Finite Group Databasebeta

← #34114238

Subgroups — #34114238

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#23758005C213yesmaximal, complemented
2#23757999C211yesmaximal, characteristic, complemented
3#23757987C213yesmaximal, complemented
4#23757969C213yesmaximal, complemented
5#23757965C213yesmaximal, complemented
6#23757942C211yesmaximal, characteristic, complemented
7#23757934C211yesmaximal, characteristic, complemented
8#23757896C213yesmaximal, complemented
9#23757876C213yesmaximal, complemented
10#23757805C213yesmaximal, complemented
11#23757802C213yesmaximal, complemented
12#23757756C211yesmaximal, characteristic, complemented
13#23757751C211yesmaximal, characteristic, complemented
14C4×#14578955C211yesmaximal, characteristic, complemented
15#548×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17#19097770C2213yes—
18#19097767C2213yes—
19#19097766C2216yes—
20#19097764C2213yes—
21#19097764C2211yescharacteristic
22#19097763C2213yes—
23#19097762C2213yes—
24#19097762C2213yes—
25#19097760C2213yes—
26#19097758C2213yes—
27#19097757C2216yes—
28#19097755C2216yes—
29#19097753C2216yes—
30#19097753C2213yes—
31#19097753C2213yes—
32#19097751C2213yes—
33#19097747C2213yes—
34#19097745C2213yes—
35#19097745C2213yes—
36#19097743C2216yes—
37#19097742C2213yes—
38#19097742C2213yes—
39Q8×Dic157C2213yes—
40Q8×Dic157C2211yescharacteristic
41Q8×Dic157C2211yescharacteristic
42Q8×Dic157C2213yes—
43Q8×Dic157C2211yescharacteristic
44C4×Dic314C2213yes—
45C4×D628C2213yes—
46D4×Dic157C2213yes—
47D4×Dic157C2213yes—
48#19097733C2213yes—
49C4×C22⋊D157C2213yes—
50C4×C22⋊D157C2213yes—
51#19097731C2216yes—
52#19097730C2213yes—
53#19097730C2213yes—
54#19097730C2213yes—
55#19097729C2213yes—
56#19097727C2216yes—
57C2×C4⋊Dic157C2213yes—
58C42×D157C2213yes—
59C2×#41×C157C2211yescharacteristic
60C22×Q8×C157C2211yescharacteristic
61#124×C157C2213yes—
62C22⋊Q8×C157C2213yes—
63#121×C157C2213yes—
64C4×Q8×C157C2211yescharacteristic
65C4×D4×C157C2213yes—
66#14578955—24no—
67C2×Dic314C2313yes—
68C2×D628C2313yes—
69C2×C22⋊D157C2316yes—
70C2×C22⋊D157C2313yes—
71C2×C22⋊D157C2313yes—
72C22×Dic157C2316yes—
73C2×C4×D157C2313yes—
74C2×C4×D157C2313yes—
75C2×C4×D157C2313yes—
76C4⋊Dic157C2313yes—
77C4⋊Dic157C2313yes—
78C4⋊Dic157C2313yes—
79C4⋊Dic157C2311yescharacteristic
80C4⋊Dic157C2313yes—
81C4⋊Dic157C2313yes—
82C4⋊Dic157C2313yes—
83C4⋊Dic157C2313yes—
84#14578933C2313yes—
85#14578933C2316yes—
86#14578933C2313yes—
87#14578933C2311yescharacteristic
88#14578933C2311yescharacteristic
89C22⋊Dic157C2313yes—
90C22⋊Dic157C2311yescharacteristic
91C22⋊Dic157C2313yes—
92C22⋊Dic157C2313yes—
93C22⋊Dic157C2313yes—
94C22⋊Dic157C2311yescharacteristic
95#14578931C2316yes—
96#14578931C2316yes—
97#14578931C2313yes—
98#14578931C2313yes—
99C4×Dic157C2311yescharacteristic
100C4×Dic157C2313yes—
101C4×Dic157C2313yes—
102C4×Dic157C2316yes—
103C4×Dic157C2313yes—
104C4×Dic157C2313yes—
105#41×C157—24no—
106C2×Q8×C157—24no—
107C2×Q8×C157C2313yes—
108C2×Q8×C157C2311yescharacteristic
109C2×D4×C157C2313yes—
110C2×D4×C157C2313yes—
111C22×C4×C157C2313yes—
112C22×C4×C157C2313yes—
113C4⋊C4×C157C2313yes—
114C4⋊C4×C157C2313yes—
115C22⋊C4×C157C2311yescharacteristic
116C22⋊C4×C157C2313yes—
117C22⋊C4×C157C2316yes—
118C42×C157C2313yes—
119C22×D157C2413yes—
120Dic314—43no—
121D628—26no—
122C22⋊D157—43no—
123C22⋊D157—43no—
124C22⋊D157—46no—
125C2×Dic157—46no—
126C2×Dic157C2411yescharacteristic
127C2×Dic157C2411yescharacteristic
128C2×Dic157C2413yes—
129C2×Dic157C2413yes—
130C2×Dic157C2411yescharacteristic
131C2×Dic157C2413yes—
132C2×Dic157C2411yescharacteristic
133C4×D157—26no—
134C4×D157—43no—
135C4×D157—43no—
136C23×C157C2413yes—
137C23×C157C2411yescharacteristic
138Q8×C157—26no—
139Q8×C157#4114yes—
140D4×C157—26no—
141D4×C157—43no—
142C2×C4×C157—26no—
143C2×C4×C157—26no—
144C2×C4×C157C2411yescharacteristic
145C2×C4×C157C2413yes—
146C2×C4×C157C2413yes—
147C2×C4×C157C2411yescharacteristic
148C2×C4×C157C2413yes—
149C22×C157—21no—
150C22×C157—23no—
151C22×C157—21no—
152C22×C157—23no—
153C22×C157—23no—
154C22×C157—21no—
155C22×C157C2511yescharacteristic
156C2×D157—23no—
157C2×D157—23no—
158C2×D157—23no—
159C4×C157—21no—
160C4×C157—23no—
161C4×C157—23no—
162C4×C157—21no—
163C4×C157C2×#4116yes—
164Dic157—21no—
165Dic157—21no—
166Dic157—23no—
167Dic157—23no—
168Dic157—21no—
169Dic157—23no—
170Dic157—21no—
171C2×C157—41no—
172C2×C157—43no—
173C2×C157C22×#4111yescharacteristic
174C2×C157#58511yescharacteristic
175C2×C157#58511yescharacteristic
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#548D15711yescharacteristic, complemented
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×#41C2×D15711yescharacteristic
196C22×Q8C2×D15711yescharacteristic
197C4⋊Q8—1573no—
198#127—1573no—
199#126—1576no—
200#126—1571no—
201#126—1573no—
202#125—1573no—
203#125—1573no—
204#125—1573no—
205#124—1573no—
206#124—1573no—
207#124—1576no—
208#124C2×D15713yes—
209#123—1576no—
210#123—1573no—
211#123—1573no—
212#123—1576no—
213C22⋊Q8—1573no—
214C22⋊Q8—1573no—
215C22⋊Q8C2×D15713yes—
216#121—1573no—
217#121—1573no—
218#121—1576no—
219#121—1573no—
220#121—1573no—
221#121C2×D15713yes—
222C4×Q8—1571no—
223C4×Q8—1573no—
224C4×Q8—1571no—
225C4×Q8—1573no—
226C4×Q8—1571no—
227C4×Q8—1573no—
228C4×Q8C2×D15711yescharacteristic
229C4×D4—1573no—
230C4×D4—1573no—
231C4×D4—1573no—
232C4×D4—1573no—
233C4×D4—1573no—
234C4×D4—1576no—
235C4×D4—1573no—
236C4×D4C2×D15713yes—
237#117—1573no—
238#117—1573no—
239#117—1573no—
240#117—1573no—
241#117—1576no—
242C2×C4⋊C4—1573no—
243C2×C42—1573no—
244#41—24no—
245#41—3144no—
246C2×Q8—24no—
247C2×Q8—1573no—
248C2×Q8C22×D15713yes—
249C2×Q8C22×D15711yescharacteristic
250C2×D4—1573no—
251C2×D4—1576no—
252C2×D4—1573no—
253C2×D4—1573no—
254C2×D4C22×D15713yes—
255C2×D4C22×D15713yes—
256C22×C4—1573no—
257C22×C4—1576no—
258C22×C4—1573no—
259C22×C4—1573no—
260C22×C4C22×D15713yes—
261C22×C4C22×D15713yes—
262C4⋊C4—1573no—
263C4⋊C4—1573no—
264C4⋊C4—1573no—
265C4⋊C4—1571no—
266C4⋊C4—1573no—
267C4⋊C4—1573no—
268C4⋊C4—1573no—
269C4⋊C4—1573no—
270C4⋊C4—1576no—
271C4⋊C4—1571no—
272C4⋊C4—1573no—
273C4⋊C4—1571no—
274C4⋊C4—1573no—
275C4⋊C4C22×D15713yes—
276C4⋊C4C22×D15713yes—
277C22⋊C4—1571no—
278C22⋊C4—1573no—
279C22⋊C4—1573no—
280C22⋊C4—1573no—
281C22⋊C4—1573no—
282C22⋊C4—1571no—
283C22⋊C4—1576no—
284C22⋊C4—1573no—
285C22⋊C4—1576no—
286C22⋊C4—1573no—
287C22⋊C4C22×D15716yes—
288C22⋊C4C22×D15713yes—
289C22⋊C4C22×D15711yescharacteristic
290C42—1571no—
291C42—1573no—
292C42—1573no—
293C42—1576no—
294C42—1573no—
295C42—1573no—
296C42C22×D15713yes—
297C23—1573no—
298C23C23×D15711yescharacteristic
299C23C23×D15713yes—
300Q8—26no—
301Q8—6283no—
302Q8#1457895314yes—
303D4—26no—
304D4—43no—
305D4—3146no—
306D4—6286no—
307D4—6283no—
308D4—6283no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—3146no—
312C2×C4—6283no—
313C2×C4—6283no—
314C2×C4—6286no—
315C2×C4—1571no—
316C2×C4—1571no—
317C2×C4—1573no—
318C2×C4—1571no—
319C2×C4—1573no—
320C2×C4—1571no—
321C2×C4—1573no—
322C2×C4C23×D15713yes—
323C2×C4C23×D15711yescharacteristic
324C2×C4C23×D15713yes—
325C2×C4C23×D15711yescharacteristic
326C2×C4C23×D15713yes—
327C22—21no—
328C22—23no—
329C22—21no—
330C22—23no—
331C22—23no—
332C22—21no—
333C22—3143no—
334C22—3143no—
335C22—3143no—
336C22C24×D15711yescharacteristic
337C4—21no—
338C4—23no—
339C4—23no—
340C4—21no—
341C4—3141no—
342C4—3141no—
343C4—3143no—
344C4—3141no—
345C4—3143no—
346C4—3143no—
347C4—3141no—
348C4C2×#1457895316yes—
349C2—41no—
350C2—43no—
351C2—6283no—
352C2C22×#1457895311yescharacteristic
353C2#2375810911yescharacteristic
354C2#2375811011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D157
commutator, socle, cosocleC22×C157C25
Fitting#548×C157C2
hypercenter#548D157
hypocenterC157#3524
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #23758005C2semidirect
2. #23757999C2semidirect
3. #23757987C2semidirect
4. #23757969C2semidirect
5. #23757965C2semidirect
6. #23757942C2semidirect
7. #23757934C2semidirect
8. #23757896C2semidirect
9. #23757876C2semidirect
10. #23757805C2semidirect
11. #23757802C2semidirect
12. #23757756C2semidirect
13. #23757751C2semidirect
14. C4×#14578955C2semidirect
15. #548×C157C2semidirect
177. C157#3524semidirect
186. #548D157semidirect

Maximal subgroups

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

subgroupindex
#237580052
#237579992
#237579872
#237579692
#237579652
#237579422
#237579342
#237578962
#237578762
#237578052
#237578022
#237577562
#237577512
C4×#145789552
#548×C1572
#3524157