Finite Group Databasebeta

← #34110280

Subgroups — #34110280

Order 20096.

430 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#23757818C211yesmaximal, characteristic, complemented
2#41×Dic157C211yesmaximal, characteristic, complemented
3#23757598C211yesmaximal, characteristic, complemented
4#23757594C211yesmaximal, characteristic, complemented
5#23757556C211yesmaximal, characteristic, complemented
6#23757551C211yesmaximal, characteristic, complemented
7#23757471C211yesmaximal, characteristic, complemented
8#23757466C211yesmaximal, characteristic, complemented
9#23757418C211yesmaximal, characteristic, complemented
10#23757385C211yesmaximal, characteristic, complemented
11#23757382C211yesmaximal, characteristic, complemented
12#23757376C211yesmaximal, characteristic, complemented
13#23757374C211yesmaximal, characteristic, complemented
14#23757327C211yesmaximal, characteristic, complemented
15#449×C157C211yesmaximal, characteristic, complemented
16C2×#14578953—21no—
17C2×#14578941C2211yescharacteristic
18C2×#14578939C2211yescharacteristic
19#19097770C2211yescharacteristic
20#19097767C2211yescharacteristic
21#19097757—21no—
22#19097745C2211yescharacteristic
23#19097743—21no—
24Q8×Dic157C2211yescharacteristic
25Q8×Dic157C2211yescharacteristic
26D4×Dic157—21no—
27D4×Dic157—21no—
28D4×Dic157C2211yescharacteristic
29D4×Dic157C2211yescharacteristic
30D4×Dic157C2211yescharacteristic
31#19097733C2211yescharacteristic
32#19097731—21no—
33#19097730—21no—
34#19097730C2211yescharacteristic
35C2×C4×Dic157—21no—
36C2×C4×Dic157C2211yescharacteristic
37#19097705C2211yescharacteristic
38#19097704C2211yescharacteristic
39#19097703C2211yescharacteristic
40#19097702C2211yescharacteristic
41#19097700C2211yescharacteristic
42#19097700C2211yescharacteristic
43#19097695C2211yescharacteristic
44#19097693C2211yescharacteristic
45#19097693C2211yescharacteristic
46#19097692C2211yescharacteristic
47#19097692C2211yescharacteristic
48#19097692C2211yescharacteristic
49C22⋊#11929434C2211yescharacteristic
50#19097681C2211yescharacteristic
51#19097681C2211yescharacteristic
52C2×#41×C157C2211yescharacteristic
53C2×SD16×C157C2211yescharacteristic
54C2×D8×C157C2211yescharacteristic
55#121×C157C2211yescharacteristic
56#103×C157C2211yescharacteristic
57#102×C157C2211yescharacteristic
58C22⋊C8×C157C2211yescharacteristic
59#14578953—44no—
60C2×Dic314—21no—
61C2×D628C2311yescharacteristic
62C2×C22⋊D157—21no—
63C2×C22⋊D157—21no—
64C2×C22⋊D157—21no—
65C22×Dic157—21no—
66C22×Dic157—21no—
67C22×Dic157—21no—
68C22×Dic157D411yescharacteristic
69C22×Dic157D411yescharacteristic
70C2×C4×D157—21no—
71#14578941—22no—
72#14578939—22no—
73C2×#11929434C2311yescharacteristic
74C2×#11929434C2311yescharacteristic
75C4⋊Dic157—21no—
76C4⋊Dic157—21no—
77C4⋊Dic157C2311yescharacteristic
78C4⋊Dic157C2311yescharacteristic
79C4⋊Dic157C2311yescharacteristic
80#14578933—21no—
81C22⋊Dic157—21no—
82C22⋊Dic157—21no—
83C22⋊Dic157—21no—
84C22⋊Dic157—21no—
85#14578931—21no—
86#14578931—21no—
87C4×Dic157—21no—
88C4×Dic157—21no—
89C4×Dic157—21no—
90C4×Dic157D411yescharacteristic
91C4×Dic157D411yescharacteristic
92C4×Dic157C2311yescharacteristic
93C4×Dic157C2311yescharacteristic
94#41×C157—22no—
95#41×C157—22no—
96C2×Q8×C157C2311yescharacteristic
97C2×D4×C157—21no—
98C2×D4×C157—21no—
99C2×D4×C157C2311yescharacteristic
100C2×D4×C157C2311yescharacteristic
101C22×C4×C157—21no—
102C22×C4×C157C2311yescharacteristic
103SD16×C157—22no—
104D8×C157—22no—
105C2×C8×C157C2311yescharacteristic
106C2×C8×C157C2311yescharacteristic
107C4⋊C4×C157C2311yescharacteristic
108C22⋊C4×C157—21no—
109C22×D157—21no—
110Dic314—42no—
111D628—21no—
112D628—21no—
113C22⋊D157—42no—
114C22⋊D157—42no—
115C22⋊D157—42no—
116C2×Dic157—22no—
117C2×Dic157—21no—
118C2×Dic157—21no—
119C2×Dic157—21no—
120C2×Dic157—41no—
121C2×Dic157—42no—
122C2×Dic157—42no—
123C2×Dic157—42no—
124C2×Dic157C2×D411yescharacteristic
125C2×Dic157C2×D411yescharacteristic
126C2×Dic157C2×D411yescharacteristic
127C2×Dic157C2×D411yescharacteristic
128C4×D157—42no—
129#11929434—21no—
130#11929434—21no—
131C23×C157—21no—
132C23×C157—21no—
133C23×C157C2×D411yescharacteristic
134Q8×C157—21no—
135Q8×C157#4112yescomplemented
136D4×C157—24no—
137D4×C157—21no—
138D4×C157—21no—
139D4×C157—21no—
140D4×C157—42no—
141D4×C157#4112yescomplemented
142C2×C4×C157—24no—
143C2×C4×C157—21no—
144C2×C4×C157—21no—
145C2×C4×C157—21no—
146C2×C4×C157—21no—
147C2×C4×C157C2×D411yescharacteristic
148C2×C4×C157C2411yescharacteristic
149C8×C157—21no—
150C8×C157—21no—
151C22×C157—21no—
152C22×C157—22no—
153C22×C157—21no—
154C22×C157—21no—
155C22×C157—41no—
156C22×C157—41no—
157C22×C157—41no—
158C22×C157—41no—
159C22×C157—41no—
160C22×C157C22×D411yescharacteristic
161C2×D157—41no—
162C2×D157—41no—
163C2×D157—41no—
164C4×C157—21no—
165C4×C157—22no—
166C4×C157—41no—
167C4×C157C2×#4111yescharacteristic
168C4×C157C2×#4111yescharacteristic
169Dic157—21no—
170Dic157—21no—
171Dic157—21no—
172Dic157—41no—
173Dic157—41no—
174Dic157—41no—
175Dic157AGL(1,ℤ/8)12yes—
176C2×C157—41no—
177C2×C157—42no—
178C2×C157—81no—
179C2×C157#52511yescharacteristic
180C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
181C2×C157#57511yescharacteristic
182D157—81no—
183C157#311511yescharacteristic, complemented, Sylow
184#3115—1571nomaximal, Sylow
185#532—1571no—
186C4×#41—1571no—
187#482—1571no—
188#482—1571no—
189#475—1571no—
190#474—1571no—
191#460—1571no—
192#459—1571no—
193#449—1571no—
194#449D15711yescharacteristic, complemented
195#442—1571no—
196#442—1571no—
197#440—1571no—
198#440—1571no—
199#432—1571no—
200C2×#41—3141no—
201C2×#41C2×D15711yescharacteristic
202C2×SD16—1571no—
203C2×SD16C2×D15711yescharacteristic
204C2×D8—1571no—
205C2×D8C2×D15711yescharacteristic
206C4⋊Q8—1571no—
207#127—1571no—
208#124—3141no—
209#121—3141no—
210#121—1571no—
211#121C2×D15711yescharacteristic
212C4×Q8—1571no—
213C4×Q8—1571no—
214C4×D4—3141no—
215C4×D4—3141no—
216C4×D4—3141no—
217C4×D4—1571no—
218C4×D4—1571no—
219C4×D4—1571no—
220C4×D4—1571no—
221#117—3141no—
222#117—1571no—
223C2×C42—3141no—
224C2×C42—1571no—
225#107—1571no—
226#107—1571no—
227#106—1571no—
228#106—1571no—
229C4⋊C8—1571no—
230C4⋊C8—1571no—
231#103—1571no—
232#103C2×D15711yescharacteristic
233#102—1571no—
234#102—1571no—
235#102—1571no—
236#102—1571no—
237#102—1571no—
238#102C2×D15711yescharacteristic
239C22⋊C8—1571no—
240C22⋊C8C2×D15711yescharacteristic
241#97—1571no—
242#97—1571no—
243#41—22no—
244#41—22no—
245#41—6284no—
246C2×Q8—3141no—
247C2×Q8C22×D15711yescharacteristic
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—3141no—
251C2×D4—3141no—
252C2×D4—3141no—
253C2×D4—1571no—
254C2×D4C22×D15711yescharacteristic
255C2×D4C22×D15711yescharacteristic
256C22×C4—21no—
257C22×C4—3141no—
258C22×C4—3141no—
259C22×C4—3141no—
260C22×C4—3141no—
261C22×C4—1571no—
262C22×C4—1571no—
263C22×C4C22×D15711yescharacteristic
264SD16—22no—
265SD16—3142no—
266D8—22no—
267D8—3142no—
268C2×C8—1571no—
269C2×C8—1571no—
270C2×C8C22×D15711yescharacteristic
271C2×C8C22×D15711yescharacteristic
272C4⋊C4—3141no—
273C4⋊C4—3141no—
274C4⋊C4—3141no—
275C4⋊C4—1571no—
276C4⋊C4—1571no—
277C4⋊C4—1571no—
278C4⋊C4C22×D15711yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—3141no—
281C22⋊C4—3141no—
282C22⋊C4—3141no—
283C22⋊C4—3141no—
284C22⋊C4—3141no—
285C22⋊C4—3141no—
286C42—3141no—
287C42—3141no—
288C42—3141no—
289C42—1571no—
290C42—1571no—
291C42—1571no—
292C42—1571no—
293C23—21no—
294C23—21no—
295C23—3141no—
296C23D4×D15711yescharacteristic
297Q8—21no—
298Q8—6282no—
299Q8#1457895312yescomplemented
300D4—24no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—42no—
305D4—3141no—
306D4—3141no—
307D4—6282no—
308D4—6282no—
309D4—6282no—
310D4#1457895312yescomplemented
311C2×C4—24no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—3142no—
317C2×C4—3141no—
318C2×C4—3141no—
319C2×C4—3141no—
320C2×C4—6282no—
321C2×C4—6281no—
322C2×C4—6282no—
323C2×C4—6282no—
324C2×C4—6282no—
325C2×C4—1571no—
326C2×C4—1571no—
327C2×C4—1571no—
328C2×C4—1571no—
329C2×C4D4×D15711yescharacteristic
330C2×C4C23×D15711yescharacteristic
331C8—21no—
332C8—21no—
333C8—3141no—
334C8—3141no—
335C22—21no—
336C22—22no—
337C22—21no—
338C22—21no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—6281no—
345C22—6281no—
346C22—6281no—
347C22C2×D4×D15711yescharacteristic
348C4—21no—
349C4—22no—
350C4—41no—
351C4—3141no—
352C4—3141no—
353C4—3141no—
354C4—6281no—
355C4—6281no—
356C4—6281no—
357C4—1572no—
358C4C2×#1457895311yescharacteristic
359C4C2×#1457895311yescharacteristic
360C2—41no—
361C2—42no—
362C2—81no—
363C2—12561no—
364C2#2375777911yescharacteristic
365C2AGL(1,ℤ/8)×D15711yescharacteristic
366C2#2375807011yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#449×C157C2
socleC22×C157C22×D4
hypercenter#449D157
hypocenterC157#3115
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. #23757818C2semidirect
2. #41×Dic157C2semidirect
3. #23757598C2semidirect
4. #23757594C2semidirect
5. #23757556C2semidirect
6. #23757551C2semidirect
7. #23757471C2semidirect
8. #23757466C2semidirect
9. #23757418C2semidirect
10. #23757385C2semidirect
11. #23757382C2semidirect
12. #23757376C2semidirect
13. #23757374C2semidirect
14. #23757327C2semidirect
15. #449×C157C2semidirect
135. Q8×C157#41semidirect
141. D4×C157#41semidirect
183. C157#3115semidirect
194. #449D157semidirect
299. Q8#14578953semidirect
310. D4#14578953semidirect

Maximal subgroups

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

subgroupindex
#237578182
#41×Dic1572
#237575982
#237575942
#237575562
#237575512
#237574712
#237574662
#237574182
#237573852
#237573822
#237573762
#237573742
#237573272
#449×C1572
#3115157