Finite Group Databasebeta

← #496×D191 #37582950

Subgroups — #496×D191 #37582950

Order 24448.

668 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 668 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×SD16×D191C212yesmaximal, complemented
2C2×D8×D191C212yesmaximal, complemented
3#127×D191C211yesmaximal, characteristic, complemented
4#124×D191C211yesmaximal, characteristic, complemented
5#27133053C211yesmaximal, characteristic, complemented
6#27133052C211yesmaximal, characteristic, complemented
7#27133051C212yesmaximal, complemented
8#27133050C211yesmaximal, characteristic, complemented
9#27133049C212yesmaximal, complemented
10#97×D191C211yesmaximal, characteristic, complemented
11#496×C191C211yesmaximal, characteristic, complemented
12C2×Q8×D191C2211yescharacteristic
13C2×D4×D191—22no—
14C2×D4×D191C2212yes—
15C2×D4×D191C2211yescharacteristic
16C2×#15690276C2212yescomplemented
17C2×#15690275C2212yescomplemented
18C2×#15690274C2212yescomplemented
19SD16×D191—28no—
20C2×D1528C2212yescomplemented
21C2×#15690272C2212yescomplemented
22C2×#15690272C2212yescomplemented
23D8×D191—28no—
24C2×C8×D191C2212yes—
25#20259319C2211yescharacteristic, complemented
26#20259318—22no—
27#20259318C2212yescomplemented
28#20259311C2211yescharacteristic, complemented
29#20259310C2211yescharacteristic, complemented
30#20259309C2211yescharacteristic, complemented
31#20259308—22no—
32C22⋊C4×D191—22no—
33C42×D191C2211yescharacteristic
34#20259233C2211yescharacteristic, complemented
35#20259232C2212yescomplemented
36C2×SD16×C191C2212yescomplemented
37C2×D8×C191C2212yescomplemented
38#127×C191C2211yescharacteristic, complemented
39#124×C191C2211yescharacteristic, complemented
40#97×C191C2211yescharacteristic, complemented
41C23×D191—22no—
42C23×D191—21no—
43C2×Dic382—21no—
44C2×Dic382C2311yescharacteristic
45Q8×D191—24no—
46C2×D764—22no—
47C2×D764C2311yescharacteristic
48C2×D764C2312yes—
49C2×C22⋊D191—24no—
50C2×C22⋊D191—21no—
51C2×C22⋊D191—22no—
52D4×D191—24no—
53D4×D191—24no—
54D4×D191—24no—
55D4×D191—48no—
56C2×C4×D191—21no—
57C2×C4×D191D412yes—
58C2×C4×D191C2311yescharacteristic
59#15690276—24no—
60#15690275—24no—
61#15690274—24no—
62D1528—24no—
63#15690272—24no—
64#15690272—24no—
65C2×C191⋊C8C2312yes—
66C8×D191D418yescomplemented
67C22⋊Dic191—22no—
68(C2×D191)⋊C4—22no—
69(C2×D191)⋊C4—22no—
70C4×Dic191D412yes—
71C4×Dic191C2311yescharacteristic
72C2×Q8×C191C2311yescharacteristic
73C2×D4×C191—22no—
74C2×D4×C191C2311yescharacteristic
75C2×D4×C191C2312yes—
76SD16×C191—24no—
77D8×C191—24no—
78C2×C8×C191C2312yes—
79C22⋊C4×C191—22no—
80C42×C191C2311yescharacteristic
81C22×D191—21no—
82C22×D191—22no—
83C22×D191—42no—
84C22×D191—44no—
85C22×D191—48no—
86C22×D191—44no—
87C22×D191C2×D411yescharacteristic
88Dic382—22no—
89Dic382—42no—
90D764—22no—
91D764—22no—
92D764—22no—
93D764—44no—
94C22⋊D191—48no—
95C22⋊D191—42no—
96C22⋊D191—44no—
97C2×Dic191—21no—
98C2×Dic191C2×D412yes—
99C2×Dic191C2×D411yescharacteristic
100C4×D191—24no—
101C4×D191—42no—
102C4×D191C2×D414yes—
103C191⋊C8C2×D414yescomplemented
104C23×C191—21no—
105C23×C191—22no—
106Q8×C191—22no—
107D4×C191—22no—
108D4×C191—22no—
109D4×C191—22no—
110D4×C191—44no—
111C2×C4×C191—21no—
112C2×C4×C191C2×D412yes—
113C2×C4×C191C2411yescharacteristic
114C8×C191C2×D414yescomplemented
115C22×C191—41no—
116C22×C191—42no—
117C22×C191—44no—
118C22×C191—42no—
119C22×C191C22×D411yescharacteristic
120C2×D191—41no—
121C2×D191—42no—
122C2×D191—44no—
123C2×D191—42no—
124C2×D191—88no—
125C2×D191—84no—
126C2×D191#12712yes—
127C2×D191AGL(1,ℤ/8)14yes—
128C4×C191—22no—
129C4×C191—41no—
130C4×C191C22×D412yes—
131Dic191—22no—
132Dic191—41no—
133Dic191#12712yes—
134C2×C191—82no—
135C2×C191—81no—
136C2×C191C2×#12711yescharacteristic
137C2×C191C2×AGL(1,ℤ/8)12yes—
138D191—82no—
139D191—81no—
140D191#49614yesdirect product factor
141C191C2×#49611yescharacteristic, complemented, Sylow
142C2×#496—1911nomaximal, Sylow
143C22×SD16—1912no—
144C22×D8—1912no—
145C2×#127—1911no—
146C2×#124—1911no—
147#496—1912no—
148#496—1912no—
149#496—1911no—
150#496—1911no—
151#496—1911no—
152#496D19111yescharacteristic, direct product factor
153C2×#97—1911no—
154C22×Q8—1911no—
155C22×D4—3822no—
156C22×D4—1911no—
157C22×D4—1912no—
158C2×SD16—3828no—
159C2×SD16—1912no—
160C2×SD16—1912no—
161C2×SD16—1912no—
162C2×SD16C2×D19112yescomplemented
163C2×D8—3828no—
164C2×D8—1912no—
165C2×D8—1912no—
166C2×D8—1912no—
167C2×D8C2×D19112yescomplemented
168C22×C8—1912no—
169#127—3822no—
170#127—1912no—
171#127—1911no—
172#127C2×D19111yescharacteristic, complemented
173#124—3822no—
174#124—1911no—
175#124—1911no—
176#124—1911no—
177#124C2×D19111yescharacteristic, complemented
178C2×C22⋊C4—3822no—
179C2×C42—1911no—
180#97—1912no—
181#97—1911no—
182#97C2×D19111yescharacteristic, complemented
183C24—3821no—
184C24—3822no—
185C2×Q8—3821no—
186C2×Q8—3824no—
187C2×Q8—1911no—
188C2×Q8C22×D19111yescharacteristic
189C2×D4—22no—
190C2×D4—3824no—
191C2×D4—3824no—
192C2×D4—3824no—
193C2×D4—3821no—
194C2×D4—3822no—
195C2×D4—3824no—
196C2×D4—3822no—
197C2×D4—1911no—
198C2×D4—1912no—
199C2×D4—7648no—
200C2×D4C22×D19111yescharacteristic
201C2×D4C22×D19112yes—
202C22×C4—3821no—
203C22×C4—1912no—
204C22×C4—1911no—
205SD16—24no—
206SD16—3824no—
207SD16—3824no—
208SD16—3824no—
209D8—24no—
210D8—3824no—
211D8—3824no—
212D8—3824no—
213C2×C8—1912no—
214C2×C8—1918no—
215C2×C8C22×D19112yes—
216C22⋊C4—22no—
217C22⋊C4—3822no—
218C22⋊C4—3822no—
219C22⋊C4—3822no—
220C42—1912no—
221C42—1911no—
222C42C22×D19111yescharacteristic
223C23—21no—
224C23—22no—
225C23—3821no—
226C23—3822no—
227C23—1911no—
228C23—7642no—
229C23—7644no—
230C23—7648no—
231C23—7644no—
232Q8—22no—
233Q8—3822no—
234Q8—7642no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—3822no—
240D4—3822no—
241D4—3822no—
242D4—7644no—
243D4—7648no—
244D4—7642no—
245D4—7644no—
246C2×C4—21no—
247C2×C4—3824no—
248C2×C4—3821no—
249C2×C4—1912no—
250C2×C4—1911no—
251C2×C4—1914no—
252C2×C4—7642no—
253C2×C4D4×D19112yes—
254C2×C4C23×D19111yescharacteristic
255C8—1914no—
256C8D4×D19114yescomplemented
257C22—44no—
258C22—42no—
259C22—41no—
260C22—42no—
261C22—1914no—
262C22—1912no—
263C22—15288no—
264C22—15284no—
265C22—7644no—
266C22—7642no—
267C22—7641no—
268C22—7642no—
269C22C2×D4×D19111yescharacteristic
270C4—22no—
271C4—41no—
272C4—3822no—
273C4—1912no—
274C4—7641no—
275C4C2×D4×D19112yes—
276C2—82no—
277C2—81no—
278C2—1914no—
279C2—15282no—
280C2—15281no—
281C2#127×D19111yescharacteristic
282C2AGL(1,ℤ/8)×D19112yes—
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#496×C191C2
socleC22×C191C22×D4
hypercenter#496D191
hypocenterC191C2×#496
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D191C2semidirect
2. C2×D8×D191C2semidirect
3. #127×D191C2semidirect
4. #124×D191C2semidirect
5. #27133053C2semidirect
6. #27133052C2semidirect
7. #27133051C2semidirect
8. #27133050C2semidirect
9. #27133049C2semidirect
10. #97×D191C2semidirect
11. #496×C191C2semidirect
16. C2×#15690276C22semidirect
17. C2×#15690275C22semidirect
18. C2×#15690274C22semidirect
20. C2×D1528C22semidirect
21. C2×#15690272C22semidirect
22. C2×#15690272C22semidirect
25. #20259319C22semidirect
27. #20259318C22semidirect
28. #20259311C22semidirect
29. #20259310C22semidirect
30. #20259309C22semidirect
34. #20259233C22semidirect
35. #20259232C22semidirect
36. C2×SD16×C191C22semidirect
37. C2×D8×C191C22semidirect
38. #127×C191C22semidirect
39. #124×C191C22semidirect
40. #97×C191C22semidirect
66. C8×D191D4semidirect
103. C191⋊C8C2×D4semidirect
114. C8×C191C2×D4semidirect
140. D191#496direct (N × H)
141. C191C2×#496semidirect
152. #496D191direct (N × H)
162. C2×SD16C2×D191semidirect
167. C2×D8C2×D191semidirect
172. #127C2×D191semidirect
177. #124C2×D191semidirect
182. #97C2×D191semidirect
256. C8D4×D191semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1912
C2×D8×D1912
#127×D1912
#124×D1912
#271330532
#271330522
#271330512
#271330502
#271330492
#97×D1912
#496×C1912
C2×#496191