Finite Group Databasebeta

← #37546044

Subgroups — #37546044

Order 23168.

430 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041948C211yesmaximal, characteristic, complemented
2#26041930C211yesmaximal, characteristic, complemented
3#26041624C211yesmaximal, characteristic, complemented
4#26041597C211yesmaximal, characteristic, complemented
5#26041596C211yesmaximal, characteristic, complemented
6#26041562C211yesmaximal, characteristic, complemented
7#26041532C211yesmaximal, characteristic, complemented
8#26041531C211yesmaximal, characteristic, complemented
9#26041473C211yesmaximal, characteristic, complemented
10#26041427C211yesmaximal, characteristic, complemented
11#26041424C211yesmaximal, characteristic, complemented
12#26041364C211yesmaximal, characteristic, complemented
13#26041355C211yesmaximal, characteristic, complemented
14#26041279C211yesmaximal, characteristic, complemented
15#417×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×#15668449—21no—
18C2×#15668436C2211yescharacteristic
19C2×D1448C2211yescharacteristic
20C2×#15668434C2211yescharacteristic
21C2×#15668432C2211yescharacteristic
22#19153815C2211yescharacteristic
23#19153812—21no—
24#19153812C2211yescharacteristic
25#19153801—21no—
26#19153799—21no—
27#19153794—21no—
28#19153791C2211yescharacteristic
29#19153790—21no—
30#19153790C2211yescharacteristic
31#19153790C2211yescharacteristic
32#19153787—21no—
33#19153787C2211yescharacteristic
34Q8×Dic181C2211yescharacteristic
35#19153778C2211yescharacteristic
36C4×C22⋊D181—21no—
37C4×C22⋊D181C2211yescharacteristic
38C4⋊C4×D181C2211yescharacteristic
39#19153748C2211yescharacteristic
40#19153747C2211yescharacteristic
41#19153747C2211yescharacteristic
42#19153745C2211yescharacteristic
43#19153740C2211yescharacteristic
44#19153738C2211yescharacteristic
45#19153738C2211yescharacteristic
46#19153737C2211yescharacteristic
47#19153736C2211yescharacteristic
48#19153736C2211yescharacteristic
49C22⋊#13089678C2211yescharacteristic
50#19153728C2211yescharacteristic
51C8×Dic181C2211yescharacteristic
52C2×#41×C181C2211yescharacteristic
53C2×M16×C181C2211yescharacteristic
54C2×C4⋊C4×C181C2211yescharacteristic
55#103×C181C2211yescharacteristic
56#103×C181C2211yescharacteristic
57#102×C181C2211yescharacteristic
58#102×C181C2211yescharacteristic
59#15668450—22no—
60#15668450—41no—
61#15668449—44no—
62C2×Dic362—21no—
63C2×D724—21no—
64C2×D724—21no—
65C2×D724C2311yescharacteristic
66C2×D724C2311yescharacteristic
67C2×C22⋊D181—21no—
68C2×C22⋊D181—21no—
69C2×C22⋊D181—21no—
70C2×C22⋊D181—21no—
71C2×C22⋊D181—21no—
72C2×C22⋊D181D411yescharacteristic
73C2×C4×D181—21no—
74C2×C4×D181—21no—
75C2×C4×D181—21no—
76C2×C4×D181C2311yescharacteristic
77#15668436—22no—
78D1448—41no—
79#15668434—41no—
80#15668432—41no—
81C2×#13089678C2311yescharacteristic
82C2×#13089678C2311yescharacteristic
83C4⋊Dic181—21no—
84C4⋊Dic181C2311yescharacteristic
85C4⋊Dic181C2311yescharacteristic
86#15668428—21no—
87#15668428—21no—
88#15668428D411yescharacteristic
89C22⋊Dic181—21no—
90C22⋊Dic181D411yescharacteristic
91#15668426—21no—
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×D181—21no—
114C22×D181C2×D411yescharacteristic
115Dic362—42no—
116D724—21no—
117D724—21no—
118D724—21no—
119D724—21no—
120D724—42no—
121D724—42no—
122C22⋊D181—22no—
123C22⋊D181—42no—
124C22⋊D181—42no—
125C22⋊D181—81no—
126C22⋊D181—81no—
127C22⋊D181—81no—
128C2×Dic181—21no—
129C2×Dic181—21no—
130C2×Dic181C2×D411yescharacteristic
131C2×Dic181C2×D411yescharacteristic
132C2×Dic181C2×D411yescharacteristic
133C4×D181—21no—
134C4×D181—21no—
135C4×D181—42no—
136C4×D181—42no—
137C4×D181—42no—
138#13089678—21no—
139#13089678—21no—
140C23×C181—21no—
141C23×C181C2×D411yescharacteristic
142Q8×C181—21no—
143Q8×C181#4112yescomplemented
144D4×C181—21no—
145D4×C181—21no—
146D4×C181—42no—
147C2×C4×C181—21no—
148C2×C4×C181—21no—
149C2×C4×C181—21no—
150C2×C4×C181—21no—
151C2×C4×C181—21no—
152C2×C4×C181—42no—
153C2×C4×C181—42no—
154C2×C4×C181C2×D411yescharacteristic
155C2×C4×C181C2411yescharacteristic
156C8×C181—21no—
157C8×C181—21no—
158C22×C181—21no—
159C22×C181—21no—
160C22×C181—21no—
161C22×C181—41no—
162C22×C181—41no—
163C22×C181—41no—
164C22×C181C22×D411yescharacteristic
165C2×D181—21no—
166C2×D181—21no—
167C2×D181—21no—
168C2×D181—41no—
169C2×D181—41no—
170C2×D181—41no—
171C2×D181—41no—
172C2×D181—41no—
173C2×D181—41no—
174C4×C181—21no—
175C4×C181—22no—
176C4×C181—41no—
177C4×C181—41no—
178C4×C181C2×#4111yescharacteristic
179C4×C181H(5,𝔽2)11yescharacteristic
180Dic181—21no—
181Dic181—21no—
182Dic181—41no—
183Dic181—41no—
184Dic181AGL(1,ℤ/8)12yes—
185C2×C181—41no—
186C2×C181—81no—
187C2×C181#54611yescharacteristic
188C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
189C2×C181#57711yescharacteristic
190D181—41no—
191D181—81no—
192D181—81no—
193C181#333211yescharacteristic, complemented, Sylow
194#3332—1811nomaximal, Sylow
195#550—1811no—
196#547—1811no—
197#486—1811no—
198#481—1811no—
199#481—1811no—
200#475—1811no—
201#469—1811no—
202#469—1811no—
203#459—1811no—
204#449—1811no—
205#449—1811no—
206C4×SD16—1811no—
207#436—1811no—
208#417—1811no—
209#417D18111yescharacteristic, complemented
210C2×#41—3621no—
211C2×#41—1811no—
212C2×#41C2×D18111yescharacteristic
213C2×SD16—1811no—
214C2×D8—1811no—
215C2×D8—1811no—
216C2×M16—1811no—
217C2×M16C2×D18111yescharacteristic
218C4⋊Q8—1811no—
219#127—3621no—
220#127—1811no—
221#123—3621no—
222#123—3621no—
223C22⋊Q8—3621no—
224#121—3621no—
225#121—3621no—
226#121—1811no—
227#121—1811no—
228#121—1811no—
229#121—1811no—
230C4×Q8—1811no—
231C4×D4—3621no—
232C4×D4—1811no—
233C4×D4—1811no—
234C2×C4⋊C4—1811no—
235C2×C4⋊C4C2×D18111yescharacteristic
236#106—1811no—
237#106—1811no—
238#106—1811no—
239C4⋊C8—1811no—
240#103—1811no—
241#103C2×D18111yescharacteristic
242#103C2×D18111yescharacteristic
243#102—1811no—
244#102—1811no—
245#102—1811no—
246#102—1811no—
247#102—1811no—
248#102C2×D18111yescharacteristic
249#102C2×D18111yescharacteristic
250C22⋊C8—1811no—
251C22⋊C8—1811no—
252C4×C8—1811no—
253#41—22no—
254#41—41no—
255#41—3622no—
256#41—7244no—
257#41—7241no—
258C2×Q8—3621no—
259C2×Q8C22×D18111yescharacteristic
260C2×D4—21no—
261C2×D4—3621no—
262C2×D4—3621no—
263C2×D4—3621no—
264C2×D4—3621no—
265C2×D4—3621no—
266C2×D4—3621no—
267C2×D4—3621no—
268C2×D4—1811no—
269C2×D4—1811no—
270C2×D4—1811no—
271C2×D4C22×D18111yescharacteristic
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—3621no—
275C22×C4—3621no—
276C22×C4—3621no—
277C22×C4—1811no—
278C22×C4C22×D18111yescharacteristic
279SD16—3622no—
280D8—7241no—
281D8—7241no—
282M16—41no—
283M16—7241no—
284C2×C8—1811no—
285C2×C8—1811no—
286C2×C8C22×D18111yescharacteristic
287C2×C8C22×D18111yescharacteristic
288C4⋊C4—21no—
289C4⋊C4—3621no—
290C4⋊C4—3621no—
291C4⋊C4—3621no—
292C4⋊C4—1811no—
293C4⋊C4—1811no—
294C4⋊C4—1811no—
295C4⋊C4C22×D18111yescharacteristic
296C4⋊C4C22×D18111yescharacteristic
297C22⋊C4—3621no—
298C22⋊C4—3621no—
299C22⋊C4—3621no—
300C22⋊C4—3621no—
301C22⋊C4—3621no—
302C22⋊C4—1811no—
303C22⋊C4—1811no—
304C42—3621no—
305C42—1811no—
306C23—21no—
307C23—3621no—
308C23—3621no—
309C23—1811no—
310C23D4×D18111yescharacteristic
311Q8—21no—
312Q8—7242no—
313Q8#1566844912yescomplemented
314D4—21no—
315D4—21no—
316D4—42no—
317D4—3622no—
318D4—3621no—
319D4—3621no—
320D4—3621no—
321D4—3621no—
322D4—14481no—
323D4—14481no—
324D4—14481no—
325D4—7242no—
326D4—7242no—
327D4—7242no—
328D4—7242no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—3621no—
337C2×C4—3621no—
338C2×C4—3621no—
339C2×C4—3621no—
340C2×C4—1811no—
341C2×C4—1811no—
342C2×C4—1811no—
343C2×C4—7242no—
344C2×C4—7242no—
345C2×C4—7242no—
346C2×C4D4×D18111yescharacteristic
347C2×C4C23×D18111yescharacteristic
348C8—21no—
349C8—21no—
350C8—3621no—
351C8—3621no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—3621no—
359C22—3621no—
360C22—3621no—
361C22—7241no—
362C22—7241no—
363C22—7241no—
364C22—7241no—
365C22—7241no—
366C22—7241no—
367C22C2×D4×D18111yescharacteristic
368C4—21no—
369C4—22no—
370C4—41no—
371C4—41no—
372C4—3621no—
373C4—3621no—
374C4—1812no—
375C4—7241no—
376C4—7241no—
377C4C2×#1566844911yescharacteristic
378C4#1915386111yescharacteristic
379C2—41no—
380C2—81no—
381C2—14481no—
382C2—14481no—
383C2—7241no—
384C2#2604191911yescharacteristic
385C2AGL(1,ℤ/8)×D18111yescharacteristic
386C2#2604208811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#417×C181C2
socleC22×C181C22×D4
hypercenter#417D181
hypocenterC181#3332
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. #26041948C2semidirect
2. #26041930C2semidirect
3. #26041624C2semidirect
4. #26041597C2semidirect
5. #26041596C2semidirect
6. #26041562C2semidirect
7. #26041532C2semidirect
8. #26041531C2semidirect
9. #26041473C2semidirect
10. #26041427C2semidirect
11. #26041424C2semidirect
12. #26041364C2semidirect
13. #26041355C2semidirect
14. #26041279C2semidirect
15. #417×C181C2semidirect
143. Q8×C181#41semidirect
193. C181#3332semidirect
209. #417D181semidirect
313. Q8#15668449semidirect

Maximal subgroups

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

subgroupindex
#260419482
#260419302
#260416242
#260415972
#260415962
#260415622
#260415322
#260415312
#260414732
#260414272
#260414242
#260413642
#260413552
#260412792
#417×C1812
#3332181