Finite Group Databasebeta

← C4×#127⋊C3 #73173

Subgroups — C4×#127⋊C3 #73173

Order 384.

194 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 27 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
1C2×#127⋊C3C211yesmaximal, characteristic
2#384⋊C3C211yesmaximal, characteristic, complemented
3C4×C42⋊C3C211yesmaximal, characteristic, complemented
4C4×#127C311yesmaximal, characteristic, complemented, Sylow
5C2×C4×A4—41nomaximal
6#127⋊C3C412yesdirect product factor
7C2×C42⋊C3C2211yescharacteristic
8C2×#127C2×C311yescharacteristic
9C2×C4×D4—31no—
10C2×C4×D4—31no—
11#390—31no—
12#390—31no—
13#384C2×C311yescharacteristic, complemented
14C43C2×C311yescharacteristic, complemented
15C22×A4—41no—
16C4×A4—41no—
17C4×A4—41no—
18C42⋊C3C2×C411yescharacteristic, complemented
19C22×D4—31no—
20C22×D4—31no—
21C23×C4—31no—
22C23×C4A411yescharacteristic
23#127—32no—
24#127C3×C412yescomplemented
25C4×D4—62no—
26C4×D4—62no—
27C4×D4—62no—
28C4×D4—62no—
29C2×C4⋊C4—31no—
30C2×C4⋊C4—31no—
31C2×C22⋊C4—32no—
32C2×C22⋊C4—31no—
33C2×C22⋊C4—31no—
34C2×C42—31no—
35C2×C42—31no—
36C2×C42C22×C311yescharacteristic
37C2×A4—42no—
38C2×A4—41no—
39C2×C3×C4—161no—
40C24—31no—
41C24C2×A411yescharacteristic
42C2×D4—32no—
43C2×D4—32no—
44C2×D4—32no—
45C2×D4—32no—
46C2×D4—62no—
47C2×D4—62no—
48C2×D4—62no—
49C2×D4—62no—
50C22×C4—31no—
51C22×C4—31no—
52C22×C4—31no—
53C22×C4—61no—
54C22×C4—61no—
55C22×C4—62no—
56C22×C4—62no—
57C22×C4—61no—
58C22×C4—61no—
59C22×C4C2×A411yescharacteristic
60C22×C4C2×A411yescharacteristic
61C4⋊C4—62no—
62C4⋊C4—62no—
63C22⋊C4—62no—
64C22⋊C4—62no—
65C22⋊C4—64no—
66C42—31no—
67C42—32no—
68C42—32no—
69C42—62no—
70C42C2×C3×C411yescharacteristic, complemented
71C22×C3—161no—
72A4—41no—
73C3×C4—161no—
74C3×C4—161no—
75C23—32no—
76C23—61no—
77C23—61no—
78C23—61no—
79C23—62no—
80C23—61no—
81C23—62no—
82C23C4×A412yes—
83C23C22×A411yescharacteristic
84D4—64no—
85D4—64no—
86D4—64no—
87D4—64no—
88C2×C4—32no—
89C2×C4—32no—
90C2×C4—31no—
91C2×C4—31no—
92C2×C4—31no—
93C2×C4—31no—
94C2×C4—32no—
95C2×C4—41no—
96C2×C4—61no—
97C2×C4—62no—
98C2×C4—61no—
99C2×C4—62no—
100C2×C4—121no—
101C2×C4—121no—
102C2×C4—122no—
103C2×C4—121no—
104C2×C3—162no—
105C2×C3—161no—
106C22—31no—
107C22—31no—
108C22—41no—
109C22—62no—
110C22—62no—
111C22—64no—
112C22—121no—
113C22—121no—
114C22—122no—
115C22—121no—
116C22C2×C4×A411yescharacteristic
117C4—32no—
118C4—32no—
119C4—31no—
120C4—41no—
121C4—62no—
122C4—121no—
123C4#127⋊C311yescharacteristic, direct product factor
124C3—161noSylow
125C2—31no—
126C2—31no—
127C2—42no—
128C2—122no—
129C2C2×#127⋊C311yescharacteristic
subgroupG/H
center, hypercenterC4#127⋊C3
commutator, hypocenterC42C2×C3×C4
Frattini, socleC23C22×A4
FittingC4×#127C3
prime radicalC2×#127⋊C3C2
cosocleC2×C42C22×C3
LCSG ▷ C42
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. #384⋊C3C2semidirect
3. C4×C42⋊C3C2semidirect
4. C4×#127C3semidirect
6. #127⋊C3C4direct (N × H)
13. #384C2×C3semidirect
14. C43C2×C3semidirect
18. C42⋊C3C2×C4semidirect
24. #127C3×C4semidirect
70. C42C2×C3×C4semidirect
123. C4#127⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#127⋊C32
#384⋊C32
C4×C42⋊C32
C4×#1273
C2×C4×A44