Finite Group Databasebeta

← #37583267

Subgroups — #37583267

Order 24448.

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

normal subgroups only

This group has 520 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#121×D191C212yesmaximal, complemented
2#27132978C212yesmaximal, complemented
3#27132975C212yesmaximal, complemented
4#27132899C211yesmaximal, characteristic, complemented
5#27132897C212yesmaximal, complemented
6#27132798C212yesmaximal, complemented
7#27132795C212yesmaximal, complemented
8C2×(C2×D191)⋊C8C211yesmaximal, characteristic, complemented
9#466×C191C211yesmaximal, characteristic, complemented
10C2×D4×D191—22no—
11C2×D4×D191C2212yescomplemented
12C22×C4×D191C2211yescharacteristic
13C2×D1528C2211yescharacteristic, complemented
14C2×#15690272C2212yescomplemented
15C22×C191⋊C8C2211yescharacteristic, complemented
16#20259297C2212yes—
17#20259296C2212yescomplemented
18#20259295—22no—
19#20259294—22no—
20#20259293C2212yes—
21C4⋊C4×D191C2212yescomplemented
22C22⋊C4×D191—22no—
23#20259256C2211yescharacteristic, complemented
24#20259255C2212yescomplemented
25#20259244C2212yescomplemented
26#20259243C2212yescomplemented
27#20259242C2212yescomplemented
28(C2×D191)⋊C8C2212yescomplemented
29(C2×D191)⋊C8C2212yescomplemented
30C2×D8×C191C2211yescharacteristic, complemented
31C22×C8×C191C2211yescharacteristic, complemented
32#121×C191C2212yescomplemented
33#107×C191C2211yescharacteristic, complemented
34#102×C191C2212yescomplemented
35C23×D191—22no—
36C23×D191D411yescharacteristic
37C2×D764—22no—
38C2×D764C2312yes—
39C2×C22⋊D191—22no—
40C2×C22⋊D191—22no—
41C2×C22⋊D191—22no—
42D4×D191—42no—
43D4×D191—42no—
44D4×D191—48no—
45C22×Dic191D411yescharacteristic
46C2×C4×D191—22no—
47C2×C4×D191—41no—
48C2×C4×D191—41no—
49C2×C4×D191D412yes—
50C2×C4×D191C2312yes—
51D1528—41no—
52#15690272—42no—
53C2×C191⋊C8—41no—
54C2×C191⋊C8C2312yes—
55C4⋊Dic191C2312yes—
56Dic191⋊C4—22no—
57C22⋊Dic191—22no—
58(C2×D191)⋊C4—22no—
59(C2×D191)⋊C4—22no—
60C2×D4×C191—22no—
61C2×D4×C191C2312yes—
62C22×C4×C191C2311yescharacteristic
63D8×C191—41no—
64C2×C8×C191—41no—
65C2×C8×C191C2311yescharacteristic
66C2×C8×C191C2311yescharacteristic
67C4⋊C4×C191C2312yes—
68C22⋊C4×C191—22no—
69C22×D191—22no—
70C22×D191—41no—
71C22×D191—41no—
72C22×D191—44no—
73C22×D191—44no—
74C22×D191—44no—
75C22×D191D814yescomplemented
76C22×D191C2×D412yes—
77D764—22no—
78D764—22no—
79D764—44no—
80C22⋊D191—44no—
81C22⋊D191—44no—
82C22⋊D191—44no—
83C2×Dic191—22no—
84C2×Dic191—41no—
85C2×Dic191C2×D412yes—
86C4×D191—22no—
87C4×D191—22no—
88C4×D191—42no—
89C4×D191—44no—
90C191⋊C8—22no—
91C23×C191—22no—
92C23×C191C2×D411yescharacteristic
93D4×C191—22no—
94D4×C191—22no—
95D4×C191—44no—
96C2×C4×C191—21no—
97C2×C4×C191—21no—
98C2×C4×C191—22no—
99C2×C4×C191C2×D411yescharacteristic
100C2×C4×C191C2411yescharacteristic
101C8×C191—21no—
102C8×C191—21no—
103C22×C191—21no—
104C22×C191—21no—
105C22×C191—42no—
106C22×C191—42no—
107C22×C191—42no—
108C22×C191C2×D812yes—
109C22×C191C22×D411yescharacteristic
110C2×D191—22no—
111C2×D191—22no—
112C2×D191—44no—
113C2×D191—42no—
114C2×D191—42no—
115C2×D191—42no—
116C2×D191—88no—
117C2×D191C2×D814yes—
118C4×C191—21no—
119C4×C191—42no—
120C4×C191H(5,𝔽2)11yescharacteristic
121C4×C191H(5,𝔽2)11yescharacteristic
122Dic191—22no—
123Dic191—42no—
124C2×C191—22no—
125C2×C191—82no—
126C2×C191#53411yescharacteristic
127C2×C191C22×D811yescharacteristic
128C2×C191#57511yescharacteristic
129D191—24no—
130D191—82no—
131C191#318611yescharacteristic, complemented, Sylow
132#3186—1911nomaximal, Sylow
133C2×#121—1912no—
134#480—1912no—
135#480—1912no—
136#466—1911no—
137#466—1912no—
138#466D19111yescharacteristic, complemented
139#447—1912no—
140#447—1912no—
141C2×C22⋊C8—1911no—
142C22×D4—3822no—
143C22×D4—1912no—
144C23×C4—1911no—
145C2×D8—1912no—
146C2×D8—1911no—
147C2×D8C2×D19111yescharacteristic, complemented
148C22×C8—1911no—
149C22×C8C2×D19111yescharacteristic, complemented
150#121—3822no—
151#121—3822no—
152#121—1912no—
153#121—1912no—
154#121—1912no—
155#121C2×D19112yescomplemented
156C2×C4⋊C4—1912no—
157C2×C22⋊C4—3822no—
158#107—1911no—
159#107—1912no—
160#107C2×D19111yescharacteristic, complemented
161#102—1912no—
162#102—1912no—
163#102—1912no—
164#102C2×D19112yescomplemented
165C22⋊C8—1912no—
166C22⋊C8—1912no—
167C24—3822no—
168C24—1911no—
169C2×D4—22no—
170C2×D4—3822no—
171C2×D4—3822no—
172C2×D4—3822no—
173C2×D4—3822no—
174C2×D4—1912no—
175C2×D4—7642no—
176C2×D4—7642no—
177C2×D4—7648no—
178C2×D4C22×D19112yes—
179C22×C4—3822no—
180C22×C4—1912no—
181C22×C4—1911no—
182C22×C4—1912no—
183C22×C4—7641no—
184C22×C4—7641no—
185C22×C4C22×D19111yescharacteristic
186D8—41no—
187D8—7641no—
188D8—7642no—
189C2×C8—41no—
190C2×C8—1912no—
191C2×C8—7641no—
192C2×C8C22×D19111yescharacteristic
193C2×C8C22×D19111yescharacteristic
194C4⋊C4—3822no—
195C4⋊C4—1912no—
196C4⋊C4C22×D19112yes—
197C22⋊C4—22no—
198C22⋊C4—3822no—
199C22⋊C4—3822no—
200C22⋊C4—3822no—
201C23—22no—
202C23—3822no—
203C23—1912no—
204C23—1914no—
205C23—7641no—
206C23—7641no—
207C23—7644no—
208C23—7644no—
209C23—7644no—
210C23D4×D19111yescharacteristic
211D4—22no—
212D4—22no—
213D4—44no—
214D4—3822no—
215D4—3822no—
216D4—7644no—
217D4—7644no—
218D4—7644no—
219D4—7644no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—3822no—
224C2×C4—3822no—
225C2×C4—3822no—
226C2×C4—1912no—
227C2×C4—7642no—
228C2×C4—7644no—
229C2×C4—7641no—
230C2×C4D4×D19111yescharacteristic
231C2×C4C23×D19111yescharacteristic
232C8—21no—
233C8—21no—
234C8—3822no—
235C22—21no—
236C22—21no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—3822no—
241C22—3822no—
242C22—1914no—
243C22—15288no—
244C22—7644no—
245C22—7642no—
246C22—7642no—
247C22—7642no—
248C22D8×D19112yes—
249C22C2×D4×D19111yescharacteristic
250C4—21no—
251C4—42no—
252C4—3822no—
253C4—7642no—
254C4#2025936711yescharacteristic
255C4#2025936811yescharacteristic
256C2—22no—
257C2—82no—
258C2—3824no—
259C2—15282no—
260C2#2713322111yescharacteristic
261C2C2×D8×D19111yescharacteristic
262C2#2713346011yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#466×C191C2
socleC22×C191C22×D4
hypercenter#466D191
hypocenterC191#3186
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #121×D191C2semidirect
2. #27132978C2semidirect
3. #27132975C2semidirect
4. #27132899C2semidirect
5. #27132897C2semidirect
6. #27132798C2semidirect
7. #27132795C2semidirect
8. C2×(C2×D191)⋊C8C2semidirect
9. #466×C191C2semidirect
11. C2×D4×D191C22semidirect
13. C2×D1528C22semidirect
14. C2×#15690272C22semidirect
15. C22×C191⋊C8C22semidirect
17. #20259296C22semidirect
21. C4⋊C4×D191C22semidirect
23. #20259256C22semidirect
24. #20259255C22semidirect
25. #20259244C22semidirect
26. #20259243C22semidirect
27. #20259242C22semidirect
28. (C2×D191)⋊C8C22semidirect
29. (C2×D191)⋊C8C22semidirect
30. C2×D8×C191C22semidirect
31. C22×C8×C191C22semidirect
32. #121×C191C22semidirect
33. #107×C191C22semidirect
34. #102×C191C22semidirect
75. C22×D191D8semidirect
131. C191#3186semidirect
138. #466D191semidirect
147. C2×D8C2×D191semidirect
149. C22×C8C2×D191semidirect
155. #121C2×D191semidirect
160. #107C2×D191semidirect
164. #102C2×D191semidirect

Maximal subgroups

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

subgroupindex
#121×D1912
#271329782
#271329752
#271328992
#271328972
#271327982
#271327952
C2×(C2×D191)⋊C82
#466×C1912
#3186191