Finite Group Databasebeta

← #37583889

Subgroups — #37583889

Order 24448.

348 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 113 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#27133218C211yesmaximal, characteristic, complemented
2#27133195C211yesmaximal, characteristic
3#27132972C212yesmaximal, complemented
4#27132959C211yesmaximal, characteristic, complemented
5#27132953C211yesmaximal, characteristic, complemented
6#27132920C212yesmaximal, complemented
7#27132906C212yesmaximal
8#27132883C212yesmaximal, complemented
9#27132870C211yesmaximal, characteristic, complemented
10#27132685C211yesmaximal, characteristic, complemented
11#461×C191C211yesmaximal, characteristic, complemented
12C2×#15690277C2212yes—
13C2×#15690270C2212yes—
14#20259322C2211yescharacteristic
15#20259315—21no—
16#20259313—21no—
17#20259313C2211yescharacteristic
18#20259309C2211yescharacteristic
19#20259304—21no—
20#20259302—21no—
21#20259302C2211yescharacteristic
22#20259298C2212yes—
23#20259293C2211yescharacteristic
24Q8×Dic191C2212yes—
25C4×Dic382C2211yescharacteristic
26#20259287—21no—
27C4×C22⋊D191—21no—
28#20259281—21no—
29#20259280—21no—
30C42×D191C2211yescharacteristic
31#20259256C2212yes—
32#20259253C2212yes—
33C4⋊(C191⋊C8)C2211yescharacteristic
34Dic191⋊C8C2212yes—
35#20259247C2212yes—
36#20259246C2212yes—
37#20259243C2212yes—
38C2×SD16×C191C2212yes—
39C4⋊Q8×C191C2211yescharacteristic
40C4×D4×C191C2211yescharacteristic
41C4⋊C8×C191C2211yescharacteristic
42#103×C191C2212yes—
43C2×Dic382—21no—
44C2×Dic382C2311yescharacteristic
45C2×C22⋊D191—21no—
46C2×C4×D191D411yescharacteristic
47C2×C4×D191D411yescharacteristic
48C2×C4×D191C2311yescharacteristic
49#15690277—42no—
50#15690270—42no—
51C2×C191⋊C8C2312yes—
52C4⋊Dic191—21no—
53C4⋊Dic191—21no—
54C4⋊Dic191—22no—
55C4⋊Dic191—21no—
56C4⋊Dic191C2312yes—
57C4⋊Dic191C2311yescharacteristic
58Dic191⋊C4—21no—
59Dic191⋊C4—22no—
60Dic191⋊C4—21no—
61C22⋊Dic191—21no—
62C22⋊Dic191—21no—
63C22⋊Dic191—21no—
64(C2×D191)⋊C4—22no—
65(C2×D191)⋊C4—21no—
66C4×Dic191—22no—
67C4×Dic191—21no—
68C4×Dic191D411yescharacteristic
69C4×Dic191D411yescharacteristic
70C4×Dic191C2311yescharacteristic
71C2×Q8×C191C2312yes—
72C2×D4×C191C2311yescharacteristic
73C22×C4×C191—21no—
74SD16×C191—42no—
75C2×C8×C191C2312yes—
76C4⋊C4×C191—21no—
77C4⋊C4×C191—21no—
78C4⋊C4×C191C2311yescharacteristic
79C22⋊C4×C191—21no—
80C42×C191C2311yescharacteristic
81C22×D191C2×D411yescharacteristic
82Dic382—21no—
83Dic382—21no—
84Dic382—42no—
85C22⋊D191—42no—
86C2×Dic191—21no—
87C2×Dic191—22no—
88C2×Dic191—21no—
89C2×Dic191C2×D411yescharacteristic
90C2×Dic191C2×D411yescharacteristic
91C2×Dic191C2×D411yescharacteristic
92C4×D191—22no—
93C4×D191—21no—
94C4×D191—21no—
95C4×D191—41no—
96C191⋊C8—22no—
97C23×C191—21no—
98Q8×C191—22no—
99Q8×C191—22no—
100D4×C191—21no—
101D4×C191—21no—
102C2×C4×C191—21no—
103C2×C4×C191—22no—
104C2×C4×C191—42no—
105C2×C4×C191C2×D411yescharacteristic
106C2×C4×C191C2×D411yescharacteristic
107C2×C4×C191C2411yescharacteristic
108C8×C191—22no—
109C22×C191—41no—
110C22×C191—41no—
111C22×C191—41no—
112C22×C191C22×D411yescharacteristic
113C2×D191—21no—
114C2×D191—21no—
115C2×D191—21no—
116C4×C191—21no—
117C4×C191—42no—
118C4×C191—41no—
119C4×C191#13712yes—
120C4×C191H(5,𝔽2)11yescharacteristic
121C4×C1912−1+411yescharacteristic
122Dic191—21no—
123Dic191—21no—
124Dic191—21no—
125Dic191—41no—
126Dic191—41no—
127Dic191—42no—
128C2×C191—81no—
129C2×C191#53711yescharacteristic
130C2×C191C2×#13711yescharacteristic
131C2×C191#57511yescharacteristic
132D191—41no—
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#461D19111yescharacteristic, complemented
145#423—1911no—
146C2×Q16—1912no—
147C2×SD16C2×D19112yes—
148C2×M16—1912no—
149C4⋊Q8—1911no—
150C4⋊Q8C2×D19111yescharacteristic
151#126—3821no—
152#125—3821no—
153#125—1911no—
154#124—1911no—
155#123—3821no—
156C22⋊Q8—3821no—
157C22⋊Q8—1912no—
158C22⋊Q8—1911no—
159#121—1911no—
160C4×Q8—3821no—
161C4×Q8—1912no—
162C4×Q8—1911no—
163C4×D4—3821no—
164C4×D4C2×D19111yescharacteristic
165#117—3821no—
166#117—3821no—
167C2×C42—1911no—
168#107—1912no—
169#106—1912no—
170C4⋊C8—1912no—
171C4⋊C8—1911no—
172C4⋊C8C2×D19111yescharacteristic
173#103—1912no—
174#103—1912no—
175#103C2×D19112yes—
176#102—1912no—
177C2×Q8—3821no—
178C2×Q8—1911no—
179C2×Q8C22×D19112yes—
180C2×D4—3821no—
181C2×D4C22×D19111yescharacteristic
182C22×C4—21no—
183C22×C4—1911no—
184C22×C4—1911no—
185C22×C4—1911no—
186Q16—7642no—
187SD16—42no—
188M16—7642no—
189C2×C8—1912no—
190C2×C8C22×D19112yes—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—3821no—
194C4⋊C4—3822no—
195C4⋊C4—3821no—
196C4⋊C4—3821no—
197C4⋊C4—3822no—
198C4⋊C4—3821no—
199C4⋊C4—3821no—
200C4⋊C4—1912no—
201C4⋊C4—1911no—
202C4⋊C4C22×D19111yescharacteristic
203C22⋊C4—21no—
204C22⋊C4—3822no—
205C22⋊C4—3821no—
206C22⋊C4—3821no—
207C22⋊C4—3821no—
208C22⋊C4—3821no—
209C42—3822no—
210C42—3821no—
211C42—1911no—
212C42—1911no—
213C42—1911no—
214C42C22×D19111yescharacteristic
215C23—21no—
216C23—1911no—
217Q8—22no—
218Q8—22no—
219Q8—3821no—
220Q8—3821no—
221Q8—7642no—
222D4—21no—
223D4—21no—
224D4—7642no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—42no—
228C2×C4—3822no—
229C2×C4—3821no—
230C2×C4—3821no—
231C2×C4—3821no—
232C2×C4—3821no—
233C2×C4—3822no—
234C2×C4—1911no—
235C2×C4—1911no—
236C2×C4—1911no—
237C2×C4—7641no—
238C2×C4D4×D19111yescharacteristic
239C2×C4D4×D19111yescharacteristic
240C2×C4C23×D19111yescharacteristic
241C8—22no—
242C8—3822no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—3821no—
247C22—3821no—
248C22—3821no—
249C22C2×D4×D19111yescharacteristic
250C4—21no—
251C4—41no—
252C4—42no—
253C4—3821no—
254C4—3821no—
255C4—3821no—
256C4—7641no—
257C4—7642no—
258C4—7641no—
259C4#2025935112yes—
260C4#2025936711yescharacteristic
261C4#2025936911yescharacteristic
262C2—81no—
263C2—7641no—
264C2#2713324211yescharacteristic
265C2C2×#2025935111yescharacteristic
266C2#2713346211yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#461×C191C2
socleC22×C191C22×D4
prime radical#20259293C22
hypercenter#461D191
hypocenterC191#3252
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #27133218C2semidirect
3. #27132972C2semidirect
4. #27132959C2semidirect
5. #27132953C2semidirect
6. #27132920C2semidirect
8. #27132883C2semidirect
9. #27132870C2semidirect
10. #27132685C2semidirect
11. #461×C191C2semidirect
133. C191#3252semidirect
144. #461D191semidirect

Maximal subgroups

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

subgroupindex
#271332182
#271331952
#271329722
#271329592
#271329532
#271329202
#271329062
#271328832
#271328702
#271326852
#461×C1912
#3252191