Finite Group Databasebeta

← #37584135

Subgroups — #37584135

Order 24448.

556 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 125 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×#20259349C212yesmaximal, complemented
2#27133335C211yesmaximal, characteristic, complemented
3#27133298C211yesmaximal, characteristic, complemented
4#27133039C211yesmaximal, characteristic, complemented
5#27132915C212yesmaximal, complemented
6#27132859C211yesmaximal, characteristic, complemented
7#27132807C212yesmaximal, complemented
8#27132798C212yesmaximal, complemented
9C4×#15690274C211yesmaximal, characteristic, complemented
10#27132740C211yesmaximal, characteristic, complemented
11#460×C191C211yesmaximal, characteristic, complemented
12C2×#15690288C2212yescomplemented
13C2×D4×D191—21no—
14C2×D4×D191C2212yes—
15#20259349—28no—
16C2×#15690274C2211yescharacteristic
17C2×D1528C2212yes—
18C2×#15690272C2212yescomplemented
19C2×#15690272C2212yescomplemented
20C2×#15690270C2212yescomplemented
21#20259319—21no—
22#20259319C2211yescharacteristic, complemented
23#20259297—22no—
24#20259297C2212yescomplemented
25#20259293—21no—
26#20259293C2212yes—
27#20259290—22no—
28C4×D764—21no—
29C4×D764C2211yescharacteristic
30C4×D764C2211yescharacteristic
31#20259253C2211yescharacteristic, complemented
32#20259246C2211yescharacteristic, complemented
33#20259244C2212yescomplemented
34#20259242C2211yescharacteristic
35(C2×D191)⋊C8C2212yescomplemented
36C4×C191⋊C8C2211yescharacteristic, complemented
37C2×SD16×C191C2212yescomplemented
38#127×C191C2211yescharacteristic
39C4×Q8×C191C2211yescharacteristic, complemented
40C4⋊C8×C191C2211yescharacteristic, complemented
41#102×C191C2212yescomplemented
42C23×D191—22no—
43#15690288—24no—
44#15690288—42no—
45C2×D764—22no—
46C2×D764—22no—
47C2×D764—21no—
48C2×D764—21no—
49C2×D764D411yescharacteristic
50C2×D764C2312yes—
51C2×D764C2311yescharacteristic
52C2×C22⋊D191—22no—
53C2×C22⋊D191—22no—
54D4×D191—24no—
55D4×D191—42no—
56D4×D191—44no—
57C2×C4×D191—22no—
58C2×C4×D191—21no—
59C2×C4×D191C2312yes—
60#15690274D414yescomplemented
61D1528—24no—
62#15690272—24no—
63#15690272—42no—
64#15690270—24no—
65C2×C191⋊C8C2311yescharacteristic
66C2×C191⋊C8C2311yescharacteristic
67C4⋊Dic191D411yescharacteristic
68C4⋊Dic191C2311yescharacteristic
69C22⋊Dic191—22no—
70(C2×D191)⋊C4—22no—
71(C2×D191)⋊C4—21no—
72(C2×D191)⋊C4D412yes—
73C2×Q8×C191C2311yescharacteristic
74C2×D4×C191—21no—
75C2×D4×C191—21no—
76C2×D4×C191C2312yes—
77SD16×C191—24no—
78C2×C8×C191C2312yes—
79C4⋊C4×C191—21no—
80C4⋊C4×C191C2311yescharacteristic
81C42×C191—21no—
82C42×C191C2311yescharacteristic
83C22×D191—21no—
84C22×D191—22no—
85C22×D191—44no—
86C22×D191—44no—
87C22×D191—82no—
88C22×D191C2×D412yes—
89D764—22no—
90D764—22no—
91D764—21no—
92D764—22no—
93D764—44no—
94D764—42no—
95D764—81no—
96D764—82no—
97D764C2×D412yes—
98C22⋊D191—44no—
99C22⋊D191—44no—
100C2×Dic191—21no—
101C2×Dic191C2×D412yes—
102C4×D191—22no—
103C4×D191—22no—
104C4×D191—44no—
105C4×D191—42no—
106C191⋊C8—21no—
107C191⋊C8C2×D412yescomplemented
108C23×C191—22no—
109Q8×C191—21no—
110Q8×C191C2×D412yescomplemented
111D4×C191—22no—
112D4×C191—22no—
113D4×C191—42no—
114D4×C191—81no—
115C2×C4×C191—21no—
116C2×C4×C191—21no—
117C2×C4×C191C2×D411yescharacteristic
118C2×C4×C191C2×D411yescharacteristic
119C2×C4×C191C2411yescharacteristic
120C8×C191—22no—
121C22×C191—42no—
122C22×C191—42no—
123C22×C191—42no—
124C22×C191C22×D411yescharacteristic
125C2×D191—22no—
126C2×D191—22no—
127C2×D191—22no—
128C2×D191—22no—
129C2×D191—41no—
130C2×D191—41no—
131C2×D191—42no—
132C2×D191—42no—
133C2×D191—42no—
134C2×D191—88no—
135C4×C191—22no—
136C4×C191—21no—
137C4×C191—41no—
138C4×C191AGL(1,ℤ/8)12yescomplemented
139C4×C191C22×D411yescharacteristic
140C4×C191H(5,𝔽2)11yescharacteristic
141Dic191—22no—
142Dic191—41no—
143C2×C191—82no—
144C2×C191D4211yescharacteristic
145C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C191#57611yescharacteristic
147D191—42no—
148D191—42no—
149D191—82no—
150C191#327411yescharacteristic, complemented, Sylow
151#3274—1911nomaximal, Sylow
152C2×AGL(1,ℤ/8)—1912no—
153#550—1911no—
154D42—1911no—
155#493—1911no—
156#469—1912no—
157#460D19111yescharacteristic, complemented
158#459—1911no—
159#449—1912no—
160#447—1912no—
161C4×SD16—1911no—
162#436—1911no—
163C2×#41—1912no—
164C22×D4—3821no—
165C22×D4—1912no—
166AGL(1,ℤ/8)—3828no—
167C2×SD16—1911no—
168C2×SD16C2×D19112yescomplemented
169C2×D8—1912no—
170C2×D8—1912no—
171C2×D8—1912no—
172C2×M16—1912no—
173#127—3821no—
174#127—1911no—
175#127C2×D19111yescharacteristic
176#121—3821no—
177#121—3822no—
178#121—1912no—
179#121—1912no—
180#120—3822no—
181C4×Q8C2×D19111yescharacteristic, complemented
182C4×D4—3821no—
183C4×D4—1911no—
184C4×D4—1911no—
185#106—1911no—
186C4⋊C8C2×D19111yescharacteristic, complemented
187#103—1911no—
188#102—1912no—
189#102—1911no—
190#102C2×D19112yescomplemented
191C22⋊C8—1912no—
192C4×C8—1911no—
193C24—3822no—
194#41—3824no—
195#41—7642no—
196C2×Q8C22×D19111yescharacteristic
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—3822no—
200C2×D4—3822no—
201C2×D4—3824no—
202C2×D4—3822no—
203C2×D4—3822no—
204C2×D4—3821no—
205C2×D4—3821no—
206C2×D4—1911no—
207C2×D4—1912no—
208C2×D4—1911no—
209C2×D4—7642no—
210C2×D4—7644no—
211C2×D4C22×D19112yes—
212C22×C4—3822no—
213C22×C4—3821no—
214C22×C4—1912no—
215SD16—24no—
216SD16—1914no—
217D8—3824no—
218D8—3824no—
219D8—7642no—
220M16—3824no—
221C2×C8—1911no—
222C2×C8—1911no—
223C2×C8C22×D19112yes—
224C4⋊C4—21no—
225C4⋊C4—1911no—
226C4⋊C4—1911no—
227C4⋊C4C22×D19111yescharacteristic
228C22⋊C4—3822no—
229C22⋊C4—3822no—
230C22⋊C4—3821no—
231C22⋊C4—1912no—
232C42—21no—
233C42C22×D19111yescharacteristic
234C23—22no—
235C23—3821no—
236C23—3822no—
237C23—1912no—
238C23—15282no—
239C23—7644no—
240C23—7644no—
241Q8—21no—
242Q8D4×D19112yescomplemented
243D4—22no—
244D4—22no—
245D4—42no—
246D4—81no—
247D4—3822no—
248D4—3822no—
249D4—3821no—
250D4—3822no—
251D4—1912no—
252D4—15281no—
253D4—15282no—
254D4—7644no—
255D4—7644no—
256D4—7644no—
257D4—7642no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—3822no—
261C2×C4—3822no—
262C2×C4—3821no—
263C2×C4—1912no—
264C2×C4—7644no—
265C2×C4—7642no—
266C2×C4D4×D19111yescharacteristic
267C2×C4D4×D19111yescharacteristic
268C2×C4C23×D19111yescharacteristic
269C8—22no—
270C8—3821no—
271C8—1912no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—3822no—
276C22—3822no—
277C22—3822no—
278C22—3822no—
279C22—15288no—
280C22—7642no—
281C22—7642no—
282C22—7642no—
283C22—7641no—
284C22—7641no—
285C22C2×D4×D19111yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—3822no—
290C4—7641no—
291C4#2025934912yescomplemented
292C4C2×D4×D19111yescharacteristic
293C4#2025936811yescharacteristic
294C2—82no—
295C2—15282no—
296C2—7642no—
297C2—7642no—
298C2#2713329711yescharacteristic
299C2C2×#2025934911yescharacteristic
300C2#2713346711yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#460×C191C2
socleC22×C191C22×D4
hypercenter#460D191
hypocenterC191#3274
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#20259349C2semidirect
2. #27133335C2semidirect
3. #27133298C2semidirect
4. #27133039C2semidirect
5. #27132915C2semidirect
6. #27132859C2semidirect
7. #27132807C2semidirect
8. #27132798C2semidirect
9. C4×#15690274C2semidirect
10. #27132740C2semidirect
11. #460×C191C2semidirect
12. C2×#15690288C22semidirect
18. C2×#15690272C22semidirect
19. C2×#15690272C22semidirect
20. C2×#15690270C22semidirect
22. #20259319C22semidirect
24. #20259297C22semidirect
31. #20259253C22semidirect
32. #20259246C22semidirect
33. #20259244C22semidirect
35. (C2×D191)⋊C8C22semidirect
36. C4×C191⋊C8C22semidirect
37. C2×SD16×C191C22semidirect
39. C4×Q8×C191C22semidirect
40. C4⋊C8×C191C22semidirect
41. #102×C191C22semidirect
60. #15690274D4semidirect
107. C191⋊C8C2×D4semidirect
110. Q8×C191C2×D4semidirect
138. C4×C191AGL(1,ℤ/8)semidirect
150. C191#3274semidirect
157. #460D191semidirect
168. C2×SD16C2×D191semidirect
181. C4×Q8C2×D191semidirect
186. C4⋊C8C2×D191semidirect
190. #102C2×D191semidirect
242. Q8D4×D191semidirect
291. C4#20259349semidirect

Maximal subgroups

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

subgroupindex
C2×#202593492
#271333352
#271332982
#271330392
#271329152
#271328592
#271328072
#271327982
C4×#156902742
#271327402
#460×C1912
#3274191