Finite Group Databasebeta

← #13054442

Subgroups — #13054442

Order 1344.

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

normal subgroups only

This group has 220 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×#6503C211yesmaximal, characteristic
2#10620950C211yesmaximal, characteristic, complemented
3#10620671C211yesmaximal, characteristic, complemented
4C7×#832C211yesmaximal, characteristic, complemented
5#10620500C211yesmaximal, characteristic, complemented
6#10620499C211yesmaximal, characteristic, complemented
7#10620487C211yesmaximal, characteristic, complemented
8#89536—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×C3×C22⋊D7—21no—
11C2×C3×C22⋊D7—21no—
12C2×C3×D28C2211yescharacteristic
13C4⋊Dic21C2211yescharacteristic
14C2×C3⋊(C7⋊C8)C2211yescharacteristic
15C7×C4⋊Dic3C2211yescharacteristic
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×C4×Dic7C2211yescharacteristic
18Dic3⋊Dic7—21no—
19#66577—21no—
20#6503D311yescharacteristic
21#6453—31no—
22C7×#102—31no—
23#6406—31no—
24#6395—31no—
25#6389—31no—
26#6383—31no—
27#5071—71nomaximal, Hall
28C23×C3×C7—21no—
29C22×C3×D7—21no—
30C3×C7×D4—21no—
31C3×C7×D4—21no—
32C2×C3×C4×C7C2311yescharacteristic
33C2×Dic21—21no—
34C2×C7×Dic3—21no—
35C3×C22⋊D7—42no—
36C3×C22⋊D7—42no—
37C2×C3×Dic7D411yescharacteristic
38C2×C3×Dic7D411yescharacteristic
39C3×D28—21no—
40C3×D28—21no—
41C3⋊(C7⋊C8)—21no—
42C7×C3⋊C8—21no—
43C2×C7×D4C2×D311yescharacteristic
44C2×C22⋊D7—21no—
45C2×C22⋊D7—21no—
46C2×D28C2×D311yescharacteristic
47C2×C7×C8—31no—
48C7×C4⋊C4—31no—
49C4⋊Dic7—31no—
50C4⋊Dic7—61no—
51Dic7⋊C4—61no—
52C4×Dic7C2×D311yescharacteristic
53C2×C7⋊C8—31no—
54C3×#127—71no—
55#870—71no—
56#832—71no—
57#832—71no—
58#832—71no—
59#832D711yescharacteristic, complemented
60#803—71no—
61C22×C3×C7—41no—
62C22×C3×C7—41no—
63C22×C3×C7—41no—
64C22×C3×C7C2×D411yescharacteristic
65C2×C3×D7—41no—
66C2×C3×D7—41no—
67C2×C3×D7—41no—
68C3×C4×C7#4111yescharacteristic
69C3×C4×C7#4111yescharacteristic
70Dic21—41no—
71C3×Dic7—21no—
72C3×Dic7—21no—
73C7×Dic3—41no—
74#490—211noSylow
75C23×C7—21no—
76C22×D7—21no—
77C7×D4—21no—
78C7×D4—21no—
79C2×C4×C7—61no—
80C2×C4×C7C22×D311yescharacteristic
81C22⋊D7—42no—
82C22⋊D7—42no—
83C2×Dic7—61no—
84C2×Dic7#6711yescharacteristic
85C2×Dic7#6711yescharacteristic
86D28—21no—
87D28—21no—
88C7×C8—61no—
89C7⋊C8—61no—
90C2×C3×D4—71no—
91C2×C3×D4—141no—
92C2×C3×D4—141no—
93C2×C3×D4C2×D711yescharacteristic
94C3×C42—71no—
95C4⋊Dic3—71no—
96C4⋊Dic3C2×D711yescharacteristic
97Dic3⋊C4—141no—
98Dic3⋊C4—141no—
99C2×C3⋊C8—71no—
100C2×C3⋊C8C2×D711yescharacteristic
101C2×C3×C7—81no—
102C2×C3×C7#12411yescharacteristic
103C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
104C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
105C3×D7—81no—
106#127—71no—
107#125—211no—
108#102—31no—
109#102—211no—
110#102—211no—
111#102—211no—
112#97—211no—
113C22×C7—41no—
114C22×C7—41no—
115C22×C7—41no—
116C22×C7C2×#6711yescharacteristic
117C2×D7—41no—
118C2×D7—41no—
119C2×D7—41no—
120C4×C7—121no—
121C4×C7#23711yescharacteristic
122C4×C7#23711yescharacteristic
123Dic7—21no—
124Dic7—21no—
125Dic7—121no—
126C23×C3—21no—
127C23×C3—141no—
128C3×D4—21no—
129C3×D4—21no—
130C3×D4—141no—
131C3×D4—141no—
132C3×D4—282no—
133C3×D4—282no—
134C2×C3×C4—71no—
135C2×C3×C4—71no—
136C2×C3×C4C22×D711yescharacteristic
137C2×Dic3—21no—
138C2×Dic3—141no—
139C3⋊C8—21no—
140C3⋊C8—141no—
141C3×C7#49011yescharacteristic, complemented, Hall
142C2×D4—71no—
143C2×D4—141no—
144C2×D4—141no—
145C2×D4D3×D711yescharacteristic
146C2×C8—31no—
147C2×C8—211no—
148C4⋊C4—31no—
149C4⋊C4—211no—
150C4⋊C4—421no—
151C4⋊C4—421no—
152C42—71no—
153C2×C7—81no—
154C2×C7#93211yescharacteristic
155C2×C7#93211yescharacteristic
156C2×C7#93611yescharacteristic
157D7—81no—
158C22×C3—41no—
159C22×C3—41no—
160C22×C3—41no—
161C22×C3—281no—
162C22×C3—281no—
163C22×C3—281no—
164C22×C3D4×D711yescharacteristic
165C3×C4—141no—
166C3×C4—141no—
167C3×C4#115711yescharacteristic
168C3×C4#115911yescharacteristic
169Dic3—41no—
170Dic3—281no—
171C23—21no—
172C23—141no—
173D4—21no—
174D4—21no—
175D4—141no—
176D4—141no—
177D4—282no—
178D4—282no—
179C2×C4—61no—
180C2×C4—71no—
181C2×C4—71no—
182C2×C4—421no—
183C2×C4C2×D3×D711yescharacteristic
184C8—61no—
185C8—421no—
186C7#507111yescharacteristic, complemented, Sylow
187C2×C3—81no—
188C2×C3—561no—
189C2×C3#644811yescharacteristic
190C2×C3#647411yescharacteristic
191C2×C3#647711yescharacteristic
192C22—41no—
193C22—41no—
194C22—41no—
195C22—281no—
196C22—281no—
197C22—281no—
198C22D7×#6711yescharacteristic
199C4—121no—
200C4—141no—
201C4—141no—
202C4—841no—
203C4#6668311yescharacteristic
204C4#6668411yescharacteristic
205C3#8953611yescharacteristic, complemented, Sylow
206C2—81no—
207C2—561no—
208C2#1062105111yescharacteristic
209C2#1062105311yescharacteristic
210C2#1062109611yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
prime radicalC3×#6503C2
hypercenterC2×D4D3×D7
hypocenterC3×C7#490
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. #10620950C2semidirect
3. #10620671C2semidirect
4. C7×#832C2semidirect
5. #10620500C2semidirect
6. #10620499C2semidirect
7. #10620487C2semidirect
59. #832D7semidirect
141. C3×C7#490semidirect
186. C7#5071semidirect
205. C3#89536semidirect

Maximal subgroups

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

subgroupindex
C3×#65032
#106209502
#106206712
C7×#8322
#106205002
#106204992
#106204872
#895363
#50717