Finite Group Databasebeta

← #27156835

Subgroups — #27156835

Order 12416.

162 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 38 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#20261722C211yesmaximal, characteristic
2#20261709C211yesmaximal, characteristic
3#20260707C211yesmaximal, characteristic
4C2×#13095521C411yescharacteristic, complemented
5C2×#13095520—21no—
6C2×#13095519C411yescharacteristic, complemented
7#15690871C412yescomplemented
8#15690870—21no—
9C22⋊#11715897—21no—
10C22⋊#11715897—21no—
11C2×#13095510C2211yescharacteristic
12#15690766—21no—
13#15690725C411yescharacteristic
14#15690724—21no—
15#99×C97C411yescharacteristic
16#13095521—41no—
17#13095520—22no—
18#13095520—41no—
19#13095519—41no—
20C2×#11715897—21no—
21C2×#11715897—21no—
22C2×#11715897—41no—
23#13095510—41no—
24#13095510—41no—
25C2×Dic194C2×C411yescharacteristic
26C2×C22⋊D97D411yescharacteristic
27C2×C22⋊D97Q811yescharacteristic
28C22×Dic97D411yescharacteristic
29C22×Dic97D411yescharacteristic
30C2×C4×D97C2×C411yescharacteristic
31#13095490—41no—
32C22⋊Dic97—21no—
33C4×Dic97—41no—
34C2×D4×C97C2×C411yescharacteristic
35C22⋊C4×C97—21no—
36C23×C97C22⋊C411yescharacteristic
37C23×C97C4⋊C411yescharacteristic
38C22×D97C4⋊C411yescharacteristic
39D4×C97—41no—
40C2×C4×C97—41no—
41C2×C4×C97C4211yescharacteristic
42C22⋊D97—41no—
43C22⋊D97—41no—
44C2×Dic97—21no—
45C2×Dic97—21no—
46C2×Dic97—21no—
47C2×Dic97—21no—
48C2×Dic97—41no—
49C2×Dic97C22⋊C411yescharacteristic
50C2×Dic97C22⋊C411yescharacteristic
51C2×Dic97C4⋊C411yescharacteristic
52C4×D97—21no—
53C4×D97—21no—
54Dic194—41no—
55#11715897—42no—
56#11715897—41no—
57#11715897—41no—
58C22×C97—21no—
59C22×C97—21no—
60C22×C97—41no—
61C22×C97—41no—
62C22×C97#9511yescharacteristic
63C2×D97—21no—
64C2×D97—41no—
65C4×C97—21no—
66C4×C97—81no—
67Dic97—21no—
68Dic97—21no—
69Dic97—81no—
70Dic97#9911yescharacteristic
71Dic97#9911yescharacteristic
72C2×C97—21no—
73C2×C97—41no—
74C2×C97—41no—
75C2×C97#34211yescharacteristic
76D97—41no—
77#1395—971nomaximal, Sylow
78C97#139511yescharacteristic, complemented, Sylow
79#413—971no—
80#410—971no—
81#408—971no—
82C2×#41—971no—
83C2×M16—971no—
84C2×M16—1941no—
85C22×C8—971no—
86#117—1941no—
87#101—972no—
88#100—1941no—
89#99—971no—
90#99—1941no—
91#99#8760411yescharacteristic
92C22⋊C8—1941no—
93C22⋊C8—1941no—
94#41—3881no—
95#41—3881no—
96C2×Q8—971no—
97C2×D4—971no—
98C2×D4—971no—
99C2×D4C2×#8760411yescharacteristic
100C22×C4—971no—
101C22×C4—971no—
102C22×C4—971no—
103M16—3881no—
104M16—3881no—
105M16—1942no—
106C2×C8—3881no—
107C2×C8—3881no—
108C2×C8—1941no—
109C2×C8—1941no—
110C4⋊C4—3881no—
111C22⋊C4—21no—
112C22⋊C4—1941no—
113C42—3881no—
114C23—971no—
115C23C22⋊#8760411yescharacteristic
116C23C4⋊#8760411yescharacteristic
117Q8—3881no—
118D4—41no—
119D4—3881no—
120D4—3881no—
121C2×C4—41no—
122C2×C4—971no—
123C2×C4—971no—
124C2×C4—971no—
125C2×C4—3881no—
126C2×C4—1941no—
127C2×C4—1941no—
128C2×C4—1941no—
129C2×C4—1941no—
130C2×C4—1941no—
131C2×C4—1941no—
132C2×C4C4×#8760411yescharacteristic
133C8—3881no—
134C8—3881no—
135C8—3882no—
136C22—21no—
137C22—21no—
138C22—41no—
139C22—41no—
140C22—3881no—
141C22—1941no—
142C22(C2×#87604)⋊C411yescharacteristic
143C4—21no—
144C4—81no—
145C4—7761no—
146C4—971no—
147C4—971no—
148C4—1941no—
149C4—1941no—
150C2—21no—
151C2—41no—
152C2—41no—
153C2—3881no—
154C2#2026161211yescharacteristic
subgroupG/H
centerC2#20261612
commutatorC2×C4×C97C42
FrattiniC2×D4C2×#87604
Fitting#99×C97C4
socleC2×C97#342
prime radical, cosocleC2×#13095510C22
hypercenter#99#87604
hypocenterC97#1395
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #99

Complements

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

NH ≅ G/Ntype
4. C2×#13095521C4semidirect
6. C2×#13095519C4semidirect
7. #15690871C4semidirect
78. C97#1395semidirect

Maximal subgroups

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

subgroupindex
#202617222
#202617092
#202607072
#139597