Finite Group Databasebeta

← C7×A4×C4⋊C4 #13060743

Subgroups — C7×A4×C4⋊C4 #13060743

Order 1344.

226 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 66 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×C4×C7×A4C212yesmaximal
2C2×C4×C7×A4C211yesmaximal, characteristic
3C22×C7×C4⋊C4C311yesmaximal, characteristic, complemented, Hall
4C22×C7×A4C2211yescharacteristic
5C4×C7×A4—22no—
6C4×C7×A4C412yescomplemented
7C3×C7×C4⋊C4—41nomaximal
8C23×C4×C7C2×C312yes—
9C23×C4×C7C2×C311yescharacteristic
10C2×C7×C4⋊C4—31no—
11C2×C7×C4⋊C4—31no—
12C2×C7×C4⋊C4—32no—
13A4×C4⋊C4C711yesmaximal, characteristic, direct product factor, Hall
14C2×C7×A4C2×C411yescharacteristic
15C2×C7×A4D411yescharacteristic
16C2×C7×A4Q811yescharacteristic
17C2×C3×C4×C7—41no—
18C2×C3×C4×C7—42no—
19C24×C7C22×C311yescharacteristic
20C22×C4×C7—22no—
21C22×C4×C7—31no—
22C22×C4×C7—31no—
23C22×C4×C7—32no—
24C22×C4×C7—32no—
25C22×C4×C7—32no—
26C22×C4×C7—62no—
27C22×C4×C7C3×C412yescomplemented
28C7×C4⋊C4—32no—
29C7×C4⋊C4—32no—
30C7×C4⋊C4—31no—
31C7×C4⋊C4A411yescharacteristic, direct product factor
32C2×C4×A4C2×C712yes—
33C2×C4×A4C2×C711yescharacteristic
34C22×C3×C7—41no—
35C7×A4C4⋊C411yescharacteristic, direct product factor
36C3×C4×C7—42no—
37C3×C4×C7—82no—
38C22×C4⋊C4C3×C711yescharacteristic, complemented, Sylow
39C23×C7—31no—
40C23×C7—31no—
41C23×C7—31no—
42C23×C7—31no—
43C23×C7C2×C3×C411yescharacteristic
44C23×C7C3×D411yescharacteristic
45C23×C7C3×Q811yescharacteristic
46C2×C4×C7—32no—
47C2×C4×C7—31no—
48C2×C4×C7—32no—
49C2×C4×C7—32no—
50C2×C4×C7—32no—
51C2×C4×C7—32no—
52C2×C4×C7—62no—
53C2×C4×C7—62no—
54C2×C4×C7—62no—
55C2×C4×C7—62no—
56C2×C4×C7C2×A412yes—
57C2×C4×C7C2×A411yescharacteristic
58C22×A4C22×C711yescharacteristic
59C4×A4—22no—
60C4×A4C4×C712yescomplemented
61C3×C4⋊C4—41no—
62C2×C3×C7—41no—
63C2×C3×C7—41no—
64C2×C3×C7—41no—
65C23×C4C2×C3×C712yes—
66C23×C4C2×C3×C711yescharacteristic
67C2×C4⋊C4—31no—
68C2×C4⋊C4—31no—
69C2×C4⋊C4—32no—
70C22×C7—31no—
71C22×C7—31no—
72C22×C7—31no—
73C22×C7—32no—
74C22×C7—31no—
75C22×C7—31no—
76C22×C7—31no—
77C22×C7—31no—
78C22×C7—31no—
79C22×C7—31no—
80C22×C7C3×C4⋊C411yescharacteristic, complemented
81C22×C7C22×A411yescharacteristic
82C4×C7—22no—
83C4×C7—32no—
84C4×C7—62no—
85C4×C7C4×A412yescomplemented
86C2×A4C2×C4×C711yescharacteristic
87C2×A4C7×D411yescharacteristic
88C2×A4C7×Q811yescharacteristic
89C2×C3×C4—41no—
90C2×C3×C4—42no—
91C3×C7—41noHall
92C24C22×C3×C711yescharacteristic
93C22×C4—22no—
94C22×C4—31no—
95C22×C4—31no—
96C22×C4—32no—
97C22×C4—32no—
98C22×C4—32no—
99C22×C4—62no—
100C22×C4C3×C4×C712yescomplemented
101C4⋊C4—32no—
102C4⋊C4—32no—
103C4⋊C4—31no—
104C4⋊C4C7×A411yescharacteristic, direct product factor
105C2×C7—31no—
106C2×C7—31no—
107C2×C7—31no—
108C2×C7—31no—
109C2×C7C2×C4×A411yescharacteristic
110C2×C7D4×A411yescharacteristic
111C2×C7Q8×A411yescharacteristic
112C22×C3—41no—
113A4C7×C4⋊C411yescharacteristic, direct product factor
114C3×C4—42no—
115C3×C4—82no—
116C23—31no—
117C23—31no—
118C23—31no—
119C23—31no—
120C23C2×C3×C4×C711yescharacteristic
121C23C3×C7×D411yescharacteristic
122C23C3×C7×Q811yescharacteristic
123C2×C4—32no—
124C2×C4—31no—
125C2×C4—32no—
126C2×C4—32no—
127C2×C4—32no—
128C2×C4—32no—
129C2×C4—62no—
130C2×C4—62no—
131C2×C4—62no—
132C2×C4—62no—
133C2×C4C2×C7×A412yes—
134C2×C4C2×C7×A411yescharacteristic
135C7A4×C4⋊C411yescharacteristic, direct product factor, Sylow
136C2×C3—41no—
137C2×C3—41no—
138C2×C3—41no—
139C22—31no—
140C22—31no—
141C22—31no—
142C22—32no—
143C22—31no—
144C22—31no—
145C22—31no—
146C22—31no—
147C22—31no—
148C22—31no—
149C22C3×C7×C4⋊C411yescharacteristic, complemented
150C22C22×C7×A411yescharacteristic
151C4—22no—
152C4—32no—
153C4—62no—
154C4C4×C7×A412yescomplemented
155C3—41noSylow
156C2—31no—
157C2—31no—
158C2—31no—
159C2—31no—
160C2C2×C4×C7×A411yescharacteristic
161C2C7×D4×A411yescharacteristic
162C2C7×Q8×A411yescharacteristic
subgroupG/H
centerC22×C7C22×A4
commutatorC23C2×C3×C4×C7
FrattiniC22C22×C7×A4
FittingC22×C7×C4⋊C4C3
socleC24×C7C22×C3
prime radicalC22×C7×A4C22
cosocleC24C22×C3×C7
hypercenterC7×C4⋊C4A4
hypocenterC22C3×C7×C4⋊C4
LCSG ▷ C23 ▷ C22
UCSC1 ◁ C22×C7 ◁ C7×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C22×C7×C4⋊C4C3semidirect
6. C4×C7×A4C4semidirect
13. A4×C4⋊C4C7direct (N × H)
27. C22×C4×C7C3×C4semidirect
31. C7×C4⋊C4A4direct (N × H)
35. C7×A4C4⋊C4direct (N × H)
38. C22×C4⋊C4C3×C7semidirect
60. C4×A4C4×C7semidirect
80. C22×C7C3×C4⋊C4semidirect
85. C4×C7C4×A4semidirect
100. C22×C4C3×C4×C7semidirect
104. C4⋊C4C7×A4direct (N × H)
113. A4C7×C4⋊C4direct (N × H)
135. C7A4×C4⋊C4direct (N × H)
149. C22C3×C7×C4⋊C4semidirect
154. C4C4×C7×A4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×A42
C22×C7×C4⋊C43
C3×C7×C4⋊C44
A4×C4⋊C47