Finite Group Databasebeta

← #37545403

Subgroups — #37545403

Order 23168.

526 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 125 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×#19153839C211yesmaximal, characteristic, complemented
2C22×D1448C211yesmaximal, characteristic, complemented
3#26041920C212yesmaximal, complemented
4#26041664C211yesmaximal, characteristic, complemented
5#26041516C211yesmaximal, characteristic, complemented
6#26041497C212yesmaximal, complemented
7#26041429C212yesmaximal, complemented
8#26041427C212yesmaximal, complemented
9#26041361C211yesmaximal, characteristic, complemented
10C8×C22⋊D181C211yesmaximal, characteristic, complemented
11#467×C181C211yesmaximal, characteristic, complemented
12C2×#15668450C2212yescomplemented
13C22×D724C2212yes—
14#19153839—24no—
15C2×#15668436C2212yescomplemented
16C2×#15668436C2212yescomplemented
17C2×D1448—24no—
18C2×D1448C2211yescharacteristic
19C2×D1448C2212yes—
20C2×D1448C2211yescharacteristic
21C2×C8×D181C2211yescharacteristic, complemented
22#19153805C2212yescomplemented
23#19153801—22no—
24#19153791—22no—
25#19153791C2212yescomplemented
26#19153786—22no—
27#19153778C2212yescomplemented
28C4×C22⋊D181C2211yescharacteristic
29C22⋊C4×D181—22no—
30#19153745C2211yescharacteristic, complemented
31#19153736C2212yescomplemented
32#19153736C2212yescomplemented
33C22⋊#13089678C2211yescharacteristic
34#19153728C2211yescharacteristic, complemented
35C8×Dic181C2211yescharacteristic, complemented
36C2×Q16×C181C2211yescharacteristic, complemented
37C22×C8×C181C2211yescharacteristic
38C22⋊Q8×C181C2212yes—
39#107×C181C2211yescharacteristic, complemented
40#103×C181C2212yescomplemented
41C23×D181—22no—
42#15668450—24no—
43#15668450—42no—
44C2×D724—24no—
45C2×D724—22no—
46C2×D724—22no—
47C2×D724—22no—
48C2×D724—42no—
49C2×D724C2312yes—
50C2×D724C2312yes—
51C2×C22⋊D181D411yescharacteristic
52C2×C4×D181—22no—
53C2×C4×D181—22no—
54C2×C4×D181C2311yescharacteristic
55#15668436—24no—
56#15668436—42no—
57D1448—24no—
58D1448—22no—
59D1448D414yescomplemented
60C2×#13089678C2311yescharacteristic
61C2×#13089678C2311yescharacteristic
62C8×D181—22no—
63#15668428D411yescharacteristic
64C22⋊Dic181D411yescharacteristic
65#15668426—22no—
66#15668426—22no—
67#15668426—22no—
68#15668426—22no—
69#15668426D411yescharacteristic
70C4×Dic181C2311yescharacteristic
71C2×Q8×C181C2312yes—
72C22×C4×C181C2311yescharacteristic
73Q16×C181—22no—
74C2×C8×C181—22no—
75C2×C8×C181C2311yescharacteristic
76C2×C8×C181C2311yescharacteristic
77C4⋊C4×C181—22no—
78C4⋊C4×C181C2312yes—
79C22⋊C4×C181—22no—
80C22×D181—22no—
81C22×D181—22no—
82C22×D181—44no—
83C22×D181—44no—
84C22×D181—82no—
85C22×D181C2×D411yescharacteristic
86D724—22no—
87D724—22no—
88D724—22no—
89D724—44no—
90D724—44no—
91D724—82no—
92D724C2×D414yes—
93C22⋊D181—22no—
94C2×Dic181C2×D411yescharacteristic
95C2×Dic181C2×D411yescharacteristic
96C2×Dic181C2×D411yescharacteristic
97C4×D181—21no—
98C4×D181—21no—
99C4×D181—44no—
100C4×D181—82no—
101#13089678—21no—
102#13089678—21no—
103C23×C181C2×D411yescharacteristic
104Q8×C181—22no—
105Q8×C181—22no—
106C2×C4×C181—21no—
107C2×C4×C181—21no—
108C2×C4×C181—22no—
109C2×C4×C181—22no—
110C2×C4×C181C2×D411yescharacteristic
111C2×C4×C181C2411yescharacteristic
112C8×C181—21no—
113C8×C181C2×D412yes—
114C22×C181—21no—
115C22×C181—21no—
116C22×C181#13512yescomplemented
117C22×C181C22×D411yescharacteristic
118C2×D181—21no—
119C2×D181—24no—
120C2×D181—21no—
121C2×D181—21no—
122C2×D181—42no—
123C2×D181—42no—
124C2×D181—42no—
125C2×D181—42no—
126C2×D181—42no—
127C2×D181—88no—
128C4×C181—21no—
129C4×C181—42no—
130C4×C181—42no—
131C4×C181C22×D411yescharacteristic
132C4×C181H(5,𝔽2)11yescharacteristic
133Dic181—21no—
134Dic181—21no—
135Dic181—21no—
136C2×C181—22no—
137C2×C181D4211yescharacteristic
138C2×C181C2×#13511yescharacteristic
139C2×C181#57611yescharacteristic
140D181—41no—
141D181—44no—
142D181—82no—
143C181#328011yescharacteristic, complemented, Sylow
144#3280—1811nomaximal, Sylow
145C2×#135—1811no—
146C22×D8—1811no—
147#546—1812no—
148#495—1811no—
149#467D18111yescharacteristic, complemented
150#466—1811no—
151#463—1812no—
152#450—1812no—
153#449—1812no—
154C4×D8—1811no—
155C8×D4—1811no—
156C2×#41—1812no—
157C22×D4—1812no—
158#135—3624no—
159C2×Q16C2×D18111yescharacteristic, complemented
160C2×SD16—1812no—
161C2×SD16—1812no—
162C2×D8—3624no—
163C2×D8—1812no—
164C2×D8—1811no—
165C2×D8—1811no—
166C22×C8—1811no—
167C22×C8C2×D18111yescharacteristic
168#124—1812no—
169#123—3622no—
170C22⋊Q8C2×D18112yes—
171#121—3622no—
172#121—1812no—
173#120—3622no—
174C4×D4—1812no—
175C4×D4—1811no—
176C2×C22⋊C4—3622no—
177#107C2×D18111yescharacteristic, complemented
178C4⋊C8—1811no—
179#103C2×D18112yescomplemented
180#102—1812no—
181#102—1812no—
182C22⋊C8—1811no—
183C22⋊C8—1811no—
184C4×C8—1811no—
185C24—3622no—
186#41—3624no—
187#41—7242no—
188C2×Q8C22×D18112yes—
189C2×D4—3624no—
190C2×D4—3622no—
191C2×D4—3622no—
192C2×D4—3622no—
193C2×D4—1811no—
194C2×D4—1812no—
195C2×D4—1812no—
196C2×D4—7242no—
197C22×C4—3622no—
198C22×C4—3622no—
199C22×C4—1811no—
200C22×C4C22×D18111yescharacteristic
201Q16—22no—
202SD16—3624no—
203SD16—7242no—
204D8—3624no—
205D8—3622no—
206D8—1814no—
207C2×C8—22no—
208C2×C8—3622no—
209C2×C8—1811no—
210C2×C8—1811no—
211C2×C8C22×D18111yescharacteristic
212C2×C8C22×D18111yescharacteristic
213C4⋊C4—22no—
214C4⋊C4—1811no—
215C4⋊C4C22×D18112yes—
216C22⋊C4—22no—
217C22⋊C4—3622no—
218C22⋊C4—3622no—
219C22⋊C4—3622no—
220C22⋊C4—3622no—
221C22⋊C4—1811no—
222C22⋊C4—1811no—
223C42—1811no—
224C23—3622no—
225C23—3622no—
226C23—14482no—
227C23—1811no—
228C23—7244no—
229C23—7244no—
230C23D4×D18111yescharacteristic
231Q8—22no—
232Q8—22no—
233D4—3622no—
234D4—3622no—
235D4—3622no—
236D4—3622no—
237D4—14482no—
238D4—1814no—
239D4—7244no—
240D4—7244no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—3621no—
246C2×C4—3621no—
247C2×C4—14482no—
248C2×C4—1811no—
249C2×C4—1811no—
250C2×C4—1811no—
251C2×C4—7244no—
252C2×C4D4×D18111yescharacteristic
253C2×C4C23×D18111yescharacteristic
254C8—21no—
255C8—3621no—
256C8—3621no—
257C8D4×D18112yes—
258C22—21no—
259C22—21no—
260C22—3621no—
261C22—3624no—
262C22—3621no—
263C22—3621no—
264C22—14488no—
265C22—7242no—
266C22—7242no—
267C22—7242no—
268C22—7242no—
269C22—7242no—
270C22#1915383912yescomplemented
271C22C2×D4×D18111yescharacteristic
272C4—21no—
273C4—42no—
274C4—42no—
275C4—3621no—
276C4—3621no—
277C4—3621no—
278C4C2×D4×D18111yescharacteristic
279C4#1915386211yescharacteristic
280C2—22no—
281C2—14482no—
282C2—7241no—
283C2—7244no—
284C2#2604191311yescharacteristic
285C2C2×#1915383911yescharacteristic
286C2#2604208111yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#467×C181C2
socleC22×C181C22×D4
hypercenter#467D181
hypocenterC181#3280
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#19153839C2semidirect
2. C22×D1448C2semidirect
3. #26041920C2semidirect
4. #26041664C2semidirect
5. #26041516C2semidirect
6. #26041497C2semidirect
7. #26041429C2semidirect
8. #26041427C2semidirect
9. #26041361C2semidirect
10. C8×C22⋊D181C2semidirect
11. #467×C181C2semidirect
12. C2×#15668450C22semidirect
15. C2×#15668436C22semidirect
16. C2×#15668436C22semidirect
21. C2×C8×D181C22semidirect
22. #19153805C22semidirect
25. #19153791C22semidirect
27. #19153778C22semidirect
30. #19153745C22semidirect
31. #19153736C22semidirect
32. #19153736C22semidirect
34. #19153728C22semidirect
35. C8×Dic181C22semidirect
36. C2×Q16×C181C22semidirect
39. #107×C181C22semidirect
40. #103×C181C22semidirect
59. D1448D4semidirect
116. C22×C181#135semidirect
143. C181#3280semidirect
149. #467D181semidirect
159. C2×Q16C2×D181semidirect
177. #107C2×D181semidirect
179. #103C2×D181semidirect
270. C22#19153839semidirect

Maximal subgroups

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

subgroupindex
C2×#191538392
C22×D14482
#260419202
#260416642
#260415162
#260414972
#260414292
#260414272
#260413612
C8×C22⋊D1812
#467×C1812
#3280181