Finite Group Databasebeta

← #37545569

Subgroups — #37545569

Order 23168.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041977C211yesmaximal, characteristic, complemented
2#26041920C211yesmaximal, characteristic, complemented
3#26041635C211yesmaximal, characteristic, complemented
4#26041606C211yesmaximal, characteristic, complemented
5#26041582C211yesmaximal, characteristic, complemented
6#26041516C211yesmaximal, characteristic, complemented
7#26041510C211yesmaximal, characteristic, complemented
8#26041497C211yesmaximal, characteristic, complemented
9#26041427C211yesmaximal, characteristic, complemented
10#26041415C211yesmaximal, characteristic, complemented
11#26041361C211yesmaximal, characteristic, complemented
12#26041342C211yesmaximal, characteristic, complemented
13#26041274C211yesmaximal, characteristic, complemented
14C2×#19153736C211yesmaximal, characteristic, complemented
15#484×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C22×D724C2211yescharacteristic
18C2×#15668436C2211yescharacteristic, complemented
19C2×D1448C2211yescharacteristic
20C2×D1448C2211yescharacteristic
21C22×#13089678C2211yescharacteristic
22C2×C8×D181C2211yescharacteristic, complemented
23#19153811—21no—
24#19153807C2211yescharacteristic, complemented
25#19153805C2211yescharacteristic
26#19153801—21no—
27#19153801—21no—
28#19153791—21no—
29#19153791C2211yescharacteristic
30#19153791C2211yescharacteristic, complemented
31#19153790—21no—
32#19153790C2211yescharacteristic
33#19153786—21no—
34#19153778C2211yescharacteristic
35#19153778C2211yescharacteristic, complemented
36C4×C22⋊D181—21no—
37C4×C22⋊D181C2211yescharacteristic
38#19153774C2211yescharacteristic, complemented
39C22⋊C4×D181—21no—
40#19153747C2211yescharacteristic, complemented
41#19153745C2211yescharacteristic, complemented
42#19153740C2211yescharacteristic, complemented
43#19153738C2211yescharacteristic
44#19153738C2211yescharacteristic
45#19153736C2211yescharacteristic
46#19153736C2211yescharacteristic
47#19153736C2211yescharacteristic
48#19153736C2211yescharacteristic
49#19153736C2211yescharacteristic, complemented
50#19153728C2211yescharacteristic, complemented
51C8×Dic181C2211yescharacteristic, complemented
52C22⋊Q8×C181C2211yescharacteristic
53C2×C4⋊C4×C181C2211yescharacteristic
54#107×C181C2211yescharacteristic, complemented
55#107×C181C2211yescharacteristic, complemented
56#103×C181C2211yescharacteristic, complemented
57#103×C181C2211yescharacteristic, complemented
58C22⋊C8×C181C2211yescharacteristic
59C23×D181—21no—
60#15668450—22no—
61#15668450—41no—
62C2×D724—22no—
63C2×D724—21no—
64C2×D724—21no—
65C2×D724—21no—
66C2×D724—21no—
67C2×D724—41no—
68C2×D724C2311yescharacteristic
69C2×D724C2311yescharacteristic
70C2×D724C2311yescharacteristic
71C2×C22⋊D181—21no—
72C2×C22⋊D181D411yescharacteristic
73C2×C4×D181—21no—
74C2×C4×D181—21no—
75C2×C4×D181—21no—
76C2×C4×D181C2311yescharacteristic
77#15668436—41no—
78D1448—22no—
79D1448—41no—
80C2×#13089678—41no—
81C2×#13089678C2311yescharacteristic
82C2×#13089678C2311yescharacteristic
83C8×D181—41no—
84C4⋊Dic181—21no—
85C4⋊Dic181C2311yescharacteristic
86#15668428—21no—
87#15668428D411yescharacteristic
88C22⋊Dic181D411yescharacteristic
89#15668426—21no—
90#15668426—21no—
91#15668426—21no—
92#15668426—21no—
93#15668426—21no—
94#15668426—21no—
95#15668426—21no—
96#15668426—21no—
97#15668426D411yescharacteristic
98C4×Dic181—21no—
99C4×Dic181C2311yescharacteristic
100C2×Q8×C181C2311yescharacteristic
101C22×C4×C181—21no—
102C22×C4×C181C2311yescharacteristic
103C2×C8×C181C2311yescharacteristic
104C2×C8×C181C2311yescharacteristic
105C4⋊C4×C181—21no—
106C4⋊C4×C181—21no—
107C4⋊C4×C181C2311yescharacteristic
108C4⋊C4×C181C2311yescharacteristic
109C4⋊C4×C181C2311yescharacteristic
110C22⋊C4×C181—21no—
111C22×D181—21no—
112C22×D181—21no—
113C22×D181—21no—
114C22×D181—42no—
115C22×D181—42no—
116C22×D181—81no—
117C22×D181C2×D411yescharacteristic
118D724—21no—
119D724—21no—
120D724—21no—
121D724—21no—
122D724—21no—
123D724—42no—
124D724—42no—
125D724—81no—
126D724—81no—
127D724#4112yes—
128C22⋊D181—22no—
129C22⋊D181—42no—
130C2×Dic181—21no—
131C2×Dic181C2×D411yescharacteristic
132C2×Dic181C2×D411yescharacteristic
133C2×Dic181C2×D411yescharacteristic
134C4×D181—21no—
135C4×D181—21no—
136C4×D181—42no—
137C4×D181—81no—
138C4×D181—81no—
139#13089678—21no—
140#13089678—21no—
141C23×C181C2×D411yescharacteristic
142Q8×C181—21no—
143Q8×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—81no—
151C2×C4×C181C2×D411yescharacteristic
152C2×C4×C181C2411yescharacteristic
153C8×C181—21no—
154C8×C181—21no—
155C22×C181—21no—
156C22×C181—21no—
157C22×C181#13512yes—
158C22×C181C22×D411yescharacteristic
159C2×D181—21no—
160C2×D181—22no—
161C2×D181—21no—
162C2×D181—21no—
163C2×D181—41no—
164C2×D181—41no—
165C2×D181—41no—
166C2×D181—41no—
167C2×D181—41no—
168C2×D181—41no—
169C2×D181—41no—
170C2×D181—41no—
171C2×D181—84no—
172C4×C181—21no—
173C4×C181—41no—
174C4×C181—41no—
175C4×C181—41no—
176C4×C181—41no—
177C4×C181C2×#4111yescharacteristic
178C4×C181H(5,𝔽2)11yescharacteristic
179Dic181—21no—
180Dic181—21no—
181Dic181—21no—
182Dic181—41no—
183C2×C181—22no—
184C2×C181#54611yescharacteristic
185C2×C181C2×#13511yescharacteristic
186C2×C181#57611yescharacteristic
187D181—41no—
188D181—42no—
189D181—81no—
190D181—81no—
191C181#329711yescharacteristic, complemented, Sylow
192#3297—1811nomaximal, Sylow
193#553—1811no—
194#546—1811no—
195#488—1811no—
196#484D18111yescharacteristic, complemented
197#482—1811no—
198#478—1811no—
199#466—1811no—
200#465—1811no—
201#463—1811no—
202#449—1811no—
203#447—1811no—
204C4×D8—1811no—
205C8×D4—1811no—
206#416—1811no—
207C2×#102—1811no—
208C2×#41—1811no—
209C22×D4—1811no—
210C2×SD16—1811no—
211C2×D8—1811no—
212C2×D8—1811no—
213C22×C8—1811no—
214C22×C8—1811no—
215#126—3621no—
216#125—1811no—
217#124—1811no—
218#123—3621no—
219#123—3621no—
220C22⋊Q8C2×D18111yescharacteristic
221#121—3621no—
222#121—3621no—
223#121—1811no—
224#121—1811no—
225#121—1811no—
226#120—3621no—
227C4×D4—3621no—
228C4×D4—1811no—
229C4×D4—1811no—
230C4×D4—1811no—
231#117—1811no—
232C2×C4⋊C4C2×D18111yescharacteristic
233C2×C22⋊C4—3621no—
234#107C2×D18111yescharacteristic, complemented
235#107C2×D18111yescharacteristic, complemented
236#106—1811no—
237C4⋊C8—1811no—
238#103—1811no—
239#103C2×D18111yescharacteristic, complemented
240#103C2×D18111yescharacteristic, complemented
241#102—1811no—
242#102—1811no—
243#102—1811no—
244#102—1811no—
245#102—1811no—
246#102—1811no—
247#102—1811no—
248C22⋊C8—1811no—
249C22⋊C8C2×D18111yescharacteristic
250C4×C8—1811no—
251C24—3621no—
252#41—3622no—
253#41—7241no—
254C2×Q8C22×D18111yescharacteristic
255C2×D4—3622no—
256C2×D4—3621no—
257C2×D4—3621no—
258C2×D4—3621no—
259C2×D4—3621no—
260C2×D4—3621no—
261C2×D4—1811no—
262C2×D4—1811no—
263C2×D4—1811no—
264C2×D4—1811no—
265C2×D4—7241no—
266C22×C4—21no—
267C22×C4—3621no—
268C22×C4—3621no—
269C22×C4—3621no—
270C22×C4—1811no—
271C22×C4C22×D18111yescharacteristic
272SD16—7241no—
273D8—3622no—
274D8—7241no—
275C2×C8—1811no—
276C2×C8—1811no—
277C2×C8—7241no—
278C2×C8—7241no—
279C2×C8C22×D18111yescharacteristic
280C2×C8C22×D18111yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—3621no—
284C4⋊C4—3621no—
285C4⋊C4—1811no—
286C4⋊C4—1811no—
287C4⋊C4C22×D18111yescharacteristic
288C4⋊C4C22×D18111yescharacteristic
289C4⋊C4C22×D18111yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—3621no—
292C22⋊C4—3621no—
293C22⋊C4—3621no—
294C22⋊C4—3621no—
295C22⋊C4—3621no—
296C22⋊C4—3621no—
297C22⋊C4—3621no—
298C22⋊C4—3621no—
299C22⋊C4—1811no—
300C22⋊C4—1811no—
301C42—3621no—
302C42—1811no—
303C23—3621no—
304C23—3621no—
305C23—3621no—
306C23—14481no—
307C23—1811no—
308C23—7242no—
309C23—7242no—
310C23D4×D18111yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—3622no—
314D4—3621no—
315D4—3621no—
316D4—3621no—
317D4—3621no—
318D4—3621no—
319D4—14481no—
320D4—14481no—
321D4—1812no—
322D4—7242no—
323D4—7242no—
324D4—7242no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—81no—
332C2×C4—3621no—
333C2×C4—3621no—
334C2×C4—3621no—
335C2×C4—14481no—
336C2×C4—14481no—
337C2×C4—1811no—
338C2×C4—1811no—
339C2×C4—1811no—
340C2×C4—7242no—
341C2×C4D4×D18111yescharacteristic
342C2×C4C23×D18111yescharacteristic
343C8—21no—
344C8—21no—
345C8—3621no—
346C8—3621no—
347C22—21no—
348C22—21no—
349C22—3621no—
350C22—3622no—
351C22—3621no—
352C22—3621no—
353C22—14484no—
354C22—7241no—
355C22—7241no—
356C22—7241no—
357C22—7241no—
358C22—7241no—
359C22—7241no—
360C22—7241no—
361C22—7241no—
362C22#1915383912yes—
363C22C2×D4×D18111yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—3621no—
370C4—3621no—
371C4—3621no—
372C4—7241no—
373C4#41×D18111yescharacteristic
374C4#1915386211yescharacteristic
375C2—22no—
376C2—14481no—
377C2—14481no—
378C2—7241no—
379C2—7242no—
380C2#2604192111yescharacteristic
381C2C2×#1915383911yescharacteristic
382C2#2604208311yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#484×C181C2
socleC22×C181C22×D4
hypercenter#484D181
hypocenterC181#3297
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #26041977C2semidirect
2. #26041920C2semidirect
3. #26041635C2semidirect
4. #26041606C2semidirect
5. #26041582C2semidirect
6. #26041516C2semidirect
7. #26041510C2semidirect
8. #26041497C2semidirect
9. #26041427C2semidirect
10. #26041415C2semidirect
11. #26041361C2semidirect
12. #26041342C2semidirect
13. #26041274C2semidirect
14. C2×#19153736C2semidirect
15. #484×C181C2semidirect
18. C2×#15668436C22semidirect
22. C2×C8×D181C22semidirect
24. #19153807C22semidirect
30. #19153791C22semidirect
35. #19153778C22semidirect
38. #19153774C22semidirect
40. #19153747C22semidirect
41. #19153745C22semidirect
42. #19153740C22semidirect
49. #19153736C22semidirect
50. #19153728C22semidirect
51. C8×Dic181C22semidirect
54. #107×C181C22semidirect
55. #107×C181C22semidirect
56. #103×C181C22semidirect
57. #103×C181C22semidirect
191. C181#3297semidirect
196. #484D181semidirect
234. #107C2×D181semidirect
235. #107C2×D181semidirect
239. #103C2×D181semidirect
240. #103C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260419772
#260419202
#260416352
#260416062
#260415822
#260415162
#260415102
#260414972
#260414272
#260414152
#260413612
#260413422
#260412742
C2×#191537362
#484×C1812
#3297181