Finite Group Databasebeta

← #37582476

Subgroups — #37582476

Order 24448.

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

normal subgroups only

This group has 418 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
1#27133230C211yesmaximal, characteristic
2#27133143C211yesmaximal, characteristic, complemented
3#27132986C212yesmaximal, complemented
4#27132985C212yesmaximal, complemented
5#27132980C212yesmaximal, complemented
6#27132975C212yesmaximal, complemented
7#27132705C211yesmaximal, characteristic, complemented
8#27132699C211yesmaximal, characteristic, complemented
9C2×#20259243C211yesmaximal, characteristic, complemented
10#27132631C211yesmaximal, characteristic, complemented
11#418×C191C211yesmaximal, characteristic, complemented
12C2×#15690287C2211yescharacteristic
13C22×C191⋊C8C2211yescharacteristic
14#20259318—21no—
15#20259309—21no—
16#20259293C2212yes—
17#20259293C2212yes—
18#20259291—22no—
19C4×Dic382—21no—
20C4×D764—21no—
21(C22⋊D191)⋊C4—22no—
22#20259281C2211yescharacteristic
23#20259280C2212yescomplemented
24C2×C4⋊Dic191C2211yescharacteristic
25C4⋊C4×D191C2212yescomplemented
26#20259256C2212yescomplemented
27#20259255C2211yescharacteristic, complemented
28#20259255C2211yescharacteristic, complemented
29#20259254C2212yescomplemented
30#20259253C2211yescharacteristic, complemented
31#20259253C2211yescharacteristic, complemented
32#20259243C2212yes—
33#20259243C2212yes—
34(C2×D191)⋊C8C2212yescomplemented
35(C2×D191)⋊C8C2212yescomplemented
36C22×D4×C191C2211yescharacteristic
37C2×M16×C191C2211yescharacteristic
38#117×C191C2211yescharacteristic
39#102×C191C2212yes—
40#102×C191C2212yes—
41#15690287—22no—
42#15690287—22no—
43C2×Dic382D411yescharacteristic
44C2×D764D411yescharacteristic
45C2×C22⋊D191—22no—
46C2×C22⋊D191—22no—
47C2×C22⋊D191D412yes—
48C22×Dic191—21no—
49C2×C4×D191—22no—
50C2×C4×D191C2312yes—
51C2×C191⋊C8—22no—
52C2×C191⋊C8C2311yescharacteristic
53C2×C191⋊C8C2311yescharacteristic
54C4⋊Dic191—21no—
55C4⋊Dic191C2312yes—
56C4⋊Dic191C2311yescharacteristic
57C4⋊Dic191C2311yescharacteristic
58Dic191⋊C4—22no—
59C22⋊Dic191—22no—
60C22⋊Dic191—22no—
61C22⋊Dic191—21no—
62(C2×D191)⋊C4—22no—
63C4×Dic191—22no—
64C4×Dic191—21no—
65C24×C191—21no—
66C2×D4×C191—21no—
67C2×D4×C191—22no—
68C2×D4×C191—22no—
69C2×D4×C191C2312yes—
70C22×C4×C191C2311yescharacteristic
71M16×C191—22no—
72C2×C8×C191C2312yes—
73C4⋊C4×C191C2311yescharacteristic
74C4⋊C4×C191C2311yescharacteristic
75C22⋊C4×C191—21no—
76C42×C191—21no—
77C22×D191C2×D412yes—
78Dic382—22no—
79D764—22no—
80C22⋊D191—24no—
81C22⋊D191—82no—
82C22⋊D191—82no—
83C2×Dic191—22no—
84C2×Dic191—21no—
85C2×Dic191—21no—
86C2×Dic191—81no—
87C2×Dic191C2×D412yes—
88C4×D191—22no—
89C4×D191—22no—
90C4×D191—82no—
91C191⋊C8—21no—
92C191⋊C8—21no—
93C23×C191—22no—
94C23×C191—42no—
95C23×C191—81no—
96C23×C191—81no—
97C23×C191C2×D411yescharacteristic
98D4×C191—22no—
99D4×C191—22no—
100D4×C191—42no—
101C2×C4×C191—21no—
102C2×C4×C191—21no—
103C2×C4×C191C2×D411yescharacteristic
104C2×C4×C191#4112yes—
105C2×C4×C191#4112yes—
106C2×C4×C191C2411yescharacteristic
107C8×C191—22no—
108C22×C191—21no—
109C22×C191—21no—
110C22×C191—42no—
111C22×C191—42no—
112C22×C191—42no—
113C22×C191—84no—
114C22×C191#12312yes—
115C22×C191C22×D411yescharacteristic
116C2×D191—22no—
117C2×D191—22no—
118C2×D191—22no—
119C4×C191—41no—
120C4×C191—41no—
121C4×C191#12312yes—
122C4×C191C2×#4111yescharacteristic
123C4×C191C2×#4111yescharacteristic
124Dic191—22no—
125Dic191—42no—
126Dic191—41no—
127Dic191—41no—
128C2×C191—22no—
129C2×C191—82no—
130C2×C191C2×#12311yescharacteristic
131C2×C191#57611yescharacteristic
132C2×C191#57711yescharacteristic
133D191—42no—
134C191#309611yescharacteristic, complemented, Sylow
135#3096—1911nomaximal, Sylow
136#535—1911no—
137#520—1911no—
138#482—1912no—
139#482—1912no—
140#481—1912no—
141#480—1912no—
142#428—1911no—
143#427—1911no—
144#418D19111yescharacteristic, complemented
145C2×#102—1911no—
146#408—1911no—
147C2×#41—1911no—
148C22×D4C2×D19111yescharacteristic
149C2×M16C2×D19111yescharacteristic
150C22×C8—1911no—
151#127—3821no—
152#124—3821no—
153#121—1912no—
154#121—1912no—
155#120—3822no—
156C4×Q8—3821no—
157C4×D4—3821no—
158C4×D4—3822no—
159#117—1911no—
160#117—1912no—
161#117C2×D19111yescharacteristic
162C2×C4⋊C4—1911no—
163C2×C4⋊C4—1912no—
164#107—1912no—
165#107—1911no—
166#107—1911no—
167#106—1912no—
168#106—1911no—
169#106—1911no—
170#102—1912no—
171#102—1912no—
172#102C2×D19112yes—
173#102C2×D19112yes—
174C22⋊C8—1912no—
175C22⋊C8—1912no—
176C24—21no—
177#41—3822no—
178#41—3822no—
179C2×Q8—1911no—
180C2×D4—21no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—3822no—
184C2×D4—3822no—
185C2×D4—1912no—
186C2×D4—1911no—
187C2×D4C22×D19112yes—
188C22×C4—3822no—
189C22×C4—3821no—
190C22×C4—1912no—
191C22×C4C22×D19111yescharacteristic
192M16—22no—
193C2×C8—3822no—
194C2×C8—1911no—
195C2×C8—1911no—
196C2×C8C22×D19112yes—
197C4⋊C4—3822no—
198C4⋊C4—3821no—
199C4⋊C4—1912no—
200C4⋊C4—1911no—
201C4⋊C4—1911no—
202C4⋊C4C22×D19111yescharacteristic
203C4⋊C4C22×D19111yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—3822no—
206C22⋊C4—3821no—
207C22⋊C4—3822no—
208C22⋊C4—3822no—
209C42—21no—
210C42—3822no—
211C42—3821no—
212C23—22no—
213C23—42no—
214C23—81no—
215C23—81no—
216C23—1912no—
217C23D4×D19111yescharacteristic
218Q8—3822no—
219D4—22no—
220D4—22no—
221D4—42no—
222D4—3824no—
223D4—3822no—
224D4—15282no—
225D4—15282no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—3822no—
229C2×C4—3822no—
230C2×C4—3821no—
231C2×C4—3821no—
232C2×C4—3822no—
233C2×C4—1912no—
234C2×C4—15282no—
235C2×C4—15281no—
236C2×C4D4×D19111yescharacteristic
237C2×C4#1569028612yes—
238C2×C4#1569028712yes—
239C2×C4C23×D19111yescharacteristic
240C8—22no—
241C8—3821no—
242C8—3821no—
243C22—21no—
244C22—21no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—84no—
249C22—3822no—
250C22—3822no—
251C22—3822no—
252C22#2025930412yes—
253C22C2×D4×D19111yescharacteristic
254C4—41no—
255C4—41no—
256C4—3822no—
257C4—7642no—
258C4—7641no—
259C4—7641no—
260C4#2025930412yes—
261C4C2×#1569028611yescharacteristic
262C4C2×#1569028711yescharacteristic
263C2—22no—
264C2—82no—
265C2—7642no—
266C2C2×#2025930411yescharacteristic
267C2#2713346711yescharacteristic
268C2#2713347311yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#418×C191C2
socleC22×C191C22×D4
prime radical#27133230C2
hypercenter#418D191
hypocenterC191#3096
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #27133143C2semidirect
3. #27132986C2semidirect
4. #27132985C2semidirect
5. #27132980C2semidirect
6. #27132975C2semidirect
7. #27132705C2semidirect
8. #27132699C2semidirect
9. C2×#20259243C2semidirect
10. #27132631C2semidirect
11. #418×C191C2semidirect
23. #20259280C22semidirect
25. C4⋊C4×D191C22semidirect
26. #20259256C22semidirect
27. #20259255C22semidirect
28. #20259255C22semidirect
29. #20259254C22semidirect
30. #20259253C22semidirect
31. #20259253C22semidirect
34. (C2×D191)⋊C8C22semidirect
35. (C2×D191)⋊C8C22semidirect
134. C191#3096semidirect
144. #418D191semidirect

Maximal subgroups

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

subgroupindex
#271332302
#271331432
#271329862
#271329852
#271329802
#271329752
#271327052
#271326992
C2×#202592432
#271326312
#418×C1912
#3096191