no—
219C22⋊Q8—1576no—
220C22⋊Q8—1573no—
221#121—1573no—
222#121—1573no—
223#121—1573no—
224#121—1573no—
225#121—1573no—
226C4×Q8—1571no—
227C4×Q8—1571no—
228C4×Q8—1573no—
229C4×Q8—1573no—
230C4×Q8—1573no—
231C4×Q8C2×D15711yescharacteristic
232C4×Q8C2×D15711yescharacteristic
233C4×D4—1573no—
234C4×D4—1573no—
235C4×D4—1576no—
236C4×D4—1573no—
237C4×D4—1573no—
238C4×D4—1573no—
239C4×D4C2×D15713yes—
240C4×D4C2×D15713yes—
241#117—1573no—
242#117C2×D15713yes—
243C2×C4⋊C4—1573no—
244C2×C4⋊C4—1573no—
245C2×C4⋊C4C2×D15713yes—
246C2×C42—1573no—
247#41—24no—
248#41—3144no—
249#41—62812no—
250C2×Q8—3144no—
251C2×Q8—1571no—
252C2×Q8—1573no—
253C2×Q8—1576no—
254C2×Q8C22×D15711yescharacteristic
255C2×D4—1573no—
256C2×D4—1573no—
257C2×D4—1576no—
258C2×D4—1573no—
259C2×D4—1573no—
260C2×D4C22×D15713yes—
261C22×C4—1571no—
262C22×C4—1573no—
263C22×C4—1573no—
264C22×C4—1573no—
265C22×C4—1576no—
266C22×C4C22×D15713yes—
267C22×C4C22×D15716yes—
268C4⋊C4—1573no—
269C4⋊C4—1573no—
270C4⋊C4—1573no—
271C4⋊C4—1576no—
272C4⋊C4—1571no—
273C4⋊C4—1571no—
274C4⋊C4—1573no—
275C4⋊C4—1573no—
276C4⋊C4—1573no—
277C4⋊C4—1573no—
278C4⋊C4—1573no—
279C4⋊C4C22×D15713yes—
280C4⋊C4C22×D15713yes—
281C4⋊C4C22×D15713yes—
282C4⋊C4C22×D15711yescharacteristic
283C22⋊C4—1573no—
284C22⋊C4—1576no—
285C22⋊C4—1573no—
286C22⋊C4—1573no—
287C22⋊C4—1573no—
288C22⋊C4—1571no—
289C22⋊C4—1571no—
290C22⋊C4—1573no—
291C22⋊C4—1576no—
292C22⋊C4—1573no—
293C22⋊C4C22×D15713yes—
294C22⋊C4C22×D15713yes—
295C42—1573no—
296C42—1573no—
297C42—1573no—
298C42—1573no—
299C42C22×D15713yes—
300C42C22×D15713yes—
301C23—1573no—
302C23—1571no—
303C23C23×D15713yes—
304Q8—3142no—
305Q8—3146no—
306Q8—6286no—
307Q8#1457895414yescomplemented
308D4—26no—
309D4—3146no—
310D4—3146no—
311D4—3146no—
312D4—3146no—
313D4—6286no—
314C2×C4—26no—
315C2×C4—46no—
316C2×C4—3146no—
317C2×C4—3142no—
318C2×C4—3146no—
319C2×C4—3146no—
320C2×C4—6286no—
321C2×C4—1571no—
322C2×C4—1573no—
323C2×C4—1573no—
324C2×C4—1573no—
325C2×C4—1571no—
326C2×C4—1571no—
327C2×C4C23×D15711yescharacteristic
328C2×C4C23×D15713yes—
329C2×C4C23×D15711yescharacteristic
330C2×C4C23×D15713yes—
331C2×C4C23×D15711yescharacteristic
332C2×C4C23×D15713yes—
333C22—23no—
334C22—23no—
335C22—23no—
336C22—3141no—
337C22—3141no—
338C22—3143no—
339C22—3143no—
340C22—3143no—
341C22—1572no—
342C22C24×D15711yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—21no—
348C4—3141no—
349C4—3143no—
350C4—3143no—
351C4—3143no—
352C4—3141no—
353C4—1572no—
354C4C2×#1457895416yes—
355C2—43no—
356C2—3142no—
357C2—6283no—
358C2C22×#1457895411yescharacteristic
359C2H(5,𝔽2)×D15711yescharacteristic
360C22−1+4×D15711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D157
commutator, socle, cosocleC22×C157C25
Fitting#520×C157C2
hypercenter#520D157
hypocenterC157#3522
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #23758006C2semidirect
2. Q8×Dic314C2semidirect
3. #23757978C2semidirect
4. #23757942C2semidirect
5. #23757935C2semidirect
6. #23757930C2semidirect
7. #23757928C2semidirect
8. #23757924C2semidirect
9. #23757894C2semidirect
10. #23757888C2semidirect
11. #23757820C2semidirect
12. #23757801C2semidirect
13. #23757755C2semidirect
14. C4×Q8×D157C2semidirect
15. #520×C157C2semidirect
142. Q8×C157#41semidirect
180. C157#3522semidirect
195. #520D157semidirect
307. Q8#14578954semidirect

Maximal subgroups

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

subgroupindex
#237580062
Q8×Dic3142
#237579782
#237579422
#237579352
#237579302
#237579282
#237579242
#237578942
#237578882
#237578202
#237578012
#237577552
C4×Q8×D1572
#520×C1572
#3522157