Finite Group Databasebeta

← #34112079

Subgroups — #34112079

Order 20096.

408 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757987C211yesmaximal, characteristic, complemented
2#23757942C211yesmaximal, characteristic, complemented
3#23757618C211yesmaximal, characteristic, complemented
4#23757598C211yesmaximal, characteristic, complemented
5#23757596C211yesmaximal, characteristic, complemented
6#23757566C211yesmaximal, characteristic, complemented
7#23757507C211yesmaximal, characteristic, complemented
8#23757506C211yesmaximal, characteristic, complemented
9#23757466C211yesmaximal, characteristic, complemented
10#23757417C211yesmaximal, characteristic, complemented
11SD16×Dic157C211yesmaximal, characteristic, complemented
12#23757336C211yesmaximal, characteristic, complemented
13#23757269C211yesmaximal, characteristic, complemented
14#23757268C211yesmaximal, characteristic, complemented
15#449×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×#14578939C2211yescharacteristic
18C22×#11929434C2211yescharacteristic
19C2×C8×D157C2211yescharacteristic
20#19097770C2211yescharacteristic
21#19097767—21no—
22#19097767C2211yescharacteristic
23#19097758—21no—
24#19097757—21no—
25#19097745—21no—
26#19097745C2211yescharacteristic
27#19097743—21no—
28#19097742—21no—
29#19097742C2211yescharacteristic
30#19097742C2211yescharacteristic
31Q8×Dic157C2211yescharacteristic
32D4×Dic157C2211yescharacteristic
33C4×C22⋊D157—21no—
34C4×C22⋊D157C2211yescharacteristic
35#19097730C2211yescharacteristic
36#19097729C2211yescharacteristic
37#19097727—21no—
38#19097705C2211yescharacteristic
39#19097704C2211yescharacteristic
40#19097703C2211yescharacteristic
41#19097700C2211yescharacteristic
42#19097695C2211yescharacteristic
43#19097695C2211yescharacteristic
44#19097693C2211yescharacteristic
45#19097693C2211yescharacteristic
46#19097692C2211yescharacteristic
47#19097692C2211yescharacteristic
48#19097692C2211yescharacteristic
49#19097691C2211yescharacteristic
50#19097683C2211yescharacteristic
51C8×Dic157C2211yescharacteristic
52C2×#41×C157C2211yescharacteristic
53C2×SD16×C157C2211yescharacteristic
54C2×D8×C157C2211yescharacteristic
55#121×C157C2211yescharacteristic
56#103×C157C2211yescharacteristic
57#102×C157C2211yescharacteristic
58C22⋊C8×C157C2211yescharacteristic
59#14578955—22no—
60#14578955—41no—
61C2×Dic314—21no—
62C2×D628—21no—
63C2×D628C2311yescharacteristic
64C2×C22⋊D157—21no—
65C2×C22⋊D157—21no—
66C2×C22⋊D157—21no—
67C2×C22⋊D157—21no—
68C2×C22⋊D157—21no—
69C2×C22⋊D157D411yescharacteristic
70C22×Dic157—21no—
71C2×C4×D157—21no—
72C2×C4×D157C2311yescharacteristic
73#14578939—41no—
74C2×#11929434—41no—
75C2×#11929434C2311yescharacteristic
76C2×#11929434C2311yescharacteristic
77C8×D157—41no—
78C4⋊Dic157—21no—
79C4⋊Dic157C2311yescharacteristic
80C4⋊Dic157C2311yescharacteristic
81C4⋊Dic157C2311yescharacteristic
82#14578933—21no—
83#14578933D411yescharacteristic
84C22⋊Dic157—21no—
85C22⋊Dic157—21no—
86C22⋊Dic157—21no—
87C22⋊Dic157—21no—
88C22⋊Dic157D411yescharacteristic
89#14578931—21no—
90#14578931—21no—
91#14578931D411yescharacteristic
92C4×Dic157—21no—
93C4×Dic157—21no—
94C4×Dic157—21no—
95C4×Dic157C2311yescharacteristic
96#41×C157—22no—
97#41×C157—41no—
98C2×Q8×C157C2311yescharacteristic
99C2×D4×C157—21no—
100C2×D4×C157—21no—
101C2×D4×C157C2311yescharacteristic
102C2×D4×C157C2311yescharacteristic
103C22×C4×C157—21no—
104C22×C4×C157C2311yescharacteristic
105SD16×C157—22no—
106D8×C157—41no—
107C2×C8×C157C2311yescharacteristic
108C2×C8×C157C2311yescharacteristic
109C4⋊C4×C157C2311yescharacteristic
110C22⋊C4×C157—21no—
111C22×D157—21no—
112C22×D157C2×D411yescharacteristic
113Dic314—81no—
114D628—21no—
115D628—21no—
116D628—42no—
117C22⋊D157—22no—
118C22⋊D157—42no—
119C22⋊D157—81no—
120C22⋊D157—81no—
121C22⋊D157—81no—
122C22⋊D157—81no—
123C2×Dic157—21no—
124C2×Dic157—21no—
125C2×Dic157—21no—
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×C157—21no—
137C23×C157C2×D411yescharacteristic
138Q8×C157—21no—
139Q8×C157#4112yes—
140D4×C157—21no—
141D4×C157—21no—
142D4×C157—21no—
143D4×C157—21no—
144D4×C157—42no—
145D4×C157—81no—
146C2×C4×C157—21no—
147C2×C4×C157—21no—
148C2×C4×C157—21no—
149C2×C4×C157—21no—
150C2×C4×C157—42no—
151C2×C4×C157C2×D411yescharacteristic
152C2×C4×C157C2411yescharacteristic
153C8×C157—21no—
154C8×C157—21no—
155C22×C157—21no—
156C22×C157—21no—
157C22×C157—21no—
158C22×C157—41no—
159C22×C157—41no—
160C22×C157—41no—
161C22×C157—41no—
162C22×C157—41no—
163C22×C157—41no—
164C22×C157C22×D411yescharacteristic
165C2×D157—21no—
166C2×D157—21no—
167C2×D157—21no—
168C2×D157—41no—
169C2×D157—41no—
170C2×D157—41no—
171C4×C157—21no—
172C4×C157—22no—
173C4×C157—41no—
174C4×C157C2×#4111yescharacteristic
175C4×C157H(5,𝔽2)11yescharacteristic
176Dic157—21no—
177Dic157—21no—
178Dic157—41no—
179Dic157—41no—
180Dic157—41no—
181Dic157#13512yes—
182C2×C157—41no—
183C2×C157—81no—
184C2×C157—81no—
185C2×C157#54611yescharacteristic
186C2×C157C2×#13511yescharacteristic
187C2×C157#57611yescharacteristic
188D157—41no—
189D157—81no—
190C157#330511yescharacteristic, complemented, Sylow
191#3305—1571nomaximal, Sylow
192#555—1571no—
193#550—1571no—
194#486—1571no—
195#482—1571no—
196#482—1571no—
197#477—1571no—
198#466—1571no—
199#466—1571no—
200#459—1571no—
201#449—1571no—
202#449D15711yescharacteristic, complemented
203C4×SD16—1571no—
204C8×D4—1571no—
205#416—1571no—
206#416—1571no—
207C2×#41—1571no—
208C2×#41C2×D15711yescharacteristic
209C2×SD16C2×D15711yescharacteristic
210C2×D8—1571no—
211C2×D8C2×D15711yescharacteristic
212C22×C8—1571no—
213C22×C8—1571no—
214C4⋊Q8—1571no—
215#127—3141no—
216#127—1571no—
217#124—3141no—
218#124—3141no—
219#121—3141no—
220#121—3141no—
221#121—3141no—
222#121—1571no—
223#121—1571no—
224#121—1571no—
225#121C2×D15711yescharacteristic
226C4×Q8—1571no—
227C4×D4—3141no—
228C4×D4—1571no—
229C4×D4—1571no—
230#117—3141no—
231#117—1571no—
232#117—1571no—
233#107—1571no—
234#107—1571no—
235#106—1571no—
236C4⋊C8—1571no—
237#103—1571no—
238#103—1571no—
239#103C2×D15711yescharacteristic
240#102—1571no—
241#102—1571no—
242#102—1571no—
243#102—1571no—
244#102—1571no—
245#102—1571no—
246#102C2×D15711yescharacteristic
247C22⋊C8—1571no—
248C22⋊C8C2×D15711yescharacteristic
249C4×C8—1571no—
250#41—22no—
251#41—41no—
252#41—3142no—
253#41—6281no—
254C2×Q8—3141no—
255C2×Q8C22×D15711yescharacteristic
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—3141no—
259C2×D4—3141no—
260C2×D4—3141no—
261C2×D4—3141no—
262C2×D4—3141no—
263C2×D4—3141no—
264C2×D4—1571no—
265C2×D4—1571no—
266C2×D4C22×D15711yescharacteristic
267C2×D4C22×D15711yescharacteristic
268C22×C4—21no—
269C22×C4—3141no—
270C22×C4—3141no—
271C22×C4—1571no—
272C22×C4C22×D15711yescharacteristic
273SD16—22no—
274D8—41no—
275D8—6281no—
276C2×C8—6281no—
277C2×C8—6281no—
278C2×C8—1571no—
279C2×C8—1571no—
280C2×C8C22×D15711yescharacteristic
281C2×C8C22×D15711yescharacteristic
282C4⋊C4—3141no—
283C4⋊C4—3141no—
284C4⋊C4—1571no—
285C4⋊C4—1571no—
286C4⋊C4—1571no—
287C4⋊C4—1571no—
288C4⋊C4C22×D15711yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—3141no—
291C22⋊C4—3141no—
292C22⋊C4—3141no—
293C22⋊C4—3141no—
294C22⋊C4—3141no—
295C22⋊C4—3141no—
296C22⋊C4—1571no—
297C22⋊C4—1571no—
298C42—3141no—
299C42—3141no—
300C42—3141no—
301C42—1571no—
302C23—21no—
303C23—21no—
304C23—3141no—
305C23—1571no—
306C23D4×D15711yescharacteristic
307Q8—21no—
308Q8—12561no—
309Q8#1457895312yes—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—81no—
316D4—3142no—
317D4—3141no—
318D4—3141no—
319D4—6282no—
320D4—6282no—
321D4—12561no—
322D4—12561no—
323D4—12561no—
324D4—12561no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—42no—
330C2×C4—3141no—
331C2×C4—3141no—
332C2×C4—3141no—
333C2×C4—3141no—
334C2×C4—3141no—
335C2×C4—6282no—
336C2×C4—6282no—
337C2×C4—1571no—
338C2×C4—1571no—
339C2×C4—1571no—
340C2×C4D4×D15711yescharacteristic
341C2×C4C23×D15711yescharacteristic
342C8—21no—
343C8—21no—
344C8—3141no—
345C8—3141no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—3141no—
356C22—3141no—
357C22—3141no—
358C22—6281no—
359C22—6281no—
360C22—6281no—
361C22C2×D4×D15711yescharacteristic
362C4—21no—
363C4—22no—
364C4—41no—
365C4—3141no—
366C4—3141no—
367C4—6281no—
368C4—6281no—
369C4—6281no—
370C4—1572no—
371C4C2×#1457895311yescharacteristic
372C4#1909781611yescharacteristic
373C2—41no—
374C2—81no—
375C2—81no—
376C2—6281no—
377C2—12561no—
378C2#2375791011yescharacteristic
379C2#135×D15711yescharacteristic
380C2#2375807711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#449×C157C2
socleC22×C157C22×D4
hypercenter#449D157
hypocenterC157#3305
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #23757987C2semidirect
2. #23757942C2semidirect
3. #23757618C2semidirect
4. #23757598C2semidirect
5. #23757596C2semidirect
6. #23757566C2semidirect
7. #23757507C2semidirect
8. #23757506C2semidirect
9. #23757466C2semidirect
10. #23757417C2semidirect
11. SD16×Dic157C2semidirect
12. #23757336C2semidirect
13. #23757269C2semidirect
14. #23757268C2semidirect
15. #449×C157C2semidirect
190. C157#3305semidirect
202. #449D157semidirect

Maximal subgroups

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

subgroupindex
#237579872
#237579422
#237576182
#237575982
#237575962
#237575662
#237575072
#237575062
#237574662
#237574172
SD16×Dic1572
#237573362
#237572692
#237572682
#449×C1572
#3305157