Finite Group Databasebeta

← #27157190

Subgroups — #27157190

Order 12416.

270 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 62 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
1C4×#13095520C212yesmaximal, complemented
2#20261591C212yesmaximal, complemented
3#20261590C212yesmaximal, complemented
4#20261281C211yesmaximal, characteristic
5C2×#13095520—22no—
6#15690865C2214yes—
7C4×#11715897—22no—
8C2×#13095511C411yescharacteristic
9#15690808C411yescharacteristic
10#15690806C411yescharacteristic
11#15690799C2211yescharacteristic
12#15690797C2211yescharacteristic
13#15690782—22no—
14C4×C22⋊D97—22no—
15C2×C4×Dic97C2211yescharacteristic
16C2×#41×C97C411yescharacteristic
17#13095520—28no—
18C2×#11715897—24no—
19#13095511—42no—
20C2×Dic194C2×C411yescharacteristic
21C2×D388C2×C411yescharacteristic
22C2×C22⋊D97—21no—
23C2×C22⋊D97—22no—
24C22×Dic97D412yes—
25C2×C4×D97—21no—
26#13095490—22no—
27C22⋊Dic97—22no—
28#13095488—22no—
29C4×Dic97C2×C411yescharacteristic
30C4×Dic97D412yes—
31C4×Dic97C2311yescharacteristic
32#41×C97—42no—
33C2×Q8×C97C2×C411yescharacteristic
34C2×D4×C97—21no—
35C2×D4×C97C2×C411yescharacteristic
36C22×C4×C97—21no—
37C22×C4×C97C2×C411yescharacteristic
38C23×C97—21no—
39C23×C97C22⋊C411yescharacteristic
40C22×D97—21no—
41Q8×C97—41no—
42D4×C97—41no—
43D4×C97—42no—
44C2×C4×C97—22no—
45C2×C4×C97—21no—
46C2×C4×C97—42no—
47C2×C4×C97C22⋊C411yescharacteristic
48C2×C4×C97C22×C411yescharacteristic
49C22⋊D97—42no—
50C22⋊D97—44no—
51C2×Dic97—24no—
52C2×Dic97—21no—
53C2×Dic97C22⋊C412yes—
54C2×Dic97C2×D412yes—
55D388—41no—
56C4×D97—42no—
57Dic194—41no—
58#11715897—28no—
59C22×C97—21no—
60C22×C97—22no—
61C22×C97—41no—
62C22×C97—42no—
63C22×C97C2×C22⋊C411yescharacteristic
64C2×D97—41no—
65C2×D97—42no—
66C4×C97—21no—
67C4×C97—41no—
68C4×C97#10012yes—
69Dic97—22no—
70Dic97—41no—
71Dic97#10414yes—
72C2×C97—41no—
73C2×C97—81no—
74C2×C97C2×#10011yescharacteristic
75C2×C97C2×#10412yes—
76D97—81no—
77#1525—971nomaximal, Sylow
78C97#152511yescharacteristic, complemented, Sylow
79#532—971no—
80C4×M16—972no—
81#329—972no—
82#329—972no—
83C2×#41—971no—
84C2×#41#8760411yescharacteristic
85C2×M16—1942no—
86C4⋊Q8—971no—
87#127—971no—
88#124—971no—
89#124—971no—
90#121—1942no—
91C4×D4—1942no—
92C2×C42—971no—
93#97—974no—
94C4×C8—1942no—
95#41—42no—
96#41—3882no—
97C2×Q8—971no—
98C2×Q8C2×#8760411yescharacteristic
99C2×D4—21no—
100C2×D4—971no—
101C2×D4—1942no—
102C2×D4—1941no—
103C2×D4C2×#8760411yescharacteristic
104C22×C4—21no—
105C22×C4—972no—
106C22×C4—1941no—
107C22×C4C2×#8760411yescharacteristic
108M16—1948no—
109C2×C8—1944no—
110C4⋊C4—1942no—
111C22⋊C4—1942no—
112C22⋊C4—1942no—
113C42—972no—
114C42—971no—
115C42—971no—
116C23—21no—
117C23—1941no—
118C23C22⋊#8760411yescharacteristic
119Q8—41no—
120Q8—3881no—
121D4—41no—
122D4—42no—
123D4—3881no—
124D4—3884no—
125D4—3882no—
126C2×C4—22no—
127C2×C4—21no—
128C2×C4—42no—
129C2×C4—972no—
130C2×C4—972no—
131C2×C4—3882no—
132C2×C4—1944no—
133C2×C4—1941no—
134C2×C4C22⋊#8760411yescharacteristic
135C2×C4C22×#8760411yescharacteristic
136C8—1948no—
137C22—21no—
138C22—22no—
139C22—41no—
140C22—42no—
141C22—3881no—
142C22—3882no—
143C22C2×C22⋊#8760411yescharacteristic
144C4—21no—
145C4—41no—
146C4—974no—
147C4—3881no—
148C4—1942no—
149C4#1569087012yes—
150C2—41no—
151C2—81no—
152C2—7761no—
153C2C2×#1569087011yescharacteristic
154C2D97⋊#10412yes—
subgroupG/H
centerC2×C4C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC2×#41×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261281C2
cosocleC4×Dic97C23
hypercenterC2×#41#87604
hypocenterC97#1525
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C4×#13095520C2semidirect
2. #20261591C2semidirect
3. #20261590C2semidirect
78. C97#1525semidirect

Maximal subgroups

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

subgroupindex
C4×#130955202
#202615912
#202615902
#202612812
#152597