Finite Group Databasebeta

← #37575835

Subgroups — #37575835

Order 24448.

1056 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 185 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×#20259290C212yesmaximal, complemented
2C22×(C2×D191)⋊C4C211yesmaximal, characteristic, complemented
3#27132574C212yesmaximal, complemented
4#27132571C212yesmaximal, complemented
5#27132543C212yesmaximal, complemented
6#27132542C214yesmaximal, complemented
7#27132526C211yesmaximal, characteristic, complemented
8C22×#41×C191C211yesmaximal, characteristic, complemented
9C24×D191C2211yescharacteristic
10C22×C22⋊D191C2214yes—
11C23×Dic191C2211yescharacteristic
12#20259290—28no—
13C2×C22⋊Dic191C2212yes—
14C2×C22⋊Dic191C2214yescomplemented
15C2×(C2×D191)⋊C4—24no—
16C2×(C2×D191)⋊C4C2212yes—
17C2×(C2×D191)⋊C4C2214yes—
18C2×(C2×D191)⋊C4C2212yes—
19C191⋊#95C2212yescomplemented
20C191⋊#95C2212yescomplemented
21C191⋊#95C2214yescomplemented
22C2×#41×C191—28no—
23C2×#41×C191—24no—
24C22×Q8×C191C2211yescharacteristic, complemented
25C22×D4×C191C2212yes—
26C22×D4×C191C2211yescharacteristic, complemented
27C23×C4×C191C2211yescharacteristic
28C23×C4×C191C2212yescomplemented
29C23×D191—24no—
30C23×D191—22no—
31C23×D191—24no—
32C23×D191—42no—
33C23×D191C2312yes—
34C2×C22⋊D191—28no—
35C2×C22⋊D191—28no—
36C2×C22⋊D191—44no—
37C22×Dic191—22no—
38C22×Dic191—42no—
39C22×Dic191C2312yes—
40C22×Dic191C2314yes—
41C22⋊Dic191—24no—
42C22⋊Dic191—44no—
43(C2×D191)⋊C4—24no—
44(C2×D191)⋊C4—24no—
45(C2×D191)⋊C4—28no—
46C24×C191C2312yes—
47C24×C191C2311yescharacteristic
48#41×C191—232no—
49C2×Q8×C191—24no—
50C2×Q8×C191—22no—
51C2×D4×C191—24no—
52C2×D4×C191—24no—
53C2×D4×C191—24no—
54C2×D4×C191—22no—
55C2×D4×C191D418yescomplemented
56C22×C4×C191—24no—
57C22×C4×C191—24no—
58C22×C4×C191—24no—
59C22×C4×C191—24no—
60C22×C4×C191D414yescomplemented
61C22×C4×C191C2311yescharacteristic
62C22×C4×C191C2311yescharacteristic
63C22×C4×C191C2312yes—
64C22×D191—28no—
65C22×D191—22no—
66C22×D191—24no—
67C22×D191—22no—
68C22×D191—24no—
69C22×D191—416no—
70C22×D191—44no—
71C22×D191—44no—
72C22×D191#4114yescomplemented
73C22⋊D191—416no—
74C2×Dic191—22no—
75C2×Dic191—24no—
76C2×Dic191—24no—
77C2×Dic191—24no—
78C2×Dic191—24no—
79C2×Dic191—44no—
80C23×C191—22no—
81C23×C191—22no—
82C23×C191—22no—
83C23×C191—22no—
84C23×C191—24no—
85C23×C191—24no—
86C23×C191C2×D414yes—
87C23×C191C2×D414yes—
88C23×C191C2×D412yes—
89C23×C191C2411yescharacteristic
90Q8×C191—28no—
91D4×C191—28no—
92D4×C191—216no—
93C2×C4×C191—216no—
94C2×C4×C191—28no—
95C2×C4×C191—22no—
96C2×C4×C191—22no—
97C2×C4×C191—22no—
98C2×C4×C191—22no—
99C2×C4×C191C2×D412yes—
100C2×C4×C191C2×D414yes—
101C2×C4×C191C2×D412yes—
102C22×C191—22no—
103C22×C191—22no—
104C22×C191—28no—
105C22×C191—24no—
106C22×C191—28no—
107C22×C191—42no—
108C22×C191—42no—
109C22×C191—44no—
110C22×C191#12014yes—
111C22×C191C22×D411yescharacteristic
112C22×C191C22×D412yes—
113C22×C191C2×#4112yes—
114C22×C191H(5,𝔽2)12yes—
115C2×D191—24no—
116C2×D191—28no—
117C2×D191—416no—
118C2×D191—42no—
119C2×D191—42no—
120C2×D191—44no—
121C4×C191—22no—
122C4×C191—24no—
123C4×C191—22no—
124Dic191—42no—
125Dic191—44no—
126C2×C191—24no—
127C2×C191—44no—
128C2×C191C2×#12011yescharacteristic
129C2×C191#52511yescharacteristic
130C2×C191#53511yescharacteristic
131C2×C191#54614yes—
132D191—44no—
133C191#240411yescharacteristic, complemented, Sylow
134#2404—1911nomaximal, Sylow
135C22×#41D19111yescharacteristic, complemented
136C2×#120—1912no—
137C22×C22⋊C4—1911no—
138#393—1912no—
139#392—1912no—
140#386—1912no—
141#386—1914no—
142#381—1911no—
143C25—1911no—
144C2×#41—28no—
145C2×#41—24no—
146C22×Q8C2×D19111yescharacteristic, complemented
147C22×D4—1914no—
148C22×D4C2×D19112yes—
149C22×D4C2×D19111yescharacteristic, complemented
150C23×C4—1911no—
151C23×C4C2×D19111yescharacteristic
152C23×C4C2×D19112yescomplemented
153#120—3828no—
154C2×C22⋊C4—3824no—
155C2×C22⋊C4—1912no—
156C2×C22⋊C4—1912no—
157C2×C22⋊C4—1912no—
158C2×C22⋊C4—1914no—
159C2×C22⋊C4—1914no—
160#95—1912no—
161#95—1912no—
162#95—1914no—
163C24—3824no—
164C24—3822no—
165C24—3824no—
166C24—1912no—
167C24—7642no—
168C24C22×D19112yes—
169C24C22×D19111yescharacteristic
170#41—232no—
171C2×Q8—24no—
172C2×Q8—22no—
173C2×D4—24no—
174C2×D4—24no—
175C2×D4—24no—
176C2×D4—22no—
177C2×D4—3828no—
178C2×D4—3828no—
179C2×D4—7644no—
180C2×D4C22⋊D19118yescomplemented
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—3822no—
186C22×C4—1914no—
187C22×C4—1912no—
188C22×C4—7642no—
189C22×C4C22⋊D19114yescomplemented
190C22×C4C22×D19111yescharacteristic
191C22×C4C22×D19111yescharacteristic
192C22×C4C22×D19112yes—
193C22⋊C4—3824no—
194C22⋊C4—3824no—
195C22⋊C4—3824no—
196C22⋊C4—3828no—
197C22⋊C4—7644no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—24no—
203C23—24no—
204C23—3828no—
205C23—3822no—
206C23—3824no—
207C23—3822no—
208C23—3824no—
209C23—1914no—
210C23—76416no—
211C23—7644no—
212C23—7644no—
213C23C2×C22⋊D19114yes—
214C23C2×C22⋊D19114yes—
215C23C2×C22⋊D19112yes—
216C23C23×D19111yescharacteristic
217Q8—28no—
218D4—28no—
219D4—216no—
220D4—76416no—
221C2×C4—216no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—28no—
227C2×C4—3822no—
228C2×C4—3824no—
229C2×C4—3824no—
230C2×C4—3824no—
231C2×C4—3824no—
232C2×C4—7644no—
233C2×C4C2×C22⋊D19112yes—
234C2×C4C2×C22⋊D19114yes—
235C2×C4C2×C22⋊D19112yes—
236C22—22no—
237C22—28no—
238C22—22no—
239C22—28no—
240C22—24no—
241C22—42no—
242C22—44no—
243C22—42no—
244C22—3828no—
245C22—3824no—
246C22—76416no—
247C22—7642no—
248C22—7644no—
249C22—7642no—
250C22#2025929114yes—
251C22C22×C22⋊D19111yescharacteristic
252C22C22×C22⋊D19112yes—
253C22#41×D19112yes—
254C22#2025936812yes—
255C4—22no—
256C4—24no—
257C4—22no—
258C4—7642no—
259C4—7644no—
260C2—24no—
261C2—44no—
262C2—7644no—
263C2C2×#2025929111yescharacteristic
264C2#2713317011yescharacteristic
265C2#2713322711yescharacteristic
266C2#2713330114yes—
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
FittingC22×#41×C191C2
hypercenterC22×#41D191
hypocenterC191#2404
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#20259290C2semidirect
2. C22×(C2×D191)⋊C4C2semidirect
3. #27132574C2semidirect
4. #27132571C2semidirect
5. #27132543C2semidirect
6. #27132542C2semidirect
7. #27132526C2semidirect
8. C22×#41×C191C2semidirect
14. C2×C22⋊Dic191C22semidirect
19. C191⋊#95C22semidirect
20. C191⋊#95C22semidirect
21. C191⋊#95C22semidirect
24. C22×Q8×C191C22semidirect
26. C22×D4×C191C22semidirect
28. C23×C4×C191C22semidirect
55. C2×D4×C191D4semidirect
60. C22×C4×C191D4semidirect
72. C22×D191#41semidirect
133. C191#2404semidirect
135. C22×#41D191semidirect
146. C22×Q8C2×D191semidirect
149. C22×D4C2×D191semidirect
152. C23×C4C2×D191semidirect
180. C2×D4C22⋊D191semidirect
189. C22×C4C22⋊D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202592902
C22×(C2×D191)⋊C42
#271325742
#271325712
#271325432
#271325422
#271325262
C22×#41×C1912
#2404191