Finite Group Databasebeta

← #27151133

Subgroups — #27151133

Order 12416.

800 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 161 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#20261294C213yesmaximal, complemented
2#20260925C214yesmaximal, complemented
3#20260922C214yesmaximal, complemented
4C2×#15690725C213yesmaximal, complemented
5C2×H(5,𝔽2)×C97C211yesmaximal, characteristic, complemented
6C22×C22⋊D97C2211yescharacteristic, complemented
7#15690793—23no—
8#15690791C2216yescomplemented
9#15690782—26no—
10#15690779—23no—
11#15690778C2216yescomplemented
12C2×C22⋊Dic97C2213yescomplemented
13#15690725C22112yescomplemented
14H(5,𝔽2)×C97—26no—
15H(5,𝔽2)×C97C2214yes—
16C2×#41×C97—23no—
17C22×D4×C97—23no—
18C22×D4×C97C2213yes—
19C23×D97C2311yescharacteristic, complemented
20C2×C22⋊D97—23no—
21C2×C22⋊D97—23no—
22C2×C22⋊D97—26no—
23C22×Dic97—23no—
24C22×Dic97C2311yescharacteristic, complemented
25#13095490—26no—
26C22⋊Dic97—23no—
27C22⋊Dic97—26no—
28C22⋊Dic97C2316yescomplemented
29#13095488—26no—
30C24×C97—21no—
31C24×C97D413yescomplemented
32C24×C97C2311yescharacteristic
33#41×C97—212no—
34#41×C97—212no—
35C2×Q8×C97—21no—
36C2×D4×C97—23no—
37C2×D4×C97—212no—
38C2×D4×C97—26no—
39C2×D4×C97—212no—
40C2×D4×C97—26no—
41C2×D4×C97D416yescomplemented
42C2×D4×C97C2316yes—
43C22×C4×C97—23no—
44C22×C4×C97D413yescomplemented
45C23×C97—24no—
46C23×C97—23no—
47C23×C97—23no—
48C23×C97—81no—
49C23×C97—83no—
50C23×C97—82no—
51C23×C97C2×D416yes—
52C23×C97C2×D413yes—
53C23×C97C2×D413yes—
54C23×C97C2411yescharacteristic
55C22×D97—23no—
56C22×D97—81no—
57Q8×C97—24no—
58D4×C97—212no—
59D4×C97—212no—
60D4×C97—212no—
61C2×C4×C97—212no—
62C2×C4×C97—23no—
63C2×C4×C97—23no—
64C2×C4×C97C2×D416yes—
65C22⋊D97—41no—
66C22⋊D97—43no—
67C22⋊D97—86no—
68C2×Dic97—26no—
69C2×Dic97—23no—
70C2×Dic97—83no—
71C22×C97—26no—
72C22×C97—43no—
73C22×C97—41no—
74C22×C97—43no—
75C22×C97—46no—
76C22×C97—46no—
77C22×C97—83no—
78C22×C97—84no—
79C22×C97#12011yescharacteristic
80C22×C97#12013yes—
81C22×C97C22×D413yes—
82C2×D97—46no—
83C2×D97—41no—
84C4×C97—26no—
85Dic97—46no—
86Dic97—41no—
87C2×C97—23no—
88C2×C97—21no—
89C2×C97—23no—
90C2×C97—46no—
91C2×C97C2×#12011yescharacteristic
92D97—81no—
93#3026—971nomaximal, Sylow
94C97#302611yescharacteristic, complemented, Sylow
95C2×H(5,𝔽2)D9711yescharacteristic, complemented
96#534—973no—
97#458—974no—
98B(4,𝔽2)—974no—
99C2×#99—973no—
100H(5,𝔽2)—26no—
101H(5,𝔽2)C2×D9714yes—
102C2×#41—23no—
103C22×D4—23no—
104C22×D4—971no—
105C22×D4C2×D9713yes—
106#123—976no—
107#123—1943no—
108#121—1946no—
109#120—976no—
110#120—1943no—
111C2×C22⋊C4—973no—
112#99—9712no—
113C24—21no—
114C24—971no—
115C24C22⋊D9713yescomplemented
116C24C22×D9711yescharacteristic
117#41—212no—
118#41—212no—
119C2×Q8—21no—
120C2×D4—23no—
121C2×D4—212no—
122C2×D4—26no—
123C2×D4—212no—
124C2×D4—26no—
125C2×D4—1946no—
126C2×D4—1943no—
127C2×D4—1943no—
128C2×D4C22⋊D9716yescomplemented
129C2×D4C22×D9716yes—
130C22×C4—23no—
131C22×C4—971no—
132C22×C4—1943no—
133C22×C4C22⋊D9713yescomplemented
134C4⋊C4—1946no—
135C22⋊C4—976no—
136C22⋊C4—1943no—
137C22⋊C4—1946no—
138C22⋊C4—1946no—
139C23—24no—
140C23—23no—
141C23—23no—
142C23—81no—
143C23—83no—
144C23—82no—
145C23—7761no—
146C23—1943no—
147C23C2×C22⋊D9716yes—
148C23C2×C22⋊D9713yes—
149C23C2×C22⋊D9713yes—
150C23C23×D9711yescharacteristic
151Q8—24no—
152D4—212no—
153D4—212no—
154D4—212no—
155D4—7766no—
156D4—3881no—
157D4—3883no—
158C2×C4—212no—
159C2×C4—23no—
160C2×C4—23no—
161C2×C4—7763no—
162C2×C4—1946no—
163C2×C4—1943no—
164C2×C4C2×C22⋊D9716yes—
165C22—26no—
166C22—43no—
167C22—43no—
168C22—41no—
169C22—46no—
170C22—46no—
171C22—83no—
172C22—84no—
173C22—3886no—
174C22—3881no—
175C22#1569077911yescharacteristic
176C22#1569077913yes—
177C22C22×C22⋊D9713yes—
178C4—26no—
179C4—3886no—
180C4—3881no—
181C2—23no—
182C2—21no—
183C2—23no—
184C2—46no—
185C2—7761no—
186C2C2×#1569077911yescharacteristic
subgroupG/H
centerC2C2×#15690779
commutator, cosocleC23×C97C24
FrattiniC23C23×D97
FittingC2×H(5,𝔽2)×C97C2
socleC2×C97C2×#120
hypercenterC2×H(5,𝔽2)D97
hypocenterC97#3026
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C24 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20261294C2semidirect
2. #20260925C2semidirect
3. #20260922C2semidirect
4. C2×#15690725C2semidirect
5. C2×H(5,𝔽2)×C97C2semidirect
6. C22×C22⋊D97C22semidirect
8. #15690791C22semidirect
11. #15690778C22semidirect
12. C2×C22⋊Dic97C22semidirect
13. #15690725C22semidirect
19. C23×D97C23semidirect
24. C22×Dic97C23semidirect
28. C22⋊Dic97C23semidirect
31. C24×C97D4semidirect
41. C2×D4×C97D4semidirect
44. C22×C4×C97D4semidirect
94. C97#3026semidirect
95. C2×H(5,𝔽2)D97semidirect
115. C24C22⋊D97semidirect
128. C2×D4C22⋊D97semidirect
133. C22×C4C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
#202612942
#202609252
#202609222
C2×#156907252
C2×H(5,𝔽2)×C972
#302697