Finite Group Databasebeta

← #27149999

Subgroups — #27149999

Order 12416.

548 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 125 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#20260672C211yesmaximal, characteristic, complemented
2#20260670C213yesmaximal, complemented
3#20260659C211yesmaximal, characteristic, complemented
4#20260658C213yesmaximal, complemented
5#20260657C213yesmaximal, complemented
6#20260590C213yesmaximal
7#396×C97C211yesmaximal, characteristic, complemented
8C24×D97C2211yescharacteristic
9C2×C4⋊Dic97C2213yescomplemented
10C2×#13095490C2213yescomplemented
11C2×#13095490C2213yescomplemented
12C2×C22⋊Dic97C2213yes—
13C2×#13095488C2213yes—
14C2×#13095488C2213yes—
15C2×#13095488C2213yes—
16C2×#13095488C2213yes—
17#15690717C2213yescomplemented
18C2×C4⋊C4×C97C2213yescomplemented
19C2×C22⋊C4×C97C2213yes—
20#95×C97C2211yescharacteristic, complemented
21C23×D97—43no—
22C23×D97—43no—
23C23×D97Q814yes—
24C23×D97C2311yescharacteristic
25C23×D97C2311yescharacteristic
26C22×Dic97C2313yes—
27C22×Dic97C2313yes—
28C4⋊Dic97—43no—
29#13095490—43no—
30#13095490—43no—
31C22⋊Dic97—43no—
32#13095488—43no—
33#13095488—43no—
34#13095488—43no—
35#13095488—43no—
36C24×C97C2311yescharacteristic
37C22×C4×C97C2313yes—
38C22×C4×C97C2313yes—
39C4⋊C4×C97—43no—
40C22⋊C4×C97—43no—
41C23×C97—23no—
42C23×C97—23no—
43C23×C97C2×Q812yes—
44C23×C97C2411yescharacteristic
45C22×D97—23no—
46C22×D97—23no—
47C22×D97—23no—
48C22×D97—23no—
49C22×D97—412no—
50C22×D97—412no—
51C22×D97—44no—
52C22×D97C2×Q812yes—
53C22×D97C2×Q812yes—
54C2×C4×C97—23no—
55C2×C4×C97—23no—
56C2×C4×C97—23no—
57C2×C4×C97—23no—
58C2×C4×C97—23no—
59C2×C4×C97—23no—
60C2×Dic97—23no—
61C2×Dic97—23no—
62C2×Dic97—23no—
63C2×Dic97—23no—
64C2×Dic97—23no—
65C2×Dic97—23no—
66C22×C97—26no—
67C22×C97—43no—
68C22×C97—41no—
69C22×C97C22×Q811yescharacteristic
70C22×C97H(5,𝔽2)13yes—
71C22×C97H(5,𝔽2)13yes—
72C2×D97—26no—
73C2×D97—26no—
74C2×D97—416no—
75C2×D97—43no—
76C2×D97—43no—
77C2×D97—41no—
78C2×D97—41no—
79C4×C97—43no—
80C4×C97—43no—
81Dic97—43no—
82Dic97—43no—
83C2×C97—42no—
84C2×C97#54313yes—
85C2×C97#56013yes—
86C2×C97#56011yescharacteristic
87D97—42no—
88D97—42no—
89#2842—971nomaximal, Sylow
90C97#284211yescharacteristic, complemented, Sylow
91#399—973no—
92#399—971no—
93#396—973no—
94#396—973no—
95#396—971no—
96#396D9711yescharacteristic, complemented
97#379—973no—
98C25—971no—
99C2×C4⋊C4—973no—
100C2×C4⋊C4—973no—
101C2×C4⋊C4—973no—
102C2×C4⋊C4C2×D9713yescomplemented
103C2×C22⋊C4—973no—
104C2×C22⋊C4—973no—
105C2×C22⋊C4—973no—
106C2×C22⋊C4—973no—
107C2×C22⋊C4—973no—
108C2×C22⋊C4C2×D9713yes—
109#95—973no—
110#95C2×D9711yescharacteristic, complemented
111C24—974no—
112C24—971no—
113C24—971no—
114C24—3883no—
115C24—3883no—
116C24C22×D9711yescharacteristic
117C22×C4—973no—
118C22×C4—973no—
119C22×C4C22×D9713yes—
120C22×C4C22×D9713yes—
121C4⋊C4—43no—
122C4⋊C4—3883no—
123C4⋊C4—3883no—
124C4⋊C4—3883no—
125C22⋊C4—43no—
126C22⋊C4—3883no—
127C22⋊C4—3883no—
128C22⋊C4—3883no—
129C22⋊C4—3883no—
130C22⋊C4—3883no—
131C23—23no—
132C23—23no—
133C23—972no—
134C23—972no—
135C23—38812no—
136C23—38812no—
137C23—3884no—
138C23—1943no—
139C23—1943no—
140C23—1943no—
141C23—1943no—
142C23Q8×D9712yes—
143C23C23×D9711yescharacteristic
144C2×C4—23no—
145C2×C4—23no—
146C2×C4—23no—
147C2×C4—23no—
148C2×C4—23no—
149C2×C4—23no—
150C2×C4—1943no—
151C2×C4—1943no—
152C2×C4—1943no—
153C2×C4—1943no—
154C2×C4—1943no—
155C2×C4—1943no—
156C22—26no—
157C22—43no—
158C22—41no—
159C22—38816no—
160C22—3883no—
161C22—3883no—
162C22—3881no—
163C22—3881no—
164C22—1946no—
165C22—1946no—
166C22C2×Q8×D9711yescharacteristic
167C22#1569085513yes—
168C22#1569085613yes—
169C4—43no—
170C4—43no—
171C4—3883no—
172C4—3883no—
173C2—42no—
174C2—3882no—
175C2—3882no—
176C2#2026136013yes—
177C2#2026147313yes—
178C2#2026147611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#396×C97C2
prime radicalC24×D97C22
hypercenter#396D97
hypocenterC97#2842
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. #20260672C2semidirect
2. #20260670C2semidirect
3. #20260659C2semidirect
4. #20260658C2semidirect
5. #20260657C2semidirect
7. #396×C97C2semidirect
9. C2×C4⋊Dic97C22semidirect
10. C2×#13095490C22semidirect
11. C2×#13095490C22semidirect
17. #15690717C22semidirect
18. C2×C4⋊C4×C97C22semidirect
20. #95×C97C22semidirect
90. C97#2842semidirect
96. #396D97semidirect
102. C2×C4⋊C4C2×D97semidirect
110. #95C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202606722
#202606702
#202606592
#202606582
#202606572
#202605902
#396×C972
#284297