Finite Group Databasebeta

← #37582131

Subgroups — #37582131

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×#20259345C211yesmaximal, characteristic, complemented
2C2×M16×D191C211yesmaximal, characteristic, complemented
3#27133176C212yesmaximal, complemented
4#27132918C211yesmaximal, characteristic, complemented
5#27132917C211yesmaximal, characteristic, complemented
6#27132916C211yesmaximal, characteristic, complemented
7#27132912C212yesmaximal, complemented
8#27132911C212yesmaximal, complemented
9#27132698C211yesmaximal, characteristic, complemented
10#27132658C212yesmaximal, complemented
11#470×C191C211yesmaximal, characteristic, complemented
12C2×#15690288C2212yes—
13C22×C4×D191C2211yescharacteristic
14#20259345—24no—
15C2×#15690274C2212yescomplemented
16C2×D1528C2211yescharacteristic, complemented
17C2×#15690271C2211yescharacteristic
18C2×#15690270C2211yescharacteristic
19C2×#15690270C2211yescharacteristic
20M16×D191—24no—
21C2×C8×D191C2211yescharacteristic
22C2×C8×D191C2211yescharacteristic
23#20259307—22no—
24#20259298C2212yes—
25#20259297C2212yes—
26#20259296C2212yes—
27#20259292—22no—
28#20259284—22no—
29(C22⋊D191)⋊C4—22no—
30#20259280C2212yes—
31#20259255C2212yescomplemented
32#20259254C2211yescharacteristic, complemented
33#20259246C2212yescomplemented
34#20259244C2212yescomplemented
35#20259242C2212yescomplemented
36C2×Q16×C191C2211yescharacteristic, complemented
37C2×M16×C191C2211yescharacteristic
38C22⋊Q8×C191C2212yes—
39#106×C191C2211yescharacteristic, complemented
40#103×C191C2212yescomplemented
41C23×D191D411yescharacteristic
42#15690288—24no—
43#15690288—24no—
44C2×D764—22no—
45C2×D764—22no—
46C2×D764C2312yes—
47C2×C22⋊D191—22no—
48C22×Dic191D411yescharacteristic
49C2×C4×D191—22no—
50C2×C4×D191—22no—
51C2×C4×D191—22no—
52C2×C4×D191D411yescharacteristic
53C2×C4×D191D411yescharacteristic
54C2×C4×D191C2311yescharacteristic
55C2×C4×D191C2311yescharacteristic
56#15690274—24no—
57D1528—22no—
58#15690271—22no—
59#15690270—22no—
60#15690270—22no—
61C2×C191⋊C8C2311yescharacteristic
62C2×C191⋊C8C2311yescharacteristic
63C8×D191—22no—
64C8×D191D414yescomplemented
65C4⋊Dic191C2312yes—
66Dic191⋊C4—22no—
67(C2×D191)⋊C4—22no—
68(C2×D191)⋊C4—22no—
69(C2×D191)⋊C4—22no—
70(C2×D191)⋊C4—22no—
71C4×Dic191—22no—
72C2×Q8×C191C2312yes—
73C22×C4×C191C2311yescharacteristic
74Q16×C191—22no—
75M16×C191—22no—
76C2×C8×C191C2311yescharacteristic
77C2×C8×C191C2311yescharacteristic
78C4⋊C4×C191—22no—
79C4⋊C4×C191C2312yes—
80C22⋊C4×C191—22no—
81C22×D191—22no—
82C22×D191—22no—
83C22×D191—42no—
84C22×D191C2×D411yescharacteristic
85C22×D191C2×D411yescharacteristic
86D764—22no—
87D764—22no—
88D764—44no—
89D764—44no—
90C22⋊D191—44no—
91C2×Dic191—22no—
92C2×Dic191—22no—
93C2×Dic191C2×D411yescharacteristic
94C2×Dic191C2×D411yescharacteristic
95C4×D191—22no—
96C4×D191—21no—
97C4×D191—21no—
98C4×D191—22no—
99C4×D191—44no—
100C4×D191C2×D412yes—
101C4×D191#4112yes—
102C191⋊C8—21no—
103C191⋊C8C2×D412yes—
104C23×C191C2×D411yescharacteristic
105Q8×C191—22no—
106Q8×C191—22no—
107C2×C4×C191—21no—
108C2×C4×C191—21no—
109C2×C4×C191—22no—
110C2×C4×C191—22no—
111C2×C4×C191C2×D411yescharacteristic
112C2×C4×C191C2411yescharacteristic
113C8×C191—21no—
114C8×C191C2×D412yes—
115C22×C191—21no—
116C22×C191—22no—
117C22×C191C22×D411yescharacteristic
118C2×D191—22no—
119C2×D191—21no—
120C2×D191—22no—
121C2×D191—44no—
122C2×D191—42no—
123C2×D191—42no—
124C2×D191—42no—
125C2×D191#12112yes—
126C4×C191—21no—
127C4×C191—42no—
128C4×C191—42no—
129C4×C191C22×D411yescharacteristic
130C4×C191C2×#4111yescharacteristic
131Dic191—21no—
132Dic191—42no—
133Dic191#12112yes—
134C2×C191—41no—
135C2×C191C2×#12111yescharacteristic
136C2×C191#57512yes—
137D191—22no—
138D191—41no—
139D191—82no—
140C191#305411yescharacteristic, complemented, Sylow
141#3054—1911nomaximal, Sylow
142C2×#135—1911no—
143C22×M16—1911no—
144#525—1912no—
145#470—1911no—
146#470D19111yescharacteristic, complemented
147#469—1911no—
148#469—1911no—
149#468—1912no—
150#468—1912no—
151#427—1911no—
152#416—1912no—
153C2×#41—1912no—
154C23×C4—1911no—
155#135—3824no—
156C2×Q16C2×D19111yescharacteristic, complemented
157C2×SD16—1912no—
158C2×D8—1911no—
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⋊Q8C2×D19112yes—
169#121—1912no—
170#121—1912no—
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#103C2×D19112yescomplemented
180#102—1912no—
181#102—1912no—
182C24—1911no—
183#41—3824no—
184#41—3824no—
185C2×Q8C22×D19112yes—
186C2×D4—3822no—
187C2×D4—3822no—
188C2×D4—3822no—
189C2×D4—1912no—
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—22no—
200SD16—3824no—
201D8—3822no—
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—22no—
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—3822no—
223C23—3822no—
224C23—1911no—
225C23—1911no—
226C23—7642no—
227C23D4×D19111yescharacteristic
228Q8—22no—
229Q8—22no—
230D4—3822no—
231D4—3822no—
232D4—7644no—
233D4—7644no—
234D4—7644no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—3822no—
240C2×C4—3822no—
241C2×C4—3822no—
242C2×C4—3821no—
243C2×C4—3821no—
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—3822no—
259C22—3822no—
260C22—3821no—
261C22—1912no—
262C22—7644no—
263C22—7642no—
264C22—7642no—
265C22—7642no—
266C22C2×D4×D19111yescharacteristic
267C4—21no—
268C4—42no—
269C4—42no—
270C4—3821no—
271C4—1912no—
272C4—7642no—
273C4C2×D4×D19111yescharacteristic
274C4#41×D19111yescharacteristic
275C2—41no—
276C2—3822no—
277C2—15282no—
278C2—7641no—
279C2#121×D19111yescharacteristic
280C2#2713346212yes—
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#470×C191C2
socleC22×C191C22×D4
hypercenter#470D191
hypocenterC191#3054
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#20259345C2semidirect
2. C2×M16×D191C2semidirect
3. #27133176C2semidirect
4. #27132918C2semidirect
5. #27132917C2semidirect
6. #27132916C2semidirect
7. #27132912C2semidirect
8. #27132911C2semidirect
9. #27132698C2semidirect
10. #27132658C2semidirect
11. #470×C191C2semidirect
15. C2×#15690274C22semidirect
16. C2×D1528C22semidirect
31. #20259255C22semidirect
32. #20259254C22semidirect
33. #20259246C22semidirect
34. #20259244C22semidirect
35. #20259242C22semidirect
36. C2×Q16×C191C22semidirect
39. #106×C191C22semidirect
40. #103×C191C22semidirect
64. C8×D191D4semidirect
140. C191#3054semidirect
146. #470D191semidirect
156. C2×Q16C2×D191semidirect
177. #106C2×D191semidirect
179. #103C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593452
C2×M16×D1912
#271331762
#271329182
#271329172
#271329162
#271329122
#271329112
#271326982
#271326582
#470×C1912
#3054191