Finite Group Databasebeta

← #34109494

Subgroups — #34109494

Order 20096.

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

normal subgroups only

This group has 762 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
1H(5,𝔽2)×D157C211yesmaximal, characteristic, complemented
2#23758072C212yesmaximal, complemented
3#23757841C211yesmaximal, characteristic, complemented
4#23757450C211yesmaximal, characteristic, complemented
5#23757449C212yesmaximal, complemented
6#23757443C211yesmaximal, characteristic, complemented
7#23757442C212yesmaximal, complemented
8#23757441C211yesmaximal, characteristic, complemented
9#23757288C212yesmaximal, complemented
10#100×D157C211yesmaximal, characteristic, complemented
11#455×C157C211yesmaximal, characteristic, complemented
12#19097817—22no—
13#19097817C2212yes—
14#19097816—22no—
15#19097816—22no—
16#19097816C2211yescharacteristic
17C2×#14578955C2211yescharacteristic, complemented
18#41×D157—22no—
19#41×D157C2212yes—
20C2×D4×D157—22no—
21C2×D4×D157—22no—
22C2×D4×D157C2211yescharacteristic
23#19097803—22no—
24#19097802C2212yescomplemented
25#19097799C2212yescomplemented
26#19097798—22no—
27#19097796C2212yescomplemented
28#19097794—22no—
29#19097792—22no—
30M16×D157C2212yescomplemented
31#19097768C2211yescharacteristic, complemented
32#19097767C2211yescharacteristic, complemented
33#19097760C2211yescharacteristic, complemented
34#19097744—22no—
35#19097741—22no—
36#19097728C2211yescharacteristic, complemented
37#19097699C2212yescomplemented
38#19097698C2212yescomplemented
39#19097697C2212yescomplemented
40#19097688C2211yescharacteristic, complemented
41#19097687C2212yescomplemented
42H(5,𝔽2)×C157C2211yescharacteristic
43#137×C157C2212yescomplemented
44#124×C157C2211yescharacteristic, complemented
45#104×C157C2212yescomplemented
46#100×C157C2211yescharacteristic, complemented
47C23×D157—22no—
48C23×D157D412yes—
49#14578955—22no—
50#14578955—22no—
51#14578955—22no—
52#14578955D412yescomplemented
53#14578954—24no—
54#14578954—22no—
55#14578954—22no—
56#14578954C2312yes—
57#14578953—24no—
58#14578953—22no—
59#14578953—22no—
60#14578953D412yescomplemented
61Q8×D157D412yescomplemented
62C2×D628—22no—
63C2×D628—22no—
64C2×D628—21no—
65C2×D628—22no—
66C2×D628C2311yescharacteristic
67C2×D628C2311yescharacteristic, complemented
68C2×C22⋊D157—22no—
69C2×C22⋊D157—24no—
70C2×C22⋊D157—22no—
71C2×C22⋊D157—22no—
72C2×C22⋊D157D412yes—
73D4×D157—24no—
74D4×D157—24no—
75D4×D157—24no—
76D4×D157—22no—
77D4×D157—24no—
78D4×D157—22no—
79D4×D157—22no—
80D4×D157—22no—
81D4×D157—22no—
82D4×D157D412yescomplemented
83C2×C4×D157—22no—
84C2×C4×D157—22no—
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#14578938C2312yescomplemented
94#14578937—22no—
95C8×D157—22no—
96#14578933—21no—
97#14578931—21no—
98#14578931—22no—
99C4×Dic157C2311yescharacteristic, complemented
100#41×C157—22no—
101#41×C157C2312yes—
102C2×Q8×C157C2311yescharacteristic, complemented
103C2×D4×C157—22no—
104C2×D4×C157—22no—
105C2×D4×C157C2311yescharacteristic
106Q16×C157—22no—
107SD16×C157—22no—
108M16×C157C2312yescomplemented
109C22⋊C4×C157—22no—
110C42×C157C2311yescharacteristic, complemented
111C22×D157—24no—
112C22×D157—22no—
113C22×D157—22no—
114C22×D157—22no—
115C22×D157—22no—
116C22×D157—21no—
117C22×D157—82no—
118C22×D157—84no—
119C22×D157C2×D412yes—
120C22×D157C2×D411yescharacteristic
121Dic314—22no—
122Dic314C2×D412yes—
123D628—24no—
124D628—22no—
125D628—22no—
126D628—22no—
127D628—41no—
128D628—41no—
129D628—82no—
130D628C2×D412yes—
131C22⋊D157—24no—
132C22⋊D157—24no—
133C22⋊D157—22no—
134C22⋊D157—22no—
135C22⋊D157—24no—
136C22⋊D157—22no—
137C22⋊D157—82no—
138C2×Dic157—22no—
139C2×Dic157—22no—
140C2×Dic157—21no—
141C2×Dic157C2×D411yescharacteristic
142C4×D157—24no—
143C4×D157—22no—
144C4×D157—22no—
145C4×D157—22no—
146C4×D157—42no—
147C4×D157C2×D412yes—
148C4×D157C2×D412yes—
149#11929434—22no—
150C23×C157—22no—
151C23×C157C2×D412yes—
152Q8×C157—22no—
153Q8×C157C2×D412yes—
154D4×C157—24no—
155D4×C157—22no—
156D4×C157—22no—
157D4×C157C2×D412yes—
158C2×C4×C157—22no—
159C2×C4×C157—22no—
160C2×C4×C157—21no—
161C2×C4×C157—21no—
162C2×C4×C157C2411yescharacteristic
163C8×C157—22no—
164C22×C157—22no—
165C22×C157—22no—
166C22×C157—42no—
167C22×C157—82no—
168C22×C157C22×D411yescharacteristic
169C2×D157—22no—
170C2×D157—22no—
171C2×D157—41no—
172C2×D157—44no—
173C2×D157—44no—
174C2×D157—42no—
175C2×D157—42no—
176C2×D157—41no—
177C2×D157—82no—
178C2×D157—84no—
179C2×D157—82no—
180C2×D157#12011yescharacteristic
181C2×D157#12011yescharacteristic
182C4×C157—22no—
183C4×C157—41no—
184C4×C157—41no—
185C4×C157C22×D412yes—
186Dic157—22no—
187Dic157—41no—
188Dic157#12012yes—
189C2×C157—21no—
190C2×C157—42no—
191C2×C157—42no—
192C2×C157C2×#12011yescharacteristic
193D157—21no—
194D157—21no—
195D157—42no—
196D157—42no—
197D157—81no—
198C157#301911yescharacteristic, complemented, Sylow
199#3019—1571nomaximal, Sylow
200C2×H(5,𝔽2)—1571no—
201#575—1572no—
202#535—1571no—
203#455—1572no—
204#455—1571no—
205#455D15711yescharacteristic, complemented
206#453—1571no—
207#453—1572no—
208#453—1571no—
209#421—1572no—
210C2×#100—1571no—
211H(5,𝔽2)—3142no—
212H(5,𝔽2)—3142no—
213H(5,𝔽2)—3142no—
214H(5,𝔽2)—1572no—
215H(5,𝔽2)—1571no—
216H(5,𝔽2)C2×D15711yescharacteristic
217C2×#41—3142no—
218C2×#41—1572no—
219C2×#41—1571no—
220C22×D4—3142no—
221C22×D4—3142no—
222C22×D4—1571no—
223#137—3142no—
224#137—1572no—
225#137C2×D15712yescomplemented
226AGL(1,ℤ/8)—3142no—
227AGL(1,ℤ/8)—1572no—
228AGL(1,ℤ/8)—1572no—
229#135—3142no—
230#135—3142no—
231C2×M16—1572no—
232#127—1571no—
233#127—1571no—
234#124—1571no—
235#124C2×D15711yescharacteristic, complemented
236#121—3142no—
237#120—3142no—
238#117—1571no—
239#104—1572no—
240#104—1572no—
241#104—1572no—
242#104C2×D15712yescomplemented
243#100—1571no—
244#100—1572no—
245#100C2×D15711yescharacteristic, complemented
246C24—3142no—
247C24—1572no—
248#41—22no—
249#41—3144no—
250#41—3142no—
251#41—3142no—
252#41—3144no—
253#41—3142no—
254#41—3142no—
255#41—3142no—
256#41—3142no—
257#41—3142no—
258#41—1572no—
259#41—1572no—
260#41—1572no—
261#41C22×D15712yes—
262C2×Q8—1572no—
263C2×Q8C22×D15711yescharacteristic, complemented
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—3142no—
267C2×D4—3142no—
268C2×D4—3141no—
269C2×D4—3142no—
270C2×D4—3144no—
271C2×D4—3144no—
272C2×D4—3144no—
273C2×D4—3142no—
274C2×D4—3144no—
275C2×D4—3142no—
276C2×D4—3142no—
277C2×D4—3144no—
278C2×D4—3142no—
279C2×D4—3142no—
280C2×D4—3142no—
281C2×D4—3142no—
282C2×D4—3142no—
283C2×D4—1572no—
284C2×D4—1572no—
285C2×D4—1571no—
286C2×D4—1571no—
287C2×D4C22×D15711yescharacteristic
288C22×C4—3142no—
289C22×C4—3142no—
290C22×C4—3141no—
291C22×C4—1571no—
292Q16—22no—
293Q16—3142no—
294SD16—22no—
295SD16—3142no—
296SD16—3142no—
297SD16—3142no—
298D8—3142no—
299D8—3142no—
300M16—3142no—
301M16—1572no—
302M16C22×D15712yescomplemented
303C2×C8—3142no—
304C4⋊C4—3141no—
305C22⋊C4—22no—
306C22⋊C4—3141no—
307C22⋊C4—3142no—
308C42—1571no—
309C42C22×D15711yescharacteristic, complemented
310C23—22no—
311C23—3142no—
312C23—3144no—
313C23—3142no—
314C23—3142no—
315C23—3142no—
316C23—3141no—
317C23—1572no—
318C23—1571no—
319C23—12562no—
320C23—12564no—
321C23D4×D15712yes—
322Q8—22no—
323Q8—3142no—
324Q8—1572no—
325Q8D4×D15712yes—
326D4—24no—
327D4—22no—
328D4—22no—
329D4—3144no—
330D4—3142no—
331D4—3144no—
332D4—3144no—
333D4—3142no—
334D4—3142no—
335D4—3144no—
336D4—3142no—
337D4—3142no—
338D4—3142no—
339D4—6281no—
340D4—6281no—
341D4—1572no—
342D4—12562no—
343D4—12562no—
344D4D4×D15712yes—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—3144no—
350C2×C4—3142no—
351C2×C4—3142no—
352C2×C4—3142no—
353C2×C4—3142no—
354C2×C4—3142no—
355C2×C4—3141no—
356C2×C4—6282no—
357C2×C4—1571no—
358C2×C4—1572no—
359C2×C4—1572no—
360C2×C4C23×D15711yescharacteristic
361C8—22no—
362C8—3142no—
363C22—22no—
364C22—22no—
365C22—42no—
366C22—82no—
367C22—3142no—
368C22—3142no—
369C22—6281no—
370C22—6282no—
371C22—6282no—
372C22—6284no—
373C22—6284no—
374C22—6281no—
375C22—1571no—
376C22—1571no—
377C22—12562no—
378C22—12562no—
379C22—12564no—
380C22C2×D4×D15711yescharacteristic
381C4—22no—
382C4—41no—
383C4—41no—
384C4—3142no—
385C4—6281no—
386C4—1572no—
387C4C2×D4×D15712yes—
388C2—21no—
389C2—42no—
390C2—42no—
391C2—3141no—
392C2—3141no—
393C2—6282no—
394C2—6282no—
395C2—12561no—
396C2#120×D15711yescharacteristic
subgroupG/H
centerC2#120×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#455×C157C2
socleC2×C157C2×#120
hypercenter#455D157
hypocenterC157#3019
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. H(5,𝔽2)×D157C2semidirect
2. #23758072C2semidirect
3. #23757841C2semidirect
4. #23757450C2semidirect
5. #23757449C2semidirect
6. #23757443C2semidirect
7. #23757442C2semidirect
8. #23757441C2semidirect
9. #23757288C2semidirect
10. #100×D157C2semidirect
11. #455×C157C2semidirect
17. C2×#14578955C22semidirect
24. #19097802C22semidirect
25. #19097799C22semidirect
27. #19097796C22semidirect
30. M16×D157C22semidirect
31. #19097768C22semidirect
32. #19097767C22semidirect
33. #19097760C22semidirect
36. #19097728C22semidirect
37. #19097699C22semidirect
38. #19097698C22semidirect
39. #19097697C22semidirect
40. #19097688C22semidirect
41. #19097687C22semidirect
43. #137×C157C22semidirect
44. #124×C157C22semidirect
45. #104×C157C22semidirect
46. #100×C157C22semidirect
52. #14578955D4semidirect
60. #14578953D4semidirect
61. Q8×D157D4semidirect
67. C2×D628C23semidirect
82. D4×D157D4semidirect
93. #14578938C23semidirect
99. C4×Dic157C23semidirect
102. C2×Q8×C157C23semidirect
108. M16×C157C23semidirect
110. C42×C157C23semidirect
198. C157#3019semidirect
205. #455D157semidirect
225. #137C2×D157semidirect
235. #124C2×D157semidirect
242. #104C2×D157semidirect
245. #100C2×D157semidirect
263. C2×Q8C22×D157semidirect
302. M16C22×D157semidirect
309. C42C22×D157semidirect

Maximal subgroups

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

subgroupindex
H(5,𝔽2)×D1572
#237580722
#237578412
#237574502
#237574492
#237574432
#237574422
#237574412
#237572882
#100×D1572
#455×C1572
#3019157