Finite Group Databasebeta

← #13053808

Subgroups — #13053808

Order 1344.

324 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 104 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×#66728C211yesmaximal, characteristic
2C2×C7×#208C211yesmaximal, characteristic, complemented
3C2×C3×C4⋊Dic7C211yesmaximal, characteristic, complemented
4#10620509C212yesmaximal, complemented
5#10620503C212yesmaximal, complemented
6#89905—31nomaximal, Hall
7C22×C3×C4×C7C2211yescharacteristic
8C2×#4268—21no—
9#66728—22no—
10#66728C414yescomplemented
11C2×D84C2211yescharacteristic
12C2×C4×D21—21no—
13C2×Dic42C2211yescharacteristic
14C22×C3×Dic7—21no—
15C7×#208—22no—
16C2×C7×C3⋊C8C2212yes—
17C3×C4⋊Dic7—21no—
18C3×C4⋊Dic7C2212yes—
19C2×#1157—31no—
20C2×C7×M16—31no—
21C2×C4⋊Dic7D311yescharacteristic, complemented
22#6395—32no—
23#6390—32no—
24#4966—71nomaximal, Hall
25C23×C3×C7D411yescharacteristic
26C22×D21—21no—
27C2×C3×C4×C7C2×C412yes—
28C2×C3×C4×C7D412yes—
29C2×C3×C4×C7D411yescharacteristic
30C2×C3×C4×C7C2311yescharacteristic
31#4268—24no—
32C2×Dic21—21no—
33D84—21no—
34D84C2×C412yescomplemented
35C4×D21—24no—
36Dic42—21no—
37Dic42C2×C412yescomplemented
38C2×C3×Dic7—22no—
39C2×C3×Dic7—42no—
40C7×C3⋊C8—22no—
41C22×C4×C7C2×D311yescharacteristic
42C2×C22⋊D7—61no—
43C22×Dic7—21no—
44#1157—34no—
45#1157—62no—
46C2×D28—31no—
47C2×C4×D7—61no—
48C2×Dic14—31no—
49C7×M16—62no—
50C2×C7×C8—32no—
51C4⋊Dic7—21no—
52C4⋊Dic7C2×D312yes—
53C2×#235—71no—
54C2×C3×C4⋊C4—71no—
55C2×#208D711yescharacteristic, complemented
56#810—72no—
57#809—72no—
58C22×C3×C7—21no—
59C22×C3×C7—21no—
60C22×C3×C7C22⋊C412yes—
61C22×C3×C7C2×D411yescharacteristic
62C2×D21—22no—
63C2×D21—41no—
64C2×D21—41no—
65C3×C4×C7C22⋊C412yes—
66C3×C4×C7C22×C411yescharacteristic
67C3×C4×C7C2×D411yescharacteristic
68Dic21—22no—
69C3×Dic7—42no—
70#417—211noSylow
71C23×C7#6711yescharacteristic
72C22×D7—61no—
73C2×C4×C7C4×D312yes—
74C2×C4×C7D1212yes—
75C2×C4×C7#6711yescharacteristic
76C2×C4×C7C22×D311yescharacteristic
77C22⋊D7—64no—
78C2×Dic7—22no—
79C2×Dic7—42no—
80C2×Dic7—61no—
81D28—32no—
82D28—61no—
83C4×D7—64no—
84Dic14—32no—
85Dic14—61no—
86C7×C8—62no—
87C22×C3×C4—141no—
88C22×C3×C4C2×D711yescharacteristic
89C2×#67—141no—
90#235—74no—
91#235—142no—
92C2×D12—71no—
93C2×C4×D3—141no—
94C2×Dic6—71no—
95C3×C4⋊C4—72no—
96C3×C4⋊C4—141no—
97#208—22no—
98C2×C3⋊C8C2×D712yes—
99C2×C3×C7—22no—
100C2×C3×C7C2×C22⋊C411yescharacteristic
101C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
102C2×C3×C7#13711yescharacteristic
103D21—42no—
104C2×#41—211no—
105C2×M16—31no—
106C2×C4⋊C4—71no—
107#103—212no—
108#102—212no—
109C22×C7—21no—
110C22×C7—21no—
111C22×C7(C2×D3)⋊C412yes—
112C22×C7C2×#6711yescharacteristic
113C2×D7—62no—
114C2×D7—121no—
115C2×D7—121no—
116C4×C7(C2×D3)⋊C412yes—
117C4×C7C2×C4×D311yescharacteristic
118C4×C7C2×D1211yescharacteristic
119Dic7—42no—
120Dic7—62no—
121C23×C3D2811yescharacteristic
122C22×D3—141no—
123C2×C3×C4—142no—
124C2×C3×C4—282no—
125C2×C3×C4C4×D712yes—
126C2×C3×C4D2811yescharacteristic
127C2×C3×C4C22⋊D712yes—
128C2×C3×C4C22×D711yescharacteristic
129#67—144no—
130C2×Dic3—141no—
131D12—72no—
132D12—141no—
133C4×D3—144no—
134Dic6—72no—
135Dic6—141no—
136C3⋊C8—22no—
137C3×C7#41711yescharacteristic, complemented, Hall
138#41—214no—
139#41—422no—
140C2×Q8—211no—
141C2×D4—211no—
142C2×D4—421no—
143C22×C4—141no—
144C22×C4—421no—
145C22×C4D3×D711yescharacteristic
146M16—62no—
147C2×C8—32no—
148C4⋊C4—72no—
149C4⋊C4—141no—
150C2×C7—22no—
151C2×C7C2×(C2×D3)⋊C411yescharacteristic
152C2×C7#93211yescharacteristic
153C2×C7#94211yescharacteristic
154D7—122no—
155C22×C3—21no—
156C22×C3—21no—
157C22×C3(C2×D7)⋊C412yes—
158C22×C3C2×D2811yescharacteristic
159C2×D3—142no—
160C2×D3—281no—
161C2×D3—281no—
162C3×C4—282no—
163C3×C4(C2×D7)⋊C412yes—
164C3×C4C2×C4×D711yescharacteristic
165C3×C4C2×C22⋊D711yescharacteristic
166Dic3—142no—
167C23—421no—
168C23#424611yescharacteristic
169Q8—212no—
170Q8—421no—
171D4—212no—
172D4—424no—
173D4—421no—
174C2×C4—142no—
175C2×C4—282no—
176C2×C4—421no—
177C2×C4—424no—
178C2×C4D21⋊C412yes—
179C2×C4#424611yescharacteristic
180C2×C4#424712yes—
181C2×C4C2×D3×D711yescharacteristic
182C8—62no—
183C7#496611yescharacteristic, complemented, Sylow
184C2×C3—22no—
185C2×C3#647111yescharacteristic
186C2×C3#647211yescharacteristic
187C2×C3C2×(C2×D7)⋊C411yescharacteristic
188D3—282no—
189C22—21no—
190C22—21no—
191C22—422no—
192C22—841no—
193C22—841no—
194C22#6657512yes—
195C22C2×#424611yescharacteristic
196C4—282no—
197C4—422no—
198C4#6657512yes—
199C4C2×D21⋊C411yescharacteristic
200C4C2×#424711yescharacteristic
201C3#8990511yescharacteristic, complemented, Sylow
202C2—22no—
203C2—842no—
204C2#1062087411yescharacteristic
205C2#1062087711yescharacteristic
206C2C2×#6657511yescharacteristic
subgroupG/H
centerC22C2×#4246
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC22×C3×C4×C7C22
socleC22×C3×C7C2×D4
prime radicalC2×#66728C2
cosocleC2×C3×C4×C7C23
hypercenterC22×C4D3×D7
hypocenterC3×C7#417
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C7×#208C2semidirect
3. C2×C3×C4⋊Dic7C2semidirect
4. #10620509C2semidirect
5. #10620503C2semidirect
10. #66728C4semidirect
21. C2×C4⋊Dic7D3semidirect
34. D84C2×C4semidirect
37. Dic42C2×C4semidirect
55. C2×#208D7semidirect
137. C3×C7#417semidirect
183. C7#4966semidirect
201. C3#89905semidirect

Maximal subgroups

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

subgroupindex
C2×#667282
C2×C7×#2082
C2×C3×C4⋊Dic72
#106205092
#106205032
#899053
#49667