Finite Group Databasebeta

← #28404559

Subgroups — #28404559

Order 14464.

1012 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20299324C211yesmaximal, characteristic, complemented
2#20299318C213yesmaximal, complemented
3#20299303C213yesmaximal, complemented
4#20299298C213yesmaximal, complemented
5#20299283C213yesmaximal, complemented
6Q8×D452C211yesmaximal, characteristic, complemented
7#20299255C211yesmaximal, characteristic, complemented
8#20299255C211yesmaximal, characteristic, complemented
9#20299213C213yesmaximal, complemented
10#20299210C213yesmaximal, complemented
11#20299130C213yesmaximal, complemented
12#20299126C213yesmaximal, complemented
13#20299074C211yesmaximal, characteristic, complemented
14C4×Q8×D113C211yesmaximal, characteristic, complemented
15#520×C113C211yesmaximal, characteristic, complemented
16C2×Q8×D113C2211yescharacteristic
17C2×Q8×D113C2211yescharacteristic
18#16813622C2213yes—
19#16813621C2213yes—
20#16813617C2213yes—
21#16813615C2216yes—
22#16813615C2213yes—
23#16813615C2216yes—
24#16813614C2213yes—
25#16813613C2213yes—
26#16813612C2216yes—
27#16813612C2213yes—
28#16813609C2213yes—
29#16813609C2213yes—
30#16813608C2213yes—
31#16813607C2211yescharacteristic
32#16813605C2216yes—
33#16813604C2213yes—
34#16813602C2213yes—
35#16813602C2213yes—
36#16813601C2213yes—
37#16813601C2216yes—
38#16813601C2213yes—
39#16813600C2213yes—
40#16813595C2213yes—
41#16813593C2213yes—
42#16813593C2213yes—
43Q8×Dic113C2211yescharacteristic
44Q8×Dic113C2211yescharacteristic
45C4×Dic226C2213yes—
46C4×Dic226C2213yes—
47C4×Dic226C2213yes—
48#16813587C2213yes—
49#16813587C2213yes—
50C4×D452C2213yes—
51C4×C22⋊D113C2213yes—
52C4×C22⋊D113C2216yes—
53C4×C22⋊D113C2213yes—
54#16813581C2216yes—
55#16813580C2213yes—
56#16813579C2213yes—
57C4⋊C4×D113C2213yes—
58C42×D113C2213yes—
59C2×#41×C113C2211yescharacteristic
60C4×Q8×C113C2211yescharacteristic
61C4×Q8×C113C2211yescharacteristic
62C4×D4×C113C2213yes—
63C4×D4×C113C2213yes—
64#117×C113C2213yes—
65C2×C4⋊C4×C113C2213yes—
66C2×Dic226C2313yes—
67C2×Dic226C2313yes—
68C2×Dic226C2313yes—
69Q8×D113—24no—
70Q8×D113—24no—
71C2×D452C2311yescharacteristic
72C2×C22⋊D113C2313yes—
73C2×C22⋊D113C2313yes—
74C2×C4×D113C2311yescharacteristic
75C2×C4×D113C2313yes—
76C2×C4×D113C2313yes—
77C2×C4×D113C2313yes—
78C4⋊Dic113C2313yes—
79C4⋊Dic113C2313yes—
80C4⋊Dic113C2313yes—
81C4⋊Dic113C2316yes—
82C4⋊Dic113C2313yes—
83#14268626C2316yes—
84#14268626C2313yes—
85#14268626C2313yes—
86#14268626C2313yes—
87#14268626C2313yes—
88#14268626C2316yes—
89#14268626C2311yescharacteristic
90#14268626C2313yes—
91C22⋊Dic113C2313yes—
92C22⋊Dic113C2316yes—
93C22⋊Dic113C2316yes—
94C22⋊Dic113C2313yes—
95#14268624C2311yescharacteristic
96#14268624C2313yes—
97#14268624C2311yescharacteristic
98#14268624C2311yescharacteristic
99#14268624C2313yes—
100#14268624C2313yes—
101C4×Dic113C2313yes—
102C4×Dic113C2313yes—
103C4×Dic113C2313yes—
104C4×Dic113C2313yes—
105C4×Dic113C2316yes—
106#41×C113—24no—
107C2×Q8×C113C2311yescharacteristic
108C2×D4×C113C2313yes—
109C22×C4×C113C2313yes—
110C22×C4×C113C2316yes—
111C4⋊C4×C113C2313yes—
112C4⋊C4×C113C2313yes—
113C4⋊C4×C113C2313yes—
114C4⋊C4×C113C2311yescharacteristic
115C22⋊C4×C113C2313yes—
116C22⋊C4×C113C2313yes—
117C42×C113C2313yes—
118C42×C113C2313yes—
119C23×C113C2413yes—
120C22×D113C2411yescharacteristic
121C22×D113C2411yescharacteristic
122Q8×C113#4114yes—
123D4×C113—26no—
124C2×C4×C113—26no—
125C2×C4×C113—46no—
126C2×C4×C113C2411yescharacteristic
127C2×C4×C113C2411yescharacteristic
128C2×C4×C113C2413yes—
129C2×C4×C113C2413yes—
130C2×C4×C113C2411yescharacteristic
131C2×C4×C113C2413yes—
132C22⋊D113—26no—
133C22⋊D113—43no—
134C2×Dic113C2411yescharacteristic
135C2×Dic113C2411yescharacteristic
136C2×Dic113C2413yes—
137C2×Dic113C2413yes—
138C2×Dic113C2413yes—
139C2×Dic113C2413yes—
140D452—22no—
141C4×D113—22no—
142C4×D113—26no—
143C4×D113—26no—
144C4×D113—43no—
145Dic226—26no—
146Dic226—26no—
147Dic226—43no—
148C22×C113—23no—
149C22×C113—23no—
150C22×C113—23no—
151C22×C113C2511yescharacteristic
152C2×D113—21no—
153C2×D113—21no—
154C2×D113—21no—
155C2×D113—21no—
156C2×D113—21no—
157C2×D1132−1+412yes—
158C4×C113—21no—
159C4×C113—21no—
160C4×C113—21no—
161C4×C113—23no—
162C4×C113—23no—
163C4×C113C2×#4116yes—
164Dic113—23no—
165Dic113—23no—
166Dic113—21no—
167Dic113—23no—
168Dic113—21no—
169Dic113—23no—
170C2×C113—43no—
171C2×C113C22×#4111yescharacteristic
172C2×C113C2×2−1+411yescharacteristic
173C2×C113#58511yescharacteristic
174D113—22no—
175D113—41no—
176#3516—1131nomaximal, Sylow
177C113#351611yescharacteristic, complemented, Sylow
178#557—1131no—
179#556—1133no—
180#554—1133no—
181#553—1133no—
182#552—1133no—
183D4×Q8—1131no—
184#548—1131no—
185#548—1131no—
186#542—1133no—
187#541—1133no—
188#529—1133no—
189#529—1133no—
190#520D11311yescharacteristic, complemented
191#519—1131no—
192C2×C4×Q8—1131no—
193C2×#41C2×D11311yescharacteristic
194C22×Q8—1131no—
195C22×Q8—1131no—
196C4⋊Q8—1133no—
197C4⋊Q8—1133no—
198#126—1133no—
199#126—1136no—
200#126—1133no—
201#126—1136no—
202#125—1133no—
203#125—1133no—
204#125—1136no—
205#125—1133no—
206#124—1133no—
207#124—1133no—
208#124—1133no—
209#123—1136no—
210#123—1131no—
211#123—1133no—
212C22⋊Q8—1133no—
213C22⋊Q8—1133no—
214C22⋊Q8—1136no—
215C22⋊Q8—1133no—
216C22⋊Q8—1133no—
217C22⋊Q8—1133no—
218#121—1133no—
219#121—1133no—
220#121—1133no—
221C4×Q8—1131no—
222C4×Q8—1131no—
223C4×Q8—1133no—
224C4×Q8—1133no—
225C4×Q8—1133no—
226C4×Q8—1133no—
227C4×Q8—1133no—
228C4×Q8C2×D11311yescharacteristic
229C4×Q8C2×D11311yescharacteristic
230C4×D4—1133no—
231C4×D4—1133no—
232C4×D4—1136no—
233C4×D4—1133no—
234C4×D4C2×D11313yes—
235C4×D4C2×D11313yes—
236#117—1136no—
237#117—1133no—
238#117—1133no—
239#117C2×D11313yes—
240C2×C4⋊C4—1133no—
241C2×C4⋊C4C2×D11313yes—
242C2×C42—1133no—
243#41—24no—
244C2×Q8—1133no—
245C2×Q8—1133no—
246C2×Q8—1133no—
247C2×Q8—2264no—
248C2×Q8—2264no—
249C2×Q8C22×D11311yescharacteristic
250C2×D4—1133no—
251C2×D4—1133no—
252C2×D4—1131no—
253C2×D4C22×D11313yes—
254C22×C4—1133no—
255C22×C4—1131no—
256C22×C4—1133no—
257C22×C4—1133no—
258C22×C4C22×D11313yes—
259C22×C4C22×D11316yes—
260C4⋊C4—1133no—
261C4⋊C4—1136no—
262C4⋊C4—1133no—
263C4⋊C4—1133no—
264C4⋊C4—1133no—
265C4⋊C4—1133no—
266C4⋊C4—1136no—
267C4⋊C4—1133no—
268C4⋊C4—1133no—
269C4⋊C4—1136no—
270C4⋊C4—1133no—
271C4⋊C4—1131no—
272C4⋊C4—1133no—
273C4⋊C4C22×D11313yes—
274C4⋊C4C22×D11313yes—
275C4⋊C4C22×D11313yes—
276C4⋊C4C22×D11311yescharacteristic
277C22⋊C4—1133no—
278C22⋊C4—1136no—
279C22⋊C4—1136no—
280C22⋊C4—1133no—
281C22⋊C4—1133no—
282C22⋊C4—1131no—
283C22⋊C4—1131no—
284C22⋊C4—1133no—
285C22⋊C4—1131no—
286C22⋊C4—1133no—
287C22⋊C4C22×D11313yes—
288C22⋊C4C22×D11313yes—
289C42—1133no—
290C42—1133no—
291C42—1133no—
292C42—1136no—
293C42—1133no—
294C42C22×D11313yes—
295C42C22×D11313yes—
296C23—1131no—
297C23—1131no—
298C23C23×D11313yes—
299Q8—4523no—
300Q8—2266no—
301Q8—2266no—
302Q8#1426864714yes—
303D4—26no—
304D4—4523no—
305D4—2262no—
306D4—2266no—
307C2×C4—26no—
308C2×C4—46no—
309C2×C4—1131no—
310C2×C4—1131no—
311C2×C4—1133no—
312C2×C4—1133no—
313C2×C4—1133no—
314C2×C4—1133no—
315C2×C4—4523no—
316C2×C4—2262no—
317C2×C4—2266no—
318C2×C4—2266no—
319C2×C4C23×D11311yescharacteristic
320C2×C4C23×D11313yes—
321C2×C4C23×D11311yescharacteristic
322C2×C4C23×D11313yes—
323C2×C4C23×D11311yescharacteristic
324C2×C4C23×D11313yes—
325C22—23no—
326C22—23no—
327C22—23no—
328C22—1132no—
329C22—2261no—
330C22—2261no—
331C22—2261no—
332C22—2261no—
333C22—2261no—
334C22C24×D11311yescharacteristic
335C4—21no—
336C4—21no—
337C4—21no—
338C4—23no—
339C4—23no—
340C4—2263no—
341C4—2263no—
342C4—2261no—
343C4—2263no—
344C4—2263no—
345C4—2261no—
346C4C2×#1426864716yes—
347C2—43no—
348C2—4521no—
349C2—2262no—
350C2C22×#1426864711yescharacteristic
351C22−1+4×D11311yescharacteristic
352C2#2029943311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D113
commutator, socle, cosocleC22×C113C25
Fitting#520×C113C2
hypercenter#520D113
hypocenterC113#3516
LCSG ▷ C22×C113 ▷ C113
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #20299324C2semidirect
2. #20299318C2semidirect
3. #20299303C2semidirect
4. #20299298C2semidirect
5. #20299283C2semidirect
6. Q8×D452C2semidirect
7. #20299255C2semidirect
8. #20299255C2semidirect
9. #20299213C2semidirect
10. #20299210C2semidirect
11. #20299130C2semidirect
12. #20299126C2semidirect
13. #20299074C2semidirect
14. C4×Q8×D113C2semidirect
15. #520×C113C2semidirect
177. C113#3516semidirect
190. #520D113semidirect

Maximal subgroups

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

subgroupindex
#202993242
#202993182
#202993032
#202992982
#202992832
Q8×D4522
#202992552
#202992132
#202992102
#202991302
#202991262
#202990742
C4×Q8×D1132
#520×C1132
#3516113