Finite Group Databasebeta

← #37582481

Subgroups — #37582481

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#27133232C211yesmaximal, characteristic
2#27133143C211yesmaximal, characteristic, complemented
3#27132990C212yesmaximal, complemented
4#27132989C212yesmaximal, complemented
5#27132983C212yesmaximal, complemented
6#27132978C212yesmaximal, complemented
7#27132706C211yesmaximal, characteristic, complemented
8#27132667C211yesmaximal, characteristic, complemented
9C2×#20259244C211yesmaximal, characteristic, complemented
10#27132629C211yesmaximal, characteristic, complemented
11#427×C191C211yesmaximal, characteristic, complemented
12C2×#15690287C2211yescharacteristic
13C22×D764C2211yescharacteristic
14C2×#15690271C2211yescharacteristic
15#20259319—21no—
16#20259310—21no—
17#20259297C2212yes—
18#20259297C2212yes—
19#20259292—22no—
20C4×Dic382—21no—
21C4×D764—21no—
22(C22⋊D191)⋊C4—22no—
23#20259280C2212yescomplemented
24C2×C4⋊Dic191C2211yescharacteristic
25C4⋊C4×D191C2212yescomplemented
26#20259255C2212yescomplemented
27#20259253C2212yescomplemented
28#20259244C2212yes—
29#20259244C2212yes—
30#20259242C2212yes—
31#20259242C2212yes—
32(C2×D191)⋊C8C2212yescomplemented
33(C2×D191)⋊C8C2212yescomplemented
34C22×C8×C191C2211yescharacteristic
35#117×C191C2211yescharacteristic
36#117×C191C2211yescharacteristic
37#107×C191C2211yescharacteristic, complemented
38#107×C191C2211yescharacteristic, complemented
39#106×C191C2211yescharacteristic, complemented
40#106×C191C2211yescharacteristic, complemented
41C23×D191—21no—
42#15690287—22no—
43#15690287—22no—
44C2×Dic382D411yescharacteristic
45C2×D764—22no—
46C2×D764—22no—
47C2×D764—21no—
48C2×D764—22no—
49C2×D764—22no—
50C2×D764D411yescharacteristic
51C2×D764C2312yes—
52C2×C22⋊D191D412yes—
53C22×Dic191—21no—
54C2×C4×D191—22no—
55C2×C4×D191C2312yes—
56#15690271—22no—
57C2×C191⋊C8C2312yes—
58C4⋊Dic191—21no—
59C4⋊Dic191C2312yes—
60Dic191⋊C4—22no—
61(C2×D191)⋊C4—22no—
62(C2×D191)⋊C4—22no—
63(C2×D191)⋊C4—22no—
64C4×Dic191—22no—
65C22×C4×C191C2311yescharacteristic
66C2×C8×C191—22no—
67C2×C8×C191C2311yescharacteristic
68C2×C8×C191C2311yescharacteristic
69C4⋊C4×C191C2311yescharacteristic
70C4⋊C4×C191C2311yescharacteristic
71C4⋊C4×C191C2311yescharacteristic
72C4⋊C4×C191C2311yescharacteristic
73C22⋊C4×C191—21no—
74C22⋊C4×C191—21no—
75C42×C191—21no—
76C42×C191—21no—
77C22×D191—22no—
78C22×D191—42no—
79C22×D191—81no—
80C22×D191—81no—
81C22×D191C2×D412yes—
82Dic382—22no—
83D764—22no—
84D764—22no—
85D764—22no—
86D764—42no—
87D764—82no—
88D764—82no—
89C22⋊D191—24no—
90C2×Dic191—22no—
91C2×Dic191—81no—
92C2×Dic191C2×D412yes—
93C4×D191—22no—
94C4×D191—22no—
95C4×D191—82no—
96C191⋊C8—22no—
97C23×C191C2×D411yescharacteristic
98C2×C4×C191—21no—
99C2×C4×C191—21no—
100C2×C4×C191—21no—
101C2×C4×C191—21no—
102C2×C4×C191C2×D411yescharacteristic
103C2×C4×C191#4112yes—
104C2×C4×C191#4112yes—
105C2×C4×C191C2411yescharacteristic
106C8×C191—21no—
107C8×C191—21no—
108C22×C191—21no—
109C22×C191—21no—
110C22×C191#12312yes—
111C22×C191C22×D411yescharacteristic
112C2×D191—22no—
113C2×D191—22no—
114C2×D191—22no—
115C2×D191—42no—
116C2×D191—42no—
117C2×D191—42no—
118C2×D191—84no—
119C4×C191—41no—
120C4×C191—41no—
121C4×C191—41no—
122C4×C191—41no—
123C4×C191#12312yes—
124C4×C191C2×#4111yescharacteristic
125C4×C191C2×#4111yescharacteristic
126Dic191—22no—
127Dic191—42no—
128C2×C191—22no—
129C2×C191C2×#12311yescharacteristic
130C2×C191#57611yescharacteristic
131C2×C191#57711yescharacteristic
132D191—42no—
133D191—82no—
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#427D19111yescharacteristic, complemented
144#418—1911no—
145C2×#102—1911no—
146#408—1911no—
147C2×#41—1911no—
148C22×D4—1911no—
149C2×M16—1911no—
150C22×C8C2×D19111yescharacteristic
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—1912no—
160#117C2×D19111yescharacteristic
161#117C2×D19111yescharacteristic
162C2×C4⋊C4—1911no—
163C2×C4⋊C4—1912no—
164#107—1912no—
165#107C2×D19111yescharacteristic, complemented
166#107C2×D19111yescharacteristic, complemented
167#106—1912no—
168#106C2×D19111yescharacteristic, complemented
169#106C2×D19111yescharacteristic, complemented
170#102—1912no—
171#102—1912no—
172#102—1912no—
173#102—1912no—
174C22⋊C8—1912no—
175C22⋊C8—1912no—
176C24—3821no—
177#41—3822no—
178#41—3822no—
179C2×Q8—1911no—
180C2×D4—3822no—
181C2×D4—3822no—
182C2×D4—3821no—
183C2×D4—3822no—
184C2×D4—3822no—
185C2×D4—1912no—
186C2×D4—1911no—
187C2×D4—1912no—
188C22×C4—3822no—
189C22×C4—3821no—
190C22×C4—1912no—
191C22×C4C22×D19111yescharacteristic
192M16—3822no—
193C2×C8—22no—
194C2×C8—1912no—
195C2×C8C22×D19111yescharacteristic
196C2×C8C22×D19111yescharacteristic
197C4⋊C4—3822no—
198C4⋊C4—3821no—
199C4⋊C4—1912no—
200C4⋊C4C22×D19111yescharacteristic
201C4⋊C4C22×D19111yescharacteristic
202C4⋊C4C22×D19111yescharacteristic
203C4⋊C4C22×D19111yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—3822no—
207C22⋊C4—3822no—
208C22⋊C4—3822no—
209C42—21no—
210C42—21no—
211C42—3822no—
212C23—3822no—
213C23—1912no—
214C23—15281no—
215C23—15281no—
216C23—7642no—
217C23D4×D19111yescharacteristic
218Q8—3822no—
219D4—3824no—
220D4—3822no—
221D4—3822no—
222D4—3822no—
223D4—15282no—
224D4—15282no—
225D4—7642no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—3822no—
231C2×C4—3822no—
232C2×C4—3822no—
233C2×C4—1912no—
234C2×C4—15282no—
235C2×C4—15281no—
236C2×C4D4×D19111yescharacteristic
237C2×C4#1569028712yes—
238C2×C4#1569028812yes—
239C2×C4C23×D19111yescharacteristic
240C8—21no—
241C8—21no—
242C8—3822no—
243C22—21no—
244C22—21no—
245C22—3822no—
246C22—3822no—
247C22—3822no—
248C22—15284no—
249C22—7642no—
250C22—7642no—
251C22—7642no—
252C22#2025930712yes—
253C22C2×D4×D19111yescharacteristic
254C4—41no—
255C4—41no—
256C4—41no—
257C4—41no—
258C4—3822no—
259C4—7642no—
260C4#2025930712yes—
261C4C2×#1569028711yescharacteristic
262C4C2×#1569028811yescharacteristic
263C2—22no—
264C2—15282no—
265C2—7642no—
266C2C2×#2025930711yescharacteristic
267C2#2713346511yescharacteristic
268C2#2713347011yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#427×C191C2
socleC22×C191C22×D4
prime radical#27133232C2
hypercenter#427D191
hypocenterC191#3096
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
2. #27133143C2semidirect
3. #27132990C2semidirect
4. #27132989C2semidirect
5. #27132983C2semidirect
6. #27132978C2semidirect
7. #27132706C2semidirect
8. #27132667C2semidirect
9. C2×#20259244C2semidirect
10. #27132629C2semidirect
11. #427×C191C2semidirect
23. #20259280C22semidirect
25. C4⋊C4×D191C22semidirect
26. #20259255C22semidirect
27. #20259253C22semidirect
32. (C2×D191)⋊C8C22semidirect
33. (C2×D191)⋊C8C22semidirect
37. #107×C191C22semidirect
38. #107×C191C22semidirect
39. #106×C191C22semidirect
40. #106×C191C22semidirect
134. C191#3096semidirect
143. #427D191semidirect
165. #107C2×D191semidirect
166. #107C2×D191semidirect
168. #106C2×D191semidirect
169. #106C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271332322
#271331432
#271329902
#271329892
#271329832
#271329782
#271327062
#271326672
C2×#202592442
#271326292
#427×C1912
#3096191