Finite Group Databasebeta

← #34113207

Subgroups — #34113207

Order 20096.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23758006C211yesmaximal, characteristic, complemented
2#23757993C211yesmaximal, characteristic, complemented
3#23757667C211yesmaximal, characteristic, complemented
4#23757661C211yesmaximal, characteristic
5#23757656C211yesmaximal, characteristic, complemented
6#23757494C211yesmaximal, characteristic, complemented
7#23757494C211yesmaximal, characteristic, complemented
8#23757488C211yesmaximal, characteristic, complemented
9#23757487C211yesmaximal, characteristic, complemented
10#23757478C211yesmaximal, characteristic, complemented
11#23757472C211yesmaximal, characteristic, complemented
12#23757401C211yesmaximal, characteristic, complemented
13#23757379C211yesmaximal, characteristic, complemented
14#23757360C211yesmaximal, characteristic, complemented
15#442×C157C211yesmaximal, characteristic, complemented
16C2×Dic628C2211yescharacteristic
17C2×Dic628C2211yescharacteristic
18C2×#14578943C2211yescharacteristic
19C2×#14578943C2211yescharacteristic
20C2×#14578942C2211yescharacteristic
21C2×#14578942C2211yescharacteristic
22C2×#14578942C2211yescharacteristic
23C2×#14578939C2211yescharacteristic
24#19097771C2211yescharacteristic
25#19097767C2211yescharacteristic
26#19097764—21no—
27#19097764—21no—
28#19097759C2211yescharacteristic
29#19097758C2211yescharacteristic
30#19097758C2211yescharacteristic
31#19097758C2211yescharacteristic
32#19097757—21no—
33#19097754—21no—
34#19097753—21no—
35#19097750—21no—
36C4×Dic314C2211yescharacteristic
37#19097736C2211yescharacteristic
38#19097736C2211yescharacteristic
39#19097736C2211yescharacteristic
40#19097733C2211yescharacteristic
41C4×C22⋊D157—21no—
42#19097730—21no—
43C4⋊#11929434C2211yescharacteristic
44#19097700C2211yescharacteristic
45#19097700C2211yescharacteristic
46#19097694C2211yescharacteristic
47#19097694C2211yescharacteristic
48#19097694C2211yescharacteristic
49#19097691C2211yescharacteristic
50#19097681C2211yescharacteristic
51C8×Dic157C2211yescharacteristic
52C2×D8×C157C2211yescharacteristic
53C4×D4×C157C2211yescharacteristic
54C4×D4×C157C2211yescharacteristic
55#106×C157C2211yescharacteristic
56#102×C157C2211yescharacteristic
57#102×C157C2211yescharacteristic
58#97×C157C2211yescharacteristic
59C2×Dic314D411yescharacteristic
60C2×Dic314C2311yescharacteristic
61C2×Dic314C2311yescharacteristic
62C2×Dic314C2311yescharacteristic
63C2×D628C2311yescharacteristic
64C2×C22⋊D157—21no—
65C2×C22⋊D157—21no—
66C2×C4×D157—21no—
67Dic628—22no—
68Dic628—41no—
69#14578943—22no—
70#14578943—41no—
71#14578942—41no—
72#14578942—41no—
73#14578942—41no—
74#14578939—41no—
75C2×#11929434C2311yescharacteristic
76C2×#11929434C2311yescharacteristic
77C4⋊Dic157—21no—
78C4⋊Dic157—21no—
79C4⋊Dic157D411yescharacteristic
80#14578933—21no—
81#14578933—21no—
82#14578933—21no—
83#14578933D411yescharacteristic
84#14578933D411yescharacteristic
85C22⋊Dic157—21no—
86C22⋊Dic157—21no—
87C22⋊Dic157—21no—
88C22⋊Dic157—21no—
89C22⋊Dic157—21no—
90C22⋊Dic157—21no—
91#14578931—21no—
92#14578931—21no—
93#14578931—21no—
94C4×Dic157—21no—
95C4×Dic157—21no—
96C4×Dic157—21no—
97C4×Dic157C2311yescharacteristic
98C4×Dic157C2311yescharacteristic
99C2×D4×C157C2311yescharacteristic
100C2×D4×C157C2311yescharacteristic
101C22×C4×C157—21no—
102C22×C4×C157—21no—
103D8×C157—22no—
104C2×C8×C157C2311yescharacteristic
105C2×C8×C157C2311yescharacteristic
106C4⋊C4×C157C2311yescharacteristic
107C4⋊C4×C157C2311yescharacteristic
108C22⋊C4×C157—21no—
109C22⋊C4×C157—21no—
110C42×C157C2311yescharacteristic
111C22×D157—21no—
112Dic314—21no—
113Dic314—21no—
114Dic314—21no—
115Dic314—21no—
116Dic314—21no—
117Dic314—21no—
118Dic314—41no—
119D628—21no—
120D628—21no—
121C22⋊D157—42no—
122C22⋊D157—81no—
123C2×Dic157—21no—
124C2×Dic157—21no—
125C2×Dic157—21no—
126C2×Dic157C2×D411yescharacteristic
127C2×Dic157C2×D411yescharacteristic
128C2×Dic157C2×D411yescharacteristic
129C2×Dic157C2×D411yescharacteristic
130C4×D157—81no—
131#11929434—21no—
132#11929434—21no—
133C23×C157—21no—
134C23×C157—21no—
135D4×C157—21no—
136D4×C157—21no—
137D4×C157—21no—
138D4×C157—21no—
139C2×C4×C157—21no—
140C2×C4×C157—21no—
141C2×C4×C157—42no—
142C2×C4×C157—81no—
143C2×C4×C157C2×D411yescharacteristic
144C2×C4×C157C2×D411yescharacteristic
145C2×C4×C157C2411yescharacteristic
146C8×C157—21no—
147C8×C157#4112yes—
148C22×C157—41no—
149C22×C157—41no—
150C22×C157—41no—
151C22×C157—41no—
152C22×C157—41no—
153C22×C157—41no—
154C22×C157C22×D411yescharacteristic
155C2×D157—41no—
156C2×D157—41no—
157C2×D157—41no—
158C4×C157—21no—
159C4×C157—21no—
160C4×C157—41no—
161C4×C157—41no—
162C4×C157C2×#4111yescharacteristic
163C4×C157H(5,𝔽2)11yescharacteristic
164Dic157—21no—
165Dic157—21no—
166Dic157—21no—
167Dic157—21no—
168Dic157—41no—
169Dic157—41no—
170Dic157—41no—
171C2×C157—81no—
172C2×C157—81no—
173C2×C157#55011yescharacteristic
174C2×C157#57511yescharacteristic
175C2×C157#57711yescharacteristic
176D157—81no—
177C157#340111yescharacteristic, complemented, Sylow
178#3401—1571nomaximal, Sylow
179#559—1571no—
180#556—1571no—
181#497—1571no—
182#496—1571no—
183#495—1571no—
184#464—1571no—
185#464—1571no—
186#463—1571no—
187#463—1571no—
188#461—1571no—
189#460—1571no—
190#446—1571no—
191#442D15711yescharacteristic, complemented
192#441—1571no—
193C4×SD16—1571no—
194C2×Q16—1571no—
195C2×Q16—1571no—
196C2×SD16—1571no—
197C2×SD16—1571no—
198C2×SD16—1571no—
199C2×SD16—1571no—
200C2×SD16—1571no—
201C2×D8—1571no—
202C2×D8C2×D15711yescharacteristic
203C4⋊Q8—1571no—
204#127—1571no—
205#126—3141no—
206#126—3141no—
207#124—3141no—
208#124—1571no—
209#124—1571no—
210#124—1571no—
211#124—1571no—
212#123—3141no—
213#123—3141no—
214C22⋊Q8—3141no—
215C4×Q8—1571no—
216C4×Q8—1571no—
217C4×Q8—1571no—
218C4×Q8—1571no—
219C4×D4—3141no—
220C4×D4—1571no—
221C4×D4C2×D15711yescharacteristic
222C4×D4C2×D15711yescharacteristic
223#117—3141no—
224#106C2×D15711yescharacteristic
225C4⋊C8—1571no—
226C4⋊C8—1571no—
227C4⋊C8—1571no—
228#103—1571no—
229#103—1571no—
230#103—1571no—
231#102—1571no—
232#102C2×D15711yescharacteristic
233#102C2×D15711yescharacteristic
234#97—1571no—
235#97C2×D15711yescharacteristic
236C4×C8—1571no—
237C2×Q8—1571no—
238C2×Q8—1571no—
239C2×Q8—1571no—
240C2×Q8—1571no—
241C2×D4—3141no—
242C2×D4—3141no—
243C2×D4—1571no—
244C2×D4C22×D15711yescharacteristic
245C2×D4C22×D15711yescharacteristic
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—3141no—
249Q16—3142no—
250Q16—6281no—
251SD16—3142no—
252SD16—6281no—
253SD16—6281no—
254SD16—6281no—
255SD16—6281no—
256D8—22no—
257D8—6281no—
258C2×C8—1571no—
259C2×C8—1571no—
260C2×C8C22×D15711yescharacteristic
261C2×C8C22×D15711yescharacteristic
262C4⋊C4—3141no—
263C4⋊C4—3141no—
264C4⋊C4—3141no—
265C4⋊C4—3141no—
266C4⋊C4—3141no—
267C4⋊C4—1571no—
268C4⋊C4—1571no—
269C4⋊C4—1571no—
270C4⋊C4C22×D15711yescharacteristic
271C4⋊C4C22×D15711yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—3141no—
275C22⋊C4—3141no—
276C22⋊C4—3141no—
277C22⋊C4—3141no—
278C22⋊C4—3141no—
279C22⋊C4—3141no—
280C22⋊C4—3141no—
281C22⋊C4—3141no—
282C22⋊C4—3141no—
283C42—3141no—
284C42—3141no—
285C42—3141no—
286C42—1571no—
287C42—1571no—
288C42C22×D15711yescharacteristic
289C23—21no—
290C23—21no—
291C23—3141no—
292Q8—3141no—
293Q8—3141no—
294Q8—3141no—
295Q8—3141no—
296Q8—3141no—
297Q8—3141no—
298Q8—6281no—
299D4—21no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—3141no—
304D4—3141no—
305D4—6282no—
306D4—12561no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—81no—
311C2×C4—3141no—
312C2×C4—3141no—
313C2×C4—3141no—
314C2×C4—1571no—
315C2×C4—1571no—
316C2×C4—1571no—
317C2×C4—1571no—
318C2×C4—12561no—
319C2×C4D4×D15711yescharacteristic
320C2×C4D4×D15711yescharacteristic
321C2×C4C23×D15711yescharacteristic
322C8—21no—
323C8—3141no—
324C8—3141no—
325C8#1457895412yes—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—6281no—
333C22—6281no—
334C22—6281no—
335C22C2×D4×D15711yescharacteristic
336C4—21no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—3141no—
341C4—3141no—
342C4—3141no—
343C4—3141no—
344C4—6281no—
345C4—6281no—
346C4—6281no—
347C4C2×#1457895411yescharacteristic
348C4#1909781711yescharacteristic
349C2—81no—
350C2—81no—
351C2—12561no—
352C2#2375794411yescharacteristic
353C2#2375806911yescharacteristic
354C2#2375808311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#442×C157C2
socleC22×C157C22×D4
prime radical#23757661C2
hypercenter#442D157
hypocenterC157#3401
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #23758006C2semidirect
2. #23757993C2semidirect
3. #23757667C2semidirect
5. #23757656C2semidirect
6. #23757494C2semidirect
7. #23757494C2semidirect
8. #23757488C2semidirect
9. #23757487C2semidirect
10. #23757478C2semidirect
11. #23757472C2semidirect
12. #23757401C2semidirect
13. #23757379C2semidirect
14. #23757360C2semidirect
15. #442×C157C2semidirect
177. C157#3401semidirect
191. #442D157semidirect

Maximal subgroups

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

subgroupindex
#237580062
#237579932
#237576672
#237576612
#237576562
#237574942
#237574882
#237574872
#237574782
#237574722
#237574012
#237573792
#237573602
#442×C1572
#3401157