Finite Group Databasebeta

← #34112040

Subgroups — #34112040

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#23757969C211yesmaximal, characteristic, complemented
2#23757932C211yesmaximal, characteristic, complemented
3#23757628C211yesmaximal, characteristic, complemented
4#23757615C211yesmaximal, characteristic, complemented
5#23757592C211yesmaximal, characteristic, complemented
6#23757581C211yesmaximal, characteristic, complemented
7#23757509C211yesmaximal, characteristic, complemented
8#23757502C211yesmaximal, characteristic, complemented
9#23757489C211yesmaximal, characteristic, complemented
10#23757420C211yesmaximal, characteristic, complemented
11#23757414C211yesmaximal, characteristic, complemented
12C4×#14578943C211yesmaximal, characteristic, complemented
13#23757270C211yesmaximal, characteristic, complemented
14C2×#19097696C211yesmaximal, characteristic, complemented
15C8×D4×C157C211yesmaximal, characteristic, complemented
16C2×#14578954C2211yescharacteristic
17C22×Dic314C2211yescharacteristic
18C2×#14578943C2211yescharacteristic
19C2×#14578943C2211yescharacteristic
20C2×D1256C2211yescharacteristic
21#19097766—21no—
22#19097763C2211yescharacteristic
23#19097759C2211yescharacteristic
24#19097757—21no—
25#19097753—21no—
26#19097750C2211yescharacteristic
27#19097749—21no—
28#19097745C2211yescharacteristic
29#19097745C2211yescharacteristic
30#19097745C2211yescharacteristic
31#19097743—21no—
32#19097743—21no—
33C4×Dic314C2211yescharacteristic
34C4×D628C2211yescharacteristic
35#19097731—21no—
36#19097731—21no—
37#19097730C2211yescharacteristic
38C2×C4⋊Dic157C2211yescharacteristic
39#19097705C2211yescharacteristic
40#19097703C2211yescharacteristic
41#19097703C2211yescharacteristic
42#19097696C2211yescharacteristic
43#19097696C2211yescharacteristic
44#19097696C2211yescharacteristic
45#19097696C2211yescharacteristic
46#19097696C2211yescharacteristic
47#19097696C2211yescharacteristic
48#19097693C2211yescharacteristic
49#19097693C2211yescharacteristic
50#19097693C2211yescharacteristic
51#19097693C2211yescharacteristic
52C22×C8×C157C2211yescharacteristic
53C22×C8×C157C2211yescharacteristic
54C4×D4×C157C2211yescharacteristic
55C4⋊C8×C157C2211yescharacteristic
56C22⋊C8×C157C2211yescharacteristic
57C22⋊C8×C157C2211yescharacteristic
58C4×C8×C157C2211yescharacteristic
59#14578954—22no—
60#14578954—41no—
61C2×Dic314—22no—
62C2×Dic314—21no—
63C2×Dic314—41no—
64C2×Dic314C2311yescharacteristic
65C2×Dic314C2311yescharacteristic
66C2×D628C2311yescharacteristic
67C2×D628C2311yescharacteristic
68C2×C22⋊D157—21no—
69C2×C22⋊D157—21no—
70C2×C22⋊D157—21no—
71C2×C22⋊D157—21no—
72C22×Dic157—21no—
73C22×Dic157—21no—
74C2×C4×D157—21no—
75C2×C4×D157—21no—
76#14578943—22no—
77#14578943—41no—
78D1256—41no—
79C4⋊Dic157—21no—
80C4⋊Dic157C2311yescharacteristic
81C4⋊Dic157C2311yescharacteristic
82C4⋊Dic157C2311yescharacteristic
83C4⋊Dic157C2311yescharacteristic
84#14578933—21no—
85#14578933—21no—
86#14578933—21no—
87#14578933—21no—
88C22⋊Dic157—21no—
89C22⋊Dic157—21no—
90#14578931—21no—
91#14578931—21no—
92#14578931—21no—
93#14578931—21no—
94#14578931—21no—
95#14578931—21no—
96C4×Dic157—21no—
97C4×Dic157—21no—
98C2×D4×C157D411yescharacteristic
99C22×C4×C157C2311yescharacteristic
100C22×C4×C157C2311yescharacteristic
101C2×C8×C157—41no—
102C2×C8×C157—41no—
103C2×C8×C157C2311yescharacteristic
104C2×C8×C157C2311yescharacteristic
105C2×C8×C157C2311yescharacteristic
106C2×C8×C157C2311yescharacteristic
107C4⋊C4×C157D411yescharacteristic
108C22⋊C4×C157D411yescharacteristic
109C22⋊C4×C157D411yescharacteristic
110C42×C157C2311yescharacteristic
111C22×D157—21no—
112C22×D157—21no—
113Dic314—21no—
114Dic314—21no—
115Dic314—21no—
116Dic314—42no—
117Dic314#4112yes—
118D628—21no—
119D628—21no—
120D628—21no—
121D628—21no—
122C22⋊D157—42no—
123C22⋊D157—42no—
124C22⋊D157—42no—
125C22⋊D157—81no—
126C2×Dic157—21no—
127C2×Dic157—21no—
128C2×Dic157—21no—
129C2×Dic157—21no—
130C2×Dic157—21no—
131C2×Dic157—21no—
132C2×Dic157—42no—
133C2×Dic157—81no—
134C4×D157—42no—
135C4×D157—81no—
136C23×C157C2×D411yescharacteristic
137C23×C157C2×D411yescharacteristic
138D4×C157—22no—
139C2×C4×C157—21no—
140C2×C4×C157—21no—
141C2×C4×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157C2×D411yescharacteristic
144C2×C4×C157C2×D411yescharacteristic
145C2×C4×C157C2×D411yescharacteristic
146C2×C4×C157C2×D411yescharacteristic
147C2×C4×C157C2411yescharacteristic
148C8×C157—21no—
149C8×C157—21no—
150C8×C157—21no—
151C8×C157—21no—
152C22×C157—21no—
153C22×C157—21no—
154C22×C157—21no—
155C22×C157—21no—
156C22×C157—21no—
157C22×C157#13512yes—
158C22×C157C22×D411yescharacteristic
159C2×D157—41no—
160C2×D157—41no—
161C2×D157—41no—
162C2×D157—41no—
163C2×D157—41no—
164C2×D157—41no—
165C4×C157—21no—
166C4×C157—21no—
167C4×C157—21no—
168C4×C157—21no—
169C4×C157C2×#4111yescharacteristic
170C4×C157H(5,𝔽2)11yescharacteristic
171Dic157—22no—
172Dic157—41no—
173Dic157—41no—
174Dic157—41no—
175Dic157—41no—
176Dic157—41no—
177C2×C157—22no—
178C2×C157—41no—
179C2×C157#54611yescharacteristic
180C2×C157C2×#13511yescharacteristic
181C2×C157#57711yescharacteristic
182D157—81no—
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#448—1571no—
197C4×SD16—1571no—
198C8×D4D15711yescharacteristic, complemented
199#416—1571no—
200C2×#103—1571no—
201C2×#41—1571no—
202C22×Q8—1571no—
203C2×SD16—1571no—
204C2×SD16—1571no—
205C2×D8—1571no—
206C22×C8C2×D15711yescharacteristic
207C22×C8C2×D15711yescharacteristic
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#121—1571no—
220C4×Q8—1571no—
221C4×D4—3141no—
222C4×D4—3141no—
223C4×D4—1571no—
224C4×D4C2×D15711yescharacteristic
225#117—1571no—
226C2×C4⋊C4—1571no—
227#107—1571no—
228#106—1571no—
229#106—1571no—
230C4⋊C8C2×D15711yescharacteristic
231#103—1571no—
232#103—1571no—
233#103—1571no—
234#103—1571no—
235#103—1571no—
236#103—1571no—
237#102—1571no—
238#102—1571no—
239#102—1571no—
240#102—1571no—
241C22⋊C8C2×D15711yescharacteristic
242C22⋊C8C2×D15711yescharacteristic
243C4×C8C2×D15711yescharacteristic
244#41—3142no—
245#41—6281no—
246C2×Q8—3142no—
247C2×Q8—3141no—
248C2×Q8—6281no—
249C2×Q8—1571no—
250C2×Q8—1571no—
251C2×D4—3141no—
252C2×D4—3141no—
253C2×D4—3141no—
254C2×D4—3141no—
255C2×D4—1571no—
256C2×D4—1571no—
257C2×D4D62811yescharacteristic
258C22×C4—3141no—
259C22×C4—3141no—
260C22×C4—3141no—
261C22×C4—3141no—
262C22×C4C22×D15711yescharacteristic
263C22×C4C22×D15711yescharacteristic
264SD16—3142no—
265SD16—6281no—
266D8—6281no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8C22×D15711yescharacteristic
270C2×C8C22×D15711yescharacteristic
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⋊C4D62811yescharacteristic
283C22⋊C4—3141no—
284C22⋊C4—3141no—
285C22⋊C4—3141no—
286C22⋊C4—3141no—
287C22⋊C4—3141no—
288C22⋊C4—3141no—
289C22⋊C4—3141no—
290C22⋊C4—3141no—
291C22⋊C4D62811yescharacteristic
292C22⋊C4D62811yescharacteristic
293C42—3141no—
294C42—3141no—
295C42C22×D15711yescharacteristic
296C23—3141no—
297C23—3141no—
298C23C2×D62811yescharacteristic
299C23C2×D62811yescharacteristic
300Q8—3141no—
301Q8—3141no—
302Q8—3141no—
303Q8—6282no—
304Q8—1572no—
305D4—22no—
306D4—3141no—
307D4—3141no—
308D4—3141no—
309D4—3141no—
310D4—6282no—
311D4—6282no—
312D4—6282no—
313D4—12561no—
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—12561no—
327C2×C4—12561no—
328C2×C4C2×D62811yescharacteristic
329C2×C4C2×D62811yescharacteristic
330C2×C4C2×D62811yescharacteristic
331C2×C4C2×D62811yescharacteristic
332C2×C4C23×D15711yescharacteristic
333C8—21no—
334C8—21no—
335C8—21no—
336C8—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—6281no—
343C22—6281no—
344C22—6281no—
345C22—6281no—
346C22—6281no—
347C22—6281no—
348C22#1909779312yes—
349C22C22×D62811yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—3142no—
355C4—6281no—
356C4—6281no—
357C4—6281no—
358C4—6281no—
359C4—6281no—
360C4#41×D15711yescharacteristic
361C4#1909781611yescharacteristic
362C2—22no—
363C2—41no—
364C2—12561no—
365C2—12561no—
366C2#2375791811yescharacteristic
367C2C2×#1909779311yescharacteristic
368C2#2375808511yescharacteristic
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC8×D4×C157C2
socleC22×C157C22×D4
hypercenterC8×D4D157
hypocenterC157#3301
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #23757969C2semidirect
2. #23757932C2semidirect
3. #23757628C2semidirect
4. #23757615C2semidirect
5. #23757592C2semidirect
6. #23757581C2semidirect
7. #23757509C2semidirect
8. #23757502C2semidirect
9. #23757489C2semidirect
10. #23757420C2semidirect
11. #23757414C2semidirect
12. C4×#14578943C2semidirect
13. #23757270C2semidirect
14. C2×#19097696C2semidirect
15. C8×D4×C157C2semidirect
184. C157#3301semidirect
198. C8×D4D157semidirect

Maximal subgroups

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

subgroupindex
#237579692
#237579322
#237576282
#237576152
#237575922
#237575812
#237575092
#237575022
#237574892
#237574202
#237574142
C4×#145789432
#237572702
C2×#190976962
C8×D4×C1572
#3301157