Finite Group Databasebeta

← A4×Dic7⋊C4 #13057607

Subgroups — A4×Dic7⋊C4 #13057607

Order 1344.

226 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 51 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×A4C211yesmaximal, characteristic
2C2×A4×Dic7C211yesmaximal, characteristic
3C2×A4×Dic7C211yesmaximal, characteristic
4C22×Dic7⋊C4C311yesmaximal, characteristic, complemented, Hall
5C22×C7×A4C2211yescharacteristic
6C4×C7×A4—21no—
7A4×Dic7—21no—
8A4×Dic7C412yescomplemented
9C3×Dic7⋊C4—41nomaximal
10C23×C4×C7C2×C311yescharacteristic
11C23×Dic7C2×C311yescharacteristic
12C23×Dic7C2×C311yescharacteristic
13C2×Dic7⋊C4—31no—
14C2×Dic7⋊C4—31no—
15C2×Dic7⋊C4—31no—
16C2×Dic7⋊C4—31no—
17A4×C4⋊C4—71nomaximal, Hall
18C2×C7×A4C2×C411yescharacteristic
19C2×C7×A4D411yescharacteristic
20C2×C7×A4Q811yescharacteristic
21C2×C3×C4×C7—41no—
22C2×C3×Dic7—41no—
23C2×C3×Dic7—41no—
24C24×C7C22×C311yescharacteristic
25C22×C4×C7—21no—
26C22×C4×C7—31no—
27C22×C4×C7—31no—
28C22×C4×C7—61no—
29C22×Dic7—21no—
30C22×Dic7—31no—
31C22×Dic7—31no—
32C22×Dic7—32no—
33C22×Dic7—31no—
34C22×Dic7—31no—
35C22×Dic7—61no—
36C22×Dic7C3×C412yescomplemented
37Dic7⋊C4—32no—
38Dic7⋊C4—31no—
39Dic7⋊C4—31no—
40Dic7⋊C4—31no—
41Dic7⋊C4A411yescharacteristic, direct product factor
42C2×C4×A4—71no—
43C2×C4×A4—71no—
44C2×C4×A4D711yescharacteristic
45C22×C3×C7—41no—
46C7×A4C4⋊C411yescharacteristic, complemented
47C3×C4×C7—81no—
48C3×Dic7—42no—
49C3×Dic7—81no—
50C22×C4⋊C4—71noSylow
51C23×C7—31no—
52C23×C7—31no—
53C23×C7—31no—
54C23×C7—31no—
55C23×C7C2×C3×C411yescharacteristic
56C23×C7C3×D411yescharacteristic
57C23×C7C3×Q811yescharacteristic
58C2×C4×C7—31no—
59C2×C4×C7—61no—
60C2×C4×C7—61no—
61C2×C4×C7—61no—
62C2×C4×C7—61no—
63C2×C4×C7C2×A411yescharacteristic
64C2×Dic7—32no—
65C2×Dic7—31no—
66C2×Dic7—32no—
67C2×Dic7—32no—
68C2×Dic7—32no—
69C2×Dic7—31no—
70C2×Dic7—61no—
71C2×Dic7—61no—
72C2×Dic7—61no—
73C2×Dic7—61no—
74C2×Dic7C2×A411yescharacteristic
75C2×Dic7C2×A411yescharacteristic
76C22×A4C2×D711yescharacteristic
77C4×A4—21no—
78C4×A4—72no—
79C4×A4—141no—
80C3×C4⋊C4—281no—
81C2×C3×C7—41no—
82C2×C3×C7—41no—
83C2×C3×C7—41no—
84C23×C4—71no—
85C23×C4—71no—
86C23×C4C3×D711yescharacteristic
87C2×C4⋊C4—211no—
88C2×C4⋊C4—211no—
89C2×C4⋊C4—211no—
90C2×C4⋊C4—211no—
91C22×C7—31no—
92C22×C7—31no—
93C22×C7—31no—
94C22×C7—32no—
95C22×C7—31no—
96C22×C7—31no—
97C22×C7—31no—
98C22×C7—31no—
99C22×C7—31no—
100C22×C7—31no—
101C22×C7C3×C4⋊C411yescharacteristic, complemented
102C22×C7C22×A411yescharacteristic
103C4×C7—21no—
104C4×C7—61no—
105Dic7—21no—
106Dic7—32no—
107Dic7—61no—
108Dic7C4×A412yescomplemented
109C2×A4Dic1411yescharacteristic
110C2×A4C4×D711yescharacteristic
111C2×A4C22⋊D711yescharacteristic
112C2×C3×C4—41no—
113C2×C3×C4—281no—
114C2×C3×C4—281no—
115C3×C7—41noHall
116C24C2×C3×D711yescharacteristic
117C22×C4—21no—
118C22×C4—31no—
119C22×C4—31no—
120C22×C4—61no—
121C22×C4—72no—
122C22×C4—141no—
123C22×C4—211no—
124C22×C4—211no—
125C22×C4—211no—
126C22×C4—212no—
127C22×C4—211no—
128C22×C4—421no—
129C4⋊C4—71no—
130C4⋊C4—211no—
131C4⋊C4—211no—
132C4⋊C4—212no—
133C4⋊C4—211no—
134C2×C7—31no—
135C2×C7—31no—
136C2×C7—31no—
137C2×C7—31no—
138C2×C7C2×C4×A411yescharacteristic
139C2×C7D4×A411yescharacteristic
140C2×C7Q8×A411yescharacteristic
141C22×C3—41no—
142A4Dic7⋊C411yescharacteristic, direct product factor
143C3×C4—81no—
144C3×C4—282no—
145C3×C4—561no—
146C23—31no—
147C23—31no—
148C23—31no—
149C23—31no—
150C23C3×Dic1411yescharacteristic
151C23C3×C4×D711yescharacteristic
152C23C3×C22⋊D711yescharacteristic
153C2×C4—31no—
154C2×C4—61no—
155C2×C4—61no—
156C2×C4—61no—
157C2×C4—61no—
158C2×C4—71no—
159C2×C4—71no—
160C2×C4—212no—
161C2×C4—212no—
162C2×C4—212no—
163C2×C4—211no—
164C2×C4—212no—
165C2×C4—211no—
166C2×C4—421no—
167C2×C4—421no—
168C2×C4—421no—
169C2×C4—421no—
170C2×C4A4×D711yescharacteristic
171C7A4×C4⋊C411yescharacteristic, complemented, Sylow
172C2×C3—41no—
173C2×C3—41no—
174C2×C3—41no—
175C22—31no—
176C22—31no—
177C22—31no—
178C22—32no—
179C22—31no—
180C22—31no—
181C22—31no—
182C22—31no—
183C22—31no—
184C22—31no—
185C22C3×Dic7⋊C411yescharacteristic, complemented
186C22C2×A4×D711yescharacteristic
187C4—21no—
188C4—61no—
189C4—72no—
190C4—141no—
191C4—212no—
192C4—421no—
193C3—41noSylow
194C2—31no—
195C2—31no—
196C2—31no—
197C2—31no—
198C2A4×Dic1411yescharacteristic
199C2C4×A4×D711yescharacteristic
200C2A4×C22⋊D711yescharacteristic
subgroupG/H
center, FrattiniC22C2×A4×D7
commutatorC23×C7C2×C3×C4
FittingC23×C4×C7C2×C3
socle, cosocleC24×C7C22×C3
prime radicalC22×C7×A4C22
hypercenterC2×C4A4×D7
hypocenterC22×C7C3×C4⋊C4
LCSG ▷ C23×C7 ▷ C22×C7
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. C22×Dic7⋊C4C3semidirect
8. A4×Dic7C4semidirect
36. C22×Dic7C3×C4semidirect
41. Dic7⋊C4A4direct (N × H)
46. C7×A4C4⋊C4semidirect
101. C22×C7C3×C4⋊C4semidirect
108. Dic7C4×A4semidirect
142. A4Dic7⋊C4direct (N × H)
171. C7A4×C4⋊C4semidirect
185. C22C3×Dic7⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×A42
C2×A4×Dic72
C22×Dic7⋊C43
C3×Dic7⋊C44
A4×C4⋊C47