Finite Group Databasebeta

← #37543663

Subgroups — #37543663

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×#19153791C211yesmaximal, characteristic
2#117×D181C211yesmaximal, characteristic, complemented
3#26041606C212yesmaximal, complemented
4#26041605C212yesmaximal, complemented
5#26041599C212yesmaximal, complemented
6#26041594C212yesmaximal, complemented
7#26041319C211yesmaximal, characteristic, complemented
8#26041285C211yesmaximal, characteristic, complemented
9#26041283C211yesmaximal, characteristic, complemented
10#26041242C211yesmaximal, characteristic, complemented
11#428×C181C211yesmaximal, characteristic, complemented
12C22×D724—21no—
13C22×D724C2211yescharacteristic
14C22×C4×D181C2211yescharacteristic
15C2×#15668433C2211yescharacteristic
16#19153791—21no—
17#19153791—21no—
18#19153791C2212yes—
19#19153791C2212yes—
20#19153775C2211yescharacteristic
21#19153774C2211yescharacteristic, complemented
22#19153774C2211yescharacteristic, complemented
23#19153773—21no—
24#19153772—21no—
25C4⋊C4×D181C2211yescharacteristic, complemented
26C4⋊C4×D181C2211yescharacteristic, complemented
27C2×#15668426—21no—
28C22⋊C4×D181—21no—
29C42×D181—21no—
30#19153749C2212yescomplemented
31#19153747C2212yescomplemented
32#19153738C2212yescomplemented
33#19153738C2212yescomplemented
34#19153736C2212yescomplemented
35#19153736C2212yescomplemented
36#19153728C2212yescomplemented
37#19153728C2212yescomplemented
38C2×M16×C181C2211yescharacteristic
39#117×C181C2211yescharacteristic
40C2×C4⋊C4×C181C2211yescharacteristic
41#107×C181C2212yescomplemented
42#106×C181C2212yescomplemented
43C23×D181—21no—
44C23×D181D411yescharacteristic
45C2×D724—22no—
46C2×D724—22no—
47C2×D724—21no—
48C2×D724—22no—
49C2×D724—22no—
50C2×D724—44no—
51C2×D724C2312yes—
52C22×Dic181D411yescharacteristic
53C2×C4×D181—21no—
54C2×C4×D181—21no—
55C2×C4×D181—22no—
56C2×C4×D181—22no—
57C2×C4×D181D411yescharacteristic
58C2×C4×D181D411yescharacteristic
59C2×C4×D181C2311yescharacteristic
60C2×C4×D181C2311yescharacteristic
61#15668433—22no—
62C2×#13089678C2312yes—
63C4⋊Dic181C2311yescharacteristic
64C4⋊Dic181C2311yescharacteristic
65#15668428—21no—
66#15668428—21no—
67C22⋊Dic181—21no—
68#15668426—21no—
69#15668426—21no—
70#15668426—22no—
71#15668426—22no—
72C4×Dic181—21no—
73C4×Dic181—21no—
74C4×Dic181—21no—
75C22×C4×C181—21no—
76C22×C4×C181C2311yescharacteristic
77M16×C181—22no—
78C2×C8×C181C2312yes—
79C4⋊C4×C181—21no—
80C4⋊C4×C181C2311yescharacteristic
81C4⋊C4×C181C2312yes—
82C4⋊C4×C181C2311yescharacteristic
83C22⋊C4×C181—21no—
84C42×C181—21no—
85C22×D181—22no—
86C22×D181—22no—
87C22×D181—22no—
88C22×D181—41no—
89C22×D181—42no—
90C22×D181—42no—
91C22×D181—42no—
92C22×D181C2×D411yescharacteristic
93C22×D181C2×D411yescharacteristic
94D724—22no—
95D724—22no—
96D724—44no—
97D724—44no—
98D724—42no—
99C2×Dic181—22no—
100C2×Dic181—21no—
101C2×Dic181—21no—
102C2×Dic181C2×D411yescharacteristic
103C2×Dic181C2×D411yescharacteristic
104C4×D181—22no—
105C4×D181—22no—
106C4×D181—21no—
107C4×D181—21no—
108C4×D181—21no—
109C4×D181—21no—
110C4×D181—42no—
111C4×D181—42no—
112#13089678—22no—
113C23×C181C2×D411yescharacteristic
114C2×C4×C181—21no—
115C2×C4×C181—21no—
116C2×C4×C181—22no—
117C2×C4×C181—42no—
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#12312yes—
126C22×C181C22×D411yescharacteristic
127C2×D181—21no—
128C2×D181—21no—
129C2×D181—21no—
130C2×D181—21no—
131C2×D181—21no—
132C2×D181—44no—
133C2×D181—42no—
134C2×D181—42no—
135C2×D181—42no—
136C2×D181—84no—
137C2×D181AGL(1,ℤ/8)12yes—
138C4×C181—42no—
139C4×C181—41no—
140C4×C181—41no—
141C4×C181#12312yes—
142C4×C181C2×#4111yescharacteristic
143C4×C181C2×#4111yescharacteristic
144Dic181—21no—
145Dic181—21no—
146Dic181—41no—
147Dic181—41no—
148C2×C181—22no—
149C2×C181C2×#12311yescharacteristic
150C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C181#57511yescharacteristic
152D181—22no—
153D181—41no—
154D181—82no—
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#428D18111yescharacteristic, complemented
165#418—1811no—
166#418—1811no—
167#407—1811no—
168C22×D4—3621no—
169C22×D4—1811no—
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⋊C4C2×D18111yescharacteristic
186C2×C22⋊C4—3621no—
187C2×C22⋊C4—3621no—
188C2×C42—3621no—
189#107—1812no—
190#107C2×D18112yescomplemented
191#106—1812no—
192#106C2×D18112yescomplemented
193#102—1812no—
194#102—1812no—
195#102—1812no—
196#102—1812no—
197C22⋊C8—1812no—
198C22⋊C8—1812no—
199C24—3621no—
200C24—1811no—
201C2×D4—3622no—
202C2×D4—3622no—
203C2×D4—3621no—
204C2×D4—3622no—
205C2×D4—3622no—
206C2×D4—1812no—
207C2×D4—7244no—
208C22×C4—21no—
209C22×C4—3621no—
210C22×C4—3621no—
211C22×C4—3622no—
212C22×C4—3622no—
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—21no—
224C4⋊C4—3621no—
225C4⋊C4—3621no—
226C4⋊C4—1811no—
227C4⋊C4—1811no—
228C4⋊C4C22×D18111yescharacteristic
229C4⋊C4C22×D18111yescharacteristic
230C4⋊C4C22×D18112yes—
231C22⋊C4—21no—
232C22⋊C4—3621no—
233C22⋊C4—3621no—
234C22⋊C4—3622no—
235C22⋊C4—3622no—
236C22⋊C4—3621no—
237C42—21no—
238C42—3621no—
239C42—3621no—
240C42—3621no—
241C23—3622no—
242C23—3622no—
243C23—3622no—
244C23—1811no—
245C23—1811no—
246C23—7241no—
247C23—7242no—
248C23—7242no—
249C23—7242no—
250C23D4×D18111yescharacteristic
251D4—3622no—
252D4—3622no—
253D4—7244no—
254D4—7244no—
255D4—7242no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—3622no—
261C2×C4—3622no—
262C2×C4—3622no—
263C2×C4—3621no—
264C2×C4—3621no—
265C2×C4—3621no—
266C2×C4—3621no—
267C2×C4—3621no—
268C2×C4—3621no—
269C2×C4—1811no—
270C2×C4—1811no—
271C2×C4—7242no—
272C2×C4—7242no—
273C2×C4D4×D18111yescharacteristic
274C2×C4#1566844912yes—
275C2×C4#1566845012yes—
276C2×C4C23×D18111yescharacteristic
277C8—22no—
278C8—3622no—
279C22—21no—
280C22—21no—
281C22—3621no—
282C22—3621no—
283C22—3621no—
284C22—3621no—
285C22—3621no—
286C22—14484no—
287C22—1812no—
288C22—7244no—
289C22—7242no—
290C22—7242no—
291C22—7242no—
292C22#1915380112yes—
293C22C2×D4×D18111yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—3621no—
298C4—3621no—
299C4—7241no—
300C4—7241no—
301C4#1915380112yes—
302C4C2×#1566844911yescharacteristic
303C4C2×#1566845011yescharacteristic
304C2—22no—
305C2—3622no—
306C2—14482no—
307C2—7241no—
308C2C2×#1915380111yescharacteristic
309C2AGL(1,ℤ/8)×D18111yescharacteristic
310C2#2604207811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#428×C181C2
socleC22×C181C22×D4
prime radicalC2×#19153791C2
hypercenter#428D181
hypocenterC181#3094
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. #117×D181C2semidirect
3. #26041606C2semidirect
4. #26041605C2semidirect
5. #26041599C2semidirect
6. #26041594C2semidirect
7. #26041319C2semidirect
8. #26041285C2semidirect
9. #26041283C2semidirect
10. #26041242C2semidirect
11. #428×C181C2semidirect
21. #19153774C22semidirect
22. #19153774C22semidirect
25. C4⋊C4×D181C22semidirect
26. C4⋊C4×D181C22semidirect
30. #19153749C22semidirect
31. #19153747C22semidirect
32. #19153738C22semidirect
33. #19153738C22semidirect
34. #19153736C22semidirect
35. #19153736C22semidirect
36. #19153728C22semidirect
37. #19153728C22semidirect
41. #107×C181C22semidirect
42. #106×C181C22semidirect
155. C181#3094semidirect
164. #428D181semidirect
190. #107C2×D181semidirect
192. #106C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191537912
#117×D1812
#260416062
#260416052
#260415992
#260415942
#260413192
#260412852
#260412832
#260412422
#428×C1812
#3094181