Finite Group Databasebeta

← #37543030

Subgroups — #37543030

Order 23168.

764 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 149 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×#19153861C211yesmaximal, characteristic, complemented
2C2×#19153842C211yesmaximal, characteristic, complemented
3C2×#19153838C211yesmaximal, characteristic, complemented
4#26041860C211yesmaximal, characteristic, complemented
5#26041442C212yesmaximal, complemented
6#26041431C212yesmaximal, complemented
7#26041425C212yesmaximal, complemented
8#26041414C212yesmaximal, complemented
9#26041278C211yesmaximal, characteristic, complemented
10#26041245C211yesmaximal, characteristic, complemented
11C2×#102×C181C211yesmaximal, characteristic, complemented
12#19153861—24no—
13#19153861—24no—
14C2×#15668449C2211yescharacteristic
15C2×#15668449C2211yescharacteristic
16C2×#15668449C2211yescharacteristic, complemented
17C2×#15668448—21no—
18C2×#15668448C2212yes—
19C22×C22⋊D181—22no—
20C2×D4×D181—21no—
21C2×D4×D181C2212yes—
22#19153842—24no—
23#19153838—24no—
24C2×Dic724C2211yescharacteristic, complemented
25C2×#15668438C2211yescharacteristic, complemented
26C2×#15668438C2211yescharacteristic, complemented
27C2×#15668437C2212yescomplemented
28C2×D1448C2211yescharacteristic, complemented
29C2×#15668434C2212yescomplemented
30C2×#15668433C2211yescharacteristic, complemented
31#19153796—21no—
32#19153796C2212yescomplemented
33#19153791—21no—
34#19153791C2212yescomplemented
35#19153790—22no—
36#19153739C2212yescomplemented
37#19153736C2212yescomplemented
38#19153728C2212yescomplemented
39#19153728C2212yescomplemented
40C22×D4×C181C2211yescharacteristic
41C22×C8×C181C2211yescharacteristic, complemented
42C2×C4⋊C4×C181C2211yescharacteristic, complemented
43#102×C181C2212yescomplemented
44#102×C181C2212yescomplemented
45C23×D181—22no—
46#15668449—22no—
47#15668449—22no—
48#15668449—22no—
49#15668449—22no—
50#15668449—22no—
51#15668449D414yescomplemented
52#15668448—28no—
53#15668448—24no—
54#15668448—24no—
55C2×Dic362D411yescharacteristic
56C2×Dic362C2311yescharacteristic
57C2×Dic362C2311yescharacteristic
58C2×D724—22no—
59C2×D724D411yescharacteristic
60C2×D724C2311yescharacteristic
61C2×D724C2311yescharacteristic
62C2×C22⋊D181—216no—
63C2×C22⋊D181—22no—
64C2×C22⋊D181—22no—
65C2×C22⋊D181—21no—
66C2×C22⋊D181—21no—
67C2×C22⋊D181D412yes—
68D4×D181—28no—
69D4×D181—24no—
70D4×D181—24no—
71C22×Dic181—22no—
72C2×C4×D181—21no—
73C2×C4×D181—21no—
74C2×C4×D181C2312yes—
75Dic724—22no—
76#15668438—22no—
77#15668438—22no—
78#15668437—24no—
79D1448—22no—
80#15668434—24no—
81#15668433—22no—
82C2×#13089678C2312yes—
83C4⋊Dic181—22no—
84#15668426—22no—
85#15668426—22no—
86C24×C181—21no—
87C2×D4×C181—21no—
88C2×D4×C181—22no—
89C2×D4×C181D414yescomplemented
90C2×D4×C181C2312yes—
91C22×C4×C181—21no—
92C22×C4×C181C2311yescharacteristic
93C2×C8×C181—22no—
94C2×C8×C181C2311yescharacteristic
95C2×C8×C181C2311yescharacteristic
96C4⋊C4×C181—21no—
97C4⋊C4×C181C2312yes—
98C22×D181—21no—
99C22×D181—28no—
100C22×D181—21no—
101C22×D181—82no—
102C22×D181—82no—
103C22×D181C2×D412yes—
104Dic362—22no—
105Dic362—21no—
106Dic362—21no—
107Dic362—21no—
108Dic362C2×D412yes—
109D724—22no—
110D724—21no—
111D724—21no—
112D724—21no—
113D724—82no—
114D724C2×D412yes—
115C22⋊D181—28no—
116C22⋊D181—28no—
117C22⋊D181—24no—
118C22⋊D181—24no—
119C22⋊D181—42no—
120C2×Dic181—28no—
121C2×Dic181—21no—
122C2×Dic181—21no—
123C2×Dic181C2×D412yes—
124C4×D181—24no—
125C4×D181—22no—
126C4×D181—22no—
127C4×D181—42no—
128#13089678—22no—
129C23×C181—22no—
130C23×C181—24no—
131C23×C181—81no—
132C23×C181—81no—
133C23×C181C2×D411yescharacteristic
134D4×C181—24no—
135D4×C181—22no—
136D4×C181C2×D414yes—
137C2×C4×C181—22no—
138C2×C4×C181—81no—
139C2×C4×C181C2×D411yescharacteristic
140C2×C4×C181C2×D412yes—
141C2×C4×C181C2×D412yes—
142C2×C4×C181C2411yescharacteristic
143C8×C181—21no—
144C8×C181—21no—
145C22×C181—21no—
146C22×C181—21no—
147C22×C181—24no—
148C22×C181—42no—
149C22×C181—42no—
150C22×C181—84no—
151C22×C181#12012yes—
152C22×C181C22×D411yescharacteristic
153C2×D181—22no—
154C2×D181—22no—
155C2×D181—22no—
156C2×D181—22no—
157C2×D181—41no—
158C2×D181—41no—
159C2×D181—41no—
160C2×D181—41no—
161C2×D181—41no—
162C2×D181—88no—
163C4×C181—42no—
164C4×C181#12012yes—
165C4×C181C22×D411yescharacteristic
166C4×C181C22×D411yescharacteristic
167Dic181—22no—
168Dic181—22no—
169Dic181—41no—
170C2×C181—22no—
171C2×C181—44no—
172C2×C181C2×#12011yescharacteristic
173C2×C181#57611yescharacteristic
174C2×C181#57711yescharacteristic
175D181—42no—
176D181—42no—
177D181—81no—
178C181#301211yescharacteristic, complemented, Sylow
179#3012—1811nomaximal, Sylow
180C2×H(5,𝔽2)—1811no—
181C2×AGL(1,ℤ/8)—1811no—
182C2×#135—1811no—
183#537—1811no—
184#452—1812no—
185#450—1812no—
186#449—1812no—
187#447—1812no—
188#417—1811no—
189C2×#102D18111yescharacteristic, complemented
190#408—1811no—
191H(5,𝔽2)—3624no—
192H(5,𝔽2)—3624no—
193C2×#41—3621no—
194C2×#41—1812no—
195C2×#41—1811no—
196C2×#41—1811no—
197C2×#41—1811no—
198C22×D4—3621no—
199C22×D4—3622no—
200C22×D4—1812no—
201C22×D4C2×D18111yescharacteristic
202AGL(1,ℤ/8)—3624no—
203#135—3624no—
204C2×Q16—1811no—
205C2×SD16—1812no—
206C2×SD16—1811no—
207C2×SD16—1811no—
208C2×D8—1812no—
209C2×D8—1811no—
210C2×M16—1811no—
211C22×C8C2×D18111yescharacteristic, complemented
212C22⋊Q8—3621no—
213C22⋊Q8—1812no—
214#121—3621no—
215#121—3622no—
216#121—1812no—
217C2×C4⋊C4C2×D18111yescharacteristic, complemented
218#103—1812no—
219#102—1812no—
220#102C2×D18112yescomplemented
221#102C2×D18112yescomplemented
222C22⋊C8—1812no—
223C22⋊C8—1812no—
224C24—21no—
225C24—3622no—
226#41—3628no—
227#41—3624no—
228#41—3624no—
229#41—3622no—
230#41—3622no—
231#41—3622no—
232#41—3622no—
233#41—3622no—
234#41—1814no—
235C2×Q8—1811no—
236C2×Q8—1811no—
237C2×Q8—1811no—
238C2×D4—21no—
239C2×D4—22no—
240C2×D4—3622no—
241C2×D4—3628no—
242C2×D4—3624no—
243C2×D4—36216no—
244C2×D4—3622no—
245C2×D4—3621no—
246C2×D4—3624no—
247C2×D4—3621no—
248C2×D4—3622no—
249C2×D4—1812no—
250C2×D4—1811no—
251C2×D4—1811no—
252C2×D4—1811no—
253C2×D4D72414yescomplemented
254C2×D4C22×D18112yes—
255C22×C4—21no—
256C22×C4—3622no—
257C22×C4—3621no—
258C22×C4—3621no—
259C22×C4—1812no—
260C22×C4C22×D18111yescharacteristic
261Q16—3622no—
262SD16—3622no—
263SD16—3622no—
264SD16—3624no—
265D8—3622no—
266D8—3624no—
267M16—3622no—
268C2×C8—22no—
269C2×C8—1812no—
270C2×C8C22×D18111yescharacteristic
271C2×C8C22×D18111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—3622no—
274C4⋊C4C22×D18112yes—
275C22⋊C4—3622no—
276C22⋊C4—3622no—
277C23—22no—
278C23—24no—
279C23—81no—
280C23—81no—
281C23—3628no—
282C23—3621no—
283C23—3621no—
284C23—14482no—
285C23—14482no—
286C23—1812no—
287C23D4×D18111yescharacteristic
288Q8—3622no—
289Q8—3621no—
290Q8—3621no—
291Q8—3621no—
292Q8—1812no—
293D4—24no—
294D4—22no—
295D4—3628no—
296D4—3628no—
297D4—3624no—
298D4—3624no—
299D4—3622no—
300D4—3621no—
301D4—3621no—
302D4—3621no—
303D4—14482no—
304D4—1812no—
305D4—7242no—
306D4C2×D72414yes—
307C2×C4—22no—
308C2×C4—81no—
309C2×C4—3628no—
310C2×C4—3621no—
311C2×C4—3624no—
312C2×C4—3622no—
313C2×C4—3622no—
314C2×C4—3621no—
315C2×C4—1812no—
316C2×C4—7242no—
317C2×C4D4×D18111yescharacteristic
318C2×C4D4×D18112yes—
319C2×C4C2×D72412yes—
320C2×C4C23×D18111yescharacteristic
321C8—21no—
322C8—21no—
323C8—3622no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—84no—
330C22—3622no—
331C22—3622no—
332C22—3622no—
333C22—3622no—
334C22—14488no—
335C22—7241no—
336C22—7241no—
337C22—7241no—
338C22—7241no—
339C22—7241no—
340C22#1915378612yes—
341C22C2×D4×D18111yescharacteristic
342C4—42no—
343C4—3622no—
344C4—3622no—
345C4—7241no—
346C4#1915378612yes—
347C4C2×D4×D18111yescharacteristic
348C4C22×D72411yescharacteristic
349C2—22no—
350C2—44no—
351C2—14481no—
352C2—7242no—
353C2—7242no—
354C2C2×#1915378611yescharacteristic
355C2#2604208011yescharacteristic
356C2#2604208511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC2×#102×C181C2
socleC22×C181C22×D4
hypercenterC2×#102D181
hypocenterC181#3012
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#19153861C2semidirect
2. C2×#19153842C2semidirect
3. C2×#19153838C2semidirect
4. #26041860C2semidirect
5. #26041442C2semidirect
6. #26041431C2semidirect
7. #26041425C2semidirect
8. #26041414C2semidirect
9. #26041278C2semidirect
10. #26041245C2semidirect
11. C2×#102×C181C2semidirect
16. C2×#15668449C22semidirect
24. C2×Dic724C22semidirect
25. C2×#15668438C22semidirect
26. C2×#15668438C22semidirect
27. C2×#15668437C22semidirect
28. C2×D1448C22semidirect
29. C2×#15668434C22semidirect
30. C2×#15668433C22semidirect
32. #19153796C22semidirect
34. #19153791C22semidirect
36. #19153739C22semidirect
37. #19153736C22semidirect
38. #19153728C22semidirect
39. #19153728C22semidirect
41. C22×C8×C181C22semidirect
42. C2×C4⋊C4×C181C22semidirect
43. #102×C181C22semidirect
44. #102×C181C22semidirect
51. #15668449D4semidirect
89. C2×D4×C181D4semidirect
178. C181#3012semidirect
189. C2×#102D181semidirect
211. C22×C8C2×D181semidirect
217. C2×C4⋊C4C2×D181semidirect
220. #102C2×D181semidirect
221. #102C2×D181semidirect
253. C2×D4D724semidirect

Maximal subgroups

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

subgroupindex
C2×#191538612
C2×#191538422
C2×#191538382
#260418602
#260414422
#260414312
#260414252
#260414142
#260412782
#260412452
C2×#102×C1812
#3012181