Finite Group Databasebeta

← #37576180

Subgroups — #37576180

Order 24448.

780 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 161 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
1C2×#20259298C212yesmaximal, complemented
2C2×C22⋊C4×D191C212yesmaximal
3#27132599C211yesmaximal, characteristic, complemented
4#27132586C212yesmaximal, complemented
5#27132537C212yesmaximal, complemented
6#27132537C212yesmaximal, complemented
7#27132522C212yesmaximal, complemented
8#27132518C211yesmaximal, characteristic
9#393×C191C211yesmaximal, characteristic, complemented
10C24×D191C2211yescharacteristic
11C22×C4×D191C2212yes—
12C22×C4×D191C2212yes—
13#20259298—28no—
14C2×C4⋊Dic191C2212yescomplemented
15C2×C4⋊Dic191C2211yescharacteristic, complemented
16C2×Dic191⋊C4C2212yescomplemented
17C2×Dic191⋊C4C2212yescomplemented
18C2×Dic191⋊C4C2212yescomplemented
19C2×C22⋊Dic191C2212yes—
20C2×C22⋊Dic191C2211yescharacteristic
21C2×(C2×D191)⋊C4C2212yes—
22C2×(C2×D191)⋊C4C2212yes—
23C2×(C2×D191)⋊C4C2212yes—
24C2×(C2×D191)⋊C4C2212yes—
25C22⋊C4×D191—28no—
26C191⋊#95C2212yescomplemented
27C191⋊#95C2211yescharacteristic, complemented
28C22×Q8×C191C2211yescharacteristic, complemented
29C2×C22⋊C4×C191C2212yes—
30C2×C22⋊C4×C191C2211yescharacteristic
31#95×C191C2211yescharacteristic, complemented
32#95×C191C2212yescomplemented
33C23×D191—24no—
34C23×D191—24no—
35C23×D191—41no—
36C23×D191—41no—
37C23×D191Q814yescomplemented
38C23×D191C2312yes—
39C22×Dic191C2312yes—
40C22×Dic191C2312yes—
41C22×Dic191C2312yes—
42C2×C4×D191—24no—
43C2×C4×D191—24no—
44C2×C4×D191—24no—
45C2×C4×D191—24no—
46C2×C4×D191—42no—
47C2×C4×D191—42no—
48C4⋊Dic191—24no—
49C4⋊Dic191—41no—
50Dic191⋊C4—24no—
51Dic191⋊C4—24no—
52Dic191⋊C4—42no—
53C22⋊Dic191—24no—
54C22⋊Dic191—41no—
55(C2×D191)⋊C4—24no—
56(C2×D191)⋊C4—24no—
57(C2×D191)⋊C4—24no—
58(C2×D191)⋊C4D418yes—
59C24×C191C2311yescharacteristic
60C2×Q8×C191—24no—
61C2×Q8×C191—41no—
62C22×C4×C191C2312yes—
63C22×C4×C191C2311yescharacteristic
64C22×C4×C191C2311yescharacteristic
65C22×C4×C191C2312yes—
66C22⋊C4×C191—24no—
67C22⋊C4×C191—41no—
68C22×D191—22no—
69C22×D191—22no—
70C22×D191—22no—
71C22×D191—22no—
72C22×D191—216no—
73C22×D191—44no—
74C22×D191—44no—
75C22×D191—48no—
76C22×D191—44no—
77C22×D191C2×D414yes—
78C22×D191C2×Q814yes—
79C22×D191#4114yes—
80C2×Dic191—22no—
81C2×Dic191—22no—
82C2×Dic191—22no—
83C2×Dic191—22no—
84C2×Dic191—22no—
85C2×Dic191—22no—
86C2×Dic191—22no—
87C2×Dic191—22no—
88C2×Dic191C2×D414yes—
89C4×D191—48no—
90C4×D191—48no—
91C23×C191—22no—
92C23×C191—22no—
93C23×C191—21no—
94C23×C191—21no—
95C23×C191C2×Q812yes—
96C23×C191C2411yescharacteristic
97Q8×C191—44no—
98C2×C4×C191—22no—
99C2×C4×C191—22no—
100C2×C4×C191—22no—
101C2×C4×C191—22no—
102C2×C4×C191—22no—
103C2×C4×C191—22no—
104C2×C4×C191—21no—
105C2×C4×C191—21no—
106C2×C4×C191—22no—
107C2×C4×C191C2×D414yes—
108C22×C191—24no—
109C22×C191—22no—
110C22×C191—41no—
111C22×C191—42no—
112C22×C191—41no—
113C22×C191C22×D412yes—
114C22×C191C22×Q811yescharacteristic
115C22×C191C2×#4112yes—
116C22×C191H(5,𝔽2)11yescharacteristic
117C22×C191H(5,𝔽2)11yescharacteristic
118C2×D191—24no—
119C2×D191—24no—
120C2×D191—24no—
121C2×D191—24no—
122C2×D191—24no—
123C2×D191—24no—
124C2×D191—416no—
125C2×D191C22⋊Q818yescomplemented
126C4×C191—24no—
127C4×C191—41no—
128C4×C191—42no—
129C4×C191—41no—
130Dic191—24no—
131Dic191—42no—
132Dic191—42no—
133C2×C191—42no—
134C2×C191C2×C22⋊Q812yes—
135C2×C191#54311yescharacteristic
136C2×C191D4211yescharacteristic
137C2×C191#54612yes—
138C2×C191#55111yescharacteristic
139D191—28no—
140C191#243711yescharacteristic, complemented, Sylow
141#2437—1911nomaximal, Sylow
142C2×C22⋊Q8—1912no—
143C22×C22⋊C4—1912no—
144#399—1911no—
145#396—1912no—
146#393D19111yescharacteristic, complemented
147#385—1912no—
148#385—1912no—
149#380—1912no—
150#379—1911no—
151C25—1911no—
152C22×Q8C2×D19111yescharacteristic, complemented
153C23×C4—1912no—
154C23×C4—1912no—
155C22⋊Q8—3828no—
156C2×C4⋊C4—1912no—
157C2×C4⋊C4—1912no—
158C2×C4⋊C4—1912no—
159C2×C4⋊C4—1912no—
160C2×C4⋊C4—1911no—
161C2×C22⋊C4—3828no—
162C2×C22⋊C4—1912no—
163C2×C22⋊C4—1912no—
164C2×C22⋊C4—1912no—
165C2×C22⋊C4—1912no—
166C2×C22⋊C4—1912no—
167C2×C22⋊C4—1911no—
168C2×C22⋊C4C2×D19112yes—
169C2×C22⋊C4C2×D19111yescharacteristic
170#95—1912no—
171#95—1911no—
172#95C2×D19111yescharacteristic, complemented
173#95C2×D19112yescomplemented
174C24—3824no—
175C24—3824no—
176C24—1912no—
177C24—1914no—
178C24—7641no—
179C24—7641no—
180C24C22×D19111yescharacteristic
181C2×Q8—24no—
182C2×Q8—41no—
183C22×C4—3824no—
184C22×C4—3824no—
185C22×C4—3824no—
186C22×C4—3824no—
187C22×C4—1912no—
188C22×C4—1912no—
189C22×C4—1912no—
190C22×C4—7642no—
191C22×C4—7642no—
192C22×C4C22×D19111yescharacteristic
193C22×C4C22×D19112yes—
194C22×C4C22×D19111yescharacteristic
195C22×C4C22×D19112yes—
196C4⋊C4—3824no—
197C4⋊C4—3824no—
198C4⋊C4—3824no—
199C4⋊C4—7642no—
200C4⋊C4—7641no—
201C22⋊C4—24no—
202C22⋊C4—41no—
203C22⋊C4—3824no—
204C22⋊C4—3824no—
205C22⋊C4—3824no—
206C22⋊C4—3824no—
207C22⋊C4—1918no—
208C22⋊C4—7641no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—21no—
213C23—38216no—
214C23—3822no—
215C23—3822no—
216C23—3822no—
217C23—3822no—
218C23—1914no—
219C23—1914no—
220C23—1914no—
221C23—7644no—
222C23—7644no—
223C23—7648no—
224C23—7644no—
225C23Q8×D19112yes—
226C23C23×D19111yescharacteristic
227Q8—44no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—3822no—
238C2×C4—3822no—
239C2×C4—3822no—
240C2×C4—3822no—
241C2×C4—3822no—
242C2×C4—3822no—
243C2×C4—3822no—
244C2×C4—3822no—
245C2×C4—1914no—
246C2×C4—7648no—
247C2×C4—7648no—
248C2×C4D4×D19114yes—
249C22—22no—
250C22—24no—
251C22—41no—
252C22—42no—
253C22—41no—
254C22—3824no—
255C22—3824no—
256C22—3824no—
257C22—3824no—
258C22—3824no—
259C22—3824no—
260C22—1918no—
261C22—76416no—
262C22C2×D4×D19112yes—
263C22C2×Q8×D19111yescharacteristic
264C22#41×D19112yes—
265C22#2025936711yescharacteristic
266C22#2025936811yescharacteristic
267C4—24no—
268C4—41no—
269C4—42no—
270C4—41no—
271C4—3824no—
272C4—7642no—
273C4—7642no—
274C2—42no—
275C2—3828no—
276C2C22⋊Q8×D19112yes—
277C2#2713328811yescharacteristic
278C2#2713329511yescharacteristic
279C2#2713330612yes—
280C2#2713334311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#393×C191C2
prime radicalC24×D191C22
hypercenter#393D191
hypocenterC191#2437
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#20259298C2semidirect
3. #27132599C2semidirect
4. #27132586C2semidirect
5. #27132537C2semidirect
6. #27132537C2semidirect
7. #27132522C2semidirect
9. #393×C191C2semidirect
14. C2×C4⋊Dic191C22semidirect
15. C2×C4⋊Dic191C22semidirect
16. C2×Dic191⋊C4C22semidirect
17. C2×Dic191⋊C4C22semidirect
18. C2×Dic191⋊C4C22semidirect
26. C191⋊#95C22semidirect
27. C191⋊#95C22semidirect
28. C22×Q8×C191C22semidirect
31. #95×C191C22semidirect
32. #95×C191C22semidirect
37. C23×D191Q8semidirect
125. C2×D191C22⋊Q8semidirect
140. C191#2437semidirect
146. #393D191semidirect
152. C22×Q8C2×D191semidirect
172. #95C2×D191semidirect
173. #95C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202592982
C2×C22⋊C4×D1912
#271325992
#271325862
#271325372
#271325222
#271325182
#393×C1912
#2437191