Finite Group Databasebeta

← #37583316

Subgroups — #37583316

Order 24448.

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

normal subgroups only

This group has 514 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#121×D191C212yesmaximal, complemented
2#27132983C212yesmaximal, complemented
3#27132980C212yesmaximal, complemented
4#27132916C211yesmaximal, characteristic, complemented
5#27132914C212yesmaximal, complemented
6#27132798C212yesmaximal, complemented
7#27132795C212yesmaximal, complemented
8#27132626C211yesmaximal, characteristic, complemented
9#469×C191C211yesmaximal, characteristic, complemented
10C2×D4×D191—22no—
11C2×D4×D191C2212yescomplemented
12C22×C4×D191C2211yescharacteristic
13C2×D1528C2211yescharacteristic, complemented
14C2×#15690272C2212yescomplemented
15C2×#15690271C2211yescharacteristic, complemented
16#20259297C2212yes—
17#20259296C2212yescomplemented
18#20259295—22no—
19#20259294—22no—
20#20259293C2212yes—
21C4⋊C4×D191C2212yescomplemented
22C22⋊C4×D191—22no—
23#20259254C2211yescharacteristic, complemented
24#20259253C2212yescomplemented
25#20259244C2212yescomplemented
26#20259243C2212yescomplemented
27#20259242C2212yescomplemented
28(C2×D191)⋊C8C2212yescomplemented
29(C2×D191)⋊C8C2212yescomplemented
30C2×D8×C191C2211yescharacteristic, complemented
31C2×M16×C191C2211yescharacteristic, complemented
32#121×C191C2212yescomplemented
33#106×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—
53#15690271—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—
64M16×C191—41no—
65C2×C8×C191C2311yescharacteristic
66C2×C8×C191C2311yescharacteristic
67C4⋊C4×C191C2312yes—
68C22⋊C4×C191—22no—
69C22×D191—24no—
70C22×D191—22no—
71C22×D191—41no—
72C22×D191—44no—
73C22×D191—44no—
74C22×D191—44no—
75C22×D191C2×D412yes—
76D764—22no—
77D764—22no—
78D764—44no—
79C22⋊D191—44no—
80C22⋊D191—44no—
81C22⋊D191—44no—
82C2×Dic191—22no—
83C2×Dic191—41no—
84C2×Dic191C2×D412yes—
85C4×D191—22no—
86C4×D191—22no—
87C4×D191—42no—
88C4×D191—44no—
89C191⋊C8—22no—
90C23×C191—22no—
91C23×C191C2×D411yescharacteristic
92D4×C191—22no—
93D4×C191—22no—
94D4×C191—44no—
95C2×C4×C191—21no—
96C2×C4×C191—21no—
97C2×C4×C191—22no—
98C2×C4×C191C2×D411yescharacteristic
99C2×C4×C191C2411yescharacteristic
100C8×C191—21no—
101C8×C191—21no—
102C22×C191—21no—
103C22×C191—22no—
104C22×C191—42no—
105C22×C191—42no—
106C22×C191—42no—
107C22×C191C22×D411yescharacteristic
108C2×D191—22no—
109C2×D191—22no—
110C2×D191—44no—
111C2×D191—42no—
112C2×D191—42no—
113C2×D191—42no—
114C2×D191—88no—
115C2×D191AGL(1,ℤ/8)14yes—
116C4×C191—21no—
117C4×C191—42no—
118C4×C191H(5,𝔽2)11yescharacteristic
119C4×C191H(5,𝔽2)11yescharacteristic
120Dic191—22no—
121Dic191—42no—
122C2×C191—41no—
123C2×C191—82no—
124C2×C191#53411yescharacteristic
125C2×C191C2×AGL(1,ℤ/8)12yes—
126D191—24no—
127D191—82no—
128C191#319211yescharacteristic, complemented, Sylow
129#3192—1911nomaximal, Sylow
130C2×#121—1912no—
131#481—1912no—
132#481—1912no—
133#469—1911no—
134#469—1912no—
135#469D19111yescharacteristic, complemented
136#447—1912no—
137#447—1912no—
138#407—1911no—
139C22×D4—3822no—
140C22×D4—1912no—
141C23×C4—1911no—
142C2×D8—1912no—
143C2×D8—1911no—
144C2×D8C2×D19111yescharacteristic, complemented
145C2×M16—1911no—
146C2×M16C2×D19111yescharacteristic, complemented
147#121—3822no—
148#121—3822no—
149#121—1912no—
150#121—1912no—
151#121—1912no—
152#121C2×D19112yescomplemented
153C2×C4⋊C4—1912no—
154C2×C22⋊C4—3822no—
155#106—1911no—
156#106—1912no—
157#106C2×D19111yescharacteristic, complemented
158#102—1912no—
159#102—1912no—
160#102—1912no—
161#102C2×D19112yescomplemented
162C22⋊C8—1912no—
163C22⋊C8—1912no—
164C24—3822no—
165C24—1911no—
166C2×D4—22no—
167C2×D4—3822no—
168C2×D4—3822no—
169C2×D4—3822no—
170C2×D4—3822no—
171C2×D4—1912no—
172C2×D4—7642no—
173C2×D4—7642no—
174C2×D4—7648no—
175C2×D4C22×D19112yes—
176C22×C4—3822no—
177C22×C4—1912no—
178C22×C4—1911no—
179C22×C4—1912no—
180C22×C4—7641no—
181C22×C4—7641no—
182C22×C4C22×D19111yescharacteristic
183D8—41no—
184D8—7641no—
185D8—7642no—
186M16—41no—
187M16—7641no—
188C2×C8—1912no—
189C2×C8C22×D19111yescharacteristic
190C2×C8C22×D19111yescharacteristic
191C4⋊C4—3822no—
192C4⋊C4—1912no—
193C4⋊C4C22×D19112yes—
194C22⋊C4—22no—
195C22⋊C4—3822no—
196C22⋊C4—3822no—
197C22⋊C4—3822no—
198C23—22no—
199C23—3824no—
200C23—3822no—
201C23—1912no—
202C23—7641no—
203C23—7644no—
204C23—7644no—
205C23—7644no—
206C23D4×D19111yescharacteristic
207D4—22no—
208D4—22no—
209D4—44no—
210D4—3822no—
211D4—3822no—
212D4—7644no—
213D4—7644no—
214D4—7644no—
215D4—7644no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—3822no—
220C2×C4—3822no—
221C2×C4—3822no—
222C2×C4—1912no—
223C2×C4—7642no—
224C2×C4—7644no—
225C2×C4—7641no—
226C2×C4D4×D19111yescharacteristic
227C2×C4C23×D19111yescharacteristic
228C8—21no—
229C8—21no—
230C8—3822no—
231C22—22no—
232C22—21no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—3822no—
237C22—3822no—
238C22—1914no—
239C22—15288no—
240C22—7644no—
241C22—7642no—
242C22—7642no—
243C22—7642no—
244C22C2×D4×D19111yescharacteristic
245C4—21no—
246C4—42no—
247C4—3822no—
248C4—7642no—
249C4#2025936711yescharacteristic
250C4#2025936811yescharacteristic
251C2—41no—
252C2—82no—
253C2—3824no—
254C2—15282no—
255C2#2713322111yescharacteristic
256C2AGL(1,ℤ/8)×D19112yes—
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#469×C191C2
socleC22×C191C22×D4
hypercenter#469D191
hypocenterC191#3192
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #121×D191C2semidirect
2. #27132983C2semidirect
3. #27132980C2semidirect
4. #27132916C2semidirect
5. #27132914C2semidirect
6. #27132798C2semidirect
7. #27132795C2semidirect
8. #27132626C2semidirect
9. #469×C191C2semidirect
11. C2×D4×D191C22semidirect
13. C2×D1528C22semidirect
14. C2×#15690272C22semidirect
15. C2×#15690271C22semidirect
17. #20259296C22semidirect
21. C4⋊C4×D191C22semidirect
23. #20259254C22semidirect
24. #20259253C22semidirect
25. #20259244C22semidirect
26. #20259243C22semidirect
27. #20259242C22semidirect
28. (C2×D191)⋊C8C22semidirect
29. (C2×D191)⋊C8C22semidirect
30. C2×D8×C191C22semidirect
31. C2×M16×C191C22semidirect
32. #121×C191C22semidirect
33. #106×C191C22semidirect
34. #102×C191C22semidirect
128. C191#3192semidirect
135. #469D191semidirect
144. C2×D8C2×D191semidirect
146. C2×M16C2×D191semidirect
152. #121C2×D191semidirect
157. #106C2×D191semidirect
161. #102C2×D191semidirect

Maximal subgroups

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

subgroupindex
#121×D1912
#271329832
#271329802
#271329162
#271329142
#271327982
#271327952
#271326262
#469×C1912
#3192191