Finite Group Databasebeta

← #37543659

Subgroups — #37543659

Order 23168.

462 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 127 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×#19153787C211yesmaximal, characteristic
2#117×D181C211yesmaximal, characteristic, complemented
3#26041602C212yesmaximal, complemented
4#26041601C212yesmaximal, complemented
5#26041596C212yesmaximal, complemented
6#26041591C212yesmaximal, complemented
7#26041322C211yesmaximal, characteristic, complemented
8#26041321C211yesmaximal, characteristic, complemented
9#26041284C211yesmaximal, characteristic, complemented
10#26041242C211yesmaximal, characteristic, complemented
11#418×C181C211yesmaximal, characteristic, complemented
12C22×C22⋊D181—21no—
13C22×C4×D181C2211yescharacteristic
14C2×#15668433C2211yescharacteristic
15#19153787—21no—
16#19153787—21no—
17#19153787C2212yes—
18#19153787C2212yes—
19#19153775C2211yescharacteristic
20#19153774C2211yescharacteristic, complemented
21#19153774C2211yescharacteristic, complemented
22#19153773—21no—
23#19153772—21no—
24C2×C4⋊Dic181C2211yescharacteristic
25C4⋊C4×D181C2211yescharacteristic, complemented
26C4⋊C4×D181C2211yescharacteristic, complemented
27C2×C22⋊Dic181—21no—
28C22⋊C4×D181—21no—
29C42×D181—21no—
30#19153750C2212yescomplemented
31#19153749C2212yescomplemented
32#19153748C2212yescomplemented
33#19153747C2212yescomplemented
34#19153737C2212yescomplemented
35#19153737C2212yescomplemented
36#19153728C2212yescomplemented
37#19153728C2212yescomplemented
38C22×D4×C181C2211yescharacteristic
39C2×M16×C181C2211yescharacteristic
40#117×C181C2211yescharacteristic
41#102×C181C2212yescomplemented
42#102×C181C2212yescomplemented
43C23×D181D411yescharacteristic
44C2×C22⋊D181—22no—
45C2×C22⋊D181—22no—
46C2×C22⋊D181—44no—
47C22×Dic181—21no—
48C22×Dic181D411yescharacteristic
49C2×C4×D181—21no—
50C2×C4×D181—21no—
51C2×C4×D181—22no—
52C2×C4×D181—22no—
53C2×C4×D181D411yescharacteristic
54C2×C4×D181D411yescharacteristic
55C2×C4×D181C2311yescharacteristic
56C2×C4×D181C2311yescharacteristic
57#15668433—22no—
58C2×#13089678C2312yes—
59C4⋊Dic181—21no—
60C4⋊Dic181C2311yescharacteristic
61C4⋊Dic181C2311yescharacteristic
62C4⋊Dic181C2312yes—
63#15668428—21no—
64#15668428—21no—
65C22⋊Dic181—22no—
66C22⋊Dic181—22no—
67C22⋊Dic181—21no—
68#15668426—21no—
69#15668426—21no—
70C4×Dic181—21no—
71C4×Dic181—21no—
72C4×Dic181—21no—
73C24×C181—21no—
74C2×D4×C181—21no—
75C2×D4×C181—22no—
76C2×D4×C181—22no—
77C2×D4×C181C2312yes—
78C22×C4×C181C2311yescharacteristic
79M16×C181—22no—
80C2×C8×C181C2312yes—
81C4⋊C4×C181C2311yescharacteristic
82C4⋊C4×C181C2311yescharacteristic
83C22⋊C4×C181—21no—
84C42×C181—21no—
85C22×D181—22no—
86C22×D181—22no—
87C22×D181—41no—
88C22×D181C2×D411yescharacteristic
89C22×D181C2×D411yescharacteristic
90C22⋊D181—44no—
91C22⋊D181—44no—
92C2×Dic181—22no—
93C2×Dic181—21no—
94C2×Dic181—21no—
95C2×Dic181—22no—
96C2×Dic181—42no—
97C2×Dic181C2×D411yescharacteristic
98C2×Dic181C2×D411yescharacteristic
99C4×D181—22no—
100C4×D181—22no—
101C4×D181—21no—
102C4×D181—21no—
103C4×D181—21no—
104C4×D181—21no—
105C4×D181—42no—
106C4×D181—42no—
107#13089678—22no—
108C23×C181—22no—
109C23×C181—42no—
110C23×C181—42no—
111C23×C181—42no—
112C23×C181C2×D411yescharacteristic
113D4×C181—22no—
114D4×C181—22no—
115D4×C181—42no—
116C2×C4×C181—21no—
117C2×C4×C181—21no—
118C2×C4×C181C2×D411yescharacteristic
119C2×C4×C181#4112yes—
120C2×C4×C181#4112yes—
121C2×C4×C181C2411yescharacteristic
122C8×C181—22no—
123C22×C181—21no—
124C22×C181—21no—
125C22×C181—42no—
126C22×C181—42no—
127C22×C181—42no—
128C22×C181—84no—
129C22×C181#12312yes—
130C22×C181C22×D411yescharacteristic
131C2×D181—21no—
132C2×D181—21no—
133C2×D181—21no—
134C2×D181—21no—
135C2×D181—21no—
136C2×D181—44no—
137C2×D181AGL(1,ℤ/8)12yes—
138C4×C181—41no—
139C4×C181—41no—
140C4×C181#12312yes—
141C4×C181C2×#4111yescharacteristic
142C4×C181C2×#4111yescharacteristic
143Dic181—21no—
144Dic181—21no—
145Dic181—41no—
146Dic181—42no—
147Dic181—41no—
148C2×C181—22no—
149C2×C181—82no—
150C2×C181C2×#12311yescharacteristic
151C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C181#57511yescharacteristic
153D181—22no—
154D181—41no—
155C181#309411yescharacteristic, complemented, Sylow
156#3094—1811nomaximal, Sylow
157C2×#121—1811no—
158C2×#117—1811no—
159#482—1812no—
160#482—1812no—
161#481—1812no—
162#480—1812no—
163#428—1811no—
164#428—1811no—
165#418—1811no—
166#418D18111yescharacteristic, complemented
167#407—1811no—
168C22×D4—3621no—
169C22×D4C2×D18111yescharacteristic
170C23×C4—1811no—
171C2×M16—1811no—
172C2×M16C2×D18111yescharacteristic
173#121—3621no—
174#121—3621no—
175#121—1812no—
176#121—1812no—
177#117—3621no—
178#117—3621no—
179#117—1811no—
180#117—1811no—
181#117—1811no—
182#117C2×D18111yescharacteristic
183C2×C4⋊C4—1811no—
184C2×C4⋊C4—1811no—
185C2×C4⋊C4—1811no—
186C2×C22⋊C4—3621no—
187C2×C22⋊C4—3621no—
188C2×C42—3621no—
189#107—1812no—
190#107—1812no—
191#106—1812no—
192#106—1812no—
193#102—1812no—
194#102—1812no—
195#102C2×D18112yescomplemented
196#102C2×D18112yescomplemented
197C22⋊C8—1812no—
198C22⋊C8—1812no—
199C24—21no—
200C24—1811no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—3622no—
205C2×D4—3622no—
206C2×D4—7244no—
207C2×D4C22×D18112yes—
208C22×C4—3621no—
209C22×C4—3621no—
210C22×C4—3622no—
211C22×C4—3622no—
212C22×C4—3621no—
213C22×C4—1811no—
214C22×C4—1811no—
215C22×C4—1811no—
216C22×C4—1811no—
217C22×C4—1811no—
218C22×C4C22×D18111yescharacteristic
219M16—22no—
220M16—3622no—
221C2×C8—1812no—
222C2×C8C22×D18112yes—
223C4⋊C4—3621no—
224C4⋊C4—3621no—
225C4⋊C4—3621no—
226C4⋊C4—1811no—
227C4⋊C4—1811no—
228C4⋊C4—1812no—
229C4⋊C4C22×D18111yescharacteristic
230C4⋊C4C22×D18111yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—3621no—
233C22⋊C4—3621no—
234C22⋊C4—3621no—
235C22⋊C4—3622no—
236C22⋊C4—3622no—
237C42—21no—
238C42—3621no—
239C42—3621no—
240C42—3621no—
241C23—22no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—3622no—
246C23—3622no—
247C23—1811no—
248C23—1811no—
249C23—7241no—
250C23D4×D18111yescharacteristic
251D4—22no—
252D4—22no—
253D4—42no—
254D4—7244no—
255D4—7244no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—3622no—
259C2×C4—3622no—
260C2×C4—3622no—
261C2×C4—3621no—
262C2×C4—3621no—
263C2×C4—3621no—
264C2×C4—3621no—
265C2×C4—3621no—
266C2×C4—3621no—
267C2×C4—3622no—
268C2×C4—1811no—
269C2×C4—1811no—
270C2×C4—7242no—
271C2×C4—7242no—
272C2×C4—7242no—
273C2×C4D4×D18111yescharacteristic
274C2×C4#1566844812yes—
275C2×C4#1566844912yes—
276C2×C4C23×D18111yescharacteristic
277C8—22no—
278C8—3622no—
279C22—21no—
280C22—21no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—84no—
285C22—3621no—
286C22—3621no—
287C22—3621no—
288C22—3621no—
289C22—3621no—
290C22—1812no—
291C22—7244no—
292C22#1915379812yes—
293C22C2×D4×D18111yescharacteristic
294C4—41no—
295C4—41no—
296C4—3621no—
297C4—3621no—
298C4—7242no—
299C4—7241no—
300C4—7241no—
301C4#1915379812yes—
302C4C2×#1566844811yescharacteristic
303C4C2×#1566844911yescharacteristic
304C2—22no—
305C2—82no—
306C2—3622no—
307C2—7241no—
308C2C2×#1915379811yescharacteristic
309C2AGL(1,ℤ/8)×D18111yescharacteristic
310C2#2604207511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#418×C181C2
socleC22×C181C22×D4
prime radicalC2×#19153787C2
hypercenter#418D181
hypocenterC181#3094
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #117×D181C2semidirect
3. #26041602C2semidirect
4. #26041601C2semidirect
5. #26041596C2semidirect
6. #26041591C2semidirect
7. #26041322C2semidirect
8. #26041321C2semidirect
9. #26041284C2semidirect
10. #26041242C2semidirect
11. #418×C181C2semidirect
20. #19153774C22semidirect
21. #19153774C22semidirect
25. C4⋊C4×D181C22semidirect
26. C4⋊C4×D181C22semidirect
30. #19153750C22semidirect
31. #19153749C22semidirect
32. #19153748C22semidirect
33. #19153747C22semidirect
34. #19153737C22semidirect
35. #19153737C22semidirect
36. #19153728C22semidirect
37. #19153728C22semidirect
41. #102×C181C22semidirect
42. #102×C181C22semidirect
155. C181#3094semidirect
166. #418D181semidirect
195. #102C2×D181semidirect
196. #102C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191537872
#117×D1812
#260416022
#260416012
#260415962
#260415912
#260413222
#260413212
#260412842
#260412422
#418×C1812
#3094181