Finite Group Databasebeta

← #37583894

Subgroups — #37583894

Order 24448.

348 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 121 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#27133220C211yesmaximal, characteristic, complemented
2#27133197C211yesmaximal, characteristic
3#27132971C212yesmaximal, complemented
4#27132959C211yesmaximal, characteristic, complemented
5#27132954C211yesmaximal, characteristic, complemented
6#27132921C212yesmaximal, complemented
7#27132909C212yesmaximal
8#27132886C212yesmaximal, complemented
9#27132874C211yesmaximal, characteristic, complemented
10#27132869C211yesmaximal, characteristic, complemented
11#423×C191C211yesmaximal, characteristic, complemented
12C2×Dic764C2212yes—
13C2×#15690276C2212yes—
14#20259322C2211yescharacteristic
15#20259322C2211yescharacteristic
16#20259316—21no—
17#20259314—21no—
18#20259314C2211yescharacteristic
19#20259310C2211yescharacteristic
20#20259305—21no—
21#20259301—21no—
22#20259301C2211yescharacteristic
23#20259301C2212yes—
24#20259296C2211yescharacteristic
25C4×Dic382—21no—
26C4×Dic382C2211yescharacteristic
27C4×Dic382C2212yes—
28C4×D764C2211yescharacteristic
29C4×C22⋊D191—21no—
30#20259281—21no—
31#20259279—21no—
32#20259256C2212yes—
33#20259254C2212yes—
34#20259247C2212yes—
35#20259247C2212yes—
36#20259247C2212yes—
37#20259244C2212yes—
38C2×M16×C191C2212yes—
39C2×C42×C191C2211yescharacteristic
40C4⋊C8×C191C2211yescharacteristic
41C4⋊C8×C191C2211yescharacteristic
42C4⋊C8×C191C2212yes—
43C2×Dic382—21no—
44C2×Dic382C2312yes—
45C2×Dic382C2311yescharacteristic
46C2×D764C2311yescharacteristic
47C2×C22⋊D191—21no—
48C2×C4×D191—21no—
49Dic764—42no—
50#15690276—42no—
51C4⋊Dic191—21no—
52C4⋊Dic191—21no—
53C4⋊Dic191—21no—
54C4⋊Dic191C2312yes—
55C4⋊Dic191C2311yescharacteristic
56C4⋊Dic191C2311yescharacteristic
57Dic191⋊C4—21no—
58Dic191⋊C4—21no—
59Dic191⋊C4—21no—
60Dic191⋊C4—22no—
61Dic191⋊C4—22no—
62Dic191⋊C4—21no—
63C22⋊Dic191—22no—
64C22⋊Dic191—21no—
65(C2×D191)⋊C4—21no—
66(C2×D191)⋊C4—21no—
67(C2×D191)⋊C4—21no—
68(C2×D191)⋊C4—21no—
69C4×Dic191—21no—
70C4×Dic191—22no—
71C22×C4×C191D411yescharacteristic
72C22×C4×C191D411yescharacteristic
73C22×C4×C191C2311yescharacteristic
74M16×C191—42no—
75C2×C8×C191C2312yes—
76C2×C8×C191C2312yes—
77C42×C191D411yescharacteristic
78C42×C191D411yescharacteristic
79C42×C191C2311yescharacteristic
80C42×C191C2311yescharacteristic
81C22×D191—21no—
82Dic382—22no—
83Dic382—21no—
84Dic382—22no—
85Dic382—21no—
86Dic382—42no—
87D764—21no—
88D764—21no—
89C22⋊D191—42no—
90C2×Dic191—21no—
91C2×Dic191—21no—
92C2×Dic191—22no—
93C2×Dic191—22no—
94C2×Dic191—21no—
95C4×D191—42no—
96C23×C191C2×D411yescharacteristic
97C2×C4×C191—22no—
98C2×C4×C191—21no—
99C2×C4×C191—21no—
100C2×C4×C191—41no—
101C2×C4×C191C2×D411yescharacteristic
102C2×C4×C191C2×D411yescharacteristic
103C2×C4×C191C2×D411yescharacteristic
104C2×C4×C191C2×D411yescharacteristic
105C2×C4×C191C2×D411yescharacteristic
106C2×C4×C191C2411yescharacteristic
107C8×C191—22no—
108C8×C191—22no—
109C22×C191—21no—
110C22×C191—21no—
111C22×C191—21no—
112C22×C191C22×D411yescharacteristic
113C2×D191—41no—
114C2×D191—41no—
115C2×D191—41no—
116C4×C191—21no—
117C4×C191—21no—
118C4×C191—21no—
119C4×C191—21no—
120C4×C191#13712yes—
121C4×C191H(5,𝔽2)11yescharacteristic
122C4×C1912−1+411yescharacteristic
123Dic191—41no—
124Dic191—42no—
125Dic191—41no—
126Dic191—41no—
127Dic191—42no—
128C2×C191—41no—
129C2×C191#53711yescharacteristic
130C2×C191C2×#13711yescharacteristic
131C2×C191#57511yescharacteristic
132D191—81no—
133C191#325211yescharacteristic, complemented, Sylow
134#3252—1911nomaximal, Sylow
135#533—1911no—
136#529—1911no—
137#479—1912no—
138#477—1911no—
139#476—1911no—
140#470—1912no—
141#468—1912no—
142#464—1912no—
143#462—1911no—
144#461—1911no—
145#423D19111yescharacteristic, complemented
146C2×Q16—1912no—
147C2×SD16—1912no—
148C2×M16C2×D19112yes—
149C4⋊Q8—1911no—
150C4⋊Q8—1911no—
151#126—3821no—
152#125—3821no—
153#125—1911no—
154#124—1911no—
155#123—3821no—
156C22⋊Q8—3821no—
157C22⋊Q8—1911no—
158C22⋊Q8—1912no—
159#121—1911no—
160C4×Q8—3821no—
161C4×Q8—1911no—
162C4×Q8—1912no—
163C4×D4—3821no—
164C4×D4—1911no—
165#117—3821no—
166#117—3821no—
167C2×C42C2×D19111yescharacteristic
168#107—1912no—
169#106—1912no—
170C4⋊C8C2×D19111yescharacteristic
171C4⋊C8C2×D19111yescharacteristic
172C4⋊C8C2×D19112yes—
173#103—1912no—
174#103—1912no—
175#103—1912no—
176#102—1912no—
177C2×Q8—3821no—
178C2×Q8—1911no—
179C2×Q8—1912no—
180C2×D4—3821no—
181C2×D4—1911no—
182C22×C4—3821no—
183C22×C4D76411yescharacteristic
184C22×C4D76411yescharacteristic
185C22×C4C22×D19111yescharacteristic
186Q16—7642no—
187SD16—7642no—
188M16—42no—
189C2×C8C22×D19112yes—
190C2×C8C22×D19112yes—
191C4⋊C4—3821no—
192C4⋊C4—3821no—
193C4⋊C4—3821no—
194C4⋊C4—3822no—
195C4⋊C4—3822no—
196C4⋊C4—3821no—
197C4⋊C4—3821no—
198C4⋊C4—3821no—
199C4⋊C4—3821no—
200C4⋊C4—1912no—
201C4⋊C4—1911no—
202C4⋊C4—1911no—
203C22⋊C4—3822no—
204C22⋊C4—3821no—
205C22⋊C4—3821no—
206C22⋊C4—3821no—
207C22⋊C4—3821no—
208C22⋊C4—3821no—
209C42—3821no—
210C42—3822no—
211C42D76411yescharacteristic
212C42D76411yescharacteristic
213C42C22×D19111yescharacteristic
214C42C22×D19111yescharacteristic
215C23—3821no—
216C23C2×D76411yescharacteristic
217Q8—3822no—
218Q8—3821no—
219Q8—3821no—
220Q8—3822no—
221Q8—7642no—
222D4—3821no—
223D4—3821no—
224D4—7642no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—41no—
229C2×C4—3821no—
230C2×C4—3822no—
231C2×C4—3821no—
232C2×C4—3822no—
233C2×C4—3821no—
234C2×C4—7642no—
235C2×C4C2×D76411yescharacteristic
236C2×C4C2×D76411yescharacteristic
237C2×C4C2×D76411yescharacteristic
238C2×C4C2×D76411yescharacteristic
239C2×C4C2×D76411yescharacteristic
240C2×C4C23×D19111yescharacteristic
241C8—22no—
242C8—22no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—7641no—
247C22—7641no—
248C22—7641no—
249C22C22×D76411yescharacteristic
250C4—21no—
251C4—21no—
252C4—21no—
253C4—21no—
254C4—7641no—
255C4—7641no—
256C4—7642no—
257C4—7641no—
258C4—7642no—
259C4#2025935512yes—
260C4#2025936711yescharacteristic
261C4#2025937011yescharacteristic
262C2—41no—
263C2—15281no—
264C2#2713324411yescharacteristic
265C2C2×#2025935511yescharacteristic
266C2#2713346311yescharacteristic
subgroupG/H
centerC22C22×D764
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#423×C191C2
socleC22×C191C22×D4
prime radical#20259296C22
hypercenter#423D191
hypocenterC191#3252
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
1. #27133220C2semidirect
3. #27132971C2semidirect
4. #27132959C2semidirect
5. #27132954C2semidirect
6. #27132921C2semidirect
8. #27132886C2semidirect
9. #27132874C2semidirect
10. #27132869C2semidirect
11. #423×C191C2semidirect
133. C191#3252semidirect
145. #423D191semidirect

Maximal subgroups

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

subgroupindex
#271332202
#271331972
#271329712
#271329592
#271329542
#271329212
#271329092
#271328862
#271328742
#271328692
#423×C1912
#3252191