Finite Group Databasebeta

← #44559720

Subgroups — #44559720

Order 32128.

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×#22548761C211yesmaximal, characteristic, complemented
2#30632485C211yesmaximal, characteristic, complemented
3#30632251C211yesmaximal, characteristic, complemented
4#30631863C214yesmaximal, complemented
5#30631853C214yesmaximal, complemented
6#30631704C211yesmaximal, characteristic, complemented
7#30631703C211yesmaximal, characteristic, complemented
8C2×#22548631C211yesmaximal, characteristic, complemented
9#575×C251C211yesmaximal, characteristic, complemented
10#22548761—22no—
11#22548761—22no—
12#22548761—22no—
13#22548761C2214yes—
14C2×#16860811—21no—
15C2×#16860810—21no—
16C2×#16860810C2211yescharacteristic
17C22×D1004—21no—
18C22×D1004C2211yescharacteristic
19C2×D4×D251—22no—
20C2×D4×D251—21no—
21#22548746C2212yescomplemented
22#22548745—21no—
23#22548741C2212yescomplemented
24#22548740—21no—
25#22548735—22no—
26C2×#16860794C2211yescharacteristic, complemented
27#22548711C2212yescomplemented
28#22548704C2212yescomplemented
29#22548689—22no—
30#22548683—22no—
31#22548674C2211yescharacteristic, complemented
32#22548642C2214yescomplemented
33#22548641C2214yescomplemented
34#22548631C2214yescomplemented
35C2×#41×C251C2211yescharacteristic
36C2×#41×C251C2211yescharacteristic, complemented
37#137×C251—21no—
38#137×C251C2212yescomplemented
39AGL(1,ℤ/8)×C251—21no—
40AGL(1,ℤ/8)×C251C2212yescomplemented
41#135×C251—22no—
42C2×M16×C251C2211yescharacteristic, complemented
43C23×D251—22no—
44C23×D251D412yes—
45#16860811—24no—
46#16860811—24no—
47#16860810—24no—
48#16860810—24no—
49#16860810—21no—
50#16860810—21no—
51#16860810D412yescomplemented
52#16860810C2312yes—
53C2×Dic502D411yescharacteristic, complemented
54C2×D1004—22no—
55C2×D1004—24no—
56C2×D1004—24no—
57C2×D1004—21no—
58C2×D1004—21no—
59C2×D1004—21no—
60C2×D1004—21no—
61C2×D1004D412yes—
62C2×D1004D411yescharacteristic, complemented
63C2×D1004C2312yes—
64C2×C22⋊D251—22no—
65C2×C22⋊D251—24no—
66C2×C22⋊D251—21no—
67D4×D251—28no—
68D4×D251—24no—
69D4×D251—28no—
70D4×D251—24no—
71C2×C4×D251—22no—
72C2×C4×D251—21no—
73#16860800—22no—
74#16860798—22no—
75#16860797—22no—
76#16860795—22no—
77#16860794—21no—
78#16860794C2312yescomplemented
79C2×C251⋊C8—21no—
80C4⋊Dic251—21no—
81C22⋊Dic251—21no—
82(C2×D251)⋊C4—24no—
83C4×Dic251C2312yescomplemented
84#41×C251—21no—
85#41×C251—21no—
86#41×C251—21no—
87#41×C251—21no—
88#41×C251—21no—
89#41×C251—21no—
90#41×C251D412yescomplemented
91#41×C251C2312yes—
92C2×Q8×C251D411yescharacteristic, complemented
93C2×Q8×C251C2311yescharacteristic, complemented
94C2×D4×C251—21no—
95C2×D4×C251—21no—
96C2×D4×C251D411yescharacteristic, complemented
97C2×D4×C251C2311yescharacteristic, complemented
98C22×C4×C251—21no—
99C22×C4×C251—21no—
100C22×C4×C251C2311yescharacteristic
101Q16×C251—22no—
102SD16×C251—22no—
103SD16×C251—22no—
104D8×C251—22no—
105M16×C251—21no—
106M16×C251C2312yescomplemented
107C2×C8×C251—21no—
108C22×D251—22no—
109C22×D251—21no—
110C22×D251—28no—
111C22×D251—22no—
112C22×D251—82no—
113C22×D251—84no—
114C22×D251C2×D414yes—
115Dic502—21no—
116Dic502C2×D412yes—
117D1004—28no—
118D1004—22no—
119D1004—22no—
120D1004—21no—
121D1004—22no—
122D1004—22no—
123D1004C2×D412yes—
124C22⋊D251—28no—
125C22⋊D251—22no—
126C22⋊D251—22no—
127C22⋊D251—84no—
128C2×Dic251—21no—
129C2×Dic251—22no—
130C4×D251—28no—
131C4×D251—22no—
132C4×D251—22no—
133C251⋊C8—22no—
134C23×C251—21no—
135C23×C251—21no—
136C23×C251C2×D411yescharacteristic
137Q8×C251—21no—
138Q8×C251—21no—
139Q8×C251—21no—
140Q8×C251C2×D412yes—
141D4×C251—22no—
142D4×C251—22no—
143D4×C251—21no—
144D4×C251—21no—
145D4×C251—21no—
146D4×C251—41no—
147D4×C251—41no—
148D4×C251C2×D412yes—
149C2×C4×C251—21no—
150C2×C4×C251—22no—
151C2×C4×C251—22no—
152C2×C4×C251—21no—
153C2×C4×C251—41no—
154C2×C4×C251—41no—
155C2×C4×C251C2×D411yescharacteristic
156C2×C4×C251C2×D411yescharacteristic
157C2×C4×C251C2×D411yescharacteristic
158C2×C4×C251C2×D411yescharacteristic
159C2×C4×C251C2×D411yescharacteristic
160C2×C4×C251C2411yescharacteristic
161C8×C251—22no—
162C22×C251—22no—
163C22×C251—41no—
164C22×C251—42no—
165C22×C251—42no—
166C22×C251—41no—
167C22×C251#12011yescharacteristic
168C22×C251#12011yescharacteristic
169C22×C251C22×D411yescharacteristic
170C2×D251—24no—
171C2×D251—22no—
172C2×D251—44no—
173C2×D251—42no—
174C2×D251—44no—
175C2×D251—82no—
176C2×D251—88no—
177C4×C251—22no—
178C4×C251—41no—
179C4×C251#12011yescharacteristic
180C4×C251#12011yescharacteristic
181C4×C251C22×D411yescharacteristic
182C4×C251C22×D411yescharacteristic
183Dic251—22no—
184Dic251—42no—
185C2×C251—21no—
186C2×C251—21no—
187C2×C251—21no—
188C2×C251—42no—
189C2×C251—81no—
190C2×C251C2×#12011yescharacteristic
191D251—44no—
192D251—42no—
193C251#301911yescharacteristic, complemented, Sylow
194#3019—2511nomaximal, Sylow
195C2×H(5,𝔽2)—2511no—
196#575—2511no—
197#575D25111yescharacteristic, complemented
198#535—2511no—
199#455—2514no—
200#453—2514no—
201#421—2511no—
202#421—2511no—
203C2×#100—2511no—
204H(5,𝔽2)—5022no—
205H(5,𝔽2)—5022no—
206H(5,𝔽2)—5022no—
207H(5,𝔽2)—2514no—
208C2×#41—5021no—
209C2×#41—5021no—
210C2×#41—2511no—
211C2×#41C2×D25111yescharacteristic
212C2×#41C2×D25111yescharacteristic, complemented
213C22×D4—5021no—
214C22×D4—5022no—
215C22×D4—5021no—
216C22×D4—2511no—
217#137—21no—
218#137—5021no—
219#137—2512no—
220#137C2×D25112yescomplemented
221AGL(1,ℤ/8)—21no—
222AGL(1,ℤ/8)—5021no—
223AGL(1,ℤ/8)—2512no—
224AGL(1,ℤ/8)C2×D25112yescomplemented
225#135—22no—
226#135—5022no—
227C2×M16—2511no—
228C2×M16C2×D25111yescharacteristic, complemented
229#127—2512no—
230#124—2512no—
231#121—5022no—
232#120—5022no—
233#117—2511no—
234#104—2514no—
235#104—2514no—
236#100—2514no—
237C24—5022no—
238C24—2512no—
239#41—21no—
240#41—21no—
241#41—21no—
242#41—21no—
243#41—21no—
244#41—21no—
245#41—5024no—
246#41—5024no—
247#41—5024no—
248#41—5024no—
249#41—5021no—
250#41—5021no—
251#41—2512no—
252#41—2512no—
253#41C22⋊D25112yescomplemented
254#41C22×D25112yes—
255C2×Q8—2511no—
256C2×Q8C22⋊D25111yescharacteristic, complemented
257C2×Q8C22×D25111yescharacteristic, complemented
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—5022no—
261C2×D4—5024no—
262C2×D4—5024no—
263C2×D4—5028no—
264C2×D4—5024no—
265C2×D4—5028no—
266C2×D4—5024no—
267C2×D4—5022no—
268C2×D4—5021no—
269C2×D4—5024no—
270C2×D4—5021no—
271C2×D4—5021no—
272C2×D4—5021no—
273C2×D4—5021no—
274C2×D4—2512no—
275C2×D4—2511no—
276C2×D4—2512no—
277C2×D4C22⋊D25111yescharacteristic, complemented
278C2×D4C22×D25111yescharacteristic, complemented
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—5022no—
282C22×C4—5021no—
283C22×C4C22×D25111yescharacteristic
284Q16—22no—
285Q16—5022no—
286SD16—22no—
287SD16—22no—
288SD16—5022no—
289SD16—5022no—
290D8—22no—
291D8—5022no—
292M16—21no—
293M16—5021no—
294M16—2512no—
295M16C22×D25112yescomplemented
296C2×C8—21no—
297C2×C8—5021no—
298C4⋊C4—5021no—
299C22⋊C4—5024no—
300C22⋊C4—5021no—
301C42—2512no—
302C23—21no—
303C23—21no—
304C23—20084no—
305C23—20082no—
306C23—5028no—
307C23—5022no—
308C23—5022no—
309C23—5021no—
310C23—2514no—
311C23D4×D25111yescharacteristic
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8—5021no—
316Q8—2512no—
317Q8C2×C22⋊D25112yes—
318D4—22no—
319D4—22no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—41no—
324D4—41no—
325D4—20084no—
326D4—5028no—
327D4—5028no—
328D4—5022no—
329D4—5022no—
330D4—5022no—
331D4—5022no—
332D4—5022no—
333D4—5021no—
334D4—5022no—
335D4—2512no—
336D4C2×C22⋊D25112yes—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—5028no—
344C2×C4—5022no—
345C2×C4—5021no—
346C2×C4—5022no—
347C2×C4—5022no—
348C2×C4D4×D25111yescharacteristic
349C2×C4D4×D25111yescharacteristic
350C2×C4D4×D25111yescharacteristic
351C2×C4C2×C22⋊D25111yescharacteristic
352C2×C4C2×C22⋊D25111yescharacteristic
353C2×C4C23×D25111yescharacteristic
354C8—22no—
355C8—5022no—
356C22—22no—
357C22—41no—
358C22—42no—
359C22—42no—
360C22—41no—
361C22—20082no—
362C22—20088no—
363C22—10044no—
364C22—10042no—
365C22—10044no—
366C22—5024no—
367C22—5022no—
368C22#2254868311yescharacteristic
369C22#2254868311yescharacteristic
370C22C2×D4×D25111yescharacteristic
371C4—22no—
372C4—41no—
373C4—10042no—
374C4—5022no—
375C4#2254868311yescharacteristic
376C4#2254868311yescharacteristic
377C4C2×D4×D25111yescharacteristic
378C4C22×C22⋊D25111yescharacteristic
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—81no—
384C2—10042no—
385C2—10044no—
386C2C2×#2254868311yescharacteristic
subgroupG/H
centerC2C2×#22548683
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#575×C251C2
socleC2×C251C2×#120
hypercenter#575D251
hypocenterC251#3019
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. C2×#22548761C2semidirect
2. #30632485C2semidirect
3. #30632251C2semidirect
4. #30631863C2semidirect
5. #30631853C2semidirect
6. #30631704C2semidirect
7. #30631703C2semidirect
8. C2×#22548631C2semidirect
9. #575×C251C2semidirect
21. #22548746C22semidirect
23. #22548741C22semidirect
26. C2×#16860794C22semidirect
27. #22548711C22semidirect
28. #22548704C22semidirect
31. #22548674C22semidirect
32. #22548642C22semidirect
33. #22548641C22semidirect
34. #22548631C22semidirect
36. C2×#41×C251C22semidirect
38. #137×C251C22semidirect
40. AGL(1,ℤ/8)×C251C22semidirect
42. C2×M16×C251C22semidirect
51. #16860810D4semidirect
53. C2×Dic502D4semidirect
62. C2×D1004D4semidirect
78. #16860794C23semidirect
83. C4×Dic251C23semidirect
90. #41×C251D4semidirect
92. C2×Q8×C251D4semidirect
93. C2×Q8×C251C23semidirect
96. C2×D4×C251D4semidirect
97. C2×D4×C251C23semidirect
106. M16×C251C23semidirect
193. C251#3019semidirect
197. #575D251semidirect
212. C2×#41C2×D251semidirect
220. #137C2×D251semidirect
224. AGL(1,ℤ/8)C2×D251semidirect
228. C2×M16C2×D251semidirect
253. #41C22⋊D251semidirect
256. C2×Q8C22⋊D251semidirect
257. C2×Q8C22×D251semidirect
277. C2×D4C22⋊D251semidirect
278. C2×D4C22×D251semidirect
295. M16C22×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487612
#306324852
#306322512
#306318632
#306318532
#306317042
#306317032
C2×#225486312
#575×C2512
#3019251