Finite Group Databasebeta

← #37544742

Subgroups — #37544742

Order 23168.

490 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 115 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×#19153812C211yesmaximal, characteristic, complemented
2#26041815C211yesmaximal, characteristic, complemented
3#26041665C212yesmaximal, complemented
4#26041642C212yesmaximal, complemented
5#26041477C212yesmaximal, complemented
6#26041473C212yesmaximal, complemented
7#26041429C211yesmaximal, characteristic, complemented
8#26041415C211yesmaximal, characteristic, complemented
9#26041413C211yesmaximal, characteristic, complemented
10#26041333C211yesmaximal, characteristic, complemented
11#450×C181C211yesmaximal, characteristic, complemented
12C22×D724C2211yescharacteristic
13C22×C22⋊D181—21no—
14C22×C22⋊D181—21no—
15C2×#15668436C2212yescomplemented
16C2×D1448C2212yescomplemented
17C2×#15668434C2212yescomplemented
18#19153812—21no—
19#19153812—21no—
20#19153812C2212yes—
21#19153812C2212yes—
22#19153811—21no—
23#19153807C2211yescharacteristic, complemented
24#19153805C2211yescharacteristic, complemented
25#19153800—21no—
26#19153790C2211yescharacteristic
27Q8×Dic181C2211yescharacteristic, complemented
28#19153778C2211yescharacteristic, complemented
29#19153776—21no—
30#19153774—21no—
31C2×C4×Dic181C2211yescharacteristic
32#19153745C2212yescomplemented
33#19153738C2212yescomplemented
34#19153737C2212yescomplemented
35#19153736C2212yescomplemented
36C22⋊#13089678C2211yescharacteristic
37#19153726C2212yescomplemented
38C22×D4×C181C2211yescharacteristic
39C2×SD16×C181C2212yescomplemented
40C22⋊Q8×C181C2211yescharacteristic
41#102×C181C2212yescomplemented
42C22⋊C8×C181C2211yescharacteristic
43C23×D181—21no—
44C2×D724—21no—
45C2×D724—42no—
46C2×D724C2312yes—
47C2×D724C2311yescharacteristic
48C2×C22⋊D181—22no—
49C2×C22⋊D181—22no—
50C2×C22⋊D181—22no—
51C2×C22⋊D181—22no—
52C2×C22⋊D181—21no—
53C2×C22⋊D181—44no—
54C2×C22⋊D181—44no—
55C22×Dic181D411yescharacteristic
56C22×Dic181D411yescharacteristic
57C2×C4×D181—21no—
58#15668436—42no—
59D1448—42no—
60#15668434—42no—
61C2×#13089678C2312yes—
62C4⋊Dic181—21no—
63C4⋊Dic181—21no—
64C4⋊Dic181C2311yescharacteristic
65#15668428—21no—
66#15668426—21no—
67#15668426—21no—
68#15668426—21no—
69#15668426—21no—
70C4×Dic181—21no—
71C4×Dic181D411yescharacteristic
72C4×Dic181D411yescharacteristic
73C4×Dic181C2311yescharacteristic
74C4×Dic181C2311yescharacteristic
75C24×C181—21no—
76C2×Q8×C181C2311yescharacteristic
77C2×D4×C181—21no—
78C2×D4×C181—42no—
79C2×D4×C181C2312yes—
80C22×C4×C181C2311yescharacteristic
81SD16×C181—42no—
82C2×C8×C181C2312yes—
83C4⋊C4×C181—21no—
84C4⋊C4×C181C2311yescharacteristic
85C22⋊C4×C181—21no—
86C22×D181—21no—
87C22×D181—22no—
88C22×D181—42no—
89C22×D181—42no—
90C22×D181—42no—
91D724—22no—
92D724—22no—
93D724—22no—
94D724—42no—
95C22⋊D181—44no—
96C22⋊D181—44no—
97C22⋊D181—44no—
98C22⋊D181—42no—
99C22⋊D181—44no—
100C2×Dic181—22no—
101C2×Dic181—22no—
102C2×Dic181—21no—
103C2×Dic181C2×D411yescharacteristic
104C2×Dic181C2×D411yescharacteristic
105C2×Dic181C2×D411yescharacteristic
106C2×Dic181C2×D411yescharacteristic
107C4×D181—42no—
108#13089678—22no—
109C23×C181—22no—
110C23×C181—42no—
111C23×C181—42no—
112C23×C181—42no—
113C23×C181C2×D411yescharacteristic
114Q8×C181—22no—
115D4×C181—22no—
116D4×C181—22no—
117D4×C181—42no—
118C2×C4×C181—22no—
119C2×C4×C181—21no—
120C2×C4×C181—21no—
121C2×C4×C181C2×D411yescharacteristic
122C2×C4×C181C2411yescharacteristic
123C8×C181—22no—
124C22×C181—21no—
125C22×C181—21no—
126C22×C181—42no—
127C22×C181—42no—
128C22×C181—42no—
129C22×C181—84no—
130C22×C181AGL(1,ℤ/8)12yes—
131C22×C181C22×D411yescharacteristic
132C2×D181—42no—
133C2×D181—41no—
134C2×D181—41no—
135C2×D181—42no—
136C2×D181—42no—
137C2×D181—41no—
138C2×D181—84no—
139C4×C181—21no—
140C4×C181—41no—
141C4×C181—41no—
142C4×C181H(5,𝔽2)12yes—
143Dic181—21no—
144Dic181—21no—
145Dic181—21no—
146Dic181—41no—
147Dic181AGL(1,ℤ/8)12yes—
148C2×C181—22no—
149C2×C181—82no—
150C2×C181#53511yescharacteristic
151C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
153D181—81no—
154D181—82no—
155C181#321711yescharacteristic, complemented, Sylow
156#3217—1811nomaximal, Sylow
157C2×#127—1811no—
158#529—1811no—
159#496—1812no—
160#490—1812no—
161#460—1812no—
162#459—1812no—
163#450—1811no—
164#450D18111yescharacteristic, complemented
165#447—1811no—
166#447—1811no—
167#432—1811no—
168C22×D4—3621no—
169C22×D4—3621no—
170C22×D4—1811no—
171C22×D4C2×D18111yescharacteristic
172C2×SD16—1812no—
173C2×SD16C2×D18112yescomplemented
174C2×D8—1812no—
175C2×D8—1812no—
176#127—3621no—
177#127—3621no—
178#127—1812no—
179#127—1812no—
180#126—3621no—
181#125—1811no—
182#124—1811no—
183#123—3621no—
184C22⋊Q8C2×D18111yescharacteristic
185#121—1811no—
186C4×Q8—1811no—
187C4×D4—3621no—
188C4×D4—1811no—
189#117—3621no—
190C2×C42—1811no—
191C4⋊C8—1812no—
192#102—1812no—
193#102—1812no—
194#102—1812no—
195#102C2×D18112yescomplemented
196C22⋊C8—1811no—
197C22⋊C8C2×D18111yescharacteristic
198#97—1812no—
199C24—21no—
200C24—3621no—
201C2×Q8C22×D18111yescharacteristic
202C2×D4—21no—
203C2×D4—42no—
204C2×D4—3621no—
205C2×D4—3622no—
206C2×D4—3622no—
207C2×D4—3622no—
208C2×D4—3622no—
209C2×D4—3621no—
210C2×D4—1812no—
211C2×D4—1811no—
212C2×D4—7242no—
213C2×D4—7244no—
214C2×D4—7244no—
215C2×D4C22×D18112yes—
216C22×C4—3621no—
217C22×C4—1811no—
218C22×C4—1811no—
219C22×C4C22×D18111yescharacteristic
220SD16—42no—
221SD16—7242no—
222D8—7242no—
223D8—7242no—
224C2×C8—1812no—
225C2×C8C22×D18112yes—
226C4⋊C4—21no—
227C4⋊C4—3621no—
228C4⋊C4—3621no—
229C4⋊C4—3621no—
230C4⋊C4—1811no—
231C4⋊C4C22×D18111yescharacteristic
232C22⋊C4—21no—
233C22⋊C4—3621no—
234C22⋊C4—3621no—
235C22⋊C4—3621no—
236C22⋊C4—3621no—
237C42—3621no—
238C42—1811no—
239C42—1811no—
240C42—1811no—
241C42—1811no—
242C23—22no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—3621no—
247C23—3622no—
248C23—7242no—
249C23—7242no—
250C23—7242no—
251C23D4×D18111yescharacteristic
252Q8—22no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—3622no—
257D4—3622no—
258D4—3622no—
259D4—7242no—
260D4—7244no—
261D4—7244no—
262D4—7244no—
263D4—7244no—
264D4—7242no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—3622no—
269C2×C4—3622no—
270C2×C4—3621no—
271C2×C4—1811no—
272C2×C4—1811no—
273C2×C4—1811no—
274C2×C4—1811no—
275C2×C4—7242no—
276C2×C4D4×D18111yescharacteristic
277C2×C4C23×D18111yescharacteristic
278C8—22no—
279C8—3622no—
280C22—21no—
281C22—21no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—84no—
286C22—14484no—
287C22—7241no—
288C22—7242no—
289C22—7242no—
290C22—7241no—
291C22—7242no—
292C22—7241no—
293C22#1915384312yes—
294C22C2×D4×D18111yescharacteristic
295C4—21no—
296C4—41no—
297C4—41no—
298C4—3621no—
299C4—3621no—
300C4—3621no—
301C4—1812no—
302C4—7241no—
303C4#1915386112yes—
304C2—22no—
305C2—82no—
306C2—14482no—
307C2—14481no—
308C2#2604184411yescharacteristic
309C2AGL(1,ℤ/8)×D18111yescharacteristic
310C2C2×#1915384311yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#450×C181C2
socleC22×C181C22×D4
hypercenter#450D181
hypocenterC181#3217
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#19153812C2semidirect
2. #26041815C2semidirect
3. #26041665C2semidirect
4. #26041642C2semidirect
5. #26041477C2semidirect
6. #26041473C2semidirect
7. #26041429C2semidirect
8. #26041415C2semidirect
9. #26041413C2semidirect
10. #26041333C2semidirect
11. #450×C181C2semidirect
15. C2×#15668436C22semidirect
16. C2×D1448C22semidirect
17. C2×#15668434C22semidirect
23. #19153807C22semidirect
24. #19153805C22semidirect
27. Q8×Dic181C22semidirect
28. #19153778C22semidirect
32. #19153745C22semidirect
33. #19153738C22semidirect
34. #19153737C22semidirect
35. #19153736C22semidirect
37. #19153726C22semidirect
39. C2×SD16×C181C22semidirect
41. #102×C181C22semidirect
155. C181#3217semidirect
164. #450D181semidirect
173. C2×SD16C2×D181semidirect
195. #102C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538122
#260418152
#260416652
#260416422
#260414772
#260414732
#260414292
#260414152
#260414132
#260413332
#450×C1812
#3217181