Finite Group Databasebeta

← #37545575

Subgroups — #37545575

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#26041981C211yesmaximal, characteristic, complemented
2#26041923C211yesmaximal, characteristic, complemented
3#26041632C211yesmaximal, characteristic, complemented
4#26041615C211yesmaximal, characteristic, complemented
5#26041601C211yesmaximal, characteristic, complemented
6#26041577C211yesmaximal, characteristic, complemented
7#26041512C211yesmaximal, characteristic, complemented
8#26041506C211yesmaximal, characteristic, complemented
9#26041492C211yesmaximal, characteristic, complemented
10#26041421C211yesmaximal, characteristic, complemented
11D8×Dic181C211yesmaximal, characteristic, complemented
12#26041342C211yesmaximal, characteristic, complemented
13#26041272C211yesmaximal, characteristic, complemented
14C2×#19153737C211yesmaximal, characteristic, complemented
15#447×C181C211yesmaximal, characteristic, complemented
16C2×#15668448C2211yescharacteristic
17C2×#15668437C2211yescharacteristic, complemented
18C22×#13089678C2211yescharacteristic
19C2×C8×D181C2211yescharacteristic, complemented
20#19153809—21no—
21#19153807C2211yescharacteristic, complemented
22#19153803C2211yescharacteristic
23#19153800—21no—
24#19153797—21no—
25#19153795C2211yescharacteristic
26#19153788—21no—
27#19153788—21no—
28#19153787C2211yescharacteristic
29#19153787C2211yescharacteristic, complemented
30#19153785—21no—
31D4×Dic181—21no—
32D4×Dic181C2211yescharacteristic
33D4×Dic181C2211yescharacteristic, complemented
34C4×C22⋊D181C2211yescharacteristic
35#19153774C2211yescharacteristic, complemented
36C2×C4⋊Dic181C2211yescharacteristic
37C2×C22⋊Dic181—21no—
38#19153750C2211yescharacteristic, complemented
39#19153750C2211yescharacteristic, complemented
40#19153747C2211yescharacteristic, complemented
41#19153745C2211yescharacteristic, complemented
42#19153741C2211yescharacteristic, complemented
43#19153741C2211yescharacteristic, complemented
44#19153739C2211yescharacteristic, complemented
45#19153737C2211yescharacteristic
46#19153737C2211yescharacteristic
47#19153737C2211yescharacteristic
48#19153737C2211yescharacteristic
49#19153737C2211yescharacteristic, complemented
50#19153728C2211yescharacteristic, complemented
51C8×Dic181C2211yescharacteristic, complemented
52C22×D4×C181C2211yescharacteristic
53C2×D8×C181C2211yescharacteristic
54C2×D8×C181C2211yescharacteristic
55#121×C181C2211yescharacteristic
56#102×C181C2211yescharacteristic
57#102×C181C2211yescharacteristic
58C22⋊C8×C181C2211yescharacteristic
59#15668448—22no—
60#15668448—41no—
61C2×Dic362C2311yescharacteristic
62C2×C22⋊D181—21no—
63C2×C22⋊D181—21no—
64C2×C22⋊D181—21no—
65C2×C22⋊D181D411yescharacteristic
66C22×Dic181—21no—
67C22×Dic181—21no—
68C22×Dic181—21no—
69C22×Dic181—21no—
70C2×C4×D181C2311yescharacteristic
71#15668437—41no—
72C2×#13089678—41no—
73C2×#13089678C2311yescharacteristic
74C2×#13089678C2311yescharacteristic
75C8×D181—41no—
76C4⋊Dic181—21no—
77C4⋊Dic181—21no—
78C4⋊Dic181C2311yescharacteristic
79C4⋊Dic181C2311yescharacteristic
80C4⋊Dic181C2311yescharacteristic
81#15668428—21no—
82#15668428—21no—
83#15668428D411yescharacteristic
84C22⋊Dic181—21no—
85C22⋊Dic181—21no—
86C22⋊Dic181—21no—
87C22⋊Dic181—21no—
88C22⋊Dic181—21no—
89C22⋊Dic181—21no—
90C22⋊Dic181—21no—
91C22⋊Dic181D411yescharacteristic
92#15668426—21no—
93#15668426D411yescharacteristic
94C4×Dic181—21no—
95C4×Dic181C2311yescharacteristic
96C24×C181—21no—
97C2×D4×C181—22no—
98C2×D4×C181—21no—
99C2×D4×C181—21no—
100C2×D4×C181—41no—
101C2×D4×C181C2311yescharacteristic
102C2×D4×C181C2311yescharacteristic
103C2×D4×C181C2311yescharacteristic
104C22×C4×C181C2311yescharacteristic
105D8×C181—22no—
106D8×C181—41no—
107C2×C8×C181C2311yescharacteristic
108C2×C8×C181C2311yescharacteristic
109C4⋊C4×C181C2311yescharacteristic
110C22⋊C4×C181—21no—
111C22×D181C2×D411yescharacteristic
112Dic362—21no—
113Dic362—21no—
114C22⋊D181—22no—
115C22⋊D181—42no—
116C22⋊D181—81no—
117C22⋊D181—81no—
118C2×Dic181—21no—
119C2×Dic181—21no—
120C2×Dic181—21no—
121C2×Dic181—21no—
122C2×Dic181—42no—
123C2×Dic181—81no—
124C2×Dic181—81no—
125C2×Dic181—81no—
126C2×Dic181C2×D411yescharacteristic
127C2×Dic181C2×D411yescharacteristic
128C2×Dic181C2×D411yescharacteristic
129C4×D181—21no—
130C4×D181—21no—
131#13089678—21no—
132#13089678—21no—
133C23×C181—21no—
134C23×C181—21no—
135C23×C181—21no—
136C23×C181—42no—
137C23×C181—42no—
138C23×C181—81no—
139C23×C181C2×D411yescharacteristic
140D4×C181—21no—
141D4×C181—21no—
142D4×C181—21no—
143D4×C181—21no—
144D4×C181—21no—
145D4×C181—42no—
146D4×C181—42no—
147D4×C181#4112yes—
148C2×C4×C181—21no—
149C2×C4×C181—21no—
150C2×C4×C181—21no—
151C2×C4×C181C2×D411yescharacteristic
152C2×C4×C181C2411yescharacteristic
153C8×C181—21no—
154C8×C181—21no—
155C22×C181—21no—
156C22×C181—21no—
157C22×C181—22no—
158C22×C181—41no—
159C22×C181—41no—
160C22×C181—41no—
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—21no—
170C2×D181—21no—
171C2×D181—21no—
172C4×C181—21no—
173C4×C181—41no—
174C4×C181C2×#4111yescharacteristic
175C4×C181H(5,𝔽2)11yescharacteristic
176Dic181—21no—
177Dic181—21no—
178Dic181—21no—
179Dic181—41no—
180Dic181—41no—
181Dic181—41no—
182Dic181—41no—
183C2×C181—22no—
184C2×C181—42no—
185C2×C181—81no—
186C2×C181—81no—
187C2×C181#54611yescharacteristic
188C2×C181C2×#13511yescharacteristic
189C2×C181#57611yescharacteristic
190D181—41no—
191C181#329711yescharacteristic, complemented, Sylow
192#3297—1811nomaximal, Sylow
193#553—1811no—
194#546—1811no—
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#447D18111yescharacteristic, complemented
204C4×D8—1811no—
205C8×D4—1811no—
206#416—1811no—
207C2×#102—1811no—
208C2×#41—1811no—
209C22×D4C2×D18111yescharacteristic
210C2×SD16—1811no—
211C2×D8C2×D18111yescharacteristic
212C2×D8C2×D18111yescharacteristic
213C22×C8—1811no—
214C22×C8—1811no—
215#126—3621no—
216#125—1811no—
217#124—1811no—
218#123—3621no—
219#123—3621no—
220C22⋊Q8—1811no—
221#121—3621no—
222#121—3621no—
223#121—1811no—
224#121—1811no—
225#121C2×D18111yescharacteristic
226#120—3621no—
227C4×D4—3621no—
228C4×D4—1811no—
229C4×D4—1811no—
230C4×D4—1811no—
231#117—1811no—
232C2×C4⋊C4—1811no—
233C2×C22⋊C4—3621no—
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#102C2×D18111yescharacteristic
247#102C2×D18111yescharacteristic
248C22⋊C8—1811no—
249C22⋊C8C2×D18111yescharacteristic
250C4×C8—1811no—
251C24—21no—
252#41—3622no—
253#41—7241no—
254C2×Q8—1811no—
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—41no—
259C2×D4—3621no—
260C2×D4—3621no—
261C2×D4—3621no—
262C2×D4—1811no—
263C2×D4C22×D18111yescharacteristic
264C2×D4C22×D18111yescharacteristic
265C2×D4C22×D18111yescharacteristic
266C22×C4—3621no—
267C22×C4—3621no—
268C22×C4—3621no—
269C22×C4—3621no—
270C22×C4—1811no—
271C22×C4C22×D18111yescharacteristic
272SD16—7241no—
273D8—22no—
274D8—41no—
275C2×C8—1811no—
276C2×C8—1811no—
277C2×C8—7241no—
278C2×C8—7241no—
279C2×C8C22×D18111yescharacteristic
280C2×C8C22×D18111yescharacteristic
281C4⋊C4—3621no—
282C4⋊C4—3621no—
283C4⋊C4—3621no—
284C4⋊C4—3621no—
285C4⋊C4—1811no—
286C4⋊C4—1811no—
287C4⋊C4—1811no—
288C4⋊C4—1811no—
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—21no—
304C23—21no—
305C23—21no—
306C23—42no—
307C23—42no—
308C23—81no—
309C23—1811no—
310C23D4×D18111yescharacteristic
311Q8—3621no—
312Q8—3621no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—42no—
319D4—42no—
320D4—3622no—
321D4—14481no—
322D4—14481no—
323D4—7242no—
324D4#1566844812yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—3621no—
329C2×C4—3621no—
330C2×C4—3621no—
331C2×C4—3621no—
332C2×C4—3621no—
333C2×C4—3621no—
334C2×C4—14481no—
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—22no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—84no—
359C22—3621no—
360C22—3621no—
361C22—3621no—
362C22#1915383512yes—
363C22C2×D4×D18111yescharacteristic
364C4—21no—
365C4—41no—
366C4—3621no—
367C4—3621no—
368C4—3621no—
369C4—7241no—
370C4—7241no—
371C4—7241no—
372C4—7241no—
373C4C2×#1566844811yescharacteristic
374C4#1915386111yescharacteristic
375C2—22no—
376C2—42no—
377C2—81no—
378C2—81no—
379C2—7241no—
380C2#2604191611yescharacteristic
381C2C2×#1915383511yescharacteristic
382C2#2604208311yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#447×C181C2
socleC22×C181C22×D4
hypercenter#447D181
hypocenterC181#3297
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #26041981C2semidirect
2. #26041923C2semidirect
3. #26041632C2semidirect
4. #26041615C2semidirect
5. #26041601C2semidirect
6. #26041577C2semidirect
7. #26041512C2semidirect
8. #26041506C2semidirect
9. #26041492C2semidirect
10. #26041421C2semidirect
11. D8×Dic181C2semidirect
12. #26041342C2semidirect
13. #26041272C2semidirect
14. C2×#19153737C2semidirect
15. #447×C181C2semidirect
17. C2×#15668437C22semidirect
19. C2×C8×D181C22semidirect
21. #19153807C22semidirect
29. #19153787C22semidirect
33. D4×Dic181C22semidirect
35. #19153774C22semidirect
38. #19153750C22semidirect
39. #19153750C22semidirect
40. #19153747C22semidirect
41. #19153745C22semidirect
42. #19153741C22semidirect
43. #19153741C22semidirect
44. #19153739C22semidirect
49. #19153737C22semidirect
50. #19153728C22semidirect
51. C8×Dic181C22semidirect
191. C181#3297semidirect
203. #447D181semidirect

Maximal subgroups

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

subgroupindex
#260419812
#260419232
#260416322
#260416152
#260416012
#260415772
#260415122
#260415062
#260414922
#260414212
D8×Dic1812
#260413422
#260412722
C2×#191537372
#447×C1812
#3297181