Finite Group Databasebeta

← #37581875

Subgroups — #37581875

Order 24448.

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×#20259368C211yesmaximal, characteristic, complemented
2#27133461C211yesmaximal, characteristic, complemented
3#27133227C211yesmaximal, characteristic, complemented
4#27132839C214yesmaximal, complemented
5#27132829C214yesmaximal, complemented
6#27132680C211yesmaximal, characteristic, complemented
7#27132679C211yesmaximal, characteristic, complemented
8C2×#20259238C211yesmaximal, characteristic, complemented
9#575×C191C211yesmaximal, characteristic, complemented
10#20259368—22no—
11#20259368—22no—
12#20259368—22no—
13#20259368C2214yes—
14C2×#15690288—21no—
15C2×#15690287—21no—
16C2×#15690287C2211yescharacteristic
17C22×D764—21no—
18C22×D764C2211yescharacteristic
19C2×D4×D191—22no—
20C2×D4×D191—21no—
21#20259353C2212yescomplemented
22#20259352—21no—
23#20259348C2212yescomplemented
24#20259347—21no—
25#20259342—22no—
26C2×#15690271C2211yescharacteristic, complemented
27#20259318C2212yescomplemented
28#20259311C2212yescomplemented
29#20259296—22no—
30#20259290—22no—
31#20259281C2211yescharacteristic, complemented
32#20259249C2214yescomplemented
33#20259248C2214yescomplemented
34#20259238C2214yescomplemented
35C2×#41×C191C2211yescharacteristic
36C2×#41×C191C2211yescharacteristic, complemented
37#137×C191—21no—
38#137×C191C2212yescomplemented
39AGL(1,ℤ/8)×C191—21no—
40AGL(1,ℤ/8)×C191C2212yescomplemented
41#135×C191—22no—
42C2×M16×C191C2211yescharacteristic, complemented
43C23×D191—22no—
44C23×D191D412yes—
45#15690288—24no—
46#15690288—24no—
47#15690287—24no—
48#15690287—24no—
49#15690287—21no—
50#15690287—21no—
51#15690287D412yescomplemented
52#15690287C2312yes—
53C2×Dic382D411yescharacteristic, complemented
54C2×D764—22no—
55C2×D764—24no—
56C2×D764—24no—
57C2×D764—21no—
58C2×D764—21no—
59C2×D764—21no—
60C2×D764—21no—
61C2×D764D412yes—
62C2×D764D411yescharacteristic, complemented
63C2×D764C2312yes—
64C2×C22⋊D191—22no—
65C2×C22⋊D191—24no—
66C2×C22⋊D191—21no—
67D4×D191—28no—
68D4×D191—24no—
69D4×D191—28no—
70D4×D191—24no—
71C2×C4×D191—22no—
72C2×C4×D191—21no—
73#15690277—22no—
74#15690275—22no—
75#15690274—22no—
76#15690272—22no—
77#15690271—21no—
78#15690271C2312yescomplemented
79C2×C191⋊C8—21no—
80C4⋊Dic191—21no—
81C22⋊Dic191—21no—
82(C2×D191)⋊C4—24no—
83C4×Dic191C2312yescomplemented
84#41×C191—21no—
85#41×C191—21no—
86#41×C191—21no—
87#41×C191—21no—
88#41×C191—21no—
89#41×C191—21no—
90#41×C191D412yescomplemented
91#41×C191C2312yes—
92C2×Q8×C191D411yescharacteristic, complemented
93C2×Q8×C191C2311yescharacteristic, complemented
94C2×D4×C191—21no—
95C2×D4×C191—21no—
96C2×D4×C191D411yescharacteristic, complemented
97C2×D4×C191C2311yescharacteristic, complemented
98C22×C4×C191—21no—
99C22×C4×C191—21no—
100C22×C4×C191C2311yescharacteristic
101Q16×C191—22no—
102SD16×C191—22no—
103SD16×C191—22no—
104D8×C191—22no—
105M16×C191—21no—
106M16×C191C2312yescomplemented
107C2×C8×C191—21no—
108C22×D191—22no—
109C22×D191—21no—
110C22×D191—28no—
111C22×D191—22no—
112C22×D191—82no—
113C22×D191—84no—
114C22×D191C2×D414yes—
115Dic382—21no—
116Dic382C2×D412yes—
117D764—28no—
118D764—22no—
119D764—22no—
120D764—21no—
121D764—22no—
122D764—22no—
123D764C2×D412yes—
124C22⋊D191—28no—
125C22⋊D191—22no—
126C22⋊D191—22no—
127C22⋊D191—84no—
128C2×Dic191—21no—
129C2×Dic191—22no—
130C4×D191—28no—
131C4×D191—22no—
132C4×D191—22no—
133C191⋊C8—22no—
134C23×C191—21no—
135C23×C191—21no—
136C23×C191C2×D411yescharacteristic
137Q8×C191—21no—
138Q8×C191—21no—
139Q8×C191—21no—
140Q8×C191C2×D412yes—
141D4×C191—22no—
142D4×C191—22no—
143D4×C191—21no—
144D4×C191—21no—
145D4×C191—21no—
146D4×C191—41no—
147D4×C191—41no—
148D4×C191C2×D412yes—
149C2×C4×C191—21no—
150C2×C4×C191—22no—
151C2×C4×C191—22no—
152C2×C4×C191—21no—
153C2×C4×C191—41no—
154C2×C4×C191—41no—
155C2×C4×C191C2×D411yescharacteristic
156C2×C4×C191C2×D411yescharacteristic
157C2×C4×C191C2×D411yescharacteristic
158C2×C4×C191C2×D411yescharacteristic
159C2×C4×C191C2×D411yescharacteristic
160C2×C4×C191C2411yescharacteristic
161C8×C191—22no—
162C22×C191—22no—
163C22×C191—41no—
164C22×C191—42no—
165C22×C191—42no—
166C22×C191—41no—
167C22×C191#12011yescharacteristic
168C22×C191#12011yescharacteristic
169C22×C191C22×D411yescharacteristic
170C2×D191—24no—
171C2×D191—22no—
172C2×D191—44no—
173C2×D191—42no—
174C2×D191—44no—
175C2×D191—82no—
176C2×D191—88no—
177C4×C191—22no—
178C4×C191—41no—
179C4×C191#12011yescharacteristic
180C4×C191#12011yescharacteristic
181C4×C191C22×D411yescharacteristic
182C4×C191C22×D411yescharacteristic
183Dic191—22no—
184Dic191—42no—
185C2×C191—21no—
186C2×C191—21no—
187C2×C191—21no—
188C2×C191—42no—
189C2×C191—81no—
190C2×C191C2×#12011yescharacteristic
191D191—44no—
192D191—42no—
193C191#301911yescharacteristic, complemented, Sylow
194#3019—1911nomaximal, Sylow
195C2×H(5,𝔽2)—1911no—
196#575—1911no—
197#575D19111yescharacteristic, complemented
198#535—1911no—
199#455—1914no—
200#453—1914no—
201#421—1911no—
202#421—1911no—
203C2×#100—1911no—
204H(5,𝔽2)—3822no—
205H(5,𝔽2)—3822no—
206H(5,𝔽2)—3822no—
207H(5,𝔽2)—1914no—
208C2×#41—3821no—
209C2×#41—3821no—
210C2×#41—1911no—
211C2×#41C2×D19111yescharacteristic
212C2×#41C2×D19111yescharacteristic, complemented
213C22×D4—3821no—
214C22×D4—3822no—
215C22×D4—3821no—
216C22×D4—1911no—
217#137—21no—
218#137—3821no—
219#137—1912no—
220#137C2×D19112yescomplemented
221AGL(1,ℤ/8)—21no—
222AGL(1,ℤ/8)—3821no—
223AGL(1,ℤ/8)—1912no—
224AGL(1,ℤ/8)C2×D19112yescomplemented
225#135—22no—
226#135—3822no—
227C2×M16—1911no—
228C2×M16C2×D19111yescharacteristic, complemented
229#127—1912no—
230#124—1912no—
231#121—3822no—
232#120—3822no—
233#117—1911no—
234#104—1914no—
235#104—1914no—
236#100—1914no—
237C24—3822no—
238C24—1912no—
239#41—21no—
240#41—21no—
241#41—21no—
242#41—21no—
243#41—21no—
244#41—21no—
245#41—3824no—
246#41—3824no—
247#41—3824no—
248#41—3824no—
249#41—3821no—
250#41—3821no—
251#41—1912no—
252#41—1912no—
253#41C22⋊D19112yescomplemented
254#41C22×D19112yes—
255C2×Q8—1911no—
256C2×Q8C22⋊D19111yescharacteristic, complemented
257C2×Q8C22×D19111yescharacteristic, complemented
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—3822no—
261C2×D4—3824no—
262C2×D4—3824no—
263C2×D4—3828no—
264C2×D4—3824no—
265C2×D4—3828no—
266C2×D4—3822no—
267C2×D4—3821no—
268C2×D4—3824no—
269C2×D4—3824no—
270C2×D4—3821no—
271C2×D4—3821no—
272C2×D4—3821no—
273C2×D4—3821no—
274C2×D4—1912no—
275C2×D4—1911no—
276C2×D4—1912no—
277C2×D4C22⋊D19111yescharacteristic, complemented
278C2×D4C22×D19111yescharacteristic, complemented
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—3822no—
282C22×C4—3821no—
283C22×C4C22×D19111yescharacteristic
284Q16—22no—
285Q16—3822no—
286SD16—22no—
287SD16—22no—
288SD16—3822no—
289SD16—3822no—
290D8—22no—
291D8—3822no—
292M16—21no—
293M16—3821no—
294M16—1912no—
295M16C22×D19112yescomplemented
296C2×C8—21no—
297C2×C8—3821no—
298C4⋊C4—3821no—
299C22⋊C4—3824no—
300C22⋊C4—3821no—
301C42—1912no—
302C23—21no—
303C23—21no—
304C23—3828no—
305C23—3822no—
306C23—3822no—
307C23—3821no—
308C23—1914no—
309C23—15284no—
310C23—15282no—
311C23D4×D19111yescharacteristic
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8—3821no—
316Q8—1912no—
317Q8C2×C22⋊D19112yes—
318D4—22no—
319D4—22no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—41no—
324D4—41no—
325D4—3828no—
326D4—3828no—
327D4—3822no—
328D4—3822no—
329D4—3822no—
330D4—3822no—
331D4—3822no—
332D4—3821no—
333D4—3822no—
334D4—1912no—
335D4—15284no—
336D4C2×C22⋊D19112yes—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—3828no—
344C2×C4—3821no—
345C2×C4—3822no—
346C2×C4—3822no—
347C2×C4—3822no—
348C2×C4D4×D19111yescharacteristic
349C2×C4D4×D19111yescharacteristic
350C2×C4D4×D19111yescharacteristic
351C2×C4C2×C22⋊D19111yescharacteristic
352C2×C4C2×C22⋊D19111yescharacteristic
353C2×C4C23×D19111yescharacteristic
354C8—22no—
355C8—3822no—
356C22—22no—
357C22—41no—
358C22—42no—
359C22—42no—
360C22—41no—
361C22—3824no—
362C22—3822no—
363C22—15282no—
364C22—15288no—
365C22—7644no—
366C22—7642no—
367C22—7644no—
368C22#2025929011yescharacteristic
369C22#2025929011yescharacteristic
370C22C2×D4×D19111yescharacteristic
371C4—22no—
372C4—41no—
373C4—3822no—
374C4—7642no—
375C4#2025929011yescharacteristic
376C4#2025929011yescharacteristic
377C4C2×D4×D19111yescharacteristic
378C4C22×C22⋊D19111yescharacteristic
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—81no—
384C2—7642no—
385C2—7644no—
386C2C2×#2025929011yescharacteristic
subgroupG/H
centerC2C2×#20259290
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#575×C191C2
socleC2×C191C2×#120
hypercenter#575D191
hypocenterC191#3019
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. C2×#20259368C2semidirect
2. #27133461C2semidirect
3. #27133227C2semidirect
4. #27132839C2semidirect
5. #27132829C2semidirect
6. #27132680C2semidirect
7. #27132679C2semidirect
8. C2×#20259238C2semidirect
9. #575×C191C2semidirect
21. #20259353C22semidirect
23. #20259348C22semidirect
26. C2×#15690271C22semidirect
27. #20259318C22semidirect
28. #20259311C22semidirect
31. #20259281C22semidirect
32. #20259249C22semidirect
33. #20259248C22semidirect
34. #20259238C22semidirect
36. C2×#41×C191C22semidirect
38. #137×C191C22semidirect
40. AGL(1,ℤ/8)×C191C22semidirect
42. C2×M16×C191C22semidirect
51. #15690287D4semidirect
53. C2×Dic382D4semidirect
62. C2×D764D4semidirect
78. #15690271C23semidirect
83. C4×Dic191C23semidirect
90. #41×C191D4semidirect
92. C2×Q8×C191D4semidirect
93. C2×Q8×C191C23semidirect
96. C2×D4×C191D4semidirect
97. C2×D4×C191C23semidirect
106. M16×C191C23semidirect
193. C191#3019semidirect
197. #575D191semidirect
212. C2×#41C2×D191semidirect
220. #137C2×D191semidirect
224. AGL(1,ℤ/8)C2×D191semidirect
228. C2×M16C2×D191semidirect
253. #41C22⋊D191semidirect
256. C2×Q8C22⋊D191semidirect
257. C2×Q8C22×D191semidirect
277. C2×D4C22⋊D191semidirect
278. C2×D4C22×D191semidirect
295. M16C22×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593682
#271334612
#271332272
#271328392
#271328292
#271326802
#271326792
C2×#202592382
#575×C1912
#3019191