Finite Group Databasebeta

← #13053653

Subgroups — #13053653

Order 1344.

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

normal subgroups only

This group has 180 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
1D8×D21C211yesmaximal, characteristic
2#10620826C211yesmaximal, characteristic, complemented
3C7×#827C211yesmaximal, characteristic, complemented
4C3×#6401C211yesmaximal, characteristic, complemented
5#10620480C211yesmaximal, characteristic, complemented
6#10620478C211yesmaximal, characteristic, complemented
7D21⋊C16C211yesmaximal, characteristic, complemented
8D7×SD32—31nomaximal, Hall
9D4×D21—21no—
10D21⋊Q8—21no—
11C3×C7×D8C2211yescharacteristic
12#66632—21no—
13D168C2211yescharacteristic
14C8×D21C2211yescharacteristic
15C7×Dic12C2211yescharacteristic, complemented
16C3×Dic28C2211yescharacteristic, complemented
17#66569—21no—
18C3×C7⋊C16C2211yescharacteristic, complemented
19C7×C3⋊C16C2211yescharacteristic, complemented
20D7×Q16—31no—
21D7×D8—31no—
22C7×SD32—31no—
23#6402—31no—
24#6401D311yescharacteristic, complemented
25#6374—31no—
26D7×C16—31no—
27D3×SD32—71nomaximal, Hall
28C22×D21—41no—
29C3×C7×D4—21no—
30#4268—41no—
31D84—21no—
32C4×D21D411yescharacteristic
33C7×Dic6—21no—
34C3×Dic14—21no—
35#4248—41no—
36D21⋊C4—41no—
37C3×C7×C8C2311yescharacteristic
38C3⋊(C7⋊C8)D411yescharacteristic
39Q8×D7—61no—
40D4×D7—61no—
41C7×Q16—31no—
42C7×D8C2×D311yescharacteristic
43#1144—61no—
44#1141—61no—
45Dic28C2×D311yescharacteristic, complemented
46D56—31no—
47C8×D7—31no—
48C7×C16—31no—
49C7⋊C16C2×D311yescharacteristic, complemented
50D3×Q16—71no—
51D3×D8—71no—
52C3×SD32—71no—
53#828—71no—
54#827D711yescharacteristic, complemented
55#800—71no—
56D3×C16—71no—
57C22×C3×C7—41no—
58C2×D21—41no—
59C2×D21—82no—
60C2×D21D811yescharacteristic
61C3×C4×C7C2×D411yescharacteristic
62Dic21D811yescharacteristic
63C3×Dic7—41no—
64C7×Dic3—41no—
65C2×SD32—211noSylow
66C22×D7—121no—
67C7×Q8—61no—
68C7×D4—21no—
69C22⋊D7—121no—
70D28—61no—
71C4×D7—31no—
72C4×D7—121no—
73Dic14—21no—
74Dic14—121no—
75C7×C8C22×D311yescharacteristic
76C7⋊C8—31no—
77D3×Q8—141no—
78D3×D4—141no—
79C3×Q16—71no—
80C3×D8C2×D711yescharacteristic
81#216—141no—
82#213—141no—
83Dic12C2×D711yescharacteristic, complemented
84D24—71no—
85D3×C8—71no—
86C3×C16—71no—
87C3⋊C16C2×D711yescharacteristic, complemented
88C2×C3×C7—81no—
89C2×C3×C7C2×D811yescharacteristic
90D21—81no—
91D21SD3212yescomplemented
92C2×Q16—211no—
93C2×D8—211no—
94SD32—31no—
95SD32—71no—
96SD32—211no—
97SD32—211no—
98C2×C16—211no—
99C22×C7—41no—
100C2×D7—31no—
101C2×D7—121no—
102C2×D7—242no—
103C4×C7—121no—
104C4×C7D3×D411yescharacteristic
105Dic7—31no—
106Dic7—41no—
107C22×D3—281no—
108C3×Q8—141no—
109C3×D4—21no—
110#67—281no—
111D12—141no—
112C4×D3—71no—
113C4×D3—281no—
114Dic6—21no—
115Dic6—281no—
116C3×C8C22×D711yescharacteristic
117C3⋊C8—71no—
118C3×C7C2×SD3211yescharacteristic, complemented, Hall
119C2×Q8—421no—
120C2×D4—421no—
121Q16—31no—
122Q16—71no—
123Q16—421no—
124D8—211no—
125D8—421no—
126D8D3×D711yescharacteristic
127C2×C8—211no—
128C16—31no—
129C16—71no—
130C2×C7—81no—
131C2×C7D3×D811yescharacteristic
132D7—32no—
133D7—241no—
134C22×C3—41no—
135C2×D3—71no—
136C2×D3—281no—
137C2×D3—562no—
138C3×C4—281no—
139C3×C4D4×D711yescharacteristic
140Dic3—41no—
141Dic3—71no—
142C23—841no—
143Q8—61no—
144Q8—141no—
145Q8—841no—
146D4—21no—
147D4—421no—
148D4—841no—
149C2×C4—211no—
150C2×C4—841no—
151C8—211no—
152C8C2×D3×D711yescharacteristic
153C7D3×SD3211yescharacteristic, complemented, Sylow
154C2×C3—81no—
155C2×C3D7×D811yescharacteristic
156D3—72no—
157D3—561no—
158C22—41no—
159C22—211no—
160C22—841no—
161C22—1682no—
162C4—121no—
163C4—211no—
164C4—281no—
165C4#6669411yescharacteristic
166C3D7×SD3211yescharacteristic, complemented, Sylow
167C2—81no—
168C2—212no—
169C2—1681no—
170C2#1062104311yescharacteristic
subgroupG/H
centerC2#10621043
commutator, cosocleC3×C7×C8C23
FrattiniC8C2×D3×D7
FittingC3×C7×D8C22
socleC2×C3×C7C2×D8
prime radicalD8×D21C2
hypercenterD8D3×D7
hypocenterC3×C7C2×SD32
LCSG ▷ C3×C7×C8 ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ D8

Complements

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

NH ≅ G/Ntype
2. #10620826C2semidirect
3. C7×#827C2semidirect
4. C3×#6401C2semidirect
5. #10620480C2semidirect
6. #10620478C2semidirect
7. D21⋊C16C2semidirect
15. C7×Dic12C22semidirect
16. C3×Dic28C22semidirect
18. C3×C7⋊C16C22semidirect
19. C7×C3⋊C16C22semidirect
24. #6401D3semidirect
45. Dic28C2×D3semidirect
49. C7⋊C16C2×D3semidirect
54. #827D7semidirect
83. Dic12C2×D7semidirect
87. C3⋊C16C2×D7semidirect
91. D21SD32semidirect
118. C3×C7C2×SD32semidirect
153. C7D3×SD32semidirect
166. C3D7×SD32semidirect

Maximal subgroups

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

subgroupindex
D8×D212
#106208262
C7×#8272
C3×#64012
#106204802
#106204782
D21⋊C162
D7×SD323
D3×SD327