Finite Group Databasebeta

← #34112042

Subgroups — #34112042

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#23757975C211yesmaximal, characteristic, complemented
2#23757934C211yesmaximal, characteristic, complemented
3#23757629C211yesmaximal, characteristic, complemented
4#23757612C211yesmaximal, characteristic, complemented
5#23757592C211yesmaximal, characteristic, complemented
6#23757578C211yesmaximal, characteristic, complemented
7#23757507C211yesmaximal, characteristic, complemented
8#23757499C211yesmaximal, characteristic, complemented
9#23757490C211yesmaximal, characteristic, complemented
10#23757417C211yesmaximal, characteristic, complemented
11SD16×Dic157C211yesmaximal, characteristic, complemented
12#23757336C211yesmaximal, characteristic, complemented
13#23757268C211yesmaximal, characteristic, complemented
14C2×#19097695C211yesmaximal, characteristic, complemented
15#448×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×#14578939C2211yescharacteristic
18C22×#11929434C2211yescharacteristic
19C2×C8×D157C2211yescharacteristic
20#19097764—21no—
21#19097762C2211yescharacteristic
22#19097760—21no—
23#19097760C2211yescharacteristic
24#19097755—21no—
25#19097747—21no—
26#19097747C2211yescharacteristic
27#19097745—21no—
28#19097745C2211yescharacteristic
29#19097743—21no—
30#19097742C2211yescharacteristic
31Q8×Dic157C2211yescharacteristic
32D4×Dic157—21no—
33D4×Dic157C2211yescharacteristic
34C4×C22⋊D157—21no—
35C4×C22⋊D157C2211yescharacteristic
36#19097729C2211yescharacteristic
37C2×C4⋊Dic157C2211yescharacteristic
38#19097704C2211yescharacteristic
39#19097703C2211yescharacteristic
40#19097703C2211yescharacteristic
41#19097700C2211yescharacteristic
42#19097695C2211yescharacteristic
43#19097695C2211yescharacteristic
44#19097695C2211yescharacteristic
45#19097695C2211yescharacteristic
46#19097693C2211yescharacteristic
47#19097693C2211yescharacteristic
48#19097692C2211yescharacteristic
49#19097691C2211yescharacteristic
50#19097683C2211yescharacteristic
51C8×Dic157C2211yescharacteristic
52C22×Q8×C157C2211yescharacteristic
53C2×SD16×C157C2211yescharacteristic
54C2×SD16×C157C2211yescharacteristic
55#121×C157C2211yescharacteristic
56#103×C157C2211yescharacteristic
57#103×C157C2211yescharacteristic
58C22⋊C8×C157C2211yescharacteristic
59#14578955—22no—
60#14578955—41no—
61C2×D628—21no—
62C2×D628C2311yescharacteristic
63C2×C22⋊D157—21no—
64C2×C22⋊D157—21no—
65C2×C22⋊D157D411yescharacteristic
66C22×Dic157—21no—
67C22×Dic157—21no—
68C2×C4×D157—21no—
69C2×C4×D157C2311yescharacteristic
70#14578939—41no—
71C2×#11929434—41no—
72C2×#11929434C2311yescharacteristic
73C2×#11929434C2311yescharacteristic
74C8×D157—41no—
75C4⋊Dic157—21no—
76C4⋊Dic157—21no—
77C4⋊Dic157—21no—
78C4⋊Dic157C2311yescharacteristic
79C4⋊Dic157C2311yescharacteristic
80C4⋊Dic157C2311yescharacteristic
81#14578933—21no—
82#14578933—21no—
83#14578933D411yescharacteristic
84C22⋊Dic157—21no—
85C22⋊Dic157—21no—
86C22⋊Dic157D411yescharacteristic
87#14578931—21no—
88#14578931—21no—
89#14578931—21no—
90#14578931—21no—
91#14578931—21no—
92#14578931D411yescharacteristic
93C4×Dic157—21no—
94C4×Dic157—21no—
95C4×Dic157C2311yescharacteristic
96C2×Q8×C157—21no—
97C2×Q8×C157—22no—
98C2×Q8×C157—41no—
99C2×Q8×C157C2311yescharacteristic
100C2×Q8×C157C2311yescharacteristic
101C2×D4×C157—21no—
102C2×D4×C157C2311yescharacteristic
103C22×C4×C157—21no—
104C22×C4×C157C2311yescharacteristic
105SD16×C157—22no—
106SD16×C157—41no—
107C2×C8×C157C2311yescharacteristic
108C2×C8×C157C2311yescharacteristic
109C4⋊C4×C157C2311yescharacteristic
110C22⋊C4×C157—21no—
111C22×D157—21no—
112C22×D157C2×D411yescharacteristic
113D628—21no—
114D628—21no—
115D628—42no—
116C22⋊D157—22no—
117C22⋊D157—42no—
118C22⋊D157—81no—
119C2×Dic157—21no—
120C2×Dic157—21no—
121C2×Dic157—21no—
122C2×Dic157—81no—
123C2×Dic157—81no—
124C2×Dic157C2×D411yescharacteristic
125C2×Dic157C2×D411yescharacteristic
126C2×Dic157C2×D411yescharacteristic
127C4×D157—21no—
128C4×D157—21no—
129C4×D157—42no—
130#11929434—21no—
131#11929434—21no—
132C23×C157—21no—
133C23×C157C2×D411yescharacteristic
134Q8×C157—21no—
135Q8×C157—21no—
136Q8×C157—21no—
137Q8×C157—42no—
138Q8×C157#4112yes—
139D4×C157—21no—
140D4×C157—21no—
141D4×C157—42no—
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—21no—
154C22×C157—41no—
155C22×C157—41no—
156C22×C157—41no—
157C22×C157#13512yes—
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—22no—
167C4×C157—41no—
168C4×C157—41no—
169C4×C157C2×#4111yescharacteristic
170C4×C157H(5,𝔽2)11yescharacteristic
171Dic157—21no—
172Dic157—21no—
173Dic157—21no—
174Dic157—41no—
175Dic157—41no—
176Dic157—41no—
177C2×C157—22no—
178C2×C157—81no—
179C2×C157#54611yescharacteristic
180C2×C157C2×#13511yescharacteristic
181C2×C157#57711yescharacteristic
182D157—41no—
183D157—81no—
184C157#330111yescharacteristic, complemented, Sylow
185#3301—1571nomaximal, Sylow
186#553—1571no—
187#548—1571no—
188#488—1571no—
189#485—1571no—
190#481—1571no—
191#479—1571no—
192#466—1571no—
193#465—1571no—
194#463—1571no—
195#449—1571no—
196#448D15711yescharacteristic, complemented
197C4×SD16—1571no—
198C8×D4—1571no—
199#416—1571no—
200C2×#103—1571no—
201C2×#41—1571no—
202C22×Q8C2×D15711yescharacteristic
203C2×SD16C2×D15711yescharacteristic
204C2×SD16C2×D15711yescharacteristic
205C2×D8—1571no—
206C22×C8—1571no—
207C22×C8—1571no—
208#126—3141no—
209#125—1571no—
210#124—3141no—
211#124—1571no—
212#123—3141no—
213C22⋊Q8—3141no—
214C22⋊Q8—1571no—
215#121—3141no—
216#121—3141no—
217#121—1571no—
218#121—1571no—
219#121C2×D15711yescharacteristic
220C4×Q8—1571no—
221C4×D4—3141no—
222C4×D4—3141no—
223C4×D4—1571no—
224C4×D4—1571no—
225#117—1571no—
226C2×C4⋊C4—1571no—
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#102—1571no—
240#102—1571no—
241C22⋊C8—1571no—
242C22⋊C8C2×D15711yescharacteristic
243C4×C8—1571no—
244#41—3142no—
245#41—6281no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8C22×D15711yescharacteristic
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—22no—
265SD16—41no—
266D8—6281no—
267C2×C8—6281no—
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⋊C4—1571no—
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—3141no—
294C42—3141no—
295C42—1571no—
296C23—21no—
297C23—3141no—
298C23—1571no—
299C23D4×D15711yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8#1457895312yes—
305D4—21no—
306D4—21no—
307D4—42no—
308D4—3142no—
309D4—3141no—
310D4—3141no—
311D4—6282no—
312D4—6282no—
313D4—12561no—
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—1571no—
327C2×C4—1571no—
328C2×C4—1571no—
329C2×C4—12561no—
330C2×C4—12561no—
331C2×C4D4×D15711yescharacteristic
332C2×C4C23×D15711yescharacteristic
333C8—21no—
334C8—21no—
335C8—3141no—
336C8—3141no—
337C22—21no—
338C22—21no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—3141no—
343C22—3141no—
344C22—3141no—
345C22—6281no—
346C22—6281no—
347C22—6281no—
348C22#1909779212yes—
349C22C2×D4×D15711yescharacteristic
350C4—21no—
351C4—22no—
352C4—41no—
353C4—41no—
354C4—3141no—
355C4—3141no—
356C4—3141no—
357C4—6281no—
358C4—6281no—
359C4—6281no—
360C4C2×#1457895311yescharacteristic
361C4#1909781611yescharacteristic
362C2—22no—
363C2—81no—
364C2—6281no—
365C2—12561no—
366C2#2375791011yescharacteristic
367C2C2×#1909779211yescharacteristic
368C2#2375808211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#448×C157C2
socleC22×C157C22×D4
hypercenter#448D157
hypocenterC157#3301
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. #23757975C2semidirect
2. #23757934C2semidirect
3. #23757629C2semidirect
4. #23757612C2semidirect
5. #23757592C2semidirect
6. #23757578C2semidirect
7. #23757507C2semidirect
8. #23757499C2semidirect
9. #23757490C2semidirect
10. #23757417C2semidirect
11. SD16×Dic157C2semidirect
12. #23757336C2semidirect
13. #23757268C2semidirect
14. C2×#19097695C2semidirect
15. #448×C157C2semidirect
184. C157#3301semidirect
196. #448D157semidirect

Maximal subgroups

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

subgroupindex
#237579752
#237579342
#237576292
#237576122
#237575922
#237575782
#237575072
#237574992
#237574902
#237574172
SD16×Dic1572
#237573362
#237572682
C2×#190976952
#448×C1572
#3301157