Finite Group Databasebeta

← #37582461

Subgroups — #37582461

Order 24448.

462 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 137 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×#20259296C211yesmaximal, characteristic
2C2×#20259281C211yesmaximal, characteristic, complemented
3#27132988C214yesmaximal, complemented
4#27132982C212yesmaximal, complemented
5#27132977C212yesmaximal, complemented
6#27132705C212yesmaximal, complemented
7#27132669C212yesmaximal, complemented
8#407×C191C211yesmaximal, characteristic, complemented
9C22×D764C2211yescharacteristic
10C22×C22⋊D191—21no—
11#20259296—21no—
12#20259296—21no—
13#20259296C2212yes—
14#20259296C2212yes—
15#20259281—21no—
16#20259281—21no—
17#20259281C2212yes—
18#20259281C2212yescomplemented
19C2×C4⋊Dic191C2211yescharacteristic
20C2×C4⋊Dic191C2211yescharacteristic, complemented
21C2×C4⋊Dic191C2211yescharacteristic, complemented
22C2×C22⋊Dic191—21no—
23C2×(C2×D191)⋊C4—21no—
24C2×C4×Dic191—21no—
25#20259256C2214yescomplemented
26#20259254C2214yescomplemented
27#20259244C2214yescomplemented
28#20259244C2214yescomplemented
29C23×C4×C191C2211yescharacteristic
30C2×M16×C191C2212yes—
31C22⋊C8×C191C2212yescomplemented
32C22⋊C8×C191C2212yescomplemented
33C23×D191—21no—
34C2×D764—21no—
35C2×D764—24no—
36C2×D764C2312yes—
37C2×C22⋊D191—22no—
38C2×C22⋊D191—22no—
39C2×C22⋊D191—44no—
40C22×Dic191—21no—
41C22×Dic191—21no—
42C22×Dic191—21no—
43C4⋊Dic191—21no—
44C4⋊Dic191—21no—
45C4⋊Dic191—21no—
46C4⋊Dic191C2312yes—
47C4⋊Dic191C2312yes—
48C4⋊Dic191C2312yes—
49C22⋊Dic191—22no—
50C22⋊Dic191—22no—
51(C2×D191)⋊C4—22no—
52(C2×D191)⋊C4—22no—
53C4×Dic191—22no—
54C4×Dic191—22no—
55C24×C191D411yescharacteristic
56C22×C4×C191—24no—
57C22×C4×C191D411yescharacteristic
58C22×C4×C191D411yescharacteristic
59C22×C4×C191D411yescharacteristic
60C22×C4×C191C2311yescharacteristic
61C22×C4×C191C2311yescharacteristic
62C22×C4×C191C2311yescharacteristic
63M16×C191—24no—
64C2×C8×C191C2314yes—
65C22×D191—22no—
66C22×D191—42no—
67C22×D191—42no—
68C22×D191—42no—
69D764—24no—
70D764—42no—
71C22⋊D191—44no—
72C22⋊D191—44no—
73C2×Dic191—22no—
74C2×Dic191—22no—
75C2×Dic191—22no—
76C2×Dic191—42no—
77C2×Dic191—42no—
78C2×Dic191—42no—
79C23×C191—22no—
80C23×C191—22no—
81C23×C191—41no—
82C23×C191C2×D411yescharacteristic
83C23×C191C2×D411yescharacteristic
84C23×C191C2×D411yescharacteristic
85C2×C4×C191—22no—
86C2×C4×C191—22no—
87C2×C4×C191—22no—
88C2×C4×C191—22no—
89C2×C4×C191—22no—
90C2×C4×C191C2×D411yescharacteristic
91C2×C4×C191C2×D411yescharacteristic
92C2×C4×C191C2×D411yescharacteristic
93C2×C4×C191#4114yes—
94C2×C4×C191C2411yescharacteristic
95C8×C191—24no—
96C22×C191—21no—
97C22×C191—21no—
98C22×C191—21no—
99C22×C191—21no—
100C22×C191—21no—
101C22×C191—21no—
102C22×C191—21no—
103C22×C191—44no—
104C22×C191#12312yes—
105C22×C191AGL(1,ℤ/8)12yes—
106C22×C191C22×D411yescharacteristic
107C2×D191—42no—
108C2×D191—42no—
109C2×D191—42no—
110C2×D191—84no—
111C4×C191—21no—
112C4×C191—21no—
113C4×C191#12312yes—
114C4×C191C2×#4112yes—
115Dic191—42no—
116Dic191—42no—
117Dic191—42no—
118C2×C191—22no—
119C2×C191—22no—
120C2×C191—41no—
121C2×C191C2×#12311yescharacteristic
122C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
123C2×C191#57511yescharacteristic
124D191—82no—
125C191#309411yescharacteristic, complemented, Sylow
126#3094—1911nomaximal, Sylow
127C2×#121—1911no—
128C2×#117—1911no—
129#482—1914no—
130#481—1912no—
131#480—1912no—
132#428—1912no—
133#418—1912no—
134#407D19111yescharacteristic, complemented
135C22×D4—3821no—
136C22×D4—1911no—
137C23×C4C2×D19111yescharacteristic
138C2×M16C2×D19112yes—
139#121—3821no—
140#121—3821no—
141#121—1912no—
142#121—1912no—
143#117—3821no—
144#117—3821no—
145#117—1912no—
146#117—1912no—
147C2×C4⋊C4—1911no—
148C2×C4⋊C4—1911no—
149C2×C4⋊C4—1911no—
150C2×C22⋊C4—3821no—
151C2×C22⋊C4—3821no—
152C2×C42—3821no—
153#107—1914no—
154#106—1914no—
155#102—1914no—
156#102—1914no—
157C22⋊C8C2×D19112yescomplemented
158C22⋊C8C2×D19112yescomplemented
159C24—3821no—
160C24D76411yescharacteristic
161C2×D4—3822no—
162C2×D4—3822no—
163C2×D4—3821no—
164C2×D4—3824no—
165C2×D4—1912no—
166C2×D4—7644no—
167C22×C4—24no—
168C22×C4—3821no—
169C22×C4—3821no—
170C22×C4—3821no—
171C22×C4D76411yescharacteristic
172C22×C4D76411yescharacteristic
173C22×C4D76411yescharacteristic
174C22×C4C22×D19111yescharacteristic
175C22×C4C22×D19111yescharacteristic
176C22×C4C22×D19111yescharacteristic
177M16—24no—
178C2×C8C22×D19114yes—
179C4⋊C4—3821no—
180C4⋊C4—3821no—
181C4⋊C4—3821no—
182C4⋊C4—1912no—
183C4⋊C4—1912no—
184C4⋊C4—1912no—
185C22⋊C4—3822no—
186C22⋊C4—3822no—
187C22⋊C4—3822no—
188C22⋊C4—3822no—
189C42—3822no—
190C42—3822no—
191C23—22no—
192C23—22no—
193C23—41no—
194C23—3822no—
195C23—7642no—
196C23—7642no—
197C23—7642no—
198C23C2×D76411yescharacteristic
199C23C2×D76411yescharacteristic
200C23C2×D76411yescharacteristic
201D4—3824no—
202D4—7644no—
203D4—7644no—
204D4—7642no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—3822no—
211C2×C4—3822no—
212C2×C4—3822no—
213C2×C4—7642no—
214C2×C4—7642no—
215C2×C4—7642no—
216C2×C4C2×D76411yescharacteristic
217C2×C4C2×D76411yescharacteristic
218C2×C4C2×D76411yescharacteristic
219C2×C4#1569028614yes—
220C2×C4C23×D19111yescharacteristic
221C8—24no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—44no—
230C22—15284no—
231C22—7642no—
232C22—7642no—
233C22—7642no—
234C22#2025930612yes—
235C22#2025935012yes—
236C22C22×D76411yescharacteristic
237C4—21no—
238C4—21no—
239C4—7642no—
240C4—7642no—
241C4—7642no—
242C4#2025930612yes—
243C4C2×#1569028612yes—
244C2—22no—
245C2—22no—
246C2—41no—
247C2—15282no—
248C2C2×#2025930611yescharacteristic
249C2C2×#2025935011yescharacteristic
250C2#2713346311yescharacteristic
subgroupG/H
centerC22C22×D764
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#407×C191C2
socleC22×C191C22×D4
prime radicalC2×#20259296C2
hypercenter#407D191
hypocenterC191#3094
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#20259281C2semidirect
3. #27132988C2semidirect
4. #27132982C2semidirect
5. #27132977C2semidirect
6. #27132705C2semidirect
7. #27132669C2semidirect
8. #407×C191C2semidirect
18. #20259281C22semidirect
20. C2×C4⋊Dic191C22semidirect
21. C2×C4⋊Dic191C22semidirect
25. #20259256C22semidirect
26. #20259254C22semidirect
27. #20259244C22semidirect
28. #20259244C22semidirect
31. C22⋊C8×C191C22semidirect
32. C22⋊C8×C191C22semidirect
125. C191#3094semidirect
134. #407D191semidirect
157. C22⋊C8C2×D191semidirect
158. C22⋊C8C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202592962
C2×#202592812
#271329882
#271329822
#271329772
#271327052
#271326692
#407×C1912
#3094191