Finite Group Databasebeta

← #36365624

Subgroups — #36365624

Order 22144.

380 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012212C211yesmaximal, characteristic, complemented
2#26012147C211yesmaximal, characteristic
3#26011849C211yesmaximal, characteristic, complemented
4#26011834C211yesmaximal, characteristic, complemented
5#26011812C211yesmaximal, characteristic, complemented
6#26011785C211yesmaximal, characteristic, complemented
7#26011731C211yesmaximal, characteristic, complemented
8#26011711C211yesmaximal, characteristic, complemented
9#26011650C211yesmaximal, characteristic, complemented
10#26011646C211yesmaximal, characteristic, complemented
11#26011595C211yesmaximal, characteristic, complemented
12#26011560C211yesmaximal, characteristic, complemented
13#26011501C211yesmaximal, characteristic, complemented
14#26011484C211yesmaximal, characteristic, complemented
15#468×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C22×Dic346C2211yescharacteristic
18C2×#14583816C2211yescharacteristic
19C2×#14583816C2211yescharacteristic
20C2×C8×D173C2211yescharacteristic
21#19149678C2211yescharacteristic
22#19149674—21no—
23#19149672—21no—
24#19149666C2211yescharacteristic
25#19149665—21no—
26#19149665—21no—
27#19149659C2211yescharacteristic
28#19149656—21no—
29#19149655C2211yescharacteristic
30#19149651—21no—
31Q8×Dic173C2211yescharacteristic
32#19149644—21no—
33#19149644C2211yescharacteristic
34D4×Dic173—21no—
35C4×C22⋊D173C2211yescharacteristic
36#19149638C2211yescharacteristic
37#19149637C2211yescharacteristic
38#19149612C2211yescharacteristic
39#19149610C2211yescharacteristic
40#19149608C2211yescharacteristic
41#19149604C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149589C2211yescharacteristic
52C2×SD16×C173C2211yescharacteristic
53C2×M16×C173C2211yescharacteristic
54C22⋊Q8×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#107×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59#14583825—22no—
60#14583825—41no—
61C2×Dic346—21no—
62C2×Dic346—22no—
63C2×Dic346—21no—
64C2×Dic346—41no—
65C2×Dic346C2311yescharacteristic
66C2×Dic346C2311yescharacteristic
67C2×C22⋊D173—21no—
68C2×C22⋊D173D411yescharacteristic
69C22×Dic173—21no—
70C22×Dic173—21no—
71C2×C4×D173C2311yescharacteristic
72#14583816—22no—
73#14583816—41no—
74C2×#13063436C2311yescharacteristic
75C2×#13063436C2311yescharacteristic
76C8×D173—41no—
77C4⋊Dic173—21no—
78C4⋊Dic173—21no—
79C4⋊Dic173C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84#14583805D411yescharacteristic
85C22⋊Dic173—21no—
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88C22⋊Dic173D411yescharacteristic
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803D411yescharacteristic
93C4×Dic173—21no—
94C4×Dic173—21no—
95C4×Dic173—21no—
96C4×Dic173—21no—
97C4×Dic173C2311yescharacteristic
98C2×Q8×C173C2311yescharacteristic
99C2×D4×C173—21no—
100C2×D4×C173C2311yescharacteristic
101C22×C4×C173C2311yescharacteristic
102SD16×C173—41no—
103M16×C173—41no—
104C2×C8×C173C2311yescharacteristic
105C2×C8×C173C2311yescharacteristic
106C4⋊C4×C173—21no—
107C4⋊C4×C173C2311yescharacteristic
108C4⋊C4×C173C2311yescharacteristic
109C22⋊C4×C173—21no—
110C22⋊C4×C173—21no—
111C22×D173C2×D411yescharacteristic
112Dic346—21no—
113Dic346—21no—
114Dic346—21no—
115Dic346—42no—
116Dic346—42no—
117Dic346#4112yes—
118C22⋊D173—41no—
119C22⋊D173—42no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—42no—
125C2×Dic173—42no—
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×C173C2×D411yescharacteristic
135Q8×C173—21no—
136Q8×C173—21no—
137D4×C173—21no—
138D4×C173—21no—
139D4×C173—42no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173C2×D411yescharacteristic
146C2×C4×C173C2411yescharacteristic
147C8×C173—21no—
148C8×C173—21no—
149C22×C173—21no—
150C22×C173—21no—
151C22×C173—21no—
152C22×C173—41no—
153C22×C173—41no—
154C22×C173—41no—
155C22×C173C22×D411yescharacteristic
156C2×D173—21no—
157C2×D173—21no—
158C2×D173—21no—
159C4×C173—21no—
160C4×C173—41no—
161C4×C173—41no—
162C4×C173—41no—
163C4×C173C2×#4111yescharacteristic
164C4×C173H(5,𝔽2)11yescharacteristic
165Dic173—21no—
166Dic173—22no—
167Dic173—21no—
168Dic173—21no—
169Dic173—41no—
170Dic173—41no—
171Dic173—41no—
172C2×C173—41no—
173C2×C173—81no—
174C2×C173#54611yescharacteristic
175C2×C173#57511yescharacteristic
176C2×C173#57811yescharacteristic
177D173—41no—
178C173#331311yescharacteristic, complemented, Sylow
179#3313—1731nomaximal, Sylow
180#556—1731no—
181#548—1731no—
182#489—1731no—
183#485—1731no—
184#482—1731no—
185#477—1731no—
186#468D17311yescharacteristic, complemented
187#467—1731no—
188#464—1731no—
189#452—1731no—
190#451—1731no—
191#441—1731no—
192#435—1731no—
193#419—1731no—
194#416—1731no—
195C2×#41—1731no—
196C22×Q8—1731no—
197C2×Q16—1731no—
198C2×Q16—1731no—
199C2×SD16C2×D17311yescharacteristic
200C2×M16C2×D17311yescharacteristic
201C22×C8—1731no—
202C4⋊Q8—1731no—
203#126—3461no—
204#126—3461no—
205#124—3461no—
206#124—3461no—
207#124—1731no—
208C22⋊Q8—3461no—
209C22⋊Q8—1731no—
210C22⋊Q8—1731no—
211C22⋊Q8C2×D17311yescharacteristic
212#121—3461no—
213#121C2×D17311yescharacteristic
214C4×Q8—3461no—
215C4×Q8—1731no—
216C4×Q8—1731no—
217C4×D4—3461no—
218C4×D4—1731no—
219#117—1731no—
220#117—1731no—
221#107—1731no—
222#107C2×D17311yescharacteristic
223#106—1731no—
224C4⋊C8—1731no—
225#103—1731no—
226#103—1731no—
227#103—1731no—
228#103—1731no—
229#103—1731no—
230#103—1731no—
231#103C2×D17311yescharacteristic
232#102—1731no—
233#102—1731no—
234#102C2×D17311yescharacteristic
235C22⋊C8—1731no—
236C22⋊C8—1731no—
237#97—1731no—
238#41—3462no—
239#41—6921no—
240C2×Q8—3461no—
241C2×Q8—3462no—
242C2×Q8—3461no—
243C2×Q8—1731no—
244C2×Q8—1731no—
245C2×Q8—6921no—
246C2×Q8C22×D17311yescharacteristic
247C2×D4—21no—
248C2×D4—3461no—
249C2×D4—1731no—
250C2×D4C22×D17311yescharacteristic
251C22×C4—3461no—
252C22×C4—3461no—
253C22×C4—1731no—
254C22×C4C22×D17311yescharacteristic
255Q16—3462no—
256Q16—6921no—
257SD16—41no—
258M16—41no—
259C2×C8—1731no—
260C2×C8—1731no—
261C2×C8—6921no—
262C2×C8C22×D17311yescharacteristic
263C2×C8C22×D17311yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—3461no—
266C4⋊C4—3461no—
267C4⋊C4—3461no—
268C4⋊C4—3461no—
269C4⋊C4—3461no—
270C4⋊C4—1731no—
271C4⋊C4—1731no—
272C4⋊C4—1731no—
273C4⋊C4C22×D17311yescharacteristic
274C4⋊C4C22×D17311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—3461no—
278C22⋊C4—3461no—
279C22⋊C4—3461no—
280C22⋊C4—3461no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C22⋊C4—1731no—
284C22⋊C4—1731no—
285C42—3461no—
286C42—3461no—
287C42—3461no—
288C42—3461no—
289C42—1731no—
290C23—21no—
291C23—1731no—
292C23D4×D17311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—3461no—
296Q8—3461no—
297Q8—3461no—
298Q8—1732no—
299Q8—6922no—
300Q8—6922no—
301D4—21no—
302D4—21no—
303D4—42no—
304D4—6921no—
305D4—6922no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—3461no—
312C2×C4—3461no—
313C2×C4—3461no—
314C2×C4—3461no—
315C2×C4—3461no—
316C2×C4—3461no—
317C2×C4—1731no—
318C2×C4—1731no—
319C2×C4—1731no—
320C2×C4—6922no—
321C2×C4—6922no—
322C2×C4D4×D17311yescharacteristic
323C2×C4C23×D17311yescharacteristic
324C8—21no—
325C8—21no—
326C8—3461no—
327C8—3461no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—3461no—
335C22—3461no—
336C22—3461no—
337C22C2×D4×D17311yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—3461no—
343C4—3461no—
344C4—3461no—
345C4—3462no—
346C4—6921no—
347C4—6921no—
348C4—6921no—
349C4#41×D17311yescharacteristic
350C4#1914972411yescharacteristic
351C2—41no—
352C2—81no—
353C2—6921no—
354C2#2601213011yescharacteristic
355C2#2601228711yescharacteristic
356C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#468×C173C2
socleC22×C173C22×D4
prime radical#26012147C2
hypercenter#468D173
hypocenterC173#3313
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #26012212C2semidirect
3. #26011849C2semidirect
4. #26011834C2semidirect
5. #26011812C2semidirect
6. #26011785C2semidirect
7. #26011731C2semidirect
8. #26011711C2semidirect
9. #26011650C2semidirect
10. #26011646C2semidirect
11. #26011595C2semidirect
12. #26011560C2semidirect
13. #26011501C2semidirect
14. #26011484C2semidirect
15. #468×C173C2semidirect
178. C173#3313semidirect
186. #468D173semidirect

Maximal subgroups

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

subgroupindex
#260122122
#260121472
#260118492
#260118342
#260118122
#260117852
#260117312
#260117112
#260116502
#260116462
#260115952
#260115602
#260115012
#260114842
#468×C1732
#3313173