Finite Group Databasebeta

← #13054370

Subgroups — #13054370

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×#214C211yesmaximal, characteristic
2C3×D4×Dic7C211yesmaximal, characteristic
3Dic6×Dic7C211yesmaximal, characteristic, complemented
4#10620671C211yesmaximal, characteristic
5#10620525C211yesmaximal, characteristic, complemented
6#10620507C211yesmaximal, characteristic, complemented
7#10620487C211yesmaximal, characteristic, complemented
8#89936—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×C7×Dic6C2211yescharacteristic
11C22×C3×Dic7—21no—
12C4⋊Dic21C2211yescharacteristic
13C2×C3⋊(C7⋊C8)C2211yescharacteristic
14C7×#214C414yescomplemented
15C2×C7×C3⋊C8C2211yescharacteristic
16C3×C22⋊Dic7—21no—
17C3×C4⋊Dic7C2211yescharacteristic
18C3×C4×Dic7C2211yescharacteristic
19Dic3⋊Dic7—21no—
20Dic3×Dic7—21no—
21C2×C7×SD16—31no—
22Q8×Dic7—31no—
23D4×Dic7D311yescharacteristic
24#6409—31no—
25#6406—31no—
26#6392—31no—
27#6389—31no—
28#5018—71nomaximal, Hall
29C23×C3×C7—21no—
30C3×C7×D4—21no—
31C3×C7×D4C2×C412yes—
32C2×C3×C4×C7C2311yescharacteristic
33C2×Dic21—21no—
34C2×C7×Dic3—21no—
35C7×Dic6—21no—
36C7×Dic6C2×C412yescomplemented
37C2×C3×Dic7—21no—
38C2×C3×Dic7—42no—
39C2×C3×Dic7D411yescharacteristic
40C2×C3×Dic7D411yescharacteristic
41C3⋊(C7⋊C8)—21no—
42C7×C3⋊C8C2×C412yescomplemented
43C2×C7×Q8—31no—
44C2×C7×D4C2×D311yescharacteristic
45C22×Dic7—21no—
46C7×SD16—34no—
47C2×C7×C8—31no—
48C22⋊Dic7—21no—
49C4⋊Dic7—31no—
50C4⋊Dic7—61no—
51C4⋊Dic7C2×D311yescharacteristic
52C4×Dic7—61no—
53C4×Dic7C2×D311yescharacteristic
54C2×C7⋊C8—31no—
55C3×C4×D4—71no—
56C2×#214D711yescharacteristic
57C4×Dic6—71no—
58#832—71no—
59#810—71no—
60#807—71no—
61#803—71no—
62C22×C3×C7—22no—
63C22×C3×C7—41no—
64C22×C3×C7—41no—
65C22×C3×C7C2×D411yescharacteristic
66C3×C4×C7C22×C411yescharacteristic
67C3×C4×C7#4111yescharacteristic
68Dic21—41no—
69C3×Dic7—21no—
70C3×Dic7—21no—
71C3×Dic7—41no—
72C7×Dic3—22no—
73#440—211noSylow
74C23×C7—21no—
75C7×Q8—32no—
76C7×Q8—61no—
77C7×D4—21no—
78C7×D4C4×D312yes—
79C2×C4×C7—61no—
80C2×C4×C7C22×D311yescharacteristic
81C2×Dic7—21no—
82C2×Dic7—42no—
83C2×Dic7—61no—
84C2×Dic7#6711yescharacteristic
85C2×Dic7#6711yescharacteristic
86C7×C8—32no—
87C7⋊C8—61no—
88C2×C3×D4C2×D711yescharacteristic
89C22×C3×C4—141no—
90C2×Dic6C2×D711yescharacteristic
91C3×C4⋊C4—71no—
92C3×C22⋊C4—141no—
93C3×C42—71no—
94#214Dic714yescomplemented
95C4⋊Dic3—71no—
96Dic3⋊C4—141no—
97C4×Dic3—141no—
98C2×C3⋊C8—71no—
99C2×C3⋊C8C2×D711yescharacteristic
100C2×C3×C7—42no—
101C2×C3×C7C4×D411yescharacteristic
102C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
103C2×C3×C7#13711yescharacteristic
104C2×SD16—31no—
105C4×Q8—211no—
106C4×D4—71no—
107#107—211no—
108#103—211no—
109#102—211no—
110#97—211no—
111C22×C7—22no—
112C22×C7—41no—
113C22×C7—41no—
114C22×C7C2×#6711yescharacteristic
115C4×C7—62no—
116C4×C7C2×C4×D311yescharacteristic
117C4×C7#23511yescharacteristic
118Dic7—21no—
119Dic7—21no—
120Dic7—41no—
121Dic7—121no—
122C23×C3—21no—
123C3×D4—21no—
124C3×D4C2×Dic712yes—
125C2×C3×C4—71no—
126C2×C3×C4—71no—
127C2×C3×C4—141no—
128C2×C3×C4—282no—
129C2×C3×C4C22×D711yescharacteristic
130C2×Dic3—21no—
131C2×Dic3—141no—
132Dic6—21no—
133Dic6C2×Dic712yescomplemented
134C3⋊C8—141no—
135C3⋊C8C2×Dic712yescomplemented
136C3×C7#44011yescharacteristic, complemented, Hall
137C2×Q8—31no—
138C2×D4D3×D711yescharacteristic
139C22×C4—141no—
140SD16—34no—
141C2×C8—31no—
142C2×C8—211no—
143C4⋊C4—71no—
144C4⋊C4—211no—
145C4⋊C4—421no—
146C22⋊C4—141no—
147C42—71no—
148C42—421no—
149C2×C7—42no—
150C2×C7C4×#6711yescharacteristic
151C2×C7#93211yescharacteristic
152C2×C7#95111yescharacteristic
153C22×C3—22no—
154C22×C3—41no—
155C22×C3—41no—
156C22×C3D4×D711yescharacteristic
157C3×C4—141no—
158C3×C4—141no—
159C3×C4—281no—
160C3×C4#115911yescharacteristic
161C3×C4C22×Dic711yescharacteristic
162Dic3—22no—
163Dic3—281no—
164C23—21no—
165Q8—32no—
166Q8—61no—
167D4—21no—
168D4D3×Dic712yes—
169C2×C4—61no—
170C2×C4—71no—
171C2×C4—71no—
172C2×C4—141no—
173C2×C4—282no—
174C2×C4—421no—
175C2×C4C2×D3×D711yescharacteristic
176C8—32no—
177C8—421no—
178C7#501811yescharacteristic, complemented, Sylow
179C2×C3—42no—
180C2×C3#647711yescharacteristic
181C2×C3#647811yescharacteristic
182C2×C3D4×Dic711yescharacteristic
183C22—22no—
184C22—41no—
185C22—41no—
186C22D7×#6711yescharacteristic
187C4—62no—
188C4—141no—
189C4—141no—
190C4—281no—
191C4—841no—
192C4C2×D3×Dic711yescharacteristic
193C4#6668611yescharacteristic
194C3#8993611yescharacteristic, complemented, Sylow
195C2—42no—
196C2#1062104611yescharacteristic
197C2#1062105311yescharacteristic
198C2#67×Dic711yescharacteristic
subgroupG/H
centerC22D7×#67
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
Fitting, prime radicalC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
cosocleC2×C3×C4×C7C23
hypercenterC2×D4D3×D7
hypocenterC3×C7#440
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
3. Dic6×Dic7C2semidirect
5. #10620525C2semidirect
6. #10620507C2semidirect
7. #10620487C2semidirect
14. C7×#214C4semidirect
36. C7×Dic6C2×C4semidirect
42. C7×C3⋊C8C2×C4semidirect
94. #214Dic7semidirect
133. Dic6C2×Dic7semidirect
135. C3⋊C8C2×Dic7semidirect
136. C3×C7#440semidirect
178. C7#5018semidirect
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×#2142
C3×D4×Dic72
Dic6×Dic72
#106206712
#106205252
#106205072
#106204872
#899363
#50187