Finite Group Databasebeta

← A4×C22⋊Dic7 #13057628

Subgroups — A4×C22⋊Dic7 #13057628

Order 1344.

338 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 57 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
1C23×C7×A4C211yesmaximal, characteristic
2C2×A4×Dic7C212yesmaximal, complemented
3C22×C22⋊Dic7C311yesmaximal, characteristic, complemented, Hall
4C22×C7×A4—22no—
5C22×C7×A4C412yescomplemented
6C22×C7×A4C2211yescharacteristic
7A4×Dic7—22no—
8C3×C22⋊Dic7—41nomaximal
9C25×C7C2×C311yescharacteristic
10C23×Dic7C2×C312yescomplemented
11C2×C22⋊Dic7—31no—
12C2×C22⋊Dic7—31no—
13C2×C22⋊Dic7—32no—
14A4×C22⋊C4—71nomaximal, Hall
15C23×C3×C7—41no—
16C2×C7×A4—22no—
17C2×C7×A4C2×C411yescharacteristic
18C2×C7×A4D412yes—
19C2×C3×Dic7—42no—
20C24×C7—22no—
21C24×C7—31no—
22C24×C7—31no—
23C24×C7—32no—
24C24×C7—62no—
25C24×C7C3×C412yescomplemented
26C24×C7C22×C311yescharacteristic
27C22×Dic7—22no—
28C22×Dic7—32no—
29C22×Dic7—32no—
30C22×Dic7—62no—
31C22⋊Dic7—32no—
32C22⋊Dic7—31no—
33C22⋊Dic7—32no—
34C22⋊Dic7A411yescharacteristic, direct product factor
35C23×A4D711yescharacteristic
36C2×C4×A4—72no—
37C22×C3×C7—42no—
38C22×C3×C7—41no—
39C22×C3×C7—82no—
40C7×A4C22⋊C411yescharacteristic, complemented
41C3×Dic7—82no—
42C22×C22⋊C4—71noSylow
43C23×C7—22no—
44C23×C7—32no—
45C23×C7—31no—
46C23×C7—31no—
47C23×C7—32no—
48C23×C7—32no—
49C23×C7—32no—
50C23×C7—32no—
51C23×C7—31no—
52C23×C7—62no—
53C23×C7—62no—
54C23×C7—62no—
55C23×C7—64no—
56C23×C7—62no—
57C23×C7—62no—
58C23×C7—64no—
59C23×C7C2×C3×C411yescharacteristic
60C23×C7C3×D412yes—
61C23×C7C2×A411yescharacteristic
62C2×Dic7—32no—
63C2×Dic7—62no—
64C2×Dic7—62no—
65C2×Dic7—62no—
66C2×Dic7—62no—
67C2×Dic7C2×A412yescomplemented
68C22×A4—22no—
69C22×A4Dic712yescomplemented
70C22×A4C2×D711yescharacteristic
71C4×A4—142no—
72C3×C22⋊C4—281no—
73C2×C3×C7—41no—
74C2×C3×C7—42no—
75C2×C3×C7—82no—
76C25C3×D711yescharacteristic
77C23×C4—72no—
78C2×C22⋊C4—211no—
79C2×C22⋊C4—212no—
80C2×C22⋊C4—211no—
81C22×C7—22no—
82C22×C7—31no—
83C22×C7—32no—
84C22×C7—32no—
85C22×C7—31no—
86C22×C7—31no—
87C22×C7—32no—
88C22×C7—32no—
89C22×C7—32no—
90C22×C7—62no—
91C22×C7—64no—
92C22×C7—62no—
93C22×C7—62no—
94C22×C7—62no—
95C22×C7—64no—
96C22×C7—62no—
97C22×C7C3×C22⋊C411yescharacteristic, complemented
98C22×C7C4×A412yescomplemented
99C22×C7C22×A411yescharacteristic
100Dic7—22no—
101Dic7—62no—
102C23×C3—41no—
103C2×A4—22no—
104C2×A4C2×Dic711yescharacteristic
105C2×A4C22⋊D712yes—
106C2×C3×C4—282no—
107C3×C7—41noHall
108C24—22no—
109C24—31no—
110C24—32no—
111C24—31no—
112C24—62no—
113C24C3×Dic712yescomplemented
114C24C2×C3×D711yescharacteristic
115C22×C4—142no—
116C22×C4—212no—
117C22×C4—212no—
118C22×C4—422no—
119C22⋊C4—71no—
120C22⋊C4—212no—
121C22⋊C4—212no—
122C22⋊C4—211no—
123C2×C7—22no—
124C2×C7—31no—
125C2×C7—32no—
126C2×C7—31no—
127C2×C7—62no—
128C2×C7C2×C4×A411yescharacteristic
129C2×C7D4×A412yes—
130C22×C3—42no—
131C22×C3—41no—
132C22×C3—82no—
133A4C22⋊Dic711yescharacteristic, direct product factor
134C3×C4—562no—
135C23—22no—
136C23—32no—
137C23—31no—
138C23—31no—
139C23—32no—
140C23—32no—
141C23—32no—
142C23—32no—
143C23—31no—
144C23—62no—
145C23—62no—
146C23—62no—
147C23—64no—
148C23—62no—
149C23—62no—
150C23—64no—
151C23C2×C3×Dic711yescharacteristic
152C23C3×C22⋊D712yes—
153C23A4×D711yescharacteristic
154C2×C4—72no—
155C2×C4—212no—
156C2×C4—422no—
157C2×C4—422no—
158C2×C4—422no—
159C2×C4—422no—
160C7A4×C22⋊C411yescharacteristic, complemented, Sylow
161C2×C3—41no—
162C2×C3—42no—
163C2×C3—82no—
164C22—22no—
165C22—31no—
166C22—32no—
167C22—32no—
168C22—31no—
169C22—31no—
170C22—32no—
171C22—32no—
172C22—32no—
173C22—62no—
174C22—64no—
175C22—62no—
176C22—62no—
177C22—62no—
178C22—64no—
179C22—62no—
180C22C3×C22⋊Dic711yescharacteristic, complemented
181C22A4×Dic712yescomplemented
182C22C2×A4×D711yescharacteristic
183C4—142no—
184C4—422no—
185C3—41noSylow
186C2—22no—
187C2—31no—
188C2—31no—
189C2—32no—
190C2—62no—
191C2C2×A4×Dic711yescharacteristic
192C2A4×C22⋊D712yes—
subgroupG/H
center, FrattiniC22C2×A4×D7
commutatorC23×C7C2×C3×C4
FittingC25×C7C2×C3
socle, cosocleC24×C7C22×C3
prime radicalC23×C7×A4C2
hypercenterC23A4×D7
hypocenterC22×C7C3×C22⋊C4
LCSG ▷ C23×C7 ▷ C22×C7
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×A4×Dic7C2semidirect
3. C22×C22⋊Dic7C3semidirect
5. C22×C7×A4C4semidirect
10. C23×Dic7C2×C3semidirect
25. C24×C7C3×C4semidirect
34. C22⋊Dic7A4direct (N × H)
40. C7×A4C22⋊C4semidirect
67. C2×Dic7C2×A4semidirect
69. C22×A4Dic7semidirect
97. C22×C7C3×C22⋊C4semidirect
98. C22×C7C4×A4semidirect
113. C24C3×Dic7semidirect
133. A4C22⋊Dic7direct (N × H)
160. C7A4×C22⋊C4semidirect
180. C22C3×C22⋊Dic7semidirect
181. C22A4×Dic7semidirect

Maximal subgroups

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

subgroupindex
C23×C7×A42
C2×A4×Dic72
C22×C22⋊Dic73
C3×C22⋊Dic74
A4×C22⋊C47