Finite Group Databasebeta

← #37545949

Subgroups — #37545949

Order 23168.

448 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 117 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#26042003C211yesmaximal, characteristic, complemented
2D4×D724C211yesmaximal, characteristic
3#26041627C211yesmaximal, characteristic, complemented
4#26041606C212yesmaximal, complemented
5#26041526C212yesmaximal, complemented
6#26041480C211yesmaximal, characteristic, complemented
7#26041431C212yesmaximal, complemented
8C4×#15668437C211yesmaximal, characteristic, complemented
9#26041356C211yesmaximal, characteristic, complemented
10#26041282C212yesmaximal, complemented
11#474×C181C211yesmaximal, characteristic, complemented
12C22×D724—21no—
13C2×D4×D181C2212yes—
14C2×#15668438C2212yescomplemented
15C2×#15668437C2211yescharacteristic
16C2×#15668432C2212yescomplemented
17#19153813C2211yescharacteristic
18#19153811—22no—
19#19153805—21no—
20#19153796C2212yescomplemented
21#19153791C2212yes—
22#19153790—21no—
23#19153786—22no—
24Q8×Dic181—21no—
25C4×Dic362C2211yescharacteristic, complemented
26C4×D724C2211yescharacteristic
27#19153774C2212yescomplemented
28#19153747C2211yescharacteristic, complemented
29#19153739C2211yescharacteristic, complemented
30#19153738C2212yescomplemented
31#19153737C2211yescharacteristic
32#19153736C2212yescomplemented
33#19153736C2212yescomplemented
34#19153728C2212yescomplemented
35C4×#13089678C2211yescharacteristic, complemented
36C4⋊Q8×C181C2211yescharacteristic, complemented
37C4×D4×C181C2211yescharacteristic
38#107×C181C2212yescomplemented
39C4⋊C8×C181C2211yescharacteristic, complemented
40#102×C181C2212yes—
41C23×D181—22no—
42C2×Dic362C2311yescharacteristic
43C2×D724—22no—
44C2×D724—21no—
45C2×D724—21no—
46C2×D724—44no—
47C2×D724D411yescharacteristic
48C2×D724C2312yes—
49C2×C22⋊D181—22no—
50D4×D181—24no—
51D4×D181—42no—
52C2×C4×D181C2312yes—
53#15668438—42no—
54#15668437—22no—
55#15668432—42no—
56C2×#13089678C2311yescharacteristic
57C2×#13089678C2311yescharacteristic
58C4⋊Dic181—22no—
59C4⋊Dic181D411yescharacteristic
60C4⋊Dic181C2311yescharacteristic
61#15668428—21no—
62#15668426—22no—
63#15668426—22no—
64#15668426—22no—
65#15668426D412yes—
66C4×Dic181—22no—
67C4×Dic181—21no—
68C2×Q8×C181—21no—
69C2×D4×C181C2311yescharacteristic
70C22×C4×C181—21no—
71C2×C8×C181C2312yes—
72C4⋊C4×C181—21no—
73C4⋊C4×C181C2312yes—
74C4⋊C4×C181C2311yescharacteristic
75C22⋊C4×C181—21no—
76C42×C181C2311yescharacteristic
77C22×D181—22no—
78C22×D181—44no—
79C22×D181—44no—
80C22×D181—82no—
81C22×D181C2×D412yes—
82Dic362—21no—
83Dic362—21no—
84D724—22no—
85D724—22no—
86D724—22no—
87D724—44no—
88D724—42no—
89D724—81no—
90C22⋊D181—44no—
91C2×Dic181—21no—
92C2×Dic181—21no—
93C2×Dic181C2×D412yes—
94C4×D181—22no—
95C4×D181—22no—
96#13089678—21no—
97#13089678—21no—
98C23×C181—21no—
99Q8×C181—42no—
100D4×C181—21no—
101D4×C181#4112yes—
102C2×C4×C181—21no—
103C2×C4×C181—22no—
104C2×C4×C181—42no—
105C2×C4×C181C2×D411yescharacteristic
106C2×C4×C181C2×D411yescharacteristic
107C2×C4×C181C2411yescharacteristic
108C8×C181—22no—
109C22×C181—22no—
110C22×C181—41no—
111C22×C181—41no—
112C22×C181C22×D411yescharacteristic
113C2×D181—22no—
114C2×D181—22no—
115C2×D181—22no—
116C2×D181—42no—
117C2×D181—42no—
118C2×D181—42no—
119C2×D181—88no—
120C4×C181—21no—
121C4×C181—41no—
122C4×C181—42no—
123C4×C181AGL(1,ℤ/8)12yes—
124C4×C181C2×#4111yescharacteristic
125C4×C181H(5,𝔽2)11yescharacteristic
126Dic181—22no—
127Dic181—41no—
128Dic181—41no—
129C2×C181—42no—
130C2×C181#54611yescharacteristic
131C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
132C2×C181#57711yescharacteristic
133D181—42no—
134D181—82no—
135C181#332411yescharacteristic, complemented, Sylow
136#3324—1811nomaximal, Sylow
137#556—1811no—
138D42—1811no—
139#486—1811no—
140#482—1812no—
141#474D18111yescharacteristic, complemented
142#468—1812no—
143#460—1811no—
144#450—1812no—
145C4×SD16—1811no—
146#436—1811no—
147#418—1812no—
148C22×D4—3621no—
149C22×D4—1812no—
150C2×SD16—1811no—
151C2×SD16—1812no—
152C2×M16—1812no—
153C4⋊Q8C2×D18111yescharacteristic, complemented
154#127—1811no—
155#126—3622no—
156#124—3621no—
157C22⋊Q8—1812no—
158#121—3621no—
159#121—1812no—
160#120—3622no—
161C4×Q8—3621no—
162C4×Q8—1811no—
163C4×D4—1811no—
164C4×D4C2×D18111yescharacteristic
165#117—1812no—
166#107C2×D18112yescomplemented
167#106—1811no—
168C4⋊C8C2×D18111yescharacteristic, complemented
169#103—1811no—
170#102—1811no—
171#102—1812no—
172#102—1812no—
173#102—1812no—
174#102C2×D18112yes—
175C22⋊C8—1812no—
176C4×C8—1811no—
177C24—3622no—
178C2×Q8—21no—
179C2×Q8—1811no—
180C2×D4—3622no—
181C2×D4—3624no—
182C2×D4—3622no—
183C2×D4—3621no—
184C2×D4—3621no—
185C2×D4—1811no—
186C2×D4—1812no—
187C2×D4—7244no—
188C2×D4—7242no—
189C2×D4C22×D18111yescharacteristic
190C22×C4—21no—
191C22×C4—1812no—
192SD16—3622no—
193SD16—7242no—
194M16—7242no—
195C2×C8—1811no—
196C2×C8—1811no—
197C2×C8C22×D18112yes—
198C4⋊C4—21no—
199C4⋊C4—3622no—
200C4⋊C4—3621no—
201C4⋊C4—1811no—
202C4⋊C4—1811no—
203C4⋊C4C22×D18112yes—
204C4⋊C4C22×D18111yescharacteristic
205C22⋊C4—21no—
206C22⋊C4—3622no—
207C22⋊C4—3622no—
208C22⋊C4—3622no—
209C22⋊C4—1812no—
210C42—3622no—
211C42—3621no—
212C42C22×D18111yescharacteristic
213C23—21no—
214C23—3622no—
215C23—14482no—
216C23—1812no—
217C23—7244no—
218C23—7244no—
219Q8—42no—
220Q8—3621no—
221Q8—3621no—
222D4—21no—
223D4—3622no—
224D4—3622no—
225D4—3622no—
226D4—14481no—
227D4—7244no—
228D4—7244no—
229D4—7242no—
230D4#1566845012yes—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—42no—
234C2×C4—3622no—
235C2×C4—3622no—
236C2×C4—3621no—
237C2×C4—3621no—
238C2×C4—1812no—
239C2×C4D4×D18111yescharacteristic
240C2×C4D4×D18111yescharacteristic
241C2×C4C23×D18111yescharacteristic
242C8—22no—
243C8—3621no—
244C8—3621no—
245C22—22no—
246C22—41no—
247C22—41no—
248C22—3622no—
249C22—3622no—
250C22—3622no—
251C22—14488no—
252C22—7242no—
253C22—7242no—
254C22—7242no—
255C22C2×D4×D18111yescharacteristic
256C4—21no—
257C4—41no—
258C4—42no—
259C4—3622no—
260C4—7241no—
261C4—7241no—
262C4#1915384012yes—
263C4C2×#1566845011yescharacteristic
264C4#1915386211yescharacteristic
265C2—42no—
266C2—14482no—
267C2—7242no—
268C2#2604192011yescharacteristic
269C2C2×#1915384011yescharacteristic
270C2#2604208811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#474×C181C2
socleC22×C181C22×D4
prime radicalD4×D724C2
hypercenter#474D181
hypocenterC181#3324
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #26042003C2semidirect
3. #26041627C2semidirect
4. #26041606C2semidirect
5. #26041526C2semidirect
6. #26041480C2semidirect
7. #26041431C2semidirect
8. C4×#15668437C2semidirect
9. #26041356C2semidirect
10. #26041282C2semidirect
11. #474×C181C2semidirect
14. C2×#15668438C22semidirect
16. C2×#15668432C22semidirect
20. #19153796C22semidirect
25. C4×Dic362C22semidirect
27. #19153774C22semidirect
28. #19153747C22semidirect
29. #19153739C22semidirect
30. #19153738C22semidirect
32. #19153736C22semidirect
33. #19153736C22semidirect
34. #19153728C22semidirect
35. C4×#13089678C22semidirect
36. C4⋊Q8×C181C22semidirect
38. #107×C181C22semidirect
39. C4⋊C8×C181C22semidirect
135. C181#3324semidirect
141. #474D181semidirect
153. C4⋊Q8C2×D181semidirect
166. #107C2×D181semidirect
168. C4⋊C8C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260420032
D4×D7242
#260416272
#260416062
#260415262
#260414802
#260414312
C4×#156684372
#260413562
#260412822
#474×C1812
#3324181