Finite Group Databasebeta

← #37582443

Subgroups — #37582443

Order 24448.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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#27133174C211yesmaximal, characteristic
2#27132998C212yesmaximal, complemented
3#27132993C212yesmaximal, complemented
4#27132981C212yesmaximal, complemented
5#27132976C212yesmaximal, complemented
6#27132703C212yesmaximal, complemented
7#27132662C212yesmaximal, complemented
8#27132627C211yesmaximal, characteristic, complemented
9C22×C4⋊C4×C191C211yesmaximal, characteristic, complemented
10C2×#15690287C2211yescharacteristic
11C2×#15690271C2212yes—
12#20259305—21no—
13#20259301C2211yescharacteristic
14#20259301C2211yescharacteristic
15#20259296C2211yescharacteristic
16#20259296C2211yescharacteristic
17#20259291—21no—
18C4×C22⋊D191—21no—
19C4×C22⋊D191—21no—
20#20259281C2211yescharacteristic
21#20259255C2214yescomplemented
22#20259253C2214yescomplemented
23#20259245C2214yescomplemented
24#20259242C2214yescomplemented
25C22⋊(C191⋊C8)C2212yes—
26C22⋊(C191⋊C8)C2212yes—
27C23×C4×C191—21no—
28C23×C4×C191C2211yescharacteristic
29C2×C4⋊C4×C191—21no—
30C2×C4⋊C4×C191—21no—
31C2×C4⋊C4×C191—21no—
32C2×C4⋊C4×C191C2212yes—
33C2×C4⋊C4×C191C2212yes—
34C2×C4⋊C4×C191C2212yes—
35#15690287—24no—
36C2×Dic382C2311yescharacteristic
37C2×D764C2311yescharacteristic
38C2×C22⋊D191—21no—
39C2×C22⋊D191—21no—
40C2×C22⋊D191—21no—
41C2×C4×D191—21no—
42#15690271—24no—
43C2×C191⋊C8C2314yes—
44C4⋊Dic191C2311yescharacteristic
45C4⋊Dic191C2311yescharacteristic
46Dic191⋊C4—21no—
47Dic191⋊C4—21no—
48C22⋊Dic191—21no—
49C22⋊Dic191—21no—
50C22⋊Dic191—21no—
51(C2×D191)⋊C4—21no—
52(C2×D191)⋊C4—21no—
53C4×Dic191—21no—
54C24×C191D411yescharacteristic
55C22×C4×C191—24no—
56C22×C4×C191—22no—
57C22×C4×C191—22no—
58C22×C4×C191—22no—
59C22×C4×C191—44no—
60C22×C4×C191D411yescharacteristic
61C22×C4×C191D411yescharacteristic
62C22×C4×C191D411yescharacteristic
63C22×C4×C191C2311yescharacteristic
64C22×C4×C191C2311yescharacteristic
65C22×C4×C191C2311yescharacteristic
66C4⋊C4×C191—22no—
67C4⋊C4×C191—22no—
68C4⋊C4×C191—22no—
69C4⋊C4×C191C2314yes—
70C22×D191—21no—
71Dic382—22no—
72D764—22no—
73C22⋊D191—42no—
74C22⋊D191—42no—
75C22⋊D191—42no—
76C2×Dic191—21no—
77C2×Dic191—21no—
78C2×Dic191—21no—
79C4×D191—42no—
80C191⋊C8—24no—
81C23×C191—22no—
82C23×C191—22no—
83C23×C191—22no—
84C23×C191C2×D411yescharacteristic
85C23×C191C2×D411yescharacteristic
86C23×C191C2×D411yescharacteristic
87C2×C4×C191—22no—
88C2×C4×C191—22no—
89C2×C4×C191—22no—
90C2×C4×C191—22no—
91C2×C4×C191—22no—
92C2×C4×C191—24no—
93C2×C4×C191—44no—
94C2×C4×C191—44no—
95C2×C4×C191—44no—
96C2×C4×C191C2×D411yescharacteristic
97C2×C4×C191C2×D411yescharacteristic
98C2×C4×C191C2×D411yescharacteristic
99C2×C4×C191#4114yes—
100C2×C4×C191C2411yescharacteristic
101C22×C191—28no—
102C22×C191—21no—
103C22×C191—21no—
104C22×C191—21no—
105C22×C191—21no—
106C22×C191—21no—
107C22×C191—21no—
108C22×C191#12312yes—
109C22×C191AGL(1,ℤ/8)12yes—
110C22×C191#13712yes—
111C22×C191C22×D411yescharacteristic
112C2×D191—41no—
113C2×D191—41no—
114C2×D191—41no—
115C4×C191—21no—
116C4×C191—21no—
117C4×C191—44no—
118C4×C191#12312yes—
119C4×C191C2×#4112yes—
120Dic191—41no—
121Dic191—41no—
122Dic191—41no—
123C2×C191—22no—
124C2×C191—22no—
125C2×C191—22no—
126C2×C191C2×#12311yescharacteristic
127C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
128C2×C191C2×#13711yescharacteristic
129D191—81no—
130C191#309211yescharacteristic, complemented, Sylow
131#3092—1911nomaximal, Sylow
132#525—1911no—
133C22×C4⋊C4D19111yescharacteristic, complemented
134#484—1912no—
135#483—1912no—
136#481—1912no—
137#480—1912no—
138#428—1912no—
139#417—1912no—
140#407—1911no—
141C2×#41—1911no—
142C23×C4—21no—
143C23×C4C2×D19111yescharacteristic
144C2×M16—1912no—
145#123—3821no—
146C22⋊Q8—1911no—
147C22⋊Q8—1911no—
148#121—1911no—
149#121—1911no—
150#120—3821no—
151C4×D4—3821no—
152C4×D4—3821no—
153#117—1911no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C2×D19112yes—
158C2×C4⋊C4C2×D19112yes—
159C2×C4⋊C4C2×D19112yes—
160#107—1914no—
161#106—1914no—
162#103—1914no—
163#102—1914no—
164C22⋊C8—1912no—
165C22⋊C8—1912no—
166C24C22⋊D19111yescharacteristic
167#41—3824no—
168C2×Q8—1911no—
169C2×D4—3821no—
170C2×D4—3821no—
171C2×D4—3821no—
172C2×D4—1911no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—44no—
178C22×C4—3821no—
179C22×C4C22⋊D19111yescharacteristic
180C22×C4C22⋊D19111yescharacteristic
181C22×C4C22⋊D19111yescharacteristic
182C22×C4C22×D19111yescharacteristic
183C22×C4C22×D19111yescharacteristic
184C22×C4C22×D19111yescharacteristic
185M16—3824no—
186C2×C8—1914no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—3821no—
191C4⋊C4—3821no—
192C4⋊C4—1911no—
193C4⋊C4—1911no—
194C4⋊C4C22×D19114yes—
195C22⋊C4—3821no—
196C22⋊C4—3821no—
197C22⋊C4—3821no—
198C22⋊C4—3821no—
199C22⋊C4—3821no—
200C42—3821no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—3821no—
205C23C2×C22⋊D19111yescharacteristic
206C23C2×C22⋊D19111yescharacteristic
207C23C2×C22⋊D19111yescharacteristic
208Q8—3822no—
209D4—3822no—
210D4—7642no—
211D4—7642no—
212D4—7642no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—3821no—
223C2×C4—3821no—
224C2×C4—3821no—
225C2×C4—7642no—
226C2×C4C2×C22⋊D19111yescharacteristic
227C2×C4C2×C22⋊D19111yescharacteristic
228C2×C4C2×C22⋊D19111yescharacteristic
229C2×C4#1569028714yes—
230C2×C4C23×D19111yescharacteristic
231C8—3824no—
232C22—28no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—7641no—
240C22—7641no—
241C22—7641no—
242C22#2025930512yes—
243C22#2025934812yes—
244C22#2025935312yes—
245C22C22×C22⋊D19111yescharacteristic
246C4—21no—
247C4—21no—
248C4—44no—
249C4—7641no—
250C4—7641no—
251C4—7641no—
252C4#2025930512yes—
253C4C2×#1569028712yes—
254C2—22no—
255C2—22no—
256C2—22no—
257C2—15281no—
258C2C2×#2025930511yescharacteristic
259C2C2×#2025934811yescharacteristic
260C2C2×#2025935311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC22×C4⋊C4×C191C2
socleC22×C191C22×D4
prime radical#27133174C2
hypercenterC22×C4⋊C4D191
hypocenterC191#3092
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #27132998C2semidirect
3. #27132993C2semidirect
4. #27132981C2semidirect
5. #27132976C2semidirect
6. #27132703C2semidirect
7. #27132662C2semidirect
8. #27132627C2semidirect
9. C22×C4⋊C4×C191C2semidirect
21. #20259255C22semidirect
22. #20259253C22semidirect
23. #20259245C22semidirect
24. #20259242C22semidirect
130. C191#3092semidirect
133. C22×C4⋊C4D191semidirect

Maximal subgroups

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

subgroupindex
#271331742
#271329982
#271329932
#271329812
#271329762
#271327032
#271326622
#271326272
C22×C4⋊C4×C1912
#3092191