Finite Group Databasebeta

← C7×SL(2,ℤ/4)⋊C4 #13057131

Subgroups — C7×SL(2,ℤ/4)⋊C4 #13057131

Order 1344.

218 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 46 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×C7×SL(2,ℤ/4)C211yesmaximal, characteristic
3C2×C7×SL(2,ℤ/4)C211yesmaximal, characteristic
4C7×#385—31nomaximal, Hall
5C22×C7×A4C2211yescharacteristic
6C4×C7×A4—21no—
7C7×SL(2,ℤ/4)—21no—
8C7×SL(2,ℤ/4)C412yescomplemented
9C7×Dic3⋊C4—41nomaximal
10C23×C4×C7D311yescharacteristic
11C2×C7×C4⋊C4—31no—
12C2×C7×C4⋊C4—31no—
13C2×C7×C22⋊C4—31no—
14C2×C7×C22⋊C4—31no—
15C7×#95—31no—
16C7×#95—31no—
17SL(2,ℤ/4)⋊C4C711yesmaximal, characteristic, direct product factor, Hall
18C2×C7×A4C2×C411yescharacteristic
19C2×C7×A4D411yescharacteristic
20C2×C7×A4Q811yescharacteristic
21C2×C3×C4×C7—41no—
22C2×C7×Dic3—41no—
23C2×C7×Dic3—41no—
24C24×C7C2×D311yescharacteristic
25C22×C4×C7—21no—
26C22×C4×C7—31no—
27C22×C4×C7—31no—
28C22×C4×C7—31no—
29C22×C4×C7—31no—
30C22×C4×C7—31no—
31C22×C4×C7—31no—
32C22×C4×C7—61no—
33C7×C4⋊C4—62no—
34C7×C4⋊C4—61no—
35C7×C4⋊C4—61no—
36C7×C22⋊C4—32no—
37C7×C22⋊C4—32no—
38C7×C22⋊C4—61no—
39C7×C22⋊C4—61no—
40C2×C4×A4C2×C711yescharacteristic
41C2×SL(2,ℤ/4)C2×C711yescharacteristic
42C2×SL(2,ℤ/4)C2×C711yescharacteristic
43C22×C3×C7—41no—
44C7×A4C4⋊C411yescharacteristic, complemented
45C3×C4×C7—81no—
46C7×Dic3—42no—
47C7×Dic3—81no—
48#385—31noSylow
49C23×C7—31no—
50C23×C7—31no—
51C23×C7—31no—
52C23×C7—31no—
53C23×C7Dic611yescharacteristic
54C23×C7C4×D311yescharacteristic
55C23×C7#6711yescharacteristic
56C2×C4×C7—32no—
57C2×C4×C7—32no—
58C2×C4×C7—32no—
59C2×C4×C7—31no—
60C2×C4×C7—61no—
61C2×C4×C7—61no—
62C2×C4×C7—61no—
63C2×C4×C7—61no—
64C2×C4×C7—61no—
65C2×C4×C7—61no—
66C2×C4×C7—61no—
67C2×C4×C7—61no—
68C2×C4×C7—61no—
69C2×C4×C7—61no—
70C2×C4×C7—61no—
71C2×C4×C7—61no—
72C2×C4×C7—61no—
73C2×C4×C7S411yescharacteristic
74C22×A4C22×C711yescharacteristic
75C4×A4—21no—
76SL(2,ℤ/4)—21no—
77SL(2,ℤ/4)C4×C712yescomplemented
78Dic3⋊C4—41no—
79C2×C3×C7—41no—
80C2×C3×C7—41no—
81C2×C3×C7—41no—
82C23×C4D3×C711yescharacteristic
83C2×C4⋊C4—31no—
84C2×C4⋊C4—31no—
85C2×C22⋊C4—31no—
86C2×C22⋊C4—31no—
87#95—31no—
88#95—31no—
89C22×C7—31no—
90C22×C7—31no—
91C22×C7—31no—
92C22×C7—31no—
93C22×C7—31no—
94C22×C7—31no—
95C22×C7—31no—
96C22×C7—31no—
97C22×C7—31no—
98C22×C7—61no—
99C22×C7Dic3⋊C411yescharacteristic, complemented
100C22×C7C2×S411yescharacteristic
101C4×C7—21no—
102C4×C7—62no—
103C4×C7—62no—
104C4×C7—61no—
105C4×C7—121no—
106C4×C7—121no—
107C2×A4C2×C4×C711yescharacteristic
108C2×A4C7×D411yescharacteristic
109C2×A4C7×Q811yescharacteristic
110C2×C3×C4—41no—
111C2×Dic3—41no—
112C2×Dic3—41no—
113C3×C7—41noHall
114C24C2×D3×C711yescharacteristic
115C22×C4—21no—
116C22×C4—31no—
117C22×C4—31no—
118C22×C4—31no—
119C22×C4—31no—
120C22×C4—31no—
121C22×C4—31no—
122C22×C4—61no—
123C4⋊C4—62no—
124C4⋊C4—61no—
125C4⋊C4—61no—
126C22⋊C4—32no—
127C22⋊C4—32no—
128C22⋊C4—61no—
129C22⋊C4—61no—
130C2×C7—31no—
131C2×C7—31no—
132C2×C7—31no—
133C2×C7—31no—
134C2×C7A4⋊Q811yescharacteristic
135C2×C7C4×S411yescharacteristic
136C2×C7GL(2,ℤ/4)11yescharacteristic
137C22×C3—41no—
138A4C7×C4⋊C411yescharacteristic, complemented
139C3×C4—81no—
140Dic3—42no—
141Dic3—81no—
142C23—31no—
143C23—31no—
144C23—31no—
145C23—31no—
146C23C7×Dic611yescharacteristic
147C23C4×D3×C711yescharacteristic
148C23C7×#6711yescharacteristic
149C2×C4—32no—
150C2×C4—32no—
151C2×C4—32no—
152C2×C4—31no—
153C2×C4—61no—
154C2×C4—61no—
155C2×C4—61no—
156C2×C4—61no—
157C2×C4—61no—
158C2×C4—61no—
159C2×C4—61no—
160C2×C4—61no—
161C2×C4—61no—
162C2×C4—61no—
163C2×C4—61no—
164C2×C4—61no—
165C2×C4—61no—
166C2×C4C7×S411yescharacteristic
167C7SL(2,ℤ/4)⋊C411yescharacteristic, direct product factor, Sylow
168C2×C3—41no—
169C2×C3—41no—
170C2×C3—41no—
171C22—31no—
172C22—31no—
173C22—31no—
174C22—31no—
175C22—31no—
176C22—31no—
177C22—31no—
178C22—31no—
179C22—31no—
180C22—61no—
181C22C7×Dic3⋊C411yescharacteristic, complemented
182C22C2×C7×S411yescharacteristic
183C4—21no—
184C4—62no—
185C4—62no—
186C4—61no—
187C4—121no—
188C4—121no—
189C3—41noSylow
190C2—31no—
191C2—31no—
192C2—31no—
193C2—31no—
194C2C7×A4⋊Q811yescharacteristic
195C2C4×C7×S411yescharacteristic
196C2C7×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22×C7C2×S4
commutatorC2×A4C2×C4×C7
FrattiniC22C2×C7×S4
FittingC23×C4×C7D3
socleC24×C7C2×D3
prime radicalC22×C7×A4C22
cosocleC22×A4C22×C7
hypercenterC2×C4×C7S4
hypocenterA4C7×C4⋊C4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22×C7 ◁ C2×C4×C7

Complements

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

NH ≅ G/Ntype
8. C7×SL(2,ℤ/4)C4semidirect
17. SL(2,ℤ/4)⋊C4C7direct (N × H)
44. C7×A4C4⋊C4semidirect
77. SL(2,ℤ/4)C4×C7semidirect
99. C22×C7Dic3⋊C4semidirect
138. A4C7×C4⋊C4semidirect
167. C7SL(2,ℤ/4)⋊C4direct (N × H)
181. C22C7×Dic3⋊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×C7×SL(2,ℤ/4)2
C7×#3853
C7×Dic3⋊C44
SL(2,ℤ/4)⋊C47