Finite Group Databasebeta

← #469×D191 #37582112

Subgroups — #469×D191 #37582112

Order 24448.

596 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 135 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×D8×D191C211yesmaximal, characteristic, complemented
2C2×M16×D191C211yesmaximal, characteristic, complemented
3#121×D191C212yesmaximal, complemented
4#27132917C211yesmaximal, characteristic, complemented
5#27132916C211yesmaximal, characteristic, complemented
6#27132915C212yesmaximal, complemented
7#27132914C212yesmaximal, complemented
8#27132913C211yesmaximal, characteristic, complemented
9#106×D191C211yesmaximal, characteristic, complemented
10#102×D191C212yesmaximal, complemented
11#469×C191C211yesmaximal, characteristic, complemented
12C2×D4×D191—22no—
13C2×D4×D191C2212yes—
14C22×C4×D191C2211yescharacteristic
15C2×D1528C2211yescharacteristic, complemented
16C2×#15690272C2212yescomplemented
17D8×D191—24no—
18C2×#15690271C2211yescharacteristic, complemented
19C2×#15690270C2211yescharacteristic, complemented
20C2×#15690270C2211yescharacteristic, complemented
21M16×D191—24no—
22C2×C8×D191C2211yescharacteristic
23C2×C8×D191C2211yescharacteristic
24#20259297C2212yescomplemented
25#20259296C2212yescomplemented
26#20259295—22no—
27#20259294—22no—
28#20259293C2212yescomplemented
29C4⋊C4×D191C2212yes—
30C22⋊C4×D191—22no—
31#20259254C2211yescharacteristic, complemented
32#20259253C2212yescomplemented
33#20259244C2212yescomplemented
34#20259243C2212yescomplemented
35#20259242C2212yescomplemented
36C2×D8×C191C2211yescharacteristic, complemented
37C2×M16×C191C2211yescharacteristic, complemented
38#121×C191C2212yescomplemented
39#106×C191C2211yescharacteristic, complemented
40#102×C191C2212yescomplemented
41C23×D191—22no—
42C23×D191D411yescharacteristic
43C2×D764—22no—
44C2×D764C2312yes—
45C2×C22⋊D191—22no—
46C2×C22⋊D191—22no—
47C2×C22⋊D191—22no—
48D4×D191—24no—
49D4×D191—24no—
50D4×D191—48no—
51C22×Dic191D411yescharacteristic
52C2×C4×D191—22no—
53C2×C4×D191—22no—
54C2×C4×D191—22no—
55C2×C4×D191D411yescharacteristic
56C2×C4×D191D411yescharacteristic
57C2×C4×D191C2311yescharacteristic
58C2×C4×D191C2311yescharacteristic
59D1528—22no—
60#15690272—24no—
61#15690271—22no—
62#15690270—22no—
63#15690270—22no—
64C2×C191⋊C8C2311yescharacteristic
65C2×C191⋊C8C2311yescharacteristic
66C8×D191—22no—
67C8×D191D414yescomplemented
68C4⋊Dic191C2312yes—
69Dic191⋊C4—22no—
70C22⋊Dic191—22no—
71(C2×D191)⋊C4—22no—
72(C2×D191)⋊C4—22no—
73C2×D4×C191—22no—
74C2×D4×C191C2312yes—
75C22×C4×C191C2311yescharacteristic
76D8×C191—22no—
77M16×C191—22no—
78C2×C8×C191C2311yescharacteristic
79C2×C8×C191C2311yescharacteristic
80C4⋊C4×C191C2312yes—
81C22⋊C4×C191—22no—
82C22×D191—22no—
83C22×D191—24no—
84C22×D191—22no—
85C22×D191—44no—
86C22×D191—44no—
87C22×D191—44no—
88C22×D191C2×D411yescharacteristic
89C22×D191C2×D411yescharacteristic
90D764—22no—
91D764—22no—
92D764—44no—
93C22⋊D191—44no—
94C22⋊D191—44no—
95C22⋊D191—44no—
96C2×Dic191—22no—
97C2×Dic191—22no—
98C2×Dic191C2×D411yescharacteristic
99C2×Dic191C2×D411yescharacteristic
100C4×D191—22no—
101C4×D191—21no—
102C4×D191—21no—
103C4×D191—22no—
104C4×D191—44no—
105C4×D191C2×D412yes—
106C4×D191#4112yes—
107C191⋊C8—21no—
108C191⋊C8C2×D412yescomplemented
109C23×C191—22no—
110C23×C191C2×D411yescharacteristic
111D4×C191—22no—
112D4×C191—22no—
113D4×C191—44no—
114C2×C4×C191—21no—
115C2×C4×C191—21no—
116C2×C4×C191—22no—
117C2×C4×C191C2×D411yescharacteristic
118C2×C4×C191C2411yescharacteristic
119C8×C191—21no—
120C8×C191C2×D412yescomplemented
121C22×C191—21no—
122C22×C191—22no—
123C22×C191—42no—
124C22×C191—42no—
125C22×C191—42no—
126C22×C191C22×D411yescharacteristic
127C2×D191—21no—
128C2×D191—22no—
129C2×D191—44no—
130C2×D191—42no—
131C2×D191—42no—
132C2×D191—42no—
133C2×D191—88no—
134C2×D191#12112yes—
135C2×D191AGL(1,ℤ/8)14yes—
136C4×C191—21no—
137C4×C191—42no—
138C4×C191C22×D411yescharacteristic
139C4×C191C2×#4111yescharacteristic
140Dic191—21no—
141Dic191—42no—
142Dic191#12112yes—
143C2×C191—41no—
144C2×C191—82no—
145C2×C191C2×#12111yescharacteristic
146C2×C191C2×AGL(1,ℤ/8)12yes—
147D191—41no—
148D191—82no—
149D191#46914yesdirect product factor
150C191C2×#46911yescharacteristic, complemented, Sylow
151C2×#469—1911nomaximal, Sylow
152C22×D8—1911no—
153C22×M16—1911no—
154C2×#121—1912no—
155#469—1911no—
156#469—1911no—
157#469—1912no—
158#469—1911no—
159#469—1912no—
160#469D19111yescharacteristic, direct product factor
161C2×#106—1911no—
162C2×#102—1912no—
163C22×D4—3822no—
164C22×D4—1912no—
165C23×C4—1911no—
166C2×D8—3824no—
167C2×D8—1912no—
168C2×D8—1911no—
169C2×D8C2×D19111yescharacteristic, complemented
170C2×M16—3824no—
171C2×M16—1911no—
172C2×M16—1911no—
173C2×M16—1911no—
174C2×M16C2×D19111yescharacteristic, complemented
175C22×C8—1911no—
176C22×C8—1911no—
177#121—3822no—
178#121—3822no—
179#121—1912no—
180#121—1912no—
181#121—1912no—
182#121C2×D19112yescomplemented
183C2×C4⋊C4—1912no—
184C2×C22⋊C4—3822no—
185#106—1911no—
186#106—1912no—
187#106C2×D19111yescharacteristic, complemented
188#102—1912no—
189#102—1912no—
190#102—1912no—
191#102C2×D19112yescomplemented
192C24—3822no—
193C24—1911no—
194C2×D4—22no—
195C2×D4—3822no—
196C2×D4—3822no—
197C2×D4—3822no—
198C2×D4—3824no—
199C2×D4—3824no—
200C2×D4—3822no—
201C2×D4—1912no—
202C2×D4—7648no—
203C2×D4C22×D19112yes—
204C22×C4—3822no—
205C22×C4—3822no—
206C22×C4—3822no—
207C22×C4—1911no—
208C22×C4—1911no—
209C22×C4—1911no—
210C22×C4—1911no—
211C22×C4—1911no—
212C22×C4C22×D19111yescharacteristic
213D8—22no—
214D8—3824no—
215D8—3822no—
216M16—22no—
217M16—3822no—
218M16—3822no—
219M16—3822no—
220C2×C8—3822no—
221C2×C8—1911no—
222C2×C8—1911no—
223C2×C8—1914no—
224C2×C8C22×D19111yescharacteristic
225C2×C8C22×D19111yescharacteristic
226C4⋊C4—3822no—
227C4⋊C4—1912no—
228C4⋊C4C22×D19112yes—
229C22⋊C4—22no—
230C22⋊C4—3822no—
231C22⋊C4—3822no—
232C22⋊C4—3822no—
233C23—22no—
234C23—3822no—
235C23—3824no—
236C23—3822no—
237C23—1911no—
238C23—1911no—
239C23—7644no—
240C23—7644no—
241C23—7644no—
242C23D4×D19111yescharacteristic
243D4—22no—
244D4—22no—
245D4—44no—
246D4—3822no—
247D4—3822no—
248D4—7644no—
249D4—7644no—
250D4—7644no—
251D4—7644no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—3822no—
256C2×C4—3822no—
257C2×C4—3822no—
258C2×C4—3821no—
259C2×C4—3821no—
260C2×C4—3822no—
261C2×C4—1911no—
262C2×C4—1911no—
263C2×C4—1912no—
264C2×C4—1912no—
265C2×C4—7644no—
266C2×C4D4×D19111yescharacteristic
267C2×C4C23×D19111yescharacteristic
268C8—21no—
269C8—3821no—
270C8—1912no—
271C8D4×D19112yescomplemented
272C22—22no—
273C22—21no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—3822no—
278C22—3821no—
279C22—1914no—
280C22—1912no—
281C22—15288no—
282C22—7644no—
283C22—7642no—
284C22—7642no—
285C22—7642no—
286C22C2×D4×D19111yescharacteristic
287C4—21no—
288C4—42no—
289C4—3821no—
290C4—1912no—
291C4—7642no—
292C4C2×D4×D19111yescharacteristic
293C4#41×D19111yescharacteristic
294C2—41no—
295C2—82no—
296C2—1914no—
297C2—15282no—
298C2—7641no—
299C2#121×D19111yescharacteristic
300C2AGL(1,ℤ/8)×D19112yes—
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#469×C191C2
socleC22×C191C22×D4
hypercenter#469D191
hypocenterC191C2×#469
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×D8×D191C2semidirect
2. C2×M16×D191C2semidirect
3. #121×D191C2semidirect
4. #27132917C2semidirect
5. #27132916C2semidirect
6. #27132915C2semidirect
7. #27132914C2semidirect
8. #27132913C2semidirect
9. #106×D191C2semidirect
10. #102×D191C2semidirect
11. #469×C191C2semidirect
15. C2×D1528C22semidirect
16. C2×#15690272C22semidirect
18. C2×#15690271C22semidirect
19. C2×#15690270C22semidirect
20. C2×#15690270C22semidirect
24. #20259297C22semidirect
25. #20259296C22semidirect
28. #20259293C22semidirect
31. #20259254C22semidirect
32. #20259253C22semidirect
33. #20259244C22semidirect
34. #20259243C22semidirect
35. #20259242C22semidirect
36. C2×D8×C191C22semidirect
37. C2×M16×C191C22semidirect
38. #121×C191C22semidirect
39. #106×C191C22semidirect
40. #102×C191C22semidirect
67. C8×D191D4semidirect
108. C191⋊C8C2×D4semidirect
120. C8×C191C2×D4semidirect
149. D191#469direct (N × H)
150. C191C2×#469semidirect
160. #469D191direct (N × H)
169. C2×D8C2×D191semidirect
174. C2×M16C2×D191semidirect
182. #121C2×D191semidirect
187. #106C2×D191semidirect
191. #102C2×D191semidirect
271. C8D4×D191semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1912
C2×M16×D1912
#121×D1912
#271329172
#271329162
#271329152
#271329142
#271329132
#106×D1912
#102×D1912
#469×C1912
C2×#469191