Finite Group Databasebeta

← #28405354

Subgroups — #28405354

Order 14464.

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#20299434C211yesmaximal, characteristic, complemented
2#20299409C218yesmaximal, complemented
3#20299401C218yesmaximal, complemented
4#20299396C216yesmaximal, complemented
5C2×#16813648C216yesmaximal, complemented
6#20299365C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)×C113C211yesmaximal, characteristic, complemented
8#16813670—21no—
9#16813669C2214yescomplemented
10#16813668C2214yescomplemented
11#16813667—23no—
12C2×#14268647C2216yescomplemented
13#41×D113—24no—
14#16813653C2214yescomplemented
15#16813652C22112yescomplemented
16#16813648C22112yescomplemented
17#16813648C22124yescomplemented
18#16813647C22112yescomplemented
19#16813642C22124yescomplemented
20C2×#14268635C22112yescomplemented
21C2×#14268632C22112yescomplemented
22#16813629C2218yescomplemented
23C2×#14268631C2216yescomplemented
24H(5,𝔽2)×C113—24no—
25H(5,𝔽2)×C113C2218yescomplemented
26C2×#41×C113—21no—
27C2×#41×C113C2211yescharacteristic
28C2×#41×C113C2213yescomplemented
29C22×D4×C113—23no—
30C22×D4×C113C2213yescomplemented
31#14268648—24no—
32#14268647—26no—
33#14268647—26no—
34#14268647C2314yes—
35#14268647C23112yescomplemented
36#14268646—212no—
37C2×Dic226C2316yescomplemented
38Q8×D113—24no—
39C2×D452C2316yescomplemented
40C2×C22⋊D113—26no—
41D4×D113—212no—
42C2×C4×D113—26no—
43#14268637C2318yescomplemented
44#14268635C23124yescomplemented
45#14268634C2318yescomplemented
46#14268632C23124yescomplemented
47#14268631C2314yes—
48#14268631C23112yescomplemented
49C2×#11744751C23112yescomplemented
50C24×C113—23no—
51#41×C113—28no—
52#41×C113—26no—
53#41×C113D414yes—
54#41×C113C23112yes—
55#41×C113C2314yes—
56C2×Q8×C113D411yescharacteristic
57C2×Q8×C113C2311yescharacteristic
58C2×D4×C113—23no—
59C2×D4×C113—224no—
60C2×D4×C113—23no—
61C2×D4×C113—26no—
62C2×D4×C113D413yes—
63C2×D4×C113C23112yes—
64C2×D4×C113C2313yes—
65C22×C4×C113—23no—
66C22×C4×C113C2313yes—
67C23×C113—23no—
68C23×C113—212no—
69C23×C113—86no—
70C23×C113C2×D413yes—
71C22×D113—26no—
72Q8×C113—21no—
73Q8×C113C2×D414yes—
74Q8×C113C2412yes—
75D4×C113—212no—
76D4×C113—212no—
77D4×C113—23no—
78D4×C113C2×D4112yes—
79D4×C113C2416yes—
80C2×C4×C113—21no—
81C2×C4×C113—212no—
82C2×C4×C113C2×D413yes—
83C2×C4×C113C2×D416yes—
84C2×C4×C113C2411yescharacteristic
85C2×C4×C113C2416yes—
86C22⋊D113—212no—
87C22⋊D113—24no—
88C2×Dic113—26no—
89D452—26no—
90D452C2414yes—
91C4×D113—24no—
92C4×D113—212no—
93Dic226—26no—
94Dic226C2414yes—
95#11744751C2418yes—
96C22×C113—26no—
97C22×C113—43no—
98C22×C113—46no—
99C22×C113—812no—
100C22×C113C22×D416yes—
101C22×C113C22×D411yescharacteristic
102C2×D113—24no—
103C2×D113—86no—
104C4×C113—22no—
105C4×C113C22×D416yes—
106C4×C113C22×D411yescharacteristic
107C4×C113C2511yescharacteristic
108Dic113—24no—
109C2×C113—26no—
110C2×C113—21no—
111C2×C113—46no—
112C2×C113C23×D411yescharacteristic
113D113—44no—
114#3585—1131nomaximal, Sylow
115C113#358511yescharacteristic, complemented, Sylow
116#585—1131no—
117C2×H(5,𝔽2)D11311yescharacteristic, complemented
118#577—1138no—
119#576—1138no—
120#575—1136no—
121C2×AGL(1,ℤ/8)—1136no—
122#568—1131no—
1232−1+4—1134no—
1242−1+4—2261no—
125H(5,𝔽2)—24no—
126H(5,𝔽2)—1134no—
127H(5,𝔽2)—2263no—
128H(5,𝔽2)C2×D11318yescomplemented
129C2×#41—21no—
130C2×#41—1136no—
131C2×#41—2264no—
132C2×#41C2×D11311yescharacteristic
133C2×#41C2×D11313yescomplemented
134C22×D4—23no—
135C22×D4C2×D11313yescomplemented
136#137—11312no—
137#137—1134no—
138AGL(1,ℤ/8)—11312no—
139AGL(1,ℤ/8)—11312no—
140AGL(1,ℤ/8)—11324no—
141#135—11324no—
142C2×SD16—11312no—
143C2×D8—11312no—
144#131—1138no—
145C2×M16—1136no—
146C24—23no—
147#41—28no—
148#41—26no—
149#41—11312no—
150#41—1134no—
151#41—2264no—
152#41—22612no—
153#41—2266no—
154#41—2266no—
155#41C22⋊D11314yes—
156#41C22×D113112yes—
157#41C22×D11314yes—
158C2×Q8—1136no—
159C2×Q8—2264no—
160C2×Q8C22⋊D11311yescharacteristic
161C2×Q8C22×D11311yescharacteristic
162C2×D4—23no—
163C2×D4—224no—
164C2×D4—23no—
165C2×D4—26no—
166C2×D4—1136no—
167C2×D4—2266no—
168C2×D4—22612no—
169C2×D4C22⋊D11313yes—
170C2×D4C22×D113112yes—
171C2×D4C22×D11313yes—
172C22×C4—23no—
173C22×C4—2266no—
174C22×C4C22×D11313yes—
175Q16—1138no—
176SD16—1138no—
177SD16—11324no—
178D8—11324no—
179M16—1134no—
180M16—11312no—
181C2×C8—11312no—
182C23—23no—
183C23—212no—
184C23—86no—
185C23—2266no—
186C23C2×C22⋊D11313yes—
187Q8—21no—
188Q8—1134no—
189Q8—2266no—
190Q8C2×C22⋊D11314yes—
191Q8C23×D11312yes—
192D4—212no—
193D4—212no—
194D4—23no—
195D4—1134no—
196D4—2264no—
197D4—22612no—
198D4—2266no—
199D4C2×C22⋊D113112yes—
200D4C23×D11316yes—
201C2×C4—21no—
202C2×C4—212no—
203C2×C4—2266no—
204C2×C4—2264no—
205C2×C4—22612no—
206C2×C4C2×C22⋊D11313yes—
207C2×C4C2×C22⋊D11316yes—
208C2×C4C23×D11316yes—
209C2×C4C23×D11311yescharacteristic
210C8—1138no—
211C22—26no—
212C22—43no—
213C22—46no—
214C22—812no—
215C22—9046no—
216C22—2264no—
217C22C22×C22⋊D11311yescharacteristic
218C22C22×C22⋊D11316yes—
219C4—22no—
220C4—2264no—
221C4C22×C22⋊D11316yes—
222C4C22×C22⋊D11311yescharacteristic
223C4C24×D11311yescharacteristic
224C2—21no—
225C2—26no—
226C2—46no—
227C2—4524no—
228C2C23×C22⋊D11311yescharacteristic
subgroupG/H
centerC2C23×C22⋊D113
commutator, cosocleC4×C113C25
FrattiniC4C24×D113
FittingC2×H(5,𝔽2)×C113C2
socleC2×C113C23×D4
hypercenterC2×H(5,𝔽2)D113
hypocenterC113#3585
LCSG ▷ C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C2 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20299434C2semidirect
2. #20299409C2semidirect
3. #20299401C2semidirect
4. #20299396C2semidirect
5. C2×#16813648C2semidirect
6. #20299365C2semidirect
7. C2×H(5,𝔽2)×C113C2semidirect
9. #16813669C22semidirect
10. #16813668C22semidirect
12. C2×#14268647C22semidirect
14. #16813653C22semidirect
15. #16813652C22semidirect
16. #16813648C22semidirect
17. #16813648C22semidirect
18. #16813647C22semidirect
19. #16813642C22semidirect
20. C2×#14268635C22semidirect
21. C2×#14268632C22semidirect
22. #16813629C22semidirect
23. C2×#14268631C22semidirect
25. H(5,𝔽2)×C113C22semidirect
28. C2×#41×C113C22semidirect
30. C22×D4×C113C22semidirect
35. #14268647C23semidirect
37. C2×Dic226C23semidirect
39. C2×D452C23semidirect
43. #14268637C23semidirect
44. #14268635C23semidirect
45. #14268634C23semidirect
46. #14268632C23semidirect
48. #14268631C23semidirect
49. C2×#11744751C23semidirect
115. C113#3585semidirect
117. C2×H(5,𝔽2)D113semidirect
128. H(5,𝔽2)C2×D113semidirect
133. C2×#41C2×D113semidirect
135. C22×D4C2×D113semidirect

Maximal subgroups

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

subgroupindex
#202994342
#202994092
#202994012
#202993962
C2×#168136482
#202993652
C2×H(5,𝔽2)×C1132
#3585113