Finite Group Databasebeta

← #36365398

Subgroups — #36365398

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#26012181C211yesmaximal, characteristic, complemented
2#26011845C211yesmaximal, characteristic, complemented
3#26011819C211yesmaximal, characteristic, complemented
4#26011812C211yesmaximal, characteristic, complemented
5#26011791C211yesmaximal, characteristic, complemented
6#26011730C211yesmaximal, characteristic, complemented
7#26011718C211yesmaximal, characteristic, complemented
8#26011710C211yesmaximal, characteristic, complemented
9#26011653C211yesmaximal, characteristic, complemented
10#26011642C211yesmaximal, characteristic, complemented
11C4×#14583814C211yesmaximal, characteristic, complemented
12D4×#13063436C211yesmaximal, characteristic, complemented
13#26011487C211yesmaximal, characteristic, complemented
14C2×#19149600C211yesmaximal, characteristic, complemented
15#546×C173C211yesmaximal, characteristic
16C2×#14583816C2211yescharacteristic, complemented
17C2×#14583814C2211yescharacteristic
18C2×#14583814C2211yescharacteristic
19C22×#13063436C2211yescharacteristic
20C22×#13063436C2211yescharacteristic, complemented
21#19149672—21no—
22#19149671C2211yescharacteristic, complemented
23#19149669—21no—
24#19149660—21no—
25#19149658C2211yescharacteristic
26#19149658C2211yescharacteristic, complemented
27C4×Dic346C2211yescharacteristic, complemented
28#19149638C2211yescharacteristic, complemented
29#19149635—21no—
30C2×C4⋊Dic173C2211yescharacteristic
31#19149612C2211yescharacteristic, complemented
32#19149610C2211yescharacteristic, complemented
33#19149610C2211yescharacteristic, complemented
34C4⋊#13063436C2211yescharacteristic, complemented
35#19149603C2211yescharacteristic, complemented
36#19149603C2211yescharacteristic, complemented
37#19149603C2211yescharacteristic, complemented
38#19149602C2211yescharacteristic, complemented
39#19149600C2211yescharacteristic
40#19149600C2211yescharacteristic
41#19149600C2211yescharacteristic
42#19149600C2211yescharacteristic
43#19149600C2211yescharacteristic
44#19149600C2211yescharacteristic
45C22⋊#13063436C2211yescharacteristic
46C22⋊#13063436C2211yescharacteristic, complemented
47C4×#13063436C2211yescharacteristic, complemented
48C2×#41×C173C2211yescharacteristic
49C22×D4×C173C2211yescharacteristic
50#124×C173C2211yescharacteristic
51#123×C173—21no—
52C22⋊Q8×C173C2211yescharacteristic
53#121×C173—21no—
54#121×C173C2211yescharacteristic
55#120×C173—21no—
56C4×D4×C173C2211yescharacteristic
57C4×D4×C173C2211yescharacteristic
58C2×C22⋊C4×C173—21no—
59C2×Dic346C2311yescharacteristic
60C22×Dic173—21no—
61#14583816—41no—
62#14583814—22no—
63#14583814—41no—
64C2×#13063436—41no—
65C2×#13063436—41no—
66C2×#13063436C2311yescharacteristic
67C2×#13063436C2311yescharacteristic
68C2×#13063436C2311yescharacteristic
69C2×#13063436C2311yescharacteristic
70C4⋊Dic173—21no—
71C4⋊Dic173C2311yescharacteristic
72C4⋊Dic173C2311yescharacteristic
73C4⋊Dic173C2311yescharacteristic
74#14583805—21no—
75#14583805—21no—
76#14583805—21no—
77#14583805—21no—
78#14583805—21no—
79C22⋊Dic173—21no—
80C22⋊Dic173—21no—
81C22⋊Dic173—21no—
82C4×Dic173—21no—
83C4×Dic173—21no—
84C24×C173—21no—
85#41×C173—22no—
86#41×C173—41no—
87C2×Q8×C173C2311yescharacteristic
88C2×D4×C173—22no—
89C2×D4×C173—21no—
90C2×D4×C173—21no—
91C2×D4×C173—21no—
92C2×D4×C173—41no—
93C2×D4×C173D411yescharacteristic
94C2×D4×C173C2311yescharacteristic
95C2×D4×C173C2311yescharacteristic
96C22×C4×C173—21no—
97C22×C4×C173—21no—
98C22×C4×C173C2311yescharacteristic
99C22×C4×C173C2311yescharacteristic
100C4⋊C4×C173—21no—
101C4⋊C4×C173D411yescharacteristic
102C4⋊C4×C173C2311yescharacteristic
103C22⋊C4×C173—21no—
104C22⋊C4×C173—21no—
105C22⋊C4×C173—21no—
106C22⋊C4×C173—21no—
107C22⋊C4×C173—21no—
108C22⋊C4×C173D411yescharacteristic
109C22⋊C4×C173D411yescharacteristic
110C42×C173C2311yescharacteristic
111Dic346—21no—
112Dic346—21no—
113C2×Dic173—21no—
114C2×Dic173—21no—
115C2×Dic173—21no—
116C2×Dic173—21no—
117C2×Dic173—42no—
118#13063436—21no—
119#13063436—21no—
120#13063436—21no—
121#13063436—21no—
122C23×C173—21no—
123C23×C173—21no—
124C23×C173—42no—
125C23×C173—42no—
126C23×C173—81no—
127C23×C173C2×D411yescharacteristic
128C23×C173C2×D411yescharacteristic
129Q8×C173—21no—
130Q8×C173—21no—
131D4×C173—22no—
132D4×C173—21no—
133D4×C173—21no—
134D4×C173—21no—
135D4×C173—42no—
136D4×C173—42no—
137D4×C173—81no—
138D4×C173#4112yes—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—42no—
146C2×C4×C173—81no—
147C2×C4×C173C2×D411yescharacteristic
148C2×C4×C173C2×D411yescharacteristic
149C2×C4×C173C2×D411yescharacteristic
150C2×C4×C173C2×D411yescharacteristic
151C2×C4×C173C2411yescharacteristic
152C22×C173—21no—
153C22×C173—21no—
154C22×C173—21no—
155C22×C173—22no—
156C22×C173—21no—
157C22×C173—21no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173—41no—
161C22×C173—41no—
162C22×C173—41no—
163C22×C173—84no—
164C22×C173#13512yes—
165C22×C173C22×D411yescharacteristic
166C4×C173—21no—
167C4×C173—21no—
168C4×C173—21no—
169C4×C173—21no—
170C4×C173—41no—
171C4×C173—41no—
172C4×C173C2×#4111yescharacteristic
173C4×C173H(5,𝔽2)11yescharacteristic
174Dic173—41no—
175Dic173—41no—
176Dic173—41no—
177Dic173—41no—
178C2×C173—22no—
179C2×C173—41no—
180C2×C173—42no—
181C2×C173—81no—
182C2×C173#54611yescharacteristic
183C2×C173C2×#13511yescharacteristic
184C2×C173#57711yescharacteristic
185C173#329611yescharacteristic, complemented, Sylow
186#3296—1731nomaximal, Sylow
187#552—1731no—
188#546D17311yescharacteristic
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×SD16—1731no—
199C8×D4—1731no—
200#416—1731no—
201C2×#102—1731no—
202C2×#41C2×D17311yescharacteristic
203C22×D4C2×D17311yescharacteristic
204C2×Q16—1731no—
205C2×SD16—1731no—
206C2×SD16—1731no—
207C22×C8—1731no—
208C22×C8—1731no—
209#126—3461no—
210#125—3461no—
211#125—1731no—
212#124C2×D17311yescharacteristic
213#123—21no—
214#123—3461no—
215C22⋊Q8—1731no—
216C22⋊Q8—1731no—
217C22⋊Q8C2×D17311yescharacteristic
218#121—21no—
219#121C2×D17311yescharacteristic
220#120—21no—
221C4×Q8—1731no—
222C4×D4C2×D17311yescharacteristic
223C4×D4C2×D17311yescharacteristic
224#117—3461no—
225#117—1731no—
226C2×C4⋊C4—1731no—
227C2×C22⋊C4—21no—
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#102—1731no—
239#102—1731no—
240#102—1731no—
241#102—1731no—
242C22⋊C8—1731no—
243C22⋊C8—1731no—
244C4×C8—1731no—
245C24—21no—
246#41—22no—
247#41—41no—
248C2×Q8—1731no—
249C2×Q8C22×D17311yescharacteristic
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—41no—
255C2×D4C22⋊D17311yescharacteristic
256C2×D4C22×D17311yescharacteristic
257C2×D4C22×D17311yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—3461no—
261C22×C4C22×D17311yescharacteristic
262C22×C4C22×D17311yescharacteristic
263Q16—6921no—
264SD16—3462no—
265SD16—6921no—
266C2×C8—1731no—
267C2×C8—1731no—
268C2×C8—1731no—
269C2×C8—1731no—
270C2×C8—6921no—
271C2×C8—6921no—
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—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4C22⋊D17311yescharacteristic
293C22⋊C4C22⋊D17311yescharacteristic
294C42—3461no—
295C42—3461no—
296C42C22×D17311yescharacteristic
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23C2×C22⋊D17311yescharacteristic
303C23C2×C22⋊D17311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—3461no—
307Q8—3461no—
308D4—22no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—42no—
313D4—42no—
314D4—81no—
315D4#1458382512yes—
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—3461no—
325C2×C4—3461no—
326C2×C4—3461no—
327C2×C4—3461no—
328C2×C4—6922no—
329C2×C4C2×C22⋊D17311yescharacteristic
330C2×C4C2×C22⋊D17311yescharacteristic
331C2×C4C2×C22⋊D17311yescharacteristic
332C2×C4C2×C22⋊D17311yescharacteristic
333C2×C4C23×D17311yescharacteristic
334C8—3461no—
335C8—3461no—
336C8—3461no—
337C8—3461no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—84no—
350C22#1914969912yes—
351C22C22×C22⋊D17311yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—6921no—
359C4—6921no—
360C4—6921no—
361C4—6921no—
362C4C2×#1458382511yescharacteristic
363C4#1914972411yescharacteristic
364C2—22no—
365C2—41no—
366C2—42no—
367C2—81no—
368C2#2601213511yescharacteristic
369C2C2×#1914969911yescharacteristic
370C2#2601230211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting, prime radical#546×C173C2
socleC22×C173C22×D4
hypercenter#546D173
hypocenterC173#3296
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #26012181C2semidirect
2. #26011845C2semidirect
3. #26011819C2semidirect
4. #26011812C2semidirect
5. #26011791C2semidirect
6. #26011730C2semidirect
7. #26011718C2semidirect
8. #26011710C2semidirect
9. #26011653C2semidirect
10. #26011642C2semidirect
11. C4×#14583814C2semidirect
12. D4×#13063436C2semidirect
13. #26011487C2semidirect
14. C2×#19149600C2semidirect
16. C2×#14583816C22semidirect
20. C22×#13063436C22semidirect
22. #19149671C22semidirect
26. #19149658C22semidirect
27. C4×Dic346C22semidirect
28. #19149638C22semidirect
31. #19149612C22semidirect
32. #19149610C22semidirect
33. #19149610C22semidirect
34. C4⋊#13063436C22semidirect
35. #19149603C22semidirect
36. #19149603C22semidirect
37. #19149603C22semidirect
38. #19149602C22semidirect
46. C22⋊#13063436C22semidirect
47. C4×#13063436C22semidirect
185. C173#3296semidirect

Maximal subgroups

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

subgroupindex
#260121812
#260118452
#260118192
#260118122
#260117912
#260117302
#260117182
#260117102
#260116532
#260116422
C4×#145838142
D4×#130634362
#260114872
C2×#191496002
#546×C1732
#3296173