Finite Group Databasebeta

← Dic7×SL(2,ℤ/4) #13057096

Subgroups — Dic7×SL(2,ℤ/4) #13057096

Order 1344.

250 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 49 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×A4×Dic7C211yesmaximal, characteristic
2C2×C7⋊SL(2,ℤ/4)C211yesmaximal, characteristic
3C2×C7×SL(2,ℤ/4)C211yesmaximal, characteristic
4C22⋊C4×Dic7—31nomaximal, Hall
5C22×C7×A4C2211yescharacteristic
6A4×Dic7C412yescomplemented
7C7⋊SL(2,ℤ/4)C412yescomplemented
8C7×SL(2,ℤ/4)C412yescomplemented
9Dic3×Dic7—41nomaximal
10C23×Dic7D311yescharacteristic
11C2×C7×C22⋊C4—31no—
12C2×C22⋊Dic7—31no—
13C2×C4×Dic7—31no—
14C2×C4×Dic7—31no—
15C7⋊#95—31no—
16C7⋊#95—31no—
17C4×SL(2,ℤ/4)—71nomaximal, Hall
18C2×C7×A4C2×C411yescharacteristic
19C2×C7×A4C2×C411yescharacteristic
20C2×C7×A4C2×C411yescharacteristic
21C2×Dic21—41no—
22C2×C7×Dic3—41no—
23C2×C3×Dic7—41no—
24C24×C7C2×D311yescharacteristic
25C22×C4×C7—31no—
26C22×C4×C7—31no—
27C22×Dic7—31no—
28C22×Dic7—31no—
29C22×Dic7—32no—
30C22×Dic7—31no—
31C22×Dic7—31no—
32C22×Dic7Dic312yescomplemented
33C7×C22⋊C4—32no—
34C7×C22⋊C4—32no—
35C22⋊Dic7—32no—
36C22⋊Dic7—32no—
37C4×Dic7—61no—
38C4×Dic7—62no—
39C4×Dic7—61no—
40C2×C4×A4—71no—
41C2×SL(2,ℤ/4)—71no—
42C2×SL(2,ℤ/4)D711yescharacteristic
43C22×C3×C7—41no—
44C7×A4C4211yescharacteristic, complemented
45Dic21—42no—
46C3×Dic7—42no—
47C7×Dic3—42no—
48C4×C22⋊C4—211noSylow
49C23×C7—31no—
50C23×C7—31no—
51C23×C7—31no—
52C23×C7—31no—
53C23×C7C4×D311yescharacteristic
54C23×C7C4×D311yescharacteristic
55C23×C7C2×Dic311yescharacteristic
56C2×C4×C7—32no—
57C2×C4×C7—32no—
58C2×C4×C7—61no—
59C2×C4×C7—61no—
60C2×C4×C7—61no—
61C2×C4×C7—61no—
62C2×Dic7—31no—
63C2×Dic7—32no—
64C2×Dic7—32no—
65C2×Dic7—32no—
66C2×Dic7—32no—
67C2×Dic7—32no—
68C2×Dic7—61no—
69C2×Dic7—61no—
70C2×Dic7—61no—
71C2×Dic7—61no—
72C2×Dic7—61no—
73C2×Dic7S411yescharacteristic
74C22×A4C2×D711yescharacteristic
75C4×A4—72no—
76SL(2,ℤ/4)—72no—
77SL(2,ℤ/4)Dic712yesdirect product factor
78C4×Dic3—281no—
79C2×C3×C7—41no—
80C2×C3×C7—41no—
81C2×C3×C7—41no—
82C23×C4—71no—
83C2×C22⋊C4—31no—
84C2×C22⋊C4—211no—
85C2×C42—211no—
86C2×C42—211no—
87#95—211no—
88#95—211no—
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×C7C4×Dic311yescharacteristic, complemented
100C22×C7C2×S411yescharacteristic
101C4×C7—62no—
102C4×C7—62no—
103Dic7—32no—
104Dic7—62no—
105Dic7—62no—
106Dic7SL(2,ℤ/4)12yesdirect product factor
107C2×A4C4×D711yescharacteristic
108C2×A4C4×D711yescharacteristic
109C2×A4C2×Dic711yescharacteristic
110C2×C3×C4—281no—
111C2×Dic3—41no—
112C2×Dic3—281no—
113C3×C7—41noHall
114C24D3×D711yescharacteristic
115C22×C4—31no—
116C22×C4—31no—
117C22×C4—72no—
118C22×C4—211no—
119C22×C4—211no—
120C22×C4—212no—
121C22×C4—211no—
122C22×C4—211no—
123C22⋊C4—32no—
124C22⋊C4—32no—
125C22⋊C4—212no—
126C22⋊C4—212no—
127C42—421no—
128C42—422no—
129C42—421no—
130C2×C7—31no—
131C2×C7—31no—
132C2×C7—31no—
133C2×C7—31no—
134C2×C7C4×S411yescharacteristic
135C2×C7C4×S411yescharacteristic
136C2×C7C2×SL(2,ℤ/4)11yescharacteristic
137C22×C3—41no—
138A4C4×Dic711yescharacteristic, complemented
139C3×C4—282no—
140Dic3—42no—
141Dic3—282no—
142C23—31no—
143C23—31no—
144C23—31no—
145C23—31no—
146C23Dic3×D711yescharacteristic
147C23D3×Dic711yescharacteristic
148C23D21⋊C411yescharacteristic
149C2×C4—32no—
150C2×C4—32no—
151C2×C4—61no—
152C2×C4—61no—
153C2×C4—61no—
154C2×C4—61no—
155C2×C4—71no—
156C2×C4—212no—
157C2×C4—212no—
158C2×C4—211no—
159C2×C4—212no—
160C2×C4—212no—
161C2×C4—212no—
162C2×C4—421no—
163C2×C4—421no—
164C2×C4—421no—
165C2×C4—421no—
166C2×C4—421no—
167C7C4×SL(2,ℤ/4)11yescharacteristic, complemented, 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—
181C22Dic3×Dic711yescharacteristic, complemented
182C22D7×S411yescharacteristic
183C4—62no—
184C4—62no—
185C4—72no—
186C4—212no—
187C4—422no—
188C4—422no—
189C3—41noSylow
190C2—31no—
191C2—31no—
192C2—31no—
193C2—31no—
194C2S4×Dic711yescharacteristic
195C2(C7⋊S4)⋊C411yescharacteristic
196C2D7×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D7×S4
commutator, hypocenterC7×A4C42
Fitting, socleC24×C7C2×D3
prime radical, cosocleC22×C7×A4C22
LCSG ▷ C7×A4
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
6. A4×Dic7C4semidirect
7. C7⋊SL(2,ℤ/4)C4semidirect
8. C7×SL(2,ℤ/4)C4semidirect
32. C22×Dic7Dic3semidirect
44. C7×A4C42semidirect
77. SL(2,ℤ/4)Dic7direct (N × H)
99. C22×C7C4×Dic3semidirect
106. Dic7SL(2,ℤ/4)direct (N × H)
138. A4C4×Dic7semidirect
167. C7C4×SL(2,ℤ/4)semidirect
181. C22Dic3×Dic7semidirect

Maximal subgroups

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

subgroupindex
C2×A4×Dic72
C2×C7⋊SL(2,ℤ/4)2
C2×C7×SL(2,ℤ/4)2
C22⋊C4×Dic73
Dic3×Dic74
C4×SL(2,ℤ/4)7