Finite Group Databasebeta

← #36364400

Subgroups — #36364400

Order 22144.

418 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012002C211yesmaximal, characteristic, complemented
2#26011832C211yesmaximal, characteristic, complemented
3#26011829C211yesmaximal, characteristic, complemented
4#26011815C211yesmaximal, characteristic, complemented
5#26011812C211yesmaximal, characteristic, complemented
6#26011747C211yesmaximal, characteristic, complemented
7#26011742C211yesmaximal, characteristic, complemented
8#26011739C211yesmaximal, characteristic, complemented
9#26011735C211yesmaximal, characteristic, complemented
10#26011653C211yesmaximal, characteristic, complemented
11#26011651C211yesmaximal, characteristic, complemented
12#26011636C211yesmaximal, characteristic, complemented
13#26011634C211yesmaximal, characteristic, complemented
14#26011455C211yesmaximal, characteristic, complemented
15#525×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×#14583816C2211yescharacteristic, complemented
18C2×#14583814C2211yescharacteristic, complemented
19C2×#14583813C2211yescharacteristic, complemented
20C2×#14583811C2211yescharacteristic, complemented
21C2×#14583810C2211yescharacteristic
22C2×#14583810C2211yescharacteristic
23#19149662—21no—
24#19149658C2211yescharacteristic
25#19149658C2211yescharacteristic
26#19149653C2211yescharacteristic
27#19149653C2211yescharacteristic
28#19149648—21no—
29C4×C22⋊D173—21no—
30C4×C22⋊D173—21no—
31#19149638C2211yescharacteristic
32#19149612C2211yescharacteristic, complemented
33#19149612C2211yescharacteristic, complemented
34#19149610C2211yescharacteristic, complemented
35#19149610C2211yescharacteristic, complemented
36#19149603C2211yescharacteristic, complemented
37#19149603C2211yescharacteristic, complemented
38#19149602C2211yescharacteristic, complemented
39#19149602C2211yescharacteristic, complemented
40#19149600C2211yescharacteristic, complemented
41#19149600C2211yescharacteristic, complemented
42#19149599C2211yescharacteristic, complemented
43#19149599C2211yescharacteristic, complemented
44C22⋊#13063436C2211yescharacteristic
45C22⋊#13063436C2211yescharacteristic
46C22⋊#13063436C2211yescharacteristic
47C22⋊#13063436C2211yescharacteristic
48C2×#41×C173C2211yescharacteristic
49C23×C4×C173C2211yescharacteristic
50#123×C173—21no—
51C22⋊Q8×C173C2211yescharacteristic
52C22⋊Q8×C173C2211yescharacteristic
53#121×C173C2211yescharacteristic
54#121×C173C2211yescharacteristic
55#120×C173—21no—
56C4×D4×C173—21no—
57C4×D4×C173—21no—
58#117×C173C2211yescharacteristic
59#14583826—41no—
60#14583826—41no—
61C2×Dic346C2311yescharacteristic
62C2×D692C2311yescharacteristic
63C2×C22⋊D173—21no—
64C2×C22⋊D173—21no—
65C2×C22⋊D173—21no—
66C2×C4×D173—21no—
67#14583816—41no—
68#14583814—41no—
69#14583813—41no—
70#14583811—41no—
71#14583810—41no—
72#14583810—41no—
73C2×#13063436C2311yescharacteristic
74C2×#13063436C2311yescharacteristic
75C2×#13063436C2311yescharacteristic
76C2×#13063436C2311yescharacteristic
77C4⋊Dic173C2311yescharacteristic
78C4⋊Dic173C2311yescharacteristic
79#14583805—21no—
80#14583805—21no—
81C22⋊Dic173—21no—
82C22⋊Dic173—21no—
83C22⋊Dic173—21no—
84#14583803—21no—
85#14583803—21no—
86C4×Dic173—21no—
87C24×C173D411yescharacteristic
88#41×C173—41no—
89#41×C173—41no—
90C2×Q8×C173C2311yescharacteristic
91C2×D4×C173—21no—
92C2×D4×C173—21no—
93C2×D4×C173—21no—
94C2×D4×C173C2311yescharacteristic
95C22×C4×C173—21no—
96C22×C4×C173—41no—
97C22×C4×C173—41no—
98C22×C4×C173D411yescharacteristic
99C22×C4×C173D411yescharacteristic
100C22×C4×C173D411yescharacteristic
101C22×C4×C173C2311yescharacteristic
102C22×C4×C173C2311yescharacteristic
103C22×C4×C173C2311yescharacteristic
104C4⋊C4×C173—21no—
105C4⋊C4×C173—21no—
106C4⋊C4×C173C2311yescharacteristic
107C4⋊C4×C173C2311yescharacteristic
108C22⋊C4×C173—21no—
109C22⋊C4×C173—21no—
110C22⋊C4×C173—21no—
111C22⋊C4×C173—21no—
112C22⋊C4×C173—21no—
113C42×C173—21no—
114C22×D173—21no—
115Dic346—21no—
116Dic346—21no—
117D692—21no—
118D692—21no—
119C22⋊D173—42no—
120C22⋊D173—42no—
121C22⋊D173—42no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C4×D173—42no—
126#13063436—21no—
127#13063436—21no—
128#13063436—21no—
129#13063436—21no—
130C23×C173—21no—
131C23×C173—41no—
132C23×C173—41no—
133C23×C173—41no—
134C23×C173C2×D411yescharacteristic
135C23×C173C2×D411yescharacteristic
136C23×C173C2×D411yescharacteristic
137Q8×C173—21no—
138Q8×C173—21no—
139D4×C173—21no—
140D4×C173—21no—
141D4×C173—42no—
142D4×C173—42no—
143D4×C173—42no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173—21no—
152C2×C4×C173—21no—
153C2×C4×C173—41no—
154C2×C4×C173—41no—
155C2×C4×C173—41no—
156C2×C4×C173—42no—
157C2×C4×C173C2×D411yescharacteristic
158C2×C4×C173C2×D411yescharacteristic
159C2×C4×C173C2×D411yescharacteristic
160C2×C4×C173C2411yescharacteristic
161C22×C173—21no—
162C22×C173—21no—
163C22×C173—21no—
164C22×C173—21no—
165C22×C173—21no—
166C22×C173—21no—
167C22×C173—21no—
168C22×C173—21no—
169C22×C173—21no—
170C22×C173—44no—
171C22×C173—41no—
172C22×C173—41no—
173C22×C173—41no—
174C22×C173C22×D411yescharacteristic
175C2×D173—41no—
176C2×D173—41no—
177C2×D173—41no—
178C4×C173—21no—
179C4×C173—21no—
180C4×C173—21no—
181C4×C173—41no—
182C4×C173—41no—
183C4×C173—41no—
184C4×C173H(5,𝔽2)11yescharacteristic
185C4×C173H(5,𝔽2)11yescharacteristic
186Dic173—41no—
187Dic173—41no—
188Dic173—41no—
189C2×C173—41no—
190C2×C173—41no—
191C2×C173—41no—
192C2×C173—81no—
193C2×C173#53411yescharacteristic
194C2×C173#57511yescharacteristic
195C2×C173#57511yescharacteristic
196D173—81no—
197C173#319711yescharacteristic, complemented, Sylow
198#3197—1731nomaximal, Sylow
199#525—1731no—
200#525D17311yescharacteristic, complemented
201#485—1731no—
202#485—1731no—
203#482—1731no—
204#482—1731no—
205#470—1731no—
206#469—1731no—
207#468—1731no—
208#468—1731no—
209#452—1731no—
210#452—1731no—
211#449—1731no—
212#449—1731no—
213#407—1731no—
214C2×#41—1731no—
215C2×#41C2×D17311yescharacteristic
216C23×C4C2×D17311yescharacteristic
217C2×Q16—1731no—
218C2×SD16—1731no—
219C2×SD16—1731no—
220C2×D8—1731no—
221C2×M16—1731no—
222C2×M16—1731no—
223#123—21no—
224#123—3461no—
225C22⋊Q8—1731no—
226C22⋊Q8—1731no—
227C22⋊Q8C2×D17311yescharacteristic
228C22⋊Q8C2×D17311yescharacteristic
229#121—1731no—
230#121—1731no—
231#121C2×D17311yescharacteristic
232#121C2×D17311yescharacteristic
233#120—21no—
234#120—3461no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—3461no—
238C4×D4—3461no—
239#117—1731no—
240#117C2×D17311yescharacteristic
241#107—1731no—
242#107—1731no—
243#106—1731no—
244#106—1731no—
245#103—1731no—
246#103—1731no—
247#103—1731no—
248#103—1731no—
249#102—1731no—
250#102—1731no—
251#102—1731no—
252#102—1731no—
253C22⋊C8—1731no—
254C22⋊C8—1731no—
255C22⋊C8—1731no—
256C22⋊C8—1731no—
257C24C22⋊D17311yescharacteristic
258#41—41no—
259#41—41no—
260#41—6921no—
261#41—6921no—
262C2×Q8—1731no—
263C2×Q8C22×D17311yescharacteristic
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—3461no—
268C2×D4—3461no—
269C2×D4—3461no—
270C2×D4—1731no—
271C2×D4C22×D17311yescharacteristic
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—3461no—
276C22×C4C22⋊D17311yescharacteristic
277C22×C4C22⋊D17311yescharacteristic
278C22×C4C22⋊D17311yescharacteristic
279C22×C4C22×D17311yescharacteristic
280C22×C4C22×D17311yescharacteristic
281C22×C4C22×D17311yescharacteristic
282Q16—6921no—
283SD16—6921no—
284SD16—6921no—
285D8—6921no—
286M16—6921no—
287M16—6921no—
288C2×C8—1731no—
289C2×C8—1731no—
290C2×C8—1731no—
291C2×C8—1731no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—3461no—
295C4⋊C4—3461no—
296C4⋊C4—1731no—
297C4⋊C4—1731no—
298C4⋊C4C22×D17311yescharacteristic
299C4⋊C4C22×D17311yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—3461no—
306C22⋊C4—3461no—
307C22⋊C4—3461no—
308C22⋊C4—3461no—
309C22⋊C4—3461no—
310C42—21no—
311C42—3461no—
312C23—21no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—3461no—
317C23C2×C22⋊D17311yescharacteristic
318C23C2×C22⋊D17311yescharacteristic
319C23C2×C22⋊D17311yescharacteristic
320Q8—21no—
321Q8—21no—
322Q8—3461no—
323Q8—3461no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—42no—
328D4—42no—
329D4—3461no—
330D4—3461no—
331D4—6922no—
332D4—6922no—
333D4—6922no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—3461no—
348C2×C4—3461no—
349C2×C4—3461no—
350C2×C4—6922no—
351C2×C4C2×C22⋊D17311yescharacteristic
352C2×C4C2×C22⋊D17311yescharacteristic
353C2×C4C2×C22⋊D17311yescharacteristic
354C2×C4C23×D17311yescharacteristic
355C8—3461no—
356C8—3461no—
357C8—3461no—
358C8—3461no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—44no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—6921no—
373C22—6921no—
374C22—6921no—
375C22C22×C22⋊D17311yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—6921no—
383C4—6921no—
384C4—6921no—
385C4#1914972411yescharacteristic
386C4#1914972411yescharacteristic
387C2—41no—
388C2—41no—
389C2—41no—
390C2—81no—
391C2—13841no—
392C2#2601205011yescharacteristic
393C2#2601228911yescharacteristic
394C2#2601228911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#525×C173C2
socleC22×C173C22×D4
hypercenter#525D173
hypocenterC173#3197
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #26012002C2semidirect
2. #26011832C2semidirect
3. #26011829C2semidirect
4. #26011815C2semidirect
5. #26011812C2semidirect
6. #26011747C2semidirect
7. #26011742C2semidirect
8. #26011739C2semidirect
9. #26011735C2semidirect
10. #26011653C2semidirect
11. #26011651C2semidirect
12. #26011636C2semidirect
13. #26011634C2semidirect
14. #26011455C2semidirect
15. #525×C173C2semidirect
17. C2×#14583816C22semidirect
18. C2×#14583814C22semidirect
19. C2×#14583813C22semidirect
20. C2×#14583811C22semidirect
32. #19149612C22semidirect
33. #19149612C22semidirect
34. #19149610C22semidirect
35. #19149610C22semidirect
36. #19149603C22semidirect
37. #19149603C22semidirect
38. #19149602C22semidirect
39. #19149602C22semidirect
40. #19149600C22semidirect
41. #19149600C22semidirect
42. #19149599C22semidirect
43. #19149599C22semidirect
197. C173#3197semidirect
200. #525D173semidirect

Maximal subgroups

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

subgroupindex
#260120022
#260118322
#260118292
#260118152
#260118122
#260117472
#260117422
#260117392
#260117352
#260116532
#260116512
#260116362
#260116342
#260114552
#525×C1732
#3197173