Finite Group Databasebeta

← #27141065

Subgroups — #27141065

Order 12416.

172 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 45 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
1C97⋊#564C211yesmaximal, characteristic
2#20261144C212yesmaximal
3#20261105C211yesmaximal, characteristic
4#20260476C212yesmaximal
5#330×C97C211yesmaximal, characteristic
6#15690810C2211yescharacteristic
7#15690802C2212yes—
8#15690801—22no—
9#15690800—22no—
10#15690800—22no—
11#15690744—22no—
12#15690742—22no—
13#15690733—22no—
14#15690721C2211yescharacteristic
15#125×C97C2211yescharacteristic
16#97×C97C2212yes—
17C2×Dic194—21no—
18C2×#11715895—21no—
19C4⋊Dic97—22no—
20C4⋊Dic97—22no—
21C4⋊Dic97D412yes—
22C4⋊Dic97D412yes—
23#13095490—22no—
24#13095490—22no—
25#13095490—22no—
26#13095490—22no—
27C4×Dic97—22no—
28C2×C8×C97—22no—
29C4⋊C4×C97—22no—
30C4⋊C4×C97D412yes—
31C42×C97C2311yescharacteristic
32C2×C4×C97—22no—
33C2×C4×C97C2×D412yes—
34C2×C4×C97C2×D411yescharacteristic
35C2×Dic97—22no—
36C2×Dic97—22no—
37C2×Dic97—41no—
38Dic194—42no—
39C8×C97—24no—
40#11715895—41no—
41C22×C97#12011yescharacteristic
42C4×C97—21no—
43C4×C97—22no—
44C4×C97—42no—
45Dic97—42no—
46Dic97—42no—
47Dic97—81no—
48C2×C97#45411yescharacteristic
49C2×C97#45612yes—
50#1663—971nomaximal, Sylow
51C97#166311yescharacteristic, complemented, Sylow
52#564—971no—
53#501—972no—
54#491—971no—
55#330—972no—
56#330D9711yescharacteristic
57C4⋊Q8—971no—
58#125—972no—
59#125—1942no—
60#125—1942no—
61#125—1942no—
62#125C2×D9711yescharacteristic
63#107—1942no—
64#106—1942no—
65#103—1942no—
66#97—971no—
67#97C2×D9712yes—
68C2×Q8—1941no—
69C2×C8—22no—
70C2×C8—1941no—
71C4⋊C4—22no—
72C4⋊C4—972no—
73C4⋊C4—972no—
74C4⋊C4—1942no—
75C4⋊C4—1942no—
76C4⋊C4—1942no—
77C4⋊C4—1942no—
78C4⋊C4—1942no—
79C4⋊C4—1942no—
80C4⋊C4D38812yes—
81C42—1942no—
82C42C22×D9711yescharacteristic
83Q8—3882no—
84C2×C4—22no—
85C2×C4—3881no—
86C2×C4—1942no—
87C2×C4—1942no—
88C2×C4D4×D9712yes—
89C2×C4C2×D38811yescharacteristic
90C8—24no—
91C8—3881no—
92C22#1569078011yescharacteristic
93C4—21no—
94C4—22no—
95C4—42no—
96C4—7761no—
97C4—3882no—
98C4—3882no—
99C2#2026091011yescharacteristic
100C2#2026091912yes—
subgroupG/H
centerC22#15690780
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#330×C97C2
socle, prime radicalC22×C97#120
hypercenter#330D97
hypocenterC97#1663
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
51. C97#1663semidirect

Maximal subgroups

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

subgroupindex
C97⋊#5642
#202611442
#202611052
#202604762
#330×C972
#166397