Finite Group Databasebeta

← #34109486

Subgroups — #34109486

Order 20096.

724 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23758108C211yesmaximal, characteristic, complemented
2#23758072C212yesmaximal, complemented
3#23757822C211yesmaximal, characteristic, complemented
4#23757454C212yesmaximal, complemented
5#23757450C211yesmaximal, characteristic, complemented
6#23757447C212yesmaximal, complemented
7#23757443C211yesmaximal, characteristic, complemented
8#23757441C211yesmaximal, characteristic, complemented
9#104×D157C212yesmaximal, complemented
10#23757254C211yesmaximal, characteristic, complemented
11#455×C157C211yesmaximal, characteristic, complemented
12#19097818—22no—
13#19097818C2212yes—
14#19097816—22no—
15#19097816—22no—
16#19097816C2211yescharacteristic
17C2×#14578955C2211yescharacteristic, complemented
18C2×#14578953—22no—
19C2×#14578953—22no—
20C2×#14578953C2211yescharacteristic
21#41×D157—22no—
22#41×D157C2212yes—
23#19097803—22no—
24#19097802C2212yescomplemented
25#19097799C2212yescomplemented
26#19097798—22no—
27#19097796C2212yescomplemented
28#19097794—22no—
29#19097792—22no—
30M16×D157C2212yescomplemented
31#19097769C2211yescharacteristic, complemented
32#19097767C2211yescharacteristic, complemented
33#19097759C2211yescharacteristic, complemented
34#19097744—22no—
35#19097731—22no—
36C42×D157C2211yescharacteristic, complemented
37#19097699C2212yescomplemented
38#19097698C2212yescomplemented
39#19097697C2212yescomplemented
40#19097689C2212yescomplemented
41#19097688C2211yescharacteristic, complemented
42H(5,𝔽2)×C157C2211yescharacteristic
43#137×C157C2212yescomplemented
44#124×C157C2211yescharacteristic, complemented
45#104×C157C2212yescomplemented
46#100×C157C2211yescharacteristic, complemented
47#14578955—22no—
48#14578955—22no—
49#14578955D412yescomplemented
50#14578954—24no—
51#14578954—22no—
52#14578954—22no—
53#14578954C2312yes—
54#14578953—24no—
55#14578953—24no—
56#14578953—24no—
57#14578953—22no—
58#14578953—22no—
59#14578953—24no—
60#14578953—22no—
61#14578953—22no—
62#14578953—22no—
63#14578953D412yescomplemented
64C2×Dic314—22no—
65C2×Dic314—22no—
66C2×Dic314C2311yescharacteristic
67Q8×D157—22no—
68Q8×D157D412yescomplemented
69C2×D628—21no—
70C2×D628C2311yescharacteristic
71C2×C22⋊D157—24no—
72C2×C22⋊D157—22no—
73C2×C22⋊D157—22no—
74C2×C22⋊D157—22no—
75C2×C22⋊D157D412yes—
76D4×D157—24no—
77D4×D157—22no—
78D4×D157—22no—
79D4×D157D412yescomplemented
80C22×Dic157—22no—
81C22×Dic157D412yes—
82C2×C4×D157—22no—
83C2×C4×D157—22no—
84C2×C4×D157—21no—
85C2×C4×D157—21no—
86C2×C4×D157C2311yescharacteristic
87#14578944—22no—
88#14578943—22no—
89#14578941—22no—
90#14578941—22no—
91D1256—22no—
92#14578939—22no—
93#14578938C2312yes—
94#14578937—22no—
95C8×D157—22no—
96#14578933—22no—
97#14578931—22no—
98C4×Dic157—21no—
99C4×Dic157C2311yescharacteristic
100#41×C157—22no—
101#41×C157C2312yes—
102C2×Q8×C157C2311yescharacteristic
103C2×D4×C157—22no—
104C2×D4×C157—22no—
105C2×D4×C157C2311yescharacteristic
106Q16×C157—22no—
107SD16×C157—22no—
108M16×C157C2312yes—
109C22⋊C4×C157—22no—
110C42×C157C2311yescharacteristic
111C22×D157—22no—
112C22×D157—22no—
113C22×D157—21no—
114C22×D157C2×D411yescharacteristic
115Dic314—24no—
116Dic314—22no—
117Dic314—22no—
118Dic314C2×D412yes—
119D628—22no—
120D628—22no—
121D628—42no—
122D628C2×D412yes—
123C22⋊D157—24no—
124C22⋊D157—24no—
125C22⋊D157—22no—
126C22⋊D157—22no—
127C22⋊D157—24no—
128C22⋊D157—22no—
129C22⋊D157—44no—
130C2×Dic157—22no—
131C2×Dic157—24no—
132C2×Dic157—21no—
133C2×Dic157—22no—
134C2×Dic157—22no—
135C2×Dic157—22no—
136C2×Dic157C2×D412yes—
137C2×Dic157C2×D411yescharacteristic
138C4×D157—24no—
139C4×D157—22no—
140C4×D157—22no—
141C4×D157—22no—
142C4×D157—42no—
143C4×D157—41no—
144C4×D157—41no—
145C4×D157C2×D412yes—
146C4×D157C2×D412yes—
147#11929434—22no—
148C23×C157—22no—
149C23×C157C2×D412yes—
150Q8×C157—22no—
151Q8×C157C2×D412yes—
152D4×C157—24no—
153D4×C157—22no—
154D4×C157—22no—
155D4×C157C2×D412yes—
156C2×C4×C157—22no—
157C2×C4×C157—22no—
158C2×C4×C157—21no—
159C2×C4×C157—21no—
160C2×C4×C157C2411yescharacteristic
161C8×C157—22no—
162C22×C157—22no—
163C22×C157—22no—
164C22×C157—42no—
165C22×C157—82no—
166C22×C157C22×D411yescharacteristic
167C2×D157—22no—
168C2×D157—41no—
169C2×D157—42no—
170C2×D157—41no—
171C2×D157—82no—
172C2×D157—82no—
173C2×D157#12012yes—
174C4×C157—22no—
175C4×C157—41no—
176C4×C157—41no—
177C4×C157C22×D412yes—
178Dic157—22no—
179Dic157—22no—
180Dic157—22no—
181Dic157#12011yescharacteristic
182Dic157#12011yescharacteristic
183C2×C157—21no—
184C2×C157—42no—
185C2×C157—42no—
186C2×C157C2×#12011yescharacteristic
187D157—22no—
188D157—42no—
189D157—81no—
190C157#301811yescharacteristic, complemented, Sylow
191#3018—1571nomaximal, Sylow
192#585—1571no—
193#575—1572no—
194#532—1571no—
195#456—1572no—
196#455—1571no—
197#455D15711yescharacteristic, complemented
198#454—1572no—
199#453—1571no—
200#453—1571no—
201C2×#104—1572no—
202#413—1571no—
2032−1+4—3142no—
2042−1+4—1572no—
205H(5,𝔽2)—3142no—
206H(5,𝔽2)—3142no—
207H(5,𝔽2)—1571no—
208H(5,𝔽2)C2×D15711yescharacteristic
209C2×#41—3142no—
210C2×#41—3142no—
211C2×#41—3142no—
212C2×#41—1572no—
213C2×#41—1571no—
214C2×#41—1571no—
215#137—3142no—
216#137—1572no—
217#137C2×D15712yescomplemented
218AGL(1,ℤ/8)—3142no—
219AGL(1,ℤ/8)—1572no—
220AGL(1,ℤ/8)—1572no—
221#135—3142no—
222#135—3142no—
223C2×M16—1572no—
224C4⋊Q8—1571no—
225#127—1571no—
226#124—1571no—
227#124C2×D15711yescharacteristic, complemented
228#121—3142no—
229C4×D4—3142no—
230C2×C42—1571no—
231#104—1572no—
232#104—1572no—
233#104—1572no—
234#104C2×D15712yescomplemented
235#101—1572no—
236#100—1571no—
237#100C2×D15711yescharacteristic, complemented
238#41—22no—
239#41—3144no—
240#41—3142no—
241#41—3142no—
242#41—3144no—
243#41—3144no—
244#41—3144no—
245#41—3142no—
246#41—3142no—
247#41—3142no—
248#41—3142no—
249#41—3144no—
250#41—3142no—
251#41—3142no—
252#41—3142no—
253#41—1572no—
254#41—1572no—
255#41—1572no—
256#41C22×D15712yes—
257C2×Q8—3142no—
258C2×Q8—3142no—
259C2×Q8—3142no—
260C2×Q8—1572no—
261C2×Q8—1571no—
262C2×Q8C22×D15711yescharacteristic
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—3141no—
266C2×D4—3144no—
267C2×D4—3142no—
268C2×D4—3142no—
269C2×D4—3144no—
270C2×D4—3142no—
271C2×D4—3142no—
272C2×D4—3142no—
273C2×D4—1572no—
274C2×D4—1572no—
275C2×D4—1571no—
276C2×D4C22×D15711yescharacteristic
277C22×C4—3142no—
278C22×C4—3142no—
279C22×C4—3141no—
280C22×C4—3141no—
281C22×C4—3142no—
282C22×C4—1572no—
283C22×C4—1571no—
284Q16—22no—
285Q16—3142no—
286SD16—22no—
287SD16—3142no—
288SD16—3142no—
289SD16—3142no—
290D8—3142no—
291D8—3142no—
292M16—3142no—
293M16—1572no—
294M16C22×D15712yes—
295C2×C8—3142no—
296C4⋊C4—3142no—
297C22⋊C4—22no—
298C22⋊C4—3142no—
299C42—3141no—
300C42—1571no—
301C42C22×D15711yescharacteristic
302C23—22no—
303C23—3142no—
304C23—3142no—
305C23—3141no—
306C23—1571no—
307C23D4×D15712yes—
308Q8—22no—
309Q8—3144no—
310Q8—3142no—
311Q8—3142no—
312Q8—1572no—
313Q8D4×D15712yes—
314D4—24no—
315D4—22no—
316D4—22no—
317D4—3142no—
318D4—3144no—
319D4—3144no—
320D4—3142no—
321D4—3144no—
322D4—3142no—
323D4—3142no—
324D4—3142no—
325D4—6282no—
326D4—6284no—
327D4—1572no—
328D4D4×D15712yes—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—3144no—
334C2×C4—3142no—
335C2×C4—3142no—
336C2×C4—3142no—
337C2×C4—3144no—
338C2×C4—3141no—
339C2×C4—3142no—
340C2×C4—3142no—
341C2×C4—3142no—
342C2×C4—3142no—
343C2×C4—6282no—
344C2×C4—6281no—
345C2×C4—6281no—
346C2×C4—1572no—
347C2×C4—1571no—
348C2×C4—1572no—
349C2×C4—1572no—
350C2×C4C23×D15711yescharacteristic
351C8—22no—
352C8—3142no—
353C22—22no—
354C22—22no—
355C22—42no—
356C22—82no—
357C22—3142no—
358C22—6281no—
359C22—6282no—
360C22—6281no—
361C22—1572no—
362C22—12562no—
363C22—12562no—
364C22C2×D4×D15711yescharacteristic
365C4—22no—
366C4—41no—
367C4—41no—
368C4—3142no—
369C4—3142no—
370C4—3142no—
371C4—1571no—
372C4—1571no—
373C4C2×D4×D15712yes—
374C2—21no—
375C2—42no—
376C2—42no—
377C2—3142no—
378C2—6282no—
379C2—12561no—
380C2#120×D15711yescharacteristic
subgroupG/H
centerC2#120×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#455×C157C2
socleC2×C157C2×#120
hypercenter#455D157
hypocenterC157#3018
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #23758108C2semidirect
2. #23758072C2semidirect
3. #23757822C2semidirect
4. #23757454C2semidirect
5. #23757450C2semidirect
6. #23757447C2semidirect
7. #23757443C2semidirect
8. #23757441C2semidirect
9. #104×D157C2semidirect
10. #23757254C2semidirect
11. #455×C157C2semidirect
17. C2×#14578955C22semidirect
24. #19097802C22semidirect
25. #19097799C22semidirect
27. #19097796C22semidirect
30. M16×D157C22semidirect
31. #19097769C22semidirect
32. #19097767C22semidirect
33. #19097759C22semidirect
36. C42×D157C22semidirect
37. #19097699C22semidirect
38. #19097698C22semidirect
39. #19097697C22semidirect
40. #19097689C22semidirect
41. #19097688C22semidirect
43. #137×C157C22semidirect
44. #124×C157C22semidirect
45. #104×C157C22semidirect
46. #100×C157C22semidirect
49. #14578955D4semidirect
63. #14578953D4semidirect
68. Q8×D157D4semidirect
79. D4×D157D4semidirect
190. C157#3018semidirect
197. #455D157semidirect
217. #137C2×D157semidirect
227. #124C2×D157semidirect
234. #104C2×D157semidirect
237. #100C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237581082
#237580722
#237578222
#237574542
#237574502
#237574472
#237574432
#237574412
#104×D1572
#237572542
#455×C1572
#3018157