Finite Group Databasebeta

← #19148591

Subgroups — #19148591

Order 5504.

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#14583622C213yesmaximal, complemented
2#14583610C213yesmaximal, complemented
3#14583602C213yesmaximal, complemented
4#14583587C213yesmaximal, complemented
5Q8×D172C211yesmaximal, characteristic, complemented
6#14583561C211yesmaximal, characteristic, complemented
7#14583561C211yesmaximal, characteristic, complemented
8#14583523C213yesmaximal, complemented
9#14583512C213yesmaximal, complemented
10#14583436C213yesmaximal, complemented
11#14583434C213yesmaximal, complemented
12#14583381C211yesmaximal, characteristic, complemented
13#14583375C211yesmaximal, characteristic, complemented
14C4×Q8×D43C211yesmaximal, characteristic, complemented
15C43×#557C211yesmaximal, characteristic, complemented
16C2×#10621902C2211yescharacteristic
17C2×Q8×D43C2211yescharacteristic
18C2×Q8×D43C2211yescharacteristic
19#13063325C2213yes—
20#13063320C2213yes—
21#13063320C2216yes—
22#13063320C2213yes—
23#13063320C2216yes—
24#13063316C2216yes—
25#13063316C2213yes—
26#13063314C2213yes—
27#13063313C2213yes—
28#13063313C2213yes—
29#13063310C2216yes—
30#13063310C2213yes—
31#13063310C2211yescharacteristic
32#13063305C2213yes—
33#13063305C2213yes—
34#13063305C2213yes—
35#13063305C2213yes—
36#13063301C2213yes—
37#13063301C2216yes—
38#13063300C2213yes—
39#13063300C2213yes—
40#13063300C2213yes—
41Q8×Dic43C2211yescharacteristic
42C4×Dic86C2213yes—
43C4×Dic86C2213yes—
44#13063290C2213yes—
45#13063290C2213yes—
46C4×D172C2213yes—
47C4×D172C2213yes—
48C4×D172C2213yes—
49C4×D172C2213yes—
50#13063287C2216yes—
51#13063287C2213yes—
52#13063283C2213yes—
53#13063283C2216yes—
54#13063282C2213yes—
55#13063282C2213yes—
56C4⋊C4×D43C2213yes—
57C4⋊C4×D43C2213yes—
58C42×D43C2213yes—
59C4⋊Q8×C43C2213yes—
60#125×C43C2213yes—
61#125×C43C2213yes—
62C4×Q8×C43C2211yescharacteristic
63C4×Q8×C43C2211yescharacteristic
64C4×Q8×C43C2211yescharacteristic
65C4×Q8×C43C2213yes—
66C2×Q8×C43C2313yes—
67C2×Q8×C43C2311yescharacteristic
68#10621902—24no—
69Q8×D43—24no—
70Q8×D43—24no—
71C2×D172C2313yes—
72C2×D172C2313yes—
73C2×D172C2311yescharacteristic
74C2×D172C2313yes—
75C2×C4×D43C2311yescharacteristic
76C2×C4×D43C2313yes—
77C2×C4×D43C2313yes—
78C2×C4×D43C2313yes—
79C2×C4×D43C2313yes—
80C2×C4×D43C2316yes—
81C2×Dic86C2313yes—
82C2×Dic86C2313yes—
83C4⋊C4×C43C2311yescharacteristic
84C4⋊C4×C43C2313yes—
85C4⋊C4×C43C2313yes—
86C4⋊C4×C43C2313yes—
87C4⋊C4×C43C2313yes—
88C4⋊C4×C43C2313yes—
89C4⋊C4×C43C2316yes—
90C42×C43C2313yes—
91C42×C43C2313yes—
92C42×C43C2313yes—
93(C2×D43)⋊C4C2311yescharacteristic
94(C2×D43)⋊C4C2313yes—
95(C2×D43)⋊C4C2313yes—
96(C2×D43)⋊C4C2313yes—
97(C2×D43)⋊C4C2313yes—
98(C2×D43)⋊C4C2316yes—
99(C2×D43)⋊C4C2313yes—
100(C2×D43)⋊C4C2316yes—
101(C2×D43)⋊C4C2311yescharacteristic
102(C2×D43)⋊C4C2311yescharacteristic
103(C2×D43)⋊C4C2313yes—
104(C2×D43)⋊C4C2313yes—
105C4⋊Dic43C2313yes—
106C4⋊Dic43C2316yes—
107C4⋊Dic43C2313yes—
108C4⋊Dic43C2313yes—
109C4⋊Dic43C2313yes—
110Dic43⋊C4C2313yes—
111Dic43⋊C4C2316yes—
112Dic43⋊C4C2311yescharacteristic
113Dic43⋊C4C2313yes—
114Dic43⋊C4C2313yes—
115C4×Dic43C2313yes—
116C4×Dic43C2313yes—
117C4×Dic43C2313yes—
118C4×Dic43C2316yes—
119C22×D43C2411yescharacteristic
120C22×D43C2413yes—
121C22×D43C2411yescharacteristic
122Q8×C43—43no—
123Q8×C43#4114yes—
124C2×C4×C43C2411yescharacteristic
125C2×C4×C43C2413yes—
126C2×C4×C43C2413yes—
127C2×C4×C43C2411yescharacteristic
128C2×C4×C43C2411yescharacteristic
129C2×C4×C43C2413yes—
130C2×C4×C43C2413yes—
131C2×Dic43C2411yescharacteristic
132C2×Dic43C2413yes—
133C2×Dic43C2413yes—
134C2×Dic43C2413yes—
135C2×Dic43C2411yescharacteristic
136D172—26no—
137D172—22no—
138D172—26no—
139D172—43no—
140C4×D43—22no—
141C4×D43—26no—
142C4×D43—26no—
143C4×D43—26no—
144C4×D43—46no—
145C4×D43—43no—
146Dic86—26no—
147Dic86—26no—
148C22×C43C2511yescharacteristic
149C2×D43—21no—
150C2×D43—21no—
151C2×D43—21no—
152C2×D43—23no—
153C2×D43—21no—
154C2×D43—23no—
155C2×D43—23no—
156C2×D43—21no—
157C2×D432−1+412yes—
158C4×C43—21no—
159C4×C43—23no—
160C4×C43—21no—
161C4×C43—23no—
162C4×C43—23no—
163C4×C43—21no—
164C4×C43C2×#4116yes—
165Dic43—23no—
166Dic43—23no—
167Dic43—21no—
168Dic43—21no—
169Dic43—23no—
170#3516—431nomaximal, Sylow
171C2×C43C22×#4111yescharacteristic
172C2×C43C2×2−1+411yescharacteristic
173C2×C43#58511yescharacteristic
174D43—22no—
175D43—41no—
176D43—43no—
177#557D4311yescharacteristic, complemented
178#556—433no—
179#554—433no—
180#553—433no—
181#552—433no—
182D4×Q8—431no—
183#548—431no—
184#548—431no—
185#542—433no—
186#541—433no—
187#529—433no—
188#529—433no—
189#520—431no—
190#519—431no—
191C2×C4×Q8—431no—
192C43#351611yescharacteristic, complemented, Sylow
193C2×#41—431no—
194C22×Q8—431no—
195C22×Q8—431no—
196C4⋊Q8—433no—
197C4⋊Q8C2×D4313yes—
198#126—436no—
199#126—433no—
200#126—433no—
201#126—436no—
202#125—433no—
203#125—436no—
204#125C2×D4313yes—
205#125C2×D4313yes—
206#124—433no—
207#124—433no—
208#124—433no—
209#123—431no—
210#123—433no—
211#123—436no—
212C22⋊Q8—433no—
213C22⋊Q8—436no—
214C22⋊Q8—433no—
215C22⋊Q8—433no—
216C22⋊Q8—433no—
217C22⋊Q8—433no—
218#121—433no—
219#121—433no—
220#121—433no—
221C4×Q8—431no—
222C4×Q8—433no—
223C4×Q8—433no—
224C4×Q8—433no—
225C4×Q8—433no—
226C4×Q8C2×D4311yescharacteristic
227C4×Q8C2×D4311yescharacteristic
228C4×Q8C2×D4311yescharacteristic
229C4×Q8C2×D4313yes—
230C4×D4—433no—
231C4×D4—436no—
232C4×D4—433no—
233C4×D4—433no—
234C4×D4—433no—
235C4×D4—433no—
236#117—433no—
237#117—436no—
238#117—433no—
239#117—433no—
240C2×C4⋊C4—433no—
241C2×C4⋊C4—433no—
242C2×C42—433no—
243#41—864no—
244C2×Q8—433no—
245C2×Q8—433no—
246C2×Q8—864no—
247C2×Q8—864no—
248C2×Q8C22×D4313yes—
249C2×Q8C22×D4311yescharacteristic
250C2×D4—433no—
251C2×D4—431no—
252C2×D4—433no—
253C2×D4—433no—
254C22×C4—431no—
255C22×C4—433no—
256C22×C4—433no—
257C22×C4—433no—
258C22×C4—433no—
259C22×C4—436no—
260C4⋊C4—433no—
261C4⋊C4—433no—
262C4⋊C4—436no—
263C4⋊C4—431no—
264C4⋊C4—433no—
265C4⋊C4—433no—
266C4⋊C4—436no—
267C4⋊C4—433no—
268C4⋊C4—433no—
269C4⋊C4—433no—
270C4⋊C4C22×D4313yes—
271C4⋊C4C22×D4313yes—
272C4⋊C4C22×D4313yes—
273C4⋊C4C22×D4313yes—
274C4⋊C4C22×D4316yes—
275C4⋊C4C22×D4311yescharacteristic
276C4⋊C4C22×D4313yes—
277C22⋊C4—433no—
278C22⋊C4—431no—
279C22⋊C4—433no—
280C22⋊C4—433no—
281C22⋊C4—433no—
282C22⋊C4—436no—
283C22⋊C4—433no—
284C22⋊C4—431no—
285C22⋊C4—431no—
286C22⋊C4—436no—
287C22⋊C4—433no—
288C22⋊C4—433no—
289C42—433no—
290C42—433no—
291C42—433no—
292C42—436no—
293C42C22×D4313yes—
294C42C22×D4313yes—
295C42C22×D4313yes—
296C23—431no—
297C23—431no—
298C23—433no—
299Q8—43no—
300Q8—866no—
301Q8—866no—
302Q8#1062190214yes—
303D4—866no—
304D4—862no—
305D4—866no—
306D4—1723no—
307C2×C4—431no—
308C2×C4—433no—
309C2×C4—433no—
310C2×C4—433no—
311C2×C4—431no—
312C2×C4—862no—
313C2×C4—866no—
314C2×C4—866no—
315C2×C4—866no—
316C2×C4—1726no—
317C2×C4—1723no—
318C2×C4C23×D4311yescharacteristic
319C2×C4C23×D4313yes—
320C2×C4C23×D4313yes—
321C2×C4C23×D4311yescharacteristic
322C2×C4C23×D4311yescharacteristic
323C2×C4C23×D4313yes—
324C2×C4C23×D4313yes—
325C22—432no—
326C22—861no—
327C22—861no—
328C22—861no—
329C22—863no—
330C22—861no—
331C22—863no—
332C22—863no—
333C22—861no—
334C22C24×D4311yescharacteristic
335C4—21no—
336C4—23no—
337C4—21no—
338C4—23no—
339C4—23no—
340C4—21no—
341C4—861no—
342C4—863no—
343C4—863no—
344C4—861no—
345C4—863no—
346C4C2×#1062190216yes—
347C2—862no—
348C2—1721no—
349C2—1723no—
350C2C22×#1062190211yescharacteristic
351C22−1+4×D4311yescharacteristic
352C2#1458373511yescharacteristic
subgroupG/H
center, FrattiniC22C24×D43
commutator, socleC22×C43C25
FittingC43×#557C2
hypercenter#557D43
hypocenterC43#3516
LCSG ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. #14583622C2semidirect
2. #14583610C2semidirect
3. #14583602C2semidirect
4. #14583587C2semidirect
5. Q8×D172C2semidirect
6. #14583561C2semidirect
7. #14583561C2semidirect
8. #14583523C2semidirect
9. #14583512C2semidirect
10. #14583436C2semidirect
11. #14583434C2semidirect
12. #14583381C2semidirect
13. #14583375C2semidirect
14. C4×Q8×D43C2semidirect
15. C43×#557C2semidirect
177. #557D43semidirect
192. C43#3516semidirect