Finite Group Databasebeta

← #27143809

Subgroups — #27143809

Order 12416.

170 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 47 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#20261171C212yesmaximal, complemented
2#20260853C211yesmaximal, characteristic, complemented
3#20260556C211yesmaximal, characteristic, complemented
4#20260548C212yesmaximal, complemented
5#493×C97C211yesmaximal, characteristic, complemented
6#15690832C2211yescharacteristic
7#15690816—21no—
8#15690754—22no—
9#15690751—22no—
10#15690750C2212yes—
11#15690746C2211yescharacteristic
12#15690738—21no—
13C2×D8×C97—21no—
14C2×D8×C97C2212yes—
15#127×C97—21no—
16C4×C8×C97C2211yescharacteristic
17#13095511—21no—
18Dic388D411yescharacteristic, complemented
19#13095500—21no—
20D776D411yescharacteristic, complemented
21#13095495—21no—
22#13095494—41no—
23C8×D97—41no—
24#13095486—22no—
25C2×D4×C97—22no—
26C2×D4×C97—42no—
27D8×C97—22no—
28D8×C97—22no—
29D8×C97—41no—
30D8×C97—41no—
31C2×C8×C97D411yescharacteristic
32C2×C8×C97C2311yescharacteristic
33C42×C97D411yescharacteristic
34C23×C97—42no—
35D4×C97—42no—
36D4×C97—42no—
37D4×C97—84no—
38C2×C4×C97—21no—
39C2×C4×C97C2×D411yescharacteristic
40C22⋊D97—41no—
41D388—21no—
42C4×D97—41no—
43Dic194—21no—
44C8×C97D811yescharacteristic
45C8×C97D811yescharacteristic
46C8×C97C2×D411yescharacteristic
47C8×C97#4111yescharacteristic
48#11715895—41no—
49C22×C97—84no—
50C22×C97—82no—
51C22×C97AGL(1,ℤ/8)11yescharacteristic
52C2×D97—41no—
53C4×C97—22no—
54C4×C97#12111yescharacteristic
55C4×C97C2×D811yescharacteristic
56Dic97—41no—
57C2×C97—21no—
58C2×C97—162no—
59C2×C97#45911yescharacteristic
60D97—81no—
61#2213—971nomaximal, Sylow
62C97#221311yescharacteristic, complemented, Sylow
63#509—972no—
64#493D9711yescharacteristic, complemented
65#443—971no—
66#364—971no—
67#361—972no—
68#135—971no—
69C2×D8—21no—
70C2×D8C2×D9712yes—
71#131—1941no—
72#127—21no—
73SD32—1942no—
74D16—1942no—
75M32—972no—
76#108—971no—
77#104—1941no—
78C4×C8C2×D9711yescharacteristic
79#41—1941no—
80C2×D4—22no—
81C2×D4—42no—
82Q16—971no—
83SD16—1941no—
84D8—22no—
85D8—22no—
86D8—41no—
87D8—41no—
88D8—971no—
89M16—3881no—
90M16—1941no—
91C2×C8—3881no—
92C2×C8C22⋊D9711yescharacteristic
93C2×C8C22×D9711yescharacteristic
94C42C22⋊D9711yescharacteristic
95C16—1942no—
96C23—42no—
97Q8—1941no—
98D4—42no—
99D4—42no—
100D4—84no—
101D4—3881no—
102D4—1941no—
103C2×C4—21no—
104C2×C4—3881no—
105C2×C4C2×C22⋊D9711yescharacteristic
106C8—3881no—
107C8#1309549611yescharacteristic
108C8#1309549611yescharacteristic
109C8D4×D9711yescharacteristic
110C8#1309551011yescharacteristic
111C22—84no—
112C22—82no—
113C22—3881no—
114C22#1569083611yescharacteristic
115C4—22no—
116C4—3881no—
117C4#1569078111yescharacteristic
118C4C2×#1309549611yescharacteristic
119C2—21no—
120C2—7761no—
121C2—162no—
122C2#2026092711yescharacteristic
subgroupG/H
centerC2#20260927
commutator, cosocleC2×C8×C97C23
FrattiniC2×C8C22×D97
Fitting#493×C97C2
socleC2×C97#459
hypercenter#493D97
hypocenterC97#2213
LCSG ▷ C2×C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #493

Complements

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

NH ≅ G/Ntype
1. #20261171C2semidirect
2. #20260853C2semidirect
3. #20260556C2semidirect
4. #20260548C2semidirect
5. #493×C97C2semidirect
18. Dic388D4semidirect
20. D776D4semidirect
62. C97#2213semidirect
64. #493D97semidirect

Maximal subgroups

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

subgroupindex
#202611712
#202608532
#202605562
#202605482
#493×C972
#221397