Finite Group Databasebeta

← #37536865

Subgroups — #37536865

Order 23168.

512 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153799C211yesmaximal, characteristic
2#26041179C211yesmaximal, characteristic
3#26041176C211yesmaximal, characteristic, complemented
4#26041174C211yesmaximal, characteristic, complemented
5#26041173C211yesmaximal, characteristic, complemented
6#26041171C211yesmaximal, characteristic, complemented
7#26041169C211yesmaximal, characteristic, complemented
8#26041167C211yesmaximal, characteristic, complemented
9#26041166C211yesmaximal, characteristic, complemented
10#26041160C211yesmaximal, characteristic, complemented
11#26041155C211yesmaximal, characteristic
12#26041153C211yesmaximal, characteristic, complemented
13#26041150C211yesmaximal, characteristic, complemented
14#26041139C211yesmaximal, characteristic, complemented
15#380×C181C211yesmaximal, characteristic, complemented
16C22×C22⋊D181C2211yescharacteristic
17C22×C4×D181C2211yescharacteristic
18#19153799C418yescomplemented
19C2×C4⋊Dic181C2211yescharacteristic
20C2×C4⋊Dic181C2211yescharacteristic
21C2×#15668428C2211yescharacteristic
22C2×#15668428C2211yescharacteristic
23C2×#15668428C2211yescharacteristic
24C2×#15668428C2211yescharacteristic
25C2×#15668428C2211yescharacteristic
26C2×C22⋊Dic181C2211yescharacteristic
27C2×C22⋊Dic181C2211yescharacteristic
28C2×C22⋊Dic181C2211yescharacteristic
29C2×#15668426C2211yescharacteristic
30C2×#15668426C2211yescharacteristic
31C2×#15668426C2211yescharacteristic
32C2×#15668426C2211yescharacteristic
33C2×#15668426C2211yescharacteristic
34C2×C4×Dic181C2211yescharacteristic
35C2×C4×Dic181C2211yescharacteristic
36C2×C4×Dic181C2211yescharacteristic
37#19153723C2211yescharacteristic
38#19153723C2211yescharacteristic
39#19153723C2211yescharacteristic
40#19153723C2211yescharacteristic
41#19153723C2211yescharacteristic
42#19153723C2211yescharacteristic
43#19153723C2211yescharacteristic
44#19153723C2211yescharacteristic
45C23×C4×C181C2211yescharacteristic
46C2×C4⋊C4×C181C2211yescharacteristic
47C2×C4⋊C4×C181C2211yescharacteristic
48C2×C22⋊C4×C181C2211yescharacteristic
49C2×C22⋊C4×C181C2211yescharacteristic
50#95×C181C2211yescharacteristic
51#95×C181C2211yescharacteristic
52C23×D181C2311yescharacteristic
53C2×C22⋊D181—41no—
54C2×C22⋊D181—41no—
55C2×C22⋊D181C2×C414yes—
56C22×Dic181C2311yescharacteristic
57C22×Dic181C2311yescharacteristic
58C22×Dic181C2311yescharacteristic
59C22×Dic181C2311yescharacteristic
60C22×Dic181C2311yescharacteristic
61C22×Dic181C2311yescharacteristic
62C22×Dic181C2311yescharacteristic
63C2×C4×D181—41no—
64C2×C4×D181—41no—
65C2×C4×D181—41no—
66C4⋊Dic181—41no—
67C4⋊Dic181—41no—
68#15668428—41no—
69#15668428—41no—
70#15668428—41no—
71#15668428C2×C414yescomplemented
72#15668428C2×C414yescomplemented
73C22⋊Dic181—41no—
74C22⋊Dic181—41no—
75C22⋊Dic181C2×C414yescomplemented
76#15668426—41no—
77#15668426—41no—
78#15668426—41no—
79#15668426C2×C414yes—
80#15668426C2×C414yes—
81C4×Dic181—41no—
82C4×Dic181—41no—
83C4×Dic181—41no—
84C24×C181C2311yescharacteristic
85C22×C4×C181—41no—
86C22×C4×C181—41no—
87C22×C4×C181C2×C414yescomplemented
88C22×C4×C181C2311yescharacteristic
89C22×C4×C181C2311yescharacteristic
90C22×C4×C181C2311yescharacteristic
91C22×C4×C181C2311yescharacteristic
92C22×C4×C181C2311yescharacteristic
93C22×C4×C181C2311yescharacteristic
94C4⋊C4×C181—41no—
95C4⋊C4×C181—41no—
96C22⋊C4×C181—41no—
97C22⋊C4×C181—41no—
98C22×D181—21no—
99C22×D181—21no—
100C22×D181—21no—
101C22×D181—21no—
102C22×D181—21no—
103C22×D181—21no—
104C22×D181C22×C412yes—
105C22⋊D181—44no—
106C2×Dic181—21no—
107C2×Dic181—21no—
108C2×Dic181—21no—
109C2×Dic181—21no—
110C2×Dic181—21no—
111C2×Dic181—21no—
112C2×Dic181—21no—
113C2×Dic181—21no—
114C2×Dic181—21no—
115C2×Dic181—21no—
116C2×Dic181—21no—
117C2×Dic181—21no—
118C2×Dic181—21no—
119C2×Dic181—21no—
120C2×Dic181—21no—
121C2×Dic181—21no—
122C2×Dic181—21no—
123C2×Dic181—21no—
124C2×Dic181C22×C412yes—
125C2×Dic181C22×C412yes—
126C2×Dic181C22×C412yes—
127C4×D181—44no—
128C23×C181—21no—
129C23×C181—21no—
130C23×C181—21no—
131C23×C181—21no—
132C23×C181—21no—
133C23×C181—21no—
134C23×C181C22×C412yes—
135C23×C181C2411yescharacteristic
136C2×C4×C181—21no—
137C2×C4×C181—21no—
138C2×C4×C181—21no—
139C2×C4×C181—21no—
140C2×C4×C181—21no—
141C2×C4×C181—21no—
142C2×C4×C181—21no—
143C2×C4×C181—21no—
144C2×C4×C181—21no—
145C2×C4×C181—21no—
146C2×C4×C181—21no—
147C2×C4×C181—21no—
148C2×C4×C181—21no—
149C2×C4×C181—21no—
150C2×C4×C181—21no—
151C2×C4×C181—21no—
152C2×C4×C181—44no—
153C2×C4×C181C22×C412yes—
154C2×C4×C181C22×C412yes—
155C22×C181—22no—
156C22×C181—22no—
157C22×C181—22no—
158C22×C181—41no—
159C22×C181—41no—
160C22×C181—41no—
161C22×C181—41no—
162C22×C181C23×C411yescharacteristic
163C22×C181H(5,𝔽2)11yescharacteristic
164C22×C181H(5,𝔽2)11yescharacteristic
165C22×C181H(5,𝔽2)11yescharacteristic
166C22×C181H(5,𝔽2)11yescharacteristic
167C22×C1812−1+411yescharacteristic
168C22×C1812−1+411yescharacteristic
169C2×D181—22no—
170C2×D181—22no—
171C2×D181—22no—
172C2×D181—41no—
173C2×D181—41no—
174C2×D181—41no—
175C2×D181—41no—
176C4×C181—22no—
177C4×C181—22no—
178C4×C181—41no—
179C4×C181—41no—
180C4×C181—41no—
181C4×C181—41no—
182Dic181—22no—
183Dic181—22no—
184Dic181—22no—
185Dic181—41no—
186Dic181—41no—
187Dic181—41no—
188Dic181—41no—
189C2×C181—42no—
190C2×C181#51811yescharacteristic
191C2×C181#52011yescharacteristic
192C2×C181#52011yescharacteristic
193C2×C181#56011yescharacteristic
194C2×C181#56211yescharacteristic
195C2×C181#56211yescharacteristic
196C2×C181#56311yescharacteristic
197D181—42no—
198C181#237611yescharacteristic, complemented, Sylow
199#2376—1811nomaximal, Sylow
200C2×#123—1811no—
201#390—1811no—
202#389—1811no—
203#389—1811no—
204#388—1811no—
205#388—1811no—
206#388—1811no—
207#388—1811no—
208#387—1811no—
209#387—1811no—
210#386—1811no—
211#385—1811no—
212#385—1811no—
213#380—1811no—
214#380D18111yescharacteristic, complemented
215C22×D4—1811no—
216C23×C4—1811no—
217C23×C4C2×D18111yescharacteristic
218#123—1818no—
219C2×C4⋊C4—1811no—
220C2×C4⋊C4—1811no—
221C2×C4⋊C4—1811no—
222C2×C4⋊C4—1811no—
223C2×C4⋊C4—1811no—
224C2×C4⋊C4—1811no—
225C2×C4⋊C4—1811no—
226C2×C4⋊C4C2×D18111yescharacteristic
227C2×C4⋊C4C2×D18111yescharacteristic
228C2×C22⋊C4—1811no—
229C2×C22⋊C4—1811no—
230C2×C22⋊C4—1811no—
231C2×C22⋊C4—1811no—
232C2×C22⋊C4—1811no—
233C2×C22⋊C4—1811no—
234C2×C22⋊C4—1811no—
235C2×C22⋊C4—1811no—
236C2×C22⋊C4C2×D18111yescharacteristic
237C2×C22⋊C4C2×D18111yescharacteristic
238C2×C42—1811no—
239C2×C42—1811no—
240C2×C42—1811no—
241#95—1811no—
242#95—1811no—
243#95—1811no—
244#95—1811no—
245#95—1811no—
246#95—1811no—
247#95—1811no—
248#95—1811no—
249#95C2×D18111yescharacteristic
250#95C2×D18111yescharacteristic
251C24—1811no—
252C24C22×D18111yescharacteristic
253C2×D4—1814no—
254C2×D4—7241no—
255C2×D4—7241no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—1811no—
259C22×C4—1811no—
260C22×C4—1811no—
261C22×C4—1811no—
262C22×C4—1811no—
263C22×C4—1811no—
264C22×C4—1811no—
265C22×C4—7241no—
266C22×C4—7241no—
267C22×C4—7241no—
268C22×C4C4×D18114yescomplemented
269C22×C4C22×D18111yescharacteristic
270C22×C4C22×D18111yescharacteristic
271C22×C4C22×D18111yescharacteristic
272C22×C4C22×D18111yescharacteristic
273C22×C4C22×D18111yescharacteristic
274C22×C4C22×D18111yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—1814no—
278C4⋊C4—1814no—
279C4⋊C4—7241no—
280C4⋊C4—7241no—
281C4⋊C4—7241no—
282C4⋊C4—7241no—
283C4⋊C4—7241no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—1814no—
287C22⋊C4—1814no—
288C22⋊C4—1814no—
289C22⋊C4—7241no—
290C22⋊C4—7241no—
291C22⋊C4—7241no—
292C22⋊C4—7241no—
293C22⋊C4—7241no—
294C42—7241no—
295C42—7241no—
296C42—7241no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—3621no—
304C23—3621no—
305C23—3621no—
306C23—3621no—
307C23—3621no—
308C23—3621no—
309C23—1812no—
310C23C2×C4×D18112yes—
311C23C23×D18111yescharacteristic
312D4—7244no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—44no—
330C2×C4—3621no—
331C2×C4—3621no—
332C2×C4—3621no—
333C2×C4—3621no—
334C2×C4—3621no—
335C2×C4—3621no—
336C2×C4—3621no—
337C2×C4—3621no—
338C2×C4—3621no—
339C2×C4—3621no—
340C2×C4—3621no—
341C2×C4—3621no—
342C2×C4—3621no—
343C2×C4—3621no—
344C2×C4—3621no—
345C2×C4—3621no—
346C2×C4—3621no—
347C2×C4—3621no—
348C2×C4—1812no—
349C2×C4—1812no—
350C2×C4—1812no—
351C2×C4—7244no—
352C2×C4C2×C4×D18112yes—
353C2×C4C2×C4×D18112yes—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—3622no—
362C22—3622no—
363C22—3622no—
364C22—7241no—
365C22—7241no—
366C22—7241no—
367C22—7241no—
368C22C22×C4×D18111yescharacteristic
369C22#1915386111yescharacteristic
370C22#1915386111yescharacteristic
371C22#1915386111yescharacteristic
372C22#1915386211yescharacteristic
373C22#1915386311yescharacteristic
374C22#1915386411yescharacteristic
375C4—22no—
376C4—22no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—3622no—
382C4—3622no—
383C4—3622no—
384C4—7241no—
385C4—7241no—
386C4—7241no—
387C4—7241no—
388C2—42no—
389C2—7242no—
390C2#2604175211yescharacteristic
391C2#2604175911yescharacteristic
392C2#2604176011yescharacteristic
393C2#2604201711yescharacteristic
394C2#2604202411yescharacteristic
395C2#2604202511yescharacteristic
396C2#2604203011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D181
commutatorC22×C181C23×C4
Fitting#380×C181C2
socle, cosocleC23×C181C24
prime radicalC22×C22⋊D181C22
hypercenter#380D181
hypocenterC181#2376
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #26041176C2semidirect
4. #26041174C2semidirect
5. #26041173C2semidirect
6. #26041171C2semidirect
7. #26041169C2semidirect
8. #26041167C2semidirect
9. #26041166C2semidirect
10. #26041160C2semidirect
12. #26041153C2semidirect
13. #26041150C2semidirect
14. #26041139C2semidirect
15. #380×C181C2semidirect
18. #19153799C4semidirect
71. #15668428C2×C4semidirect
72. #15668428C2×C4semidirect
75. C22⋊Dic181C2×C4semidirect
87. C22×C4×C181C2×C4semidirect
198. C181#2376semidirect
214. #380D181semidirect
268. C22×C4C4×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191537992
#260411792
#260411762
#260411742
#260411732
#260411712
#260411692
#260411672
#260411662
#260411602
#260411552
#260411532
#260411502
#260411392
#380×C1812
#2376181