Finite Group Databasebeta

← #28405355

Subgroups — #28405355

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
2C2×#16813668C211yesmaximal, characteristic, complemented
3#20299410C218yesmaximal, complemented
4#20299403C218yesmaximal, complemented
5#20299398C216yesmaximal, complemented
6C2×#16813650C216yesmaximal, complemented
7#568×C113C211yesmaximal, characteristic, complemented
8#16813670—21no—
9#16813669C2214yescomplemented
10#16813668—24no—
11#16813668C2218yescomplemented
12#16813668C2214yescomplemented
13#16813667—23no—
14C2×#14268648—21no—
15C2×#14268647C2213yescomplemented
16C2×#14268647C2216yescomplemented
17#41×D113—24no—
18C22×D452C2213yescomplemented
19C2×D4×D113—23no—
20#16813655C2214yescomplemented
21#16813655C22112yescomplemented
22#16813650C22112yescomplemented
23#16813650C22124yescomplemented
24#16813650C22112yescomplemented
25#16813644C22124yescomplemented
26C2×#14268636C22112yescomplemented
27C2×D904C22112yescomplemented
28C2×#41×C113C2211yescharacteristic
29#131×C113C2218yescomplemented
30C2×M16×C113C2216yescomplemented
31C23×D113—23no—
32#14268648—24no—
33#14268648—28no—
34#14268647—26no—
35#14268647—26no—
36#14268647—26no—
37#14268647C23112yes—
38#14268647C2314yes—
39#14268647C23112yescomplemented
40#14268646—212no—
41C2×Dic226C2311yescharacteristic
42C2×Dic226C2316yescomplemented
43Q8×D113—24no—
44C2×D452—23no—
45C2×D452—26no—
46C2×D452C23112yes—
47C2×D452C2313yes—
48C2×D452C2316yescomplemented
49C2×C22⋊D113—23no—
50C2×C22⋊D113—26no—
51D4×D113—224no—
52D4×D113—212no—
53C2×C4×D113—23no—
54C2×C4×D113—26no—
55Dic452C2318yescomplemented
56#14268636C23124yescomplemented
57#14268636C2318yescomplemented
58D904C23124yescomplemented
59#41×C113D414yes—
60#41×C113C2314yes—
61C2×Q8×C113D411yescharacteristic
62C2×D4×C113D413yes—
63C22×C4×C113C2313yes—
64M16×C113C2314yes—
65M16×C113C23112yescomplemented
66C2×C8×C113C23112yescomplemented
67C23×C113C2×D413yes—
68C22×D113—23no—
69C22×D113—26no—
70C22×D113—212no—
71C22×D113—86no—
72Q8×C113C2×D414yes—
73D4×C113C2×D4112yes—
74C2×C4×C113C2×D413yes—
75C2×C4×C113C2×D416yes—
76C2×C4×C113C2416yes—
77C2×C4×C113C2411yescharacteristic
78C22⋊D113—212no—
79C22⋊D113—212no—
80C22⋊D113—24no—
81C2×Dic113—21no—
82C2×Dic113—26no—
83D452—212no—
84D452—26no—
85D452—23no—
86D452C2414yes—
87D452C2416yes—
88C4×D113—24no—
89C4×D113—212no—
90C4×D113—212no—
91Dic226—26no—
92Dic226—21no—
93Dic226C2414yes—
94Dic226C2412yes—
95C8×C113C2418yes—
96C22×C113—43no—
97C22×C113C22×D411yescharacteristic
98C22×C113C22×D416yes—
99C2×D113—24no—
100C2×D113—26no—
101C2×D113—46no—
102C2×D113—86no—
103C2×D113—812no—
104C4×C113C22×D411yescharacteristic
105C4×C113C22×D416yes—
106C4×C113C2511yescharacteristic
107Dic113—24no—
108Dic113—22no—
109C2×C113—21no—
110C2×C113—26no—
111C2×C113C23×D411yescharacteristic
112D113—46no—
113D113—44no—
114#3585—1131nomaximal, Sylow
115C113#358511yescharacteristic, complemented, Sylow
116#585—1131no—
117C2×H(5,𝔽2)—1131no—
118#577—1138no—
119#576—1138no—
120#575—1136no—
121C2×AGL(1,ℤ/8)—1136no—
122#568D11311yescharacteristic, complemented
1232−1+4—1134no—
1242−1+4—2261no—
125H(5,𝔽2)—1134no—
126H(5,𝔽2)—1138no—
127H(5,𝔽2)—2264no—
128H(5,𝔽2)—2263no—
129C2×#41—1136no—
130C2×#41—1133no—
131C2×#41—2264no—
132C2×#41—2261no—
133C2×#41C2×D11311yescharacteristic
134C22×D4—1133no—
135C22×D4—2263no—
136#137—1134no—
137#137—11312no—
138AGL(1,ℤ/8)—11312no—
139AGL(1,ℤ/8)—11324no—
140AGL(1,ℤ/8)—11312no—
141#135—11324no—
142C2×SD16—11312no—
143C2×D8—11312no—
144#131C2×D11318yescomplemented
145C2×M16C2×D11316yescomplemented
146C24—2263no—
147#41—11312no—
148#41—11312no—
149#41—1134no—
150#41—2268no—
151#41—2264no—
152#41—22612no—
153#41—2266no—
154#41—2266no—
155#41—2266no—
156#41D45214yes—
157#41C22×D11314yes—
158C2×Q8—1136no—
159C2×Q8—1131no—
160C2×Q8—2264no—
161C2×Q8D45211yescharacteristic
162C2×D4—1136no—
163C2×D4—11312no—
164C2×D4—1133no—
165C2×D4—2263no—
166C2×D4—2266no—
167C2×D4—22624no—
168C2×D4—22612no—
169C2×D4—2263no—
170C2×D4—2266no—
171C2×D4D45213yes—
172C22×C4—2263no—
173C22×C4—2266no—
174C22×C4C22×D11313yes—
175Q16—1138no—
176SD16—11324no—
177SD16—1138no—
178D8—11324no—
179M16C22×D11314yes—
180M16C22×D113112yescomplemented
181C2×C8C22×D113112yescomplemented
182C23—9046no—
183C23—2266no—
184C23—22612no—
185C23—2263no—
186C23C2×D45213yes—
187Q8—1134no—
188Q8—1132no—
189Q8—2266no—
190Q8—2261no—
191Q8C2×D45214yes—
192D4—1134no—
193D4—1136no—
194D4—22612no—
195D4—2264no—
196D4—22612no—
197D4—2263no—
198D4—22612no—
199D4—2266no—
200D4C2×D452112yes—
201C2×C4—2261no—
202C2×C4—2266no—
203C2×C4—2264no—
204C2×C4—22612no—
205C2×C4—22612no—
206C2×C4C2×D45213yes—
207C2×C4C2×D45216yes—
208C2×C4C23×D11316yes—
209C2×C4C23×D11311yescharacteristic
210C8C23×D11318yes—
211C22—43no—
212C22—90412no—
213C22—9046no—
214C22—4526no—
215C22—2264no—
216C22—2266no—
217C22C22×D45211yescharacteristic
218C22C22×D45216yes—
219C4—2262no—
220C4—2264no—
221C4C22×D45211yescharacteristic
222C4C22×D45216yes—
223C4C24×D11311yescharacteristic
224C2—21no—
225C2—26no—
226C2—4526no—
227C2—4524no—
228C2C23×D45211yescharacteristic
subgroupG/H
centerC2C23×D452
commutator, cosocleC4×C113C25
FrattiniC4C24×D113
Fitting#568×C113C2
socleC2×C113C23×D4
hypercenter#568D113
hypocenterC113#3585
LCSG ▷ C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C2 ◁ C2×#41 ◁ #568

Complements

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

NH ≅ G/Ntype
1. #20299434C2semidirect
2. C2×#16813668C2semidirect
3. #20299410C2semidirect
4. #20299403C2semidirect
5. #20299398C2semidirect
6. C2×#16813650C2semidirect
7. #568×C113C2semidirect
9. #16813669C22semidirect
11. #16813668C22semidirect
12. #16813668C22semidirect
15. C2×#14268647C22semidirect
16. C2×#14268647C22semidirect
18. C22×D452C22semidirect
20. #16813655C22semidirect
21. #16813655C22semidirect
22. #16813650C22semidirect
23. #16813650C22semidirect
24. #16813650C22semidirect
25. #16813644C22semidirect
26. C2×#14268636C22semidirect
27. C2×D904C22semidirect
29. #131×C113C22semidirect
30. C2×M16×C113C22semidirect
39. #14268647C23semidirect
42. C2×Dic226C23semidirect
48. C2×D452C23semidirect
55. Dic452C23semidirect
56. #14268636C23semidirect
57. #14268636C23semidirect
58. D904C23semidirect
65. M16×C113C23semidirect
66. C2×C8×C113C23semidirect
115. C113#3585semidirect
122. #568D113semidirect
144. #131C2×D113semidirect
145. C2×M16C2×D113semidirect
180. M16C22×D113semidirect
181. C2×C8C22×D113semidirect

Maximal subgroups

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

subgroupindex
#202994342
C2×#168136682
#202994102
#202994032
#202993982
C2×#168136502
#568×C1132
#3585113