Finite Group Databasebeta

← #34112037

Subgroups — #34112037

Order 20096.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757976C211yesmaximal, characteristic, complemented
2#23757931C211yesmaximal, characteristic, complemented
3#23757612C211yesmaximal, characteristic, complemented
4#23757590C211yesmaximal, characteristic, complemented
5#23757577C211yesmaximal, characteristic, complemented
6#23757508C211yesmaximal, characteristic, complemented
7#23757505C211yesmaximal, characteristic, complemented
8#23757485C211yesmaximal, characteristic, complemented
9#23757419C211yesmaximal, characteristic, complemented
10#23757413C211yesmaximal, characteristic, complemented
11C4×#14578943C211yesmaximal, characteristic, complemented
12C8×D628C211yesmaximal, characteristic, complemented
13#23757266C211yesmaximal, characteristic, complemented
14#103×D157C211yesmaximal, characteristic, complemented
15#488×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic
17C2×Q8×D157C2211yescharacteristic
18C2×#14578943C2211yescharacteristic
19C2×#14578941C2211yescharacteristic
20C2×#14578939C2211yescharacteristic
21C2×C8×D157C2211yescharacteristic
22C2×C8×D157C2211yescharacteristic
23#19097766—21no—
24#19097757—21no—
25#19097756—21no—
26#19097748—21no—
27#19097747C2211yescharacteristic
28#19097746—21no—
29#19097746C2211yescharacteristic
30#19097746C2211yescharacteristic
31#19097744—21no—
32#19097742C2211yescharacteristic
33C4×Dic314C2211yescharacteristic
34C4×D628C2211yescharacteristic
35C4×D628C2211yescharacteristic
36#19097733—21no—
37#19097731—21no—
38#19097729C2211yescharacteristic
39C4⋊C4×D157C2211yescharacteristic
40#19097704C2211yescharacteristic
41#19097703C2211yescharacteristic
42#19097702C2211yescharacteristic
43C4⋊#11929434C2211yescharacteristic
44#19097696C2211yescharacteristic
45#19097695C2211yescharacteristic
46#19097694C2211yescharacteristic
47#19097694C2211yescharacteristic
48#19097693C2211yescharacteristic
49#19097692C2211yescharacteristic
50#19097683C2211yescharacteristic
51#19097683C2211yescharacteristic
52#125×C157C2211yescharacteristic
53#124×C157C2211yescharacteristic
54#103×C157C2211yescharacteristic
55#103×C157C2211yescharacteristic
56#102×C157C2211yescharacteristic
57#102×C157C2211yescharacteristic
58C4×C8×C157C2211yescharacteristic
59#14578953—22no—
60#14578953—41no—
61C2×Dic314—21no—
62C2×Dic314C2311yescharacteristic
63Q8×D157—22no—
64Q8×D157—41no—
65C2×D628—21no—
66C2×D628—21no—
67C2×D628D411yescharacteristic
68C2×D628C2311yescharacteristic
69C2×C22⋊D157—21no—
70C2×C22⋊D157—21no—
71C22×Dic157—21no—
72C2×C4×D157—21no—
73C2×C4×D157—21no—
74C2×C4×D157—21no—
75C2×C4×D157C2311yescharacteristic
76C2×C4×D157C2311yescharacteristic
77#14578943—22no—
78#14578941—41no—
79#14578939—41no—
80C2×#11929434C2311yescharacteristic
81C2×#11929434C2311yescharacteristic
82C8×D157—41no—
83C8×D157—41no—
84C4⋊Dic157D411yescharacteristic
85C4⋊Dic157C2311yescharacteristic
86C4⋊Dic157C2311yescharacteristic
87#14578933—21no—
88#14578933—21no—
89#14578933—21no—
90C22⋊Dic157—21no—
91#14578931—21no—
92#14578931—21no—
93#14578931—21no—
94#14578931—21no—
95#14578931—21no—
96#14578931D411yescharacteristic
97#14578931D411yescharacteristic
98C4×Dic157—21no—
99C4×Dic157—21no—
100C2×Q8×C157C2311yescharacteristic
101C2×D4×C157C2311yescharacteristic
102C2×C8×C157C2311yescharacteristic
103C2×C8×C157C2311yescharacteristic
104C4⋊C4×C157—21no—
105C4⋊C4×C157—21no—
106C4⋊C4×C157C2311yescharacteristic
107C4⋊C4×C157C2311yescharacteristic
108C22⋊C4×C157—21no—
109C22⋊C4×C157—21no—
110C42×C157C2311yescharacteristic
111C22×D157—21no—
112C22×D157C2×D411yescharacteristic
113C22×D157C2×D411yescharacteristic
114Dic314—21no—
115Dic314—42no—
116Dic314#4112yes—
117D628—22no—
118D628—21no—
119D628—21no—
120D628—42no—
121D628—81no—
122C22⋊D157—42no—
123C22⋊D157—42no—
124C2×Dic157—21no—
125C2×Dic157—21no—
126C2×Dic157—21no—
127C2×Dic157—42no—
128C2×Dic157C2×D411yescharacteristic
129C2×Dic157C2×D411yescharacteristic
130C4×D157—21no—
131C4×D157—21no—
132C4×D157—21no—
133C4×D157—21no—
134C4×D157—42no—
135C4×D157—81no—
136C4×D157—81no—
137#11929434—21no—
138#11929434—21no—
139C23×C157—21no—
140Q8×C157—21no—
141Q8×C157—21no—
142D4×C157—21no—
143D4×C157—21no—
144C2×C4×C157—21no—
145C2×C4×C157—21no—
146C2×C4×C157—21no—
147C2×C4×C157C2×D411yescharacteristic
148C2×C4×C157C2×D411yescharacteristic
149C2×C4×C157C2411yescharacteristic
150C8×C157—21no—
151C8×C157—21no—
152C22×C157—41no—
153C22×C157—41no—
154C22×C157—41no—
155C22×C157C22×D411yescharacteristic
156C2×D157—21no—
157C2×D157—21no—
158C2×D157—21no—
159C2×D157—21no—
160C2×D157—21no—
161C2×D157—41no—
162C2×D157—41no—
163C2×D157—41no—
164C2×D157#13512yes—
165C4×C157—21no—
166C4×C157—21no—
167C4×C157—41no—
168C4×C157—41no—
169C4×C157—41no—
170C4×C157C2×#4111yescharacteristic
171C4×C157H(5,𝔽2)11yescharacteristic
172Dic157—21no—
173Dic157—21no—
174Dic157—22no—
175Dic157—41no—
176Dic157—41no—
177C2×C157—81no—
178C2×C157#54611yescharacteristic
179C2×C157C2×#13511yescharacteristic
180C2×C157#57711yescharacteristic
181D157—22no—
182D157—41no—
183D157—81no—
184C157#330111yescharacteristic, complemented, Sylow
185#3301—1571nomaximal, Sylow
186#553—1571no—
187#548—1571no—
188#488D15711yescharacteristic, complemented
189#485—1571no—
190#481—1571no—
191#479—1571no—
192#466—1571no—
193#465—1571no—
194#463—1571no—
195#449—1571no—
196#448—1571no—
197C4×SD16—1571no—
198C8×D4—1571no—
199#416—1571no—
200C2×#103—1571no—
201C2×#41—1571no—
202C22×Q8—1571no—
203C2×SD16—1571no—
204C2×SD16—1571no—
205C2×D8—1571no—
206C22×C8—1571no—
207C22×C8—1571no—
208#126—3141no—
209#125C2×D15711yescharacteristic
210#124—3141no—
211#124C2×D15711yescharacteristic
212#123—3141no—
213C22⋊Q8—3141no—
214C22⋊Q8—1571no—
215#121—3141no—
216#121—3141no—
217#121—1571no—
218#121—1571no—
219#121—1571no—
220C4×Q8—1571no—
221C4×D4—3141no—
222C4×D4—3141no—
223C4×D4—1571no—
224C4×D4—1571no—
225#117—1571no—
226C2×C4⋊C4—1571no—
227#107—1571no—
228#106—1571no—
229#106—1571no—
230C4⋊C8—1571no—
231#103—1571no—
232#103—1571no—
233#103—1571no—
234#103—1571no—
235#103C2×D15711yescharacteristic
236#103C2×D15711yescharacteristic
237#102—1571no—
238#102—1571no—
239#102C2×D15711yescharacteristic
240#102C2×D15711yescharacteristic
241C22⋊C8—1571no—
242C22⋊C8—1571no—
243C4×C8C2×D15711yescharacteristic
244#41—3142no—
245#41—6281no—
246C2×Q8—3141no—
247C2×Q8—3142no—
248C2×Q8—6281no—
249C2×Q8—1571no—
250C2×Q8C22×D15711yescharacteristic
251C2×D4—3141no—
252C2×D4—3141no—
253C2×D4—3141no—
254C2×D4—3141no—
255C2×D4—1571no—
256C2×D4—1571no—
257C2×D4C22×D15711yescharacteristic
258C22×C4—3141no—
259C22×C4—3141no—
260C22×C4—3141no—
261C22×C4—3141no—
262C22×C4—1571no—
263C22×C4—1571no—
264SD16—3142no—
265SD16—6281no—
266D8—6281no—
267C2×C8—6281no—
268C2×C8—6281no—
269C2×C8—1571no—
270C2×C8—1571no—
271C2×C8C22×D15711yescharacteristic
272C2×C8C22×D15711yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—3141no—
276C4⋊C4—3141no—
277C4⋊C4—3141no—
278C4⋊C4—1571no—
279C4⋊C4—1571no—
280C4⋊C4—1571no—
281C4⋊C4C22×D15711yescharacteristic
282C4⋊C4C22×D15711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—3141no—
286C22⋊C4—3141no—
287C22⋊C4—3141no—
288C22⋊C4—3141no—
289C22⋊C4—3141no—
290C22⋊C4—3141no—
291C22⋊C4—1571no—
292C22⋊C4—1571no—
293C42—3141no—
294C42—3141no—
295C42C22×D15711yescharacteristic
296C23—21no—
297C23—3141no—
298C23—1571no—
299C23—1571no—
300Q8—21no—
301Q8—21no—
302Q8—3141no—
303Q8—6282no—
304Q8—1572no—
305D4—21no—
306D4—21no—
307D4—3142no—
308D4—3141no—
309D4—3141no—
310D4—6282no—
311D4—6282no—
312D4—6282no—
313D4—12561no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—3141no—
318C2×C4—3141no—
319C2×C4—3141no—
320C2×C4—3141no—
321C2×C4—3141no—
322C2×C4—3141no—
323C2×C4—3141no—
324C2×C4—6282no—
325C2×C4—6282no—
326C2×C4—1571no—
327C2×C4—1571no—
328C2×C4—12561no—
329C2×C4—12561no—
330C2×C4D4×D15711yescharacteristic
331C2×C4D4×D15711yescharacteristic
332C2×C4C23×D15711yescharacteristic
333C8—21no—
334C8—21no—
335C8—3141no—
336C8—3141no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—3141no—
341C22—3141no—
342C22—3141no—
343C22—3141no—
344C22—3141no—
345C22—6281no—
346C22—6281no—
347C22—6281no—
348C22—1572no—
349C22C2×D4×D15711yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—3141no—
356C4—3141no—
357C4—3142no—
358C4—6281no—
359C4—6281no—
360C4#41×D15711yescharacteristic
361C4#1909781711yescharacteristic
362C2—81no—
363C2—3142no—
364C2—6281no—
365C2—12561no—
366C2#2375791611yescharacteristic
367C2#135×D15711yescharacteristic
368C2#2375808011yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#488×C157C2
socleC22×C157C22×D4
hypercenter#488D157
hypocenterC157#3301
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #23757976C2semidirect
2. #23757931C2semidirect
3. #23757612C2semidirect
4. #23757590C2semidirect
5. #23757577C2semidirect
6. #23757508C2semidirect
7. #23757505C2semidirect
8. #23757485C2semidirect
9. #23757419C2semidirect
10. #23757413C2semidirect
11. C4×#14578943C2semidirect
12. C8×D628C2semidirect
13. #23757266C2semidirect
14. #103×D157C2semidirect
15. #488×C157C2semidirect
184. C157#3301semidirect
188. #488D157semidirect

Maximal subgroups

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

subgroupindex
#237579762
#237579312
#237576122
#237575902
#237575772
#237575082
#237575052
#237574852
#237574192
#237574132
C4×#145789432
C8×D6282
#237572662
#103×D1572
#488×C1572
#3301157