Finite Group Databasebeta

← #34110398

Subgroups — #34110398

Order 20096.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757894C211yesmaximal, characteristic, complemented
2#23757608C211yesmaximal, characteristic, complemented
3#23757586C211yesmaximal, characteristic, complemented
4#23757568C211yesmaximal, characteristic, complemented
5#23757553C211yesmaximal, characteristic, complemented
6#23757495C211yesmaximal, characteristic, complemented
7#23757488C211yesmaximal, characteristic, complemented
8#23757435C211yesmaximal, characteristic, complemented
9#23757418C211yesmaximal, characteristic
10#23757373C211yesmaximal, characteristic, complemented
11#23757371C211yesmaximal, characteristic, complemented
12C4×#14578944C211yesmaximal, characteristic, complemented
13C4×#14578939C211yesmaximal, characteristic, complemented
14#23757334C211yesmaximal, characteristic, complemented
15#520×C157C211yesmaximal, characteristic, complemented
16C2×#14578944C2211yescharacteristic
17C2×#14578942C2211yescharacteristic
18C2×#14578941C2211yescharacteristic
19C2×#14578939C2211yescharacteristic
20#19097771C2211yescharacteristic
21#19097766—21no—
22#19097759C2211yescharacteristic
23#19097757—21no—
24#19097753—21no—
25#19097750C2211yescharacteristic
26#19097749—21no—
27#19097745C2211yescharacteristic
28C4×Dic314C2211yescharacteristic
29C4×D628C2211yescharacteristic
30#19097704C2211yescharacteristic
31#19097704C2211yescharacteristic
32#19097704C2211yescharacteristic
33#19097704C2211yescharacteristic
34C4⋊#11929434C2211yescharacteristic
35C4⋊#11929434C2211yescharacteristic
36#19097695C2211yescharacteristic
37#19097695C2211yescharacteristic
38#19097694C2211yescharacteristic
39#19097694C2211yescharacteristic
40#19097692C2211yescharacteristic
41#19097692C2211yescharacteristic
42#19097691C2211yescharacteristic
43#19097691C2211yescharacteristic
44C22⋊#11929434C2211yescharacteristic
45C22⋊#11929434C2211yescharacteristic
46#19097682C2211yescharacteristic
47C4×#11929434C2211yescharacteristic
48C2×#41×C157C2211yescharacteristic
49C4×Q8×C157C2211yescharacteristic
50C4×Q8×C157C2211yescharacteristic
51C4×D4×C157—21no—
52C4×D4×C157—21no—
53C4×D4×C157C2211yescharacteristic
54C4×D4×C157C2211yescharacteristic
55#117×C157—21no—
56#117×C157C2211yescharacteristic
57C2×C4⋊C4×C157—21no—
58C2×C4⋊C4×C157C2211yescharacteristic
59C2×Dic314—21no—
60C2×Dic314C2311yescharacteristic
61C2×D628C2311yescharacteristic
62C2×C22⋊D157—21no—
63C2×C4×D157—21no—
64#14578944—22no—
65#14578942—22no—
66#14578941—22no—
67#14578939—22no—
68C2×#11929434C2311yescharacteristic
69C2×#11929434C2311yescharacteristic
70C2×#11929434C2311yescharacteristic
71C2×#11929434C2311yescharacteristic
72C4⋊Dic157—21no—
73C4⋊Dic157C2311yescharacteristic
74C4⋊Dic157C2311yescharacteristic
75#14578933—21no—
76#14578933—21no—
77C22⋊Dic157—21no—
78#14578931—21no—
79#14578931—21no—
80#14578931—21no—
81#14578931—21no—
82C4×Dic157—21no—
83#41×C157—22no—
84#41×C157—22no—
85C2×Q8×C157C2311yescharacteristic
86C2×D4×C157—21no—
87C2×D4×C157C2311yescharacteristic
88C22×C4×C157—21no—
89C22×C4×C157—21no—
90C22×C4×C157—21no—
91C22×C4×C157—21no—
92C22×C4×C157C2311yescharacteristic
93C4⋊C4×C157—21no—
94C4⋊C4×C157—21no—
95C4⋊C4×C157—21no—
96C4⋊C4×C157D411yescharacteristic
97C4⋊C4×C157D411yescharacteristic
98C4⋊C4×C157C2311yescharacteristic
99C4⋊C4×C157C2311yescharacteristic
100C22⋊C4×C157—21no—
101C22⋊C4×C157—21no—
102C22⋊C4×C157D411yescharacteristic
103C22⋊C4×C157D411yescharacteristic
104C42×C157—21no—
105C42×C157—21no—
106C42×C157C2311yescharacteristic
107C42×C157C2311yescharacteristic
108C22×D157—21no—
109Dic314—21no—
110Dic314—21no—
111Dic314—81no—
112D628—21no—
113D628—21no—
114C22⋊D157—81no—
115C2×Dic157—21no—
116C2×Dic157—21no—
117C2×Dic157—21no—
118C4×D157—81no—
119#11929434—21no—
120#11929434—21no—
121#11929434—21no—
122#11929434—21no—
123C23×C157—21no—
124C23×C157C2×D411yescharacteristic
125Q8×C157—21no—
126Q8×C157#4112yes—
127D4×C157—24no—
128D4×C157—21no—
129D4×C157#4112yes—
130C2×C4×C157—24no—
131C2×C4×C157—21no—
132C2×C4×C157—21no—
133C2×C4×C157—21no—
134C2×C4×C157—21no—
135C2×C4×C157—21no—
136C2×C4×C157—81no—
137C2×C4×C157—81no—
138C2×C4×C157—81no—
139C2×C4×C157C2×D411yescharacteristic
140C2×C4×C157C2×D411yescharacteristic
141C2×C4×C157C2×D411yescharacteristic
142C2×C4×C157C2×D411yescharacteristic
143C2×C4×C157C2×D411yescharacteristic
144C2×C4×C157C2411yescharacteristic
145C22×C157—21no—
146C22×C157—22no—
147C22×C157—21no—
148C22×C157—21no—
149C22×C157—41no—
150C22×C157—41no—
151C22×C157C22×D411yescharacteristic
152C2×D157—41no—
153C2×D157—41no—
154C2×D157—41no—
155C4×C157—21no—
156C4×C157—22no—
157C4×C157—21no—
158C4×C157—21no—
159C4×C157—21no—
160C4×C157—21no—
161C4×C157—41no—
162C4×C157—41no—
163C4×C157C2×#4111yescharacteristic
164C4×C157C2×#4111yescharacteristic
165Dic157—41no—
166Dic157—41no—
167Dic157—41no—
168C2×C157—41no—
169C2×C157—42no—
170C2×C157#52511yescharacteristic
171C2×C157#57611yescharacteristic
172C2×C157#57811yescharacteristic
173D157—81no—
174C157#312411yescharacteristic, complemented, Sylow
175#3124—1571nomaximal, Sylow
176#542—1571no—
177#520D15711yescharacteristic, complemented
178#484—1571no—
179#480—1571no—
180#477—1571no—
181#474—1571no—
182#464—1571no—
183#463—1571no—
184#452—1571no—
185#449—1571no—
186#440—1571no—
187#440—1571no—
188C4×Q16—1571no—
189C4×D8—1571no—
190#433—1571no—
191C2×#41C2×D15711yescharacteristic
192C2×Q16—1571no—
193C2×SD16—1571no—
194C2×SD16—1571no—
195C2×D8—1571no—
196C4⋊Q8—1571no—
197#126—3141no—
198#124—3141no—
199#124—1571no—
200#123—3141no—
201C22⋊Q8—3141no—
202C22⋊Q8—1571no—
203#121—1571no—
204C4×Q8—1571no—
205C4×Q8C2×D15711yescharacteristic
206C4×Q8C2×D15711yescharacteristic
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—1571no—
210C4×D4C2×D15711yescharacteristic
211C4×D4C2×D15711yescharacteristic
212#117—21no—
213#117C2×D15711yescharacteristic
214C2×C4⋊C4—21no—
215C2×C4⋊C4C2×D15711yescharacteristic
216#107—1571no—
217#107—1571no—
218#107—1571no—
219#107—1571no—
220C4⋊C8—1571no—
221C4⋊C8—1571no—
222#103—1571no—
223#103—1571no—
224#103—1571no—
225#103—1571no—
226#102—1571no—
227#102—1571no—
228#102—1571no—
229#102—1571no—
230C22⋊C8—1571no—
231C22⋊C8—1571no—
232#97—1571no—
233C4×C8—1571no—
234#41—22no—
235#41—22no—
236C2×Q8—3141no—
237C2×Q8—1571no—
238C2×Q8C22×D15711yescharacteristic
239C2×D4—21no—
240C2×D4—3141no—
241C2×D4—1571no—
242C2×D4C22×D15711yescharacteristic
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—3141no—
248C22×C4C22×D15711yescharacteristic
249Q16—3142no—
250SD16—3142no—
251SD16—3142no—
252D8—3142no—
253C2×C8—1571no—
254C2×C8—1571no—
255C2×C8—1571no—
256C2×C8—1571no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—3141no—
261C4⋊C4—3141no—
262C4⋊C4—3141no—
263C4⋊C4—1571no—
264C4⋊C4—1571no—
265C4⋊C4C22⋊D15711yescharacteristic
266C4⋊C4C22⋊D15711yescharacteristic
267C4⋊C4C22×D15711yescharacteristic
268C4⋊C4C22×D15711yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—3141no—
272C22⋊C4—3141no—
273C22⋊C4—3141no—
274C22⋊C4—3141no—
275C22⋊C4—3141no—
276C22⋊C4C22⋊D15711yescharacteristic
277C22⋊C4C22⋊D15711yescharacteristic
278C42—21no—
279C42—21no—
280C42—3141no—
281C42C22×D15711yescharacteristic
282C42C22×D15711yescharacteristic
283C23—21no—
284C23—3141no—
285C23C2×C22⋊D15711yescharacteristic
286Q8—21no—
287Q8—3141no—
288Q8—3141no—
289Q8—12561no—
290Q8#1457895412yes—
291D4—24no—
292D4—21no—
293D4—3141no—
294D4—3141no—
295D4—12561no—
296D4#1457895412yes—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—3141no—
307C2×C4—3141no—
308C2×C4—3141no—
309C2×C4—12561no—
310C2×C4C2×C22⋊D15711yescharacteristic
311C2×C4C2×C22⋊D15711yescharacteristic
312C2×C4C2×C22⋊D15711yescharacteristic
313C2×C4C2×C22⋊D15711yescharacteristic
314C2×C4C2×C22⋊D15711yescharacteristic
315C2×C4C23×D15711yescharacteristic
316C8—3141no—
317C8—3141no—
318C8—3141no—
319C8—3141no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—6281no—
327C22—6281no—
328C22—6281no—
329C22C22×C22⋊D15711yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—22no—
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—6281no—
339C4—6281no—
340C4—6281no—
341C4C2×#1457895411yescharacteristic
342C4C2×#1457895411yescharacteristic
343C2—41no—
344C2—42no—
345C2—12561no—
346C2#2375778411yescharacteristic
347C2#2375807611yescharacteristic
348C2#2375808811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#520×C157C2
socleC22×C157C22×D4
prime radical#23757418C2
hypercenter#520D157
hypocenterC157#3124
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #23757894C2semidirect
2. #23757608C2semidirect
3. #23757586C2semidirect
4. #23757568C2semidirect
5. #23757553C2semidirect
6. #23757495C2semidirect
7. #23757488C2semidirect
8. #23757435C2semidirect
10. #23757373C2semidirect
11. #23757371C2semidirect
12. C4×#14578944C2semidirect
13. C4×#14578939C2semidirect
14. #23757334C2semidirect
15. #520×C157C2semidirect
174. C157#3124semidirect
177. #520D157semidirect

Maximal subgroups

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

subgroupindex
#237578942
#237576082
#237575862
#237575682
#237575532
#237574952
#237574882
#237574352
#237574182
#237573732
#237573712
C4×#145789442
C4×#145789392
#237573342
#520×C1572
#3124157