Finite Group Databasebeta

← #13052624

Subgroups — #13052624

Order 1344.

212 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 58 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
1#10621375C211yesmaximal, characteristic, complemented
2#10621028C211yesmaximal, characteristic
3C7×#809C211yesmaximal, characteristic, complemented
4C3×#6382C211yesmaximal, characteristic, complemented
5#10620525C211yesmaximal, characteristic, complemented
6#10620502C211yesmaximal, characteristic, complemented
7#10620492C211yesmaximal, characteristic, complemented
8#89556—31nomaximal, Hall
9C2×C4×D21C2211yescharacteristic
10C2×C7×D12C2211yescharacteristic
11C2×#4247—21no—
12C3×C7×C4⋊C4C2211yescharacteristic
13#66631—21no—
14C4⋊Dic21C2211yescharacteristic
15C4×Dic21—21no—
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×C4⋊Dic7C2211yescharacteristic
18C2×C3×C7⋊C8C2211yescharacteristic
19(C2×D3)⋊Dic7—21no—
20#6501—31no—
21#6455—31no—
22C7×#102—31no—
23#6406—31no—
24(C2×D7)⋊C8—31no—
25#6392—31no—
26#6382D311yescharacteristic, complemented
27#4866—71nomaximal, Hall
28C22×D21D411yescharacteristic
29C22×D3×C7—21no—
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7C2311yescharacteristic
32C2×Dic21—21no—
33C2×Dic21D411yescharacteristic
34C4×D21—21no—
35C4×D21—21no—
36C7×D12—21no—
37C7×D12—21no—
38C2×C3×Dic7—21no—
39#4247—42no—
40C3×C7⋊C8—21no—
41C7×C3⋊C8—21no—
42C2×C7×D4—31no—
43C2×C22⋊D7—61no—
44C2×C4×D7—31no—
45C2×C7×C8—31no—
46C7×C4⋊C4C2×D311yescharacteristic
47C22⋊Dic7—61no—
48(C2×D7)⋊C4—61no—
49C4⋊Dic7—31no—
50C4⋊Dic7C2×D311yescharacteristic
51C4×Dic7—61no—
52C2×C7⋊C8C2×D311yescharacteristic
53#895—71no—
54#892—71no—
55C3×#106—71no—
56#821—71no—
57(C2×D3)⋊C8—71no—
58#809D711yescharacteristic, complemented
59#807—71no—
60C22×C3×C7C2×D411yescharacteristic
61C2×D21—21no—
62C2×D21—21no—
63C2×D21—21no—
64C2×D3×C7—41no—
65C2×D3×C7—41no—
66C2×D3×C7—41no—
67C3×C4×C7—41no—
68C3×C4×C7#4111yescharacteristic
69C3×C4×C7#4111yescharacteristic
70Dic21—21no—
71Dic21—41no—
72C3×Dic7—41no—
73#482—211noSylow
74C23×C7—61no—
75C22×D7—31no—
76C7×D4—61no—
77C7×D4—61no—
78C2×C4×C7—21no—
79C2×C4×C7C22×D311yescharacteristic
80C22⋊D7—122no—
81C2×Dic7—21no—
82C2×Dic7—31no—
83C2×Dic7—61no—
84C4×D7—61no—
85C4×D7—61no—
86C7×C8—61no—
87C7⋊C8—21no—
88C2×D12—141no—
89C2×D12C2×D711yescharacteristic
90C2×C4×D3—71no—
91C2×C3×C8—71no—
92C3×C4⋊C4—71no—
93C3×C4⋊C4C2×D711yescharacteristic
94(C2×D3)⋊C4—141no—
95(C2×D3)⋊C4—141no—
96C4⋊Dic3—71no—
97C4×Dic3—141no—
98C2×C3⋊C8C2×D711yescharacteristic
99C2×C3×C7#12311yescharacteristic
100C2×C3×C7#13511yescharacteristic
101C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
102D21—41no—
103D3×C7—81no—
104#121—211no—
105#117—211no—
106#107—211no—
107#106—71no—
108#102—31no—
109#102—211no—
110C22⋊C8—211no—
111C22×C7—121no—
112C22×C7—121no—
113C22×C7—121no—
114C22×C7D3×D411yescharacteristic
115C2×D7—61no—
116C2×D7—61no—
117C2×D7—61no—
118C4×C7—41no—
119C4×C7#23511yescharacteristic
120C4×C7#23911yescharacteristic
121Dic7—41no—
122Dic7—61no—
123Dic7—121no—
124C22×D3—21no—
125C22×D3—71no—
126C2×C3×C4—21no—
127C2×C3×C4—141no—
128C2×C3×C4C22×D711yescharacteristic
129C2×Dic3—71no—
130C2×Dic3—141no—
131D12—21no—
132D12—21no—
133D12—282no—
134C4×D3—141no—
135C4×D3—141no—
136C3×C8—141no—
137C3⋊C8—21no—
138C3×C7#48211yescharacteristic, complemented, Hall
139C2×D4—31no—
140C2×D4—421no—
141C22×C4—211no—
142C2×C8—31no—
143C2×C8—71no—
144C4⋊C4—71no—
145C4⋊C4—211no—
146C4⋊C4D3×D711yescharacteristic
147C22⋊C4—421no—
148C22⋊C4—421no—
149C42—421no—
150C2×C7—241no—
151C2×C7#89411yescharacteristic
152C2×C7#91411yescharacteristic
153C2×C7#91611yescharacteristic
154D7—121no—
155C22×C3D4×D711yescharacteristic
156C2×D3—41no—
157C2×D3—41no—
158C2×D3—41no—
159C2×D3—141no—
160C2×D3—141no—
161C2×D3—141no—
162C3×C4—41no—
163C3×C4—281no—
164C3×C4#115711yescharacteristic
165C3×C4#115911yescharacteristic
166Dic3—141no—
167Dic3—281no—
168C23—61no—
169C23—211no—
170D4—61no—
171D4—61no—
172D4—842no—
173C2×C4—21no—
174C2×C4—141no—
175C2×C4—211no—
176C2×C4—421no—
177C2×C4—421no—
178C2×C4—421no—
179C2×C4C2×D3×D711yescharacteristic
180C8—61no—
181C8—141no—
182C7#486611yescharacteristic, complemented, Sylow
183C2×C3#644611yescharacteristic
184C2×C3#647511yescharacteristic
185C2×C3#647711yescharacteristic
186D3—81no—
187D3—281no—
188C22—121no—
189C22—121no—
190C22—121no—
191C22—421no—
192C22—421no—
193C22—421no—
194C22#6669411yescharacteristic
195C4—41no—
196C4—281no—
197C4—421no—
198C4—841no—
199C4#6667211yescharacteristic
200C4#6667511yescharacteristic
201C3#8955611yescharacteristic, complemented, Sylow
202C2—241no—
203C2—841no—
204C2#1062091711yescharacteristic
205C2#1062106111yescharacteristic
206C2#1062106811yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radical#10621028C2
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#482
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #10621375C2semidirect
3. C7×#809C2semidirect
4. C3×#6382C2semidirect
5. #10620525C2semidirect
6. #10620502C2semidirect
7. #10620492C2semidirect
26. #6382D3semidirect
58. #809D7semidirect
138. C3×C7#482semidirect
182. C7#4866semidirect
201. C3#89556semidirect

Maximal subgroups

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

subgroupindex
#106213752
#106210282
C7×#8092
C3×#63822
#106205252
#106205022
#106204922
#895563
#48667