Finite Group Databasebeta

← #37575839

Subgroups — #37575839

Order 24448.

1056 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C22×#15690287C211yesmaximal, characteristic, complemented
2C2×#20259291C212yesmaximal, complemented
3#27132575C212yesmaximal, complemented
4#27132572C212yesmaximal, complemented
5#27132542C212yesmaximal, complemented
6#27132539C214yesmaximal, complemented
7#27132526C211yesmaximal, characteristic, complemented
8C22×C22⋊C4×C191C211yesmaximal, characteristic, complemented
9C2×#15690287—28no—
10C2×#15690287—24no—
11C22×Dic382C2211yescharacteristic, complemented
12C22×D764C2211yescharacteristic, complemented
13C22×C22⋊D191C2214yes—
14C22×C22⋊D191C2212yes—
15C22×C4×D191C2212yescomplemented
16#20259291—28no—
17C2×C22⋊Dic191C2212yes—
18C2×C22⋊Dic191C2214yes—
19C2×(C2×D191)⋊C4C2214yescomplemented
20C191⋊#95C2212yescomplemented
21C191⋊#95C2212yescomplemented
22C191⋊#95C2214yescomplemented
23C25×C191C2211yescharacteristic
24C23×C4×C191C2211yescharacteristic
25C23×C4×C191C2211yescharacteristic
26C2×C22⋊C4×C191—24no—
27C2×C22⋊C4×C191C2212yes—
28C2×C22⋊C4×C191C2212yes—
29C23×D191C2312yes—
30#15690287—232no—
31C2×Dic382—24no—
32C2×Dic382—22no—
33C2×D764—24no—
34C2×D764—22no—
35C2×C22⋊D191—28no—
36C2×C22⋊D191—24no—
37C2×C22⋊D191—24no—
38C2×C22⋊D191—28no—
39C2×C22⋊D191—44no—
40C2×C22⋊D191D418yescomplemented
41C22×Dic191C2314yes—
42C22×Dic191C2312yes—
43C2×C4×D191—24no—
44C2×C4×D191—24no—
45C2×C4×D191—24no—
46C22⋊Dic191—28no—
47C22⋊Dic191—24no—
48(C2×D191)⋊C4—44no—
49C24×C191—22no—
50C24×C191—24no—
51C24×C191—24no—
52C24×C191—42no—
53C24×C191C2312yes—
54C24×C191C2311yescharacteristic
55C22×C4×C191—24no—
56C22×C4×C191—22no—
57C22×C4×C191—42no—
58C22×C4×C191D414yescomplemented
59C22×C4×C191C2312yes—
60C22×C4×C191C2311yescharacteristic
61C22×C4×C191C2311yescharacteristic
62C22⋊C4×C191—24no—
63C22⋊C4×C191—24no—
64C22×D191—22no—
65C22×D191—22no—
66C22×D191—24no—
67C22×D191—24no—
68C22×D191C2×D414yes—
69Dic382—28no—
70D764—28no—
71C22⋊D191—216no—
72C22⋊D191—416no—
73C2×Dic191—22no—
74C2×Dic191—22no—
75C2×Dic191—24no—
76C2×Dic191—24no—
77C2×Dic191—24no—
78C2×Dic191C2×D414yes—
79C4×D191—216no—
80C23×C191—22no—
81C23×C191—22no—
82C23×C191—28no—
83C23×C191—22no—
84C23×C191—24no—
85C23×C191—22no—
86C23×C191—24no—
87C23×C191—416no—
88C23×C191—44no—
89C23×C191—44no—
90C23×C191C2×D414yes—
91C23×C191C2×D412yes—
92C23×C191#4114yescomplemented
93C23×C191C2411yescharacteristic
94C2×C4×C191—28no—
95C2×C4×C191—22no—
96C2×C4×C191—22no—
97C2×C4×C191—22no—
98C2×C4×C191—24no—
99C2×C4×C191—44no—
100C2×C4×C191C2×D412yes—
101C2×C4×C191C2×D412yes—
102C22×C191—22no—
103C22×C191—22no—
104C22×C191—28no—
105C22×C191—24no—
106C22×C191—28no—
107C22×C191—416no—
108C22×C191—42no—
109C22×C191—42no—
110C22×C191—44no—
111C22×C191#12014yes—
112C22×C191C22×D412yes—
113C22×C191C22×D411yescharacteristic
114C22×C191C2×#4112yes—
115C22×C191H(5,𝔽2)12yes—
116C2×D191—24no—
117C2×D191—28no—
118C2×D191—42no—
119C2×D191—42no—
120C2×D191—44no—
121C4×C191—22no—
122C4×C191—22no—
123C4×C191—42no—
124Dic191—24no—
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×#41—1911no—
136C2×#120—1912no—
137C22×C22⋊C4D19111yescharacteristic, complemented
138#393—1912no—
139#392—1912no—
140#386—1912no—
141#386—1914no—
142#381—1911no—
143C25C2×D19111yescharacteristic
144C2×#41—3828no—
145C2×#41—3824no—
146C22×Q8—1911no—
147C22×D4—1912no—
148C22×D4—1914no—
149C22×D4—1911no—
150C23×C4—1912no—
151C23×C4C2×D19111yescharacteristic
152C23×C4C2×D19111yescharacteristic
153#120—3828no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—1912no—
156C2×C22⋊C4—1914no—
157C2×C22⋊C4—1914no—
158C2×C22⋊C4C2×D19112yes—
159C2×C22⋊C4C2×D19112yes—
160#95—1912no—
161#95—1912no—
162#95—1914no—
163C24—22no—
164C24—24no—
165C24—24no—
166C24—42no—
167C24—1912no—
168C24C22×D19112yes—
169C24C22×D19111yescharacteristic
170#41—38232no—
171C2×Q8—3824no—
172C2×Q8—3822no—
173C2×D4—3824no—
174C2×D4—3824no—
175C2×D4—3824no—
176C2×D4—3828no—
177C2×D4—3828no—
178C2×D4—3822no—
179C2×D4—1918no—
180C2×D4—7644no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—42no—
184C22×C4—3824no—
185C22×C4—3824no—
186C22×C4—3824no—
187C22×C4—1914no—
188C22×C4—1912no—
189C22×C4C22⋊D19114yescomplemented
190C22×C4C22×D19112yes—
191C22×C4C22×D19111yescharacteristic
192C22×C4C22×D19111yescharacteristic
193C22⋊C4—24no—
194C22⋊C4—24no—
195C22⋊C4—3824no—
196C22⋊C4—3828no—
197C22⋊C4—7644no—
198C23—22no—
199C23—22no—
200C23—28no—
201C23—24no—
202C23—22no—
203C23—22no—
204C23—24no—
205C23—416no—
206C23—44no—
207C23—44no—
208C23—3822no—
209C23—3822no—
210C23—3824no—
211C23—3824no—
212C23—1914no—
213C23D4×D19114yes—
214C23C2×C22⋊D19112yes—
215C23#1569028714yescomplemented
216C23C23×D19111yescharacteristic
217Q8—3828no—
218D4—3828no—
219D4—38216no—
220D4—76416no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—28no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4—44no—
227C2×C4—38216no—
228C2×C4—3822no—
229C2×C4—3822no—
230C2×C4—3824no—
231C2×C4—3824no—
232C2×C4—3824no—
233C2×C4—1914no—
234C2×C4C2×C22⋊D19112yes—
235C2×C4C2×C22⋊D19112yes—
236C22—22no—
237C22—28no—
238C22—22no—
239C22—28no—
240C22—24no—
241C22—416no—
242C22—42no—
243C22—44no—
244C22—42no—
245C22—3828no—
246C22—3824no—
247C22—7642no—
248C22—7644no—
249C22—7642no—
250C22#2025929014yes—
251C22C2×D4×D19112yes—
252C22C22×C22⋊D19111yescharacteristic
253C22C2×#1569028712yes—
254C22#2025936712yes—
255C4—22no—
256C4—22no—
257C4—42no—
258C4—3824no—
259C4—7644no—
260C2—24no—
261C2—44no—
262C2—7644no—
263C2C2×#2025929011yescharacteristic
264C2#2713317411yescharacteristic
265C2#2713323011yescharacteristic
266C2#2713330314yes—
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
FittingC22×C22⋊C4×C191C2
hypercenterC22×C22⋊C4D191
hypocenterC191#2404
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#15690287C2semidirect
2. C2×#20259291C2semidirect
3. #27132575C2semidirect
4. #27132572C2semidirect
5. #27132542C2semidirect
6. #27132539C2semidirect
7. #27132526C2semidirect
8. C22×C22⋊C4×C191C2semidirect
11. C22×Dic382C22semidirect
12. C22×D764C22semidirect
15. C22×C4×D191C22semidirect
19. C2×(C2×D191)⋊C4C22semidirect
20. C191⋊#95C22semidirect
21. C191⋊#95C22semidirect
22. C191⋊#95C22semidirect
40. C2×C22⋊D191D4semidirect
58. C22×C4×C191D4semidirect
92. C23×C191#41semidirect
133. C191#2404semidirect
137. C22×C22⋊C4D191semidirect
189. C22×C4C22⋊D191semidirect
215. C23#15690287semidirect

Maximal subgroups

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

subgroupindex
C22×#156902872
C2×#202592912
#271325752
#271325722
#271325422
#271325392
#271325262
C22×C22⋊C4×C1912
#2404191