Finite Group Databasebeta

← C7×A4×S4 #72244506

Subgroups — C7×A4×S4 #72244506

Order 2016.

172 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 24 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
1C7×A42C211yesmaximal, characteristic, complemented
2C22×C7×S4C311yesmaximal, characteristic, complemented
3C7×D4×A4—31nomaximal
4D3×C7×A4—41nomaximal
5C3×C7×S4—41nomaximal
6C7×C22⋊A4—21no—
7C22×C7×A4—31no—
8C22×C7×A4D311yescharacteristic, complemented
9C22×C7×A4C2×C311yescharacteristic, complemented
10C2×C7×S4—31no—
11C2×C7×S4—31no—
12C4×C7×A4—31no—
13A4×S4C711yesmaximal, characteristic, direct product factor, Hall
14C3×C7×A4—41no—
15C3×C7×A4—41no—
16C22×C7×D4—31noHall
17C22×D3×C7—41no—
18C2×C7×A4—31no—
19C2×C7×A4—31no—
20C2×C7×A4—61no—
21C7×S4—31no—
22C7×S4A411yescharacteristic, direct product factor
23C3×C7×D4—121no—
24A42C2×C711yescharacteristic, complemented
25C3×D3×C7—161no—
26C24×C7—31no—
27C24×C7C3×D311yescharacteristic, complemented
28C2×C7×D4—91no—
29C2×C7×D4—91no—
30C2×C7×D4—91no—
31C2×C7×D4—91no—
32C22×C4×C7—31no—
33C22×S4C3×C711yescharacteristic, complemented
34D4×A4—31no—
35C22×C3×C7—41no—
36C22×C3×C7—41no—
37C22×C3×C7—121no—
38C2×D3×C7—121no—
39C2×D3×C7—121no—
40C7×A4—81no—
41C7×A4—81no—
42C7×A4—241no—
43C7×A4S411yescharacteristic, direct product factor
44C7×A4C2×A411yescharacteristic, complemented
45C3×C4×C7—121no—
46D3×A4—41no—
47C3×S4—41no—
48C32×C7—161noHall
49C23×C7—31no—
50C23×C7—31no—
51C23×C7—61no—
52C23×C7—91no—
53C23×C7—91no—
54C23×C7—91no—
55C23×C7—181no—
56C7×D4—31no—
57C7×D4—91no—
58C7×D4—91no—
59C7×D4—92no—
60C7×D4—91no—
61C2×C4×C7—91no—
62C2×C4×C7—91no—
63C22⋊A4—21no—
64C22×A4—31no—
65C22×A4D3×C711yescharacteristic, complemented
66C22×A4C2×C3×C711yescharacteristic, complemented
67C2×S4—31no—
68C2×S4—31no—
69C4×A4—31no—
70C2×C3×C7—121no—
71C2×C3×C7—121no—
72C2×C3×C7—241no—
73D3×C7—41no—
74D3×C7—121no—
75C3×A4—41no—
76C3×A4—41no—
77C22×D4—31noSylow
78C22×C7—31no—
79C22×C7—61no—
80C22×C7—91no—
81C22×C7—91no—
82C22×C7—91no—
83C22×C7—91no—
84C22×C7—181no—
85C22×C7—181no—
86C22×C7—181no—
87C22×C7—181no—
88C22×C7C3×S411yescharacteristic, complemented
89C22×C7D3×A411yescharacteristic, complemented
90C4×C7—31no—
91C4×C7—91no—
92C22×D3—41no—
93C2×A4—31no—
94C2×A4—31no—
95C2×A4—61no—
96S4—31no—
97S4C7×A411yescharacteristic, direct product factor
98C3×D4—121no—
99C3×C7—41no—
100C3×C7—41no—
101C3×C7—321no—
102C3×D3—161no—
103C24—31no—
104C24C3×D3×C711yescharacteristic, complemented
105C2×D4—91no—
106C2×D4—91no—
107C2×D4—91no—
108C2×D4—91no—
109C22×C4—31no—
110C2×C7—31no—
111C2×C7—31no—
112C2×C7—61no—
113C2×C7—91no—
114C2×C7—181no—
115C22×C3—41no—
116C22×C3—41no—
117C22×C3—121no—
118C2×D3—121no—
119C2×D3—121no—
120A4—81no—
121A4—81no—
122A4—241no—
123A4C7×S411yescharacteristic, direct product factor
124A4C2×C7×A411yescharacteristic, complemented
125C3×C4—121no—
126C32—161noSylow
127C23—31no—
128C23—31no—
129C23—61no—
130C23—91no—
131C23—91no—
132C23—91no—
133C23—181no—
134D4—31no—
135D4—91no—
136D4—91no—
137D4—92no—
138D4—91no—
139C2×C4—91no—
140C2×C4—91no—
141C7A4×S411yescharacteristic, direct product factor, Sylow
142C2×C3—121no—
143C2×C3—121no—
144C2×C3—241no—
145D3—41no—
146D3—121no—
147C22—31no—
148C22—61no—
149C22—91no—
150C22—91no—
151C22—91no—
152C22—91no—
153C22—181no—
154C22—181no—
155C22—181no—
156C22—181no—
157C22C3×C7×S411yescharacteristic, complemented
158C22D3×C7×A411yescharacteristic, complemented
159C4—31no—
160C4—91no—
161C3—41no—
162C3—41no—
163C3—321no—
164C2—31no—
165C2—31no—
166C2—61no—
167C2—91no—
168C2—181no—
subgroupG/H
center, hypercenterC7A4×S4
commutator, cosocle, hypocenterC22×A4C2×C3×C7
FrattiniC1G
Fitting, socleC24×C7C3×D3
LCSG ▷ C22×A4
UCSC1 ◁ C7

Complements

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

NH ≅ G/Ntype
1. C7×A42C2semidirect
2. C22×C7×S4C3semidirect
8. C22×C7×A4D3semidirect
9. C22×C7×A4C2×C3semidirect
13. A4×S4C7direct (N × H)
22. C7×S4A4direct (N × H)
24. A42C2×C7semidirect
27. C24×C7C3×D3semidirect
33. C22×S4C3×C7semidirect
43. C7×A4S4direct (N × H)
44. C7×A4C2×A4semidirect
65. C22×A4D3×C7semidirect
66. C22×A4C2×C3×C7semidirect
88. C22×C7C3×S4semidirect
89. C22×C7D3×A4semidirect
97. S4C7×A4direct (N × H)
104. C24C3×D3×C7semidirect
123. A4C7×S4direct (N × H)
124. A4C2×C7×A4semidirect
141. C7A4×S4direct (N × H)
157. C22C3×C7×S4semidirect
158. C22D3×C7×A4semidirect

Maximal subgroups

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

subgroupindex
C7×A422
C22×C7×S43
C7×D4×A43
D3×C7×A44
C3×C7×S44
A4×S47