Finite Group Databasebeta

← #36367661

Subgroups — #36367661

Order 22144.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012227C213yesmaximal, complemented
2#26012217C211yesmaximal, characteristic, complemented
3#26012205C213yesmaximal, complemented
4#26012193C213yesmaximal, complemented
5#26012181C213yesmaximal, complemented
6#26012148C211yesmaximal, characteristic, complemented
7#26012148C211yesmaximal, characteristic, complemented
8#26012108C213yesmaximal, complemented
9#26012095C213yesmaximal, complemented
10#26012022C213yesmaximal, complemented
11#26012021C213yesmaximal, complemented
12#26011974C211yesmaximal, characteristic, complemented
13#26011967C211yesmaximal, characteristic, complemented
14#41×Dic173C211yesmaximal, characteristic, complemented
15#550×C173C211yesmaximal, characteristic, complemented
16C2×Q8×D173C2211yescharacteristic
17#19149678C2213yes—
18#19149674C2211yescharacteristic
19#19149672C2216yes—
20#19149672C2213yes—
21#19149671C2213yes—
22#19149670C2213yes—
23#19149669C2213yes—
24#19149666C2216yes—
25#19149666C2213yes—
26#19149666C2213yes—
27#19149665C2213yes—
28#19149663C2213yes—
29#19149660C2216yes—
30#19149660C2213yes—
31#19149660C2216yes—
32#19149658C2213yes—
33#19149658C2213yes—
34#19149656C2213yes—
35#19149651C2216yes—
36#19149651C2213yes—
37#19149650C2213yes—
38#19149650C2213yes—
39Q8×Dic173C2211yescharacteristic
40Q8×Dic173C2211yescharacteristic
41Q8×Dic173C2211yescharacteristic
42C4×Dic346C2213yes—
43C4×Dic346C2213yes—
44#19149644C2213yes—
45D4×Dic173C2213yes—
46D4×Dic173C2216yes—
47D4×Dic173C2213yes—
48D4×Dic173C2213yes—
49D4×Dic173C2213yes—
50C4×C22⋊D173C2213yes—
51C4×C22⋊D173C2213yes—
52#19149638C2213yes—
53#19149638C2213yes—
54#19149637C2213yes—
55#19149636C2213yes—
56#19149635C2216yes—
57C2×C4⋊Dic173C2213yes—
58C2×C4×Dic173C2213yes—
59C2×#41×C173C2211yescharacteristic
60C2×#41×C173C2211yescharacteristic
61#127×C173C2213yes—
62#121×C173C2213yes—
63#121×C173C2213yes—
64C4×Q8×C173C2211yescharacteristic
65C4×D4×C173C2213yes—
66C2×Dic346C2313yes—
67C2×Dic346C2313yes—
68Q8×D173—24no—
69C2×C22⋊D173C2313yes—
70C2×C22⋊D173C2313yes—
71C22×Dic173C2313yes—
72C22×Dic173C2313yes—
73C22×Dic173C2316yes—
74C2×C4×D173C2313yes—
75C4⋊Dic173C2313yes—
76C4⋊Dic173C2313yes—
77C4⋊Dic173C2313yes—
78C4⋊Dic173C2311yescharacteristic
79C4⋊Dic173C2313yes—
80C4⋊Dic173C2313yes—
81#14583805C2313yes—
82#14583805C2316yes—
83#14583805C2313yes—
84#14583805C2313yes—
85#14583805C2313yes—
86#14583805C2311yescharacteristic
87#14583805C2313yes—
88C22⋊Dic173C2316yes—
89C22⋊Dic173C2313yes—
90C22⋊Dic173C2316yes—
91C22⋊Dic173C2313yes—
92C22⋊Dic173C2313yes—
93C22⋊Dic173C2313yes—
94C22⋊Dic173C2316yes—
95#14583803C2311yescharacteristic
96#14583803C2313yes—
97#14583803C2311yescharacteristic
98#14583803C2311yescharacteristic
99C4×Dic173C2311yescharacteristic
100C4×Dic173C2313yes—
101C4×Dic173C2313yes—
102C4×Dic173C2313yes—
103C4×Dic173C2316yes—
104C4×Dic173C2313yes—
105#41×C173—24no—
106#41×C173—24no—
107C2×Q8×C173C2311yescharacteristic
108C2×D4×C173C2313yes—
109C2×D4×C173C2313yes—
110C2×D4×C173C2316yes—
111C2×D4×C173C2313yes—
112C22×C4×C173C2313yes—
113C22×C4×C173C2313yes—
114C4⋊C4×C173C2313yes—
115C4⋊C4×C173C2311yescharacteristic
116C22⋊C4×C173C2313yes—
117C22⋊C4×C173C2313yes—
118C42×C173C2313yes—
119C22×D173C2411yescharacteristic
120Dic346—26no—
121Dic346—43no—
122C22⋊D173—43no—
123C22⋊D173—43no—
124C2×Dic173—43no—
125C2×Dic173—46no—
126C2×Dic173—43no—
127C2×Dic173C2413yes—
128C2×Dic173C2411yescharacteristic
129C2×Dic173C2411yescharacteristic
130C2×Dic173C2413yes—
131C2×Dic173C2413yes—
132C2×Dic173C2413yes—
133C2×Dic173C2411yescharacteristic
134C4×D173—26no—
135C23×C173C2413yes—
136C23×C173C2413yes—
137Q8×C173#4114yes—
138D4×C173—26no—
139D4×C173—26no—
140D4×C173—43no—
141D4×C173—46no—
142C2×C4×C173—26no—
143C2×C4×C173—26no—
144C2×C4×C173C2413yes—
145C2×C4×C173C2411yescharacteristic
146C2×C4×C173C2411yescharacteristic
147C2×C4×C173C2411yescharacteristic
148C2×C4×C173C2413yes—
149C22×C173—23no—
150C22×C173—23no—
151C22×C173—23no—
152C22×C173—23no—
153C22×C173—23no—
154C22×C173—23no—
155C22×C173C2511yescharacteristic
156C2×D173—21no—
157C2×D173—21no—
158C2×D173—21no—
159C4×C173—21no—
160C4×C173—21no—
161C4×C173—21no—
162C4×C173—23no—
163C4×C173C2×#4116yes—
164Dic173—23no—
165Dic173—21no—
166Dic173—21no—
167Dic173—23no—
168Dic173—23no—
169Dic173—23no—
170Dic173—21no—
171C2×C173—43no—
172C2×C173—43no—
173C2×C173C22×#4111yescharacteristic
174C2×C173#58511yescharacteristic
175C2×C173#58511yescharacteristic
176D173—41no—
177C173#352411yescharacteristic, complemented, Sylow
178#3524—1731nomaximal, Sylow
179#559—1733no—
180#557—1731no—
181#555—1733no—
182#553—1733no—
183#552—1733no—
184#550D17311yescharacteristic, complemented
185#548—1731no—
186#548—1731no—
187#542—1733no—
188#540—1733no—
189#529—1733no—
190#529—1733no—
191#520—1731no—
192#519—1731no—
193C4×#41—1731no—
194C2×#41C2×D17311yescharacteristic
195C2×#41C2×D17311yescharacteristic
196C22×Q8—1731no—
197C4⋊Q8—1733no—
198#127C2×D17313yes—
199#126—1731no—
200#126—1733no—
201#126—1736no—
202#125—1733no—
203#125—1733no—
204#125—1733no—
205#124—1733no—
206#124—1733no—
207#124—1736no—
208#124—1733no—
209#123—1736no—
210#123—1733no—
211#123—1736no—
212#123—1733no—
213C22⋊Q8—1733no—
214C22⋊Q8—1733no—
215C22⋊Q8—1733no—
216#121—1733no—
217#121—1733no—
218#121—1733no—
219#121—1736no—
220#121C2×D17313yes—
221#121C2×D17313yes—
222C4×Q8—1731no—
223C4×Q8—1731no—
224C4×Q8—1731no—
225C4×Q8—1733no—
226C4×Q8—1733no—
227C4×Q8—1733no—
228C4×Q8C2×D17311yescharacteristic
229C4×D4—1733no—
230C4×D4—1733no—
231C4×D4—1733no—
232C4×D4—1733no—
233C4×D4—1736no—
234C4×D4—1733no—
235C4×D4—1733no—
236C4×D4C2×D17313yes—
237#117—1733no—
238#117—1733no—
239#117—1733no—
240#117—1736no—
241#117—1733no—
242C2×C4⋊C4—1733no—
243C2×C42—1733no—
244#41—24no—
245#41—24no—
246C2×Q8—3464no—
247C2×Q8—1733no—
248C2×Q8—1733no—
249C2×Q8C22×D17311yescharacteristic
250C2×D4—1733no—
251C2×D4—1733no—
252C2×D4C22×D17313yes—
253C2×D4C22×D17313yes—
254C2×D4C22×D17316yes—
255C2×D4C22×D17313yes—
256C22×C4—1733no—
257C22×C4—1733no—
258C22×C4—1733no—
259C22×C4—1736no—
260C22×C4C22×D17313yes—
261C22×C4C22×D17313yes—
262C4⋊C4—1733no—
263C4⋊C4—1733no—
264C4⋊C4—1733no—
265C4⋊C4—1731no—
266C4⋊C4—1733no—
267C4⋊C4—1733no—
268C4⋊C4—1733no—
269C4⋊C4—1733no—
270C4⋊C4—1736no—
271C4⋊C4—1733no—
272C4⋊C4—1733no—
273C4⋊C4—1731no—
274C4⋊C4—1733no—
275C4⋊C4C22×D17313yes—
276C4⋊C4C22×D17311yescharacteristic
277C22⋊C4—1733no—
278C22⋊C4—1736no—
279C22⋊C4—1733no—
280C22⋊C4—1736no—
281C22⋊C4—1733no—
282C22⋊C4—1736no—
283C22⋊C4—1733no—
284C22⋊C4—1731no—
285C22⋊C4—1733no—
286C22⋊C4—1731no—
287C22⋊C4—1731no—
288C22⋊C4C22×D17313yes—
289C22⋊C4C22×D17313yes—
290C42—1731no—
291C42—1733no—
292C42—1733no—
293C42—1733no—
294C42—1736no—
295C42—1733no—
296C42C22×D17313yes—
297C23—1731no—
298C23C23×D17313yes—
299C23C23×D17313yes—
300Q8—3466no—
301Q8—6923no—
302Q8#1458382514yes—
303D4—26no—
304D4—26no—
305D4—46no—
306D4—43no—
307D4—6923no—
308D4—6923no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—3466no—
312C2×C4—1731no—
313C2×C4—1731no—
314C2×C4—1733no—
315C2×C4—1731no—
316C2×C4—1733no—
317C2×C4—1733no—
318C2×C4—1733no—
319C2×C4—6923no—
320C2×C4—6923no—
321C2×C4—6926no—
322C2×C4C23×D17311yescharacteristic
323C2×C4C23×D17313yes—
324C2×C4C23×D17311yescharacteristic
325C2×C4C23×D17311yescharacteristic
326C2×C4C23×D17313yes—
327C22—23no—
328C22—23no—
329C22—23no—
330C22—23no—
331C22—23no—
332C22—23no—
333C22—3461no—
334C22—3461no—
335C22—3461no—
336C22C24×D17311yescharacteristic
337C4—21no—
338C4—21no—
339C4—23no—
340C4—21no—
341C4—3463no—
342C4—3461no—
343C4—3463no—
344C4—3461no—
345C4—3461no—
346C4—3463no—
347C4—3463no—
348C4C2×#1458382516yes—
349C2—43no—
350C2—43no—
351C2—6921no—
352C2C22×#1458382511yescharacteristic
353C2#2601232611yescharacteristic
354C2#2601232711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D173
commutator, socle, cosocleC22×C173C25
Fitting#550×C173C2
hypercenter#550D173
hypocenterC173#3524
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #26012227C2semidirect
2. #26012217C2semidirect
3. #26012205C2semidirect
4. #26012193C2semidirect
5. #26012181C2semidirect
6. #26012148C2semidirect
7. #26012148C2semidirect
8. #26012108C2semidirect
9. #26012095C2semidirect
10. #26012022C2semidirect
11. #26012021C2semidirect
12. #26011974C2semidirect
13. #26011967C2semidirect
14. #41×Dic173C2semidirect
15. #550×C173C2semidirect
177. C173#3524semidirect
184. #550D173semidirect

Maximal subgroups

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

subgroupindex
#260122272
#260122172
#260122052
#260121932
#260121812
#260121482
#260121082
#260120952
#260120222
#260120212
#260119742
#260119672
#41×Dic1732
#550×C1732
#3524173