Finite Group Databasebeta

← #37582127

Subgroups — #37582127

Order 24448.

500 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 127 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×#20259341C211yesmaximal, characteristic, complemented
2C2×M16×D191C211yesmaximal, characteristic, complemented
3#27133173C212yesmaximal, complemented
4#27132922C211yesmaximal, characteristic, complemented
5#27132921C211yesmaximal, characteristic, complemented
6#27132913C211yesmaximal, characteristic, complemented
7#27132908C212yesmaximal, complemented
8#27132907C212yesmaximal, complemented
9#27132698C211yesmaximal, characteristic, complemented
10#27132656C212yesmaximal, complemented
11#469×C191C211yesmaximal, characteristic, complemented
12C2×#15690286C2212yes—
13C22×C4×D191C2211yescharacteristic
14#20259341—24no—
15C2×Dic764C2211yescharacteristic, complemented
16C2×#15690275C2212yescomplemented
17C2×#15690271C2211yescharacteristic
18C2×#15690270C2211yescharacteristic
19C2×#15690270C2211yescharacteristic
20M16×D191—24no—
21C2×C8×D191C2211yescharacteristic
22C2×C8×D191C2211yescharacteristic
23#20259304—22no—
24#20259302C2212yes—
25#20259301C2212yes—
26#20259293C2212yes—
27#20259290—22no—
28D4×Dic191—22no—
29(C22⋊D191)⋊C4—22no—
30#20259280C2212yes—
31#20259255C2212yescomplemented
32#20259254C2211yescharacteristic, complemented
33#20259247C2212yescomplemented
34#20259245C2212yescomplemented
35#20259243C2212yescomplemented
36C2×D8×C191C2211yescharacteristic, complemented
37C2×M16×C191C2211yescharacteristic
38#121×C191C2212yes—
39#106×C191C2211yescharacteristic, complemented
40#102×C191C2212yescomplemented
41C23×D191D411yescharacteristic
42#15690286—24no—
43#15690286—24no—
44C2×Dic382C2312yes—
45C2×C22⋊D191—22no—
46C2×C22⋊D191—22no—
47C22×Dic191—22no—
48C22×Dic191D411yescharacteristic
49C2×C4×D191—22no—
50C2×C4×D191—22no—
51C2×C4×D191D411yescharacteristic
52C2×C4×D191D411yescharacteristic
53C2×C4×D191C2311yescharacteristic
54C2×C4×D191C2311yescharacteristic
55Dic764—22no—
56#15690275—24no—
57#15690271—22no—
58#15690270—22no—
59#15690270—22no—
60C2×C191⋊C8C2311yescharacteristic
61C2×C191⋊C8C2311yescharacteristic
62C8×D191—22no—
63C8×D191D414yescomplemented
64C4⋊Dic191—22no—
65C4⋊Dic191C2312yes—
66Dic191⋊C4—22no—
67C22⋊Dic191—22no—
68C22⋊Dic191—22no—
69(C2×D191)⋊C4—22no—
70(C2×D191)⋊C4—22no—
71C4×Dic191—22no—
72C2×D4×C191—22no—
73C2×D4×C191C2312yes—
74C22×C4×C191C2311yescharacteristic
75D8×C191—22no—
76M16×C191—22no—
77C2×C8×C191C2311yescharacteristic
78C2×C8×C191C2311yescharacteristic
79C4⋊C4×C191C2312yes—
80C22⋊C4×C191—22no—
81C22×D191—22no—
82C22×D191—42no—
83C22×D191C2×D411yescharacteristic
84C22×D191C2×D411yescharacteristic
85Dic382—22no—
86Dic382—22no—
87C22⋊D191—44no—
88C22⋊D191—44no—
89C2×Dic191—22no—
90C2×Dic191—22no—
91C2×Dic191—22no—
92C2×Dic191—44no—
93C2×Dic191C2×D411yescharacteristic
94C2×Dic191C2×D411yescharacteristic
95C4×D191—22no—
96C4×D191—21no—
97C4×D191—21no—
98C4×D191—22no—
99C4×D191C2×D412yes—
100C4×D191#4112yes—
101C191⋊C8—21no—
102C191⋊C8C2×D412yes—
103C23×C191—22no—
104C23×C191C2×D411yescharacteristic
105D4×C191—22no—
106D4×C191—22no—
107D4×C191—44no—
108C2×C4×C191—21no—
109C2×C4×C191—21no—
110C2×C4×C191—22no—
111C2×C4×C191C2×D411yescharacteristic
112C2×C4×C191C2411yescharacteristic
113C8×C191—21no—
114C8×C191C2×D412yes—
115C22×C191—21no—
116C22×C191—22no—
117C22×C191—42no—
118C22×C191—42no—
119C22×C191—42no—
120C22×C191C22×D411yescharacteristic
121C2×D191—22no—
122C2×D191—21no—
123C2×D191—22no—
124C2×D191—44no—
125C2×D191#12112yes—
126C4×C191—21no—
127C4×C191—42no—
128C4×C191C22×D411yescharacteristic
129C4×C191C2×#4111yescharacteristic
130Dic191—21no—
131Dic191—42no—
132Dic191—42no—
133Dic191#12112yes—
134C2×C191—41no—
135C2×C191—82no—
136C2×C191C2×#12111yescharacteristic
137C2×C191#57512yes—
138D191—22no—
139D191—41no—
140C191#305411yescharacteristic, complemented, Sylow
141#3054—1911nomaximal, Sylow
142C2×#135—1911no—
143C22×M16—1911no—
144#525—1912no—
145#470—1911no—
146#470—1911no—
147#469—1911no—
148#469D19111yescharacteristic, complemented
149#468—1912no—
150#468—1912no—
151#427—1911no—
152#416—1912no—
153C2×#41—1912no—
154C23×C4—1911no—
155#135—3824no—
156C2×Q16—1911no—
157C2×SD16—1912no—
158C2×D8C2×D19111yescharacteristic, complemented
159C2×M16—3824no—
160C2×M16—1911no—
161C2×M16—1911no—
162C2×M16—1911no—
163C2×M16C2×D19111yescharacteristic
164C22×C8—1911no—
165C22×C8—1911no—
166#123—3822no—
167C22⋊Q8—1912no—
168C22⋊Q8—1912no—
169#121—1912no—
170#121C2×D19112yes—
171#120—3822no—
172C4×D4—3822no—
173C4×D4—3822no—
174#117—1912no—
175#107—1912no—
176#106—1911no—
177#106C2×D19111yescharacteristic, complemented
178#103—1912no—
179#103—1912no—
180#102—1912no—
181#102C2×D19112yescomplemented
182C24—1911no—
183#41—3824no—
184#41—3824no—
185C2×Q8—1912no—
186C2×D4—22no—
187C2×D4—3822no—
188C2×D4—3822no—
189C2×D4C22×D19112yes—
190C22×C4—3822no—
191C22×C4—3822no—
192C22×C4—3822no—
193C22×C4—1911no—
194C22×C4—1911no—
195C22×C4—1911no—
196C22×C4—1911no—
197C22×C4—1911no—
198C22×C4C22×D19111yescharacteristic
199Q16—3822no—
200SD16—3824no—
201D8—22no—
202M16—22no—
203M16—3822no—
204M16—3822no—
205M16—3822no—
206C2×C8—3822no—
207C2×C8—1911no—
208C2×C8—1911no—
209C2×C8—1914no—
210C2×C8C22×D19111yescharacteristic
211C2×C8C22×D19111yescharacteristic
212C4⋊C4—3822no—
213C4⋊C4—3822no—
214C4⋊C4—1912no—
215C4⋊C4C22×D19112yes—
216C22⋊C4—22no—
217C22⋊C4—3822no—
218C22⋊C4—3822no—
219C22⋊C4—3822no—
220C22⋊C4—3822no—
221C42—3822no—
222C23—22no—
223C23—3822no—
224C23—1911no—
225C23—1911no—
226C23—7642no—
227C23D4×D19111yescharacteristic
228Q8—3822no—
229Q8—3822no—
230D4—22no—
231D4—22no—
232D4—44no—
233D4—7644no—
234D4—7644no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—3822no—
239C2×C4—3822no—
240C2×C4—3822no—
241C2×C4—3821no—
242C2×C4—3821no—
243C2×C4—3822no—
244C2×C4—3822no—
245C2×C4—1911no—
246C2×C4—1911no—
247C2×C4—1912no—
248C2×C4—1912no—
249C2×C4—7644no—
250C2×C4D4×D19111yescharacteristic
251C2×C4C23×D19111yescharacteristic
252C8—21no—
253C8—3821no—
254C8—1912no—
255C8D4×D19112yes—
256C22—22no—
257C22—21no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—3822no—
262C22—3822no—
263C22—3821no—
264C22—1912no—
265C22—7644no—
266C22C2×D4×D19111yescharacteristic
267C4—21no—
268C4—42no—
269C4—3821no—
270C4—1912no—
271C4—7642no—
272C4—7642no—
273C4C2×D4×D19111yescharacteristic
274C4#41×D19111yescharacteristic
275C2—41no—
276C2—82no—
277C2—3822no—
278C2—7641no—
279C2#121×D19111yescharacteristic
280C2#2713345912yes—
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#469×C191C2
socleC22×C191C22×D4
hypercenter#469D191
hypocenterC191#3054
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×#20259341C2semidirect
2. C2×M16×D191C2semidirect
3. #27133173C2semidirect
4. #27132922C2semidirect
5. #27132921C2semidirect
6. #27132913C2semidirect
7. #27132908C2semidirect
8. #27132907C2semidirect
9. #27132698C2semidirect
10. #27132656C2semidirect
11. #469×C191C2semidirect
15. C2×Dic764C22semidirect
16. C2×#15690275C22semidirect
31. #20259255C22semidirect
32. #20259254C22semidirect
33. #20259247C22semidirect
34. #20259245C22semidirect
35. #20259243C22semidirect
36. C2×D8×C191C22semidirect
39. #106×C191C22semidirect
40. #102×C191C22semidirect
63. C8×D191D4semidirect
140. C191#3054semidirect
148. #469D191semidirect
158. C2×D8C2×D191semidirect
177. #106C2×D191semidirect
181. #102C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593412
C2×M16×D1912
#271331732
#271329222
#271329212
#271329132
#271329082
#271329072
#271326982
#271326562
#469×C1912
#3054191