Finite Group Databasebeta

← #24918928

Subgroups — #24918928

Order 10624.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128322C211yesmaximal, characteristic, complemented
2#19127991C211yesmaximal, characteristic, complemented
3#19127988C211yesmaximal, characteristic, complemented
4#19127986C211yesmaximal, characteristic, complemented
5#19127983C211yesmaximal, characteristic, complemented
6#19127957C211yesmaximal, characteristic, complemented
7#19127952C211yesmaximal, characteristic
8#19127932C211yesmaximal, characteristic, complemented
9#19127928C211yesmaximal, characteristic, complemented
10#19127923C211yesmaximal, characteristic, complemented
11#19127920C211yesmaximal, characteristic, complemented
12#19127916C211yesmaximal, characteristic, complemented
13#19127913C211yesmaximal, characteristic, complemented
14#19127728C211yesmaximal, characteristic, complemented
15#542×C83C211yesmaximal, characteristic, complemented
16C2×#13050580C2211yescharacteristic
17C2×#13050578C2211yescharacteristic
18C2×#13050577C2211yescharacteristic
19C2×#13050575C2211yescharacteristic
20C2×#13050574C2211yescharacteristic
21C22×C83⋊C8C2211yescharacteristic
22#14581792C2211yescharacteristic
23#14581787—21no—
24#14581780C2211yescharacteristic
25#14581778—21no—
26#14581774—21no—
27#14581771C2211yescharacteristic
28#14581770—21no—
29#14581766C2211yescharacteristic
30C4×Dic166C2211yescharacteristic
31C4×D332C2211yescharacteristic
32#14581723C2211yescharacteristic
33#14581723C2211yescharacteristic
34#14581723C2211yescharacteristic
35#14581723C2211yescharacteristic
36C4⋊(C83⋊C8)C2211yescharacteristic
37C4⋊(C83⋊C8)C2211yescharacteristic
38C4⋊(C83⋊C8)C2211yescharacteristic
39C4⋊(C83⋊C8)C2211yescharacteristic
40#14581716C2211yescharacteristic
41#14581716C2211yescharacteristic
42#14581715C2211yescharacteristic
43#14581715C2211yescharacteristic
44#14581713C2211yescharacteristic
45#14581713C2211yescharacteristic
46#14581712C2211yescharacteristic
47#14581712C2211yescharacteristic
48C4⋊Q8×C83C2211yescharacteristic
49#126×C83—21no—
50#124×C83—21no—
51#124×C83C2211yescharacteristic
52#123×C83—21no—
53C22⋊Q8×C83—21no—
54C22⋊Q8×C83C2211yescharacteristic
55#121×C83C2211yescharacteristic
56C4×Q8×C83C2211yescharacteristic
57C4×D4×C83C2211yescharacteristic
58#117×C83C2211yescharacteristic
59C2×Dic166—21no—
60C2×Dic166C2311yescharacteristic
61C2×D332C2311yescharacteristic
62C2×C22⋊D83—21no—
63C2×C4×D83—21no—
64#13050580—41no—
65#13050578—41no—
66#13050577—41no—
67#13050575—41no—
68#13050574—41no—
69C2×C83⋊C8—41no—
70C2×C83⋊C8C2311yescharacteristic
71C2×C83⋊C8C2311yescharacteristic
72C2×C83⋊C8C2311yescharacteristic
73C2×C83⋊C8C2311yescharacteristic
74C4⋊Dic83—21no—
75C4⋊Dic83C2311yescharacteristic
76C4⋊Dic83C2311yescharacteristic
77Dic83⋊C4—21no—
78Dic83⋊C4—21no—
79C22⋊Dic83—21no—
80(C2×D83)⋊C4—21no—
81(C2×D83)⋊C4—21no—
82(C2×D83)⋊C4—21no—
83(C2×D83)⋊C4—21no—
84C4×Dic83—21no—
85C2×Q8×C83—21no—
86C2×Q8×C83C2311yescharacteristic
87C2×D4×C83—21no—
88C2×D4×C83C2311yescharacteristic
89C22×C4×C83—21no—
90C22×C4×C83C2311yescharacteristic
91C4⋊C4×C83—21no—
92C4⋊C4×C83—21no—
93C4⋊C4×C83—21no—
94C4⋊C4×C83D411yescharacteristic
95C4⋊C4×C83D411yescharacteristic
96C4⋊C4×C83C2311yescharacteristic
97C4⋊C4×C83C2311yescharacteristic
98C22⋊C4×C83—21no—
99C22⋊C4×C83—21no—
100C22⋊C4×C83—21no—
101C22⋊C4×C83—21no—
102C22⋊C4×C83—21no—
103C22⋊C4×C83D411yescharacteristic
104C22⋊C4×C83D411yescharacteristic
105C42×C83—21no—
106C42×C83C2311yescharacteristic
107C42×C83C2311yescharacteristic
108C23×C83—21no—
109C23×C83C2×D411yescharacteristic
110C22×D83—21no—
111Q8×C83—21no—
112Q8×C83—21no—
113Q8×C83—81no—
114D4×C83—21no—
115D4×C83—21no—
116D4×C83—81no—
117C2×C4×C83—21no—
118C2×C4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—81no—
123C2×C4×C83C2×D411yescharacteristic
124C2×C4×C83C2×D411yescharacteristic
125C2×C4×C83C2×D411yescharacteristic
126C2×C4×C83C2×D411yescharacteristic
127C2×C4×C83C2×D411yescharacteristic
128C2×C4×C83C2411yescharacteristic
129C22⋊D83—81no—
130C2×Dic83—21no—
131C2×Dic83—21no—
132C2×Dic83—21no—
133D332—21no—
134D332—21no—
135C4×D83—81no—
136Dic166—21no—
137Dic166—21no—
138Dic166—81no—
139C83⋊C8—21no—
140C83⋊C8—21no—
141C83⋊C8—21no—
142C83⋊C8—21no—
143C22×C83—21no—
144C22×C83—21no—
145C22×C83—21no—
146C22×C83—41no—
147C22×C83—41no—
148C22×C83—41no—
149C22×C83C22×D411yescharacteristic
150C2×D83—41no—
151C2×D83—41no—
152C2×D83—41no—
153C4×C83—21no—
154C4×C83—21no—
155C4×C83—21no—
156C4×C83—21no—
157C4×C83—21no—
158C4×C83—41no—
159C4×C83—41no—
160C4×C83—41no—
161C4×C83H(5,𝔽2)11yescharacteristic
162C4×C832−1+411yescharacteristic
163Dic83—41no—
164Dic83—41no—
165Dic83—41no—
166C2×C83—41no—
167C2×C83—81no—
168C2×C83#53711yescharacteristic
169C2×C83#57711yescharacteristic
170C2×C83#57711yescharacteristic
171D83—81no—
172#3263—831nomaximal, Sylow
173C83#326311yescharacteristic, complemented, Sylow
174#542—831no—
175#542D8311yescharacteristic, complemented
176#476—831no—
177#476—831no—
178#475—831no—
179#475—831no—
180#470—831no—
181#469—831no—
182#465—831no—
183#465—831no—
184#464—831no—
185#464—831no—
186#463—831no—
187#463—831no—
188#424—831no—
189C2×Q16—831no—
190C2×SD16—831no—
191C2×SD16—831no—
192C2×D8—831no—
193C2×M16—831no—
194C22×C8—831no—
195C4⋊Q8—831no—
196C4⋊Q8C2×D8311yescharacteristic
197#126—21no—
198#126—1661no—
199#124—21no—
200#124—831no—
201#124—1661no—
202#124C2×D8311yescharacteristic
203#123—21no—
204#123—1661no—
205C22⋊Q8—21no—
206C22⋊Q8—831no—
207C22⋊Q8—1661no—
208C22⋊Q8C2×D8311yescharacteristic
209#121—831no—
210#121C2×D8311yescharacteristic
211C4×Q8—831no—
212C4×Q8C2×D8311yescharacteristic
213C4×D4—831no—
214C4×D4C2×D8311yescharacteristic
215#117C2×D8311yescharacteristic
216#106—831no—
217#106—831no—
218#106—831no—
219#106—831no—
220C4⋊C8—831no—
221C4⋊C8—831no—
222C4⋊C8—831no—
223C4⋊C8—831no—
224#103—831no—
225#103—831no—
226#103—831no—
227#103—831no—
228#102—831no—
229#102—831no—
230#102—831no—
231#102—831no—
232C2×Q8—21no—
233C2×Q8—831no—
234C2×Q8—1661no—
235C2×Q8C22×D8311yescharacteristic
236C2×D4—21no—
237C2×D4—831no—
238C2×D4—1661no—
239C2×D4C22×D8311yescharacteristic
240C22×C4—21no—
241C22×C4—1661no—
242C22×C4C22×D8311yescharacteristic
243Q16—3321no—
244SD16—3321no—
245SD16—3321no—
246D8—3321no—
247M16—3321no—
248C2×C8—3321no—
249C2×C8—831no—
250C2×C8—831no—
251C2×C8—831no—
252C2×C8—831no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—831no—
257C4⋊C4—831no—
258C4⋊C4—1661no—
259C4⋊C4—1661no—
260C4⋊C4—1661no—
261C4⋊C4C22⋊D8311yescharacteristic
262C4⋊C4C22⋊D8311yescharacteristic
263C4⋊C4C22×D8311yescharacteristic
264C4⋊C4C22×D8311yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—1661no—
271C22⋊C4—1661no—
272C22⋊C4—1661no—
273C22⋊C4—1661no—
274C22⋊C4—1661no—
275C22⋊C4C22⋊D8311yescharacteristic
276C22⋊C4C22⋊D8311yescharacteristic
277C42—21no—
278C42—1661no—
279C42C22×D8311yescharacteristic
280C42C22×D8311yescharacteristic
281C23—21no—
282C23—1661no—
283C23C2×C22⋊D8311yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—81no—
287Q8—6641no—
288Q8—1661no—
289Q8—1661no—
290D4—21no—
291D4—21no—
292D4—81no—
293D4—6641no—
294D4—1661no—
295D4—1661no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—81no—
302C2×C4—6641no—
303C2×C4—1661no—
304C2×C4—1661no—
305C2×C4—1661no—
306C2×C4C2×C22⋊D8311yescharacteristic
307C2×C4C2×C22⋊D8311yescharacteristic
308C2×C4C2×C22⋊D8311yescharacteristic
309C2×C4C2×C22⋊D8311yescharacteristic
310C2×C4C2×C22⋊D8311yescharacteristic
311C2×C4C23×D8311yescharacteristic
312C8—1661no—
313C8—1661no—
314C8—1661no—
315C8—1661no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—3321no—
323C22—3321no—
324C22—3321no—
325C22C22×C22⋊D8311yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—3321no—
335C4—3321no—
336C4—3321no—
337C4#1458183811yescharacteristic
338C4#1458183911yescharacteristic
339C2—41no—
340C2—81no—
341C2—6641no—
342C2#1912827911yescharacteristic
343C2#1912850911yescharacteristic
344C2#1912851211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#542×C83C2
socleC22×C83C22×D4
prime radical#19127952C2
hypercenter#542D83
hypocenterC83#3263
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #19128322C2semidirect
2. #19127991C2semidirect
3. #19127988C2semidirect
4. #19127986C2semidirect
5. #19127983C2semidirect
6. #19127957C2semidirect
8. #19127932C2semidirect
9. #19127928C2semidirect
10. #19127923C2semidirect
11. #19127920C2semidirect
12. #19127916C2semidirect
13. #19127913C2semidirect
14. #19127728C2semidirect
15. #542×C83C2semidirect
173. C83#3263semidirect
175. #542D83semidirect

Maximal subgroups

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

subgroupindex
#191283222
#191279912
#191279882
#191279862
#191279832
#191279572
#191279522
#191279322
#191279282
#191279232
#191279202
#191279162
#191279132
#191277282
#542×C832
#326383