Finite Group Databasebeta

← #37536222

Subgroups — #37536222

Order 23168.

1224 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×D181C211yesmaximal, characteristic, complemented
2C2×#19153786C211yesmaximal, characteristic
3C2×#19153778C212yesmaximal, complemented
4C2×#19153776C211yesmaximal, characteristic, complemented
5#26041158C211yesmaximal, characteristic, complemented
6#26041157C212yesmaximal, complemented
7#26041150C212yesmaximal, complemented
8#26041134C211yesmaximal, characteristic, complemented
9C22⋊C4×Dic181C211yesmaximal, characteristic, complemented
10#26041126C212yesmaximal, complemented
11#385×C181C211yesmaximal, characteristic, complemented
12C24×D181C2211yescharacteristic
13C22×D724C2212yes—
14C22×C22⋊D181C2211yescharacteristic
15C23×Dic181C2211yescharacteristic
16C22×C4×D181—24no—
17C22×C4×D181—28no—
18C22×C4×D181C2212yes—
19C22×C4×D181C2212yes—
20C22×C4×D181C2211yescharacteristic, complemented
21#19153786C418yescomplemented
22#19153778—28no—
23#19153776—24no—
24C2×#15668428C2211yescharacteristic, complemented
25C2×C22⋊Dic181C2211yescharacteristic
26C2×#15668426C2212yes—
27C2×#15668426C2212yes—
28C2×#15668426C2212yes—
29C2×#15668426C2211yescharacteristic, complemented
30C2×#15668426C2212yescomplemented
31C2×C4×Dic181C2211yescharacteristic, complemented
32C2×C4×Dic181C2212yescomplemented
33#19153723C2212yescomplemented
34#19153723C2212yescomplemented
35C23×C4×C181C2211yescharacteristic
36C2×C4⋊C4×C181C2212yescomplemented
37C2×C22⋊C4×C181C2211yescharacteristic
38C2×C22⋊C4×C181C2211yescharacteristic
39#95×C181C2212yescomplemented
40C23×D181—24no—
41C23×D181—24no—
42C23×D181—22no—
43C23×D181—22no—
44C23×D181C2×C414yescomplemented
45C23×D181C2311yescharacteristic
46C23×D181C2312yes—
47C2×D724—24no—
48C2×D724—24no—
49C2×D724C2×C418yescomplemented
50C2×C22⋊D181—22no—
51C2×C22⋊D181—22no—
52C2×C22⋊D181C2×C414yescomplemented
53C22×Dic181—24no—
54C22×Dic181D414yescomplemented
55C22×Dic181C2312yes—
56C22×Dic181C2311yescharacteristic
57C22×Dic181C2311yescharacteristic
58C22×Dic181C2311yescharacteristic
59C2×C4×D181—232no—
60C2×C4×D181—24no—
61C2×C4×D181—24no—
62C2×C4×D181—24no—
63C2×C4×D181—24no—
64C2×C4×D181—22no—
65C2×C4×D181—24no—
66C2×C4×D181—24no—
67C2×C4×D181D418yescomplemented
68#15668428—22no—
69C22⋊Dic181—22no—
70#15668426—24no—
71#15668426—24no—
72#15668426—22no—
73#15668426—24no—
74#15668426C2×C418yescomplemented
75C4×Dic181—22no—
76C4×Dic181—24no—
77C24×C181C2311yescharacteristic
78C22×C4×C181—22no—
79C22×C4×C181—24no—
80C22×C4×C181C2311yescharacteristic
81C22×C4×C181C2311yescharacteristic
82C22×C4×C181C2312yes—
83C22×C4×C181C2312yes—
84C4⋊C4×C181—24no—
85C22⋊C4×C181—22no—
86C22⋊C4×C181C2×C414yescomplemented
87C22×D181—28no—
88C22×D181—216no—
89C22×D181—28no—
90C22×D181—28no—
91C22×D181—22no—
92C22×D181—22no—
93C22×D181—22no—
94C22×D181—22no—
95C22×D181—216no—
96C22×D181—21no—
97C22×D181—22no—
98C22×D181—21no—
99C22×D181—22no—
100C22×D181C22×C414yes—
101C22×D181C22×C412yescomplemented
102C22×D181C2×D414yes—
103C22×D181#4114yes—
104D724—216no—
105C22⋊D181—28no—
106C2×Dic181—21no—
107C2×Dic181—22no—
108C2×Dic181—21no—
109C2×Dic181—28no—
110C2×Dic181—22no—
111C2×Dic181—22no—
112C2×Dic181—22no—
113C2×Dic181C22×C412yescomplemented
114C2×Dic181C2×D412yes—
115C2×Dic181C2×D412yes—
116C2×Dic181C2×D414yes—
117C4×D181—216no—
118C4×D181—216no—
119C4×D181—44no—
120C23×C181—22no—
121C23×C181—22no—
122C23×C181C22×C412yes—
123C23×C181C2×D412yes—
124C23×C181#4112yes—
125C23×C181C2411yescharacteristic
126C2×C4×C181—28no—
127C2×C4×C181—22no—
128C2×C4×C181—22no—
129C2×C4×C181—22no—
130C2×C4×C181—22no—
131C2×C4×C181—21no—
132C2×C4×C181—21no—
133C2×C4×C181—22no—
134C2×C4×C181—22no—
135C2×C4×C181C22×C414yescomplemented
136C2×C4×C181C2×D414yes—
137C22×C181—24no—
138C22×C181—22no—
139C22×C181—22no—
140C22×C181—24no—
141C22×C181C4×D414yescomplemented
142C22×C181C23×C411yescharacteristic
143C22×C181C22×D411yescharacteristic
144C22×C181C22×D412yes—
145C22×C181C2×#4112yes—
146C22×C181C2×#4111yescharacteristic
147C2×D181—232no—
148C2×D181—24no—
149C2×D181—24no—
150C2×D181—24no—
151C2×D181—24no—
152C2×D181—24no—
153C2×D181—24no—
154C2×D181—22no—
155C2×D181—24no—
156C2×D181—41no—
157C2×D181—41no—
158C2×D181—42no—
159C2×D181C4×D418yescomplemented
160C4×C181—24no—
161C4×C181—24no—
162C4×C181—41no—
163C4×C181—41no—
164Dic181—22no—
165Dic181—22no—
166Dic181—42no—
167Dic181#12014yescomplemented
168C2×C181—24no—
169C2×C181C2×C4×D412yes—
170C2×C181C2×C4×D411yescharacteristic
171C2×C181C2×#12011yescharacteristic
172C2×C181#52511yescharacteristic
173C2×C181#52512yes—
174D181—28no—
175D181—42no—
176C181C4×#12011yescharacteristic, complemented, Sylow
177C4×#120—1811nomaximal, Sylow
178C24×C4—1811no—
179C2×#120—1811no—
180C2×C4×D4—1812no—
181C2×C4×D4—1811no—
182#386—1811no—
183#386—1812no—
184#385—1812no—
185#385D18111yescharacteristic, complemented
186#379—1811no—
187C4×C22⋊C4—1811no—
188C4×C22⋊C4—1812no—
189C25—1811no—
190C22×D4—1811no—
191C22×D4—1812no—
192C23×C4—3624no—
193C23×C4—3628no—
194C23×C4—1811no—
195C23×C4—1812no—
196C23×C4—1811no—
197C23×C4—1812no—
198C23×C4C2×D18111yescharacteristic
199#120—1818no—
200C4×D4—3624no—
201C4×D4—3628no—
202C2×C4⋊C4—1811no—
203C2×C4⋊C4C2×D18112yescomplemented
204C2×C22⋊C4—1811no—
205C2×C22⋊C4—1812no—
206C2×C22⋊C4—1811no—
207C2×C22⋊C4—1812no—
208C2×C22⋊C4—1812no—
209C2×C22⋊C4—1812no—
210C2×C22⋊C4C2×D18111yescharacteristic
211C2×C22⋊C4C2×D18111yescharacteristic
212C2×C42—1812no—
213C2×C42—1811no—
214#95—1812no—
215#95—1812no—
216#95C2×D18112yescomplemented
217C24—3624no—
218C24—3624no—
219C24—3622no—
220C24—3622no—
221C24—1814no—
222C24—1811no—
223C24—1812no—
224C24C22×D18111yescharacteristic
225C2×D4—3624no—
226C2×D4—3624no—
227C2×D4—3622no—
228C2×D4—3622no—
229C2×D4—1814no—
230C2×D4—1818no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—36232no—
234C22×C4—3624no—
235C22×C4—3624no—
236C22×C4—3624no—
237C22×C4—3624no—
238C22×C4—3622no—
239C22×C4—3624no—
240C22×C4—3624no—
241C22×C4—3624no—
242C22×C4—1812no—
243C22×C4—1814no—
244C22×C4—1811no—
245C22×C4—1811no—
246C22×C4—1818no—
247C22×C4—1811no—
248C22×C4C22×D18111yescharacteristic
249C22×C4C22×D18111yescharacteristic
250C22×C4C22×D18112yes—
251C22×C4C22×D18112yes—
252C4⋊C4—24no—
253C4⋊C4—3622no—
254C22⋊C4—22no—
255C22⋊C4—3624no—
256C22⋊C4—3624no—
257C22⋊C4—3622no—
258C22⋊C4—3622no—
259C22⋊C4—3624no—
260C22⋊C4—1818no—
261C22⋊C4C4×D18114yescomplemented
262C42—3624no—
263C42—3622no—
264C23—22no—
265C23—22no—
266C23—3628no—
267C23—3628no—
268C23—36216no—
269C23—36216no—
270C23—3628no—
271C23—3622no—
272C23—3622no—
273C23—3622no—
274C23—3622no—
275C23—3622no—
276C23—3621no—
277C23—3622no—
278C23—3621no—
279C23—1812no—
280C23—1814no—
281C23—1814no—
282C23—1814no—
283C23C2×C4×D18112yes—
284C23D4×D18112yes—
285C23#1566845012yes—
286C23C23×D18111yescharacteristic
287D4—36216no—
288D4—3628no—
289C2×C4—28no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—36216no—
299C2×C4—36216no—
300C2×C4—3621no—
301C2×C4—3622no—
302C2×C4—3621no—
303C2×C4—3628no—
304C2×C4—3622no—
305C2×C4—3622no—
306C2×C4—3622no—
307C2×C4—1812no—
308C2×C4—1812no—
309C2×C4—1814no—
310C2×C4—1812no—
311C2×C4—7244no—
312C2×C4C2×C4×D18114yescomplemented
313C2×C4D4×D18114yes—
314C22—24no—
315C22—22no—
316C22—22no—
317C22—24no—
318C22—36232no—
319C22—3624no—
320C22—3624no—
321C22—3624no—
322C22—3624no—
323C22—3624no—
324C22—3624no—
325C22—3624no—
326C22—3622no—
327C22—1818no—
328C22—7241no—
329C22—7241no—
330C22—7242no—
331C22#1915377814yescomplemented
332C22C22×C4×D18111yescharacteristic
333C22C2×D4×D18112yes—
334C22C2×D4×D18111yescharacteristic
335C22#41×D18112yes—
336C22C2×#1566845011yescharacteristic
337C4—24no—
338C4—24no—
339C4—41no—
340C4—41no—
341C4—3622no—
342C4—3622no—
343C4—1814no—
344C4—7242no—
345C2—24no—
346C2—3628no—
347C2—7242no—
348C2C4×D4×D18112yes—
349C2C2×#1915377811yescharacteristic
350C2#120×D18111yescharacteristic
351C2#2604178711yescharacteristic
352C2#2604179212yes—
subgroupG/H
center, FrattiniC23C23×D181
commutatorC22×C181C23×C4
Fitting#385×C181C2
socle, cosocleC23×C181C24
prime radicalC2×#19153786C2
hypercenter#385D181
hypocenterC181C4×#120
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C23×C4×D181C2semidirect
3. C2×#19153778C2semidirect
4. C2×#19153776C2semidirect
5. #26041158C2semidirect
6. #26041157C2semidirect
7. #26041150C2semidirect
8. #26041134C2semidirect
9. C22⋊C4×Dic181C2semidirect
10. #26041126C2semidirect
11. #385×C181C2semidirect
20. C22×C4×D181C22semidirect
21. #19153786C4semidirect
24. C2×#15668428C22semidirect
29. C2×#15668426C22semidirect
30. C2×#15668426C22semidirect
31. C2×C4×Dic181C22semidirect
32. C2×C4×Dic181C22semidirect
33. #19153723C22semidirect
34. #19153723C22semidirect
36. C2×C4⋊C4×C181C22semidirect
39. #95×C181C22semidirect
44. C23×D181C2×C4semidirect
49. C2×D724C2×C4semidirect
52. C2×C22⋊D181C2×C4semidirect
54. C22×Dic181D4semidirect
67. C2×C4×D181D4semidirect
74. #15668426C2×C4semidirect
86. C22⋊C4×C181C2×C4semidirect
101. C22×D181C22×C4semidirect
113. C2×Dic181C22×C4semidirect
135. C2×C4×C181C22×C4semidirect
141. C22×C181C4×D4semidirect
159. C2×D181C4×D4semidirect
167. Dic181#120semidirect
176. C181C4×#120semidirect
185. #385D181semidirect
203. C2×C4⋊C4C2×D181semidirect
216. #95C2×D181semidirect
261. C22⋊C4C4×D181semidirect
312. C2×C4C2×C4×D181semidirect
331. C22#19153778semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D1812
C2×#191537862
C2×#191537782
C2×#191537762
#260411582
#260411572
#260411502
#260411342
C22⋊C4×Dic1812
#260411262
#385×C1812
C4×#120181