Finite Group Databasebeta

← #27144929

Subgroups — #27144929

Order 12416.

456 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×#15690804C211yesmaximal, characteristic
2#20260630C213yesmaximal, complemented
3#20260625C213yesmaximal
4#20260616C213yesmaximal, complemented
5#20260606C213yesmaximal
6#20260603C211yesmaximal, characteristic, complemented
7#384×C97C211yesmaximal, characteristic, complemented
8C22×C4×D97C2211yescharacteristic
9#15690804C418yescomplemented
10C2×#13095490C2213yes—
11C2×#13095490C2213yes—
12C2×#13095488C2213yes—
13C2×#13095488C2213yes—
14C2×C4×Dic97C2213yes—
15#15690717C2213yes—
16#15690717C2213yes—
17#15690717C2213yes—
18#15690717C2213yes—
19C2×C4⋊C4×C97C2213yes—
20C2×C4⋊C4×C97C2213yes—
21C2×C42×C97C2211yescharacteristic
22C23×D97C2311yescharacteristic
23C22×Dic97C2313yes—
24C22×Dic97C2311yescharacteristic
25C22×Dic97C2313yes—
26C2×C4×D97—43no—
27#13095490—43no—
28#13095490C2×C4112yescomplemented
29#13095488—43no—
30#13095488C2×C4112yes—
31C4×Dic97—43no—
32C22×C4×C97C2313yes—
33C22×C4×C97C2311yescharacteristic
34C22×C4×C97C2313yes—
35C4⋊C4×C97—43no—
36C4⋊C4×C97—43no—
37C42×C97C2×C414yescomplemented
38C23×C97C2411yescharacteristic
39C22×D97—23no—
40C22×D97—23no—
41C22×D97C22×C412yes—
42C2×C4×C97—23no—
43C2×C4×C97—23no—
44C2×C4×C97—23no—
45C2×C4×C97—23no—
46C2×C4×C97—23no—
47C2×C4×C97—23no—
48C2×C4×C97C22×C416yes—
49C2×Dic97—23no—
50C2×Dic97—23no—
51C2×Dic97—23no—
52C2×Dic97—23no—
53C2×Dic97—26no—
54C2×Dic97C22×C416yes—
55C4×D97—44no—
56C22×C97C23×C411yescharacteristic
57C22×C97H(5,𝔽2)13yes—
58C22×C972−1+413yes—
59C2×D97—26no—
60C2×D97—41no—
61C2×D97—43no—
62C4×C97—26no—
63C4×C97—41no—
64C4×C97—43no—
65Dic97—26no—
66Dic97—41no—
67Dic97—43no—
68C2×C97#52013yes—
69C2×C97#56011yescharacteristic
70C2×C97#56313yes—
71D97—42no—
72#2383—971nomaximal, Sylow
73C97#238311yescharacteristic, complemented, Sylow
74C2×#126—971no—
75#389—973no—
76#388—973no—
77#387—973no—
78#385—973no—
79#384D9711yescharacteristic, complemented
80#383—971no—
81C23×C4—971no—
82#126—978no—
83C2×C4⋊C4—973no—
84C2×C4⋊C4—973no—
85C2×C4⋊C4C2×D9713yes—
86C2×C4⋊C4C2×D9713yes—
87C2×C22⋊C4—973no—
88C2×C22⋊C4—973no—
89C2×C42—973no—
90C2×C42C2×D9711yescharacteristic
91#95—973no—
92#95—973no—
93#95—973no—
94#95—973no—
95C24—971no—
96C22×C4—973no—
97C22×C4—973no—
98C22×C4—971no—
99C22×C4—3883no—
100C22×C4C22×D9711yescharacteristic
101C22×C4C22×D9713yes—
102C22×C4C22×D9713yes—
103C4⋊C4—43no—
104C4⋊C4—43no—
105C4⋊C4—9712no—
106C4⋊C4—3883no—
107C22⋊C4—9712no—
108C22⋊C4—3883no—
109C42—3883no—
110C42C4×D9714yescomplemented
111C23—972no—
112C23—1943no—
113C23—1943no—
114C23C23×D9711yescharacteristic
115C2×C4—23no—
116C2×C4—23no—
117C2×C4—23no—
118C2×C4—23no—
119C2×C4—23no—
120C2×C4—23no—
121C2×C4—976no—
122C2×C4—3884no—
123C2×C4—1943no—
124C2×C4—1943no—
125C2×C4—1946no—
126C2×C4—1943no—
127C2×C4—1943no—
128C2×C4C2×C4×D9716yes—
129C22—3881no—
130C22—3883no—
131C22—1946no—
132C22C22×C4×D9711yescharacteristic
133C22#1569085513yes—
134C22#1569085813yes—
135C4—26no—
136C4—41no—
137C4—43no—
138C4—3881no—
139C4—3883no—
140C4—1946no—
141C2—3882no—
142C2#2026121613yes—
143C2#2026147211yescharacteristic
144C2#2026148413yes—
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#384×C97C2
socle, cosocleC23×C97C24
prime radicalC23×D97C23
hypercenter#384D97
hypocenterC97#2383
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
2. #20260630C2semidirect
4. #20260616C2semidirect
6. #20260603C2semidirect
7. #384×C97C2semidirect
9. #15690804C4semidirect
28. #13095490C2×C4semidirect
37. C42×C97C2×C4semidirect
73. C97#2383semidirect
79. #384D97semidirect
110. C42C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908042
#202606302
#202606252
#202606162
#202606062
#202606032
#384×C972
#238397