Finite Group Databasebeta

← #37545713

Subgroups — #37545713

Order 23168.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041936C211yesmaximal, characteristic, complemented
2#26041930C211yesmaximal, characteristic, complemented
3#26041637C211yesmaximal, characteristic, complemented
4#26041614C211yesmaximal, characteristic, complemented
5#26041591C211yesmaximal, characteristic, complemented
6#26041572C211yesmaximal, characteristic, complemented
7#26041525C211yesmaximal, characteristic, complemented
8#26041514C211yesmaximal, characteristic, complemented
9#26041493C211yesmaximal, characteristic, complemented
10#26041424C211yesmaximal, characteristic, complemented
11#26041419C211yesmaximal, characteristic, complemented
12#26041382C211yesmaximal, characteristic, complemented
13#26041348C211yesmaximal, characteristic, complemented
14#103×D181C211yesmaximal, characteristic, complemented
15#417×C181C211yesmaximal, characteristic, complemented
16C2×#15668449—21no—
17C2×Q8×D181C2211yescharacteristic
18C2×#15668438C2211yescharacteristic
19C2×#15668436C2211yescharacteristic
20C2×#15668434C2211yescharacteristic
21C2×C8×D181C2211yescharacteristic
22#19153815C2211yescharacteristic
23#19153803—21no—
24#19153803C2211yescharacteristic
25#19153801—21no—
26#19153799—21no—
27#19153795—21no—
28#19153795C2211yescharacteristic
29#19153794—21no—
30#19153791C2211yescharacteristic
31#19153790C2211yescharacteristic
32#19153787—21no—
33#19153787C2211yescharacteristic
34Q8×Dic181C2211yescharacteristic
35#19153778C2211yescharacteristic
36C4×C22⋊D181—21no—
37C4×C22⋊D181C2211yescharacteristic
38C4⋊C4×D181C2211yescharacteristic
39#19153750C2211yescharacteristic
40#19153749C2211yescharacteristic
41#19153749C2211yescharacteristic
42#19153745C2211yescharacteristic
43#19153741C2211yescharacteristic
44#19153740C2211yescharacteristic
45#19153739C2211yescharacteristic
46#19153739C2211yescharacteristic
47#19153738C2211yescharacteristic
48#19153737C2211yescharacteristic
49C22⋊#13089678C2211yescharacteristic
50#19153728C2211yescharacteristic
51#19153726C2211yescharacteristic
52C2×#41×C181C2211yescharacteristic
53C2×M16×C181C2211yescharacteristic
54C2×C4⋊C4×C181C2211yescharacteristic
55#103×C181C2211yescharacteristic
56#103×C181C2211yescharacteristic
57#102×C181C2211yescharacteristic
58#102×C181C2211yescharacteristic
59#15668449—44no—
60C2×Dic362—21no—
61C2×Dic362—21no—
62C2×Dic362C2311yescharacteristic
63Q8×D181—22no—
64Q8×D181—41no—
65C2×D724—21no—
66C2×D724C2311yescharacteristic
67C2×C22⋊D181—21no—
68C2×C22⋊D181—21no—
69C2×C22⋊D181D411yescharacteristic
70C2×C4×D181—21no—
71C2×C4×D181—21no—
72C2×C4×D181—21no—
73C2×C4×D181C2311yescharacteristic
74#15668438—41no—
75#15668436—22no—
76#15668434—41no—
77C2×#13089678C2311yescharacteristic
78C2×#13089678C2311yescharacteristic
79C8×D181—41no—
80C4⋊Dic181—21no—
81C4⋊Dic181C2311yescharacteristic
82C4⋊Dic181C2311yescharacteristic
83#15668428—21no—
84#15668428—21no—
85#15668428—21no—
86#15668428D411yescharacteristic
87C22⋊Dic181—21no—
88C22⋊Dic181—21no—
89C22⋊Dic181—21no—
90C22⋊Dic181—21no—
91C22⋊Dic181D411yescharacteristic
92#15668426—21no—
93#15668426—21no—
94#15668426—21no—
95#15668426D411yescharacteristic
96C4×Dic181—21no—
97C4×Dic181C2311yescharacteristic
98#41×C181—22no—
99#41×C181—41no—
100C2×Q8×C181C2311yescharacteristic
101C2×D4×C181—21no—
102C2×D4×C181C2311yescharacteristic
103C22×C4×C181—21no—
104C22×C4×C181—21no—
105C22×C4×C181C2311yescharacteristic
106M16×C181—41no—
107C2×C8×C181C2311yescharacteristic
108C2×C8×C181C2311yescharacteristic
109C4⋊C4×C181—21no—
110C4⋊C4×C181C2311yescharacteristic
111C4⋊C4×C181C2311yescharacteristic
112C22×D181—21no—
113C22×D181C2×D411yescharacteristic
114Dic362—21no—
115Dic362—21no—
116Dic362—42no—
117Dic362—42no—
118D724—21no—
119D724—21no—
120D724—42no—
121C22⋊D181—22no—
122C22⋊D181—42no—
123C22⋊D181—42no—
124C2×Dic181—21no—
125C2×Dic181—21no—
126C2×Dic181—21no—
127C2×Dic181C2×D411yescharacteristic
128C2×Dic181C2×D411yescharacteristic
129C2×Dic181C2×D411yescharacteristic
130C4×D181—21no—
131C4×D181—21no—
132C4×D181—42no—
133C4×D181—42no—
134C4×D181—42no—
135#13089678—21no—
136#13089678—21no—
137C23×C181—21no—
138C23×C181C2×D411yescharacteristic
139Q8×C181—21no—
140Q8×C181#4112yescomplemented
141D4×C181—21no—
142D4×C181—21no—
143D4×C181—42no—
144C2×C4×C181—21no—
145C2×C4×C181—21no—
146C2×C4×C181—21no—
147C2×C4×C181—21no—
148C2×C4×C181—21no—
149C2×C4×C181—42no—
150C2×C4×C181—42no—
151C2×C4×C181C2×D411yescharacteristic
152C2×C4×C181C2411yescharacteristic
153C8×C181—21no—
154C8×C181—21no—
155C22×C181—21no—
156C22×C181—21no—
157C22×C181—21no—
158C22×C181—41no—
159C22×C181—41no—
160C22×C181—41no—
161C22×C181C22×D411yescharacteristic
162C2×D181—21no—
163C2×D181—21no—
164C2×D181—41no—
165C2×D181—41no—
166C2×D181—41no—
167C2×D181AGL(1,ℤ/8)12yes—
168C4×C181—21no—
169C4×C181—22no—
170C4×C181—41no—
171C4×C181—41no—
172C4×C181C2×#4111yescharacteristic
173C4×C181H(5,𝔽2)11yescharacteristic
174Dic181—21no—
175Dic181—21no—
176Dic181—21no—
177Dic181—41no—
178Dic181—41no—
179Dic181—41no—
180C2×C181—41no—
181C2×C181—81no—
182C2×C181#54611yescharacteristic
183C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
184C2×C181#57811yescharacteristic
185D181—22no—
186D181—81no—
187C181#330811yescharacteristic, complemented, Sylow
188#3308—1811nomaximal, Sylow
189#548—1811no—
190#547—1811no—
191#489—1811no—
192#484—1811no—
193#480—1811no—
194#477—1811no—
195#468—1811no—
196#466—1811no—
197#463—1811no—
198#449—1811no—
199#448—1811no—
200#440—1811no—
201#435—1811no—
202#417D18111yescharacteristic, complemented
203C2×#103—1811no—
204C2×#41—3621no—
205C2×#41C2×D18111yescharacteristic
206C22×Q8—1811no—
207C2×SD16—1811no—
208C2×SD16—1811no—
209C2×D8—1811no—
210C2×M16C2×D18111yescharacteristic
211C22×C8—1811no—
212C4⋊Q8—1811no—
213#124—3621no—
214#124—1811no—
215#123—3621no—
216#123—3621no—
217C22⋊Q8—3621no—
218C22⋊Q8—3621no—
219C22⋊Q8—1811no—
220#121—3621no—
221#121—1811no—
222#121—1811no—
223#121—1811no—
224C4×Q8—1811no—
225C4×D4—3621no—
226C4×D4—1811no—
227C4×D4—1811no—
228C2×C4⋊C4—1811no—
229C2×C4⋊C4C2×D18111yescharacteristic
230#107—1811no—
231#107—1811no—
232#107—1811no—
233C4⋊C8—1811no—
234#103—1811no—
235#103—1811no—
236#103—1811no—
237#103—1811no—
238#103C2×D18111yescharacteristic
239#103C2×D18111yescharacteristic
240#102—1811no—
241#102—1811no—
242#102C2×D18111yescharacteristic
243#102C2×D18111yescharacteristic
244C22⋊C8—1811no—
245C22⋊C8—1811no—
246#97—1811no—
247#41—22no—
248#41—41no—
249#41—7244no—
250C2×Q8—3621no—
251C2×Q8—3621no—
252C2×Q8—3622no—
253C2×Q8—1811no—
254C2×Q8—7241no—
255C2×Q8C22×D18111yescharacteristic
256C2×D4—21no—
257C2×D4—3621no—
258C2×D4—3621no—
259C2×D4—3621no—
260C2×D4—1811no—
261C2×D4—1811no—
262C2×D4C22×D18111yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—3621no—
266C22×C4—3621no—
267C22×C4—3621no—
268C22×C4—1811no—
269C22×C4C22×D18111yescharacteristic
270SD16—3622no—
271SD16—7241no—
272D8—7241no—
273M16—41no—
274C2×C8—1811no—
275C2×C8—1811no—
276C2×C8—7241no—
277C2×C8C22×D18111yescharacteristic
278C2×C8C22×D18111yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—3621no—
281C4⋊C4—3621no—
282C4⋊C4—3621no—
283C4⋊C4—3621no—
284C4⋊C4—1811no—
285C4⋊C4—1811no—
286C4⋊C4—1811no—
287C4⋊C4C22×D18111yescharacteristic
288C4⋊C4C22×D18111yescharacteristic
289C22⋊C4—3621no—
290C22⋊C4—3621no—
291C22⋊C4—3621no—
292C22⋊C4—3621no—
293C22⋊C4—3621no—
294C22⋊C4—3621no—
295C22⋊C4—3621no—
296C22⋊C4—1811no—
297C22⋊C4—1811no—
298C42—3621no—
299C42—1811no—
300C23—21no—
301C23—3621no—
302C23—1811no—
303C23D4×D18111yescharacteristic
304Q8—21no—
305Q8—3621no—
306Q8—3621no—
307Q8—7242no—
308Q8—7242no—
309Q8#1566844912yescomplemented
310D4—21no—
311D4—21no—
312D4—42no—
313D4—3622no—
314D4—3621no—
315D4—3621no—
316D4—7242no—
317D4—7242no—
318D4—7242no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—3621no—
327C2×C4—3621no—
328C2×C4—3621no—
329C2×C4—3621no—
330C2×C4—3621no—
331C2×C4—1811no—
332C2×C4—1811no—
333C2×C4—1811no—
334C2×C4—7242no—
335C2×C4—7242no—
336C2×C4—7242no—
337C2×C4D4×D18111yescharacteristic
338C2×C4C23×D18111yescharacteristic
339C8—21no—
340C8—21no—
341C8—3621no—
342C8—3621no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—3621no—
350C22—3621no—
351C22—1812no—
352C22—7241no—
353C22—7241no—
354C22—7241no—
355C22C2×D4×D18111yescharacteristic
356C4—21no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—3621no—
361C4—3621no—
362C4—3621no—
363C4—7241no—
364C4—7241no—
365C4—7241no—
366C4C2×#1566844911yescharacteristic
367C4#1915386111yescharacteristic
368C2—41no—
369C2—81no—
370C2—3622no—
371C2—14481no—
372C2#2604191911yescharacteristic
373C2AGL(1,ℤ/8)×D18111yescharacteristic
374C2#2604209511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#417×C181C2
socleC22×C181C22×D4
hypercenter#417D181
hypocenterC181#3308
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #26041936C2semidirect
2. #26041930C2semidirect
3. #26041637C2semidirect
4. #26041614C2semidirect
5. #26041591C2semidirect
6. #26041572C2semidirect
7. #26041525C2semidirect
8. #26041514C2semidirect
9. #26041493C2semidirect
10. #26041424C2semidirect
11. #26041419C2semidirect
12. #26041382C2semidirect
13. #26041348C2semidirect
14. #103×D181C2semidirect
15. #417×C181C2semidirect
140. Q8×C181#41semidirect
187. C181#3308semidirect
202. #417D181semidirect
309. Q8#15668449semidirect

Maximal subgroups

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

subgroupindex
#260419362
#260419302
#260416372
#260416142
#260415912
#260415722
#260415252
#260415142
#260414932
#260414242
#260414192
#260413822
#260413482
#103×D1812
#417×C1812
#3308181