Finite Group Databasebeta

← #34108836

Subgroups — #34108836

Order 20096.

492 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097779C211yesmaximal, characteristic, complemented
2C2×M16×D157C211yesmaximal, characteristic, complemented
3#23757782C211yesmaximal, characteristic
4#23757352C211yesmaximal, characteristic, complemented
5#23757349C211yesmaximal, characteristic, complemented
6#23757349C211yesmaximal, characteristic, complemented
7#23757347C211yesmaximal, characteristic, complemented
8#23757345C211yesmaximal, characteristic, complemented
9#23757343C211yesmaximal, characteristic, complemented
10#23757341C211yesmaximal, characteristic, complemented
11#23757298C211yesmaximal, characteristic, complemented
12#23757238C211yesmaximal, characteristic, complemented
13#23757235C211yesmaximal, characteristic, complemented
14#23757227C211yesmaximal, characteristic, complemented
15#436×C157C211yesmaximal, characteristic, complemented
16C2×#14578954C2211yescharacteristic
17C22×C4×D157C2211yescharacteristic
18#19097779—24no—
19C2×#14578938C2211yescharacteristic
20C2×#14578937C2211yescharacteristic
21C2×#14578937C2211yescharacteristic
22C2×#14578937C2211yescharacteristic
23C2×#14578937C2211yescharacteristic
24C2×#14578937C2211yescharacteristic, complemented
25M16×D157—24no—
26C22×#11929434C2211yescharacteristic, complemented
27C2×C8×D157C2211yescharacteristic
28C2×C8×D157C2211yescharacteristic
29C2×C8×D157C2211yescharacteristic, complemented
30#19097756C411yescharacteristic
31#19097753C411yescharacteristic
32#19097749C411yescharacteristic
33#19097748C411yescharacteristic
34#19097744C411yescharacteristic
35#19097743C411yescharacteristic
36#19097741C411yescharacteristic
37#19097739C411yescharacteristic
38C4×D628C2211yescharacteristic
39C4×C22⋊D157C2211yescharacteristic
40C4×C22⋊D157C2211yescharacteristic
41#19097728C2211yescharacteristic
42C4⋊#11929434C2211yescharacteristic, complemented
43#19097700C2211yescharacteristic, complemented
44#19097700C2211yescharacteristic, complemented
45C22⋊#11929434C2211yescharacteristic
46C22⋊#11929434C2211yescharacteristic, complemented
47#19097683C2211yescharacteristic
48#19097683C2211yescharacteristic
49#19097683C2211yescharacteristic, complemented
50#19097683C2211yescharacteristic, complemented
51#19097682C2211yescharacteristic, complemented
52#19097681C2211yescharacteristic, complemented
53C8×Dic157C2211yescharacteristic, complemented
54C2×M16×C157C2211yescharacteristic
55C2×M16×C157C2211yescharacteristic, complemented
56C4×D4×C157C2211yescharacteristic
57C4⋊C8×C157C2211yescharacteristic, complemented
58C22⋊C8×C157C2211yescharacteristic
59C22⋊C8×C157C2211yescharacteristic, complemented
60C4×C8×C157C2211yescharacteristic, complemented
61C23×D157C2×C411yescharacteristic
62#14578954—22no—
63#14578954—22no—
64C2×Dic314C2×C411yescharacteristic
65C2×D628C2×C411yescharacteristic
66C2×D628C2×C411yescharacteristic
67C2×C22⋊D157C2×C411yescharacteristic
68C2×C22⋊D157C2×C411yescharacteristic
69C2×C22⋊D157C2×C411yescharacteristic
70C2×C22⋊D157C2×C411yescharacteristic
71C22×Dic157C2×C411yescharacteristic
72C2×C4×D157—22no—
73C2×C4×D157—22no—
74C2×C4×D157C2×C411yescharacteristic
75C2×C4×D157C2×C411yescharacteristic
76C2×C4×D157C2311yescharacteristic
77C2×C4×D157C2311yescharacteristic
78C2×C4×D157C2311yescharacteristic
79#14578938—22no—
80#14578937—22no—
81#14578937—22no—
82#14578937—22no—
83#14578937—22no—
84#14578937—22no—
85C2×#11929434—22no—
86C2×#11929434C2311yescharacteristic
87C2×#11929434C2311yescharacteristic
88C2×#11929434C2311yescharacteristic
89C2×#11929434C2311yescharacteristic
90C8×D157—22no—
91C8×D157—22no—
92C8×D157D414yescomplemented
93C4⋊Dic157C2×C411yescharacteristic
94#14578933C2×C411yescharacteristic
95#14578933C2×C411yescharacteristic
96#14578933C2×C411yescharacteristic
97#14578933C2×C411yescharacteristic
98C22⋊Dic157C2×C411yescharacteristic
99C22⋊Dic157C2×C411yescharacteristic
100#14578931C2×C411yescharacteristic
101#14578931C2×C411yescharacteristic
102#14578931C2×C411yescharacteristic
103#14578931C2×C411yescharacteristic
104#14578931C2×C411yescharacteristic
105#14578931C2×C411yescharacteristic
106C4×Dic157C2311yescharacteristic
107C2×D4×C157C2×C411yescharacteristic
108C22×C4×C157C2311yescharacteristic
109C22×C4×C157C2311yescharacteristic
110M16×C157—22no—
111M16×C157—22no—
112C2×C8×C157C2311yescharacteristic
113C2×C8×C157C2311yescharacteristic
114C2×C8×C157C2311yescharacteristic
115C2×C8×C157C2311yescharacteristic
116C4⋊C4×C157C2×C411yescharacteristic
117C22⋊C4×C157C2×C411yescharacteristic
118C22⋊C4×C157C2×C411yescharacteristic
119C42×C157C2311yescharacteristic
120C22×D157—22no—
121C22×D157—41no—
122C22×D157—41no—
123C22×D157C22×C411yescharacteristic
124C22×D157C22×C411yescharacteristic
125C22×D157C22×C411yescharacteristic
126Dic314—22no—
127D628—22no—
128D628—41no—
129C22⋊D157—22no—
130C22⋊D157—22no—
131C22⋊D157—41no—
132C22⋊D157—41no—
133C2×Dic157—22no—
134C2×Dic157C22×C411yescharacteristic
135C2×Dic157C22×C411yescharacteristic
136C2×Dic157C22×C411yescharacteristic
137C2×Dic157C22×C411yescharacteristic
138C2×Dic157C22×C411yescharacteristic
139C4×D157—22no—
140C4×D157—21no—
141C4×D157—21no—
142C4×D157—21no—
143C4×D157—21no—
144C4×D157—22no—
145C4×D157C2×D412yes—
146C4×D157#4112yes—
147#11929434—21no—
148#11929434—21no—
149#11929434—21no—
150#11929434C2×D412yes—
151C23×C157C22×C411yescharacteristic
152C23×C157C22×C411yescharacteristic
153D4×C157—41no—
154C2×C4×C157—21no—
155C2×C4×C157—21no—
156C2×C4×C157—21no—
157C2×C4×C157—21no—
158C2×C4×C157C22×C411yescharacteristic
159C2×C4×C157C22×C411yescharacteristic
160C2×C4×C157C22×C411yescharacteristic
161C2×C4×C157C22×C411yescharacteristic
162C2×C4×C157C2411yescharacteristic
163C8×C157—21no—
164C8×C157—21no—
165C8×C157—21no—
166C8×C157C2×D412yes—
167C22×C157—21no—
168C22×C157—21no—
169C22×C157—21no—
170C22×C157—21no—
171C22×C157—21no—
172C22×C157—21no—
173C22×C157C23×C411yescharacteristic
174C2×D157—21no—
175C2×D157—21no—
176C2×D157—21no—
177C2×D157—21no—
178C2×D157—21no—
179C2×D157—21no—
180C2×D157—21no—
181C2×D157—21no—
182C2×D157—44no—
183C2×D157C4×D412yes—
184C4×C157—21no—
185C4×C157—21no—
186C4×C157—21no—
187C4×C157—21no—
188C4×C157C22×D411yescharacteristic
189C4×C157C2×#4111yescharacteristic
190Dic157—21no—
191Dic157—21no—
192Dic157—21no—
193Dic157—21no—
194Dic157C4×D412yes—
195C2×C157—41no—
196C2×C157—41no—
197C2×C157C2×C4×D411yescharacteristic
198C2×C157#56811yescharacteristic
199C2×C157#56811yescharacteristic
200D157—22no—
201D157—41no—
202D157—41no—
203C157#293011yescharacteristic, complemented, Sylow
204#2930—1571nomaximal, Sylow
205C2×#131—1571no—
206C22×M16—1571no—
207#525—1571no—
208#436—1571no—
209#436—1571no—
210#436—1571no—
211#436D15711yescharacteristic, complemented
212#435—1571no—
213#435—1571no—
214#435—1571no—
215#435—1571no—
216#424—1571no—
217#408—1571no—
218#407—1571no—
219#405—1571no—
220C2×#41—1571no—
221C23×C4—1571no—
222#131—3144no—
223C2×M16—3144no—
224C2×M16—1571no—
225C2×M16—1571no—
226C2×M16—1571no—
227C2×M16—1571no—
228C2×M16—1571no—
229C2×M16—1571no—
230C2×M16C2×D15711yescharacteristic
231C2×M16C2×D15711yescharacteristic, complemented
232C22×C8—1571no—
233C22×C8—1571no—
234C22×C8—1571no—
235C22×C8—1571no—
236#123—1571no—
237#123—1571no—
238C22⋊Q8—1571no—
239C22⋊Q8—1571no—
240#121—1571no—
241#121—1571no—
242#120—1571no—
243#120—1571no—
244C4×D4—1571no—
245C4×D4—1571no—
246C4×D4—1571no—
247C4×D4C2×D15711yescharacteristic
248#117—1571no—
249C4⋊C8—1571no—
250C4⋊C8—1571no—
251C4⋊C8—1571no—
252C4⋊C8C2×D15711yescharacteristic, complemented
253C22⋊C8—1571no—
254C22⋊C8—1571no—
255C22⋊C8—1571no—
256C22⋊C8—1571no—
257C22⋊C8—1571no—
258C22⋊C8—1571no—
259C22⋊C8C2×D15711yescharacteristic
260C22⋊C8C2×D15711yescharacteristic, complemented
261#97—1571no—
262#97—1571no—
263C4×C8—1571no—
264C4×C8C2×D15711yescharacteristic, complemented
265C24—1571no—
266#41—3142no—
267#41—3142no—
268C2×Q8—1571no—
269C2×D4—1571no—
270C2×D4—1571no—
271C2×D4—1571no—
272C2×D4—1571no—
273C2×D4—1571no—
274C2×D4—1571no—
275C2×D4C4×D15711yescharacteristic
276C22×C4—3142no—
277C22×C4—3142no—
278C22×C4—1571no—
279C22×C4—1571no—
280C22×C4—1571no—
281C22×C4—1571no—
282C22×C4—1571no—
283C22×C4—1571no—
284C22×C4C22×D15711yescharacteristic
285C22×C4C22×D15711yescharacteristic
286M16—22no—
287M16—22no—
288M16—3142no—
289M16—3142no—
290M16—3142no—
291M16—3142no—
292M16—3142no—
293M16—3142no—
294C2×C8—3142no—
295C2×C8—3142no—
296C2×C8—3142no—
297C2×C8—1571no—
298C2×C8—1571no—
299C2×C8—1571no—
300C2×C8—1571no—
301C2×C8—1574no—
302C2×C8C22×D15711yescharacteristic
303C2×C8C22×D15711yescharacteristic
304C2×C8C22×D15711yescharacteristic
305C2×C8C22×D15711yescharacteristic
306C4⋊C4—1571no—
307C4⋊C4—1571no—
308C4⋊C4—1571no—
309C4⋊C4—1571no—
310C4⋊C4—1571no—
311C4⋊C4C4×D15711yescharacteristic
312C22⋊C4—1571no—
313C22⋊C4—1571no—
314C22⋊C4—1571no—
315C22⋊C4—1571no—
316C22⋊C4—1571no—
317C22⋊C4—1571no—
318C22⋊C4—1571no—
319C22⋊C4—1571no—
320C22⋊C4C4×D15711yescharacteristic
321C22⋊C4C4×D15711yescharacteristic
322C42—1571no—
323C42C22×D15711yescharacteristic
324C23—3142no—
325C23—6281no—
326C23—6281no—
327C23—1571no—
328C23—1571no—
329C23—1571no—
330C23C2×C4×D15711yescharacteristic
331C23C2×C4×D15711yescharacteristic
332Q8—3142no—
333D4—41no—
334D4—3142no—
335D4—3142no—
336D4—3142no—
337D4—6281no—
338D4—6281no—
339D4—6281no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—3142no—
345C2×C4—3142no—
346C2×C4—3142no—
347C2×C4—3141no—
348C2×C4—3141no—
349C2×C4—3141no—
350C2×C4—3141no—
351C2×C4—1571no—
352C2×C4—1571no—
353C2×C4—1571no—
354C2×C4—1571no—
355C2×C4—1571no—
356C2×C4—1572no—
357C2×C4—1572no—
358C2×C4C2×C4×D15711yescharacteristic
359C2×C4C2×C4×D15711yescharacteristic
360C2×C4C2×C4×D15711yescharacteristic
361C2×C4C2×C4×D15711yescharacteristic
362C2×C4C23×D15711yescharacteristic
363C8—21no—
364C8—21no—
365C8—21no—
366C8—3141no—
367C8—3141no—
368C8—3141no—
369C8—1572no—
370C8D4×D15712yes—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—3141no—
378C22—3141no—
379C22—3141no—
380C22—3141no—
381C22—3141no—
382C22—3141no—
383C22—3141no—
384C22—3141no—
385C22—6284no—
386C22—1572no—
387C22C22×C4×D15711yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—3141no—
393C4—3141no—
394C4—3141no—
395C4—3141no—
396C4—1572no—
397C4C2×D4×D15711yescharacteristic
398C4#41×D15711yescharacteristic
399C2—41no—
400C2—41no—
401C2—3142no—
402C2—6281no—
403C2—6281no—
404C2C4×D4×D15711yescharacteristic
405C2#2375803811yescharacteristic
406C2#2375803911yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D157
commutator, socleC22×C157C23×C4
Fitting#436×C157C2
prime radical#23757782C2
cosocleC2×C4×C157C24
hypercenter#436D157
hypocenterC157#2930
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C2×C4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#19097779C2semidirect
2. C2×M16×D157C2semidirect
4. #23757352C2semidirect
5. #23757349C2semidirect
6. #23757349C2semidirect
7. #23757347C2semidirect
8. #23757345C2semidirect
9. #23757343C2semidirect
10. #23757341C2semidirect
11. #23757298C2semidirect
12. #23757238C2semidirect
13. #23757235C2semidirect
14. #23757227C2semidirect
15. #436×C157C2semidirect
24. C2×#14578937C22semidirect
26. C22×#11929434C22semidirect
29. C2×C8×D157C22semidirect
42. C4⋊#11929434C22semidirect
43. #19097700C22semidirect
44. #19097700C22semidirect
46. C22⋊#11929434C22semidirect
49. #19097683C22semidirect
50. #19097683C22semidirect
51. #19097682C22semidirect
52. #19097681C22semidirect
53. C8×Dic157C22semidirect
55. C2×M16×C157C22semidirect
57. C4⋊C8×C157C22semidirect
59. C22⋊C8×C157C22semidirect
60. C4×C8×C157C22semidirect
92. C8×D157D4semidirect
203. C157#2930semidirect
211. #436D157semidirect
231. C2×M16C2×D157semidirect
252. C4⋊C8C2×D157semidirect
260. C22⋊C8C2×D157semidirect
264. C4×C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977792
C2×M16×D1572
#237577822
#237573522
#237573492
#237573472
#237573452
#237573432
#237573412
#237572982
#237572382
#237572352
#237572272
#436×C1572
#2930157