Finite Group Databasebeta

← #34110283

Subgroups — #34110283

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#23757820C211yesmaximal, characteristic, complemented
2C4×#14578953C211yesmaximal, characteristic, complemented
3#23757599C211yesmaximal, characteristic, complemented
4#23757595C211yesmaximal, characteristic, complemented
5#23757557C211yesmaximal, characteristic, complemented
6#23757551C211yesmaximal, characteristic, complemented
7#23757472C211yesmaximal, characteristic, complemented
8#23757466C211yesmaximal, characteristic, complemented
9#23757419C211yesmaximal, characteristic, complemented
10#23757415C211yesmaximal, characteristic, complemented
11#23757383C211yesmaximal, characteristic, complemented
12#23757375C211yesmaximal, characteristic, complemented
13#23757373C211yesmaximal, characteristic, complemented
14#23757328C211yesmaximal, characteristic, complemented
15#442×C157C211yesmaximal, characteristic, complemented
16C2×#14578954—21no—
17C2×#14578953C2211yescharacteristic
18C2×#14578943C2211yescharacteristic
19C2×#14578942C2211yescharacteristic
20C2×#14578939C2211yescharacteristic
21#19097770C2211yescharacteristic
22#19097767C2211yescharacteristic
23#19097757—21no—
24#19097746C2211yescharacteristic
25#19097744—21no—
26#19097742C2211yescharacteristic
27C4×Dic314C2211yescharacteristic
28#19097736C2211yescharacteristic
29C4×D628C2211yescharacteristic
30D4×Dic157C2211yescharacteristic
31C4×C22⋊D157—21no—
32C4×C22⋊D157—21no—
33#19097731—21no—
34#19097729C2211yescharacteristic
35#19097727—21no—
36C2×C4×Dic157—21no—
37C42×D157C2211yescharacteristic
38#19097705C2211yescharacteristic
39#19097704C2211yescharacteristic
40#19097702C2211yescharacteristic
41#19097700C2211yescharacteristic
42#19097700C2211yescharacteristic
43#19097696C2211yescharacteristic
44#19097694C2211yescharacteristic
45#19097693C2211yescharacteristic
46#19097692C2211yescharacteristic
47#19097692C2211yescharacteristic
48#19097691C2211yescharacteristic
49#19097683C2211yescharacteristic
50#19097683C2211yescharacteristic
51#19097682C2211yescharacteristic
52C2×D8×C157C2211yescharacteristic
53C4×D4×C157C2211yescharacteristic
54C4×D4×C157C2211yescharacteristic
55#106×C157C2211yescharacteristic
56#102×C157C2211yescharacteristic
57#102×C157C2211yescharacteristic
58#97×C157C2211yescharacteristic
59#14578954—44no—
60#14578953—22no—
61#14578953—22no—
62C2×Dic314—21no—
63C2×Dic314C2311yescharacteristic
64C2×D628—21no—
65C2×D628C2311yescharacteristic
66C2×C22⋊D157—21no—
67C2×C22⋊D157—21no—
68C2×C22⋊D157—21no—
69C2×C22⋊D157—21no—
70C22×Dic157—21no—
71C22×Dic157—21no—
72C2×C4×D157—21no—
73C2×C4×D157D411yescharacteristic
74C2×C4×D157D411yescharacteristic
75C2×C4×D157C2311yescharacteristic
76#14578943—22no—
77#14578942—22no—
78#14578939—22no—
79C2×#11929434C2311yescharacteristic
80C2×#11929434C2311yescharacteristic
81C4⋊Dic157C2311yescharacteristic
82C4⋊Dic157C2311yescharacteristic
83#14578933—21no—
84#14578933—21no—
85#14578933—21no—
86C22⋊Dic157—21no—
87C22⋊Dic157—21no—
88#14578931—21no—
89#14578931—21no—
90#14578931—21no—
91C4×Dic157—21no—
92C4×Dic157—21no—
93C4×Dic157—21no—
94C4×Dic157D411yescharacteristic
95C4×Dic157D411yescharacteristic
96C4×Dic157C2311yescharacteristic
97C2×D4×C157C2311yescharacteristic
98C2×D4×C157C2311yescharacteristic
99C22×C4×C157—21no—
100C22×C4×C157—21no—
101D8×C157—22no—
102C2×C8×C157C2311yescharacteristic
103C2×C8×C157C2311yescharacteristic
104C4⋊C4×C157C2311yescharacteristic
105C4⋊C4×C157C2311yescharacteristic
106C22⋊C4×C157—21no—
107C22⋊C4×C157—21no—
108C42×C157C2311yescharacteristic
109C22×D157—21no—
110C22×D157C2×D411yescharacteristic
111Dic314—21no—
112Dic314—42no—
113Dic314#4112yescomplemented
114D628—21no—
115D628—21no—
116D628—42no—
117C22⋊D157—24no—
118C22⋊D157—42no—
119C22⋊D157—42no—
120C22⋊D157—42no—
121C2×Dic157—24no—
122C2×Dic157—21no—
123C2×Dic157—21no—
124C2×Dic157—21no—
125C2×Dic157—42no—
126C2×Dic157C2×D411yescharacteristic
127C2×Dic157C2×D411yescharacteristic
128C2×Dic157C2×D411yescharacteristic
129C4×D157—21no—
130C4×D157—21no—
131C4×D157—22no—
132C4×D157—41no—
133C4×D157—42no—
134#11929434—21no—
135#11929434—21no—
136C23×C157—21no—
137C23×C157—21no—
138D4×C157—21no—
139D4×C157—21no—
140D4×C157—21no—
141D4×C157#4112yescomplemented
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157—42no—
145C2×C4×C157—42no—
146C2×C4×C157C2×D411yescharacteristic
147C2×C4×C157C2×D411yescharacteristic
148C2×C4×C157C2411yescharacteristic
149C8×C157—21no—
150C8×C157—21no—
151C22×C157—22no—
152C22×C157—41no—
153C22×C157—41no—
154C22×C157—41no—
155C22×C157—41no—
156C22×C157—41no—
157C22×C157C22×D411yescharacteristic
158C2×D157—21no—
159C2×D157—21no—
160C2×D157—21no—
161C2×D157—41no—
162C2×D157—41no—
163C2×D157—41no—
164C4×C157—21no—
165C4×C157—21no—
166C4×C157—41no—
167C4×C157—41no—
168C4×C157C2×#4111yescharacteristic
169C4×C157C2×#4111yescharacteristic
170Dic157—21no—
171Dic157—22no—
172Dic157—21no—
173Dic157—41no—
174Dic157—41no—
175Dic157AGL(1,ℤ/8)12yes—
176C2×C157—42no—
177C2×C157—81no—
178C2×C157#52511yescharacteristic
179C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
180C2×C157#57511yescharacteristic
181D157—41no—
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#449—1571no—
195#442—1571no—
196#442D15711yescharacteristic, complemented
197#440—1571no—
198#440—1571no—
199#432—1571no—
200C2×#41—3141no—
201C2×#41—1571no—
202C2×SD16—1571no—
203C2×SD16—1571no—
204C2×D8—1571no—
205C2×D8C2×D15711yescharacteristic
206C4⋊Q8—1571no—
207#127—1571no—
208#124—3141no—
209#121—3141no—
210#121—1571no—
211#121—1571no—
212C4×Q8—1571no—
213C4×Q8—1571no—
214C4×D4—3141no—
215C4×D4—3141no—
216C4×D4—3141no—
217C4×D4—1571no—
218C4×D4—1571no—
219C4×D4C2×D15711yescharacteristic
220C4×D4C2×D15711yescharacteristic
221#117—3141no—
222#117—1571no—
223C2×C42—3141no—
224C2×C42—1571no—
225#107—1571no—
226#107—1571no—
227#106—1571no—
228#106C2×D15711yescharacteristic
229C4⋊C8—1571no—
230C4⋊C8—1571no—
231#103—1571no—
232#103—1571no—
233#102—1571no—
234#102—1571no—
235#102—1571no—
236#102—1571no—
237#102C2×D15711yescharacteristic
238#102C2×D15711yescharacteristic
239C22⋊C8—1571no—
240C22⋊C8—1571no—
241#97—1571no—
242#97C2×D15711yescharacteristic
243#41—3142no—
244#41—3142no—
245#41—6284no—
246C2×Q8—3141no—
247C2×Q8—1571no—
248C2×D4—3141no—
249C2×D4—3141no—
250C2×D4—3141no—
251C2×D4—3141no—
252C2×D4—3141no—
253C2×D4—1571no—
254C2×D4C22×D15711yescharacteristic
255C2×D4C22×D15711yescharacteristic
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—3141no—
259C22×C4—3141no—
260C22×C4—3141no—
261C22×C4—1571no—
262C22×C4—1571no—
263C22×C4—1571no—
264SD16—3142no—
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⋊C4C22×D15711yescharacteristic
278C4⋊C4C22×D15711yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—21no—
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—
292C42C22×D15711yescharacteristic
293C23—21no—
294C23—21no—
295C23—3141no—
296C23—1571no—
297Q8—3141no—
298Q8—6282no—
299Q8—1572no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—3144no—
304D4—3141no—
305D4—3141no—
306D4—6282no—
307D4—6282no—
308D4—6282no—
309D4—6282no—
310D4#1457895412yescomplemented
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—3142no—
316C2×C4—3144no—
317C2×C4—3141no—
318C2×C4—3141no—
319C2×C4—3141no—
320C2×C4—3141no—
321C2×C4—3141no—
322C2×C4—6281no—
323C2×C4—6282no—
324C2×C4—6282no—
325C2×C4—1571no—
326C2×C4—1571no—
327C2×C4—1571no—
328C2×C4D4×D15711yescharacteristic
329C2×C4D4×D15711yescharacteristic
330C2×C4C23×D15711yescharacteristic
331C8—21no—
332C8—21no—
333C8—3141no—
334C8—3141no—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—3141no—
342C22—3141no—
343C22—3141no—
344C22—6281no—
345C22—6281no—
346C22—6281no—
347C22C2×D4×D15711yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—3141no—
353C4—3142no—
354C4—3141no—
355C4—6281no—
356C4—6281no—
357C4—1572no—
358C4#41×D15711yescharacteristic
359C4C2×#1457895411yescharacteristic
360C2—42no—
361C2—81no—
362C2—6281no—
363C2—12561no—
364C2#2375778211yescharacteristic
365C2AGL(1,ℤ/8)×D15711yescharacteristic
366C2#2375806911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#442×C157C2
socleC22×C157C22×D4
hypercenter#442D157
hypocenterC157#3115
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. #23757820C2semidirect
2. C4×#14578953C2semidirect
3. #23757599C2semidirect
4. #23757595C2semidirect
5. #23757557C2semidirect
6. #23757551C2semidirect
7. #23757472C2semidirect
8. #23757466C2semidirect
9. #23757419C2semidirect
10. #23757415C2semidirect
11. #23757383C2semidirect
12. #23757375C2semidirect
13. #23757373C2semidirect
14. #23757328C2semidirect
15. #442×C157C2semidirect
113. Dic314#41semidirect
141. D4×C157#41semidirect
183. C157#3115semidirect
196. #442D157semidirect
310. D4#14578954semidirect

Maximal subgroups

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

subgroupindex
#237578202
C4×#145789532
#237575992
#237575952
#237575572
#237575512
#237574722
#237574662
#237574192
#237574152
#237573832
#237573752
#237573732
#237573282
#442×C1572
#3115157