Finite Group Databasebeta

← #34112060

Subgroups — #34112060

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#23757986C211yesmaximal, characteristic, complemented
2#23757922C211yesmaximal, characteristic, complemented
3#23757622C211yesmaximal, characteristic, complemented
4#23757614C211yesmaximal, characteristic, complemented
5#23757612C211yesmaximal, characteristic, complemented
6#23757561C211yesmaximal, characteristic, complemented
7#23757505C211yesmaximal, characteristic, complemented
8#23757502C211yesmaximal, characteristic, complemented
9#23757477C211yesmaximal, characteristic, complemented
10#23757419C211yesmaximal, characteristic, complemented
11#23757418C211yesmaximal, characteristic, complemented
12#23757353C211yesmaximal, characteristic, complemented
13C8×C22⋊D157C211yesmaximal, characteristic, complemented
14#23757266C211yesmaximal, characteristic, complemented
15#416×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic
17C2×#14578943C2211yescharacteristic
18C2×#14578941C2211yescharacteristic
19C2×#14578939C2211yescharacteristic
20C2×C8×D157C2211yescharacteristic
21#19097770C2211yescharacteristic
22#19097769C2211yescharacteristic
23#19097759—21no—
24#19097757—21no—
25#19097754—21no—
26#19097752—21no—
27#19097750C2211yescharacteristic
28#19097747C2211yescharacteristic
29#19097746C2211yescharacteristic
30#19097745C2211yescharacteristic
31#19097744—21no—
32#19097743—21no—
33D4×Dic157C2211yescharacteristic
34#19097733C2211yescharacteristic
35C4×C22⋊D157C2211yescharacteristic
36#19097729C2211yescharacteristic
37#19097728—21no—
38C2×#14578933—21no—
39#19097704C2211yescharacteristic
40#19097703C2211yescharacteristic
41#19097702C2211yescharacteristic
42#19097700C2211yescharacteristic
43#19097696C2211yescharacteristic
44#19097695C2211yescharacteristic
45#19097694C2211yescharacteristic
46#19097694C2211yescharacteristic
47#19097693C2211yescharacteristic
48#19097692C2211yescharacteristic
49C22⋊#11929434C2211yescharacteristic
50#19097683C2211yescharacteristic
51C8×Dic157C2211yescharacteristic
52C2×#41×C157C2211yescharacteristic
53C22×C8×C157C2211yescharacteristic
54#117×C157C2211yescharacteristic
55#103×C157C2211yescharacteristic
56#103×C157C2211yescharacteristic
57#102×C157C2211yescharacteristic
58#102×C157C2211yescharacteristic
59#14578953—22no—
60#14578953—41no—
61C2×Dic314—21no—
62C2×Dic314C2311yescharacteristic
63C2×D628—21no—
64C2×D628C2311yescharacteristic
65C2×C22⋊D157—21no—
66C2×C22⋊D157—21no—
67C2×C22⋊D157D411yescharacteristic
68C22×Dic157—21no—
69C22×Dic157—21no—
70C2×C4×D157—21no—
71C2×C4×D157C2311yescharacteristic
72#14578943—41no—
73#14578941—41no—
74#14578939—22no—
75C2×#11929434C2311yescharacteristic
76C2×#11929434C2311yescharacteristic
77C8×D157—41no—
78C4⋊Dic157C2311yescharacteristic
79C4⋊Dic157C2311yescharacteristic
80#14578933—21no—
81#14578933—21no—
82#14578933—21no—
83#14578933—21no—
84#14578933—21no—
85#14578933D411yescharacteristic
86C22⋊Dic157—21no—
87C22⋊Dic157—21no—
88C22⋊Dic157D411yescharacteristic
89#14578931—21no—
90#14578931—21no—
91#14578931—21no—
92#14578931—21no—
93#14578931—21no—
94#14578931D411yescharacteristic
95C4×Dic157—21no—
96C4×Dic157C2311yescharacteristic
97#41×C157—22no—
98#41×C157—41no—
99C2×Q8×C157C2311yescharacteristic
100C2×D4×C157—21no—
101C2×D4×C157C2311yescharacteristic
102C22×C4×C157—21no—
103C22×C4×C157C2311yescharacteristic
104C2×C8×C157—41no—
105C2×C8×C157C2311yescharacteristic
106C2×C8×C157C2311yescharacteristic
107C4⋊C4×C157C2311yescharacteristic
108C4⋊C4×C157C2311yescharacteristic
109C22⋊C4×C157—21no—
110C42×C157—21no—
111C22×D157—21no—
112C22×D157C2×D411yescharacteristic
113Dic314—21no—
114Dic314—21no—
115Dic314—81no—
116D628—21no—
117D628—21no—
118D628—81no—
119C22⋊D157—22no—
120C22⋊D157—42no—
121C22⋊D157—81no—
122C2×Dic157—21no—
123C2×Dic157—21no—
124C2×Dic157—21no—
125C2×Dic157—42no—
126C2×Dic157—42no—
127C2×Dic157C2×D411yescharacteristic
128C2×Dic157C2×D411yescharacteristic
129C2×Dic157C2×D411yescharacteristic
130C4×D157—21no—
131C4×D157—21no—
132C4×D157—42no—
133#11929434—21no—
134#11929434—21no—
135C23×C157—21no—
136C23×C157C2×D411yescharacteristic
137Q8×C157—21no—
138Q8×C157—21no—
139D4×C157—21no—
140D4×C157—42no—
141D4×C157#4112yes—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157—21no—
145C2×C4×C157—21no—
146C2×C4×C157—21no—
147C2×C4×C157—42no—
148C2×C4×C157C2×D411yescharacteristic
149C2×C4×C157C2411yescharacteristic
150C8×C157—21no—
151C8×C157—21no—
152C22×C157—21no—
153C22×C157—22no—
154C22×C157—21no—
155C22×C157—21no—
156C22×C157—41no—
157C22×C157—41no—
158C22×C157C22×D411yescharacteristic
159C2×D157—21no—
160C2×D157—21no—
161C2×D157—21no—
162C2×D157—41no—
163C2×D157—41no—
164C2×D157—41no—
165C4×C157—21no—
166C4×C157—41no—
167C4×C157—41no—
168C4×C157—41no—
169C4×C157C2×#4111yescharacteristic
170C4×C157H(5,𝔽2)11yescharacteristic
171Dic157—21no—
172Dic157—21no—
173Dic157—41no—
174Dic157—41no—
175Dic157—41no—
176Dic157#13512yes—
177C2×C157—41no—
178C2×C157—42no—
179C2×C157#54611yescharacteristic
180C2×C157C2×#13511yescharacteristic
181C2×C157#57711yescharacteristic
182D157—41no—
183D157—81no—
184C157#330311yescharacteristic, complemented, Sylow
185#3303—1571nomaximal, Sylow
186#555—1571no—
187#547—1571no—
188#487—1571no—
189#485—1571no—
190#485—1571no—
191#476—1571no—
192#465—1571no—
193#465—1571no—
194#461—1571no—
195#449—1571no—
196#449—1571no—
197C4×D8—1571no—
198C8×D4—1571no—
199#416—1571no—
200#416D15711yescharacteristic, complemented
201C2×#41—1571no—
202C2×#41C2×D15711yescharacteristic
203C2×SD16—1571no—
204C2×SD16—1571no—
205C2×D8—1571no—
206C22×C8—1571no—
207C22×C8C2×D15711yescharacteristic
208C4⋊Q8—1571no—
209C4⋊Q8—1571no—
210#124—3141no—
211#124—3141no—
212#123—3141no—
213#123—3141no—
214C22⋊Q8—1571no—
215C22⋊Q8—1571no—
216#121—3141no—
217#121—3141no—
218#121—1571no—
219#121—1571no—
220C4×D4—1571no—
221C4×D4—1571no—
222C4×D4—1571no—
223#117—3141no—
224#117—1571no—
225#117C2×D15711yescharacteristic
226C2×C4⋊C4—3141no—
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×C8—1571no—
244#41—22no—
245#41—41no—
246#41—3142no—
247#41—6281no—
248C2×Q8—3141no—
249C2×Q8—1571no—
250C2×Q8C22×D15711yescharacteristic
251C2×D4—21no—
252C2×D4—3141no—
253C2×D4—3141no—
254C2×D4—3141no—
255C2×D4—1571no—
256C2×D4—1571no—
257C2×D4C22×D15711yescharacteristic
258C22×C4—21no—
259C22×C4—3141no—
260C22×C4—3141no—
261C22×C4—3141no—
262C22×C4—1571no—
263C22×C4C22×D15711yescharacteristic
264SD16—6281no—
265SD16—6281no—
266D8—3142no—
267C2×C8—41no—
268C2×C8—6281no—
269C2×C8—1571no—
270C2×C8—1571no—
271C2×C8C22×D15711yescharacteristic
272C2×C8C22×D15711yescharacteristic
273C4⋊C4—3141no—
274C4⋊C4—3141no—
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—3141no—
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—21no—
294C42—3141no—
295C42—1571no—
296C23—21no—
297C23—3141no—
298C23—1571no—
299C23D4×D15711yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—3141no—
303Q8—3141no—
304Q8—12561no—
305D4—21no—
306D4—42no—
307D4—3142no—
308D4—3141no—
309D4—3141no—
310D4—6282no—
311D4—12561no—
312D4—12561no—
313D4#1457895412yes—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—3141no—
321C2×C4—3141no—
322C2×C4—3141no—
323C2×C4—3141no—
324C2×C4—3141no—
325C2×C4—6282no—
326C2×C4—6282no—
327C2×C4—6282no—
328C2×C4—1571no—
329C2×C4—1571no—
330C2×C4—1571no—
331C2×C4D4×D15711yescharacteristic
332C2×C4C23×D15711yescharacteristic
333C8—21no—
334C8—21no—
335C8—3141no—
336C8—3141no—
337C22—21no—
338C22—21no—
339C22—22no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—3141no—
344C22—3141no—
345C22—3141no—
346C22—6281no—
347C22—6281no—
348C22—6281no—
349C22C2×D4×D15711yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—3141no—
355C4—3141no—
356C4—6281no—
357C4—6281no—
358C4—6281no—
359C4—1572no—
360C4C2×#1457895411yescharacteristic
361C4#1909781611yescharacteristic
362C2—41no—
363C2—42no—
364C2—6281no—
365C2—12561no—
366C2#2375791311yescharacteristic
367C2#135×D15711yescharacteristic
368C2#2375808011yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#416×C157C2
socleC22×C157C22×D4
hypercenter#416D157
hypocenterC157#3303
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #23757986C2semidirect
2. #23757922C2semidirect
3. #23757622C2semidirect
4. #23757614C2semidirect
5. #23757612C2semidirect
6. #23757561C2semidirect
7. #23757505C2semidirect
8. #23757502C2semidirect
9. #23757477C2semidirect
10. #23757419C2semidirect
11. #23757418C2semidirect
12. #23757353C2semidirect
13. C8×C22⋊D157C2semidirect
14. #23757266C2semidirect
15. #416×C157C2semidirect
184. C157#3303semidirect
200. #416D157semidirect

Maximal subgroups

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

subgroupindex
#237579862
#237579222
#237576222
#237576142
#237576122
#237575612
#237575052
#237575022
#237574772
#237574192
#237574182
#237573532
C8×C22⋊D1572
#237572662
#416×C1572
#3303157