Finite Group Databasebeta

← #34111996

Subgroups — #34111996

Order 20096.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757936C211yesmaximal, characteristic, complemented
2#23757923C211yesmaximal, characteristic
3#23757622C211yesmaximal, characteristic, complemented
4#23757605C211yesmaximal, characteristic, complemented
5#23757601C211yesmaximal, characteristic, complemented
6#23757557C211yesmaximal, characteristic, complemented
7#23757501C211yesmaximal, characteristic, complemented
8#23757476C211yesmaximal, characteristic, complemented
9#23757411C211yesmaximal, characteristic, complemented
10#23757410C211yesmaximal, characteristic, complemented
11#23757364C211yesmaximal, characteristic, complemented
12C8×C22⋊D157C211yesmaximal, characteristic, complemented
13C2×#19097696C211yesmaximal, characteristic, complemented
14#103×D157C211yesmaximal, characteristic, complemented
15#465×C157C211yesmaximal, characteristic, complemented
16C2×#14578953—21no—
17C22×Dic314C2211yescharacteristic
18C2×Q8×D157C2211yescharacteristic
19C2×#14578942C2211yescharacteristic
20C2×#14578942C2211yescharacteristic
21C2×C8×D157C2211yescharacteristic
22#19097770—21no—
23#19097770C2211yescharacteristic
24#19097769C2211yescharacteristic
25#19097753—21no—
26#19097752—21no—
27#19097751C2211yescharacteristic
28#19097749—21no—
29#19097748—21no—
30#19097748—21no—
31#19097742C2211yescharacteristic
32#19097736C2211yescharacteristic
33D4×Dic157C2211yescharacteristic
34C4×C22⋊D157C2211yescharacteristic
35#19097731—21no—
36C2×C4⋊Dic157C2211yescharacteristic
37C4⋊C4×D157C2211yescharacteristic
38#19097702C2211yescharacteristic
39#19097702C2211yescharacteristic
40#19097700C2211yescharacteristic
41#19097696C2211yescharacteristic
42#19097696C2211yescharacteristic
43#19097696C2211yescharacteristic
44#19097696C2211yescharacteristic
45#19097695C2211yescharacteristic
46#19097695C2211yescharacteristic
47#19097694C2211yescharacteristic
48#19097694C2211yescharacteristic
49C22⋊#11929434C2211yescharacteristic
50#19097683C2211yescharacteristic
51C8×Dic157C2211yescharacteristic
52C2×SD16×C157C2211yescharacteristic
53C22×C8×C157C2211yescharacteristic
54C22⋊Q8×C157C2211yescharacteristic
55#121×C157C2211yescharacteristic
56#106×C157C2211yescharacteristic
57#103×C157C2211yescharacteristic
58#102×C157C2211yescharacteristic
59#14578953—44no—
60C2×Dic314—21no—
61C2×Dic314—21no—
62C2×Dic314—22no—
63C2×Dic314—21no—
64C2×Dic314—41no—
65C2×Dic314C2311yescharacteristic
66C2×Dic314C2311yescharacteristic
67Q8×D157—22no—
68Q8×D157—41no—
69C2×C22⋊D157—21no—
70C2×C22⋊D157D411yescharacteristic
71C22×Dic157—21no—
72C22×Dic157—21no—
73C22×Dic157—21no—
74C2×C4×D157—21no—
75C2×C4×D157—21no—
76C2×C4×D157C2311yescharacteristic
77#14578942—22no—
78#14578942—41no—
79C2×#11929434C2311yescharacteristic
80C2×#11929434C2311yescharacteristic
81C8×D157—41no—
82C4⋊Dic157—21no—
83C4⋊Dic157—21no—
84C4⋊Dic157C2311yescharacteristic
85C4⋊Dic157C2311yescharacteristic
86#14578933—21no—
87#14578933—21no—
88#14578933—21no—
89#14578933—21no—
90#14578933—21no—
91#14578933D411yescharacteristic
92C22⋊Dic157—21no—
93C22⋊Dic157—21no—
94C22⋊Dic157D411yescharacteristic
95#14578931—21no—
96#14578931D411yescharacteristic
97C4×Dic157—21no—
98C4×Dic157C2311yescharacteristic
99C2×Q8×C157C2311yescharacteristic
100C2×D4×C157—21no—
101C2×D4×C157C2311yescharacteristic
102C22×C4×C157C2311yescharacteristic
103SD16×C157—41no—
104C2×C8×C157—41no—
105C2×C8×C157C2311yescharacteristic
106C2×C8×C157C2311yescharacteristic
107C4⋊C4×C157—21no—
108C4⋊C4×C157C2311yescharacteristic
109C4⋊C4×C157C2311yescharacteristic
110C22⋊C4×C157—21no—
111C22⋊C4×C157—21no—
112C22×D157C2×D411yescharacteristic
113Dic314—21no—
114Dic314—21no—
115Dic314—21no—
116Dic314—42no—
117Dic314—42no—
118Dic314—42no—
119Dic314#4112yescomplemented
120C22⋊D157—42no—
121C22⋊D157SD1614yescomplemented
122C2×Dic157—21no—
123C2×Dic157—21no—
124C2×Dic157—21no—
125C2×Dic157—21no—
126C2×Dic157—42no—
127C2×Dic157—42no—
128C2×Dic157—42no—
129C2×Dic157C2×D411yescharacteristic
130C2×Dic157C2×D411yescharacteristic
131C2×Dic157C2×D411yescharacteristic
132C4×D157—21no—
133C4×D157—21no—
134C4×D157—42no—
135C4×D157—42no—
136#11929434—21no—
137#11929434—21no—
138C23×C157—21no—
139C23×C157C2×D411yescharacteristic
140Q8×C157—21no—
141Q8×C157—21no—
142D4×C157—21no—
143D4×C157—21no—
144D4×C157—42no—
145C2×C4×C157—21no—
146C2×C4×C157—21no—
147C2×C4×C157—21no—
148C2×C4×C157—21no—
149C2×C4×C157—21no—
150C2×C4×C157C2×D411yescharacteristic
151C2×C4×C157C2411yescharacteristic
152C8×C157—21no—
153C8×C157—21no—
154C22×C157—21no—
155C22×C157—21no—
156C22×C157—41no—
157C22×C157—41no—
158C22×C157—41no—
159C22×C157C2×SD1612yes—
160C22×C157C22×D411yescharacteristic
161C2×D157—21no—
162C2×D157—21no—
163C2×D157C2×SD1612yes—
164C4×C157—21no—
165C4×C157—41no—
166C4×C157—41no—
167C4×C157—41no—
168C4×C157C2×#4111yescharacteristic
169C4×C157H(5,𝔽2)11yescharacteristic
170Dic157—21no—
171Dic157—22no—
172Dic157—21no—
173Dic157—41no—
174Dic157—41no—
175Dic157—41no—
176Dic157C2×SD1612yes—
177C2×C157—22no—
178C2×C157—81no—
179C2×C157#54611yescharacteristic
180C2×C157C22×SD1611yescharacteristic
181C2×C157#57811yescharacteristic
182D157—22no—
183C157#329811yescharacteristic, complemented, Sylow
184#3298—1571nomaximal, Sylow
185D4×Q8—1571no—
186#547—1571no—
187#487—1571no—
188#483—1571no—
189#483—1571no—
190#475—1571no—
191#465—1571no—
192#465D15711yescharacteristic, complemented
193#461—1571no—
194#448—1571no—
195#448—1571no—
196C4×SD16—1571no—
197C8×D4—1571no—
198C2×#103—1571no—
199C2×#103—1571no—
200C2×#41—3141no—
201C22×Q8—1571no—
202C22×Q8—1571no—
203C2×SD16—1571no—
204C2×SD16—1571no—
205C2×SD16C2×D15711yescharacteristic
206C22×C8—1571no—
207C22×C8C2×D15711yescharacteristic
208C4⋊Q8—3141no—
209C4⋊Q8—1571no—
210C4⋊Q8—1571no—
211#123—3141no—
212#123—3141no—
213C22⋊Q8—3141no—
214C22⋊Q8—3141no—
215C22⋊Q8—3141no—
216C22⋊Q8—1571no—
217C22⋊Q8C2×D15711yescharacteristic
218#121—1571no—
219#121C2×D15711yescharacteristic
220C4×Q8—1571no—
221C4×D4—3141no—
222C4×D4—1571no—
223C4×D4—1571no—
224C2×C4⋊C4—1571no—
225C2×C4⋊C4—1571no—
226#106—1571no—
227#106—1571no—
228#106C2×D15711yescharacteristic
229C4⋊C8—1571no—
230#103—1571no—
231#103—1571no—
232#103—1571no—
233#103—1571no—
234#103—1571no—
235#103—1571no—
236#103—1571no—
237#103—1571no—
238#103C2×D15711yescharacteristic
239#102C2×D15711yescharacteristic
240C22⋊C8—1571no—
241C22⋊C8—1571no—
242C4×C8—1571no—
243#41—6284no—
244C2×Q8—3141no—
245C2×Q8—3141no—
246C2×Q8—3142no—
247C2×Q8—3141no—
248C2×Q8—3142no—
249C2×Q8—6281no—
250C2×Q8—6281no—
251C2×Q8—1571no—
252C2×Q8—1571no—
253C2×Q8C22×D15711yescharacteristic
254C2×D4—21no—
255C2×D4—3141no—
256C2×D4—1571no—
257C2×D4C22×D15711yescharacteristic
258C22×C4—3141no—
259C22×C4—3141no—
260C22×C4—3141no—
261C22×C4—3141no—
262C22×C4—3141no—
263C22×C4—1571no—
264C22×C4C22×D15711yescharacteristic
265SD16—41no—
266SD16—3142no—
267SD16—6281no—
268C2×C8—41no—
269C2×C8—6281no—
270C2×C8—1571no—
271C2×C8—1571no—
272C2×C8C22×D15711yescharacteristic
273C2×C8C22×D15711yescharacteristic
274C4⋊C4—21no—
275C4⋊C4—3141no—
276C4⋊C4—3141no—
277C4⋊C4—3141no—
278C4⋊C4—3141no—
279C4⋊C4—3141no—
280C4⋊C4—3141no—
281C4⋊C4—3141no—
282C4⋊C4—1571no—
283C4⋊C4—1571no—
284C4⋊C4—1571no—
285C4⋊C4C22×D15711yescharacteristic
286C4⋊C4C22×D15711yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—3141no—
290C22⋊C4—3141no—
291C22⋊C4—3141no—
292C22⋊C4—1571no—
293C22⋊C4—1571no—
294C42—3141no—
295C42—1571no—
296C23—21no—
297C23—1571no—
298C23D4×D15711yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—3141no—
302Q8—3141no—
303Q8—3141no—
304Q8—6282no—
305Q8—6282no—
306Q8—6282no—
307Q8—1572no—
308D4—21no—
309D4—21no—
310D4—42no—
311D4—6282no—
312D4—1574no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
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—6282no—
327C2×C4—6282no—
328C2×C4—6282no—
329C2×C4—1571no—
330C2×C4—1571no—
331C2×C4—1571no—
332C2×C4D4×D15711yescharacteristic
333C2×C4C23×D15711yescharacteristic
334C8—21no—
335C8—21no—
336C8—3141no—
337C8—3141no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—3141no—
344C22—3141no—
345C22—1572no—
346C22SD16×D15712yes—
347C22C2×D4×D15711yescharacteristic
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—3141no—
353C4—3141no—
354C4—3142no—
355C4—6281no—
356C4—6281no—
357C4—6281no—
358C4—1572no—
359C4#41×D15711yescharacteristic
360C4#1909781611yescharacteristic
361C2—22no—
362C2—81no—
363C2—3142no—
364C2#2375791211yescharacteristic
365C2C2×SD16×D15711yescharacteristic
366C2#2375808711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#465×C157C2
socleC22×C157C22×D4
prime radical#23757923C2
hypercenter#465D157
hypocenterC157#3298
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #23757936C2semidirect
3. #23757622C2semidirect
4. #23757605C2semidirect
5. #23757601C2semidirect
6. #23757557C2semidirect
7. #23757501C2semidirect
8. #23757476C2semidirect
9. #23757411C2semidirect
10. #23757410C2semidirect
11. #23757364C2semidirect
12. C8×C22⋊D157C2semidirect
13. C2×#19097696C2semidirect
14. #103×D157C2semidirect
15. #465×C157C2semidirect
119. Dic314#41semidirect
121. C22⋊D157SD16semidirect
183. C157#3298semidirect
192. #465D157semidirect

Maximal subgroups

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

subgroupindex
#237579362
#237579232
#237576222
#237576052
#237576012
#237575572
#237575012
#237574762
#237574112
#237574102
#237573642
C8×C22⋊D1572
C2×#190976962
#103×D1572
#465×C1572
#3298157