Finite Group Databasebeta

← #36365600

Subgroups — #36365600

Order 22144.

402 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#26012209C211yesmaximal, characteristic, complemented
2#26012135C211yesmaximal, characteristic
3#26011849C211yesmaximal, characteristic, complemented
4#26011832C211yesmaximal, characteristic, complemented
5#26011814C211yesmaximal, characteristic, complemented
6#26011779C211yesmaximal, characteristic, complemented
7#26011749C211yesmaximal, characteristic, complemented
8#26011719C211yesmaximal, characteristic, complemented
9#26011712C211yesmaximal, characteristic, complemented
10#26011654C211yesmaximal, characteristic, complemented
11#26011642C211yesmaximal, characteristic, complemented
12#26011588C211yesmaximal, characteristic, complemented
13#26011560C211yesmaximal, characteristic, complemented
14#26011484C211yesmaximal, characteristic, complemented
15#418×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×Dic692C2211yescharacteristic, complemented
18C2×#14583814C2211yescharacteristic
19C2×#14583814C2211yescharacteristic
20C2×C8×D173C2211yescharacteristic, complemented
21#19149678C2211yescharacteristic, complemented
22#19149673—21no—
23#19149672—21no—
24#19149666C2211yescharacteristic
25#19149665—21no—
26#19149660—21no—
27#19149659C2211yescharacteristic, complemented
28#19149658C2211yescharacteristic
29#19149658C2211yescharacteristic
30#19149651—21no—
31#19149650C2211yescharacteristic
32#19149648—21no—
33C4×Dic346—21no—
34#19149644C2211yescharacteristic, complemented
35D4×Dic173C2211yescharacteristic
36C4×C22⋊D173C2211yescharacteristic
37#19149637C2211yescharacteristic, complemented
38C2×C22⋊Dic173—21no—
39#19149612C2211yescharacteristic, complemented
40#19149611C2211yescharacteristic, complemented
41#19149610C2211yescharacteristic, complemented
42#19149608C2211yescharacteristic, complemented
43#19149604C2211yescharacteristic, complemented
44#19149604C2211yescharacteristic, complemented
45#19149602C2211yescharacteristic, complemented
46#19149602C2211yescharacteristic, complemented
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic, complemented
51#19149589C2211yescharacteristic, complemented
52C22×D4×C173C2211yescharacteristic
53C2×M16×C173C2211yescharacteristic
54#117×C173C2211yescharacteristic
55#102×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59#14583825—22no—
60#14583825—41no—
61C2×Dic346—21no—
62C2×Dic346C2311yescharacteristic
63C2×Dic346C2311yescharacteristic
64C2×C22⋊D173—21no—
65C2×C22⋊D173—21no—
66C2×C22⋊D173D411yescharacteristic
67C22×Dic173—21no—
68C22×Dic173—21no—
69C2×C4×D173C2311yescharacteristic
70Dic692—41no—
71#14583814—22no—
72#14583814—41no—
73C2×#13063436C2311yescharacteristic
74C2×#13063436C2311yescharacteristic
75C8×D173—41no—
76C4⋊Dic173—21no—
77C4⋊Dic173C2311yescharacteristic
78C4⋊Dic173C2311yescharacteristic
79#14583805—21no—
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805D411yescharacteristic
84C22⋊Dic173—21no—
85C22⋊Dic173—21no—
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173D411yescharacteristic
91#14583803—21no—
92#14583803—21no—
93#14583803D411yescharacteristic
94C4×Dic173—21no—
95C4×Dic173—21no—
96C4×Dic173C2311yescharacteristic
97C24×C173—21no—
98C2×D4×C173—22no—
99C2×D4×C173—21no—
100C2×D4×C173—41no—
101C2×D4×C173C2311yescharacteristic
102C2×D4×C173C2311yescharacteristic
103C22×C4×C173C2311yescharacteristic
104M16×C173—41no—
105C2×C8×C173C2311yescharacteristic
106C2×C8×C173C2311yescharacteristic
107C4⋊C4×C173C2311yescharacteristic
108C4⋊C4×C173C2311yescharacteristic
109C22⋊C4×C173—21no—
110C42×C173—21no—
111C22×D173C2×D411yescharacteristic
112Dic346—21no—
113Dic346—21no—
114Dic346—21no—
115Dic346—21no—
116Dic346—42no—
117C22⋊D173—41no—
118C22⋊D173—42no—
119C22⋊D173—81no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—42no—
125C2×Dic173—81no—
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C4×D173—21no—
130C4×D173—21no—
131#13063436—21no—
132#13063436—21no—
133C23×C173—21no—
134C23×C173—21no—
135C23×C173—42no—
136C23×C173—42no—
137C23×C173—81no—
138C23×C173C2×D411yescharacteristic
139D4×C173—21no—
140D4×C173—21no—
141D4×C173—21no—
142D4×C173—42no—
143D4×C173#4112yes—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173C2×D411yescharacteristic
149C2×C4×C173C2411yescharacteristic
150C8×C173—21no—
151C8×C173—21no—
152C22×C173—21no—
153C22×C173—22no—
154C22×C173—21no—
155C22×C173—21no—
156C22×C173—41no—
157C22×C173—41no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173—41no—
161C22×C173—84no—
162C22×C173C22×D411yescharacteristic
163C2×D173—21no—
164C2×D173—21no—
165C2×D173—21no—
166C4×C173—21no—
167C4×C173—41no—
168C4×C173—41no—
169C4×C173C2×#4111yescharacteristic
170C4×C173H(5,𝔽2)11yescharacteristic
171Dic173—21no—
172Dic173—21no—
173Dic173—21no—
174Dic173—41no—
175Dic173—41no—
176Dic173—41no—
177Dic173—41no—
178C2×C173—41no—
179C2×C173—42no—
180C2×C173—81no—
181C2×C173#54611yescharacteristic
182C2×C173#57511yescharacteristic
183C2×C173#57711yescharacteristic
184D173—41no—
185C173#331111yescharacteristic, complemented, Sylow
186#3311—1731nomaximal, Sylow
187#556—1731no—
188#546—1731no—
189#489—1731no—
190#485—1731no—
191#482—1731no—
192#476—1731no—
193#470—1731no—
194#465—1731no—
195#464—1731no—
196#452—1731no—
197#450—1731no—
198#440—1731no—
199#435—1731no—
200#418D17311yescharacteristic, complemented
201#416—1731no—
202C2×#41—1731no—
203C22×D4C2×D17311yescharacteristic
204C2×Q16—1731no—
205C2×SD16—1731no—
206C2×SD16—1731no—
207C2×M16C2×D17311yescharacteristic
208C22×C8—1731no—
209C4⋊Q8—1731no—
210#126—3461no—
211#126—3461no—
212#124—3461no—
213#124—1731no—
214#123—3461no—
215C22⋊Q8—1731no—
216C22⋊Q8—1731no—
217C22⋊Q8—1731no—
218#121—3461no—
219#121—1731no—
220#120—3461no—
221C4×Q8—3461no—
222C4×Q8—1731no—
223C4×D4—1731no—
224C4×D4—1731no—
225#117—1731no—
226#117C2×D17311yescharacteristic
227C2×C22⋊C4—3461no—
228#107—1731no—
229#106—1731no—
230#106—1731no—
231C4⋊C8—1731no—
232#103—1731no—
233#103—1731no—
234#103—1731no—
235#103—1731no—
236#102—1731no—
237#102—1731no—
238#102C2×D17311yescharacteristic
239#102C2×D17311yescharacteristic
240#102C2×D17311yescharacteristic
241#102C2×D17311yescharacteristic
242C22⋊C8—1731no—
243C22⋊C8—1731no—
244#97—1731no—
245C24—21no—
246#41—3462no—
247#41—6921no—
248C2×Q8—3461no—
249C2×Q8—1731no—
250C2×Q8—1731no—
251C2×D4—22no—
252C2×D4—21no—
253C2×D4—41no—
254C2×D4—3461no—
255C2×D4—3461no—
256C2×D4—1731no—
257C2×D4C22×D17311yescharacteristic
258C2×D4C22×D17311yescharacteristic
259C22×C4—3461no—
260C22×C4—3461no—
261C22×C4—1731no—
262C22×C4C22×D17311yescharacteristic
263Q16—6921no—
264SD16—3462no—
265SD16—6921no—
266M16—41no—
267C2×C8—1731no—
268C2×C8—1731no—
269C2×C8—6921no—
270C2×C8C22×D17311yescharacteristic
271C2×C8C22×D17311yescharacteristic
272C4⋊C4—3461no—
273C4⋊C4—3461no—
274C4⋊C4—3461no—
275C4⋊C4—3461no—
276C4⋊C4—3461no—
277C4⋊C4—1731no—
278C4⋊C4—1731no—
279C4⋊C4—1731no—
280C4⋊C4C22×D17311yescharacteristic
281C4⋊C4C22×D17311yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—3461no—
284C22⋊C4—3461no—
285C22⋊C4—3461no—
286C22⋊C4—3461no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—1731no—
292C22⋊C4—1731no—
293C42—21no—
294C42—3461no—
295C42—3461no—
296C42—1731no—
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23—1731no—
303C23D4×D17311yescharacteristic
304Q8—3461no—
305Q8—3461no—
306Q8—3461no—
307Q8—3461no—
308Q8—6922no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—42no—
313D4—13841no—
314D4—6921no—
315D4—6922no—
316D4#1458382612yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—3461no—
322C2×C4—3461no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—3461no—
326C2×C4—3461no—
327C2×C4—13841no—
328C2×C4—1731no—
329C2×C4—1731no—
330C2×C4—1731no—
331C2×C4—6922no—
332C2×C4D4×D17311yescharacteristic
333C2×C4C23×D17311yescharacteristic
334C8—21no—
335C8—21no—
336C8—3461no—
337C8—3461no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—84no—
348C22—3461no—
349C22—3461no—
350C22—3461no—
351C22C2×D4×D17311yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—3461no—
356C4—3461no—
357C4—3461no—
358C4—6921no—
359C4—6921no—
360C4—6921no—
361C4—6921no—
362C4C2×#1458382611yescharacteristic
363C4#1914972411yescharacteristic
364C2—41no—
365C2—42no—
366C2—81no—
367C2—6921no—
368C2#2601213111yescharacteristic
369C2#2601228711yescharacteristic
370C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#418×C173C2
socleC22×C173C22×D4
prime radical#26012135C2
hypercenter#418D173
hypocenterC173#3311
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #26012209C2semidirect
3. #26011849C2semidirect
4. #26011832C2semidirect
5. #26011814C2semidirect
6. #26011779C2semidirect
7. #26011749C2semidirect
8. #26011719C2semidirect
9. #26011712C2semidirect
10. #26011654C2semidirect
11. #26011642C2semidirect
12. #26011588C2semidirect
13. #26011560C2semidirect
14. #26011484C2semidirect
15. #418×C173C2semidirect
17. C2×Dic692C22semidirect
20. C2×C8×D173C22semidirect
21. #19149678C22semidirect
27. #19149659C22semidirect
34. #19149644C22semidirect
37. #19149637C22semidirect
39. #19149612C22semidirect
40. #19149611C22semidirect
41. #19149610C22semidirect
42. #19149608C22semidirect
43. #19149604C22semidirect
44. #19149604C22semidirect
45. #19149602C22semidirect
46. #19149602C22semidirect
50. #19149591C22semidirect
51. #19149589C22semidirect
185. C173#3311semidirect
200. #418D173semidirect

Maximal subgroups

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

subgroupindex
#260122092
#260121352
#260118492
#260118322
#260118142
#260117792
#260117492
#260117192
#260117122
#260116542
#260116422
#260115882
#260115602
#260114842
#418×C1732
#3311173