Finite Group Databasebeta

← #24922243

Subgroups — #24922243

Order 10624.

1464 conjugacy classes of subgroups; only the 545 normal subgroups are shown.

normal subgroups only

This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 545 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#19128537C211yesmaximal, characteristic, complemented
2#19128512C218yesmaximal, complemented
3#19128504C218yesmaximal, complemented
4#19128499C216yesmaximal, complemented
5C2×#14581818C216yesmaximal, complemented
6#19128468C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)×C83C211yesmaximal, characteristic, complemented
8#14581840—21no—
9#14581839C2214yescomplemented
10#14581838C2214yescomplemented
11#14581837—23no—
12C2×#13050590C2216yescomplemented
13#41×D83—24no—
14#14581823C2214yescomplemented
15#14581822C22112yescomplemented
16#14581818C22112yescomplemented
17#14581818C22124yescomplemented
18#14581817C22112yescomplemented
19#14581812C22124yescomplemented
20C2×#13050578C22112yescomplemented
21C2×#13050575C22112yescomplemented
22#14581799C2218yescomplemented
23C2×#13050574C2216yescomplemented
24H(5,𝔽2)×C83—24no—
25H(5,𝔽2)×C83C2218yescomplemented
26C2×#41×C83—21no—
27C2×#41×C83C2211yescharacteristic
28C2×#41×C83C2213yescomplemented
29C22×D4×C83—23no—
30C22×D4×C83C2213yescomplemented
31#13050591—24no—
32#13050590—26no—
33#13050590—26no—
34#13050590C2314yes—
35#13050590C23112yescomplemented
36#13050589—212no—
37C2×Dic166C2316yescomplemented
38Q8×D83—24no—
39C2×D332C2316yescomplemented
40C2×C22⋊D83—26no—
41D4×D83—212no—
42C2×C4×D83—26no—
43#13050580C2318yescomplemented
44#13050578C23124yescomplemented
45#13050577C2318yescomplemented
46#13050575C23124yescomplemented
47#13050574C2314yes—
48#13050574C23112yescomplemented
49C2×C83⋊C8C23112yescomplemented
50C24×C83—23no—
51#41×C83—28no—
52#41×C83—26no—
53#41×C83D414yes—
54#41×C83C23112yes—
55#41×C83C2314yes—
56C2×Q8×C83D411yescharacteristic
57C2×Q8×C83C2311yescharacteristic
58C2×D4×C83—23no—
59C2×D4×C83—224no—
60C2×D4×C83—23no—
61C2×D4×C83—26no—
62C2×D4×C83D413yes—
63C2×D4×C83C23112yes—
64C2×D4×C83C2313yes—
65C22×C4×C83—23no—
66C22×C4×C83C2313yes—
67C23×C83—23no—
68C23×C83—212no—
69C23×C83—86no—
70C23×C83C2×D413yes—
71C22×D83—26no—
72Q8×C83—21no—
73Q8×C83C2×D414yes—
74Q8×C83C2412yes—
75D4×C83—212no—
76D4×C83—212no—
77D4×C83—23no—
78D4×C83C2×D4112yes—
79D4×C83C2416yes—
80C2×C4×C83—21no—
81C2×C4×C83—212no—
82C2×C4×C83C2×D413yes—
83C2×C4×C83C2×D416yes—
84C2×C4×C83C2411yescharacteristic
85C2×C4×C83C2416yes—
86C22⋊D83—212no—
87C22⋊D83—24no—
88C2×Dic83—26no—
89D332—26no—
90D332C2414yes—
91C4×D83—24no—
92C4×D83—212no—
93Dic166—26no—
94Dic166C2414yes—
95C83⋊C8C2418yes—
96C22×C83—26no—
97C22×C83—43no—
98C22×C83—46no—
99C22×C83—812no—
100C22×C83C22×D416yes—
101C22×C83C22×D411yescharacteristic
102C2×D83—24no—
103C2×D83—86no—
104C4×C83—22no—
105C4×C83C22×D416yes—
106C4×C83C22×D411yescharacteristic
107C4×C83C2511yescharacteristic
108Dic83—24no—
109C2×C83—26no—
110C2×C83—21no—
111C2×C83—46no—
112C2×C83C23×D411yescharacteristic
113D83—44no—
114#3585—831nomaximal, Sylow
115C83#358511yescharacteristic, complemented, Sylow
116#585—831no—
117C2×H(5,𝔽2)D8311yescharacteristic, complemented
118#577—838no—
119#576—838no—
120#575—836no—
121C2×AGL(1,ℤ/8)—836no—
122#568—831no—
1232−1+4—834no—
1242−1+4—1661no—
125H(5,𝔽2)—24no—
126H(5,𝔽2)—834no—
127H(5,𝔽2)—1663no—
128H(5,𝔽2)C2×D8318yescomplemented
129C2×#41—21no—
130C2×#41—836no—
131C2×#41—1664no—
132C2×#41C2×D8311yescharacteristic
133C2×#41C2×D8313yescomplemented
134C22×D4—23no—
135C22×D4C2×D8313yescomplemented
136#137—8312no—
137#137—834no—
138AGL(1,ℤ/8)—8312no—
139AGL(1,ℤ/8)—8312no—
140AGL(1,ℤ/8)—8324no—
141#135—8324no—
142C2×SD16—8312no—
143C2×D8—8312no—
144#131—838no—
145C2×M16—836no—
146C24—23no—
147#41—28no—
148#41—26no—
149#41—8312no—
150#41—834no—
151#41—1664no—
152#41—16612no—
153#41—1666no—
154#41—1666no—
155#41C22⋊D8314yes—
156#41C22×D83112yes—
157#41C22×D8314yes—
158C2×Q8—836no—
159C2×Q8—1664no—
160C2×Q8C22⋊D8311yescharacteristic
161C2×Q8C22×D8311yescharacteristic
162C2×D4—23no—
163C2×D4—224no—
164C2×D4—23no—
165C2×D4—26no—
166C2×D4—836no—
167C2×D4—1666no—
168C2×D4—16612no—
169C2×D4C22⋊D8313yes—
170C2×D4C22×D83112yes—
171C2×D4C22×D8313yes—
172C22×C4—23no—
173C22×C4—1666no—
174C22×C4C22×D8313yes—
175Q16—838no—
176SD16—838no—
177SD16—8324no—
178D8—8324no—
179M16—834no—
180M16—8312no—
181C2×C8—8312no—
182C23—23no—
183C23—212no—
184C23—86no—
185C23—1666no—
186C23C2×C22⋊D8313yes—
187Q8—21no—
188Q8—834no—
189Q8—1666no—
190Q8C2×C22⋊D8314yes—
191Q8C23×D8312yes—
192D4—212no—
193D4—212no—
194D4—23no—
195D4—834no—
196D4—1664no—
197D4—16612no—
198D4—1666no—
199D4C2×C22⋊D83112yes—
200D4C23×D8316yes—
201C2×C4—21no—
202C2×C4—212no—
203C2×C4—1666no—
204C2×C4—1664no—
205C2×C4—16612no—
206C2×C4C2×C22⋊D8313yes—
207C2×C4C2×C22⋊D8316yes—
208C2×C4C23×D8316yes—
209C2×C4C23×D8311yescharacteristic
210C8—838no—
211C22—26no—
212C22—43no—
213C22—46no—
214C22—812no—
215C22—6646no—
216C22—1664no—
217C22C22×C22⋊D8311yescharacteristic
218C22C22×C22⋊D8316yes—
219C4—22no—
220C4—1664no—
221C4C22×C22⋊D8316yes—
222C4C22×C22⋊D8311yescharacteristic
223C4C24×D8311yescharacteristic
224C2—21no—
225C2—26no—
226C2—46no—
227C2—3324no—
228C2C23×C22⋊D8311yescharacteristic
subgroupG/H
centerC2C23×C22⋊D83
commutator, cosocleC4×C83C25
FrattiniC4C24×D83
FittingC2×H(5,𝔽2)×C83C2
socleC2×C83C23×D4
hypercenterC2×H(5,𝔽2)D83
hypocenterC83#3585
LCSG ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #19128537C2semidirect
2. #19128512C2semidirect
3. #19128504C2semidirect
4. #19128499C2semidirect
5. C2×#14581818C2semidirect
6. #19128468C2semidirect
7. C2×H(5,𝔽2)×C83C2semidirect
9. #14581839C22semidirect
10. #14581838C22semidirect
12. C2×#13050590C22semidirect
14. #14581823C22semidirect
15. #14581822C22semidirect
16. #14581818C22semidirect
17. #14581818C22semidirect
18. #14581817C22semidirect
19. #14581812C22semidirect
20. C2×#13050578C22semidirect
21. C2×#13050575C22semidirect
22. #14581799C22semidirect
23. C2×#13050574C22semidirect
25. H(5,𝔽2)×C83C22semidirect
28. C2×#41×C83C22semidirect
30. C22×D4×C83C22semidirect
35. #13050590C23semidirect
37. C2×Dic166C23semidirect
39. C2×D332C23semidirect
43. #13050580C23semidirect
44. #13050578C23semidirect
45. #13050577C23semidirect
46. #13050575C23semidirect
48. #13050574C23semidirect
49. C2×C83⋊C8C23semidirect
115. C83#3585semidirect
117. C2×H(5,𝔽2)D83semidirect
128. H(5,𝔽2)C2×D83semidirect
133. C2×#41C2×D83semidirect
135. C22×D4C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191285372
#191285122
#191285042
#191284992
C2×#145818182
#191284682
C2×H(5,𝔽2)×C832
#358583