Finite Group Databasebeta

← #37583270

Subgroups — #37583270

Order 24448.

520 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 121 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×#20259297C211yesmaximal, characteristic, complemented
2#121×D191C211yesmaximal, characteristic, complemented
3#27132978C212yesmaximal, complemented
4#27132976C211yesmaximal, characteristic, complemented
5#27132900C212yesmaximal, complemented
6#27132898C212yesmaximal, complemented
7#27132799C211yesmaximal, characteristic, complemented
8#27132798C212yesmaximal, complemented
9#27132796C211yesmaximal, characteristic, complemented
10C22⋊C8×D191C211yesmaximal, characteristic, complemented
11#480×C191C211yesmaximal, characteristic, complemented
12C22×D764—21no—
13C22×D764C2211yescharacteristic, complemented
14C2×D4×D191—21no—
15C2×D4×D191C2211yescharacteristic, complemented
16C22×C4×D191C2211yescharacteristic
17C2×D1528C2212yescomplemented
18C2×#15690272C2212yescomplemented
19C2×C8×D191C2212yescomplemented
20#20259297—21no—
21#20259297—21no—
22#20259297C2212yes—
23#20259297C2211yescharacteristic, complemented
24#20259297C2212yescomplemented
25#20259296C2211yescharacteristic
26#20259295—21no—
27#20259294—21no—
28#20259293C2211yescharacteristic, complemented
29C4⋊C4×D191C2211yescharacteristic, complemented
30C2×(C2×D191)⋊C4—21no—
31C22⋊C4×D191—21no—
32#20259255C2212yescomplemented
33#20259244C2212yescomplemented
34#20259242C2212yescomplemented
35#20259242C2212yescomplemented
36C22⋊(C191⋊C8)C2211yescharacteristic, complemented
37(C2×D191)⋊C8C2212yescomplemented
38#121×C191C2211yescharacteristic
39C2×C4⋊C4×C191C2211yescharacteristic, complemented
40#107×C191C2212yescomplemented
41#102×C191C2212yescomplemented
42C22⋊C8×C191C2211yescharacteristic, complemented
43C23×D191—21no—
44C23×D191—21no—
45C23×D191D411yescharacteristic
46C2×D764—21no—
47C2×D764—22no—
48C2×D764—22no—
49C2×D764—21no—
50C2×D764—42no—
51C2×D764—44no—
52C2×D764C2312yes—
53C2×D764C2311yescharacteristic
54C2×C22⋊D191—21no—
55C2×C22⋊D191—21no—
56C2×C22⋊D191—21no—
57D4×D191—42no—
58D4×D191—44no—
59C22×Dic191D411yescharacteristic
60C2×C4×D191—21no—
61C2×C4×D191—42no—
62C2×C4×D191D411yescharacteristic
63C2×C4×D191D411yescharacteristic
64C2×C4×D191C2311yescharacteristic
65C2×C4×D191C2311yescharacteristic
66D1528—42no—
67#15690272—42no—
68C2×C191⋊C8C2312yes—
69C8×D191—42no—
70C4⋊Dic191C2311yescharacteristic
71Dic191⋊C4—21no—
72C22⋊Dic191—21no—
73(C2×D191)⋊C4—21no—
74(C2×D191)⋊C4—22no—
75(C2×D191)⋊C4—22no—
76(C2×D191)⋊C4—21no—
77C2×D4×C191—21no—
78C2×D4×C191C2311yescharacteristic
79C22×C4×C191—21no—
80C22×C4×C191C2311yescharacteristic
81C2×C8×C191C2312yes—
82C4⋊C4×C191—21no—
83C4⋊C4×C191C2312yes—
84C4⋊C4×C191C2311yescharacteristic
85C22⋊C4×C191—21no—
86C22×D191—21no—
87C22×D191—22no—
88C22×D191—41no—
89C22×D191—41no—
90C22×D191—42no—
91C22×D191—42no—
92C22×D191—42no—
93C22×D191—42no—
94C22×D191—42no—
95C22×D191—42no—
96C22×D191D814yescomplemented
97C22×D191C2×D411yescharacteristic
98C22×D191C2×D411yescharacteristic
99D764—22no—
100D764—22no—
101D764—22no—
102D764—42no—
103D764—42no—
104D764—44no—
105D764—44no—
106C22⋊D191—42no—
107C22⋊D191—42no—
108C22⋊D191—42no—
109C2×Dic191—21no—
110C2×Dic191—41no—
111C2×Dic191C2×D411yescharacteristic
112C2×Dic191C2×D411yescharacteristic
113C4×D191—22no—
114C4×D191—22no—
115C4×D191—41no—
116C4×D191—41no—
117C4×D191—42no—
118C191⋊C8—22no—
119C23×C191—21no—
120C23×C191C2×D411yescharacteristic
121D4×C191—22no—
122D4×C191—42no—
123C2×C4×C191—22no—
124C2×C4×C191—21no—
125C2×C4×C191—22no—
126C2×C4×C191—42no—
127C2×C4×C191C2×D411yescharacteristic
128C2×C4×C191C2411yescharacteristic
129C8×C191—22no—
130C22×C191—21no—
131C22×C191—21no—
132C22×C191—41no—
133C22×C191—41no—
134C22×C191—41no—
135C22×C191C2×D812yes—
136C22×C191C22×D411yescharacteristic
137C2×D191—21no—
138C2×D191—21no—
139C2×D191—21no—
140C2×D191—21no—
141C2×D191—44no—
142C2×D191—41no—
143C2×D191—42no—
144C2×D191—41no—
145C2×D191—42no—
146C2×D191—41no—
147C2×D191—42no—
148C2×D191—84no—
149C2×D191—84no—
150C2×D191C2×D812yes—
151C2×D191C2×D812yes—
152C4×C191—21no—
153C4×C191—42no—
154C4×C191—41no—
155C4×C191H(5,𝔽2)12yes—
156Dic191—21no—
157Dic191—21no—
158Dic191—41no—
159C2×C191—22no—
160C2×C191—81no—
161C2×C191#53411yescharacteristic
162C2×C191C22×D811yescharacteristic
163C2×C191#57511yescharacteristic
164D191—22no—
165D191—22no—
166D191—81no—
167D191—82no—
168C191#318611yescharacteristic, complemented, Sylow
169#3186—1911nomaximal, Sylow
170C2×#121—1911no—
171C2×#121—1911no—
172#480—1911no—
173#480—1912no—
174#480D19111yescharacteristic, complemented
175#466—1912no—
176#466—1912no—
177#447—1911no—
178#447—1912no—
179#447—1911no—
180C2×C22⋊C8—1911no—
181C22×D4—3821no—
182C22×D4—3821no—
183C22×D4—1911no—
184C22×D4—1911no—
185C23×C4—1911no—
186C2×D8—1912no—
187C2×D8—1912no—
188C22×C8—1912no—
189#121—3821no—
190#121—3821no—
191#121—3821no—
192#121—3821no—
193#121—1911no—
194#121—1911no—
195#121—1912no—
196#121—1911no—
197#121—1912no—
198#121C2×D19111yescharacteristic
199C2×C4⋊C4—1911no—
200C2×C4⋊C4C2×D19111yescharacteristic, complemented
201C2×C22⋊C4—3821no—
202C2×C22⋊C4—3821no—
203#107—1912no—
204#107C2×D19112yescomplemented
205#102—1912no—
206#102—1912no—
207#102—1912no—
208#102C2×D19112yescomplemented
209C22⋊C8—1911no—
210C22⋊C8—1912no—
211C22⋊C8C2×D19111yescharacteristic, complemented
212C24—3821no—
213C24—3821no—
214C24—1911no—
215C2×D4—21no—
216C2×D4—3821no—
217C2×D4—3822no—
218C2×D4—3822no—
219C2×D4—3821no—
220C2×D4—3821no—
221C2×D4—3821no—
222C2×D4—3821no—
223C2×D4—1912no—
224C2×D4—1911no—
225C2×D4—7642no—
226C2×D4—7644no—
227C2×D4—7642no—
228C2×D4—7644no—
229C2×D4C22×D19111yescharacteristic
230C22×C4—21no—
231C22×C4—3821no—
232C22×C4—1911no—
233C22×C4—1911no—
234C22×C4—1911no—
235C22×C4—1911no—
236C22×C4—1911no—
237C22×C4—7642no—
238C22×C4C22×D19111yescharacteristic
239D8—7642no—
240D8—7642no—
241C2×C8—1912no—
242C2×C8—7642no—
243C2×C8C22×D19112yes—
244C4⋊C4—21no—
245C4⋊C4—3821no—
246C4⋊C4—1911no—
247C4⋊C4C22×D19111yescharacteristic
248C4⋊C4C22×D19112yes—
249C22⋊C4—21no—
250C22⋊C4—3821no—
251C22⋊C4—3822no—
252C22⋊C4—3822no—
253C22⋊C4—3821no—
254C22⋊C4—3821no—
255C23—21no—
256C23—3822no—
257C23—3821no—
258C23—1911no—
259C23—1911no—
260C23—1914no—
261C23—7641no—
262C23—7641no—
263C23—7642no—
264C23—7642no—
265C23—7642no—
266C23—7642no—
267C23—7642no—
268C23—7642no—
269C23D4×D19111yescharacteristic
270D4—22no—
271D4—42no—
272D4—3822no—
273D4—3822no—
274D4—3822no—
275D4—7642no—
276D4—7644no—
277D4—7644no—
278D4—7642no—
279D4—7642no—
280D4—7642no—
281D4—7642no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—42no—
286C2×C4—3822no—
287C2×C4—3822no—
288C2×C4—3821no—
289C2×C4—1911no—
290C2×C4—1911no—
291C2×C4—7641no—
292C2×C4—7641no—
293C2×C4—7642no—
294C2×C4—7641no—
295C2×C4D4×D19111yescharacteristic
296C2×C4C23×D19111yescharacteristic
297C8—22no—
298C8—3822no—
299C22—21no—
300C22—21no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—3821no—
305C22—3821no—
306C22—3821no—
307C22—3821no—
308C22—1912no—
309C22—1912no—
310C22—15284no—
311C22—15284no—
312C22—7644no—
313C22—7642no—
314C22—7641no—
315C22—7642no—
316C22—7641no—
317C22—7642no—
318C22—7641no—
319C22D8×D19112yes—
320C22C2×D4×D19111yescharacteristic
321C4—21no—
322C4—42no—
323C4—41no—
324C4—3821no—
325C4—3821no—
326C4—7641no—
327C4#2025936812yes—
328C2—22no—
329C2—81no—
330C2—3822no—
331C2—3822no—
332C2—15282no—
333C2—15281no—
334C2#2713322411yescharacteristic
335C2C2×D8×D19111yescharacteristic
336C2#2713346211yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#480×C191C2
socleC22×C191C22×D4
hypercenter#480D191
hypocenterC191#3186
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. C2×#20259297C2semidirect
2. #121×D191C2semidirect
3. #27132978C2semidirect
4. #27132976C2semidirect
5. #27132900C2semidirect
6. #27132898C2semidirect
7. #27132799C2semidirect
8. #27132798C2semidirect
9. #27132796C2semidirect
10. C22⋊C8×D191C2semidirect
11. #480×C191C2semidirect
13. C22×D764C22semidirect
15. C2×D4×D191C22semidirect
17. C2×D1528C22semidirect
18. C2×#15690272C22semidirect
19. C2×C8×D191C22semidirect
23. #20259297C22semidirect
24. #20259297C22semidirect
28. #20259293C22semidirect
29. C4⋊C4×D191C22semidirect
32. #20259255C22semidirect
33. #20259244C22semidirect
34. #20259242C22semidirect
35. #20259242C22semidirect
36. C22⋊(C191⋊C8)C22semidirect
37. (C2×D191)⋊C8C22semidirect
39. C2×C4⋊C4×C191C22semidirect
40. #107×C191C22semidirect
41. #102×C191C22semidirect
42. C22⋊C8×C191C22semidirect
96. C22×D191D8semidirect
168. C191#3186semidirect
174. #480D191semidirect
200. C2×C4⋊C4C2×D191semidirect
204. #107C2×D191semidirect
208. #102C2×D191semidirect
211. C22⋊C8C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202592972
#121×D1912
#271329782
#271329762
#271329002
#271328982
#271327992
#271327982
#271327962
C22⋊C8×D1912
#480×C1912
#3186191