Finite Group Databasebeta

← #34112028

Subgroups — #34112028

Order 20096.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757964C211yesmaximal, characteristic, complemented
2#23757935C211yesmaximal, characteristic
3#23757628C211yesmaximal, characteristic, complemented
4#23757616C211yesmaximal, characteristic, complemented
5#23757588C211yesmaximal, characteristic, complemented
6#23757580C211yesmaximal, characteristic, complemented
7#23757513C211yesmaximal, characteristic, complemented
8#23757505C211yesmaximal, characteristic, complemented
9#23757495C211yesmaximal, characteristic, complemented
10#23757434C211yesmaximal, characteristic, complemented
11#23757427C211yesmaximal, characteristic, complemented
12C8×D628C211yesmaximal, characteristic, complemented
13#23757268C211yesmaximal, characteristic, complemented
14#103×D157C211yesmaximal, characteristic, complemented
15C4×Q16×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×Q8×D157C2211yescharacteristic
18C2×#14578944C2211yescharacteristic
19C2×#14578941C2211yescharacteristic
20C2×C8×D157C2211yescharacteristic
21C2×C8×D157C2211yescharacteristic
22#19097765—21no—
23#19097763C2211yescharacteristic
24#19097761—21no—
25#19097759—21no—
26#19097759C2211yescharacteristic
27#19097756—21no—
28#19097751—21no—
29#19097751C2211yescharacteristic
30#19097747C2211yescharacteristic
31#19097747C2211yescharacteristic
32#19097746—21no—
33#19097746C2211yescharacteristic
34C4×D628—21no—
35C4×D628C2211yescharacteristic
36#19097729C2211yescharacteristic
37#19097728—21no—
38C4⋊C4×D157C2211yescharacteristic
39#19097704C2211yescharacteristic
40#19097702C2211yescharacteristic
41C4⋊#11929434C2211yescharacteristic
42#19097696C2211yescharacteristic
43#19097696C2211yescharacteristic
44#19097695C2211yescharacteristic
45#19097695C2211yescharacteristic
46#19097694C2211yescharacteristic
47#19097693C2211yescharacteristic
48#19097693C2211yescharacteristic
49#19097691C2211yescharacteristic
50#19097683C2211yescharacteristic
51#19097683C2211yescharacteristic
52C2×Q16×C157C2211yescharacteristic
53C4×Q8×C157C2211yescharacteristic
54C4×Q8×C157C2211yescharacteristic
55#107×C157C2211yescharacteristic
56#103×C157C2211yescharacteristic
57#103×C157C2211yescharacteristic
58C4×C8×C157C2211yescharacteristic
59#14578955—22no—
60#14578955—41no—
61C2×Dic314—21no—
62C2×Dic314C2311yescharacteristic
63Q8×D157—22no—
64Q8×D157—41no—
65C2×D628—21no—
66C2×D628—21no—
67C2×D628D411yescharacteristic
68C2×D628C2311yescharacteristic
69C2×C4×D157—21no—
70C2×C4×D157—21no—
71C2×C4×D157—21no—
72C2×C4×D157C2311yescharacteristic
73C2×C4×D157C2311yescharacteristic
74#14578944—41no—
75#14578941—41no—
76C2×#11929434C2311yescharacteristic
77C2×#11929434C2311yescharacteristic
78C8×D157—41no—
79C8×D157—41no—
80C4⋊Dic157—21no—
81C4⋊Dic157D411yescharacteristic
82C4⋊Dic157C2311yescharacteristic
83C4⋊Dic157C2311yescharacteristic
84#14578933—21no—
85#14578933—21no—
86#14578933—21no—
87#14578933—21no—
88#14578933—21no—
89#14578931—21no—
90#14578931—21no—
91#14578931—21no—
92#14578931—21no—
93#14578931—21no—
94#14578931—21no—
95#14578931—21no—
96#14578931D411yescharacteristic
97#14578931D411yescharacteristic
98C4×Dic157—21no—
99C2×Q8×C157C2311yescharacteristic
100C2×Q8×C157C2311yescharacteristic
101Q16×C157—22no—
102C2×C8×C157C2311yescharacteristic
103C2×C8×C157C2311yescharacteristic
104C4⋊C4×C157—21no—
105C4⋊C4×C157—21no—
106C4⋊C4×C157C2311yescharacteristic
107C4⋊C4×C157C2311yescharacteristic
108C42×C157—21no—
109C42×C157—21no—
110C42×C157C2311yescharacteristic
111C22×D157—21no—
112C22×D157C2×D411yescharacteristic
113C22×D157C2×D411yescharacteristic
114Dic314—21no—
115Dic314—21no—
116Dic314—42no—
117D628—22no—
118D628—21no—
119D628—21no—
120D628—42no—
121D628—42no—
122C2×Dic157—21no—
123C2×Dic157—21no—
124C2×Dic157—21no—
125C2×Dic157C2×D411yescharacteristic
126C2×Dic157C2×D411yescharacteristic
127C4×D157—21no—
128C4×D157—21no—
129C4×D157—21no—
130C4×D157—21no—
131C4×D157—42no—
132C4×D157—42no—
133C4×D157—42no—
134#11929434—21no—
135#11929434—21no—
136Q8×C157—21no—
137Q8×C157—21no—
138Q8×C157—21no—
139Q8×C157#4112yes—
140C2×C4×C157—21no—
141C2×C4×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157C2×D411yescharacteristic
145C2×C4×C157C2×D411yescharacteristic
146C2×C4×C157C2411yescharacteristic
147C8×C157—21no—
148C8×C157—21no—
149C22×C157C22×D411yescharacteristic
150C2×D157—21no—
151C2×D157—21no—
152C2×D157—21no—
153C2×D157—21no—
154C2×D157—21no—
155C2×D157—41no—
156C2×D157—41no—
157C2×D157—41no—
158C2×D157#13512yes—
159C4×C157—22no—
160C4×C157—21no—
161C4×C157—21no—
162C4×C157—41no—
163C4×C157—41no—
164C4×C157—41no—
165C4×C157C2×#4111yescharacteristic
166C4×C157H(5,𝔽2)11yescharacteristic
167Dic157—21no—
168Dic157—21no—
169Dic157—41no—
170Dic157—41no—
171Dic157—41no—
172C2×C157#54611yescharacteristic
173C2×C157C2×#13511yescharacteristic
174C2×C157#57811yescharacteristic
175D157—22no—
176D157—41no—
177D157—81no—
178C157#330011yescharacteristic, complemented, Sylow
179#3300—1571nomaximal, Sylow
180#552—1571no—
181#548—1571no—
182#488—1571no—
183#485—1571no—
184#480—1571no—
185#479—1571no—
186#467—1571no—
187#465—1571no—
188#464—1571no—
189#452—1571no—
190#451—1571no—
191C4×Q16D15711yescharacteristic, complemented
192C8×D4—1571no—
193#416—1571no—
194C2×#103—1571no—
195C2×#41—1571no—
196C22×Q8—1571no—
197C2×Q16—1571no—
198C2×Q16C2×D15711yescharacteristic
199C2×SD16—1571no—
200C22×C8—1571no—
201C22×C8—1571no—
202#126—3141no—
203#125—3141no—
204#125—1571no—
205#124—3141no—
206#124—1571no—
207#123—3141no—
208C22⋊Q8—3141no—
209C22⋊Q8—1571no—
210C22⋊Q8—1571no—
211C22⋊Q8—1571no—
212#121—3141no—
213#121—1571no—
214C4×Q8C2×D15711yescharacteristic
215C4×Q8C2×D15711yescharacteristic
216C4×D4—3141no—
217C4×D4—1571no—
218#117—3141no—
219#117—1571no—
220C2×C4⋊C4—1571no—
221#107—1571no—
222#107C2×D15711yescharacteristic
223#106—1571no—
224C4⋊C8—1571no—
225#103—1571no—
226#103—1571no—
227#103—1571no—
228#103—1571no—
229#103—1571no—
230#103C2×D15711yescharacteristic
231#103C2×D15711yescharacteristic
232#102—1571no—
233#102—1571no—
234#102—1571no—
235C22⋊C8—1571no—
236C22⋊C8—1571no—
237C4×C8C2×D15711yescharacteristic
238#41—3142no—
239#41—6281no—
240C2×Q8—3141no—
241C2×Q8—3142no—
242C2×Q8—6281no—
243C2×Q8—1571no—
244C2×Q8C22×D15711yescharacteristic
245C2×Q8C22×D15711yescharacteristic
246C2×D4—3141no—
247C2×D4—3141no—
248C2×D4—1571no—
249C2×D4—1571no—
250C22×C4—3141no—
251C22×C4—3141no—
252C22×C4—3141no—
253C22×C4—1571no—
254C22×C4—1571no—
255Q16—22no—
256Q16—6281no—
257SD16—6281no—
258C2×C8—6281no—
259C2×C8—6281no—
260C2×C8—1571no—
261C2×C8—1571no—
262C2×C8C22×D15711yescharacteristic
263C2×C8C22×D15711yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—3141no—
267C4⋊C4—3141no—
268C4⋊C4—3141no—
269C4⋊C4—3141no—
270C4⋊C4—3141no—
271C4⋊C4—3141no—
272C4⋊C4—1571no—
273C4⋊C4—1571no—
274C4⋊C4—1571no—
275C4⋊C4C22×D15711yescharacteristic
276C4⋊C4C22×D15711yescharacteristic
277C22⋊C4—3141no—
278C22⋊C4—3141no—
279C22⋊C4—3141no—
280C22⋊C4—3141no—
281C22⋊C4—3141no—
282C22⋊C4—3141no—
283C22⋊C4—3141no—
284C22⋊C4—1571no—
285C22⋊C4—1571no—
286C42—21no—
287C42—21no—
288C42—3141no—
289C42C22×D15711yescharacteristic
290C23—3141no—
291C23—1571no—
292C23—1571no—
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—3141no—
297Q8—3141no—
298Q8—6282no—
299Q8#1457895412yes—
300D4—3142no—
301D4—3141no—
302D4—3141no—
303D4—6282no—
304D4—6282no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—3141no—
310C2×C4—3141no—
311C2×C4—3141no—
312C2×C4—3141no—
313C2×C4—3141no—
314C2×C4—3141no—
315C2×C4—3141no—
316C2×C4—6282no—
317C2×C4—6282no—
318C2×C4—6282no—
319C2×C4—1571no—
320C2×C4—1571no—
321C2×C4D4×D15711yescharacteristic
322C2×C4D4×D15711yescharacteristic
323C2×C4C23×D15711yescharacteristic
324C8—21no—
325C8—21no—
326C8—3141no—
327C8—3141no—
328C22—3141no—
329C22—3141no—
330C22—3141no—
331C22—3141no—
332C22—3141no—
333C22—6281no—
334C22—6281no—
335C22—6281no—
336C22—1572no—
337C22C2×D4×D15711yescharacteristic
338C4—21no—
339C4—22no—
340C4—21no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—3141no—
345C4—3141no—
346C4—6281no—
347C4—6281no—
348C4—6281no—
349C4C2×#1457895411yescharacteristic
350C4#1909781711yescharacteristic
351C2—3142no—
352C2—6281no—
353C2—12561no—
354C2#2375791911yescharacteristic
355C2#135×D15711yescharacteristic
356C2#2375808611yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC4×Q16×C157C2
socleC22×C157C22×D4
prime radical#23757935C2
hypercenterC4×Q16D157
hypocenterC157#3300
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
1. #23757964C2semidirect
3. #23757628C2semidirect
4. #23757616C2semidirect
5. #23757588C2semidirect
6. #23757580C2semidirect
7. #23757513C2semidirect
8. #23757505C2semidirect
9. #23757495C2semidirect
10. #23757434C2semidirect
11. #23757427C2semidirect
12. C8×D628C2semidirect
13. #23757268C2semidirect
14. #103×D157C2semidirect
15. C4×Q16×C157C2semidirect
178. C157#3300semidirect
191. C4×Q16D157semidirect

Maximal subgroups

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

subgroupindex
#237579642
#237579352
#237576282
#237576162
#237575882
#237575802
#237575132
#237575052
#237574952
#237574342
#237574272
C8×D6282
#237572682
#103×D1572
C4×Q16×C1572
#3300157