Finite Group Databasebeta

← #13054337

Subgroups — #13054337

Order 1344.

212 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C3×#6501C211yesmaximal, characteristic
2#10620920C211yesmaximal, characteristic, complemented
3#10620671C211yesmaximal, characteristic, complemented
4C7×#832C211yesmaximal, characteristic, complemented
5#10620525C211yesmaximal, characteristic, complemented
6#10620521C211yesmaximal, characteristic, complemented
7(C2×D7)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#89556—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×C3×C22⋊D7—21no—
11C2×C3×C4×D7C2211yescharacteristic
12C4⋊Dic21C2211yescharacteristic
13C2×C3⋊(C7⋊C8)C2211yescharacteristic
14C7×C4⋊Dic3C2211yescharacteristic
15C2×C7×C3⋊C8C2211yescharacteristic
16C3×C22⋊Dic7—21no—
17C3×C4⋊Dic7C2211yescharacteristic
18(C2×D7)⋊Dic3—21no—
19Dic3×Dic7—21no—
20#6501D311yescharacteristic
21#6455—31no—
22C7×#102—31no—
23#6406—31no—
24(C2×D7)⋊C8—31no—
25#6392—31no—
26#6382—31no—
27#5035—71nomaximal, Hall
28C23×C3×C7—21no—
29C22×C3×D7D411yescharacteristic
30C3×C7×D4—21no—
31C3×C7×D4—21no—
32C2×C3×C4×C7C2311yescharacteristic
33C2×Dic21—21no—
34C2×C7×Dic3—21no—
35C3×C22⋊D7—42no—
36C2×C3×Dic7—21no—
37C2×C3×Dic7D411yescharacteristic
38C3×C4×D7—21no—
39C3×C4×D7—21no—
40C3⋊(C7⋊C8)—21no—
41C7×C3⋊C8—21no—
42C2×C7×D4C2×D311yescharacteristic
43C2×C22⋊D7—21no—
44C2×C4×D7C2×D311yescharacteristic
45C2×C7×C8—31no—
46C7×C4⋊C4—31no—
47C22⋊Dic7—21no—
48(C2×D7)⋊C4—61no—
49C4⋊Dic7—31no—
50C4⋊Dic7C2×D311yescharacteristic
51C4×Dic7—61no—
52C2×C7⋊C8—31no—
53C3×#121—71no—
54#926—71no—
55#832—71no—
56#832D711yescharacteristic, complemented
57#830—71no—
58#808—71no—
59#807—71no—
60C22×C3×C7—41no—
61C22×C3×C7—41no—
62C22×C3×C7—41no—
63C22×C3×C7C2×D411yescharacteristic
64C2×C3×D7—21no—
65C2×C3×D7—21no—
66C2×C3×D7—21no—
67C3×C4×C7#4111yescharacteristic
68C3×C4×C7#4111yescharacteristic
69Dic21—41no—
70C3×Dic7—21no—
71C3×Dic7—41no—
72C7×Dic3—41no—
73#482—211noSylow
74C23×C7—21no—
75C22×D7#6711yescharacteristic
76C7×D4—21no—
77C7×D4—21no—
78C2×C4×C7—61no—
79C2×C4×C7C22×D311yescharacteristic
80C22⋊D7—42no—
81C2×Dic7—21no—
82C2×Dic7—61no—
83C2×Dic7#6711yescharacteristic
84C4×D7—21no—
85C4×D7—21no—
86C7×C8—61no—
87C7⋊C8—61no—
88C2×C3×D4—141no—
89C2×C3×D4C2×D711yescharacteristic
90C22×C3×C4—71no—
91C3×C4⋊C4—71no—
92C3×C22⋊C4—141no—
93#217—141no—
94C4⋊Dic3—71no—
95C4⋊Dic3C2×D711yescharacteristic
96C4×Dic3—141no—
97C2×C3⋊C8—71no—
98C2×C3⋊C8C2×D711yescharacteristic
99C2×C3×C7—81no—
100C2×C3×C7#12311yescharacteristic
101C2×C3×C7#13511yescharacteristic
102C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
103C3×D7—41no—
104#121—71no—
105#117—211no—
106#107—211no—
107#106—211no—
108#102—31no—
109#102—211no—
110C22⋊C8—211no—
111C22×C7—41no—
112C22×C7—41no—
113C22×C7—41no—
114C22×C7C2×#6711yescharacteristic
115C2×D7—21no—
116C2×D7—21no—
117C2×D7—21no—
118C4×C7—121no—
119C4×C7#23711yescharacteristic
120C4×C7#23711yescharacteristic
121Dic7—21no—
122Dic7—41no—
123Dic7—121no—
124C23×C3—21no—
125C23×C3—71no—
126C3×D4—21no—
127C3×D4—21no—
128C3×D4—282no—
129C2×C3×C4—71no—
130C2×C3×C4—141no—
131C2×C3×C4—141no—
132C2×C3×C4—141no—
133C2×C3×C4C22×D711yescharacteristic
134C2×Dic3—21no—
135C2×Dic3—141no—
136C3⋊C8—21no—
137C3⋊C8—141no—
138C3×C7#48211yescharacteristic, complemented, Hall
139C2×D4—141no—
140C2×D4D3×D711yescharacteristic
141C22×C4—71no—
142C2×C8—31no—
143C2×C8—211no—
144C4⋊C4—31no—
145C4⋊C4—71no—
146C4⋊C4—211no—
147C22⋊C4—141no—
148C22⋊C4—421no—
149C42—421no—
150C2×C7—81no—
151C2×C7#93211yescharacteristic
152C2×C7#93511yescharacteristic
153C2×C7#95011yescharacteristic
154D7—41no—
155C22×C3—41no—
156C22×C3—41no—
157C22×C3—41no—
158C22×C3—141no—
159C22×C3—141no—
160C22×C3—141no—
161C22×C3D4×D711yescharacteristic
162C3×C4—141no—
163C3×C4—281no—
164C3×C4#115711yescharacteristic
165C3×C4#115911yescharacteristic
166Dic3—41no—
167Dic3—281no—
168C23—21no—
169C23—71no—
170D4—21no—
171D4—21no—
172D4—282no—
173C2×C4—61no—
174C2×C4—71no—
175C2×C4—141no—
176C2×C4—141no—
177C2×C4—141no—
178C2×C4—421no—
179C2×C4C2×D3×D711yescharacteristic
180C8—61no—
181C8—421no—
182C7#503511yescharacteristic, complemented, Sylow
183C2×C3—81no—
184C2×C3—281no—
185C2×C3#644611yescharacteristic
186C2×C3#647511yescharacteristic
187C2×C3#647711yescharacteristic
188C22—41no—
189C22—41no—
190C22—41no—
191C22—141no—
192C22—141no—
193C22—141no—
194C22D7×#6711yescharacteristic
195C4—121no—
196C4—141no—
197C4—281no—
198C4—841no—
199C4#6668311yescharacteristic
200C4#6668411yescharacteristic
201C3#8955611yescharacteristic, complemented, Sylow
202C2—81no—
203C2—281no—
204C2#1062105211yescharacteristic
205C2#1062105311yescharacteristic
206C2#1062111011yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
prime radicalC3×#6501C2
hypercenterC2×D4D3×D7
hypocenterC3×C7#482
LCSG ▷ C2×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
2. #10620920C2semidirect
3. #10620671C2semidirect
4. C7×#832C2semidirect
5. #10620525C2semidirect
6. #10620521C2semidirect
7. (C2×D7)⋊(C3⋊C8)C2semidirect
56. #832D7semidirect
138. C3×C7#482semidirect
182. C7#5035semidirect
201. C3#89556semidirect

Maximal subgroups

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

subgroupindex
C3×#65012
#106209202
#106206712
C7×#8322
#106205252
#106205212
(C2×D7)⋊(C3⋊C8)2
#895563
#50357