Finite Group Databasebeta

← #37545340

Subgroups — #37545340

Order 23168.

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

normal subgroups only

This group has 556 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×#19153840C212yesmaximal, complemented
2#26041949C211yesmaximal, characteristic, complemented
3#26041914C211yesmaximal, characteristic, complemented
4#26041529C212yesmaximal, complemented
5#26041476C211yesmaximal, characteristic, complemented
6#26041471C211yesmaximal, characteristic, complemented
7#26041421C212yesmaximal, complemented
8#26041411C212yesmaximal, complemented
9C4×#15668438C211yesmaximal, characteristic, complemented
10#26041358C211yesmaximal, characteristic, complemented
11#493×C181C211yesmaximal, characteristic, complemented
12C2×#15668448C2212yescomplemented
13C2×D4×D181—21no—
14C2×D4×D181C2212yes—
15#19153840—28no—
16C2×#15668438C2211yescharacteristic
17C2×#15668437C2212yescomplemented
18C2×#15668434C2212yes—
19C2×#15668432C2212yescomplemented
20#19153812—21no—
21#19153788—22no—
22#19153787—21no—
23#19153787C2212yes—
24#19153787C2212yescomplemented
25#19153784—22no—
26C4×Dic362C2211yescharacteristic, complemented
27C4×D724C2211yescharacteristic
28C4×D724C2211yescharacteristic
29D4×Dic181—21no—
30#19153748C2211yescharacteristic, complemented
31C4⋊#13089678C2211yescharacteristic, complemented
32#19153741C2211yescharacteristic, complemented
33#19153738C2211yescharacteristic
34#19153737C2212yescomplemented
35#19153737C2212yescomplemented
36#19153728C2212yescomplemented
37C2×D8×C181C2212yescomplemented
38C2×D8×C181C2212yescomplemented
39#127×C181C2211yescharacteristic
40#127×C181C2211yescharacteristic, complemented
41C4×C8×C181C2211yescharacteristic, complemented
42C23×D181—22no—
43#15668448—24no—
44#15668448—42no—
45C2×Dic362C2311yescharacteristic
46C2×D724D411yescharacteristic
47C2×D724C2311yescharacteristic
48C2×C22⋊D181—22no—
49C2×C22⋊D181—22no—
50C2×C22⋊D181—22no—
51C2×C22⋊D181—22no—
52D4×D181—24no—
53D4×D181—42no—
54D4×D181—44no—
55C22×Dic181—22no—
56C2×C4×D181—21no—
57C2×C4×D181C2312yes—
58#15668438D414yescomplemented
59#15668437—24no—
60#15668434—24no—
61#15668432—24no—
62C2×#13089678C2312yes—
63C4⋊Dic181D411yescharacteristic
64C4⋊Dic181C2311yescharacteristic
65C4⋊Dic181C2311yescharacteristic
66#15668428—21no—
67C22⋊Dic181—22no—
68C22⋊Dic181—22no—
69#15668426—21no—
70#15668426D412yes—
71C4×Dic181—21no—
72C2×D4×C181—21no—
73C2×D4×C181—21no—
74C2×D4×C181—21no—
75C2×D4×C181—21no—
76C2×D4×C181C2312yes—
77C2×D4×C181C2312yes—
78D8×C181—24no—
79D8×C181—42no—
80C2×C8×C181C2311yescharacteristic
81C2×C8×C181C2311yescharacteristic
82C42×C181C2311yescharacteristic
83C22×D181—21no—
84C22×D181—44no—
85C22×D181—44no—
86C22×D181—82no—
87C22×D181C2×D412yes—
88Dic362—21no—
89Dic362C2×D412yescomplemented
90D724—22no—
91D724—21no—
92D724C2×D412yes—
93C22⋊D181—44no—
94C22⋊D181—44no—
95C22⋊D181—44no—
96C22⋊D181—82no—
97C2×Dic181—21no—
98C2×Dic181—21no—
99C2×Dic181—21no—
100C2×Dic181—44no—
101C2×Dic181C2×D412yes—
102C4×D181—22no—
103C4×D181—22no—
104C4×D181—42no—
105#13089678—22no—
106C23×C181—22no—
107C23×C181—22no—
108D4×C181—22no—
109D4×C181—22no—
110D4×C181—22no—
111D4×C181—22no—
112D4×C181—42no—
113D4×C181—42no—
114D4×C181—81no—
115D4×C181—81no—
116C2×C4×C181C2×D411yescharacteristic
117C2×C4×C181C2×D411yescharacteristic
118C2×C4×C181C2411yescharacteristic
119C8×C181—21no—
120C8×C181C2×D412yescomplemented
121C22×C181—42no—
122C22×C181—42no—
123C22×C181—42no—
124C22×C181—42no—
125C22×C181—42no—
126C22×C181—42no—
127C22×C181C22×D411yescharacteristic
128C2×D181—22no—
129C2×D181—22no—
130C2×D181—22no—
131C2×D181—22no—
132C2×D181—41no—
133C2×D181—41no—
134C2×D181—88no—
135C4×C181—21no—
136C4×C181AGL(1,ℤ/8)12yescomplemented
137C4×C181C22×D411yescharacteristic
138C4×C181H(5,𝔽2)11yescharacteristic
139Dic181—22no—
140Dic181—22no—
141Dic181—41no—
142Dic181—41no—
143C2×C181—82no—
144C2×C181—82no—
145C2×C181D4211yescharacteristic
146C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C181#57611yescharacteristic
148D181—42no—
149D181—42no—
150C181#327411yescharacteristic, complemented, Sylow
151#3274—1811nomaximal, Sylow
152C2×AGL(1,ℤ/8)—1812no—
153#550—1811no—
154D42—1811no—
155#493D18111yescharacteristic, complemented
156#469—1812no—
157#460—1811no—
158#459—1811no—
159#449—1812no—
160#447—1812no—
161C4×SD16—1811no—
162#436—1811no—
163C2×#41—1812no—
164C22×D4—3621no—
165C22×D4—1812no—
166AGL(1,ℤ/8)—3628no—
167C2×SD16—1812no—
168C2×SD16—1811no—
169C2×D8—1812no—
170C2×D8C2×D18112yescomplemented
171C2×D8C2×D18112yescomplemented
172C2×M16—1812no—
173#127—3621no—
174#127C2×D18111yescharacteristic
175#127C2×D18111yescharacteristic, complemented
176#121—3621no—
177#121—3622no—
178#121—1812no—
179#121—1812no—
180#120—3622no—
181C4×Q8—1811no—
182C4×D4—3621no—
183C4×D4—1811no—
184C4×D4—1811no—
185#106—1811no—
186C4⋊C8—1811no—
187#103—1811no—
188#102—1812no—
189#102—1812no—
190#102—1811no—
191C22⋊C8—1812no—
192C4×C8C2×D18111yescharacteristic, complemented
193C24—3622no—
194#41—3624no—
195#41—7242no—
196C2×Q8—1811no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—3624no—
202C2×D4—3622no—
203C2×D4—3622no—
204C2×D4—3622no—
205C2×D4—3622no—
206C2×D4—1811no—
207C2×D4—1811no—
208C2×D4—7242no—
209C2×D4—7244no—
210C2×D4C22×D18112yes—
211C2×D4C22×D18112yes—
212C22×C4—3622no—
213C22×C4—3621no—
214C22×C4—1812no—
215SD16—3624no—
216SD16—1814no—
217D8—24no—
218D8—42no—
219D8—3624no—
220M16—3624no—
221C2×C8—1812no—
222C2×C8C22×D18111yescharacteristic
223C2×C8C22×D18111yescharacteristic
224C4⋊C4—3621no—
225C4⋊C4—1811no—
226C4⋊C4—1811no—
227C4⋊C4—1811no—
228C22⋊C4—3622no—
229C22⋊C4—3622no—
230C22⋊C4—3621no—
231C22⋊C4—1812no—
232C42—3621no—
233C42C22×D18111yescharacteristic
234C23—22no—
235C23—22no—
236C23—3621no—
237C23—14482no—
238C23—1812no—
239C23—7244no—
240C23—7244no—
241Q8—3621no—
242Q8—1812no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—22no—
247D4—42no—
248D4—42no—
249D4—81no—
250D4—81no—
251D4—3622no—
252D4—3621no—
253D4—14482no—
254D4—1812no—
255D4—7244no—
256D4—7244no—
257D4—7244no—
258C2×C4—3621no—
259C2×C4—3622no—
260C2×C4—3622no—
261C2×C4—3621no—
262C2×C4—3621no—
263C2×C4—1812no—
264C2×C4—7244no—
265C2×C4—7242no—
266C2×C4D4×D18111yescharacteristic
267C2×C4D4×D18111yescharacteristic
268C2×C4C23×D18111yescharacteristic
269C8—21no—
270C8—3622no—
271C8D4×D18112yescomplemented
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—3622no—
279C22—3622no—
280C22—3622no—
281C22—3622no—
282C22—14488no—
283C22—7241no—
284C22—7241no—
285C22C2×D4×D18111yescharacteristic
286C4—21no—
287C4—3622no—
288C4—3622no—
289C4—7241no—
290C4—7241no—
291C4#1915384012yescomplemented
292C4C2×D4×D18111yescharacteristic
293C4#1915386111yescharacteristic
294C2—82no—
295C2—82no—
296C2—7242no—
297C2—7242no—
298C2#2604191411yescharacteristic
299C2C2×#1915384011yescharacteristic
300C2#2604208011yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#493×C181C2
socleC22×C181C22×D4
hypercenter#493D181
hypocenterC181#3274
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#19153840C2semidirect
2. #26041949C2semidirect
3. #26041914C2semidirect
4. #26041529C2semidirect
5. #26041476C2semidirect
6. #26041471C2semidirect
7. #26041421C2semidirect
8. #26041411C2semidirect
9. C4×#15668438C2semidirect
10. #26041358C2semidirect
11. #493×C181C2semidirect
12. C2×#15668448C22semidirect
17. C2×#15668437C22semidirect
19. C2×#15668432C22semidirect
24. #19153787C22semidirect
26. C4×Dic362C22semidirect
30. #19153748C22semidirect
31. C4⋊#13089678C22semidirect
32. #19153741C22semidirect
34. #19153737C22semidirect
35. #19153737C22semidirect
36. #19153728C22semidirect
37. C2×D8×C181C22semidirect
38. C2×D8×C181C22semidirect
40. #127×C181C22semidirect
41. C4×C8×C181C22semidirect
58. #15668438D4semidirect
89. Dic362C2×D4semidirect
120. C8×C181C2×D4semidirect
136. C4×C181AGL(1,ℤ/8)semidirect
150. C181#3274semidirect
155. #493D181semidirect
170. C2×D8C2×D181semidirect
171. C2×D8C2×D181semidirect
175. #127C2×D181semidirect
192. C4×C8C2×D181semidirect
271. C8D4×D181semidirect
291. C4#19153840semidirect

Maximal subgroups

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

subgroupindex
C2×#191538402
#260419492
#260419142
#260415292
#260414762
#260414712
#260414212
#260414112
C4×#156684382
#260413582
#493×C1812
#3274181