Finite Group Databasebeta

← #34110321

Subgroups — #34110321

Order 20096.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757896C211yesmaximal, characteristic, complemented
2#23757748C211yesmaximal, characteristic, complemented
3#23757611C211yesmaximal, characteristic, complemented
4#23757598C211yesmaximal, characteristic, complemented
5#23757561C211yesmaximal, characteristic, complemented
6#23757551C211yesmaximal, characteristic, complemented
7#23757490C211yesmaximal, characteristic, complemented
8#23757490C211yesmaximal, characteristic, complemented
9#23757407C211yesmaximal, characteristic
10#23757382C211yesmaximal, characteristic, complemented
11#23757374C211yesmaximal, characteristic, complemented
12SD16×Dic157C211yesmaximal, characteristic, complemented
13#23757353C211yesmaximal, characteristic, complemented
14#23757331C211yesmaximal, characteristic, complemented
15#450×C157C211yesmaximal, characteristic, complemented
16C2×#14578939C2211yescharacteristic
17C2×#14578939C2211yescharacteristic
18#19097770C2211yescharacteristic, complemented
19#19097766—21no—
20#19097760C2211yescharacteristic, complemented
21#19097757—21no—
22#19097753—21no—
23#19097751—21no—
24#19097745C2211yescharacteristic
25Q8×Dic157C2211yescharacteristic, complemented
26D4×Dic157—21no—
27D4×Dic157—21no—
28D4×Dic157C2211yescharacteristic
29D4×Dic157C2211yescharacteristic
30D4×Dic157C2211yescharacteristic
31D4×Dic157C2211yescharacteristic
32#19097733C2211yescharacteristic, complemented
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#19097695C2211yescharacteristic, complemented
44#19097695C2211yescharacteristic, complemented
45#19097693C2211yescharacteristic, complemented
46#19097693C2211yescharacteristic, 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×Dic314—21no—
60C2×D628C2311yescharacteristic
61C2×C22⋊D157—21no—
62C22×Dic157—21no—
63C22×Dic157—21no—
64C22×Dic157—21no—
65C22×Dic157—21no—
66C2×C4×D157—21no—
67#14578939—22no—
68#14578939—22no—
69C2×#11929434C2311yescharacteristic
70C2×#11929434C2311yescharacteristic
71C4⋊Dic157—21no—
72C4⋊Dic157D411yescharacteristic
73C4⋊Dic157D411yescharacteristic
74C4⋊Dic157C2311yescharacteristic
75C4⋊Dic157C2311yescharacteristic
76C4⋊Dic157C2311yescharacteristic
77#14578933—21no—
78C22⋊Dic157—21no—
79C22⋊Dic157—21no—
80C22⋊Dic157—21no—
81C22⋊Dic157—21no—
82C22⋊Dic157—21no—
83C22⋊Dic157D411yescharacteristic
84C22⋊Dic157D411yescharacteristic
85#14578931—21no—
86#14578931—21no—
87#14578931—21no—
88#14578931—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—
108C22×D157—21no—
109Dic314—81no—
110D628—21no—
111D628—21no—
112C22⋊D157—81no—
113C2×Dic157—21no—
114C2×Dic157—21no—
115C2×Dic157—21no—
116C2×Dic157—81no—
117C2×Dic157—81no—
118C2×Dic157—81no—
119C2×Dic157—81no—
120C2×Dic157C2×D411yescharacteristic
121C2×Dic157C2×D411yescharacteristic
122C2×Dic157C2×D411yescharacteristic
123C2×Dic157C2×D411yescharacteristic
124C4×D157—81no—
125#11929434—21no—
126#11929434—21no—
127C23×C157—24no—
128C23×C157—21no—
129C23×C157—21no—
130C23×C157—81no—
131C23×C157—81no—
132C23×C157C2×D411yescharacteristic
133Q8×C157—21no—
134Q8×C157—21no—
135D4×C157—24no—
136D4×C157—21no—
137D4×C157—21no—
138D4×C157#4112yes—
139D4×C157#4112yes—
140C2×C4×C157—21no—
141C2×C4×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157C2×D411yescharacteristic
145C2×C4×C157C2411yescharacteristic
146C8×C157—21no—
147C8×C157—21no—
148C22×C157—21no—
149C22×C157—22no—
150C22×C157—22no—
151C22×C157—21no—
152C22×C157—21no—
153C22×C157—41no—
154C22×C157—41no—
155C22×C157—41no—
156C22×C157—41no—
157C22×C157—84no—
158C22×C157C22×D411yescharacteristic
159C2×D157—41no—
160C2×D157—41no—
161C2×D157—41no—
162C4×C157—21no—
163C4×C157—41no—
164C4×C157—41no—
165C4×C157C2×#4111yescharacteristic
166C4×C157C2×#4111yescharacteristic
167Dic157—21no—
168Dic157—21no—
169Dic157—21no—
170Dic157—21no—
171Dic157—41no—
172Dic157—41no—
173Dic157—41no—
174C2×C157—41no—
175C2×C157—42no—
176C2×C157—42no—
177C2×C157#52511yescharacteristic
178C2×C157#57611yescharacteristic
179C2×C157#57711yescharacteristic
180D157—81no—
181C157#311811yescharacteristic, complemented, Sylow
182#3118—1571nomaximal, Sylow
183#542—1571no—
184#518—1571no—
185#485—1571no—
186#482—1571no—
187#476—1571no—
188#474—1571no—
189#463—1571no—
190#463—1571no—
191#450D15711yescharacteristic, complemented
192#447—1571no—
193#442—1571no—
194#440—1571no—
195C4×SD16—1571no—
196C4×D8—1571no—
197#433—1571no—
198C22×D4C2×D15711yescharacteristic
199C2×SD16C2×D15711yescharacteristic
200C2×SD16C2×D15711yescharacteristic
201C2×D8—1571no—
202C2×D8—1571no—
203C4⋊Q8—1571no—
204#126—3141no—
205#124—3141no—
206#124—1571no—
207#123—3141no—
208C22⋊Q8—3141no—
209C22⋊Q8C2×D15711yescharacteristic
210#121—1571no—
211C4×Q8—1571no—
212C4×D4—3141no—
213C4×D4—3141no—
214C4×D4—1571no—
215C4×D4—1571no—
216C4×D4—1571no—
217C4×D4—1571no—
218C4×D4—1571no—
219#117—1571no—
220#117—1571no—
221C2×C22⋊C4—3141no—
222C2×C22⋊C4—3141no—
223#107—1571no—
224#107—1571no—
225#106—1571no—
226#106—1571no—
227C4⋊C8—1571no—
228C4⋊C8—1571no—
229#103—1571no—
230#103—1571no—
231#102—1571no—
232#102—1571no—
233#102—1571no—
234#102—1571no—
235#102C2×D15711yescharacteristic
236#102C2×D15711yescharacteristic
237C22⋊C8—1571no—
238C22⋊C8C2×D15711yescharacteristic
239#97—1571no—
240C4×C8—1571no—
241C24—21no—
242C2×Q8—3141no—
243C2×Q8C22×D15711yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—3141no—
248C2×D4—1571no—
249C2×D4C22×D15711yescharacteristic
250C2×D4C22×D15711yescharacteristic
251C22×C4—3141no—
252C22×C4—3141no—
253C22×C4—3141no—
254C22×C4—3141no—
255C22×C4—3141no—
256C22×C4C22×D15711yescharacteristic
257SD16—22no—
258SD16—22no—
259D8—3142no—
260D8—3142no—
261C2×C8—1571no—
262C2×C8—1571no—
263C2×C8C22×D15711yescharacteristic
264C2×C8C22×D15711yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—3141no—
267C4⋊C4—3141no—
268C4⋊C4—1571no—
269C4⋊C4—1571no—
270C4⋊C4—1571no—
271C4⋊C4—1571no—
272C4⋊C4—1571no—
273C4⋊C4C22×D15711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—3141no—
276C22⋊C4—3141no—
277C22⋊C4—3141no—
278C22⋊C4—3141no—
279C22⋊C4—3141no—
280C22⋊C4—3141no—
281C22⋊C4—3141no—
282C22⋊C4—3141no—
283C22⋊C4—3141no—
284C22⋊C4—1571no—
285C22⋊C4—1571no—
286C42—3141no—
287C42—3141no—
288C42—1571no—
289C42—1571no—
290C23—24no—
291C23—21no—
292C23—21no—
293C23—81no—
294C23—81no—
295C23—3141no—
296C23D4×D15711yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—12561no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—3141no—
304D4—3141no—
305D4—12561no—
306D4#1457895312yes—
307D4#1457895312yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—3141no—
313C2×C4—3141no—
314C2×C4—3141no—
315C2×C4—1571no—
316C2×C4—1571no—
317C2×C4—1571no—
318C2×C4—1571no—
319C2×C4—12561no—
320C2×C4—12561no—
321C2×C4—12561no—
322C2×C4—12561no—
323C2×C4—12561no—
324C2×C4D4×D15711yescharacteristic
325C2×C4C23×D15711yescharacteristic
326C8—21no—
327C8—21no—
328C8—3141no—
329C8—3141no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—84no—
340C22—6281no—
341C22—6281no—
342C22—6281no—
343C22C2×D4×D15711yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—3141no—
348C4—3141no—
349C4—3141no—
350C4—3141no—
351C4—6281no—
352C4—6281no—
353C4—6281no—
354C4C2×#1457895311yescharacteristic
355C4C2×#1457895311yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—12561no—
360C2#2375777911yescharacteristic
361C2#2375807711yescharacteristic
362C2#2375807911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#450×C157C2
socleC22×C157C22×D4
prime radical#23757407C2
hypercenter#450D157
hypocenterC157#3118
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. #23757896C2semidirect
2. #23757748C2semidirect
3. #23757611C2semidirect
4. #23757598C2semidirect
5. #23757561C2semidirect
6. #23757551C2semidirect
7. #23757490C2semidirect
8. #23757490C2semidirect
10. #23757382C2semidirect
11. #23757374C2semidirect
12. SD16×Dic157C2semidirect
13. #23757353C2semidirect
14. #23757331C2semidirect
15. #450×C157C2semidirect
18. #19097770C22semidirect
20. #19097760C22semidirect
25. Q8×Dic157C22semidirect
32. #19097733C22semidirect
37. #19097705C22semidirect
38. #19097704C22semidirect
39. #19097703C22semidirect
40. #19097702C22semidirect
41. #19097700C22semidirect
42. #19097700C22semidirect
43. #19097695C22semidirect
44. #19097695C22semidirect
45. #19097693C22semidirect
46. #19097693C22semidirect
50. #19097681C22semidirect
51. C8×Dic157C22semidirect
181. C157#3118semidirect
191. #450D157semidirect

Maximal subgroups

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

subgroupindex
#237578962
#237577482
#237576112
#237575982
#237575612
#237575512
#237574902
#237574072
#237573822
#237573742
SD16×Dic1572
#237573532
#237573312
#450×C1572
#3118157