Finite Group Databasebeta

← #34109495

Subgroups — #34109495

Order 20096.

762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×#19097817C211yesmaximal, characteristic, complemented
2#23758073C211yesmaximal, characteristic, complemented
3#23758071C211yesmaximal, characteristic, complemented
4#23757842C211yesmaximal, characteristic, complemented
5#23757452C214yesmaximal, complemented
6#23757442C214yesmaximal, complemented
7#23757291C211yesmaximal, characteristic, complemented
8C2×#19097687C211yesmaximal, characteristic, complemented
9#421×C157C211yesmaximal, characteristic, complemented
10#19097817—22no—
11#19097817—22no—
12#19097817—22no—
13#19097817C2214yes—
14C2×#14578955—21no—
15C2×#14578954—21no—
16C2×#14578954C2211yescharacteristic
17C2×#14578954C2211yescharacteristic, complemented
18C22×D628—21no—
19C22×D628C2211yescharacteristic
20C2×D4×D157—21no—
21C2×D4×D157—22no—
22#19097804—21no—
23#19097804C2212yescomplemented
24#19097802—21no—
25#19097801C2212yescomplemented
26#19097799—21no—
27#19097799C2212yescomplemented
28#19097797—21no—
29#19097796C2212yescomplemented
30#19097793—22no—
31#19097791—22no—
32C2×#14578938C2211yescharacteristic, complemented
33#19097768C2212yescomplemented
34#19097759C2212yescomplemented
35#19097746—22no—
36#19097741—22no—
37#19097699C2214yescomplemented
38#19097687C2214yescomplemented
39C2×#41×C157C2211yescharacteristic
40C2×M16×C157C2211yescharacteristic, complemented
41#117×C157C2211yescharacteristic, complemented
42#104×C157C2214yescomplemented
43C23×D157—22no—
44C23×D157D412yes—
45#14578955—24no—
46#14578955—24no—
47#14578954—24no—
48#14578954—24no—
49#14578954—21no—
50#14578954—21no—
51#14578954—21no—
52#14578954—21no—
53#14578954—21no—
54#14578954—21no—
55#14578954D412yescomplemented
56#14578954C2312yes—
57C2×Dic314D411yescharacteristic, complemented
58C2×Dic314C2311yescharacteristic, complemented
59C2×D628—22no—
60C2×D628—24no—
61C2×D628—24no—
62C2×D628—24no—
63C2×D628—21no—
64C2×D628—21no—
65C2×D628—21no—
66C2×D628—21no—
67C2×D628D412yes—
68C2×D628D411yescharacteristic, complemented
69C2×D628C2312yes—
70C2×D628C2311yescharacteristic, complemented
71C2×C22⋊D157—22no—
72C2×C22⋊D157—21no—
73C2×C22⋊D157—21no—
74D4×D157—28no—
75D4×D157—24no—
76D4×D157—28no—
77D4×D157—24no—
78C2×C4×D157—22no—
79C2×C4×D157—21no—
80C2×C4×D157—21no—
81Dic628—22no—
82#14578944—22no—
83#14578943—22no—
84#14578943—22no—
85#14578942—22no—
86#14578941—22no—
87D1256—22no—
88#14578939—22no—
89#14578938—21no—
90#14578938C2312yescomplemented
91C2×#11929434—21no—
92#14578931—24no—
93#41×C157—21no—
94#41×C157—21no—
95#41×C157D412yescomplemented
96#41×C157C2312yes—
97C2×Q8×C157D411yescharacteristic, complemented
98C2×D4×C157—21no—
99C2×D4×C157D411yescharacteristic, complemented
100C22×C4×C157—21no—
101C22×C4×C157C2311yescharacteristic
102M16×C157—21no—
103M16×C157C2312yescomplemented
104C2×C8×C157—21no—
105C4⋊C4×C157—21no—
106C22⋊C4×C157—21no—
107C42×C157C2312yescomplemented
108C22×D157—22no—
109C22×D157—21no—
110C22×D157—28no—
111C22×D157—22no—
112C22×D157—21no—
113C22×D157—82no—
114C22×D157—84no—
115C22×D157C2×D414yes—
116Dic314—21no—
117Dic314—21no—
118Dic314—21no—
119Dic314C2×D412yes—
120D628—28no—
121D628—22no—
122D628—22no—
123D628—21no—
124D628—22no—
125D628—22no—
126D628—21no—
127D628—21no—
128D628—84no—
129D628C2×D412yes—
130C22⋊D157—28no—
131C22⋊D157—22no—
132C22⋊D157—22no—
133C22⋊D157—41no—
134C22⋊D157—41no—
135C2×Dic157—21no—
136C2×Dic157—21no—
137C4×D157—28no—
138C4×D157—22no—
139C4×D157—22no—
140C4×D157—41no—
141C4×D157—41no—
142#11929434—22no—
143C23×C157—21no—
144C23×C157C2×D411yescharacteristic
145Q8×C157—21no—
146Q8×C157C2×D412yes—
147D4×C157—22no—
148D4×C157—22no—
149D4×C157—21no—
150D4×C157C2×D412yes—
151C2×C4×C157—21no—
152C2×C4×C157—22no—
153C2×C4×C157—22no—
154C2×C4×C157—22no—
155C2×C4×C157C2×D411yescharacteristic
156C2×C4×C157C2×D411yescharacteristic
157C2×C4×C157C2×D411yescharacteristic
158C2×C4×C157C2×D411yescharacteristic
159C2×C4×C157C2×D411yescharacteristic
160C2×C4×C157C2411yescharacteristic
161C8×C157—22no—
162C22×C157—22no—
163C22×C157—41no—
164C22×C157—42no—
165C22×C157#12011yescharacteristic
166C22×C157#12011yescharacteristic
167C22×C157C22×D411yescharacteristic
168C2×D157—24no—
169C2×D157—22no—
170C2×D157—44no—
171C2×D157—42no—
172C2×D157—44no—
173C2×D157—42no—
174C2×D157—41no—
175C2×D157—82no—
176C2×D157—88no—
177C4×C157—22no—
178C4×C157—42no—
179C4×C157#12011yescharacteristic
180C4×C157#12011yescharacteristic
181C4×C157C22×D411yescharacteristic
182C4×C157C22×D411yescharacteristic
183Dic157—22no—
184Dic157—41no—
185C2×C157—21no—
186C2×C157—21no—
187C2×C157—21no—
188C2×C157—42no—
189C2×C157C2×#12011yescharacteristic
190D157—44no—
191D157—42no—
192D157—81no—
193C157#301911yescharacteristic, complemented, Sylow
194#3019—1571nomaximal, Sylow
195C2×H(5,𝔽2)—1571no—
196#575—1571no—
197#575—1571no—
198#535—1571no—
199#455—1574no—
200#453—1574no—
201#421—1571no—
202#421D15711yescharacteristic, complemented
203C2×#100—1571no—
204H(5,𝔽2)—3142no—
205H(5,𝔽2)—3142no—
206H(5,𝔽2)—3142no—
207H(5,𝔽2)—1574no—
208C2×#41—3141no—
209C2×#41—3141no—
210C2×#41—1571no—
211C2×#41—1571no—
212C2×#41C2×D15711yescharacteristic
213C22×D4—3141no—
214C22×D4—3142no—
215C22×D4—3141no—
216C22×D4—1571no—
217#137—3141no—
218#137—3141no—
219#137—1572no—
220#137—1572no—
221AGL(1,ℤ/8)—3141no—
222AGL(1,ℤ/8)—3141no—
223AGL(1,ℤ/8)—1572no—
224AGL(1,ℤ/8)—1572no—
225#135—3142no—
226#135—3142no—
227C2×M16—1571no—
228C2×M16C2×D15711yescharacteristic, complemented
229#127—1572no—
230#124—1572no—
231#121—3142no—
232#120—3142no—
233#117C2×D15711yescharacteristic, complemented
234#104—1574no—
235#104C2×D15714yescomplemented
236#100—1574no—
237C24—3142no—
238C24—1572no—
239#41—21no—
240#41—21no—
241#41—3144no—
242#41—3144no—
243#41—3144no—
244#41—3144no—
245#41—3141no—
246#41—3141no—
247#41—3141no—
248#41—3141no—
249#41—3141no—
250#41—3141no—
251#41—1572no—
252#41—1572no—
253#41D62812yescomplemented
254#41C22×D15712yes—
255C2×Q8—1571no—
256C2×Q8—1571no—
257C2×Q8D62811yescharacteristic, complemented
258C2×D4—21no—
259C2×D4—3142no—
260C2×D4—3144no—
261C2×D4—3144no—
262C2×D4—3144no—
263C2×D4—3148no—
264C2×D4—3144no—
265C2×D4—3148no—
266C2×D4—3142no—
267C2×D4—3141no—
268C2×D4—3144no—
269C2×D4—3141no—
270C2×D4—3141no—
271C2×D4—3141no—
272C2×D4—3141no—
273C2×D4—3141no—
274C2×D4—1572no—
275C2×D4—1571no—
276C2×D4—1571no—
277C2×D4—1572no—
278C2×D4D62811yescharacteristic, complemented
279C22×C4—21no—
280C22×C4—3142no—
281C22×C4—3141no—
282C22×C4—3141no—
283C22×C4C22×D15711yescharacteristic
284Q16—3142no—
285Q16—3142no—
286SD16—3142no—
287SD16—3142no—
288SD16—3142no—
289SD16—3142no—
290D8—3142no—
291D8—3142no—
292M16—21no—
293M16—3141no—
294M16—1572no—
295M16C22×D15712yescomplemented
296C2×C8—21no—
297C2×C8—3141no—
298C4⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—3144no—
301C42C22×D15712yescomplemented
302C23—21no—
303C23—3148no—
304C23—3142no—
305C23—3141no—
306C23—3142no—
307C23—3141no—
308C23—1574no—
309C23—12562no—
310C23—12564no—
311C23D4×D15711yescharacteristic
312Q8—21no—
313Q8—3141no—
314Q8—3141no—
315Q8—3141no—
316Q8—1572no—
317Q8C2×D62812yes—
318D4—22no—
319D4—22no—
320D4—21no—
321D4—3148no—
322D4—3148no—
323D4—3142no—
324D4—3142no—
325D4—3142no—
326D4—3141no—
327D4—3142no—
328D4—3142no—
329D4—3141no—
330D4—3142no—
331D4—3141no—
332D4—6281no—
333D4—6281no—
334D4—1572no—
335D4—12564no—
336D4C2×D62812yes—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—3148no—
342C2×C4—3141no—
343C2×C4—3142no—
344C2×C4—3142no—
345C2×C4—3141no—
346C2×C4—6281no—
347C2×C4—6281no—
348C2×C4D4×D15711yescharacteristic
349C2×C4D4×D15711yescharacteristic
350C2×C4D4×D15711yescharacteristic
351C2×C4C2×D62811yescharacteristic
352C2×C4C2×D62811yescharacteristic
353C2×C4C23×D15711yescharacteristic
354C8—22no—
355C8—3142no—
356C22—22no—
357C22—41no—
358C22—42no—
359C22—3144no—
360C22—3142no—
361C22—6284no—
362C22—6284no—
363C22—6282no—
364C22—6282no—
365C22—6281no—
366C22—12562no—
367C22—12568no—
368C22#1909774111yescharacteristic
369C22#1909774111yescharacteristic
370C22C2×D4×D15711yescharacteristic
371C4—22no—
372C4—42no—
373C4—3142no—
374C4—6281no—
375C4#1909774111yescharacteristic
376C4#1909774111yescharacteristic
377C4C2×D4×D15711yescharacteristic
378C4C22×D62811yescharacteristic
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—6284no—
384C2—6282no—
385C2—12561no—
386C2C2×#1909774111yescharacteristic
subgroupG/H
centerC2C2×#19097741
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#421×C157C2
socleC2×C157C2×#120
hypercenter#421D157
hypocenterC157#3019
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. C2×#19097817C2semidirect
2. #23758073C2semidirect
3. #23758071C2semidirect
4. #23757842C2semidirect
5. #23757452C2semidirect
6. #23757442C2semidirect
7. #23757291C2semidirect
8. C2×#19097687C2semidirect
9. #421×C157C2semidirect
17. C2×#14578954C22semidirect
23. #19097804C22semidirect
25. #19097801C22semidirect
27. #19097799C22semidirect
29. #19097796C22semidirect
32. C2×#14578938C22semidirect
33. #19097768C22semidirect
34. #19097759C22semidirect
37. #19097699C22semidirect
38. #19097687C22semidirect
40. C2×M16×C157C22semidirect
41. #117×C157C22semidirect
42. #104×C157C22semidirect
55. #14578954D4semidirect
57. C2×Dic314D4semidirect
58. C2×Dic314C23semidirect
68. C2×D628D4semidirect
70. C2×D628C23semidirect
90. #14578938C23semidirect
95. #41×C157D4semidirect
97. C2×Q8×C157D4semidirect
99. C2×D4×C157D4semidirect
103. M16×C157C23semidirect
107. C42×C157C23semidirect
193. C157#3019semidirect
202. #421D157semidirect
228. C2×M16C2×D157semidirect
233. #117C2×D157semidirect
235. #104C2×D157semidirect
253. #41D628semidirect
257. C2×Q8D628semidirect
278. C2×D4D628semidirect
295. M16C22×D157semidirect
301. C42C22×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978172
#237580732
#237580712
#237578422
#237574522
#237574422
#237572912
C2×#190976872
#421×C1572
#3019157