Finite Group Databasebeta

← #36367642

Subgroups — #36367642

Order 22144.

1104 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012224C213yesmaximal, complemented
2Q8×Dic346C211yesmaximal, characteristic, complemented
3#26012196C213yesmaximal, complemented
4#26012160C211yesmaximal, characteristic, complemented
5#26012153C211yesmaximal, characteristic, complemented
6#26012148C211yesmaximal, characteristic, complemented
7#26012146C213yesmaximal, complemented
8#26012142C213yesmaximal, complemented
9#26012112C213yesmaximal, complemented
10#26012106C213yesmaximal, complemented
11#26012038C213yesmaximal, complemented
12#26012019C213yesmaximal, complemented
13#26011973C211yesmaximal, characteristic, complemented
14C4×Q8×D173C211yesmaximal, characteristic, complemented
15#520×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583826C2216yes—
18C2×Q8×D173C2211yescharacteristic
19#19149679C2213yes—
20#19149678C2213yes—
21#19149678C2213yes—
22#19149675C2213yes—
23#19149674C2213yes—
24#19149672C2213yes—
25#19149669C2213yes—
26#19149667C2213yes—
27#19149666C2213yes—
28#19149665C2213yes—
29#19149665C2216yes—
30#19149664C2213yes—
31#19149662C2216yes—
32#19149661C2216yes—
33#19149661C2213yes—
34#19149659C2213yes—
35#19149658C2216yes—
36#19149658C2213yes—
37#19149657C2216yes—
38#19149657C2211yescharacteristic
39#19149654C2213yes—
40#19149653C2213yes—
41#19149652C2213yes—
42#19149650C2213yes—
43#19149650C2213yes—
44Q8×Dic173C2211yescharacteristic
45Q8×Dic173C2211yescharacteristic
46C4×Dic346C2213yes—
47C4×Dic346C2213yes—
48#19149644C2213yes—
49C4×D692C2213yes—
50C4×D692C2213yes—
51#19149641C2213yes—
52C4×C22⋊D173C2213yes—
53C4×C22⋊D173C2216yes—
54C4×C22⋊D173C2213yes—
55#19149636C2213yes—
56C4⋊C4×D173C2213yes—
57C4⋊C4×D173C2213yes—
58C42×D173C2213yes—
59C2×#41×C173C2211yescharacteristic
60C4×Q8×C173C2211yescharacteristic
61C4×Q8×C173C2211yescharacteristic
62C4×D4×C173C2213yes—
63C4×D4×C173C2213yes—
64#117×C173C2213yes—
65C2×C4⋊C4×C173C2213yes—
66#14583827—24no—
67#14583826—412no—
68C2×Dic346C2313yes—
69C2×Dic346C2311yescharacteristic
70C2×Dic346C2316yes—
71Q8×D173—24no—
72C2×D692C2313yes—
73C2×D692C2313yes—
74C2×C22⋊D173C2313yes—
75C2×C22⋊D173C2313yes—
76C2×C22⋊D173C2316yes—
77C2×C4×D173C2311yescharacteristic
78C2×C4×D173C2313yes—
79C2×C4×D173C2313yes—
80C2×C4×D173C2313yes—
81C2×C4×D173C2316yes—
82C4⋊Dic173C2313yes—
83C4⋊Dic173C2313yes—
84C4⋊Dic173C2313yes—
85C4⋊Dic173C2313yes—
86#14583805C2313yes—
87#14583805C2316yes—
88#14583805C2311yescharacteristic
89#14583805C2311yescharacteristic
90#14583805C2313yes—
91#14583805C2313yes—
92#14583805C2313yes—
93C22⋊Dic173C2313yes—
94C22⋊Dic173C2316yes—
95C22⋊Dic173C2313yes—
96#14583803C2313yes—
97#14583803C2313yes—
98#14583803C2311yescharacteristic
99#14583803C2313yes—
100#14583803C2316yes—
101#14583803C2311yescharacteristic
102#14583803C2313yes—
103C4×Dic173C2313yes—
104C4×Dic173C2313yes—
105C4×Dic173C2313yes—
106C4×Dic173C2313yes—
107#41×C173—24no—
108C2×Q8×C173C2311yescharacteristic
109C2×D4×C173C2313yes—
110C22×C4×C173C2313yes—
111C22×C4×C173C2316yes—
112C4⋊C4×C173C2313yes—
113C4⋊C4×C173C2313yes—
114C4⋊C4×C173C2313yes—
115C4⋊C4×C173C2311yescharacteristic
116C22⋊C4×C173C2313yes—
117C22⋊C4×C173C2313yes—
118C42×C173C2313yes—
119C42×C173C2313yes—
120C22×D173C2413yes—
121C22×D173C2411yescharacteristic
122Dic346—26no—
123Dic346—22no—
124Dic346—46no—
125D692—26no—
126D692—26no—
127C22⋊D173—26no—
128C22⋊D173—26no—
129C22⋊D173—46no—
130C2×Dic173C2411yescharacteristic
131C2×Dic173C2413yes—
132C2×Dic173C2413yes—
133C2×Dic173C2413yes—
134C2×Dic173C2411yescharacteristic
135C2×Dic173C2411yescharacteristic
136C4×D173—26no—
137C4×D173—26no—
138C4×D173—26no—
139C4×D173—22no—
140C4×D173—46no—
141C23×C173C2413yes—
142Q8×C173#4114yescomplemented
143D4×C173—26no—
144C2×C4×C173—26no—
145C2×C4×C173—46no—
146C2×C4×C173C2413yes—
147C2×C4×C173C2411yescharacteristic
148C2×C4×C173C2413yes—
149C2×C4×C173C2411yescharacteristic
150C2×C4×C173C2411yescharacteristic
151C2×C4×C173C2413yes—
152C22×C173—23no—
153C22×C173—23no—
154C22×C173—23no—
155C22×C173C2511yescharacteristic
156C2×D173—21no—
157C2×D173—21no—
158C2×D173—23no—
159C2×D173—23no—
160C2×D173—23no—
161C2×D173H(5,𝔽2)12yes—
162C4×C173—21no—
163C4×C173—23no—
164C4×C173—23no—
165C4×C173—21no—
166C4×C173—21no—
167C4×C173C2×#4116yes—
168Dic173—21no—
169Dic173—23no—
170Dic173—23no—
171Dic173—23no—
172Dic173—21no—
173Dic1732−1+412yes—
174C2×C173—43no—
175C2×C173C22×#4111yescharacteristic
176C2×C173C2×H(5,𝔽2)11yescharacteristic
177C2×C173C2×2−1+411yescharacteristic
178D173—22no—
179D173—43no—
180C173#352211yescharacteristic, complemented, Sylow
181#3522—1731nomaximal, Sylow
182#559—1733no—
183Q82—1731no—
184#554—1733no—
185#550—1731no—
186#548—1731no—
187#548—1731no—
188#547—1733no—
189#547—1733no—
190#542—1733no—
191#542—1733no—
192#532—1733no—
193#529—1733no—
194#520—1731no—
195#520D17311yescharacteristic, complemented
196C2×C4×Q8—1731no—
197C2×#41—1731no—
198C2×#41—1736no—
199C2×#41C2×D17311yescharacteristic
200C22×Q8—1731no—
201C4⋊Q8—1733no—
202C4⋊Q8—1733no—
203C4⋊Q8—1733no—
204#127—1733no—
205#126—1733no—
206#126—1733no—
207#125—1733no—
208#124—1733no—
209#124—1733no—
210#124—1736no—
211#124—1733no—
212#123—1736no—
213#123—1733no—
214#123—1733no—
215#123—1736no—
216C22⋊Q8—1736no—
217C22⋊Q8—1733no—
218C22⋊Q8—1731no—
219C22⋊Q8—1736no—
220C22⋊Q8—1733no—
221#121—1733no—
222#121—1733no—
223#121—1733no—
224#121—1733no—
225#121—1733no—
226C4×Q8—1731no—
227C4×Q8—1731no—
228C4×Q8—1733no—
229C4×Q8—1733no—
230C4×Q8—1733no—
231C4×Q8C2×D17311yescharacteristic
232C4×Q8C2×D17311yescharacteristic
233C4×D4—1733no—
234C4×D4—1733no—
235C4×D4—1736no—
236C4×D4—1733no—
237C4×D4—1733no—
238C4×D4—1733no—
239C4×D4C2×D17313yes—
240C4×D4C2×D17313yes—
241#117—1733no—
242#117C2×D17313yes—
243C2×C4⋊C4—1733no—
244C2×C4⋊C4—1733no—
245C2×C4⋊C4C2×D17313yes—
246C2×C42—1733no—
247#41—24no—
248#41—3464no—
249#41—69212no—
250C2×Q8—3464no—
251C2×Q8—1731no—
252C2×Q8—1733no—
253C2×Q8—1736no—
254C2×Q8C22×D17311yescharacteristic
255C2×D4—1733no—
256C2×D4—1733no—
257C2×D4—1736no—
258C2×D4—1733no—
259C2×D4—1733no—
260C2×D4C22×D17313yes—
261C22×C4—1731no—
262C22×C4—1733no—
263C22×C4—1733no—
264C22×C4—1733no—
265C22×C4—1736no—
266C22×C4C22×D17313yes—
267C22×C4C22×D17316yes—
268C4⋊C4—1733no—
269C4⋊C4—1733no—
270C4⋊C4—1733no—
271C4⋊C4—1736no—
272C4⋊C4—1731no—
273C4⋊C4—1731no—
274C4⋊C4—1733no—
275C4⋊C4—1733no—
276C4⋊C4—1733no—
277C4⋊C4—1733no—
278C4⋊C4—1733no—
279C4⋊C4C22×D17313yes—
280C4⋊C4C22×D17313yes—
281C4⋊C4C22×D17313yes—
282C4⋊C4C22×D17311yescharacteristic
283C22⋊C4—1733no—
284C22⋊C4—1736no—
285C22⋊C4—1733no—
286C22⋊C4—1733no—
287C22⋊C4—1733no—
288C22⋊C4—1731no—
289C22⋊C4—1731no—
290C22⋊C4—1733no—
291C22⋊C4—1736no—
292C22⋊C4—1733no—
293C22⋊C4C22×D17313yes—
294C22⋊C4C22×D17313yes—
295C42—1733no—
296C42—1733no—
297C42—1733no—
298C42—1733no—
299C42C22×D17313yes—
300C42C22×D17313yes—
301C23—1733no—
302C23—1731no—
303C23C23×D17313yes—
304Q8—3462no—
305Q8—3466no—
306Q8—6926no—
307Q8#1458382614yescomplemented
308D4—26no—
309D4—3466no—
310D4—3466no—
311D4—3466no—
312D4—3466no—
313D4—6926no—
314C2×C4—26no—
315C2×C4—46no—
316C2×C4—3466no—
317C2×C4—3462no—
318C2×C4—3466no—
319C2×C4—3466no—
320C2×C4—1731no—
321C2×C4—1733no—
322C2×C4—1733no—
323C2×C4—1733no—
324C2×C4—1731no—
325C2×C4—1731no—
326C2×C4—6926no—
327C2×C4C23×D17311yescharacteristic
328C2×C4C23×D17313yes—
329C2×C4C23×D17311yescharacteristic
330C2×C4C23×D17313yes—
331C2×C4C23×D17311yescharacteristic
332C2×C4C23×D17313yes—
333C22—23no—
334C22—23no—
335C22—23no—
336C22—3461no—
337C22—3461no—
338C22—3463no—
339C22—3463no—
340C22—3463no—
341C22—1732no—
342C22C24×D17311yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—21no—
348C4—3461no—
349C4—3463no—
350C4—3463no—
351C4—3463no—
352C4—3461no—
353C4—1732no—
354C4C2×#1458382616yes—
355C2—43no—
356C2—3462no—
357C2—6923no—
358C2C22×#1458382611yescharacteristic
359C2H(5,𝔽2)×D17311yescharacteristic
360C22−1+4×D17311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D173
commutator, socle, cosocleC22×C173C25
Fitting#520×C173C2
hypercenter#520D173
hypocenterC173#3522
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #26012224C2semidirect
2. Q8×Dic346C2semidirect
3. #26012196C2semidirect
4. #26012160C2semidirect
5. #26012153C2semidirect
6. #26012148C2semidirect
7. #26012146C2semidirect
8. #26012142C2semidirect
9. #26012112C2semidirect
10. #26012106C2semidirect
11. #26012038C2semidirect
12. #26012019C2semidirect
13. #26011973C2semidirect
14. C4×Q8×D173C2semidirect
15. #520×C173C2semidirect
142. Q8×C173#41semidirect
180. C173#3522semidirect
195. #520D173semidirect
307. Q8#14583826semidirect

Maximal subgroups

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

subgroupindex
#260122242
Q8×Dic3462
#260121962
#260121602
#260121532
#260121482
#260121462
#260121422
#260121122
#260121062
#260120382
#260120192
#260119732
C4×Q8×D1732
#520×C1732
#3522173