Finite Group Databasebeta

← #37543338

Subgroups — #37543338

Order 23168.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 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×#19153842C211yesmaximal, characteristic, complemented
2C2×M16×D181C211yesmaximal, characteristic, complemented
3#26041791C211yesmaximal, characteristic, complemented
4C2×#19153787C211yesmaximal, characteristic, complemented
5#26041531C212yesmaximal, complemented
6#26041524C212yesmaximal, complemented
7#26041514C212yesmaximal, complemented
8#26041507C212yesmaximal, complemented
9#26041322C211yesmaximal, characteristic, complemented
10#26041281C211yesmaximal, characteristic, complemented
11#418×C181C211yesmaximal, characteristic, complemented
12C2×#15668449C2211yescharacteristic
13C22×C22⋊D181—21no—
14C22×C4×D181C2211yescharacteristic
15#19153842—24no—
16C2×#15668437C2212yescomplemented
17C2×#15668434C2212yescomplemented
18C2×#15668433C2211yescharacteristic
19C2×#15668432C2212yescomplemented
20M16×D181—24no—
21C2×C8×D181C2212yescomplemented
22#19153800—21no—
23#19153796C2211yescharacteristic, complemented
24#19153796C2211yescharacteristic, complemented
25#19153791C2211yescharacteristic, complemented
26#19153791C2211yescharacteristic, complemented
27#19153787—21no—
28#19153787—21no—
29#19153787C2212yes—
30#19153787C2212yes—
31#19153786—21no—
32C4×D724—21no—
33C4×D724—21no—
34C2×C4⋊Dic181C2211yescharacteristic
35C2×C22⋊Dic181—21no—
36#19153749C2212yescomplemented
37#19153747C2212yescomplemented
38#19153741C2212yescomplemented
39#19153738C2212yescomplemented
40C22×D4×C181C2211yescharacteristic
41C2×M16×C181C2211yescharacteristic
42#117×C181C2211yescharacteristic
43#102×C181C2212yescomplemented
44#102×C181C2212yescomplemented
45C23×D181D411yescharacteristic
46#15668449—22no—
47#15668449—22no—
48C2×Dic362C2311yescharacteristic
49C2×D724—21no—
50C2×D724—21no—
51C2×D724C2311yescharacteristic
52C2×C22⋊D181—22no—
53C2×C22⋊D181—22no—
54C2×C22⋊D181—21no—
55C2×C22⋊D181—44no—
56C22×Dic181—21no—
57C22×Dic181D411yescharacteristic
58C2×C4×D181—22no—
59C2×C4×D181—21no—
60C2×C4×D181—22no—
61C2×C4×D181D411yescharacteristic
62C2×C4×D181D411yescharacteristic
63C2×C4×D181C2311yescharacteristic
64C2×C4×D181C2311yescharacteristic
65#15668437—24no—
66#15668434—24no—
67#15668433D414yescomplemented
68#15668432—24no—
69C2×#13089678C2312yes—
70C8×D181—24no—
71C4⋊Dic181—21no—
72C4⋊Dic181—21no—
73C4⋊Dic181—21no—
74C4⋊Dic181C2312yes—
75C22⋊Dic181—22no—
76C22⋊Dic181—22no—
77#15668426—21no—
78#15668426—21no—
79#15668426—21no—
80#15668426—21no—
81C24×C181—21no—
82C2×D4×C181—22no—
83C2×D4×C181—21no—
84C2×D4×C181—22no—
85C2×D4×C181C2312yes—
86C22×C4×C181C2311yescharacteristic
87M16×C181—22no—
88C2×C8×C181C2312yes—
89C4⋊C4×C181C2311yescharacteristic
90C4⋊C4×C181C2311yescharacteristic
91C22⋊C4×C181—21no—
92C42×C181—21no—
93C22×D181—22no—
94C22×D181—22no—
95C22×D181—21no—
96C22×D181—41no—
97C22×D181C2×D411yescharacteristic
98C22×D181C2×D411yescharacteristic
99Dic362—21no—
100Dic362—21no—
101D724—21no—
102D724—21no—
103D724—42no—
104D724—42no—
105C22⋊D181—44no—
106C22⋊D181—42no—
107C22⋊D181—44no—
108C2×Dic181—22no—
109C2×Dic181—21no—
110C2×Dic181—22no—
111C2×Dic181—42no—
112C2×Dic181C2×D411yescharacteristic
113C2×Dic181C2×D411yescharacteristic
114C4×D181—22no—
115C4×D181—22no—
116C4×D181—21no—
117C4×D181—21no—
118C4×D181—21no—
119C4×D181—21no—
120C4×D181—42no—
121#13089678C2×D414yescomplemented
122C23×C181—22no—
123C23×C181—42no—
124C23×C181—42no—
125C23×C181—42no—
126C23×C181C2×D411yescharacteristic
127D4×C181—22no—
128D4×C181—22no—
129D4×C181—42no—
130C2×C4×C181—21no—
131C2×C4×C181—21no—
132C2×C4×C181C2×D411yescharacteristic
133C2×C4×C181C2×D412yes—
134C2×C4×C181#4112yes—
135C2×C4×C181C2411yescharacteristic
136C8×C181—22no—
137C22×C181—21no—
138C22×C181—21no—
139C22×C181—42no—
140C22×C181—42no—
141C22×C181—42no—
142C22×C181—84no—
143C22×C181#12112yes—
144C22×C181C22×D411yescharacteristic
145C2×D181—21no—
146C2×D181—21no—
147C2×D181—21no—
148C2×D181—21no—
149C2×D181—21no—
150C2×D181—44no—
151C2×D181—41no—
152C2×D181—41no—
153C2×D181—41no—
154C2×D181AGL(1,ℤ/8)12yescomplemented
155C4×C181—41no—
156C4×C181—41no—
157C4×C181#12112yes—
158C4×C181C22×D411yescharacteristic
159C4×C181C2×#4111yescharacteristic
160Dic181—21no—
161Dic181—21no—
162Dic181—41no—
163Dic181—42no—
164C2×C181—22no—
165C2×C181—82no—
166C2×C181C2×#12111yescharacteristic
167C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C181#57511yescharacteristic
169D181—22no—
170D181—41no—
171D181—81no—
172C181#305511yescharacteristic, complemented, Sylow
173#3055—1811nomaximal, Sylow
174C2×AGL(1,ℤ/8)—1811no—
175C22×M16—1811no—
176#525—1811no—
177C2×#121—1811no—
178#469—1812no—
179#468—1812no—
180#466—1812no—
181#465—1812no—
182#428—1811no—
183#418D18111yescharacteristic, complemented
184#417—1811no—
185C2×#41—1811no—
186C22×D4—3621no—
187C22×D4C2×D18111yescharacteristic
188C23×C4—1811no—
189AGL(1,ℤ/8)—3624no—
190C2×SD16—1812no—
191C2×D8—1812no—
192C2×M16—3624no—
193C2×M16—1811no—
194C2×M16—1812no—
195C2×M16C2×D18111yescharacteristic
196C22×C8—1812no—
197#123—3621no—
198C22⋊Q8—1811no—
199C22⋊Q8—1811no—
200#121—3621no—
201#121—3621no—
202#121—1812no—
203#121—1812no—
204#121—1811no—
205#121—1811no—
206#120—3621no—
207C4×D4—3621no—
208C4×D4—3621no—
209#117C2×D18111yescharacteristic
210C2×C4⋊C4—1811no—
211C2×C22⋊C4—3621no—
212#107—1812no—
213#106—1812no—
214#103—1812no—
215#102—1812no—
216#102C2×D18112yescomplemented
217#102C2×D18112yescomplemented
218C24—21no—
219C24—1811no—
220#41—3622no—
221#41—3622no—
222C2×Q8—1811no—
223C2×D4—22no—
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—3621no—
227C2×D4—3621no—
228C2×D4—3622no—
229C2×D4—3622no—
230C2×D4—3621no—
231C2×D4—1811no—
232C2×D4—7244no—
233C2×D4C22×D18112yes—
234C22×C4—3622no—
235C22×C4—3622no—
236C22×C4—3621no—
237C22×C4—3621no—
238C22×C4—1811no—
239C22×C4—1811no—
240C22×C4—1811no—
241C22×C4—1811no—
242C22×C4—1811no—
243C22×C4C22×D18111yescharacteristic
244SD16—3624no—
245D8—3624no—
246M16—22no—
247M16—3624no—
248M16—1814no—
249C2×C8—3624no—
250C2×C8—1812no—
251C2×C8C22×D18112yes—
252C4⋊C4—3621no—
253C4⋊C4—3621no—
254C4⋊C4—3621no—
255C4⋊C4—1812no—
256C4⋊C4C22×D18111yescharacteristic
257C4⋊C4C22×D18111yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—3621no—
260C22⋊C4—3621no—
261C22⋊C4—3621no—
262C22⋊C4—3621no—
263C22⋊C4—3622no—
264C22⋊C4—3622no—
265C42—21no—
266C23—22no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—3622no—
271C23—3622no—
272C23—3621no—
273C23—1811no—
274C23—1811no—
275C23—7241no—
276C23D4×D18111yescharacteristic
277Q8—3621no—
278Q8—3621no—
279D4—22no—
280D4—22no—
281D4—42no—
282D4—3621no—
283D4—3621no—
284D4—7242no—
285D4—7242no—
286D4—7244no—
287D4—7244no—
288D4—7242no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—3622no—
292C2×C4—3622no—
293C2×C4—3622no—
294C2×C4—3621no—
295C2×C4—3621no—
296C2×C4—3621no—
297C2×C4—3621no—
298C2×C4—3621no—
299C2×C4—3622no—
300C2×C4—1811no—
301C2×C4—1811no—
302C2×C4—7242no—
303C2×C4—7242no—
304C2×C4D4×D18111yescharacteristic
305C2×C4D4×D18112yes—
306C2×C4#1566844912yes—
307C2×C4C23×D18111yescharacteristic
308C8—22no—
309C8—1814no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—84no—
316C22—3621no—
317C22—3621no—
318C22—3621no—
319C22—3621no—
320C22—3621no—
321C22—1812no—
322C22—7244no—
323C22—7241no—
324C22—7241no—
325C22—7241no—
326C22#1915378912yes—
327C22C2×D4×D18111yescharacteristic
328C4—41no—
329C4—41no—
330C4—3621no—
331C4—3621no—
332C4—7241no—
333C4—7242no—
334C4#1915378912yes—
335C4C2×D4×D18111yescharacteristic
336C4C2×#1566844911yescharacteristic
337C2—22no—
338C2—82no—
339C2—3622no—
340C2—14481no—
341C2—7241no—
342C2C2×#1915378911yescharacteristic
343C2AGL(1,ℤ/8)×D18111yescharacteristic
344C2#2604207511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#418×C181C2
socleC22×C181C22×D4
hypercenter#418D181
hypocenterC181#3055
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
1. C2×#19153842C2semidirect
2. C2×M16×D181C2semidirect
3. #26041791C2semidirect
4. C2×#19153787C2semidirect
5. #26041531C2semidirect
6. #26041524C2semidirect
7. #26041514C2semidirect
8. #26041507C2semidirect
9. #26041322C2semidirect
10. #26041281C2semidirect
11. #418×C181C2semidirect
16. C2×#15668437C22semidirect
17. C2×#15668434C22semidirect
19. C2×#15668432C22semidirect
21. C2×C8×D181C22semidirect
23. #19153796C22semidirect
24. #19153796C22semidirect
25. #19153791C22semidirect
26. #19153791C22semidirect
36. #19153749C22semidirect
37. #19153747C22semidirect
38. #19153741C22semidirect
39. #19153738C22semidirect
43. #102×C181C22semidirect
44. #102×C181C22semidirect
67. #15668433D4semidirect
121. #13089678C2×D4semidirect
154. C2×D181AGL(1,ℤ/8)semidirect
172. C181#3055semidirect
183. #418D181semidirect
216. #102C2×D181semidirect
217. #102C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538422
C2×M16×D1812
#260417912
C2×#191537872
#260415312
#260415242
#260415142
#260415072
#260413222
#260412812
#418×C1812
#3055181