Finite Group Databasebeta

← #24921867

Subgroups — #24921867

Order 10624.

1258 conjugacy classes of subgroups; only the 455 normal subgroups are shown.

normal subgroups only

This group has 1258 conjugacy classes of subgroups — too many to draw. Only its 455 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128443C213yesmaximal, complemented
2#19128413C216yesmaximal, complemented
3#19128333C216yesmaximal, complemented
4#19128269C216yesmaximal, complemented
5#19128266C216yesmaximal, complemented
6#19128250C213yesmaximal, complemented
7H(3,𝔽4)×C83C211yesmaximal, characteristic, complemented
8C2×#13050591C2213yescomplemented
9C2×#13050589C2213yescomplemented
10C22×C22⋊D83C2216yescomplemented
11C2×D4×D83C2216yescomplemented
12#14581790C2213yescomplemented
13#14581789C2212yescomplemented
14#14581788C2213yes—
15#14581788C2216yescomplemented
16#14581781C2216yescomplemented
17#14581780C2213yescomplemented
18#14581778C22112yescomplemented
19#14581765C22112yescomplemented
20#14581765C22112yescomplemented
21#14581764C22112yescomplemented
22#14581762C22112yescomplemented
23#14581762C2216yescomplemented
24#14581761C2212yescomplemented
25#14581760C2216yescomplemented
26(C22⋊D83)⋊C4C22112yescomplemented
27#14581749C2216yescomplemented
28#14581748C2216yescomplemented
29C42×D83C2211yescharacteristic, complemented
30#124×C83C2216yescomplemented
31#124×C83C2213yescomplemented
32#120×C83C2216yescomplemented
33C23×D83C2316yes—
34#13050591—46no—
35#13050589—46no—
36C2×Dic166C2313yes—
37C2×D332C2313yes—
38C2×D332C2316yes—
39C2×D332C2316yes—
40C2×C22⋊D83—412no—
41C2×C22⋊D83C23112yes—
42C2×C22⋊D83C23112yes—
43C2×C22⋊D83C2316yes—
44D4×D83—412no—
45C22×Dic83C2316yes—
46C2×C4×D83C2313yes—
47C2×C4×D83C2316yes—
48Dic83⋊C4C23112yes—
49C22⋊Dic83C2316yes—
50(C2×D83)⋊C4C23112yes—
51(C2×D83)⋊C4C23112yes—
52C4×Dic83C2316yes—
53C4×Dic83C2313yes—
54C24×C83C2312yes—
55C2×Q8×C83C2313yes—
56C2×D4×C83C2313yes—
57C2×D4×C83C2316yes—
58C22⋊C4×C83C2316yes—
59C22⋊C4×C83C23112yes—
60C42×C83C2311yescharacteristic
61C42×C83C2312yes—
62C23×C83—46no—
63C23×C83C2416yes—
64C22×D83—412no—
65C22×D83—46no—
66C22×D83C2416yes—
67C22×D83C2411yescharacteristic
68Q8×C83—43no—
69D4×C83—46no—
70D4×C83—43no—
71C2×C4×C83C2413yes—
72C2×C4×C83C2416yes—
73C22⋊D83—224no—
74C22⋊D83—412no—
75C22⋊D83—46no—
76C2×Dic83—212no—
77C2×Dic83C2416yes—
78C2×Dic83C2413yes—
79D332—46no—
80D332—43no—
81D332—46no—
82C4×D83—212no—
83C4×D83—26no—
84Dic166—43no—
85C22×C83—26no—
86C22×C83—212no—
87C22×C83—84no—
88C22×C83C2511yescharacteristic
89C2×D83—23no—
90C2×D83—26no—
91C2×D83—212no—
92C2×D83—812no—
93C4×C83—23no—
94C4×C83—26no—
95Dic83—26no—
96Dic83H(5,𝔽2)16yes—
97C2×C83—46no—
98C2×C83C2×H(5,𝔽2)13yes—
99D83—41no—
100D83—46no—
101#3549—831nomaximal, Sylow
102C83#354911yescharacteristic, complemented, Sylow
103H(3,𝔽4)—833no—
104H(3,𝔽4)D8311yescharacteristic, complemented
105#555—836no—
106D42—836no—
107#535—836no—
108#535—836no—
109#532—833no—
110C2×#41—833no—
111C2×#41—833no—
112C22×D4—836no—
113C22×D4—836no—
114C4⋊Q8—833no—
115#127—833no—
116#127—836no—
117#127—832no—
118#124—836no—
119#124—8312no—
120#124—833no—
121#124C2×D8313yescomplemented
122#124C2×D8316yescomplemented
123#121—8312no—
124#121—8312no—
125#121—8312no—
126#120—832no—
127#120—836no—
128#120—836no—
129#120—8312no—
130#120C2×D8316yescomplemented
131C4×D4—8312no—
132#117—836no—
133#117—836no—
134C2×C42—831no—
135C24—836no—
136C24C22×D8312yes—
137#41—3326no—
138#41—3326no—
139C2×Q8—833no—
140C2×Q8C22×D8313yes—
141C2×D4—33212no—
142C2×D4—33212no—
143C2×D4—8312no—
144C2×D4—836no—
145C2×D4—8312no—
146C2×D4—833no—
147C2×D4—836no—
148C2×D4—836no—
149C2×D4C22×D8313yes—
150C2×D4C22×D8316yes—
151C22×C4—836no—
152C22×C4—833no—
153C22×C4—836no—
154C4⋊C4—8312no—
155C22⋊C4—836no—
156C22⋊C4—8312no—
157C22⋊C4—8312no—
158C22⋊C4C22×D8316yes—
159C22⋊C4C22×D83112yes—
160C42—833no—
161C42—836no—
162C42C22×D8311yescharacteristic
163C42C22×D8312yes—
164C23—46no—
165C23—3326no—
166C23—33212no—
167C23—836no—
168C23—831no—
169C23C23×D8316yes—
170Q8—43no—
171Q8—3323no—
172D4—46no—
173D4—43no—
174D4—3326no—
175D4—3326no—
176D4—3323no—
177D4—33212no—
178D4—3326no—
179D4—16624no—
180C2×C4—833no—
181C2×C4—836no—
182C2×C4—16612no—
183C2×C4—1666no—
184C2×C4—16612no—
185C2×C4C23×D8313yes—
186C2×C4C23×D8316yes—
187C22—26no—
188C22—212no—
189C22—84no—
190C22—66412no—
191C22—1663no—
192C22—1666no—
193C22—16612no—
194C22C24×D8311yescharacteristic
195C4—26no—
196C4—23no—
197C4—836no—
198C4—1666no—
199C2—46no—
200C2—3326no—
201C2—3321no—
202C2H(5,𝔽2)×D8313yes—
subgroupG/H
center, FrattiniC22C24×D83
commutator, socle, cosocleC22×C83C25
FittingH(3,𝔽4)×C83C2
hypercenterH(3,𝔽4)D83
hypocenterC83#3549
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #19128443C2semidirect
2. #19128413C2semidirect
3. #19128333C2semidirect
4. #19128269C2semidirect
5. #19128266C2semidirect
6. #19128250C2semidirect
7. H(3,𝔽4)×C83C2semidirect
8. C2×#13050591C22semidirect
9. C2×#13050589C22semidirect
10. C22×C22⋊D83C22semidirect
11. C2×D4×D83C22semidirect
12. #14581790C22semidirect
13. #14581789C22semidirect
15. #14581788C22semidirect
16. #14581781C22semidirect
17. #14581780C22semidirect
18. #14581778C22semidirect
19. #14581765C22semidirect
20. #14581765C22semidirect
21. #14581764C22semidirect
22. #14581762C22semidirect
23. #14581762C22semidirect
24. #14581761C22semidirect
25. #14581760C22semidirect
26. (C22⋊D83)⋊C4C22semidirect
27. #14581749C22semidirect
28. #14581748C22semidirect
29. C42×D83C22semidirect
30. #124×C83C22semidirect
31. #124×C83C22semidirect
32. #120×C83C22semidirect
102. C83#3549semidirect
104. H(3,𝔽4)D83semidirect
121. #124C2×D83semidirect
122. #124C2×D83semidirect
130. #120C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191284432
#191284132
#191283332
#191282692
#191282662
#191282502
H(3,𝔽4)×C832
#354983