Finite Group Databasebeta

← #34109504

Subgroups — #34109504

Order 20096.

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

normal subgroups only

This group has 736 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
2AGL(1,ℤ/8)×D157C212yesmaximal, complemented
3#23757836C211yesmaximal, characteristic, complemented
4#23757447C212yesmaximal, complemented
5#23757444C212yesmaximal, complemented
6#23757443C211yesmaximal, characteristic, complemented
7#23757441C211yesmaximal, characteristic, complemented
8#23757440C211yesmaximal, characteristic, complemented
9#23757288C212yesmaximal, complemented
10#23757254C211yesmaximal, characteristic, complemented
11#453×C157C211yesmaximal, characteristic, complemented
12#19097818—22no—
13#19097818C2212yescomplemented
14#19097816—22no—
15#19097816—22no—
16#19097816C2211yescharacteristic, complemented
17C2×#14578953—22no—
18C2×#14578953—22no—
19C2×#14578953C2211yescharacteristic
20#41×D157—22no—
21#41×D157C2212yes—
22C2×D4×D157C2211yescharacteristic, complemented
23#19097799C2212yescomplemented
24#19097798—22no—
25#19097797C2212yescomplemented
26#19097796C2212yescomplemented
27#19097795—22no—
28SD16×D157—22no—
29D8×D157—22no—
30M16×D157C2212yescomplemented
31#19097767C2211yescharacteristic, complemented
32#19097759C2211yescharacteristic, complemented
33#19097758C2211yescharacteristic, complemented
34#19097743—22no—
35#19097739—22no—
36#19097728C2211yescharacteristic, complemented
37#19097699C2212yescomplemented
38#19097698C2212yescomplemented
39#19097697C2212yescomplemented
40#19097689C2212yescomplemented
41#19097688C2211yescharacteristic, complemented
42H(5,𝔽2)×C157C2211yescharacteristic, complemented
43AGL(1,ℤ/8)×C157C2212yescomplemented
44#127×C157C2211yescharacteristic, complemented
45#104×C157C2212yescomplemented
46#100×C157C2211yescharacteristic, complemented
47C23×D157—21no—
48#14578955D412yescomplemented
49#14578954—24no—
50#14578954—22no—
51#14578954—22no—
52#14578954C2312yes—
53#14578953—24no—
54#14578953—24no—
55#14578953—24no—
56#14578953—22no—
57#14578953—22no—
58#14578953—24no—
59#14578953—22no—
60#14578953—22no—
61#14578953—22no—
62#14578953D412yescomplemented
63C2×Dic314—22no—
64C2×Dic314—22no—
65C2×Dic314C2311yescharacteristic
66Q8×D157—22no—
67Q8×D157D412yescomplemented
68C2×D628C2311yescharacteristic
69C2×C22⋊D157—24no—
70C2×C22⋊D157—22no—
71C2×C22⋊D157—21no—
72C2×C22⋊D157—22no—
73C2×C22⋊D157—22no—
74C2×C22⋊D157D412yes—
75D4×D157—24no—
76D4×D157—22no—
77D4×D157—22no—
78D4×D157—22no—
79D4×D157—22no—
80D4×D157D412yescomplemented
81C22×Dic157—22no—
82C22×Dic157D412yes—
83C2×C4×D157—22no—
84C2×C4×D157—22no—
85C2×C4×D157C2311yescharacteristic
86#14578943—22no—
87#14578942—22no—
88#14578941—22no—
89D1256—22no—
90#14578939—22no—
91#14578939—22no—
92#14578938C2312yes—
93#14578937—22no—
94C8×D157—22no—
95#14578933—21no—
96C22⋊Dic157—22no—
97#14578931—21no—
98#14578931—22no—
99C4×Dic157C2311yescharacteristic
100#41×C157—22no—
101#41×C157C2312yes—
102C2×D4×C157—22no—
103C2×D4×C157—22no—
104C2×D4×C157—22no—
105C2×D4×C157C2311yescharacteristic
106C2×D4×C157C2311yescharacteristic
107SD16×C157—22no—
108D8×C157—22no—
109M16×C157C2312yes—
110C42×C157C2311yescharacteristic
111C22×D157—22no—
112C22×D157—22no—
113C22×D157—21no—
114C22×D157—42no—
115C22×D157—82no—
116C22×D157C2×D411yescharacteristic
117Dic314—24no—
118Dic314—22no—
119Dic314—22no—
120Dic314C2×D412yes—
121D628—22no—
122D628—22no—
123D628C2×D412yes—
124C22⋊D157—24no—
125C22⋊D157—24no—
126C22⋊D157—22no—
127C22⋊D157—22no—
128C22⋊D157—24no—
129C22⋊D157—22no—
130C22⋊D157—41no—
131C22⋊D157—41no—
132C22⋊D157—82no—
133C2×Dic157—22no—
134C2×Dic157—24no—
135C2×Dic157—22no—
136C2×Dic157—22no—
137C2×Dic157—22no—
138C2×Dic157—21no—
139C2×Dic157C2×D412yes—
140C2×Dic157C2×D411yescharacteristic
141C4×D157—24no—
142C4×D157—22no—
143C4×D157—22no—
144C4×D157—22no—
145C4×D157C2×D412yes—
146C4×D157C2×D412yes—
147#11929434—22no—
148C23×C157—22no—
149C23×C157—21no—
150C23×C157C2×D412yes—
151Q8×C157C2×D412yes—
152D4×C157—24no—
153D4×C157—22no—
154D4×C157—22no—
155D4×C157—22no—
156D4×C157—82no—
157D4×C157C2×D412yes—
158C2×C4×C157—22no—
159C2×C4×C157—22no—
160C2×C4×C157—21no—
161C2×C4×C157C2411yescharacteristic
162C8×C157—22no—
163C22×C157—22no—
164C22×C157—22no—
165C22×C157—42no—
166C22×C157—42no—
167C22×C157—41no—
168C22×C157—82no—
169C22×C157C22×D411yescharacteristic
170C2×D157—22no—
171C2×D157—41no—
172C2×D157—42no—
173C2×D157—41no—
174C2×D157—82no—
175C2×D157—84no—
176C2×D157—82no—
177C2×D157#12012yes—
178C4×C157—22no—
179C4×C157—41no—
180C4×C157C22×D412yes—
181Dic157—22no—
182Dic157—22no—
183Dic157—41no—
184Dic157#12011yescharacteristic
185Dic157#12011yescharacteristic
186C2×C157—21no—
187C2×C157—42no—
188C2×C157—42no—
189C2×C157—81no—
190C2×C157C2×#12011yescharacteristic
191D157—22no—
192D157—42no—
193D157—81no—
194C157#302111yescharacteristic, complemented, Sylow
195#3021—1571nomaximal, Sylow
196#585—1571no—
197C2×AGL(1,ℤ/8)—1572no—
198#535—1571no—
199#454—1572no—
200#454—1572no—
201#453—1571no—
202#453—1571no—
203#453—1571no—
204#453D15711yescharacteristic, complemented
205#421—1572no—
206#413—1571no—
2072−1+4—3142no—
2082−1+4—1572no—
209H(5,𝔽2)—3142no—
210H(5,𝔽2)—3142no—
211H(5,𝔽2)—1571no—
212H(5,𝔽2)C2×D15711yescharacteristic, complemented
213C2×#41—3142no—
214C2×#41—3142no—
215C2×#41—3142no—
216C2×#41—1572no—
217C2×#41—1571no—
218C22×D4—1571no—
219AGL(1,ℤ/8)—3142no—
220AGL(1,ℤ/8)—3142no—
221AGL(1,ℤ/8)—1572no—
222AGL(1,ℤ/8)—1572no—
223AGL(1,ℤ/8)—1572no—
224AGL(1,ℤ/8)C2×D15712yescomplemented
225C2×SD16—3142no—
226C2×D8—3142no—
227C2×M16—1572no—
228#127—1571no—
229#127C2×D15711yescharacteristic, complemented
230#124—1571no—
231#124—1571no—
232#121—3142no—
233#120—3142no—
234#117—1571no—
235#104—1572no—
236#104—1572no—
237#104—1572no—
238#104C2×D15712yescomplemented
239#101—1572no—
240#100—1571no—
241#100C2×D15711yescharacteristic, complemented
242C24—3141no—
243#41—22no—
244#41—3144no—
245#41—3142no—
246#41—3142no—
247#41—3144no—
248#41—3144no—
249#41—3144no—
250#41—3142no—
251#41—3142no—
252#41—3142no—
253#41—3142no—
254#41—3144no—
255#41—3142no—
256#41—1572no—
257#41—1572no—
258#41—1572no—
259#41C22×D15712yes—
260C2×Q8—3142no—
261C2×Q8—3142no—
262C2×Q8—3142no—
263C2×Q8—1572no—
264C2×Q8—1571no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—3144no—
269C2×D4—3142no—
270C2×D4—3142no—
271C2×D4—3144no—
272C2×D4—3142no—
273C2×D4—3142no—
274C2×D4—3142no—
275C2×D4—3142no—
276C2×D4—3142no—
277C2×D4—3141no—
278C2×D4—1572no—
279C2×D4—1572no—
280C2×D4—1571no—
281C2×D4C22×D15711yescharacteristic
282C2×D4C22×D15711yescharacteristic
283C22×C4—3142no—
284C22×C4—3142no—
285C22×C4—3142no—
286C22×C4—1572no—
287C22×C4—1571no—
288SD16—22no—
289SD16—3142no—
290SD16—3142no—
291SD16—3142no—
292D8—22no—
293D8—3142no—
294D8—3142no—
295D8—3142no—
296M16—3142no—
297M16—1572no—
298M16C22×D15712yes—
299C2×C8—3142no—
300C4⋊C4—3141no—
301C22⋊C4—3141no—
302C22⋊C4—3142no—
303C22⋊C4—3142no—
304C42—1571no—
305C42C22×D15711yescharacteristic
306C23—22no—
307C23—21no—
308C23—3142no—
309C23—3142no—
310C23—3141no—
311C23—6282no—
312C23—1571no—
313C23—12562no—
314C23D4×D15712yes—
315Q8—3144no—
316Q8—3142no—
317Q8—3142no—
318Q8—1572no—
319Q8D4×D15712yes—
320D4—24no—
321D4—22no—
322D4—22no—
323D4—22no—
324D4—82no—
325D4—3142no—
326D4—3144no—
327D4—3144no—
328D4—3142no—
329D4—3144no—
330D4—3142no—
331D4—3142no—
332D4—3142no—
333D4—6281no—
334D4—6281no—
335D4—1572no—
336D4—12562no—
337D4D4×D15712yes—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—3144no—
342C2×C4—3142no—
343C2×C4—3142no—
344C2×C4—3142no—
345C2×C4—3144no—
346C2×C4—3142no—
347C2×C4—3142no—
348C2×C4—3142no—
349C2×C4—3142no—
350C2×C4—3141no—
351C2×C4—1572no—
352C2×C4—1571no—
353C2×C4—1572no—
354C2×C4—1572no—
355C2×C4C23×D15711yescharacteristic
356C8—22no—
357C8—3142no—
358C22—22no—
359C22—22no—
360C22—42no—
361C22—42no—
362C22—41no—
363C22—82no—
364C22—3142no—
365C22—6281no—
366C22—6282no—
367C22—6281no—
368C22—1572no—
369C22—12562no—
370C22—12562no—
371C22—12564no—
372C22C2×D4×D15711yescharacteristic
373C4—22no—
374C4—41no—
375C4—3142no—
376C4—3142no—
377C4—6281no—
378C4—1571no—
379C4—1571no—
380C4C2×D4×D15712yes—
381C2—21no—
382C2—42no—
383C2—42no—
384C2—81no—
385C2—3142no—
386C2—6282no—
387C2—12561no—
388C2#120×D15711yescharacteristic
subgroupG/H
centerC2#120×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#453×C157C2
socleC2×C157C2×#120
hypercenter#453D157
hypocenterC157#3021
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #23758108C2semidirect
2. AGL(1,ℤ/8)×D157C2semidirect
3. #23757836C2semidirect
4. #23757447C2semidirect
5. #23757444C2semidirect
6. #23757443C2semidirect
7. #23757441C2semidirect
8. #23757440C2semidirect
9. #23757288C2semidirect
10. #23757254C2semidirect
11. #453×C157C2semidirect
13. #19097818C22semidirect
16. #19097816C22semidirect
22. C2×D4×D157C22semidirect
23. #19097799C22semidirect
25. #19097797C22semidirect
26. #19097796C22semidirect
30. M16×D157C22semidirect
31. #19097767C22semidirect
32. #19097759C22semidirect
33. #19097758C22semidirect
36. #19097728C22semidirect
37. #19097699C22semidirect
38. #19097698C22semidirect
39. #19097697C22semidirect
40. #19097689C22semidirect
41. #19097688C22semidirect
42. H(5,𝔽2)×C157C22semidirect
43. AGL(1,ℤ/8)×C157C22semidirect
44. #127×C157C22semidirect
45. #104×C157C22semidirect
46. #100×C157C22semidirect
48. #14578955D4semidirect
62. #14578953D4semidirect
67. Q8×D157D4semidirect
80. D4×D157D4semidirect
194. C157#3021semidirect
204. #453D157semidirect
212. H(5,𝔽2)C2×D157semidirect
224. AGL(1,ℤ/8)C2×D157semidirect
229. #127C2×D157semidirect
238. #104C2×D157semidirect
241. #100C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237581082
AGL(1,ℤ/8)×D1572
#237578362
#237574472
#237574442
#237574432
#237574412
#237574402
#237572882
#237572542
#453×C1572
#3021157