Finite Group Databasebeta

← #34110382

Subgroups — #34110382

Order 20096.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757883C211yesmaximal, characteristic, complemented
2#23757748C211yesmaximal, characteristic
3#23757615C211yesmaximal, characteristic, complemented
4#23757611C211yesmaximal, characteristic, complemented
5#23757575C211yesmaximal, characteristic, complemented
6#23757571C211yesmaximal, characteristic, complemented
7#23757476C211yesmaximal, characteristic, complemented
8#23757476C211yesmaximal, characteristic, complemented
9#23757424C211yesmaximal, characteristic
10#23757374C211yesmaximal, characteristic, complemented
11#23757370C211yesmaximal, characteristic, complemented
12#23757364C211yesmaximal, characteristic, complemented
13SD16×Dic157C211yesmaximal, characteristic, complemented
14#23757331C211yesmaximal, characteristic, complemented
15#450×C157C211yesmaximal, characteristic
16C2×#14578942C2211yescharacteristic
17C2×#14578942C2211yescharacteristic
18#19097769C2211yescharacteristic, complemented
19#19097764—21no—
20#19097764—21no—
21#19097762—21no—
22#19097762C2211yescharacteristic, complemented
23#19097761—21no—
24#19097750C2211yescharacteristic
25Q8×Dic157C2211yescharacteristic, complemented
26#19097736C2211yescharacteristic, complemented
27D4×Dic157—21no—
28D4×Dic157—21no—
29D4×Dic157C2211yescharacteristic
30D4×Dic157C2211yescharacteristic
31D4×Dic157C2211yescharacteristic
32D4×Dic157C2211yescharacteristic
33#19097730C2211yescharacteristic
34#19097730C2211yescharacteristic
35C2×C22⋊Dic157—21no—
36C2×C22⋊Dic157—21no—
37#19097705C2211yescharacteristic, complemented
38#19097704C2211yescharacteristic, complemented
39#19097703C2211yescharacteristic, complemented
40#19097702C2211yescharacteristic, complemented
41#19097700C2211yescharacteristic, complemented
42#19097700C2211yescharacteristic, complemented
43#19097696C2211yescharacteristic, complemented
44#19097696C2211yescharacteristic, complemented
45#19097695C2211yescharacteristic, complemented
46#19097695C2211yescharacteristic, complemented
47#19097692C2211yescharacteristic
48#19097692C2211yescharacteristic
49C22⋊#11929434C2211yescharacteristic
50#19097681C2211yescharacteristic, complemented
51C8×Dic157C2211yescharacteristic, complemented
52C22×D4×C157C2211yescharacteristic
53C2×SD16×C157C2211yescharacteristic
54C2×SD16×C157C2211yescharacteristic
55C22⋊Q8×C157C2211yescharacteristic
56#102×C157C2211yescharacteristic
57#102×C157C2211yescharacteristic
58C22⋊C8×C157C2211yescharacteristic
59C2×Dic314C2311yescharacteristic
60C22×Dic157—21no—
61C22×Dic157—21no—
62C22×Dic157—21no—
63C22×Dic157—21no—
64#14578942—22no—
65#14578942—22no—
66C2×#11929434C2311yescharacteristic
67C2×#11929434C2311yescharacteristic
68C4⋊Dic157—21no—
69C4⋊Dic157—21no—
70C4⋊Dic157D411yescharacteristic
71C4⋊Dic157D411yescharacteristic
72C4⋊Dic157C2311yescharacteristic
73C4⋊Dic157C2311yescharacteristic
74C4⋊Dic157C2311yescharacteristic
75#14578933—21no—
76#14578933—21no—
77#14578933—21no—
78#14578933—21no—
79#14578933—21no—
80C22⋊Dic157—21no—
81C22⋊Dic157—21no—
82C22⋊Dic157—21no—
83C22⋊Dic157—21no—
84C22⋊Dic157—21no—
85C22⋊Dic157—21no—
86C22⋊Dic157D411yescharacteristic
87C22⋊Dic157D411yescharacteristic
88C4×Dic157—21no—
89C4×Dic157—21no—
90C4×Dic157—21no—
91C4×Dic157C2311yescharacteristic
92C4×Dic157C2311yescharacteristic
93C24×C157—21no—
94C2×Q8×C157C2311yescharacteristic
95C2×D4×C157—22no—
96C2×D4×C157—22no—
97C2×D4×C157—21no—
98C2×D4×C157C2311yescharacteristic
99C2×D4×C157C2311yescharacteristic
100C22×C4×C157C2311yescharacteristic
101SD16×C157—22no—
102SD16×C157—22no—
103C2×C8×C157C2311yescharacteristic
104C2×C8×C157C2311yescharacteristic
105C4⋊C4×C157—21no—
106C4⋊C4×C157C2311yescharacteristic
107C22⋊C4×C157—21no—
108Dic314—21no—
109Dic314—21no—
110C2×Dic157—21no—
111C2×Dic157—21no—
112C2×Dic157—21no—
113C2×Dic157—21no—
114C2×Dic157—81no—
115C2×Dic157—81no—
116C2×Dic157—81no—
117C2×Dic157—81no—
118C2×Dic157C2×D411yescharacteristic
119C2×Dic157C2×D411yescharacteristic
120C2×Dic157C2×D411yescharacteristic
121C2×Dic157C2×D411yescharacteristic
122#11929434—21no—
123#11929434—21no—
124C23×C157—24no—
125C23×C157—21no—
126C23×C157—21no—
127C23×C157—81no—
128C23×C157—81no—
129C23×C157C2×D411yescharacteristic
130Q8×C157—21no—
131Q8×C157—21no—
132D4×C157—24no—
133D4×C157—21no—
134D4×C157—21no—
135D4×C157#4112yes—
136D4×C157#4112yes—
137C2×C4×C157—21no—
138C2×C4×C157—21no—
139C2×C4×C157—21no—
140C2×C4×C157—21no—
141C2×C4×C157C2×D411yescharacteristic
142C2×C4×C157C2411yescharacteristic
143C8×C157—21no—
144C8×C157—21no—
145C22×C157—21no—
146C22×C157—22no—
147C22×C157—22no—
148C22×C157—21no—
149C22×C157—21no—
150C22×C157—41no—
151C22×C157—41no—
152C22×C157—41no—
153C22×C157—41no—
154C22×C157—84no—
155C22×C157C22×D411yescharacteristic
156C4×C157—21no—
157C4×C157—41no—
158C4×C157—41no—
159C4×C157C2×#4111yescharacteristic
160C4×C157C2×#4111yescharacteristic
161Dic157—21no—
162Dic157—21no—
163Dic157—21no—
164Dic157—21no—
165Dic157—41no—
166Dic157—41no—
167Dic157—41no—
168Dic157—41no—
169C2×C157—41no—
170C2×C157—42no—
171C2×C157—42no—
172C2×C157#52511yescharacteristic
173C2×C157#57711yescharacteristic
174C2×C157#57711yescharacteristic
175C157#312311yescharacteristic, complemented, Sylow
176#3123—1571nomaximal, Sylow
177#541—1571no—
178#518—1571no—
179#485—1571no—
180#485—1571no—
181#478—1571no—
182#478—1571no—
183#461—1571no—
184#461—1571no—
185#450—1571no—
186#450D15711yescharacteristic
187#440—1571no—
188#440—1571no—
189C4×SD16—1571no—
190C4×SD16—1571no—
191#433—1571no—
192C22×D4C2×D15711yescharacteristic
193C2×SD16—1571no—
194C2×SD16—1571no—
195C2×SD16C2×D15711yescharacteristic
196C2×SD16C2×D15711yescharacteristic
197C4⋊Q8—1571no—
198#126—3141no—
199#126—3141no—
200#125—3141no—
201#125—3141no—
202#125—1571no—
203C22⋊Q8—1571no—
204C22⋊Q8C2×D15711yescharacteristic
205C4×Q8—1571no—
206C4×Q8—1571no—
207C4×D4—3141no—
208C4×D4—3141no—
209C4×D4—1571no—
210C4×D4—1571no—
211C4×D4—1571no—
212C4×D4—1571no—
213#117—1571no—
214#117—1571no—
215C2×C22⋊C4—3141no—
216C2×C22⋊C4—3141no—
217#107—1571no—
218#107—1571no—
219#106—1571no—
220#106—1571no—
221C4⋊C8—1571no—
222C4⋊C8—1571no—
223#103—1571no—
224#103—1571no—
225#103—1571no—
226#103—1571no—
227#102—1571no—
228#102—1571no—
229#102C2×D15711yescharacteristic
230#102C2×D15711yescharacteristic
231C22⋊C8—1571no—
232C22⋊C8C2×D15711yescharacteristic
233#97—1571no—
234C4×C8—1571no—
235C24—21no—
236C2×Q8—1571no—
237C2×Q8C22×D15711yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—21no—
241C2×D4C22×D15711yescharacteristic
242C2×D4C22×D15711yescharacteristic
243C22×C4—3141no—
244C22×C4—3141no—
245C22×C4—3141no—
246C22×C4—3141no—
247C22×C4C22×D15711yescharacteristic
248SD16—22no—
249SD16—22no—
250SD16—3142no—
251SD16—3142no—
252C2×C8—1571no—
253C2×C8—1571no—
254C2×C8C22×D15711yescharacteristic
255C2×C8C22×D15711yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—3141no—
258C4⋊C4—3141no—
259C4⋊C4—3141no—
260C4⋊C4—3141no—
261C4⋊C4—3141no—
262C4⋊C4—3141no—
263C4⋊C4—3141no—
264C4⋊C4—1571no—
265C4⋊C4—1571no—
266C4⋊C4—1571no—
267C4⋊C4—1571no—
268C4⋊C4—1571no—
269C4⋊C4C22×D15711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—3141no—
272C22⋊C4—3141no—
273C22⋊C4—3141no—
274C22⋊C4—3141no—
275C22⋊C4—3141no—
276C22⋊C4—3141no—
277C22⋊C4—1571no—
278C22⋊C4—1571no—
279C42—3141no—
280C42—3141no—
281C42—3141no—
282C42—1571no—
283C42—1571no—
284C23—24no—
285C23—21no—
286C23—21no—
287C23—81no—
288C23—81no—
289C23D4×D15711yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—3141no—
293Q8—3141no—
294D4—24no—
295D4—21no—
296D4—21no—
297D4#1457895312yes—
298D4#1457895312yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—3141no—
304C2×C4—3141no—
305C2×C4—3141no—
306C2×C4—3141no—
307C2×C4—1571no—
308C2×C4—1571no—
309C2×C4—1571no—
310C2×C4—1571no—
311C2×C4—12561no—
312C2×C4—12561no—
313C2×C4—12561no—
314C2×C4—12561no—
315C2×C4D4×D15711yescharacteristic
316C2×C4C23×D15711yescharacteristic
317C8—21no—
318C8—21no—
319C8—3141no—
320C8—3141no—
321C22—21no—
322C22—21no—
323C22—22no—
324C22—22no—
325C22—21no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—84no—
331C22C2×D4×D15711yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—3141no—
336C4—3141no—
337C4—3141no—
338C4—3141no—
339C4—6281no—
340C4—6281no—
341C4—6281no—
342C4—6281no—
343C4C2×#1457895311yescharacteristic
344C4C2×#1457895311yescharacteristic
345C2—41no—
346C2—42no—
347C2—42no—
348C2#2375777911yescharacteristic
349C2#2375807911yescharacteristic
350C2#2375808311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#450×C157C2
socleC22×C157C22×D4
prime radicalC22×D4×C157C22
hypercenter#450D157
hypocenterC157#3123
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #23757883C2semidirect
3. #23757615C2semidirect
4. #23757611C2semidirect
5. #23757575C2semidirect
6. #23757571C2semidirect
7. #23757476C2semidirect
8. #23757476C2semidirect
10. #23757374C2semidirect
11. #23757370C2semidirect
12. #23757364C2semidirect
13. SD16×Dic157C2semidirect
14. #23757331C2semidirect
18. #19097769C22semidirect
22. #19097762C22semidirect
25. Q8×Dic157C22semidirect
26. #19097736C22semidirect
37. #19097705C22semidirect
38. #19097704C22semidirect
39. #19097703C22semidirect
40. #19097702C22semidirect
41. #19097700C22semidirect
42. #19097700C22semidirect
43. #19097696C22semidirect
44. #19097696C22semidirect
45. #19097695C22semidirect
46. #19097695C22semidirect
50. #19097681C22semidirect
51. C8×Dic157C22semidirect
175. C157#3123semidirect

Maximal subgroups

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

subgroupindex
#237578832
#237577482
#237576152
#237576112
#237575752
#237575712
#237574762
#237574242
#237573742
#237573702
#237573642
SD16×Dic1572
#237573312
#450×C1572
#3123157