Finite Group Databasebeta

← #35372849

Subgroups — #35372849

Order 21248.

1308 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#24916574C211yesmaximal, characteristic, complemented
2#24916031C211yesmaximal, characteristic, complemented
3#24906140C213yesmaximal, complemented
4#24906131C213yesmaximal, complemented
5#24906112C213yesmaximal, complemented
6C2×#19127445C213yesmaximal, complemented
7C83×#3498C211yesmaximal, characteristic
8C2×#14581799—21no—
9Q8×C83⋊C8—21no—
10D4×C83⋊C8—23no—
11#19127754C2213yescomplemented
12#19127728—23no—
13#19127725C2213yescomplemented
14C2×C4⋊(C83⋊C8)C2213yescomplemented
15C2×C4⋊(C83⋊C8)C2213yescomplemented
16#19127447C2211yescharacteristic, complemented
17#19127447C2213yescomplemented
18#19127445C2213yescomplemented
19#19127445C2216yescomplemented
20#19127445C2213yescomplemented
21C2×H(5,𝔽2)×C83—21no—
22#550×C83—21no—
23#550×C83C412yes—
24D42×C83—23no—
25D42×C83C416yes—
26#535×C83—23no—
27#532×C83C2213yes—
28C2×#127×C83C2213yes—
29C4×#41×C83C2211yescharacteristic
30#14581799—44no—
31C2×#13050574—23no—
32C22×C83⋊C8—23no—
33C22×C83⋊C8—23no—
34C4⋊(C83⋊C8)—23no—
35C4⋊(C83⋊C8)—23no—
36C4⋊(C83⋊C8)—23no—
37C4⋊(C83⋊C8)C2311yescharacteristic
38C4⋊(C83⋊C8)C2313yes—
39C4⋊(C83⋊C8)C2313yes—
40C4⋊(C83⋊C8)C2311yescharacteristic
41C22⋊(C83⋊C8)—23no—
42C4×C83⋊C8—23no—
43H(5,𝔽2)×C83—44no—
44H(5,𝔽2)×C83—44no—
45C2×#41×C83—22no—
46C2×#41×C83—23no—
47C2×#41×C83D411yescharacteristic
48C22×D4×C83—26no—
49C22×D4×C83—23no—
50C22×D4×C83—23no—
51C22×D4×C83C2×C416yes—
52C4⋊Q8×C83C2311yescharacteristic
53#127×C83—23no—
54#127×C83—23no—
55#127×C83C2×C416yes—
56#127×C83C2313yes—
57#124×C83—23no—
58#121×C83—23no—
59#121×C83—26no—
60#121×C83—26no—
61#121×C83C2×C416yes—
62#120×C83—212no—
63C4×Q8×C83C2×C411yescharacteristic
64C4×Q8×C83D411yescharacteristic
65C4×D4×C83—23no—
66C4×D4×C83C2×C416yes—
67C4×D4×C83C2×C413yes—
68C4×D4×C83D413yes—
69#117×C83D413yes—
70C2×C42×C83C2313yes—
71#13050574—46no—
72C2×C83⋊C8—21no—
73C2×C83⋊C8—23no—
74C2×C83⋊C8—23no—
75C2×C83⋊C8—21no—
76C2×C83⋊C8—46no—
77C2×C83⋊C8—83no—
78C24×C83—26no—
79#41×C83—48no—
80#41×C83—46no—
81#41×C83—46no—
82#41×C83D814yes—
83#41×C83SD1614yes—
84C2×Q8×C83—21no—
85C2×Q8×C83C22⋊C411yescharacteristic
86C2×D4×C83—26no—
87C2×D4×C83—26no—
88C2×D4×C83—26no—
89C2×D4×C83—26no—
90C2×D4×C83—23no—
91C2×D4×C83—23no—
92C2×D4×C83—424no—
93C2×D4×C83—46no—
94C2×D4×C83—46no—
95C2×D4×C83—83no—
96C2×D4×C83—83no—
97C2×D4×C83—86no—
98C2×D4×C83C22⋊C413yes—
99C2×D4×C83C22×C416yes—
100C22×C4×C83—26no—
101C22×C4×C83C22×C413yes—
102C22×C4×C83C2×D413yes—
103C22×C4×C83C2×D413yes—
104C4⋊C4×C83—21no—
105C4⋊C4×C83C22⋊C413yes—
106C4⋊C4×C83C22⋊C413yes—
107C4⋊C4×C83C22×C412yes—
108C22⋊C4×C83—26no—
109C22⋊C4×C83—26no—
110C22⋊C4×C83C22⋊C416yes—
111C42×C83C22×C413yes—
112C42×C83C2×D413yes—
113C42×C83C2×D413yes—
114C42×C83C2411yescharacteristic
115C23×C83—26no—
116C23×C83—412no—
117C23×C83—86no—
118C23×C83—86no—
119C23×C83C2×C22⋊C413yes—
120Q8×C83—41no—
121Q8×C83—41no—
122Q8×C83#10214yes—
123D4×C83—46no—
124D4×C83—412no—
125D4×C83—43no—
126D4×C83—43no—
127D4×C83—43no—
128D4×C83—43no—
129D4×C83—412no—
130D4×C83—86no—
131D4×C83—86no—
132D4×C83#102112yes—
133C2×C4×C83—23no—
134C2×C4×C83—23no—
135C2×C4×C83—22no—
136C2×C4×C83—43no—
137C2×C4×C83—412no—
138C2×C4×C83C2×C22⋊C413yes—
139C2×C4×C83C2×C22⋊C413yes—
140C2×C4×C83C2×C22⋊C413yes—
141C2×C4×C83C2×D816yes—
142C2×C4×C83C2×SD1616yes—
143C2×C4×C83C23×C411yescharacteristic
144C2×C4×C83C22×D411yescharacteristic
145C2×C4×C83C22×D411yescharacteristic
146C83⋊C8—41no—
147C83⋊C8—43no—
148C83⋊C8—43no—
149C83⋊C8—41no—
150C22×C83—23no—
151C22×C83—23no—
152C22×C83—46no—
153C22×C83—46no—
154C22×C83—46no—
155C22×C83—1612no—
156C22×C83C2×#10216yes—
157C22×C83C22×C22⋊C411yescharacteristic
158C4×C83—21no—
159C4×C83—23no—
160C4×C83—23no—
161C4×C83—42no—
162C4×C83C2×#10216yes—
163C4×C83C22×D811yescharacteristic
164C4×C83C22×SD1611yescharacteristic
165C4×C83#57611yescharacteristic
166C4×C83#57711yescharacteristic
167#17720—831nomaximal, Sylow
168C2×C83—26no—
169C2×C83—86no—
170C2×C83#288811yescharacteristic
171C2×C83C22×#10211yescharacteristic
172C2×C83#289711yescharacteristic
173#3498D8311yescharacteristic
174#2988—831no—
175#2906—831no—
176#1547—833no—
177#1545—833no—
178#1540—833no—
179C2×#331—833no—
180C83#1772011yescharacteristic, complemented, Sylow
181C2×H(5,𝔽2)—21no—
182C2×#131—1661no—
183#550—21no—
184#550Dic8312yes—
185D42—23no—
186D42Dic8316yes—
187#535—23no—
188#532C2×D8313yes—
189C2×#127C2×D8313yes—
190C4×#41C2×D8311yescharacteristic
191C8×Q8—1661no—
192C8×D4—1663no—
193#431—833no—
194#424—1663no—
195#423—833no—
196C2×C4⋊C8—833no—
197C2×C4⋊C8—833no—
198#332—831no—
199#332—833no—
200#331—833no—
201#331—836no—
202#331—833no—
203H(5,𝔽2)—44no—
204H(5,𝔽2)—44no—
205C2×#41—22no—
206C2×#41—23no—
207C2×#41C22⋊D8311yescharacteristic
208C22×D4—26no—
209C22×D4—23no—
210C22×D4—23no—
211C22×D4C2×Dic8316yes—
212#131—3324no—
213C2×M16—1663no—
214C22×C8—1663no—
215C22×C8—1663no—
216C4⋊Q8C22×D8311yescharacteristic
217#127—23no—
218#127—23no—
219#127C2×Dic8316yes—
220#127C22×D8313yes—
221#124—23no—
222#121—23no—
223#121—26no—
224#121—26no—
225#121C2×Dic8316yes—
226#120—212no—
227C4×Q8C2×Dic8311yescharacteristic
228C4×Q8C22⋊D8311yescharacteristic
229C4×D4—23no—
230C4×D4C2×Dic8316yes—
231C4×D4C2×Dic8313yes—
232C4×D4C22⋊D8313yes—
233#117C22⋊D8313yes—
234C2×C42C22×D8313yes—
235C4⋊C8—831no—
236C4⋊C8—833no—
237C4⋊C8—833no—
238C4⋊C8—831no—
239C4⋊C8—1663no—
240C4⋊C8—1663no—
241C4⋊C8—1663no—
242C22⋊C8—1663no—
243C4×C8—1663no—
244C24—26no—
245#41—48no—
246#41—46no—
247#41—46no—
248#41#1305057514yes—
249#41#1305057814yes—
250C2×Q8—21no—
251C2×Q8C22⋊Dic8311yescharacteristic
252C2×D4—26no—
253C2×D4—23no—
254C2×D4—23no—
255C2×D4—26no—
256C2×D4—26no—
257C2×D4—26no—
258C2×D4—424no—
259C2×D4—46no—
260C2×D4—46no—
261C2×D4—83no—
262C2×D4—83no—
263C2×D4—86no—
264C2×D4C22⋊Dic8313yes—
265C2×D4C22×Dic8316yes—
266C22×C4—26no—
267C22×C4C22×Dic8313yes—
268C22×C4C2×C22⋊D8313yes—
269C22×C4C2×C22⋊D8313yes—
270M16—3326no—
271C2×C8—3326no—
272C2×C8—6643no—
273C2×C8—1661no—
274C2×C8—1663no—
275C2×C8—1661no—
276C2×C8—1663no—
277C4⋊C4—21no—
278C4⋊C4C22⋊Dic8313yes—
279C4⋊C4C22⋊Dic8313yes—
280C4⋊C4C22×Dic8312yes—
281C22⋊C4—26no—
282C22⋊C4—26no—
283C22⋊C4C22⋊Dic8316yes—
284C42C22×Dic8313yes—
285C42C2×C22⋊D8313yes—
286C42C2×C22⋊D8313yes—
287C42C23×D8311yescharacteristic
288C23—26no—
289C23—412no—
290C23—86no—
291C23—86no—
292C23C2×C22⋊Dic8313yes—
293Q8—41no—
294Q8—41no—
295Q8#1458171314yes—
296D4—46no—
297D4—412no—
298D4—412no—
299D4—43no—
300D4—43no—
301D4—43no—
302D4—43no—
303D4—86no—
304D4—86no—
305D4#14581713112yes—
306C2×C4—23no—
307C2×C4—23no—
308C2×C4—22no—
309C2×C4—43no—
310C2×C4—412no—
311C2×C4C2×C22⋊Dic8313yes—
312C2×C4C2×C22⋊Dic8313yes—
313C2×C4C2×C22⋊Dic8313yes—
314C2×C4C2×#1305057516yes—
315C2×C4C2×#1305057816yes—
316C2×C4C23×Dic8311yescharacteristic
317C2×C4C22×C22⋊D8311yescharacteristic
318C2×C4C22×C22⋊D8311yescharacteristic
319C8—3321no—
320C8—3323no—
321C8—3323no—
322C8—3321no—
323C22—23no—
324C22—23no—
325C22—46no—
326C22—46no—
327C22—46no—
328C22—1612no—
329C22C2×#1458171316yes—
330C22C22×C22⋊Dic8311yescharacteristic
331C4—23no—
332C4—21no—
333C4—23no—
334C4—42no—
335C4C2×#1458171316yes—
336C4C22×#1305057511yescharacteristic
337C4C22×#1305057811yescharacteristic
338C4#1912850411yescharacteristic
339C4#1912851211yescharacteristic
340C2—26no—
341C2—86no—
342C2#2491594211yescharacteristic
343C2C22×#1458171311yescharacteristic
344C2#2491598911yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
Fitting, prime radicalC83×#3498C2
socleC22×C83C22×C22⋊C4
cosocleC42×C83C24
hypercenter#3498D83
hypocenterC83#17720
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3498

Complements

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

NH ≅ G/Ntype
1. #24916574C2semidirect
2. #24916031C2semidirect
3. #24906140C2semidirect
4. #24906131C2semidirect
5. #24906112C2semidirect
6. C2×#19127445C2semidirect
11. #19127754C22semidirect
13. #19127725C22semidirect
14. C2×C4⋊(C83⋊C8)C22semidirect
15. C2×C4⋊(C83⋊C8)C22semidirect
16. #19127447C22semidirect
17. #19127447C22semidirect
18. #19127445C22semidirect
19. #19127445C22semidirect
20. #19127445C22semidirect
180. C83#17720semidirect

Maximal subgroups

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

subgroupindex
#249165742
#249160312
#249061402
#249061312
#249061122
C2×#191274452
C83×#34982
#1772083