Finite Group Databasebeta

← #34109509

Subgroups — #34109509

Order 20096.

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

normal subgroups only

This group has 736 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
1#23758109C211yesmaximal, characteristic, complemented
2C2×#19097799C211yesmaximal, characteristic, complemented
3C2×#19097796C211yesmaximal, characteristic, complemented
4#23757842C211yesmaximal, characteristic, complemented
5#23757447C214yesmaximal, complemented
6#23757442C214yesmaximal, complemented
7#23757291C211yesmaximal, characteristic, complemented
8#23757255C211yesmaximal, characteristic, complemented
9#421×C157C211yesmaximal, characteristic, complemented
10#19097819—21no—
11#19097818—21no—
12#19097818C2212yescomplemented
13#19097817—21no—
14#19097817C2212yescomplemented
15#19097816—21no—
16#19097816—22no—
17C2×#14578954—22no—
18C2×#14578954C2211yescharacteristic
19C2×#14578954C2211yescharacteristic
20#41×D157—22no—
21#41×D157—22no—
22C22×D628C2211yescharacteristic, complemented
23#19097799—22no—
24#19097799C2214yescomplemented
25#19097796—22no—
26#19097796C2214yescomplemented
27C2×#14578943—21no—
28C2×#14578941—21no—
29C2×D1256—21no—
30C2×#14578939—21no—
31C2×#14578938C2211yescharacteristic, complemented
32#19097768C2212yescomplemented
33#19097759C2212yescomplemented
34#19097746—22no—
35#19097741—22no—
36#19097699C2214yescomplemented
37#19097689C2212yescomplemented
38#19097687C2212yescomplemented
39C2×#41×C157C2211yescharacteristic
40C2×M16×C157C2211yescharacteristic, complemented
41#117×C157C2211yescharacteristic, complemented
42#104×C157C2214yescomplemented
43C23×D157—21no—
44#14578955—22no—
45#14578955—22no—
46#14578954—24no—
47#14578954—24no—
48#14578954—22no—
49#14578954—22no—
50#14578954—21no—
51#14578954—21no—
52#14578954—21no—
53#14578954—21no—
54#14578954—21no—
55#14578954—21no—
56#14578954D412yescomplemented
57#14578954C2312yes—
58#14578953—24no—
59#14578953—22no—
60#14578953—24no—
61#14578953—22no—
62C2×Dic314—22no—
63C2×Dic314D411yescharacteristic, complemented
64C2×Dic314C2311yescharacteristic
65Q8×D157—22no—
66Q8×D157—22no—
67C2×D628—22no—
68C2×D628—24no—
69C2×D628—21no—
70C2×D628—21no—
71C2×D628—21no—
72C2×D628—21no—
73C2×D628—21no—
74C2×D628D411yescharacteristic, complemented
75C2×D628C2312yes—
76C2×D628C2311yescharacteristic
77C2×C22⋊D157—24no—
78C2×C22⋊D157—21no—
79C2×C22⋊D157D412yes—
80D4×D157—24no—
81D4×D157—22no—
82D4×D157—24no—
83D4×D157—22no—
84C2×C4×D157—24no—
85C2×C4×D157—21no—
86C2×C4×D157D412yes—
87#14578943—24no—
88#14578941—24no—
89D1256—24no—
90#14578939—24no—
91#14578938—21no—
92#14578938C2312yes—
93C2×#11929434—21no—
94#14578931—24no—
95#41×C157—21no—
96#41×C157—21no—
97#41×C157D412yescomplemented
98#41×C157C2312yes—
99C2×Q8×C157D411yescharacteristic, complemented
100C2×D4×C157—21no—
101C2×D4×C157D411yescharacteristic, complemented
102C22×C4×C157—21no—
103C22×C4×C157C2311yescharacteristic
104M16×C157—21no—
105M16×C157C2312yes—
106C2×C8×C157—21no—
107C4⋊C4×C157—21no—
108C22⋊C4×C157—21no—
109C42×C157C2312yes—
110C22×D157—24no—
111C22×D157—21no—
112C22×D157—22no—
113C22×D157—41no—
114C22×D157—41no—
115C22×D157—82no—
116C22×D157C2×D412yes—
117Dic314—24no—
118Dic314—21no—
119Dic314—21no—
120Dic314—21no—
121Dic314C2×D412yes—
122D628—24no—
123D628—21no—
124D628—21no—
125D628—21no—
126D628—22no—
127D628—22no—
128D628—41no—
129D628—41no—
130D628—84no—
131D628C2×D412yes—
132C22⋊D157—24no—
133C22⋊D157—24no—
134C22⋊D157—22no—
135C22⋊D157—22no—
136C22⋊D157—22no—
137C22⋊D157—22no—
138C2×Dic157—24no—
139C2×Dic157—21no—
140C2×Dic157C2×D412yes—
141C4×D157—24no—
142C4×D157—24no—
143C4×D157—22no—
144C4×D157—22no—
145C4×D157—22no—
146C4×D157—22no—
147#11929434—22no—
148C23×C157—21no—
149C23×C157C2×D411yescharacteristic
150Q8×C157—21no—
151Q8×C157C2×D412yes—
152D4×C157—22no—
153D4×C157—22no—
154D4×C157—21no—
155D4×C157C2×D412yes—
156C2×C4×C157—21no—
157C2×C4×C157—22no—
158C2×C4×C157—22no—
159C2×C4×C157—22no—
160C2×C4×C157C2×D411yescharacteristic
161C2×C4×C157C2×D411yescharacteristic
162C2×C4×C157C2×D411yescharacteristic
163C2×C4×C157C2×D411yescharacteristic
164C2×C4×C157C2×D411yescharacteristic
165C2×C4×C157C2411yescharacteristic
166C8×C157—22no—
167C22×C157—22no—
168C22×C157—41no—
169C22×C157—81no—
170C22×C157#12011yescharacteristic
171C22×C157#12011yescharacteristic
172C22×C157C22×D411yescharacteristic
173C2×D157—22no—
174C2×D157—22no—
175C2×D157—42no—
176C2×D157—44no—
177C2×D157—42no—
178C2×D157—81no—
179C2×D157—84no—
180C2×D157—82no—
181C2×D157—82no—
182C4×C157—22no—
183C4×C157—42no—
184C4×C157#12011yescharacteristic
185C4×C157#12011yescharacteristic
186C4×C157C22×D411yescharacteristic
187C4×C157C22×D411yescharacteristic
188Dic157—22no—
189Dic157—22no—
190C2×C157—21no—
191C2×C157—21no—
192C2×C157—21no—
193C2×C157—42no—
194C2×C157C2×#12011yescharacteristic
195D157—42no—
196D157—42no—
197D157—82no—
198C157#302111yescharacteristic, complemented, Sylow
199#3021—1571nomaximal, Sylow
200#585—1571no—
201C2×AGL(1,ℤ/8)—1571no—
202C2×AGL(1,ℤ/8)—1571no—
203#535—1571no—
204#454—1574no—
205#453—1574no—
206#421—1571no—
207#421D15711yescharacteristic, complemented
208#413—1571no—
2092−1+4—3141no—
2102−1+4—3141no—
2112−1+4—1572no—
212H(5,𝔽2)—3142no—
213H(5,𝔽2)—3141no—
214H(5,𝔽2)—3141no—
215H(5,𝔽2)—1572no—
216C2×#41—3142no—
217C2×#41—3142no—
218C2×#41—3142no—
219C2×#41—1571no—
220C2×#41—1571no—
221C2×#41C2×D15711yescharacteristic
222C22×D4—1571no—
223AGL(1,ℤ/8)—3142no—
224AGL(1,ℤ/8)—3142no—
225AGL(1,ℤ/8)—1574no—
226AGL(1,ℤ/8)—1574no—
227C2×SD16—3141no—
228C2×SD16—3141no—
229C2×D8—3141no—
230C2×D8—3141no—
231C2×M16—1571no—
232C2×M16C2×D15711yescharacteristic, complemented
233#127—1572no—
234#124—1572no—
235#121—3142no—
236#120—3142no—
237#117C2×D15711yescharacteristic, complemented
238#104—1574no—
239#104C2×D15714yescomplemented
240#101—1572no—
241#100—1572no—
242C24—3141no—
243#41—21no—
244#41—21no—
245#41—3144no—
246#41—3144no—
247#41—3142no—
248#41—3142no—
249#41—3144no—
250#41—3142no—
251#41—3142no—
252#41—3142no—
253#41—3142no—
254#41—3144no—
255#41—3141no—
256#41—3141no—
257#41—3141no—
258#41—3141no—
259#41—3141no—
260#41—3141no—
261#41—1572no—
262#41—1572no—
263#41D62812yescomplemented
264#41C22×D15712yes—
265C2×Q8—3142no—
266C2×Q8—3142no—
267C2×Q8—3142no—
268C2×Q8—1571no—
269C2×Q8—1571no—
270C2×Q8D62811yescharacteristic, complemented
271C2×D4—21no—
272C2×D4—3142no—
273C2×D4—3144no—
274C2×D4—3144no—
275C2×D4—3142no—
276C2×D4—3144no—
277C2×D4—3144no—
278C2×D4—3141no—
279C2×D4—3142no—
280C2×D4—3141no—
281C2×D4—3141no—
282C2×D4—3141no—
283C2×D4—3141no—
284C2×D4—3141no—
285C2×D4—1572no—
286C2×D4—1571no—
287C2×D4—1572no—
288C2×D4—1571no—
289C2×D4D62811yescharacteristic, complemented
290C22×C4—21no—
291C22×C4—3144no—
292C22×C4—3141no—
293C22×C4—1572no—
294C22×C4C22×D15711yescharacteristic
295SD16—3144no—
296SD16—3144no—
297D8—3144no—
298D8—3144no—
299M16—21no—
300M16—3141no—
301M16—1572no—
302M16C22×D15712yes—
303C2×C8—21no—
304C2×C8—3141no—
305C4⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—3144no—
308C42C22×D15712yes—
309C23—21no—
310C23—3144no—
311C23—3141no—
312C23—3142no—
313C23—6281no—
314C23—6281no—
315C23—1572no—
316C23—12562no—
317C23D4×D15711yescharacteristic
318Q8—21no—
319Q8—3144no—
320Q8—3141no—
321Q8—3141no—
322Q8—3141no—
323Q8—1572no—
324Q8C2×D62812yes—
325D4—22no—
326D4—22no—
327D4—21no—
328D4—3144no—
329D4—3144no—
330D4—3144no—
331D4—3142no—
332D4—3142no—
333D4—3142no—
334D4—3142no—
335D4—3141no—
336D4—3141no—
337D4—3142no—
338D4—3141no—
339D4—3142no—
340D4—6281no—
341D4—6281no—
342D4—1572no—
343D4—12564no—
344D4C2×D62812yes—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—3144no—
350C2×C4—3144no—
351C2×C4—3144no—
352C2×C4—3141no—
353C2×C4—3142no—
354C2×C4—3142no—
355C2×C4—3142no—
356C2×C4—3142no—
357C2×C4—1572no—
358C2×C4D4×D15711yescharacteristic
359C2×C4D4×D15711yescharacteristic
360C2×C4D4×D15711yescharacteristic
361C2×C4C2×D62811yescharacteristic
362C2×C4C2×D62811yescharacteristic
363C2×C4C23×D15711yescharacteristic
364C8—22no—
365C8—3142no—
366C22—22no—
367C22—41no—
368C22—81no—
369C22—3142no—
370C22—3142no—
371C22—6282no—
372C22—6284no—
373C22—6282no—
374C22—12561no—
375C22—12562no—
376C22—12562no—
377C22—12564no—
378C22#1909774111yescharacteristic
379C22#1909774111yescharacteristic
380C22C2×D4×D15711yescharacteristic
381C4—22no—
382C4—42no—
383C4—3142no—
384C4—3142no—
385C4#1909774111yescharacteristic
386C4#1909774111yescharacteristic
387C4C2×D4×D15711yescharacteristic
388C4C22×D62811yescharacteristic
389C2—21no—
390C2—21no—
391C2—21no—
392C2—42no—
393C2—6282no—
394C2—6282no—
395C2—12562no—
396C2C2×#1909774111yescharacteristic
subgroupG/H
centerC2C2×#19097741
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#421×C157C2
socleC2×C157C2×#120
hypercenter#421D157
hypocenterC157#3021
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. #23758109C2semidirect
2. C2×#19097799C2semidirect
3. C2×#19097796C2semidirect
4. #23757842C2semidirect
5. #23757447C2semidirect
6. #23757442C2semidirect
7. #23757291C2semidirect
8. #23757255C2semidirect
9. #421×C157C2semidirect
12. #19097818C22semidirect
14. #19097817C22semidirect
22. C22×D628C22semidirect
24. #19097799C22semidirect
26. #19097796C22semidirect
31. C2×#14578938C22semidirect
32. #19097768C22semidirect
33. #19097759C22semidirect
36. #19097699C22semidirect
37. #19097689C22semidirect
38. #19097687C22semidirect
40. C2×M16×C157C22semidirect
41. #117×C157C22semidirect
42. #104×C157C22semidirect
56. #14578954D4semidirect
63. C2×Dic314D4semidirect
74. C2×D628D4semidirect
97. #41×C157D4semidirect
99. C2×Q8×C157D4semidirect
101. C2×D4×C157D4semidirect
198. C157#3021semidirect
207. #421D157semidirect
232. C2×M16C2×D157semidirect
237. #117C2×D157semidirect
239. #104C2×D157semidirect
263. #41D628semidirect
270. C2×Q8D628semidirect
289. C2×D4D628semidirect

Maximal subgroups

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

subgroupindex
#237581092
C2×#190977992
C2×#190977962
#237578422
#237574472
#237574422
#237572912
#237572552
#421×C1572
#3021157