Finite Group Databasebeta

← #27149988

Subgroups — #27149988

Order 12416.

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

normal subgroups only

This group has 580 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#20260648C216yesmaximal, complemented
2#20260647C213yesmaximal, complemented
3#20260646C212yesmaximal, complemented
4#20260590C213yesmaximal
5#393×C97C211yesmaximal, characteristic, complemented
6C24×D97C2211yescharacteristic
7C22×Dic194C2213yescomplemented
8C2×C22⋊Dic97C2213yes—
9C2×#13095488C2216yes—
10C2×#13095488C2216yes—
11#15690717C2216yescomplemented
12#15690717C2213yescomplemented
13C22×Q8×C97C2211yescharacteristic, complemented
14C2×C22⋊C4×C97C2213yes—
15#95×C97C2213yescomplemented
16C23×D97—43no—
17C23×D97—43no—
18C23×D97Q814yescomplemented
19C23×D97C2312yes—
20C2×Dic194—43no—
21C2×Dic194—46no—
22C22×Dic97C2316yes—
23C22⋊Dic97—43no—
24#13095488—46no—
25#13095488—46no—
26C24×C97C2311yescharacteristic
27C2×Q8×C97—43no—
28C22×C4×C97C2313yes—
29C22×C4×C97C2313yes—
30C22⋊C4×C97—43no—
31C23×C97—23no—
32C23×C97—23no—
33C23×C97C2×Q812yes—
34C23×C97C2411yescharacteristic
35C22×D97—26no—
36C22×D97—26no—
37C22×D97—412no—
38C22×D97—412no—
39C22×D97—44no—
40C22×D97C2×Q814yes—
41Q8×C97—44no—
42C2×C4×C97—23no—
43C2×C4×C97—23no—
44C2×C4×C97—26no—
45C2×C4×C97—26no—
46C2×Dic97—26no—
47C2×Dic97—26no—
48C2×Dic97—26no—
49Dic194—412no—
50C22×C97—26no—
51C22×C97—43no—
52C22×C97—41no—
53C22×C97C22×Q811yescharacteristic
54C22×C97H(5,𝔽2)13yes—
55C22×C97H(5,𝔽2)13yes—
56C2×D97—212no—
57C2×D97—416no—
58C2×D97—46no—
59C2×D97—42no—
60C4×C97—43no—
61C4×C97—43no—
62Dic97—46no—
63C2×C97—42no—
64C2×C97#54313yes—
65C2×C97H(3,𝔽4)13yes—
66C2×C97H(3,𝔽4)11yescharacteristic
67D97—44no—
68#2840—971nomaximal, Sylow
69C97#284011yescharacteristic, complemented, Sylow
70#393—972no—
71#393—973no—
72#393—976no—
73#393D9711yescharacteristic, complemented
74#379—973no—
75C25—971no—
76C22×Q8—973no—
77C22×Q8C2×D9711yescharacteristic, complemented
78C2×C22⋊C4—973no—
79C2×C22⋊C4—976no—
80C2×C22⋊C4—976no—
81C2×C22⋊C4C2×D9713yes—
82#95—976no—
83#95—973no—
84#95C2×D9713yescomplemented
85C24—974no—
86C24—972no—
87C24—3883no—
88C24—3883no—
89C24C22×D9711yescharacteristic
90C2×Q8—43no—
91C2×Q8—3883no—
92C2×Q8—3886no—
93C22×C4—976no—
94C22×C4C22×D9713yes—
95C22×C4C22×D9713yes—
96C22⋊C4—43no—
97C22⋊C4—3886no—
98C22⋊C4—3886no—
99C22⋊C4—3883no—
100C23—23no—
101C23—23no—
102C23—974no—
103C23—38812no—
104C23—38812no—
105C23—3884no—
106C23—1946no—
107C23—1946no—
108C23Q8×D9712yes—
109C23C23×D9711yescharacteristic
110Q8—44no—
111Q8—38812no—
112C2×C4—23no—
113C2×C4—23no—
114C2×C4—26no—
115C2×C4—26no—
116C2×C4—1946no—
117C2×C4—1946no—
118C2×C4—1946no—
119C22—26no—
120C22—43no—
121C22—41no—
122C22—38816no—
123C22—3886no—
124C22—3882no—
125C22—19412no—
126C22C2×Q8×D9711yescharacteristic
127C22#1569085513yes—
128C22#1569085613yes—
129C4—43no—
130C4—43no—
131C4—3886no—
132C2—42no—
133C2—3884no—
134C2#2026136013yes—
135C2#2026147713yes—
136C2#2026147811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#393×C97C2
prime radicalC24×D97C22
hypercenter#393D97
hypocenterC97#2840
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. #20260648C2semidirect
2. #20260647C2semidirect
3. #20260646C2semidirect
5. #393×C97C2semidirect
7. C22×Dic194C22semidirect
11. #15690717C22semidirect
12. #15690717C22semidirect
13. C22×Q8×C97C22semidirect
15. #95×C97C22semidirect
18. C23×D97Q8semidirect
69. C97#2840semidirect
73. #393D97semidirect
77. C22×Q8C2×D97semidirect
84. #95C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202606482
#202606472
#202606462
#202605902
#393×C972
#284097