Finite Group Databasebeta

← #37545556

Subgroups — #37545556

Order 23168.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041969C211yesmaximal, characteristic, complemented
2#26041633C211yesmaximal, characteristic, complemented
3#26041607C211yesmaximal, characteristic, complemented
4#26041600C211yesmaximal, characteristic, complemented
5#26041579C211yesmaximal, characteristic, complemented
6#26041518C211yesmaximal, characteristic, complemented
7#26041506C211yesmaximal, characteristic, complemented
8#26041498C211yesmaximal, characteristic, complemented
9#26041441C211yesmaximal, characteristic, complemented
10#26041430C211yesmaximal, characteristic, complemented
11C4×#15668437C211yesmaximal, characteristic, complemented
12D4×#13089678C211yesmaximal, characteristic, complemented
13#26041275C211yesmaximal, characteristic, complemented
14C2×#19153737C211yesmaximal, characteristic, complemented
15#546×C181C211yesmaximal, characteristic
16C2×#15668439C2211yescharacteristic, complemented
17C2×#15668437C2211yescharacteristic
18C2×#15668437C2211yescharacteristic
19C22×#13089678C2211yescharacteristic
20C22×#13089678C2211yescharacteristic, complemented
21#19153809—21no—
22#19153808C2211yescharacteristic, complemented
23#19153806—21no—
24#19153797—21no—
25#19153795C2211yescharacteristic
26#19153795C2211yescharacteristic, complemented
27C4×Dic362C2211yescharacteristic, complemented
28#19153775C2211yescharacteristic, complemented
29#19153772—21no—
30C2×C4⋊Dic181C2211yescharacteristic
31#19153749C2211yescharacteristic, complemented
32#19153747C2211yescharacteristic, complemented
33#19153747C2211yescharacteristic, complemented
34C4⋊#13089678C2211yescharacteristic, complemented
35#19153740C2211yescharacteristic, complemented
36#19153740C2211yescharacteristic, complemented
37#19153740C2211yescharacteristic, complemented
38#19153739C2211yescharacteristic, complemented
39#19153737C2211yescharacteristic
40#19153737C2211yescharacteristic
41#19153737C2211yescharacteristic
42#19153737C2211yescharacteristic
43#19153737C2211yescharacteristic
44#19153737C2211yescharacteristic
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×Dic362C2311yescharacteristic
60C22×Dic181—21no—
61#15668439—41no—
62#15668437—22no—
63#15668437—41no—
64C2×#13089678—41no—
65C2×#13089678—41no—
66C2×#13089678C2311yescharacteristic
67C2×#13089678C2311yescharacteristic
68C2×#13089678C2311yescharacteristic
69C2×#13089678C2311yescharacteristic
70C4⋊Dic181—21no—
71C4⋊Dic181C2311yescharacteristic
72C4⋊Dic181C2311yescharacteristic
73C4⋊Dic181C2311yescharacteristic
74#15668428—21no—
75#15668428—21no—
76#15668428—21no—
77#15668428—21no—
78#15668428—21no—
79C22⋊Dic181—21no—
80C22⋊Dic181—21no—
81C22⋊Dic181—21no—
82C4×Dic181—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
111Dic362—21no—
112Dic362—21no—
113C2×Dic181—21no—
114C2×Dic181—21no—
115C2×Dic181—21no—
116C2×Dic181—21no—
117C2×Dic181—42no—
118#13089678—21no—
119#13089678—21no—
120#13089678—21no—
121#13089678—21no—
122C23×C181—21no—
123C23×C181—21no—
124C23×C181—42no—
125C23×C181—42no—
126C23×C181—81no—
127C23×C181C2×D411yescharacteristic
128C23×C181C2×D411yescharacteristic
129Q8×C181—21no—
130Q8×C181—21no—
131D4×C181—22no—
132D4×C181—21no—
133D4×C181—21no—
134D4×C181—21no—
135D4×C181—42no—
136D4×C181—42no—
137D4×C181—81no—
138D4×C181#4112yes—
139C2×C4×C181—21no—
140C2×C4×C181—21no—
141C2×C4×C181—21no—
142C2×C4×C181—21no—
143C2×C4×C181—21no—
144C2×C4×C181—21no—
145C2×C4×C181—42no—
146C2×C4×C181—81no—
147C2×C4×C181C2×D411yescharacteristic
148C2×C4×C181C2×D411yescharacteristic
149C2×C4×C181C2×D411yescharacteristic
150C2×C4×C181C2×D411yescharacteristic
151C2×C4×C181C2411yescharacteristic
152C22×C181—21no—
153C22×C181—21no—
154C22×C181—21no—
155C22×C181—22no—
156C22×C181—21no—
157C22×C181—21no—
158C22×C181—41no—
159C22×C181—41no—
160C22×C181—41no—
161C22×C181—41no—
162C22×C181—41no—
163C22×C181—84no—
164C22×C181#13512yes—
165C22×C181C22×D411yescharacteristic
166C4×C181—21no—
167C4×C181—21no—
168C4×C181—21no—
169C4×C181—21no—
170C4×C181—41no—
171C4×C181—41no—
172C4×C181C2×#4111yescharacteristic
173C4×C181H(5,𝔽2)11yescharacteristic
174Dic181—41no—
175Dic181—41no—
176Dic181—41no—
177Dic181—41no—
178C2×C181—22no—
179C2×C181—41no—
180C2×C181—42no—
181C2×C181—81no—
182C2×C181#54611yescharacteristic
183C2×C181C2×#13511yescharacteristic
184C2×C181#57711yescharacteristic
185C181#329611yescharacteristic, complemented, Sylow
186#3296—1811nomaximal, Sylow
187#552—1811no—
188#546D18111yescharacteristic
189#488—1811no—
190#483—1811no—
191#482—1811no—
192#478—1811no—
193#467—1811no—
194#465—1811no—
195#464—1811no—
196#452—1811no—
197#450—1811no—
198C4×SD16—1811no—
199C8×D4—1811no—
200#416—1811no—
201C2×#102—1811no—
202C2×#41C2×D18111yescharacteristic
203C22×D4C2×D18111yescharacteristic
204C2×Q16—1811no—
205C2×SD16—1811no—
206C2×SD16—1811no—
207C22×C8—1811no—
208C22×C8—1811no—
209#126—3621no—
210#125—3621no—
211#125—1811no—
212#124C2×D18111yescharacteristic
213#123—21no—
214#123—3621no—
215C22⋊Q8—1811no—
216C22⋊Q8—1811no—
217C22⋊Q8C2×D18111yescharacteristic
218#121—21no—
219#121C2×D18111yescharacteristic
220#120—21no—
221C4×Q8—1811no—
222C4×D4C2×D18111yescharacteristic
223C4×D4C2×D18111yescharacteristic
224#117—3621no—
225#117—1811no—
226C2×C4⋊C4—1811no—
227C2×C22⋊C4—21no—
228#107—1811no—
229#106—1811no—
230#106—1811no—
231C4⋊C8—1811no—
232#103—1811no—
233#103—1811no—
234#103—1811no—
235#103—1811no—
236#102—1811no—
237#102—1811no—
238#102—1811no—
239#102—1811no—
240#102—1811no—
241#102—1811no—
242C22⋊C8—1811no—
243C22⋊C8—1811no—
244C4×C8—1811no—
245C24—21no—
246#41—22no—
247#41—41no—
248C2×Q8—1811no—
249C2×Q8C22×D18111yescharacteristic
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—41no—
255C2×D4C22⋊D18111yescharacteristic
256C2×D4C22×D18111yescharacteristic
257C2×D4C22×D18111yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—3621no—
261C22×C4C22×D18111yescharacteristic
262C22×C4C22×D18111yescharacteristic
263Q16—7241no—
264SD16—3622no—
265SD16—7241no—
266C2×C8—1811no—
267C2×C8—1811no—
268C2×C8—1811no—
269C2×C8—1811no—
270C2×C8—7241no—
271C2×C8—7241no—
272C4⋊C4—21no—
273C4⋊C4—3621no—
274C4⋊C4—3621no—
275C4⋊C4—3621no—
276C4⋊C4—3621no—
277C4⋊C4—3621no—
278C4⋊C4—3621no—
279C4⋊C4—1811no—
280C4⋊C4—1811no—
281C4⋊C4—1811no—
282C4⋊C4C22⋊D18111yescharacteristic
283C4⋊C4C22×D18111yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—3621no—
290C22⋊C4—3621no—
291C22⋊C4—3621no—
292C22⋊C4C22⋊D18111yescharacteristic
293C22⋊C4C22⋊D18111yescharacteristic
294C42—3621no—
295C42—3621no—
296C42C22×D18111yescharacteristic
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23C2×C22⋊D18111yescharacteristic
303C23C2×C22⋊D18111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—3621no—
307Q8—3621no—
308D4—22no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—42no—
313D4—42no—
314D4—81no—
315D4#1566844812yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—81no—
324C2×C4—3621no—
325C2×C4—3621no—
326C2×C4—3621no—
327C2×C4—3621no—
328C2×C4—7242no—
329C2×C4C2×C22⋊D18111yescharacteristic
330C2×C4C2×C22⋊D18111yescharacteristic
331C2×C4C2×C22⋊D18111yescharacteristic
332C2×C4C2×C22⋊D18111yescharacteristic
333C2×C4C23×D18111yescharacteristic
334C8—3621no—
335C8—3621no—
336C8—3621no—
337C8—3621no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—84no—
350C22#1915383612yes—
351C22C22×C22⋊D18111yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—7241no—
359C4—7241no—
360C4—7241no—
361C4—7241no—
362C4C2×#1566844811yescharacteristic
363C4#1915386111yescharacteristic
364C2—22no—
365C2—41no—
366C2—42no—
367C2—81no—
368C2#2604192311yescharacteristic
369C2C2×#1915383611yescharacteristic
370C2#2604209011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting, prime radical#546×C181C2
socleC22×C181C22×D4
hypercenter#546D181
hypocenterC181#3296
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. #26041969C2semidirect
2. #26041633C2semidirect
3. #26041607C2semidirect
4. #26041600C2semidirect
5. #26041579C2semidirect
6. #26041518C2semidirect
7. #26041506C2semidirect
8. #26041498C2semidirect
9. #26041441C2semidirect
10. #26041430C2semidirect
11. C4×#15668437C2semidirect
12. D4×#13089678C2semidirect
13. #26041275C2semidirect
14. C2×#19153737C2semidirect
16. C2×#15668439C22semidirect
20. C22×#13089678C22semidirect
22. #19153808C22semidirect
26. #19153795C22semidirect
27. C4×Dic362C22semidirect
28. #19153775C22semidirect
31. #19153749C22semidirect
32. #19153747C22semidirect
33. #19153747C22semidirect
34. C4⋊#13089678C22semidirect
35. #19153740C22semidirect
36. #19153740C22semidirect
37. #19153740C22semidirect
38. #19153739C22semidirect
46. C22⋊#13089678C22semidirect
47. C4×#13089678C22semidirect
185. C181#3296semidirect

Maximal subgroups

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

subgroupindex
#260419692
#260416332
#260416072
#260416002
#260415792
#260415182
#260415062
#260414982
#260414412
#260414302
C4×#156684372
D4×#130896782
#260412752
C2×#191537372
#546×C1812
#3296181