Finite Group Databasebeta

← #13054592

Subgroups — #13054592

Order 1344.

240 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 84 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×C7×#215C211yesmaximal, characteristic
2C3×Q8×Dic7C211yesmaximal, characteristic, complemented
3D12×Dic7C211yesmaximal, characteristic
4#10620674C211yesmaximal, characteristic, complemented
5#10620525C211yesmaximal, characteristic, complemented
6#10620501C211yesmaximal, characteristic
7#10620487C211yesmaximal, characteristic, complemented
8#89936—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×C7×D12C2211yescharacteristic
11C2×D3×Dic7—21no—
12C4⋊Dic21C2211yescharacteristic
13C2×C3⋊(C7⋊C8)C2211yescharacteristic
14C7×#215C414yescomplemented
15C2×C7×C3⋊C8C2211yescharacteristic
16C3×C4⋊Dic7—21no—
17C3×C4⋊Dic7C2211yescharacteristic
18C3×C4×Dic7—21no—
19C3×C4×Dic7C2211yescharacteristic
20(C2×D3)⋊Dic7—21no—
21C2×C7×SD16—31no—
22Q8×Dic7D311yescharacteristic, complemented
23D4×Dic7—31no—
24#6409—31no—
25#6406—31no—
26#6392—31no—
27#6389—31no—
28#5026—71nomaximal, Hall
29C22×D3×C7—21no—
30C3×C7×Q8—21no—
31C3×C7×Q8C2×C412yescomplemented
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7C2311yescharacteristic
34C2×Dic21—21no—
35C7×D12—21no—
36C7×D12C2×C412yes—
37C2×C3×Dic7—21no—
38C2×C3×Dic7D411yescharacteristic
39C2×C3×Dic7D411yescharacteristic
40D3×Dic7—42no—
41C3⋊(C7⋊C8)—21no—
42C7×C3⋊C8C2×C412yescomplemented
43C2×C7×Q8C2×D311yescharacteristic
44C2×C7×D4—31no—
45C22×Dic7—61no—
46C7×SD16—34no—
47C2×C7×C8—31no—
48C22⋊Dic7—61no—
49C4⋊Dic7—21no—
50C4⋊Dic7—31no—
51C4⋊Dic7C2×D311yescharacteristic
52C4×Dic7—21no—
53C4×Dic7C2×D311yescharacteristic
54C2×C7⋊C8—31no—
55C3×C4×Q8—71no—
56C2×#215D711yescharacteristic
57C4×D12—71no—
58#835—71no—
59#809—71no—
60#807—71no—
61#803—71no—
62C22×C3×C7C2×D411yescharacteristic
63C2×D3×C7—22no—
64C2×D3×C7—41no—
65C2×D3×C7—41no—
66C3×C4×C7—22no—
67C3×C4×C7C22×C411yescharacteristic
68C3×C4×C7#4111yescharacteristic
69Dic21—41no—
70C3×Dic7—21no—
71C3×Dic7—21no—
72C3×Dic7—41no—
73#440—211noSylow
74C23×C7—61no—
75C7×Q8—21no—
76C7×Q8C4×D312yescomplemented
77C7×D4—32no—
78C7×D4—61no—
79C2×C4×C7—21no—
80C2×C4×C7C22×D311yescharacteristic
81C2×Dic7—21no—
82C2×Dic7—61no—
83C2×Dic7—122no—
84C2×Dic7#6711yescharacteristic
85C2×Dic7#6711yescharacteristic
86C7×C8—32no—
87C7⋊C8—61no—
88C2×C3×Q8C2×D711yescharacteristic
89C2×D12C2×D711yescharacteristic
90C2×C4×D3—141no—
91C3×C4⋊C4—71no—
92C3×C4⋊C4—141no—
93C3×C42—71no—
94C3×C42—141no—
95#215Dic714yescomplemented
96(C2×D3)⋊C4—141no—
97C4⋊Dic3—71no—
98C2×C3⋊C8—71no—
99C2×C3⋊C8C2×D711yescharacteristic
100C2×C3×C7C4×D411yescharacteristic
101C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
102C2×C3×C7#13711yescharacteristic
103D3×C7—42no—
104C2×SD16—31no—
105C4×Q8—71no—
106C4×D4—211no—
107#107—211no—
108#103—211no—
109#102—211no—
110#97—211no—
111C22×C7—62no—
112C22×C7—121no—
113C22×C7—121no—
114C22×C7C2×#6711yescharacteristic
115C4×C7—22no—
116C4×C7C2×C4×D311yescharacteristic
117C4×C7#23511yescharacteristic
118Dic7—21no—
119Dic7—21no—
120Dic7—41no—
121Dic7—121no—
122C22×D3—21no—
123C3×Q8—21no—
124C3×Q8C2×Dic712yescomplemented
125C2×C3×C4—21no—
126C2×C3×C4—71no—
127C2×C3×C4—71no—
128C2×C3×C4—141no—
129C2×C3×C4C22×D711yescharacteristic
130C2×Dic3—141no—
131D12—21no—
132D12C2×Dic712yes—
133C4×D3—282no—
134C3⋊C8—141no—
135C3⋊C8C2×Dic712yescomplemented
136C3×C7#44011yescharacteristic, complemented, Hall
137C2×Q8D3×D711yescharacteristic
138C2×D4—31no—
139C22×C4—421no—
140SD16—34no—
141C2×C8—31no—
142C2×C8—211no—
143C4⋊C4—71no—
144C4⋊C4—141no—
145C4⋊C4—211no—
146C22⋊C4—421no—
147C42—71no—
148C42—141no—
149C2×C7—122no—
150C2×C7C4×#6711yescharacteristic
151C2×C7#94211yescharacteristic
152C2×C7#94911yescharacteristic
153C22×C3D4×D711yescharacteristic
154C2×D3—22no—
155C2×D3—41no—
156C2×D3—41no—
157C3×C4—22no—
158C3×C4—141no—
159C3×C4—141no—
160C3×C4—281no—
161C3×C4#115911yescharacteristic
162C3×C4C22×Dic711yescharacteristic
163Dic3—281no—
164C23—61no—
165Q8—21no—
166Q8D3×Dic712yescomplemented
167D4—32no—
168D4—61no—
169C2×C4—21no—
170C2×C4—71no—
171C2×C4—71no—
172C2×C4—141no—
173C2×C4—421no—
174C2×C4—842no—
175C2×C4C2×D3×D711yescharacteristic
176C8—32no—
177C8—421no—
178C7#502611yescharacteristic, complemented, Sylow
179C2×C3#647711yescharacteristic
180C2×C3#647811yescharacteristic
181C2×C3D4×Dic711yescharacteristic
182D3—42no—
183C22—62no—
184C22—121no—
185C22—121no—
186C22D7×#6711yescharacteristic
187C4—22no—
188C4—141no—
189C4—141no—
190C4—281no—
191C4—841no—
192C4C2×D3×Dic711yescharacteristic
193C4#6668611yescharacteristic
194C3#8993611yescharacteristic, complemented, Sylow
195C2—122no—
196C2#1062106411yescharacteristic
197C2#1062107511yescharacteristic
198C2#67×Dic711yescharacteristic
subgroupG/H
centerC22D7×#67
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radicalC2×C7×D12C22
cosocleC2×C3×C4×C7C23
hypercenterC2×Q8D3×D7
hypocenterC3×C7#440
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C3×Q8×Dic7C2semidirect
4. #10620674C2semidirect
5. #10620525C2semidirect
7. #10620487C2semidirect
14. C7×#215C4semidirect
22. Q8×Dic7D3semidirect
31. C3×C7×Q8C2×C4semidirect
42. C7×C3⋊C8C2×C4semidirect
76. C7×Q8C4×D3semidirect
95. #215Dic7semidirect
124. C3×Q8C2×Dic7semidirect
135. C3⋊C8C2×Dic7semidirect
136. C3×C7#440semidirect
166. Q8D3×Dic7semidirect
178. C7#5026semidirect
194. C3#89936semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#2152
C3×Q8×Dic72
D12×Dic72
#106206742
#106205252
#106205012
#106204872
#899363
#50267