Finite Group Databasebeta

← #28404570

Subgroups — #28404570

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#20299321C213yesmaximal, complemented
2#20299309C213yesmaximal, complemented
3#20299301C213yesmaximal, complemented
4#20299286C213yesmaximal, complemented
5Q8×D452C211yesmaximal, characteristic, complemented
6#20299260C211yesmaximal, characteristic, complemented
7#20299260C211yesmaximal, characteristic, complemented
8#20299222C213yesmaximal, complemented
9#20299211C213yesmaximal, complemented
10#20299135C213yesmaximal, complemented
11#20299133C213yesmaximal, complemented
12#20299080C211yesmaximal, characteristic, complemented
13#20299074C211yesmaximal, characteristic, complemented
14C4×Q8×D113C211yesmaximal, characteristic, complemented
15#557×C113C211yesmaximal, characteristic, complemented
16C2×#14268648C2211yescharacteristic
17C2×Q8×D113C2211yescharacteristic
18C2×Q8×D113C2211yescharacteristic
19#16813622C2213yes—
20#16813617C2213yes—
21#16813617C2216yes—
22#16813617C2213yes—
23#16813617C2216yes—
24#16813613C2216yes—
25#16813613C2213yes—
26#16813611C2213yes—
27#16813610C2213yes—
28#16813610C2213yes—
29#16813607C2216yes—
30#16813607C2213yes—
31#16813607C2211yescharacteristic
32#16813602C2213yes—
33#16813602C2213yes—
34#16813602C2213yes—
35#16813602C2213yes—
36#16813598C2213yes—
37#16813598C2216yes—
38#16813597C2213yes—
39#16813597C2213yes—
40#16813597C2213yes—
41Q8×Dic113C2211yescharacteristic
42C4×Dic226C2213yes—
43C4×Dic226C2213yes—
44#16813587C2213yes—
45#16813587C2213yes—
46C4×D452C2213yes—
47C4×D452C2213yes—
48C4×D452C2213yes—
49C4×D452C2213yes—
50#16813584C2216yes—
51#16813584C2213yes—
52#16813580C2213yes—
53#16813580C2216yes—
54#16813579C2213yes—
55#16813579C2213yes—
56C4⋊C4×D113C2213yes—
57C4⋊C4×D113C2213yes—
58C42×D113C2213yes—
59C4⋊Q8×C113C2213yes—
60#125×C113C2213yes—
61#125×C113C2213yes—
62C4×Q8×C113C2211yescharacteristic
63C4×Q8×C113C2211yescharacteristic
64C4×Q8×C113C2213yes—
65C4×Q8×C113C2211yescharacteristic
66#14268648—24no—
67C2×Dic226C2313yes—
68C2×Dic226C2313yes—
69Q8×D113—24no—
70Q8×D113—24no—
71C2×D452C2313yes—
72C2×D452C2313yes—
73C2×D452C2311yescharacteristic
74C2×D452C2313yes—
75C2×C4×D113C2311yescharacteristic
76C2×C4×D113C2313yes—
77C2×C4×D113C2313yes—
78C2×C4×D113C2313yes—
79C2×C4×D113C2313yes—
80C2×C4×D113C2316yes—
81C4⋊Dic113C2313yes—
82C4⋊Dic113C2316yes—
83C4⋊Dic113C2313yes—
84C4⋊Dic113C2313yes—
85C4⋊Dic113C2313yes—
86#14268626C2313yes—
87#14268626C2316yes—
88#14268626C2311yescharacteristic
89#14268626C2313yes—
90#14268626C2313yes—
91#14268624C2311yescharacteristic
92#14268624C2313yes—
93#14268624C2313yes—
94#14268624C2313yes—
95#14268624C2313yes—
96#14268624C2316yes—
97#14268624C2313yes—
98#14268624C2316yes—
99#14268624C2311yescharacteristic
100#14268624C2311yescharacteristic
101#14268624C2313yes—
102#14268624C2313yes—
103C4×Dic113C2313yes—
104C4×Dic113C2313yes—
105C4×Dic113C2313yes—
106C4×Dic113C2316yes—
107C2×Q8×C113C2313yes—
108C2×Q8×C113C2311yescharacteristic
109C4⋊C4×C113C2313yes—
110C4⋊C4×C113C2313yes—
111C4⋊C4×C113C2313yes—
112C4⋊C4×C113C2313yes—
113C4⋊C4×C113C2316yes—
114C4⋊C4×C113C2313yes—
115C4⋊C4×C113C2311yescharacteristic
116C42×C113C2313yes—
117C42×C113C2313yes—
118C42×C113C2313yes—
119C22×D113C2411yescharacteristic
120C22×D113C2413yes—
121C22×D113C2411yescharacteristic
122Q8×C113—43no—
123Q8×C113#4114yes—
124C2×C4×C113C2411yescharacteristic
125C2×C4×C113C2411yescharacteristic
126C2×C4×C113C2411yescharacteristic
127C2×C4×C113C2413yes—
128C2×C4×C113C2413yes—
129C2×C4×C113C2413yes—
130C2×C4×C113C2413yes—
131C2×Dic113C2411yescharacteristic
132C2×Dic113C2413yes—
133C2×Dic113C2413yes—
134C2×Dic113C2413yes—
135C2×Dic113C2411yescharacteristic
136D452—26no—
137D452—22no—
138D452—26no—
139D452—43no—
140C4×D113—22no—
141C4×D113—26no—
142C4×D113—26no—
143C4×D113—26no—
144C4×D113—46no—
145C4×D113—43no—
146Dic226—26no—
147Dic226—26no—
148C22×C113C2511yescharacteristic
149C2×D113—21no—
150C2×D113—21no—
151C2×D113—21no—
152C2×D113—23no—
153C2×D113—21no—
154C2×D113—23no—
155C2×D113—23no—
156C2×D113—21no—
157C2×D1132−1+412yes—
158C4×C113—21no—
159C4×C113—21no—
160C4×C113—23no—
161C4×C113—23no—
162C4×C113—23no—
163C4×C113—21no—
164C4×C113C2×#4116yes—
165Dic113—23no—
166Dic113—23no—
167Dic113—21no—
168Dic113—21no—
169Dic113—23no—
170C2×C113C22×#4111yescharacteristic
171C2×C113C2×2−1+411yescharacteristic
172C2×C113#58511yescharacteristic
173D113—22no—
174D113—41no—
175D113—43no—
176#3516—1131nomaximal, Sylow
177C113#351611yescharacteristic, complemented, Sylow
178#557D11311yescharacteristic, complemented
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#520—1131no—
191#519—1131no—
192C2×C4×Q8—1131no—
193C2×#41—1131no—
194C22×Q8—1131no—
195C22×Q8—1131no—
196C4⋊Q8—1133no—
197C4⋊Q8C2×D11313yes—
198#126—1136no—
199#126—1133no—
200#126—1133no—
201#126—1136no—
202#125—1133no—
203#125—1136no—
204#125C2×D11313yes—
205#125C2×D11313yes—
206#124—1133no—
207#124—1133no—
208#124—1133no—
209#123—1131no—
210#123—1133no—
211#123—1136no—
212C22⋊Q8—1133no—
213C22⋊Q8—1136no—
214C22⋊Q8—1133no—
215C22⋊Q8—1133no—
216C22⋊Q8—1133no—
217C22⋊Q8—1133no—
218#121—1133no—
219#121—1133no—
220#121—1133no—
221C4×Q8—1131no—
222C4×Q8—1133no—
223C4×Q8—1133no—
224C4×Q8—1133no—
225C4×Q8—1133no—
226C4×Q8C2×D11311yescharacteristic
227C4×Q8C2×D11311yescharacteristic
228C4×Q8C2×D11311yescharacteristic
229C4×Q8C2×D11313yes—
230C4×D4—1133no—
231C4×D4—1136no—
232C4×D4—1133no—
233C4×D4—1133no—
234C4×D4—1133no—
235C4×D4—1133no—
236#117—1133no—
237#117—1136no—
238#117—1133no—
239#117—1133no—
240C2×C4⋊C4—1133no—
241C2×C4⋊C4—1133no—
242C2×C42—1133no—
243#41—2264no—
244C2×Q8—1133no—
245C2×Q8—1133no—
246C2×Q8—2264no—
247C2×Q8—2264no—
248C2×Q8C22×D11313yes—
249C2×Q8C22×D11311yescharacteristic
250C2×D4—1133no—
251C2×D4—1133no—
252C2×D4—1131no—
253C2×D4—1133no—
254C22×C4—1133no—
255C22×C4—1136no—
256C22×C4—1131no—
257C22×C4—1133no—
258C22×C4—1133no—
259C22×C4—1133no—
260C4⋊C4—1133no—
261C4⋊C4—1133no—
262C4⋊C4—1136no—
263C4⋊C4—1131no—
264C4⋊C4—1133no—
265C4⋊C4—1133no—
266C4⋊C4—1136no—
267C4⋊C4—1133no—
268C4⋊C4—1133no—
269C4⋊C4—1133no—
270C4⋊C4C22×D11313yes—
271C4⋊C4C22×D11313yes—
272C4⋊C4C22×D11313yes—
273C4⋊C4C22×D11313yes—
274C4⋊C4C22×D11316yes—
275C4⋊C4C22×D11311yescharacteristic
276C4⋊C4C22×D11313yes—
277C22⋊C4—1133no—
278C22⋊C4—1131no—
279C22⋊C4—1133no—
280C22⋊C4—1133no—
281C22⋊C4—1133no—
282C22⋊C4—1136no—
283C22⋊C4—1133no—
284C22⋊C4—1131no—
285C22⋊C4—1131no—
286C22⋊C4—1136no—
287C22⋊C4—1133no—
288C22⋊C4—1133no—
289C42—1133no—
290C42—1133no—
291C42—1133no—
292C42—1136no—
293C42C22×D11313yes—
294C42C22×D11313yes—
295C42C22×D11313yes—
296C23—1131no—
297C23—1131no—
298C23—1133no—
299Q8—43no—
300Q8—2266no—
301Q8—2266no—
302Q8#1426864814yes—
303D4—4523no—
304D4—2266no—
305D4—2262no—
306D4—2266no—
307C2×C4—1131no—
308C2×C4—1133no—
309C2×C4—1133no—
310C2×C4—1133no—
311C2×C4—1131no—
312C2×C4—4526no—
313C2×C4—4523no—
314C2×C4—2262no—
315C2×C4—2266no—
316C2×C4—2266no—
317C2×C4—2266no—
318C2×C4C23×D11311yescharacteristic
319C2×C4C23×D11313yes—
320C2×C4C23×D11313yes—
321C2×C4C23×D11311yescharacteristic
322C2×C4C23×D11311yescharacteristic
323C2×C4C23×D11313yes—
324C2×C4C23×D11313yes—
325C22—1132no—
326C22—2261no—
327C22—2263no—
328C22—2261no—
329C22—2263no—
330C22—2263no—
331C22—2261no—
332C22—2261no—
333C22—2261no—
334C22C24×D11311yescharacteristic
335C4—21no—
336C4—23no—
337C4—21no—
338C4—23no—
339C4—23no—
340C4—21no—
341C4—2261no—
342C4—2263no—
343C4—2263no—
344C4—2261no—
345C4—2263no—
346C4C2×#1426864816yes—
347C2—4521no—
348C2—4523no—
349C2—2262no—
350C2C22×#1426864811yescharacteristic
351C22−1+4×D11311yescharacteristic
352C2#2029943411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D113
commutator, socle, cosocleC22×C113C25
Fitting#557×C113C2
hypercenter#557D113
hypocenterC113#3516
LCSG ▷ C22×C113 ▷ C113
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. #20299321C2semidirect
2. #20299309C2semidirect
3. #20299301C2semidirect
4. #20299286C2semidirect
5. Q8×D452C2semidirect
6. #20299260C2semidirect
7. #20299260C2semidirect
8. #20299222C2semidirect
9. #20299211C2semidirect
10. #20299135C2semidirect
11. #20299133C2semidirect
12. #20299080C2semidirect
13. #20299074C2semidirect
14. C4×Q8×D113C2semidirect
15. #557×C113C2semidirect
177. C113#3516semidirect
178. #557D113semidirect

Maximal subgroups

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

subgroupindex
#202993212
#202993092
#202993012
#202992862
Q8×D4522
#202992602
#202992222
#202992112
#202991352
#202991332
#202990802
#202990742
C4×Q8×D1132
#557×C1132
#3516113