Finite Group Databasebeta

← #37545683

Subgroups — #37545683

Order 23168.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041930C211yesmaximal, characteristic, complemented
2#26041917C211yesmaximal, characteristic, complemented
3#26041640C211yesmaximal, characteristic, complemented
4#26041608C211yesmaximal, characteristic, complemented
5#26041597C211yesmaximal, characteristic, complemented
6#26041567C211yesmaximal, characteristic, complemented
7#26041532C211yesmaximal, characteristic, complemented
8#26041511C211yesmaximal, characteristic, complemented
9#26041497C211yesmaximal, characteristic, complemented
10#26041424C211yesmaximal, characteristic, complemented
11#26041414C211yesmaximal, characteristic, complemented
12#26041388C211yesmaximal, characteristic, complemented
13#26041348C211yesmaximal, characteristic, complemented
14#102×D181C211yesmaximal, characteristic, complemented
15#417×C181C211yesmaximal, characteristic, complemented
16C2×#15668449—21no—
17C2×D4×D181C2211yescharacteristic
18C2×#15668436C2211yescharacteristic, complemented
19C2×D1448C2211yescharacteristic
20C2×#15668434C2211yescharacteristic
21C2×C8×D181C2211yescharacteristic
22#19153815C2211yescharacteristic, complemented
23#19153805C2211yescharacteristic
24#19153801—21no—
25#19153799—21no—
26#19153799—21no—
27#19153794—21no—
28#19153792C2211yescharacteristic
29#19153791C2211yescharacteristic
30#19153790C2211yescharacteristic
31#19153790C2211yescharacteristic, complemented
32#19153788—21no—
33#19153784—21no—
34D4×Dic181C2211yescharacteristic
35#19153778C2211yescharacteristic, complemented
36C4×C22⋊D181C2211yescharacteristic
37C4⋊C4×D181C2211yescharacteristic
38C2×#15668426—21no—
39#19153748C2211yescharacteristic, complemented
40#19153747C2211yescharacteristic, complemented
41#19153747C2211yescharacteristic, complemented
42#19153745C2211yescharacteristic, complemented
43#19153740C2211yescharacteristic, complemented
44#19153738C2211yescharacteristic
45#19153738C2211yescharacteristic, complemented
46#19153737C2211yescharacteristic
47#19153736C2211yescharacteristic
48#19153736C2211yescharacteristic
49C22⋊#13089678C2211yescharacteristic, complemented
50#19153728C2211yescharacteristic
51#19153726C2211yescharacteristic, complemented
52C2×#41×C181C2211yescharacteristic
53C2×M16×C181C2211yescharacteristic, complemented
54C2×C4⋊C4×C181C2211yescharacteristic, complemented
55#103×C181C2211yescharacteristic, complemented
56#103×C181C2211yescharacteristic, complemented
57#102×C181C2211yescharacteristic
58#102×C181C2211yescharacteristic
59C23×D181—21no—
60#15668449—44no—
61C2×Dic362—21no—
62C2×D724—21no—
63C2×D724C2311yescharacteristic
64C2×D724C2311yescharacteristic
65C2×C22⋊D181—21no—
66C2×C22⋊D181—21no—
67C2×C22⋊D181—21no—
68C2×C22⋊D181D411yescharacteristic
69D4×D181—22no—
70D4×D181—41no—
71C22×Dic181—21no—
72C2×C4×D181—21no—
73C2×C4×D181—21no—
74C2×C4×D181C2311yescharacteristic
75#15668436—41no—
76D1448—41no—
77#15668434—22no—
78C2×#13089678C2311yescharacteristic
79C2×#13089678C2311yescharacteristic
80C8×D181—41no—
81C4⋊Dic181C2311yescharacteristic
82C4⋊Dic181C2311yescharacteristic
83#15668428—21no—
84#15668428—21no—
85#15668428—21no—
86#15668428D411yescharacteristic
87C22⋊Dic181—21no—
88C22⋊Dic181D411yescharacteristic
89#15668426—21no—
90#15668426—21no—
91#15668426—21no—
92#15668426—21no—
93#15668426—21no—
94#15668426—21no—
95#15668426—21no—
96#15668426D411yescharacteristic
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×D181—42no—
115C22×D181—42no—
116C22×D181—81no—
117C22×D181C2×D411yescharacteristic
118Dic362—42no—
119D724—21no—
120D724—21no—
121D724—21no—
122D724—21no—
123D724—42no—
124C22⋊D181—22no—
125C22⋊D181—42no—
126C22⋊D181—42no—
127C22⋊D181—81no—
128C2×Dic181—21no—
129C2×Dic181—21no—
130C2×Dic181—81no—
131C2×Dic181C2×D411yescharacteristic
132C2×Dic181C2×D411yescharacteristic
133C2×Dic181C2×D411yescharacteristic
134C4×D181—21no—
135C4×D181—21no—
136C4×D181—42no—
137C4×D181—42no—
138#13089678—21no—
139#13089678—21no—
140C23×C181—21no—
141C23×C181C2×D411yescharacteristic
142Q8×C181—21no—
143Q8×C181—21no—
144D4×C181—21no—
145D4×C181—42no—
146D4×C181#4112yescomplemented
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—22no—
160C22×C181—21no—
161C22×C181—21no—
162C22×C181—41no—
163C22×C181—41no—
164C22×C181C22×D411yescharacteristic
165C2×D181—21no—
166C2×D181—21no—
167C2×D181—41no—
168C2×D181—41no—
169C2×D181—41no—
170C2×D181—41no—
171C2×D181—41no—
172C2×D181—41no—
173C2×D181—84no—
174C2×D181AGL(1,ℤ/8)12yes—
175C4×C181—21no—
176C4×C181—41no—
177C4×C181—41no—
178C4×C181—41no—
179C4×C181C2×#4111yescharacteristic
180C4×C181H(5,𝔽2)11yescharacteristic
181Dic181—21no—
182Dic181—21no—
183Dic181—21no—
184Dic181—41no—
185Dic181—41no—
186C2×C181—42no—
187C2×C181—41no—
188C2×C181#54611yescharacteristic
189C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C181#57711yescharacteristic
191D181—22no—
192D181—81no—
193D181—81no—
194C181#330611yescharacteristic, complemented, Sylow
195#3306—1811nomaximal, Sylow
196#547—1811no—
197#546—1811no—
198#489—1811no—
199#483—1811no—
200#481—1811no—
201#476—1811no—
202#469—1811no—
203#465—1811no—
204#463—1811no—
205#449—1811no—
206#447—1811no—
207#442—1811no—
208#435—1811no—
209#417D18111yescharacteristic, complemented
210C2×#102—1811no—
211C2×#41—3621no—
212C2×#41C2×D18111yescharacteristic
213C22×D4—1811no—
214C2×SD16—1811no—
215C2×D8—1811no—
216C2×D8—1811no—
217C2×M16C2×D18111yescharacteristic, complemented
218C22×C8—1811no—
219C4⋊Q8—1811no—
220#124—1811no—
221#123—3621no—
222#123—3621no—
223#123—3621no—
224C22⋊Q8—3621no—
225C22⋊Q8—1811no—
226#121—3621no—
227#121—1811no—
228#121—1811no—
229#121—1811no—
230#120—3621no—
231C4×D4—1811no—
232C4×D4—1811no—
233C4×D4—1811no—
234C2×C4⋊C4—1811no—
235C2×C4⋊C4C2×D18111yescharacteristic, complemented
236C2×C22⋊C4—3621no—
237#106—1811no—
238#106—1811no—
239#106—1811no—
240C4⋊C8—1811no—
241#103—1811no—
242#103C2×D18111yescharacteristic, complemented
243#103C2×D18111yescharacteristic, complemented
244#102—1811no—
245#102—1811no—
246#102—1811no—
247#102—1811no—
248#102—1811no—
249#102C2×D18111yescharacteristic
250#102C2×D18111yescharacteristic
251C22⋊C8—1811no—
252C22⋊C8—1811no—
253#97—1811no—
254C24—3621no—
255#41—22no—
256#41—41no—
257#41—7244no—
258C2×Q8—3621no—
259C2×Q8C22×D18111yescharacteristic
260C2×D4—21no—
261C2×D4—3621no—
262C2×D4—3622no—
263C2×D4—3621no—
264C2×D4—3621no—
265C2×D4—3621no—
266C2×D4—1811no—
267C2×D4—1811no—
268C2×D4—1811no—
269C2×D4—7241no—
270C2×D4C22×D18111yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—3621no—
274C22×C4—3621no—
275C22×C4—3621no—
276C22×C4—1811no—
277C22×C4C22×D18111yescharacteristic
278SD16—7241no—
279D8—3622no—
280D8—7241no—
281M16—41no—
282C2×C8—1811no—
283C2×C8—1811no—
284C2×C8—7241no—
285C2×C8C22×D18111yescharacteristic
286C2×C8C22×D18111yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—3621no—
289C4⋊C4—3621no—
290C4⋊C4—3621no—
291C4⋊C4—1811no—
292C4⋊C4—1811no—
293C4⋊C4—1811no—
294C4⋊C4C22×D18111yescharacteristic
295C4⋊C4C22×D18111yescharacteristic
296C22⋊C4—3621no—
297C22⋊C4—3621no—
298C22⋊C4—3621no—
299C22⋊C4—3621no—
300C22⋊C4—3621no—
301C22⋊C4—3621no—
302C22⋊C4—3621no—
303C22⋊C4—3621no—
304C22⋊C4—1811no—
305C22⋊C4—1811no—
306C42—1811no—
307C23—21no—
308C23—3621no—
309C23—3621no—
310C23—14481no—
311C23—1811no—
312C23—7242no—
313C23—7242no—
314C23D4×D18111yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—7242no—
318D4—21no—
319D4—42no—
320D4—3622no—
321D4—3621no—
322D4—3621no—
323D4—3621no—
324D4—3621no—
325D4—14481no—
326D4—7242no—
327D4—7242no—
328D4—7242no—
329D4#1566844912yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—3621no—
338C2×C4—3621no—
339C2×C4—3621no—
340C2×C4—3621no—
341C2×C4—14481no—
342C2×C4—1811no—
343C2×C4—1811no—
344C2×C4—1811no—
345C2×C4—7242no—
346C2×C4—7242no—
347C2×C4D4×D18111yescharacteristic
348C2×C4C23×D18111yescharacteristic
349C8—21no—
350C8—21no—
351C8—3621no—
352C8—3621no—
353C22—21no—
354C22—22no—
355C22—21no—
356C22—21no—
357C22—41no—
358C22—41no—
359C22—3621no—
360C22—3621no—
361C22—14484no—
362C22—1812no—
363C22—7241no—
364C22—7241no—
365C22—7241no—
366C22—7241no—
367C22—7241no—
368C22—7241no—
369C22C2×D4×D18111yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—3621no—
375C4—3621no—
376C4—3621no—
377C4—7241no—
378C4—7241no—
379C4C2×#1566844911yescharacteristic
380C4#1915386111yescharacteristic
381C2—41no—
382C2—42no—
383C2—3622no—
384C2—14481no—
385C2—14481no—
386C2#2604191911yescharacteristic
387C2AGL(1,ℤ/8)×D18111yescharacteristic
388C2#2604208811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#417×C181C2
socleC22×C181C22×D4
hypercenter#417D181
hypocenterC181#3306
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. #26041930C2semidirect
2. #26041917C2semidirect
3. #26041640C2semidirect
4. #26041608C2semidirect
5. #26041597C2semidirect
6. #26041567C2semidirect
7. #26041532C2semidirect
8. #26041511C2semidirect
9. #26041497C2semidirect
10. #26041424C2semidirect
11. #26041414C2semidirect
12. #26041388C2semidirect
13. #26041348C2semidirect
14. #102×D181C2semidirect
15. #417×C181C2semidirect
18. C2×#15668436C22semidirect
22. #19153815C22semidirect
31. #19153790C22semidirect
35. #19153778C22semidirect
39. #19153748C22semidirect
40. #19153747C22semidirect
41. #19153747C22semidirect
42. #19153745C22semidirect
43. #19153740C22semidirect
45. #19153738C22semidirect
49. C22⋊#13089678C22semidirect
51. #19153726C22semidirect
53. C2×M16×C181C22semidirect
54. C2×C4⋊C4×C181C22semidirect
55. #103×C181C22semidirect
56. #103×C181C22semidirect
146. D4×C181#41semidirect
194. C181#3306semidirect
209. #417D181semidirect
217. C2×M16C2×D181semidirect
235. C2×C4⋊C4C2×D181semidirect
242. #103C2×D181semidirect
243. #103C2×D181semidirect
329. D4#15668449semidirect

Maximal subgroups

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

subgroupindex
#260419302
#260419172
#260416402
#260416082
#260415972
#260415672
#260415322
#260415112
#260414972
#260414242
#260414142
#260413882
#260413482
#102×D1812
#417×C1812
#3306181