Finite Group Databasebeta

← #34114234

Subgroups — #34114234

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#23758009C213yesmaximal, complemented
2#23757999C211yesmaximal, characteristic, complemented
3#23757987C213yesmaximal, complemented
4#23757975C213yesmaximal, complemented
5#23757963C213yesmaximal, complemented
6#23757930C211yesmaximal, characteristic, complemented
7#23757930C211yesmaximal, characteristic, complemented
8#23757890C213yesmaximal, complemented
9#23757877C213yesmaximal, complemented
10#23757804C213yesmaximal, complemented
11#23757803C213yesmaximal, complemented
12#23757756C211yesmaximal, characteristic, complemented
13#23757749C211yesmaximal, characteristic, complemented
14#41×Dic157C211yesmaximal, characteristic, complemented
15#550×C157C211yesmaximal, characteristic, complemented
16C2×Q8×D157C2211yescharacteristic
17#19097770C2213yes—
18#19097766C2211yescharacteristic
19#19097764C2216yes—
20#19097764C2213yes—
21#19097763C2213yes—
22#19097762C2213yes—
23#19097761C2213yes—
24#19097758C2213yes—
25#19097758C2216yes—
26#19097758C2213yes—
27#19097757C2213yes—
28#19097755C2213yes—
29#19097752C2216yes—
30#19097752C2216yes—
31#19097752C2213yes—
32#19097750C2213yes—
33#19097750C2213yes—
34#19097748C2213yes—
35#19097743C2213yes—
36#19097743C2216yes—
37#19097742C2213yes—
38#19097742C2213yes—
39Q8×Dic157C2211yescharacteristic
40Q8×Dic157C2211yescharacteristic
41Q8×Dic157C2211yescharacteristic
42C4×Dic314C2213yes—
43C4×Dic314C2213yes—
44#19097736C2213yes—
45D4×Dic157C2213yes—
46D4×Dic157C2213yes—
47D4×Dic157C2213yes—
48D4×Dic157C2216yes—
49D4×Dic157C2213yes—
50C4×C22⋊D157C2213yes—
51C4×C22⋊D157C2213yes—
52#19097730C2213yes—
53#19097730C2213yes—
54#19097729C2213yes—
55#19097728C2213yes—
56#19097727C2216yes—
57C2×C4⋊Dic157C2213yes—
58C2×C4×Dic157C2213yes—
59C2×#41×C157C2211yescharacteristic
60C2×#41×C157C2211yescharacteristic
61#127×C157C2213yes—
62#121×C157C2213yes—
63#121×C157C2213yes—
64C4×Q8×C157C2211yescharacteristic
65C4×D4×C157C2213yes—
66C2×Dic314C2313yes—
67C2×Dic314C2313yes—
68Q8×D157—24no—
69C2×C22⋊D157C2313yes—
70C2×C22⋊D157C2313yes—
71C22×Dic157C2313yes—
72C22×Dic157C2313yes—
73C22×Dic157C2316yes—
74C2×C4×D157C2313yes—
75C4⋊Dic157C2313yes—
76C4⋊Dic157C2313yes—
77C4⋊Dic157C2313yes—
78C4⋊Dic157C2311yescharacteristic
79C4⋊Dic157C2313yes—
80C4⋊Dic157C2313yes—
81#14578933C2313yes—
82#14578933C2316yes—
83#14578933C2313yes—
84#14578933C2313yes—
85#14578933C2313yes—
86#14578933C2311yescharacteristic
87#14578933C2313yes—
88C22⋊Dic157C2316yes—
89C22⋊Dic157C2313yes—
90C22⋊Dic157C2316yes—
91C22⋊Dic157C2313yes—
92C22⋊Dic157C2313yes—
93C22⋊Dic157C2313yes—
94C22⋊Dic157C2316yes—
95#14578931C2311yescharacteristic
96#14578931C2313yes—
97#14578931C2311yescharacteristic
98#14578931C2311yescharacteristic
99C4×Dic157C2311yescharacteristic
100C4×Dic157C2313yes—
101C4×Dic157C2313yes—
102C4×Dic157C2313yes—
103C4×Dic157C2316yes—
104C4×Dic157C2313yes—
105#41×C157—24no—
106#41×C157—24no—
107C2×Q8×C157C2311yescharacteristic
108C2×D4×C157C2313yes—
109C2×D4×C157C2313yes—
110C2×D4×C157C2313yes—
111C2×D4×C157C2316yes—
112C22×C4×C157C2313yes—
113C22×C4×C157C2313yes—
114C4⋊C4×C157C2311yescharacteristic
115C4⋊C4×C157C2313yes—
116C22⋊C4×C157C2313yes—
117C22⋊C4×C157C2313yes—
118C42×C157C2313yes—
119C22×D157C2411yescharacteristic
120Dic314—26no—
121Dic314—43no—
122C22⋊D157—43no—
123C22⋊D157—43no—
124C2×Dic157—43no—
125C2×Dic157—46no—
126C2×Dic157—43no—
127C2×Dic157C2413yes—
128C2×Dic157C2411yescharacteristic
129C2×Dic157C2411yescharacteristic
130C2×Dic157C2413yes—
131C2×Dic157C2413yes—
132C2×Dic157C2413yes—
133C2×Dic157C2411yescharacteristic
134C4×D157—26no—
135C23×C157C2413yes—
136C23×C157C2413yes—
137Q8×C157#4114yes—
138D4×C157—26no—
139D4×C157—26no—
140D4×C157—43no—
141D4×C157—46no—
142C2×C4×C157—26no—
143C2×C4×C157—26no—
144C2×C4×C157C2413yes—
145C2×C4×C157C2411yescharacteristic
146C2×C4×C157C2411yescharacteristic
147C2×C4×C157C2411yescharacteristic
148C2×C4×C157C2413yes—
149C22×C157—23no—
150C22×C157—23no—
151C22×C157—23no—
152C22×C157—23no—
153C22×C157—23no—
154C22×C157—23no—
155C22×C157C2511yescharacteristic
156C2×D157—21no—
157C2×D157—21no—
158C2×D157—21no—
159C4×C157—21no—
160C4×C157—21no—
161C4×C157—21no—
162C4×C157—23no—
163C4×C157C2×#4116yes—
164Dic157—23no—
165Dic157—21no—
166Dic157—21no—
167Dic157—23no—
168Dic157—23no—
169Dic157—23no—
170Dic157—21no—
171C2×C157—43no—
172C2×C157—43no—
173C2×C157C22×#4111yescharacteristic
174C2×C157#58511yescharacteristic
175C2×C157#58511yescharacteristic
176D157—41no—
177C157#352411yescharacteristic, complemented, Sylow
178#3524—1571nomaximal, Sylow
179#559—1573no—
180#557—1571no—
181#555—1573no—
182#553—1573no—
183#552—1573no—
184#550D15711yescharacteristic, complemented
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×#41C2×D15711yescharacteristic
195C2×#41C2×D15711yescharacteristic
196C22×Q8—1571no—
197C4⋊Q8—1573no—
198#127C2×D15713yes—
199#126—1571no—
200#126—1573no—
201#126—1576no—
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—1573no—
219#121—1576no—
220#121C2×D15713yes—
221#121C2×D15713yes—
222C4×Q8—1571no—
223C4×Q8—1571no—
224C4×Q8—1571no—
225C4×Q8—1573no—
226C4×Q8—1573no—
227C4×Q8—1573no—
228C4×Q8C2×D15711yescharacteristic
229C4×D4—1573no—
230C4×D4—1573no—
231C4×D4—1573no—
232C4×D4—1573no—
233C4×D4—1576no—
234C4×D4—1573no—
235C4×D4—1573no—
236C4×D4C2×D15713yes—
237#117—1573no—
238#117—1573no—
239#117—1573no—
240#117—1576no—
241#117—1573no—
242C2×C4⋊C4—1573no—
243C2×C42—1573no—
244#41—24no—
245#41—24no—
246C2×Q8—3144no—
247C2×Q8—1573no—
248C2×Q8—1573no—
249C2×Q8C22×D15711yescharacteristic
250C2×D4—1573no—
251C2×D4—1573no—
252C2×D4C22×D15713yes—
253C2×D4C22×D15713yes—
254C2×D4C22×D15716yes—
255C2×D4C22×D15713yes—
256C22×C4—1573no—
257C22×C4—1573no—
258C22×C4—1573no—
259C22×C4—1576no—
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—1573no—
272C4⋊C4—1573no—
273C4⋊C4—1571no—
274C4⋊C4—1573no—
275C4⋊C4C22×D15713yes—
276C4⋊C4C22×D15711yescharacteristic
277C22⋊C4—1573no—
278C22⋊C4—1576no—
279C22⋊C4—1573no—
280C22⋊C4—1576no—
281C22⋊C4—1573no—
282C22⋊C4—1576no—
283C22⋊C4—1573no—
284C22⋊C4—1571no—
285C22⋊C4—1573no—
286C22⋊C4—1571no—
287C22⋊C4—1571no—
288C22⋊C4C22×D15713yes—
289C22⋊C4C22×D15713yes—
290C42—1571no—
291C42—1573no—
292C42—1573no—
293C42—1573no—
294C42—1576no—
295C42—1573no—
296C42C22×D15713yes—
297C23—1571no—
298C23C23×D15713yes—
299C23C23×D15713yes—
300Q8—3146no—
301Q8—6283no—
302Q8#1457895314yes—
303D4—26no—
304D4—26no—
305D4—46no—
306D4—43no—
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—1573no—
321C2×C4—1573no—
322C2×C4C23×D15711yescharacteristic
323C2×C4C23×D15713yes—
324C2×C4C23×D15711yescharacteristic
325C2×C4C23×D15711yescharacteristic
326C2×C4C23×D15713yes—
327C22—23no—
328C22—23no—
329C22—23no—
330C22—23no—
331C22—23no—
332C22—23no—
333C22—3141no—
334C22—3141no—
335C22—3141no—
336C22C24×D15711yescharacteristic
337C4—21no—
338C4—21no—
339C4—23no—
340C4—21no—
341C4—3143no—
342C4—3141no—
343C4—3143no—
344C4—3141no—
345C4—3141no—
346C4—3143no—
347C4—3143no—
348C4C2×#1457895316yes—
349C2—43no—
350C2—43no—
351C2—6281no—
352C2C22×#1457895311yescharacteristic
353C2#2375810811yescharacteristic
354C2#2375810911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D157
commutator, socle, cosocleC22×C157C25
Fitting#550×C157C2
hypercenter#550D157
hypocenterC157#3524
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #23758009C2semidirect
2. #23757999C2semidirect
3. #23757987C2semidirect
4. #23757975C2semidirect
5. #23757963C2semidirect
6. #23757930C2semidirect
7. #23757930C2semidirect
8. #23757890C2semidirect
9. #23757877C2semidirect
10. #23757804C2semidirect
11. #23757803C2semidirect
12. #23757756C2semidirect
13. #23757749C2semidirect
14. #41×Dic157C2semidirect
15. #550×C157C2semidirect
177. C157#3524semidirect
184. #550D157semidirect

Maximal subgroups

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

subgroupindex
#237580092
#237579992
#237579872
#237579752
#237579632
#237579302
#237578902
#237578772
#237578042
#237578032
#237577562
#237577492
#41×Dic1572
#550×C1572
#3524157