Finite Group Databasebeta

← #34109179

Subgroups — #34109179

Order 20096.

412 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757780C211yesmaximal, characteristic
2#23757349C211yesmaximal, characteristic, complemented
3#23757348C211yesmaximal, characteristic, complemented
4#23757347C211yesmaximal, characteristic, complemented
5#23757344C211yesmaximal, characteristic, complemented
6#23757342C211yesmaximal, characteristic, complemented
7#23757341C211yesmaximal, characteristic, complemented
8C8×C22⋊D157C211yesmaximal, characteristic, complemented
9#23757336C211yesmaximal, characteristic, complemented
10#23757331C211yesmaximal, characteristic, complemented
11#23757331C211yesmaximal, characteristic, complemented
12#23757240C211yesmaximal, characteristic, complemented
13#23757235C211yesmaximal, characteristic, complemented
14C22⋊C8×D157C211yesmaximal, characteristic, complemented
15#408×C157C211yesmaximal, characteristic, complemented
16C22×C4×D157C2211yescharacteristic
17C2×#14578938C2211yescharacteristic, complemented
18C2×#14578937C2211yescharacteristic
19C2×#14578937C2211yescharacteristic
20C22×#11929434C2211yescharacteristic, complemented
21C2×C8×D157C2211yescharacteristic
22C2×C8×D157C2211yescharacteristic
23#19097752C411yescharacteristic
24#19097752C411yescharacteristic
25#19097747C411yescharacteristic
26#19097747C411yescharacteristic
27#19097742C411yescharacteristic
28#19097742C411yescharacteristic
29#19097739C411yescharacteristic
30#19097739C411yescharacteristic
31C4×C22⋊D157C2211yescharacteristic
32C4×C22⋊D157C2211yescharacteristic
33C4×C22⋊D157C2211yescharacteristic
34C4×C22⋊D157C2211yescharacteristic
35#19097730C2211yescharacteristic
36#19097700C2211yescharacteristic, complemented
37#19097700C2211yescharacteristic, complemented
38#19097700C2211yescharacteristic, complemented
39#19097700C2211yescharacteristic, complemented
40C22⋊#11929434C2211yescharacteristic
41C22⋊#11929434C2211yescharacteristic
42C22⋊#11929434C2211yescharacteristic, complemented
43C22⋊#11929434C2211yescharacteristic, complemented
44#19097683C2211yescharacteristic
45#19097683C2211yescharacteristic
46#19097683C2211yescharacteristic
47#19097683C2211yescharacteristic
48#19097681C2211yescharacteristic, complemented
49#19097681C2211yescharacteristic, complemented
50C8×Dic157C2211yescharacteristic, complemented
51C8×Dic157C2211yescharacteristic, complemented
52C2×#41×C157C2211yescharacteristic
53C2×M16×C157C2211yescharacteristic, complemented
54C22×C8×C157C2211yescharacteristic, complemented
55C22⋊C8×C157C2211yescharacteristic
56C22⋊C8×C157C2211yescharacteristic
57C22⋊C8×C157C2211yescharacteristic, complemented
58C22⋊C8×C157C2211yescharacteristic, complemented
59C23×D157C2×C411yescharacteristic
60C2×C22⋊D157C2×C411yescharacteristic
61C2×C22⋊D157C2×C411yescharacteristic
62C2×C22⋊D157C2×C411yescharacteristic
63C2×C22⋊D157C2×C411yescharacteristic
64C22×Dic157C2×C411yescharacteristic
65C2×C4×D157—41no—
66C2×C4×D157—41no—
67C2×C4×D157C2×C411yescharacteristic
68C2×C4×D157C2×C411yescharacteristic
69C2×C4×D157C2311yescharacteristic
70C2×C4×D157C2311yescharacteristic
71#14578938—41no—
72#14578937—41no—
73#14578937—41no—
74C2×#11929434—41no—
75C2×#11929434C2311yescharacteristic
76C2×#11929434C2311yescharacteristic
77C2×#11929434C2311yescharacteristic
78C2×#11929434C2311yescharacteristic
79C8×D157—41no—
80C8×D157—41no—
81C4⋊Dic157C2×C411yescharacteristic
82C4⋊Dic157C2×C411yescharacteristic
83#14578933C2×C411yescharacteristic
84#14578933C2×C411yescharacteristic
85#14578933C2×C411yescharacteristic
86#14578933C2×C411yescharacteristic
87C22⋊Dic157C2×C411yescharacteristic
88C22⋊Dic157C2×C411yescharacteristic
89C22⋊Dic157C2×C411yescharacteristic
90C22⋊Dic157C2×C411yescharacteristic
91C22⋊Dic157C2×C411yescharacteristic
92C22⋊Dic157C2×C411yescharacteristic
93#14578931C2×C411yescharacteristic
94#14578931C2×C411yescharacteristic
95#14578931C2×C411yescharacteristic
96#14578931C2×C411yescharacteristic
97C4×Dic157C2311yescharacteristic
98C4×Dic157C2311yescharacteristic
99#41×C157—41no—
100#41×C157—41no—
101C2×Q8×C157C2×C411yescharacteristic
102C2×D4×C157C2×C411yescharacteristic
103C2×D4×C157C2×C411yescharacteristic
104C2×D4×C157C2×C411yescharacteristic
105C22×C4×C157C2311yescharacteristic
106C22×C4×C157C2311yescharacteristic
107C22×C4×C157C2311yescharacteristic
108M16×C157—41no—
109C2×C8×C157—41no—
110C2×C8×C157C2311yescharacteristic
111C2×C8×C157C2311yescharacteristic
112C2×C8×C157C2311yescharacteristic
113C2×C8×C157C2311yescharacteristic
114C22×D157—22no—
115C22×D157—41no—
116C22×D157—41no—
117C22×D157C22×C411yescharacteristic
118C22×D157C22×C411yescharacteristic
119C22⋊D157—22no—
120C22⋊D157—22no—
121C22⋊D157—41no—
122C22⋊D157—41no—
123C2×Dic157—41no—
124C2×Dic157C22×C411yescharacteristic
125C2×Dic157C22×C411yescharacteristic
126C2×Dic157C22×C411yescharacteristic
127C2×Dic157C22×C411yescharacteristic
128C2×Dic157C22×C411yescharacteristic
129C2×Dic157C22×C411yescharacteristic
130C4×D157—21no—
131C4×D157—21no—
132C4×D157—21no—
133C4×D157—21no—
134C4×D157—41no—
135C4×D157—41no—
136#11929434—21no—
137#11929434—21no—
138#11929434—21no—
139#11929434—21no—
140C23×C157C22×C411yescharacteristic
141C23×C157C22×C411yescharacteristic
142C23×C157C22×C411yescharacteristic
143Q8×C157—41no—
144D4×C157—41no—
145D4×C157—41no—
146D4×C157—41no—
147C2×C4×C157—21no—
148C2×C4×C157—21no—
149C2×C4×C157—21no—
150C2×C4×C157—21no—
151C2×C4×C157—21no—
152C2×C4×C157—21no—
153C2×C4×C157C22×C411yescharacteristic
154C2×C4×C157C22×C411yescharacteristic
155C2×C4×C157C22×C411yescharacteristic
156C2×C4×C157C2411yescharacteristic
157C8×C157—21no—
158C8×C157—21no—
159C8×C157—21no—
160C8×C157—21no—
161C22×C157—21no—
162C22×C157—21no—
163C22×C157—21no—
164C22×C157—21no—
165C22×C157—21no—
166C22×C157—21no—
167C22×C157—21no—
168C22×C157—21no—
169C22×C157—21no—
170C22×C157C23×C411yescharacteristic
171C2×D157—21no—
172C2×D157—21no—
173C2×D157—21no—
174C2×D157—21no—
175C2×D157—21no—
176C2×D157—44no—
177C2×D157#13112yes—
178C4×C157—21no—
179C4×C157—21no—
180C4×C157—21no—
181C4×C157H(5,𝔽2)11yescharacteristic
182C4×C157H(5,𝔽2)11yescharacteristic
183Dic157—21no—
184Dic157—21no—
185Dic157—21no—
186Dic157—21no—
187Dic157—21no—
188Dic157—21no—
189C2×C157—41no—
190C2×C157—41no—
191C2×C157—41no—
192C2×C157#51811yescharacteristic
193C2×C157C2×#13111yescharacteristic
194C2×C157#56811yescharacteristic
195D157—22no—
196D157—41no—
197C157#297611yescharacteristic, complemented, Sylow
198#2976—1571nomaximal, Sylow
199#525—1571no—
200#436—1571no—
201#436—1571no—
202#435—1571no—
203#435—1571no—
204#435—1571no—
205#435—1571no—
206C8×D4—1571no—
207C8×D4—1571no—
208#433—1571no—
209#433—1571no—
210#408—1571no—
211#408D15711yescharacteristic, complemented
212#407—1571no—
213C2×C22⋊C8—1571no—
214C2×#41C2×D15711yescharacteristic
215C23×C4—1571no—
216C2×M16—1571no—
217C2×M16—1571no—
218C2×M16—1571no—
219C2×M16C2×D15711yescharacteristic, complemented
220C22×C8—1571no—
221C22×C8—1571no—
222C22×C8—1571no—
223C22×C8C2×D15711yescharacteristic, complemented
224#123—1571no—
225#123—1571no—
226C22⋊Q8—1571no—
227C22⋊Q8—1571no—
228#121—1571no—
229#121—1571no—
230#120—1571no—
231#120—1571no—
232C4×D4—1571no—
233C4×D4—1571no—
234C4×D4—1571no—
235C4×D4—1571no—
236#117—1571no—
237C4⋊C8—1571no—
238C4⋊C8—1571no—
239C4⋊C8—1571no—
240C4⋊C8—1571no—
241C22⋊C8—1571no—
242C22⋊C8—1571no—
243C22⋊C8—1571no—
244C22⋊C8—1571no—
245C22⋊C8—1571no—
246C22⋊C8—1571no—
247C22⋊C8—1571no—
248C22⋊C8—1571no—
249C22⋊C8C2×D15711yescharacteristic
250C22⋊C8C2×D15711yescharacteristic
251C22⋊C8C2×D15711yescharacteristic, complemented
252C22⋊C8C2×D15711yescharacteristic, complemented
253#97—1571no—
254#97—1571no—
255C4×C8—1571no—
256C4×C8—1571no—
257C24—1571no—
258#41—41no—
259#41—41no—
260C2×Q8C4×D15711yescharacteristic
261C2×D4—1571no—
262C2×D4—1571no—
263C2×D4—1571no—
264C2×D4—1571no—
265C2×D4C4×D15711yescharacteristic
266C2×D4C4×D15711yescharacteristic
267C2×D4C4×D15711yescharacteristic
268C22×C4—6281no—
269C22×C4—6281no—
270C22×C4—1571no—
271C22×C4—1571no—
272C22×C4—1571no—
273C22×C4—1571no—
274C22×C4—1571no—
275C22×C4C22×D15711yescharacteristic
276C22×C4C22×D15711yescharacteristic
277C22×C4C22×D15711yescharacteristic
278M16—41no—
279M16—6281no—
280M16—6281no—
281M16—6281no—
282C2×C8—41no—
283C2×C8—6281no—
284C2×C8—6281no—
285C2×C8—6281no—
286C2×C8—1571no—
287C2×C8—1571no—
288C2×C8—1571no—
289C2×C8—1571no—
290C2×C8C22×D15711yescharacteristic
291C2×C8C22×D15711yescharacteristic
292C2×C8C22×D15711yescharacteristic
293C2×C8C22×D15711yescharacteristic
294C4⋊C4—1571no—
295C4⋊C4—1571no—
296C4⋊C4—1571no—
297C4⋊C4—1571no—
298C4⋊C4—1571no—
299C4⋊C4—1571no—
300C22⋊C4—1571no—
301C22⋊C4—1571no—
302C22⋊C4—1571no—
303C22⋊C4—1571no—
304C22⋊C4—1571no—
305C22⋊C4—1571no—
306C22⋊C4—1571no—
307C22⋊C4—1571no—
308C22⋊C4—1571no—
309C22⋊C4—1571no—
310C42—1571no—
311C42—1571no—
312C23—3142no—
313C23—6281no—
314C23—6281no—
315C23—1571no—
316C23—1571no—
317C23C2×C4×D15711yescharacteristic
318C23C2×C4×D15711yescharacteristic
319C23C2×C4×D15711yescharacteristic
320Q8—41no—
321D4—41no—
322D4—41no—
323D4—41no—
324D4—3142no—
325D4—3142no—
326D4—6281no—
327D4—6281no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—3141no—
335C2×C4—3141no—
336C2×C4—3141no—
337C2×C4—3141no—
338C2×C4—6281no—
339C2×C4—6281no—
340C2×C4—6281no—
341C2×C4—1571no—
342C2×C4—1571no—
343C2×C4—1571no—
344C2×C4—1571no—
345C2×C4—1571no—
346C2×C4—1571no—
347C2×C4C2×C4×D15711yescharacteristic
348C2×C4C2×C4×D15711yescharacteristic
349C2×C4C2×C4×D15711yescharacteristic
350C2×C4C23×D15711yescharacteristic
351C8—21no—
352C8—21no—
353C8—21no—
354C8—21no—
355C8—3141no—
356C8—3141no—
357C8—3141no—
358C8—3141no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—3141no—
369C22—3141no—
370C22—3141no—
371C22—3141no—
372C22—3141no—
373C22—6284no—
374C22—1572no—
375C22C22×C4×D15711yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—3141no—
380C4—3141no—
381C4—3141no—
382C4—3141no—
383C4—3141no—
384C4—3141no—
385C4#1909781611yescharacteristic
386C4#1909781611yescharacteristic
387C2—41no—
388C2—41no—
389C2—41no—
390C2—3142no—
391C2—6281no—
392C2#2375774611yescharacteristic
393C2#131×D15711yescharacteristic
394C2#2375803811yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D157
commutator, socleC22×C157C23×C4
Fitting#408×C157C2
prime radical#23757780C2
cosocleC2×C4×C157C24
hypercenter#408D157
hypocenterC157#2976
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C2×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #23757349C2semidirect
3. #23757348C2semidirect
4. #23757347C2semidirect
5. #23757344C2semidirect
6. #23757342C2semidirect
7. #23757341C2semidirect
8. C8×C22⋊D157C2semidirect
9. #23757336C2semidirect
10. #23757331C2semidirect
11. #23757331C2semidirect
12. #23757240C2semidirect
13. #23757235C2semidirect
14. C22⋊C8×D157C2semidirect
15. #408×C157C2semidirect
17. C2×#14578938C22semidirect
20. C22×#11929434C22semidirect
36. #19097700C22semidirect
37. #19097700C22semidirect
38. #19097700C22semidirect
39. #19097700C22semidirect
42. C22⋊#11929434C22semidirect
43. C22⋊#11929434C22semidirect
48. #19097681C22semidirect
49. #19097681C22semidirect
50. C8×Dic157C22semidirect
51. C8×Dic157C22semidirect
53. C2×M16×C157C22semidirect
54. C22×C8×C157C22semidirect
57. C22⋊C8×C157C22semidirect
58. C22⋊C8×C157C22semidirect
197. C157#2976semidirect
211. #408D157semidirect
219. C2×M16C2×D157semidirect
223. C22×C8C2×D157semidirect
251. C22⋊C8C2×D157semidirect
252. C22⋊C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237577802
#237573492
#237573482
#237573472
#237573442
#237573422
#237573412
C8×C22⋊D1572
#237573362
#237573312
#237572402
#237572352
C22⋊C8×D1572
#408×C1572
#2976157