Finite Group Databasebeta

← #24919469

Subgroups — #24919469

Order 10624.

404 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128397C211yesmaximal, characteristic, complemented
2#19128065C211yesmaximal, characteristic, complemented
3#19128022C211yesmaximal, characteristic, complemented
4#19128012C211yesmaximal, characteristic, complemented
5#19128008C211yesmaximal, characteristic, complemented
6#19127952C211yesmaximal, characteristic, complemented
7#19127935C211yesmaximal, characteristic, complemented
8#19127893C211yesmaximal, characteristic, complemented
9#19127846C211yesmaximal, characteristic, complemented
10#19127846C211yesmaximal, characteristic, complemented
11#19127799C211yesmaximal, characteristic, complemented
12#19127773C211yesmaximal, characteristic, complemented
13#19127702C211yesmaximal, characteristic, complemented
14#19127697C211yesmaximal, characteristic, complemented
15#550×C83C211yesmaximal, characteristic, complemented
16C2×#13050577C2211yescharacteristic
17C2×#13050575C2211yescharacteristic
18C2×#13050575C2211yescharacteristic
19C2×#13050574C2211yescharacteristic
20C22×C83⋊C8C2211yescharacteristic
21#14581787—21no—
22#14581784C2211yescharacteristic
23#14581774—21no—
24#14581766C2211yescharacteristic
25#14581766C2211yescharacteristic
26#14581764—21no—
27C4×D332C2211yescharacteristic
28(C22⋊D83)⋊C4—21no—
29#14581751C2211yescharacteristic
30C2×C4⋊Dic83C2211yescharacteristic
31#14581725C2211yescharacteristic
32#14581725C2211yescharacteristic
33#14581723C2211yescharacteristic
34C4⋊(C83⋊C8)C2211yescharacteristic
35#14581716C2211yescharacteristic
36#14581716C2211yescharacteristic
37#14581716C2211yescharacteristic
38#14581713C2211yescharacteristic
39#14581713C2211yescharacteristic
40#14581713C2211yescharacteristic
41#14581713C2211yescharacteristic
42#14581713C2211yescharacteristic
43#14581713C2211yescharacteristic
44#14581712C2211yescharacteristic
45C22⋊(C83⋊C8)C2211yescharacteristic
46C22⋊(C83⋊C8)C2211yescharacteristic
47#14581703C2211yescharacteristic
48C2×#41×C83C2211yescharacteristic
49C2×#41×C83C2211yescharacteristic
50#127×C83—21no—
51#127×C83C2211yescharacteristic
52#121×C83—21no—
53#121×C83—21no—
54#121×C83C2211yescharacteristic
55#121×C83C2211yescharacteristic
56C4×Q8×C83C2211yescharacteristic
57C4×D4×C83—21no—
58C4×D4×C83C2211yescharacteristic
59C2×D332C2311yescharacteristic
60C2×C22⋊D83—21no—
61C2×C22⋊D83—21no—
62C22×Dic83—21no—
63C2×C4×D83—21no—
64#13050577—22no—
65#13050575—41no—
66#13050575—41no—
67#13050574—41no—
68C2×C83⋊C8—41no—
69C2×C83⋊C8C2311yescharacteristic
70C2×C83⋊C8C2311yescharacteristic
71C2×C83⋊C8C2311yescharacteristic
72C2×C83⋊C8C2311yescharacteristic
73C4⋊Dic83—21no—
74C4⋊Dic83C2311yescharacteristic
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—
83C4×Dic83—21no—
84#41×C83—22no—
85#41×C83—22no—
86#41×C83—41no—
87#41×C83—41no—
88C2×Q8×C83C2311yescharacteristic
89C2×D4×C83—21no—
90C2×D4×C83—21no—
91C2×D4×C83—21no—
92C2×D4×C83—21no—
93C2×D4×C83—21no—
94C2×D4×C83—21no—
95C2×D4×C83D411yescharacteristic
96C2×D4×C83C2311yescharacteristic
97C2×D4×C83C2311yescharacteristic
98C22×C4×C83—21no—
99C22×C4×C83—21no—
100C22×C4×C83C2311yescharacteristic
101C22×C4×C83C2311yescharacteristic
102C4⋊C4×C83—21no—
103C4⋊C4×C83D411yescharacteristic
104C4⋊C4×C83C2311yescharacteristic
105C22⋊C4×C83—21no—
106C22⋊C4×C83—21no—
107C22⋊C4×C83D411yescharacteristic
108C22⋊C4×C83D411yescharacteristic
109C42×C83—21no—
110C42×C83C2311yescharacteristic
111C23×C83—21no—
112C23×C83—21no—
113C23×C83C2×D411yescharacteristic
114C23×C83C2×D411yescharacteristic
115C22×D83—21no—
116Q8×C83—21no—
117Q8×C83#4112yes—
118D4×C83—21no—
119D4×C83—21no—
120D4×C83—21no—
121D4×C83—21no—
122D4×C83—41no—
123D4×C83—42no—
124D4×C83—42no—
125D4×C83—42no—
126D4×C83—81no—
127D4×C83—81no—
128D4×C83—81no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83—21no—
135C2×C4×C83—42no—
136C2×C4×C83—42no—
137C2×C4×C83C2×D411yescharacteristic
138C2×C4×C83C2×D411yescharacteristic
139C2×C4×C83C2×D411yescharacteristic
140C2×C4×C83C2×D411yescharacteristic
141C2×C4×C83C2411yescharacteristic
142C22⋊D83—42no—
143C22⋊D83—81no—
144C2×Dic83—21no—
145C2×Dic83—21no—
146C2×Dic83—21no—
147C2×Dic83—81no—
148D332—21no—
149D332—21no—
150C4×D83—81no—
151C83⋊C8—21no—
152C83⋊C8—21no—
153C83⋊C8—21no—
154C83⋊C8—21no—
155C22×C83—21no—
156C22×C83—21no—
157C22×C83—21no—
158C22×C83—21no—
159C22×C83—21no—
160C22×C83—21no—
161C22×C83—41no—
162C22×C83—41no—
163C22×C83—41no—
164C22×C83—41no—
165C22×C83—41no—
166C22×C83—41no—
167C22×C83C22×D411yescharacteristic
168C2×D83—41no—
169C2×D83—41no—
170C2×D83—41no—
171C4×C83—21no—
172C4×C83—21no—
173C4×C83—22no—
174C4×C83—21no—
175C4×C83—21no—
176C4×C83—41no—
177C4×C83C2×#4111yescharacteristic
178C4×C83H(5,𝔽2)11yescharacteristic
179Dic83—41no—
180Dic83—41no—
181Dic83—41no—
182C2×C83—41no—
183C2×C83—41no—
184C2×C83—81no—
185C2×C83—81no—
186C2×C83#54611yescharacteristic
187C2×C83#57511yescharacteristic
188C2×C83#57611yescharacteristic
189D83—81no—
190#3309—831nomaximal, Sylow
191C83#330911yescharacteristic, complemented, Sylow
192#553—831no—
193#550D8311yescharacteristic, complemented
194#490—831no—
195#482—831no—
196#480—831no—
197#479—831no—
198#469—831no—
199#466—831no—
200#459—831no—
201#449—831no—
202#449—831no—
203#440—831no—
204#435—831no—
205#417—831no—
206#416—831no—
207C2×#41C2×D8311yescharacteristic
208C2×#41C2×D8311yescharacteristic
209C2×SD16—831no—
210C2×D8—831no—
211C2×D8—831no—
212C2×M16—831no—
213C22×C8—831no—
214#127—21no—
215#127C2×D8311yescharacteristic
216#126—1661no—
217#125—831no—
218#123—1661no—
219#121—21no—
220#121—21no—
221#121—831no—
222#121—831no—
223#121—1661no—
224#121C2×D8311yescharacteristic
225#121C2×D8311yescharacteristic
226C4×Q8C2×D8311yescharacteristic
227C4×D4—21no—
228C4×D4—831no—
229C4×D4—1661no—
230C4×D4C2×D8311yescharacteristic
231#117—831no—
232C2×C4⋊C4—831no—
233#107—831no—
234#107—831no—
235#106—831no—
236C4⋊C8—831no—
237#103—831no—
238#103—831no—
239#103—831no—
240#102—831no—
241#102—831no—
242#102—831no—
243#102—831no—
244#102—831no—
245#102—831no—
246#102—831no—
247C22⋊C8—831no—
248C22⋊C8—831no—
249#97—831no—
250#41—22no—
251#41—22no—
252#41—41no—
253#41—41no—
254C2×Q8C22×D8311yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—831no—
262C2×D4—1661no—
263C2×D4—1661no—
264C2×D4C22⋊D8311yescharacteristic
265C2×D4C22×D8311yescharacteristic
266C2×D4C22×D8311yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—1661no—
270C22×C4—1661no—
271C22×C4C22×D8311yescharacteristic
272C22×C4C22×D8311yescharacteristic
273SD16—1662no—
274D8—3321no—
275D8—3321no—
276M16—3321no—
277C2×C8—3321no—
278C2×C8—831no—
279C2×C8—831no—
280C2×C8—831no—
281C2×C8—831no—
282C4⋊C4—21no—
283C4⋊C4—831no—
284C4⋊C4—831no—
285C4⋊C4—831no—
286C4⋊C4—1661no—
287C4⋊C4—1661no—
288C4⋊C4—1661no—
289C4⋊C4C22⋊D8311yescharacteristic
290C4⋊C4C22×D8311yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—1661no—
294C22⋊C4—1661no—
295C22⋊C4—1661no—
296C22⋊C4—1661no—
297C22⋊C4C22⋊D8311yescharacteristic
298C22⋊C4C22⋊D8311yescharacteristic
299C42—21no—
300C42—1661no—
301C42C22×D8311yescharacteristic
302C23—21no—
303C23—21no—
304C23—1661no—
305C23C2×C22⋊D8311yescharacteristic
306C23C2×C22⋊D8311yescharacteristic
307Q8—21no—
308Q8#1305058912yes—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—41no—
314D4—42no—
315D4—42no—
316D4—42no—
317D4—81no—
318D4—81no—
319D4—81no—
320D4—3322no—
321D4—6641no—
322D4—1661no—
323D4—1661no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—6641no—
333C2×C4—6641no—
334C2×C4—1661no—
335C2×C4—1661no—
336C2×C4—1661no—
337C2×C4C2×C22⋊D8311yescharacteristic
338C2×C4C2×C22⋊D8311yescharacteristic
339C2×C4C2×C22⋊D8311yescharacteristic
340C2×C4C2×C22⋊D8311yescharacteristic
341C2×C4C23×D8311yescharacteristic
342C8—1661no—
343C8—1661no—
344C8—1661no—
345C8—1661no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—3321no—
359C22—3321no—
360C22—3321no—
361C22C22×C22⋊D8311yescharacteristic
362C4—21no—
363C4—21no—
364C4—22no—
365C4—21no—
366C4—21no—
367C4—41no—
368C4—3321no—
369C4—3321no—
370C4—3321no—
371C4C2×#1305058911yescharacteristic
372C4#1458183711yescharacteristic
373C2—41no—
374C2—41no—
375C2—81no—
376C2—81no—
377C2—6641no—
378C2#1912834511yescharacteristic
379C2#1912849911yescharacteristic
380C2#1912850411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#550×C83C2
socleC22×C83C22×D4
hypercenter#550D83
hypocenterC83#3309
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #19128397C2semidirect
2. #19128065C2semidirect
3. #19128022C2semidirect
4. #19128012C2semidirect
5. #19128008C2semidirect
6. #19127952C2semidirect
7. #19127935C2semidirect
8. #19127893C2semidirect
9. #19127846C2semidirect
10. #19127846C2semidirect
11. #19127799C2semidirect
12. #19127773C2semidirect
13. #19127702C2semidirect
14. #19127697C2semidirect
15. #550×C83C2semidirect
191. C83#3309semidirect
193. #550D83semidirect

Maximal subgroups

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

subgroupindex
#191283972
#191280652
#191280222
#191280122
#191280082
#191279522
#191279352
#191278932
#191278462
#191277992
#191277732
#191277022
#191276972
#550×C832
#330983