Finite Group Databasebeta

← #36367674

Subgroups — #36367674

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#26012226C213yesmaximal, complemented
2#26012217C211yesmaximal, characteristic, complemented
3#26012204C213yesmaximal, complemented
4#26012185C213yesmaximal, complemented
5#26012182C213yesmaximal, complemented
6#26012163C211yesmaximal, characteristic, complemented
7#26012152C211yesmaximal, characteristic, complemented
8#26012152C211yesmaximal, characteristic, complemented
9#26012113C213yesmaximal, complemented
10#26012096C213yesmaximal, complemented
11#26012028C213yesmaximal, complemented
12#26012019C213yesmaximal, complemented
13#26011973C211yesmaximal, characteristic, complemented
14C4×#14583827C211yesmaximal, characteristic, complemented
15#519×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583827C2211yescharacteristic
18#19149678C2213yes—
19#19149676C2213yes—
20#19149674C2213yes—
21#19149673C2216yes—
22#19149672C2211yescharacteristic
23#19149671C2213yes—
24#19149670C2213yes—
25#19149669C2213yes—
26#19149668C2213yes—
27#19149668C2213yes—
28#19149667C2216yes—
29#19149665C2213yes—
30#19149664C2216yes—
31#19149664C2213yes—
32#19149664C2216yes—
33#19149661C2213yes—
34#19149657C2213yes—
35#19149655C2213yes—
36#19149655C2213yes—
37#19149654C2213yes—
38#19149654C2213yes—
39#19149653C2213yes—
40#19149653C2213yes—
41#19149652C2216yes—
42#19149652C2213yes—
43Q8×Dic173C2211yescharacteristic
44Q8×Dic173C2211yescharacteristic
45C4×Dic346C2213yes—
46C4×D692C2213yes—
47C4×D692C2216yes—
48C4×D692C2213yes—
49C4×D692C2213yes—
50#19149641C2213yes—
51#19149641C2213yes—
52C4×C22⋊D173C2213yes—
53C4×C22⋊D173C2213yes—
54#19149637C2213yes—
55#19149636C2216yes—
56#19149636C2213yes—
57C4⋊C4×D173C2213yes—
58C42×D173C2213yes—
59C22×Q8×C173C2211yescharacteristic
60C4×Q8×C173C2211yescharacteristic
61C4×Q8×C173C2213yes—
62C4×Q8×C173C2211yescharacteristic
63C4×Q8×C173C2213yes—
64#117×C173C2213yes—
65#117×C173C2213yes—
66#14583827—24no—
67#14583827—24no—
68C2×Dic346C2313yes—
69C2×D692C2313yes—
70C2×D692C2313yes—
71C2×D692C2316yes—
72C2×D692C2313yes—
73C2×C22⋊D173C2313yes—
74C2×C22⋊D173C2313yes—
75C2×C4×D173C2313yes—
76C2×C4×D173C2313yes—
77C2×C4×D173C2313yes—
78C2×C4×D173C2313yes—
79C2×C4×D173C2316yes—
80C4⋊Dic173C2313yes—
81C4⋊Dic173C2313yes—
82C4⋊Dic173C2311yescharacteristic
83C4⋊Dic173C2313yes—
84#14583805C2313yes—
85#14583805C2311yescharacteristic
86#14583805C2313yes—
87#14583805C2316yes—
88#14583805C2313yes—
89#14583805C2311yescharacteristic
90#14583805C2313yes—
91C22⋊Dic173C2311yescharacteristic
92C22⋊Dic173C2311yescharacteristic
93#14583803C2313yes—
94#14583803C2313yes—
95#14583803C2313yes—
96#14583803C2316yes—
97#14583803C2316yes—
98#14583803C2313yes—
99#14583803C2313yes—
100#14583803C2316yes—
101#14583803C2313yes—
102C4×Dic173C2311yescharacteristic
103C4×Dic173C2313yes—
104C4×Dic173C2313yes—
105C4×Dic173C2313yes—
106C2×Q8×C173—24no—
107C2×Q8×C173C2313yes—
108C2×Q8×C173C2311yescharacteristic
109C22×C4×C173C2313yes—
110C4⋊C4×C173C2313yes—
111C4⋊C4×C173C2313yes—
112C4⋊C4×C173C2313yes—
113C4⋊C4×C173C2313yes—
114C22⋊C4×C173C2311yescharacteristic
115C22⋊C4×C173C2313yes—
116C42×C173C2313yes—
117C42×C173C2316yes—
118C42×C173C2313yes—
119C22×D173C2413yes—
120C22×D173C2413yes—
121Dic346—43no—
122D692—26no—
123D692—26no—
124D692—43no—
125D692—46no—
126C22⋊D173—43no—
127C22⋊D173—43no—
128C2×Dic173C2411yescharacteristic
129C2×Dic173C2411yescharacteristic
130C2×Dic173C2411yescharacteristic
131C2×Dic173C2413yes—
132C2×Dic173C2413yes—
133C2×Dic173C2411yescharacteristic
134C4×D173—26no—
135C4×D173—26no—
136C4×D173—43no—
137C4×D173—46no—
138C4×D173—43no—
139C23×C173C2411yescharacteristic
140Q8×C173—26no—
141Q8×C173#4114yes—
142C2×C4×C173—26no—
143C2×C4×C173C2413yes—
144C2×C4×C173C2411yescharacteristic
145C2×C4×C173C2411yescharacteristic
146C2×C4×C173C2413yes—
147C2×C4×C173C2413yes—
148C2×C4×C173C2413yes—
149C22×C173—21no—
150C22×C173—21no—
151C22×C173—21no—
152C22×C173C2511yescharacteristic
153C2×D173—23no—
154C2×D173—23no—
155C2×D173—23no—
156C2×D173—23no—
157C2×D173—23no—
158C2×D173—23no—
159C4×C173—23no—
160C4×C173—23no—
161C4×C173—21no—
162C4×C173—21no—
163C4×C173—23no—
164C4×C173C2×#4116yes—
165Dic173—21no—
166Dic173—21no—
167Dic173—21no—
168Dic173—23no—
169Dic173—23no—
170Dic173—21no—
171C2×C173—41no—
172C2×C173C22×#4111yescharacteristic
173C2×C173#58511yescharacteristic
174C2×C173#58511yescharacteristic
175D173—43no—
176D173—43no—
177C173#352411yescharacteristic, complemented, Sylow
178#3524—1731nomaximal, Sylow
179#559—1733no—
180#557—1731no—
181#555—1733no—
182#553—1733no—
183#552—1733no—
184#550—1731no—
185#548—1731no—
186#548—1731no—
187#542—1733no—
188#540—1733no—
189#529—1733no—
190#529—1733no—
191#520—1731no—
192#519D17311yescharacteristic, complemented
193C4×#41—1731no—
194C2×#41—1731no—
195C2×#41—1731no—
196C22×Q8C2×D17311yescharacteristic
197C4⋊Q8—1733no—
198#127—1733no—
199#126—1733no—
200#126—1731no—
201#126—1736no—
202#125—1733no—
203#125—1733no—
204#125—1733no—
205#124—1733no—
206#124—1733no—
207#124—1733no—
208#124—1736no—
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—1733no—
220#121—1733no—
221#121—1736no—
222C4×Q8—1731no—
223C4×Q8—1731no—
224C4×Q8—1733no—
225C4×Q8C2×D17311yescharacteristic
226C4×Q8C2×D17313yes—
227C4×Q8C2×D17311yescharacteristic
228C4×Q8C2×D17313yes—
229C4×D4—1733no—
230C4×D4—1733no—
231C4×D4—1733no—
232C4×D4—1733no—
233C4×D4—1733no—
234C4×D4—1733no—
235C4×D4—1733no—
236C4×D4—1736no—
237#117—1733no—
238#117—1733no—
239#117—1736no—
240#117C2×D17313yes—
241#117C2×D17313yes—
242C2×C4⋊C4—1733no—
243C2×C42—1733no—
244#41—3464no—
245#41—3464no—
246C2×Q8—24no—
247C2×Q8—1733no—
248C2×Q8C22×D17313yes—
249C2×Q8C22×D17311yescharacteristic
250C2×D4—1733no—
251C2×D4—1733no—
252C2×D4—1733no—
253C2×D4—1733no—
254C2×D4—1736no—
255C2×D4—1733no—
256C22×C4—1733no—
257C22×C4—1733no—
258C22×C4—1733no—
259C22×C4—1733no—
260C22×C4—1736no—
261C22×C4C22×D17313yes—
262C4⋊C4—1733no—
263C4⋊C4—1733no—
264C4⋊C4—1731no—
265C4⋊C4—1733no—
266C4⋊C4—1733no—
267C4⋊C4—1736no—
268C4⋊C4—1731no—
269C4⋊C4—1733no—
270C4⋊C4—1733no—
271C4⋊C4—1731no—
272C4⋊C4—1733no—
273C4⋊C4C22×D17313yes—
274C4⋊C4C22×D17313yes—
275C4⋊C4C22×D17313yes—
276C4⋊C4C22×D17313yes—
277C22⋊C4—1731no—
278C22⋊C4—1731no—
279C22⋊C4—1733no—
280C22⋊C4—1733no—
281C22⋊C4—1736no—
282C22⋊C4—1733no—
283C22⋊C4—1736no—
284C22⋊C4—1733no—
285C22⋊C4—1733no—
286C22⋊C4—1736no—
287C22⋊C4—1733no—
288C22⋊C4C22×D17311yescharacteristic
289C22⋊C4C22×D17313yes—
290C42—1731no—
291C42—1733no—
292C42—1733no—
293C42—1733no—
294C42C22×D17313yes—
295C42C22×D17316yes—
296C42C22×D17313yes—
297C23—1733no—
298C23—1733no—
299C23C23×D17311yescharacteristic
300Q8—26no—
301Q8—6923no—
302Q8#1458382614yes—
303D4—3466no—
304D4—3466no—
305D4—6923no—
306D4—6926no—
307D4—6923no—
308D4—6923no—
309C2×C4—26no—
310C2×C4—3466no—
311C2×C4—3466no—
312C2×C4—1731no—
313C2×C4—1731no—
314C2×C4—1731no—
315C2×C4—1733no—
316C2×C4—1733no—
317C2×C4—1731no—
318C2×C4—6923no—
319C2×C4—6926no—
320C2×C4—6923no—
321C2×C4C23×D17311yescharacteristic
322C2×C4C23×D17313yes—
323C2×C4C23×D17313yes—
324C2×C4C23×D17311yescharacteristic
325C2×C4C23×D17313yes—
326C2×C4C23×D17313yes—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—3463no—
331C22—3463no—
332C22—3463no—
333C22—3463no—
334C22—3463no—
335C22—3463no—
336C22C24×D17311yescharacteristic
337C4—23no—
338C4—23no—
339C4—21no—
340C4—21no—
341C4—23no—
342C4—3461no—
343C4—3461no—
344C4—3461no—
345C4—3463no—
346C4—3463no—
347C4—3461no—
348C4C2×#1458382616yes—
349C2—41no—
350C2—6923no—
351C2—6923no—
352C2C22×#1458382611yescharacteristic
353C2#2601232811yescharacteristic
354C2#2601232811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D173
commutator, socle, cosocleC22×C173C25
Fitting#519×C173C2
hypercenter#519D173
hypocenterC173#3524
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
1. #26012226C2semidirect
2. #26012217C2semidirect
3. #26012204C2semidirect
4. #26012185C2semidirect
5. #26012182C2semidirect
6. #26012163C2semidirect
7. #26012152C2semidirect
8. #26012152C2semidirect
9. #26012113C2semidirect
10. #26012096C2semidirect
11. #26012028C2semidirect
12. #26012019C2semidirect
13. #26011973C2semidirect
14. C4×#14583827C2semidirect
15. #519×C173C2semidirect
177. C173#3524semidirect
192. #519D173semidirect

Maximal subgroups

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

subgroupindex
#260122262
#260122172
#260122042
#260121852
#260121822
#260121632
#260121522
#260121132
#260120962
#260120282
#260120192
#260119732
C4×#145838272
#519×C1732
#3524173