Finite Group Databasebeta

← #37545571

Subgroups — #37545571

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#26041975C211yesmaximal, characteristic, complemented
2#26041633C211yesmaximal, characteristic, complemented
3#26041612C211yesmaximal, characteristic, complemented
4#26041600C211yesmaximal, characteristic, complemented
5#26041579C211yesmaximal, characteristic, complemented
6#26041513C211yesmaximal, characteristic, complemented
7#26041510C211yesmaximal, characteristic, complemented
8#26041491C211yesmaximal, characteristic, complemented
9#26041424C211yesmaximal, characteristic, complemented
10#26041413C211yesmaximal, characteristic, complemented
11C4×#15668434C211yesmaximal, characteristic, complemented
12D4×#13089678C211yesmaximal, characteristic, complemented
13#26041275C211yesmaximal, characteristic, complemented
14C2×#19153737C211yesmaximal, characteristic, complemented
15#546×C181C211yesmaximal, characteristic, complemented
16C2×#15668436C2211yescharacteristic, complemented
17C2×#15668434C2211yescharacteristic
18C2×#15668434C2211yescharacteristic
19C22×#13089678C2211yescharacteristic
20C22×#13089678C2211yescharacteristic, complemented
21#19153811—21no—
22#19153808C2211yescharacteristic, complemented
23#19153798—21no—
24#19153790C2211yescharacteristic
25#19153790C2211yescharacteristic, complemented
26#19153788—21no—
27C4×D724C2211yescharacteristic, complemented
28#19153776—21no—
29#19153775C2211yescharacteristic, complemented
30C2×C4⋊Dic181C2211yescharacteristic
31#19153749C2211yescharacteristic, complemented
32#19153749C2211yescharacteristic, complemented
33#19153747C2211yescharacteristic, complemented
34C4⋊#13089678C2211yescharacteristic, complemented
35#19153740C2211yescharacteristic, complemented
36#19153740C2211yescharacteristic, complemented
37#19153740C2211yescharacteristic, complemented
38#19153737C2211yescharacteristic
39#19153737C2211yescharacteristic
40#19153737C2211yescharacteristic
41#19153737C2211yescharacteristic
42#19153737C2211yescharacteristic
43#19153737C2211yescharacteristic
44#19153736C2211yescharacteristic, complemented
45C22⋊#13089678C2211yescharacteristic
46C22⋊#13089678C2211yescharacteristic, complemented
47C4×#13089678C2211yescharacteristic, complemented
48C2×#41×C181C2211yescharacteristic
49C22×D4×C181C2211yescharacteristic
50#124×C181C2211yescharacteristic
51#123×C181—21no—
52C22⋊Q8×C181C2211yescharacteristic
53#121×C181—21no—
54#121×C181C2211yescharacteristic
55#120×C181—21no—
56C4×D4×C181C2211yescharacteristic
57C4×D4×C181C2211yescharacteristic
58C2×C22⋊C4×C181—21no—
59C2×D724C2311yescharacteristic
60C2×C22⋊D181—21no—
61C2×C22⋊D181—21no—
62C22×Dic181—21no—
63C2×C4×D181—21no—
64#15668436—41no—
65#15668434—22no—
66#15668434—41no—
67C2×#13089678—41no—
68C2×#13089678—41no—
69C2×#13089678C2311yescharacteristic
70C2×#13089678C2311yescharacteristic
71C2×#13089678C2311yescharacteristic
72C2×#13089678C2311yescharacteristic
73C4⋊Dic181—21no—
74C4⋊Dic181C2311yescharacteristic
75C4⋊Dic181C2311yescharacteristic
76C4⋊Dic181C2311yescharacteristic
77#15668428—21no—
78#15668428—21no—
79C22⋊Dic181—21no—
80#15668426—21no—
81#15668426—21no—
82#15668426—21no—
83C4×Dic181—21no—
84C24×C181—21no—
85#41×C181—22no—
86#41×C181—41no—
87C2×Q8×C181C2311yescharacteristic
88C2×D4×C181—22no—
89C2×D4×C181—21no—
90C2×D4×C181—21no—
91C2×D4×C181—21no—
92C2×D4×C181—41no—
93C2×D4×C181D411yescharacteristic
94C2×D4×C181C2311yescharacteristic
95C2×D4×C181C2311yescharacteristic
96C22×C4×C181—21no—
97C22×C4×C181—21no—
98C22×C4×C181C2311yescharacteristic
99C22×C4×C181C2311yescharacteristic
100C4⋊C4×C181—21no—
101C4⋊C4×C181D411yescharacteristic
102C4⋊C4×C181C2311yescharacteristic
103C22⋊C4×C181—21no—
104C22⋊C4×C181—21no—
105C22⋊C4×C181—21no—
106C22⋊C4×C181—21no—
107C22⋊C4×C181—21no—
108C22⋊C4×C181D411yescharacteristic
109C22⋊C4×C181D411yescharacteristic
110C42×C181C2311yescharacteristic
111C22×D181—21no—
112D724—21no—
113D724—21no—
114C22⋊D181—42no—
115C22⋊D181—81no—
116C2×Dic181—21no—
117C2×Dic181—21no—
118C2×Dic181—21no—
119C2×Dic181—81no—
120C4×D181—81no—
121#13089678—21no—
122#13089678—21no—
123#13089678—21no—
124#13089678—21no—
125C23×C181—21no—
126C23×C181—21no—
127C23×C181—42no—
128C23×C181—42no—
129C23×C181—81no—
130C23×C181C2×D411yescharacteristic
131C23×C181C2×D411yescharacteristic
132Q8×C181—21no—
133Q8×C181—21no—
134D4×C181—22no—
135D4×C181—21no—
136D4×C181—21no—
137D4×C181—21no—
138D4×C181—42no—
139D4×C181—42no—
140D4×C181—81no—
141D4×C181#4112yes—
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—42no—
149C2×C4×C181—81no—
150C2×C4×C181C2×D411yescharacteristic
151C2×C4×C181C2×D411yescharacteristic
152C2×C4×C181C2×D411yescharacteristic
153C2×C4×C181C2×D411yescharacteristic
154C2×C4×C181C2411yescharacteristic
155C22×C181—21no—
156C22×C181—21no—
157C22×C181—21no—
158C22×C181—22no—
159C22×C181—21no—
160C22×C181—21no—
161C22×C181—41no—
162C22×C181—41no—
163C22×C181—41no—
164C22×C181—41no—
165C22×C181—41no—
166C22×C181—84no—
167C22×C181#13512yes—
168C22×C181C22×D411yescharacteristic
169C2×D181—41no—
170C2×D181—41no—
171C2×D181—41no—
172C4×C181—21no—
173C4×C181—21no—
174C4×C181—21no—
175C4×C181—21no—
176C4×C181—41no—
177C4×C181—41no—
178C4×C181C2×#4111yescharacteristic
179C4×C181H(5,𝔽2)11yescharacteristic
180Dic181—41no—
181Dic181—41no—
182Dic181—41no—
183C2×C181—22no—
184C2×C181—41no—
185C2×C181—42no—
186C2×C181—81no—
187C2×C181#54611yescharacteristic
188C2×C181C2×#13511yescharacteristic
189C2×C181#57611yescharacteristic
190D181—81no—
191C181#329711yescharacteristic, complemented, Sylow
192#3297—1811nomaximal, Sylow
193#553—1811no—
194#546D18111yescharacteristic, complemented
195#488—1811no—
196#484—1811no—
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×#41C2×D18111yescharacteristic
209C22×D4C2×D18111yescharacteristic
210C2×SD16—1811no—
211C2×D8—1811no—
212C2×D8—1811no—
213C22×C8—1811no—
214C22×C8—1811no—
215#126—3621no—
216#125—1811no—
217#124C2×D18111yescharacteristic
218#123—21no—
219#123—3621no—
220C22⋊Q8C2×D18111yescharacteristic
221#121—21no—
222#121—3621no—
223#121—1811no—
224#121—1811no—
225#121C2×D18111yescharacteristic
226#120—21no—
227C4×D4—3621no—
228C4×D4—1811no—
229C4×D4C2×D18111yescharacteristic
230C4×D4C2×D18111yescharacteristic
231#117—1811no—
232C2×C4⋊C4—1811no—
233C2×C22⋊C4—21no—
234#107—1811no—
235#107—1811no—
236#106—1811no—
237C4⋊C8—1811no—
238#103—1811no—
239#103—1811no—
240#103—1811no—
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⋊C8—1811no—
250C4×C8—1811no—
251C24—21no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D18111yescharacteristic
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—41no—
260C2×D4—3621no—
261C2×D4—3621no—
262C2×D4—1811no—
263C2×D4C22⋊D18111yescharacteristic
264C2×D4C22×D18111yescharacteristic
265C2×D4C22×D18111yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—3621no—
269C22×C4—3621no—
270C22×C4C22×D18111yescharacteristic
271C22×C4C22×D18111yescharacteristic
272SD16—7241no—
273D8—3622no—
274D8—7241no—
275C2×C8—1811no—
276C2×C8—1811no—
277C2×C8—1811no—
278C2×C8—1811no—
279C2×C8—7241no—
280C2×C8—7241no—
281C4⋊C4—21no—
282C4⋊C4—3621no—
283C4⋊C4—3621no—
284C4⋊C4—3621no—
285C4⋊C4—1811no—
286C4⋊C4—1811no—
287C4⋊C4—1811no—
288C4⋊C4C22⋊D18111yescharacteristic
289C4⋊C4C22×D18111yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—3621no—
296C22⋊C4—3621no—
297C22⋊C4—3621no—
298C22⋊C4—3621no—
299C22⋊C4C22⋊D18111yescharacteristic
300C22⋊C4C22⋊D18111yescharacteristic
301C42—3621no—
302C42C22×D18111yescharacteristic
303C23—21no—
304C23—21no—
305C23—42no—
306C23—42no—
307C23—81no—
308C23—3621no—
309C23C2×C22⋊D18111yescharacteristic
310C23C2×C22⋊D18111yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—42no—
318D4—42no—
319D4—81no—
320D4—3621no—
321D4—3621no—
322D4—14481no—
323D4—7242no—
324D4#1566844812yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—81no—
333C2×C4—3621no—
334C2×C4—3621no—
335C2×C4—3621no—
336C2×C4—14481no—
337C2×C4—14481no—
338C2×C4C2×C22⋊D18111yescharacteristic
339C2×C4C2×C22⋊D18111yescharacteristic
340C2×C4C2×C22⋊D18111yescharacteristic
341C2×C4C2×C22⋊D18111yescharacteristic
342C2×C4C23×D18111yescharacteristic
343C8—3621no—
344C8—3621no—
345C8—3621no—
346C8—3621no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—22no—
351C22—21no—
352C22—21no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—84no—
359C22—7241no—
360C22—7241no—
361C22—7241no—
362C22#1915383612yes—
363C22C22×C22⋊D18111yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—7241no—
371C4—7241no—
372C4—7241no—
373C4C2×#1566844811yescharacteristic
374C4#1915386111yescharacteristic
375C2—22no—
376C2—41no—
377C2—42no—
378C2—81no—
379C2—14481no—
380C2#2604192311yescharacteristic
381C2C2×#1915383611yescharacteristic
382C2#2604208211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#546×C181C2
socleC22×C181C22×D4
hypercenter#546D181
hypocenterC181#3297
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #26041975C2semidirect
2. #26041633C2semidirect
3. #26041612C2semidirect
4. #26041600C2semidirect
5. #26041579C2semidirect
6. #26041513C2semidirect
7. #26041510C2semidirect
8. #26041491C2semidirect
9. #26041424C2semidirect
10. #26041413C2semidirect
11. C4×#15668434C2semidirect
12. D4×#13089678C2semidirect
13. #26041275C2semidirect
14. C2×#19153737C2semidirect
15. #546×C181C2semidirect
16. C2×#15668436C22semidirect
20. C22×#13089678C22semidirect
22. #19153808C22semidirect
25. #19153790C22semidirect
27. C4×D724C22semidirect
29. #19153775C22semidirect
31. #19153749C22semidirect
32. #19153749C22semidirect
33. #19153747C22semidirect
34. C4⋊#13089678C22semidirect
35. #19153740C22semidirect
36. #19153740C22semidirect
37. #19153740C22semidirect
44. #19153736C22semidirect
46. C22⋊#13089678C22semidirect
47. C4×#13089678C22semidirect
191. C181#3297semidirect
194. #546D181semidirect

Maximal subgroups

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

subgroupindex
#260419752
#260416332
#260416122
#260416002
#260415792
#260415132
#260415102
#260414912
#260414242
#260414132
C4×#156684342
D4×#130896782
#260412752
C2×#191537372
#546×C1812
#3297181