Finite Group Databasebeta

← #37543124

Subgroups — #37543124

Order 23168.

772 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 145 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#26042115C211yesmaximal, characteristic, complemented
2#26041848C212yesmaximal, complemented
3#26041843C211yesmaximal, characteristic, complemented
4#26041467C218yesmaximal, complemented
5#26041252C212yesmaximal, complemented
6#410×C181C211yesmaximal, characteristic, complemented
7#19153864—21no—
8#19153863—22no—
9#19153862—22no—
10#19153861—21no—
11#19153861C2214yes—
12C2×#15668449—22no—
13C2×#15668449C2212yes—
14#41×D181—24no—
15C22×D724C2211yescharacteristic, complemented
16#19153813—22no—
17#19153812—21no—
18#19153805—21no—
19#19153804—22no—
20#19153791—24no—
21#19153790—22no—
22#19153786C2218yescomplemented
23#19153784C2214yescomplemented
24#19153775C2211yescharacteristic, complemented
25#19153730C2218yescomplemented
26C2×#41×C181C2211yescharacteristic
27#117×C181C2212yescomplemented
28#99×C181C2214yescomplemented
29C23×D181C2312yes—
30#15668450—24no—
31#15668449—24no—
32#15668449—24no—
33#15668449—22no—
34#15668449—22no—
35#15668449—24no—
36#15668449—22no—
37#15668449—22no—
38#15668448—24no—
39#15668448—28no—
40C2×Dic362—22no—
41C2×Dic362D412yescomplemented
42Q8×D181—24no—
43C2×D724—22no—
44C2×D724—28no—
45C2×D724—21no—
46C2×D724—22no—
47C2×D724—22no—
48C2×D724—21no—
49C2×D724D412yescomplemented
50C2×C22⋊D181—24no—
51C2×C22⋊D181—24no—
52C2×C22⋊D181C2314yes—
53D4×D181—24no—
54D4×D181—28no—
55C2×C4×D181—24no—
56C2×C4×D181D414yescomplemented
57C4⋊Dic181—21no—
58C22⋊Dic181C2312yes—
59#15668426—28no—
60#15668426—24no—
61C4×Dic181—21no—
62#41×C181—22no—
63#41×C181—22no—
64C2×Q8×C181D411yescharacteristic, complemented
65C2×D4×C181D411yescharacteristic, complemented
66C2×D4×C181C2312yes—
67C22×C4×C181D412yescomplemented
68C22×C4×C181C2311yescharacteristic
69C4⋊C4×C181—22no—
70C22⋊C4×C181C2314yes—
71C42×C181—22no—
72C22×D181—24no—
73C22×D181—22no—
74C22×D181—24no—
75C22×D181—82no—
76C22×D181C2×D414yes—
77Dic362—24no—
78Dic362—22no—
79Dic362—22no—
80D724—24no—
81D724—22no—
82D724—22no—
83D724—41no—
84D724—41no—
85D724—42no—
86D724—88no—
87C22⋊D181—28no—
88C22⋊D181—24no—
89C22⋊D181—24no—
90C22⋊D181—84no—
91C2×Dic181—24no—
92C2×Dic181—22no—
93C2×Dic181C2×D414yes—
94C4×D181—28no—
95C4×D181—24no—
96C4×D181—24no—
97C23×C181C2×D412yes—
98C23×C181C2411yescharacteristic
99Q8×C181—22no—
100D4×C181—24no—
101D4×C181—22no—
102C2×C4×C181—22no—
103C2×C4×C181—22no—
104C2×C4×C181—24no—
105C2×C4×C181C2×D412yes—
106C2×C4×C181C2×D412yes—
107C2×C4×C181C2×D412yes—
108C2×C4×C181C2×D411yescharacteristic
109C2×C4×C181C2×D411yescharacteristic
110C22×C181—22no—
111C22×C181—41no—
112C22×C181—42no—
113C22×C181C22×D412yes—
114C22×C181C22×D411yescharacteristic
115C2×D181—24no—
116C2×D181—44no—
117C2×D181—44no—
118C2×D181—48no—
119C2×D181—42no—
120C2×D181—84no—
121C4×C181—22no—
122C4×C181—44no—
123C4×C181#12011yescharacteristic
124C4×C181#12011yescharacteristic
125C4×C181#12012yes—
126Dic181—24no—
127Dic181—42no—
128C2×C181—21no—
129C2×C181—22no—
130C2×C181—42no—
131C2×C181C2×#12011yescharacteristic
132D181—44no—
133D181—82no—
134C181#302711yescharacteristic, complemented, Sylow
135#3027—1811nomaximal, Sylow
136#585—1811no—
137#535—1811no—
138#535—1812no—
139B(4,𝔽2)—1818no—
140#410—1812no—
141#410D18111yescharacteristic, complemented
1422−1+4—3622no—
1432−1+4—3621no—
144H(5,𝔽2)—3622no—
145H(5,𝔽2)—3621no—
146H(5,𝔽2)—1814no—
147C2×#41—3624no—
148C2×#41—3622no—
149C2×#41—1812no—
150C2×#41C2×D18111yescharacteristic
151C22×D4—1811no—
152#127—3622no—
153#127—3621no—
154#124—3622no—
155#124—3621no—
156#121—3624no—
157#121—3622no—
158#120—1814no—
159#120—1818no—
160#117—1811no—
161#117C2×D18112yescomplemented
162#99—1818no—
163#99C2×D18114yescomplemented
164C24—1812no—
165#41—22no—
166#41—22no—
167#41—3624no—
168#41—3624no—
169#41—3624no—
170#41—3624no—
171#41—3628no—
172#41—3622no—
173#41—3622no—
174#41—3622no—
175#41—3624no—
176#41—3622no—
177C2×Q8—3622no—
178C2×Q8—3624no—
179C2×Q8—1812no—
180C2×Q8D72411yescharacteristic, complemented
181C2×D4—3622no—
182C2×D4—3628no—
183C2×D4—3624no—
184C2×D4—3628no—
185C2×D4—3624no—
186C2×D4—3624no—
187C2×D4—3622no—
188C2×D4—3622no—
189C2×D4—3621no—
190C2×D4—3621no—
191C2×D4—1814no—
192C2×D4—1812no—
193C2×D4D72411yescharacteristic, complemented
194C2×D4C22×D18112yes—
195C22×C4—3624no—
196C22×C4—1814no—
197C22×C4D72412yescomplemented
198C22×C4C22×D18111yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—3621no—
201C22⋊C4—3628no—
202C22⋊C4—3624no—
203C22⋊C4—1812no—
204C22⋊C4C22×D18114yes—
205C42—22no—
206C42—3621no—
207C23—3624no—
208C23—3624no—
209C23—3622no—
210C23—14482no—
211C23—1814no—
212C23C2×D72412yes—
213C23C23×D18111yescharacteristic
214Q8—22no—
215Q8—3624no—
216Q8—3622no—
217Q8—3622no—
218D4—24no—
219D4—22no—
220D4—3624no—
221D4—3628no—
222D4—3624no—
223D4—3624no—
224D4—3622no—
225D4—3622no—
226D4—14488no—
227D4—14484no—
228D4—7241no—
229D4—7241no—
230D4—7242no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—3628no—
235C2×C4—3624no—
236C2×C4—3624no—
237C2×C4—3624no—
238C2×C4—3622no—
239C2×C4—1814no—
240C2×C4D4×D18112yes—
241C2×C4D4×D18112yes—
242C2×C4C2×D72412yes—
243C2×C4C2×D72411yescharacteristic
244C2×C4C2×D72411yescharacteristic
245C22—22no—
246C22—41no—
247C22—42no—
248C22—3624no—
249C22—14484no—
250C22—7248no—
251C22—7244no—
252C22—7244no—
253C22—7242no—
254C22C2×D4×D18112yes—
255C22C22×D72411yescharacteristic
256C4—22no—
257C4—44no—
258C4—3624no—
259C4—7242no—
260C4#1915378611yescharacteristic
261C4#1915378611yescharacteristic
262C4#1915378612yes—
263C2—22no—
264C2—21no—
265C2—42no—
266C2—14482no—
267C2—7244no—
268C2C2×#1915378611yescharacteristic
subgroupG/H
centerC2C2×#19153786
commutator, cosocleC23×C181C24
FrattiniC23C23×D181
Fitting#410×C181C2
socleC2×C181C2×#120
hypercenter#410D181
hypocenterC181#3027
LCSG ▷ C23×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #26042115C2semidirect
2. #26041848C2semidirect
3. #26041843C2semidirect
4. #26041467C2semidirect
5. #26041252C2semidirect
6. #410×C181C2semidirect
15. C22×D724C22semidirect
22. #19153786C22semidirect
23. #19153784C22semidirect
24. #19153775C22semidirect
25. #19153730C22semidirect
27. #117×C181C22semidirect
28. #99×C181C22semidirect
41. C2×Dic362D4semidirect
49. C2×D724D4semidirect
56. C2×C4×D181D4semidirect
64. C2×Q8×C181D4semidirect
65. C2×D4×C181D4semidirect
67. C22×C4×C181D4semidirect
134. C181#3027semidirect
141. #410D181semidirect
161. #117C2×D181semidirect
163. #99C2×D181semidirect
180. C2×Q8D724semidirect
193. C2×D4D724semidirect
197. C22×C4D724semidirect

Maximal subgroups

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

subgroupindex
#260421152
#260418482
#260418432
#260414672
#260412522
#410×C1812
#3027181