Finite Group Databasebeta

← #27141045

Subgroups — #27141045

Order 12416.

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

normal subgroups only

This group has 196 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
1#20261485C211yesmaximal, characteristic
2#20261141C212yesmaximal, complemented
3#20260476C212yesmaximal, complemented
4#20260475C211yesmaximal, characteristic
5#497×C97C211yesmaximal, characteristic
6#15690808—21no—
7#15690803—22no—
8#15690802C2212yes—
9#15690791—21no—
10#15690787—22no—
11#15690743—22no—
12#15690741—22no—
13#15690721C2212yes—
14C2×Q16×C97—21no—
15C2×SD16×C97—21no—
16C4⋊Q8×C97C2211yescharacteristic
17#124×C97C2211yescharacteristic
18#97×C97C2211yescharacteristic
19C2×Dic194—21no—
20C22×Dic97—21no—
21C2×#11715895—22no—
22C4⋊Dic97D412yes—
23C4⋊Dic97D412yes—
24#13095490—22no—
25#13095490—22no—
26#13095490—22no—
27C22⋊Dic97—22no—
28C22⋊Dic97—21no—
29C4×Dic97—21no—
30C2×Q8×C97—21no—
31C2×Q8×C97D411yescharacteristic
32C2×D4×C97D411yescharacteristic
33Q16×C97—42no—
34SD16×C97—42no—
35C2×C8×C97—21no—
36C4⋊C4×C97—22no—
37C22⋊C4×C97—22no—
38C42×C97C2311yescharacteristic
39C23×C97—21no—
40Q8×C97—41no—
41Q8×C97—41no—
42Q8×C97—41no—
43D4×C97—41no—
44C2×C4×C97—21no—
45C2×C4×C97—41no—
46C2×C4×C97C2×D412yes—
47C2×C4×C97C2×D411yescharacteristic
48C2×Dic97—22no—
49C2×Dic97—22no—
50C2×Dic97—81no—
51Dic194—81no—
52C8×C97—22no—
53#11715895—24no—
54C22×C97—41no—
55C22×C97—42no—
56C22×C97#12011yescharacteristic
57C4×C97—21no—
58C4×C97—22no—
59C4×C97—41no—
60C4×C97—81no—
61Dic97—42no—
62Dic97—42no—
63C2×C97—81no—
64C2×C97#45311yescharacteristic
65C2×C97#45612yes—
66#1660—971nomaximal, Sylow
67C97#166011yescharacteristic, complemented, Sylow
68#563—971no—
69#501—972no—
70#497D9711yescharacteristic
71#330—972no—
72#329—971no—
73C2×Q16—21no—
74C2×SD16—21no—
75C4⋊Q8—1941no—
76C4⋊Q8C2×D9711yescharacteristic
77#126—1942no—
78#125—972no—
79#124C2×D9711yescharacteristic
80#123—1941no—
81C22⋊Q8—1942no—
82#107—1942no—
83#106—1942no—
84#97—972no—
85#97C2×D9711yescharacteristic
86C2×Q8—21no—
87C2×Q8—1941no—
88C2×Q8C22⋊D9711yescharacteristic
89C2×D4C22⋊D9711yescharacteristic
90C22×C4—1941no—
91Q16—42no—
92SD16—42no—
93C2×C8—21no—
94C2×C8—1942no—
95C4⋊C4—22no—
96C4⋊C4—972no—
97C4⋊C4—972no—
98C4⋊C4—1942no—
99C4⋊C4—1942no—
100C4⋊C4—1942no—
101C22⋊C4—22no—
102C22⋊C4—1942no—
103C22⋊C4—1941no—
104C42—1941no—
105C42C22×D9711yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—41no—
109Q8—41no—
110Q8—7761no—
111D4—41no—
112C2×C4—21no—
113C2×C4—41no—
114C2×C4—7761no—
115C2×C4—1942no—
116C2×C4—1942no—
117C2×C4D4×D9712yes—
118C2×C4C2×C22⋊D9711yescharacteristic
119C8—22no—
120C8—1944no—
121C22—41no—
122C22—42no—
123C22#1569077811yescharacteristic
124C4—21no—
125C4—22no—
126C4—41no—
127C4—81no—
128C4—3882no—
129C4—3882no—
130C2—81no—
131C2#2026090211yescharacteristic
132C2#2026091612yes—
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#497×C97C2
socleC22×C97#120
prime radicalC2×D4×C97D4
hypercenter#497D97
hypocenterC97#1660
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
2. #20261141C2semidirect
3. #20260476C2semidirect
67. C97#1660semidirect

Maximal subgroups

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

subgroupindex
#202614852
#202611412
#202604762
#202604752
#497×C972
#166097