Finite Group Databasebeta

← #24919725

Subgroups — #24919725

Order 10624.

380 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128360C211yesmaximal, characteristic
2#19128058C211yesmaximal, characteristic, complemented
3#19128036C211yesmaximal, characteristic, complemented
4#19128025C211yesmaximal, characteristic, complemented
5#19128005C211yesmaximal, characteristic, complemented
6#19127957C211yesmaximal, characteristic, complemented
7#19127949C211yesmaximal, characteristic, complemented
8#19127923C211yesmaximal, characteristic, complemented
9#19127861C211yesmaximal, characteristic, complemented
10#19127856C211yesmaximal, characteristic, complemented
11C4×#13050580C211yesmaximal, characteristic, complemented
12#19127779C211yesmaximal, characteristic, complemented
13#19127709C211yesmaximal, characteristic, complemented
14#19127700C211yesmaximal, characteristic, complemented
15#552×C83C211yesmaximal, characteristic, complemented
16C2×#13050590C2211yescharacteristic
17C22×Dic166C2211yescharacteristic
18C2×#13050580C2211yescharacteristic
19C2×#13050580C2211yescharacteristic
20C2×#13050577C2211yescharacteristic
21C2×#13050574C2211yescharacteristic
22C2×#13050574C2211yescharacteristic
23#14581780C2211yescharacteristic
24#14581778—21no—
25#14581771C2211yescharacteristic
26#14581770—21no—
27#14581766C2211yescharacteristic
28#14581764—21no—
29C4×Dic166C2211yescharacteristic
30(C22⋊D83)⋊C4—21no—
31#14581725C2211yescharacteristic
32#14581725C2211yescharacteristic
33#14581723C2211yescharacteristic
34C4⋊(C83⋊C8)C2211yescharacteristic
35#14581716C2211yescharacteristic
36#14581715C2211yescharacteristic
37#14581715C2211yescharacteristic
38#14581715C2211yescharacteristic
39#14581715C2211yescharacteristic
40#14581715C2211yescharacteristic
41#14581715C2211yescharacteristic
42#14581712C2211yescharacteristic
43#14581712C2211yescharacteristic
44#14581712C2211yescharacteristic
45C22⋊(C83⋊C8)C2211yescharacteristic
46C22⋊(C83⋊C8)C2211yescharacteristic
47C4×C83⋊C8C2211yescharacteristic
48#126×C83—21no—
49#125×C83—21no—
50#125×C83C2211yescharacteristic
51#123×C83—21no—
52C22⋊Q8×C83C2211yescharacteristic
53C22⋊Q8×C83C2211yescharacteristic
54C4×Q8×C83C2211yescharacteristic
55C4×D4×C83C2211yescharacteristic
56#117×C83—21no—
57#117×C83C2211yescharacteristic
58C2×C4⋊C4×C83C2211yescharacteristic
59#13050590—22no—
60#13050590—41no—
61C2×Dic166—22no—
62C2×Dic166—21no—
63C2×Dic166—41no—
64C2×Dic166C2311yescharacteristic
65C2×Dic166C2311yescharacteristic
66C2×D332C2311yescharacteristic
67C2×C22⋊D83—21no—
68C2×C22⋊D83—21no—
69C22×Dic83—21no—
70C2×C4×D83—21no—
71#13050580—22no—
72#13050580—41no—
73#13050577—41no—
74#13050574—41no—
75#13050574—41no—
76C2×C83⋊C8C2311yescharacteristic
77C2×C83⋊C8C2311yescharacteristic
78C2×C83⋊C8C2311yescharacteristic
79C2×C83⋊C8C2311yescharacteristic
80C4⋊Dic83C2311yescharacteristic
81Dic83⋊C4—21no—
82Dic83⋊C4—21no—
83C22⋊Dic83—21no—
84(C2×D83)⋊C4—21no—
85(C2×D83)⋊C4—21no—
86(C2×D83)⋊C4—21no—
87C4×Dic83—21no—
88C2×Q8×C83C2311yescharacteristic
89C2×D4×C83D411yescharacteristic
90C22×C4×C83—21no—
91C22×C4×C83C2311yescharacteristic
92C22×C4×C83C2311yescharacteristic
93C4⋊C4×C83—21no—
94C4⋊C4×C83—21no—
95C4⋊C4×C83—21no—
96C4⋊C4×C83—21no—
97C4⋊C4×C83—21no—
98C4⋊C4×C83—21no—
99C4⋊C4×C83D411yescharacteristic
100C4⋊C4×C83C2311yescharacteristic
101C4⋊C4×C83C2311yescharacteristic
102C4⋊C4×C83C2311yescharacteristic
103C22⋊C4×C83—21no—
104C22⋊C4×C83—21no—
105C22⋊C4×C83—21no—
106C22⋊C4×C83D411yescharacteristic
107C22⋊C4×C83D411yescharacteristic
108C42×C83—21no—
109C42×C83—21no—
110C42×C83C2311yescharacteristic
111C23×C83C2×D411yescharacteristic
112C23×C83C2×D411yescharacteristic
113C22×D83—21no—
114Q8×C83—21no—
115Q8×C83—21no—
116D4×C83—41no—
117C2×C4×C83—21no—
118C2×C4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—42no—
126C2×C4×C83C2×D411yescharacteristic
127C2×C4×C83C2×D411yescharacteristic
128C2×C4×C83C2×D411yescharacteristic
129C2×C4×C83C2×D411yescharacteristic
130C2×C4×C83C2411yescharacteristic
131C22⋊D83—42no—
132C22⋊D83—42no—
133C2×Dic83—21no—
134C2×Dic83—21no—
135C2×Dic83—21no—
136C2×Dic83—42no—
137D332—21no—
138D332—21no—
139C4×D83—42no—
140Dic166—21no—
141Dic166—21no—
142Dic166—21no—
143Dic166—42no—
144Dic166#4112yes—
145C83⋊C8—21no—
146C83⋊C8—21no—
147C83⋊C8—21no—
148C83⋊C8—21no—
149C22×C83—21no—
150C22×C83—21no—
151C22×C83—21no—
152C22×C83—21no—
153C22×C83—21no—
154C22×C83—21no—
155C22×C83C22×D411yescharacteristic
156C2×D83—41no—
157C2×D83—41no—
158C2×D83—41no—
159C4×C83—21no—
160C4×C83—21no—
161C4×C83—21no—
162C4×C83—21no—
163C4×C83—41no—
164C4×C83—41no—
165C4×C83—41no—
166C4×C83—41no—
167C4×C83C2×#4111yescharacteristic
168C4×C83H(5,𝔽2)11yescharacteristic
169Dic83—22no—
170Dic83—41no—
171Dic83—41no—
172C2×C83—41no—
173C2×C83—41no—
174C2×C83#54611yescharacteristic
175C2×C83#57511yescharacteristic
176C2×C83#57811yescharacteristic
177D83—81no—
178#3327—831nomaximal, Sylow
179C83#332711yescharacteristic, complemented, Sylow
180#552D8311yescharacteristic, complemented
181#548—831no—
182#488—831no—
183#484—831no—
184#482—831no—
185#479—831no—
186#470—831no—
187#468—831no—
188#464—831no—
189#452—831no—
190#451—831no—
191C4×Q16—831no—
192#436—831no—
193#419—831no—
194#417—831no—
195C2×#41—831no—
196C22×Q8—831no—
197C2×Q16—831no—
198C2×Q16—831no—
199C2×SD16—831no—
200C2×M16—831no—
201C2×M16—831no—
202#126—21no—
203#125—21no—
204#125C2×D8311yescharacteristic
205#124—831no—
206#124—1661no—
207#123—21no—
208C22⋊Q8—831no—
209C22⋊Q8—1661no—
210C22⋊Q8C2×D8311yescharacteristic
211C22⋊Q8C2×D8311yescharacteristic
212#121—831no—
213#121—1661no—
214C4×Q8—831no—
215C4×Q8C2×D8311yescharacteristic
216C4×D4—1661no—
217C4×D4C2×D8311yescharacteristic
218#117—21no—
219#117C2×D8311yescharacteristic
220C2×C4⋊C4C2×D8311yescharacteristic
221#107—831no—
222#107—831no—
223#106—831no—
224C4⋊C8—831no—
225#103—831no—
226#103—831no—
227#103—831no—
228#103—831no—
229#103—831no—
230#103—831no—
231#103—831no—
232#102—831no—
233#102—831no—
234#102—831no—
235C22⋊C8—831no—
236C22⋊C8—831no—
237C4×C8—831no—
238#41—3321no—
239#41—1662no—
240C2×Q8—3321no—
241C2×Q8—831no—
242C2×Q8—831no—
243C2×Q8—1662no—
244C2×Q8—1661no—
245C2×Q8C22×D8311yescharacteristic
246C2×D4—831no—
247C2×D4—1661no—
248C2×D4—1661no—
249C2×D4C22⋊D8311yescharacteristic
250C22×C4—21no—
251C22×C4—1661no—
252C22×C4—1661no—
253C22×C4C22×D8311yescharacteristic
254C22×C4C22×D8311yescharacteristic
255Q16—3321no—
256Q16—1662no—
257SD16—3321no—
258M16—3321no—
259M16—3321no—
260C2×C8—831no—
261C2×C8—831no—
262C2×C8—831no—
263C2×C8—831no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—831no—
271C4⋊C4—1661no—
272C4⋊C4—1661no—
273C4⋊C4C22⋊D8311yescharacteristic
274C4⋊C4C22×D8311yescharacteristic
275C4⋊C4C22×D8311yescharacteristic
276C4⋊C4C22×D8311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—1661no—
281C22⋊C4—1661no—
282C22⋊C4—1661no—
283C22⋊C4—1661no—
284C22⋊C4C22⋊D8311yescharacteristic
285C22⋊C4C22⋊D8311yescharacteristic
286C42—21no—
287C42—21no—
288C42—1661no—
289C42C22×D8311yescharacteristic
290C23—1661no—
291C23C2×C22⋊D8311yescharacteristic
292C23C2×C22⋊D8311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—3322no—
296Q8—832no—
297Q8—1661no—
298Q8—1661no—
299Q8—1661no—
300D4—41no—
301D4—3322no—
302D4—3322no—
303D4—1661no—
304D4—1661no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—3322no—
315C2×C4—3322no—
316C2×C4—1661no—
317C2×C4—1661no—
318C2×C4—1661no—
319C2×C4C2×C22⋊D8311yescharacteristic
320C2×C4C2×C22⋊D8311yescharacteristic
321C2×C4C2×C22⋊D8311yescharacteristic
322C2×C4C2×C22⋊D8311yescharacteristic
323C2×C4C23×D8311yescharacteristic
324C8—1661no—
325C8—1661no—
326C8—1661no—
327C8—1661no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—3321no—
335C22—3321no—
336C22—3321no—
337C22C22×C22⋊D8311yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—3321no—
347C4—3321no—
348C4—1662no—
349C4#41×D8311yescharacteristic
350C4#1458183811yescharacteristic
351C2—41no—
352C2—41no—
353C2—6641no—
354C2#1912833911yescharacteristic
355C2#1912849911yescharacteristic
356C2#1912851611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#552×C83C2
socleC22×C83C22×D4
prime radical#19128360C2
hypercenter#552D83
hypocenterC83#3327
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #19128058C2semidirect
3. #19128036C2semidirect
4. #19128025C2semidirect
5. #19128005C2semidirect
6. #19127957C2semidirect
7. #19127949C2semidirect
8. #19127923C2semidirect
9. #19127861C2semidirect
10. #19127856C2semidirect
11. C4×#13050580C2semidirect
12. #19127779C2semidirect
13. #19127709C2semidirect
14. #19127700C2semidirect
15. #552×C83C2semidirect
179. C83#3327semidirect
180. #552D83semidirect

Maximal subgroups

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

subgroupindex
#191283602
#191280582
#191280362
#191280252
#191280052
#191279572
#191279492
#191279232
#191278612
#191278562
C4×#130505802
#191277792
#191277092
#191277002
#552×C832
#332783