Finite Group Databasebeta

← #36365403

Subgroups — #36365403

Order 22144.

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#26012182C211yesmaximal, characteristic, complemented
2#26012136C211yesmaximal, characteristic
3#26011846C211yesmaximal, characteristic, complemented
4#26011823C211yesmaximal, characteristic, complemented
5#26011817C211yesmaximal, characteristic, complemented
6#26011791C211yesmaximal, characteristic, complemented
7#26011731C211yesmaximal, characteristic, complemented
8#26011723C211yesmaximal, characteristic, complemented
9#26011713C211yesmaximal, characteristic, complemented
10#26011650C211yesmaximal, characteristic, complemented
11#26011640C211yesmaximal, characteristic, complemented
12C8×D692C211yesmaximal, characteristic, complemented
13#26011484C211yesmaximal, characteristic, complemented
14#102×D173C211yesmaximal, characteristic, complemented
15C4×SD16×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×D4×D173C2211yescharacteristic
18C2×#14583816C2211yescharacteristic, complemented
19C2×#14583813C2211yescharacteristic
20C2×C8×D173C2211yescharacteristic
21C2×C8×D173C2211yescharacteristic, complemented
22#19149673—21no—
23#19149671C2211yescharacteristic, complemented
24#19149669—21no—
25#19149667C2211yescharacteristic
26#19149664—21no—
27#19149663—21no—
28#19149659C2211yescharacteristic
29#19149655C2211yescharacteristic
30#19149655C2211yescharacteristic, complemented
31#19149654C2211yescharacteristic
32#19149653—21no—
33#19149649—21no—
34C4×D692C2211yescharacteristic
35#19149637C2211yescharacteristic, complemented
36#19149636—21no—
37C4⋊C4×D173C2211yescharacteristic
38C2×#14583803—21no—
39#19149612C2211yescharacteristic, complemented
40#19149610C2211yescharacteristic, complemented
41C4⋊#13063436C2211yescharacteristic, complemented
42#19149604C2211yescharacteristic, complemented
43#19149604C2211yescharacteristic, complemented
44#19149602C2211yescharacteristic, complemented
45#19149601C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49#19149599C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149591C2211yescharacteristic, complemented
52C2×SD16×C173C2211yescharacteristic
53C4×Q8×C173C2211yescharacteristic, complemented
54C4×D4×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic, complemented
56#103×C173C2211yescharacteristic, complemented
57#102×C173C2211yescharacteristic
58C4×C8×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583825—22no—
61#14583825—41no—
62C2×Dic346C2311yescharacteristic
63C2×D692—21no—
64C2×D692D411yescharacteristic
65C2×D692C2311yescharacteristic
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68D4×D173—22no—
69D4×D173—41no—
70C22×Dic173—21no—
71C2×C4×D173—21no—
72C2×C4×D173C2311yescharacteristic
73C2×C4×D173C2311yescharacteristic
74#14583816—41no—
75#14583813—41no—
76C2×#13063436C2311yescharacteristic
77C2×#13063436C2311yescharacteristic
78C8×D173—41no—
79C8×D173—41no—
80C4⋊Dic173—21no—
81C4⋊Dic173D411yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83C4⋊Dic173C2311yescharacteristic
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803D411yescharacteristic
97#14583803D411yescharacteristic
98C4×Dic173—21no—
99C2×Q8×C173C2311yescharacteristic
100C2×D4×C173C2311yescharacteristic
101C22×C4×C173—21no—
102SD16×C173—22no—
103C2×C8×C173C2311yescharacteristic
104C2×C8×C173C2311yescharacteristic
105C4⋊C4×C173—21no—
106C4⋊C4×C173C2311yescharacteristic
107C4⋊C4×C173C2311yescharacteristic
108C22⋊C4×C173—21no—
109C42×C173—21no—
110C42×C173C2311yescharacteristic
111C22×D173—21no—
112C22×D173—42no—
113C22×D173—42no—
114C22×D173—81no—
115C22×D173C2×D411yescharacteristic
116C22×D173C2×D411yescharacteristic
117Dic346—21no—
118Dic346—21no—
119D692—22no—
120D692—21no—
121D692—21no—
122D692—81no—
123C22⋊D173—42no—
124C22⋊D173—42no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—42no—
129C2×Dic173C2×D411yescharacteristic
130C2×Dic173C2×D411yescharacteristic
131C4×D173—21no—
132C4×D173—21no—
133C4×D173—21no—
134C4×D173—21no—
135C4×D173—42no—
136#13063436—21no—
137#13063436—21no—
138C23×C173—21no—
139Q8×C173—21no—
140Q8×C173—21no—
141D4×C173—21no—
142D4×C173#4112yes—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—81no—
147C2×C4×C173C2×D411yescharacteristic
148C2×C4×C173C2×D411yescharacteristic
149C2×C4×C173C2411yescharacteristic
150C8×C173—21no—
151C8×C173—21no—
152C22×C173—22no—
153C22×C173—41no—
154C22×C173—41no—
155C22×C173C22×D411yescharacteristic
156C2×D173—21no—
157C2×D173—21no—
158C2×D173—21no—
159C2×D173—21no—
160C2×D173—21no—
161C2×D173—41no—
162C2×D173—41no—
163C2×D173—41no—
164C2×D173—84no—
165C2×D173#13512yes—
166C4×C173—21no—
167C4×C173—21no—
168C4×C173—41no—
169C4×C173—41no—
170C4×C173—41no—
171C4×C173C2×#4111yescharacteristic
172C4×C173H(5,𝔽2)11yescharacteristic
173Dic173—21no—
174Dic173—21no—
175Dic173—41no—
176Dic173—41no—
177Dic173—41no—
178C2×C173—42no—
179C2×C173#54611yescharacteristic
180C2×C173C2×#13511yescharacteristic
181C2×C173#57711yescharacteristic
182D173—22no—
183D173—41no—
184D173—81no—
185C173#329611yescharacteristic, complemented, Sylow
186#3296—1731nomaximal, Sylow
187#552—1731no—
188#546—1731no—
189#488—1731no—
190#483—1731no—
191#482—1731no—
192#478—1731no—
193#467—1731no—
194#465—1731no—
195#464—1731no—
196#452—1731no—
197#450—1731no—
198C4×SD16D17311yescharacteristic, complemented
199C8×D4—1731no—
200#416—1731no—
201C2×#102—1731no—
202C2×#41—1731no—
203C22×D4—1731no—
204C2×Q16—1731no—
205C2×SD16—1731no—
206C2×SD16C2×D17311yescharacteristic
207C22×C8—1731no—
208C22×C8—1731no—
209#126—3461no—
210#125—3461no—
211#125—1731no—
212#124—1731no—
213#123—3461no—
214#123—3461no—
215C22⋊Q8—1731no—
216C22⋊Q8—1731no—
217C22⋊Q8—1731no—
218#121—3461no—
219#121—1731no—
220#120—3461no—
221C4×Q8C2×D17311yescharacteristic, complemented
222C4×D4—1731no—
223C4×D4C2×D17311yescharacteristic
224#117—3461no—
225#117—1731no—
226C2×C4⋊C4—1731no—
227C2×C22⋊C4—3461no—
228#107—1731no—
229#106—1731no—
230#106C2×D17311yescharacteristic, complemented
231C4⋊C8—1731no—
232#103—1731no—
233#103—1731no—
234#103—1731no—
235#103C2×D17311yescharacteristic, complemented
236#102—1731no—
237#102—1731no—
238#102—1731no—
239#102—1731no—
240#102—1731no—
241#102C2×D17311yescharacteristic
242C22⋊C8—1731no—
243C22⋊C8—1731no—
244C4×C8C2×D17311yescharacteristic, complemented
245C24—3461no—
246#41—3462no—
247#41—6921no—
248C2×Q8—1731no—
249C2×Q8C22×D17311yescharacteristic
250C2×D4—3461no—
251C2×D4—3462no—
252C2×D4—3461no—
253C2×D4—3461no—
254C2×D4—1731no—
255C2×D4—1731no—
256C2×D4—6921no—
257C2×D4C22×D17311yescharacteristic
258C22×C4—21no—
259C22×C4—3461no—
260C22×C4—3461no—
261C22×C4—1731no—
262C22×C4—1731no—
263Q16—6921no—
264SD16—22no—
265SD16—6921no—
266C2×C8—1731no—
267C2×C8—1731no—
268C2×C8—6921no—
269C2×C8—6921no—
270C2×C8C22×D17311yescharacteristic
271C2×C8C22×D17311yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—3461no—
274C4⋊C4—3461no—
275C4⋊C4—3461no—
276C4⋊C4—3461no—
277C4⋊C4—3461no—
278C4⋊C4—3461no—
279C4⋊C4—1731no—
280C4⋊C4—1731no—
281C4⋊C4—1731no—
282C4⋊C4C22×D17311yescharacteristic
283C4⋊C4C22×D17311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—3461no—
286C22⋊C4—3461no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4—1731no—
293C22⋊C4—1731no—
294C42—21no—
295C42—3461no—
296C42C22×D17311yescharacteristic
297C23—21no—
298C23—3461no—
299C23—13841no—
300C23—1731no—
301C23—1731no—
302C23—6922no—
303C23—6922no—
304Q8—21no—
305Q8—21no—
306Q8—3461no—
307Q8—3461no—
308D4—21no—
309D4—3462no—
310D4—3461no—
311D4—3461no—
312D4—13841no—
313D4—6922no—
314D4—6922no—
315D4#1458382612yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—81no—
320C2×C4—3461no—
321C2×C4—3461no—
322C2×C4—3461no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—3461no—
326C2×C4—3461no—
327C2×C4—1731no—
328C2×C4—1731no—
329C2×C4—6922no—
330C2×C4—6922no—
331C2×C4D4×D17311yescharacteristic
332C2×C4D4×D17311yescharacteristic
333C2×C4C23×D17311yescharacteristic
334C8—21no—
335C8—21no—
336C8—3461no—
337C8—3461no—
338C22—22no—
339C22—41no—
340C22—41no—
341C22—3461no—
342C22—3461no—
343C22—3461no—
344C22—3461no—
345C22—3461no—
346C22—13844no—
347C22—1732no—
348C22—6921no—
349C22—6921no—
350C22—6921no—
351C22C2×D4×D17311yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—3461no—
358C4—3461no—
359C4—6921no—
360C4—6921no—
361C4—6921no—
362C4C2×#1458382611yescharacteristic
363C4#1914972511yescharacteristic
364C2—42no—
365C2—3462no—
366C2—13841no—
367C2—6921no—
368C2#2601213711yescharacteristic
369C2#135×D17311yescharacteristic
370C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC4×SD16×C173C2
socleC22×C173C22×D4
prime radical#26012136C2
hypercenterC4×SD16D173
hypocenterC173#3296
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #26012182C2semidirect
3. #26011846C2semidirect
4. #26011823C2semidirect
5. #26011817C2semidirect
6. #26011791C2semidirect
7. #26011731C2semidirect
8. #26011723C2semidirect
9. #26011713C2semidirect
10. #26011650C2semidirect
11. #26011640C2semidirect
12. C8×D692C2semidirect
13. #26011484C2semidirect
14. #102×D173C2semidirect
15. C4×SD16×C173C2semidirect
18. C2×#14583816C22semidirect
21. C2×C8×D173C22semidirect
23. #19149671C22semidirect
30. #19149655C22semidirect
35. #19149637C22semidirect
39. #19149612C22semidirect
40. #19149610C22semidirect
41. C4⋊#13063436C22semidirect
42. #19149604C22semidirect
43. #19149604C22semidirect
44. #19149602C22semidirect
51. #19149591C22semidirect
53. C4×Q8×C173C22semidirect
55. #106×C173C22semidirect
56. #103×C173C22semidirect
58. C4×C8×C173C22semidirect
185. C173#3296semidirect
198. C4×SD16D173semidirect
221. C4×Q8C2×D173semidirect
230. #106C2×D173semidirect
235. #103C2×D173semidirect
244. C4×C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121822
#260121362
#260118462
#260118232
#260118172
#260117912
#260117312
#260117232
#260117132
#260116502
#260116402
C8×D6922
#260114842
#102×D1732
C4×SD16×C1732
#3296173