Finite Group Databasebeta

← #13054448

Subgroups — #13054448

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
1C7×#940C211yesmaximal, characteristic
2#10620943C211yesmaximal, characteristic, complemented
3#10620671C211yesmaximal, characteristic, complemented
4C3×#6406C211yesmaximal, characteristic, complemented
5#10620502C211yesmaximal, characteristic, complemented
6#10620501C211yesmaximal, characteristic, complemented
7#10620488C211yesmaximal, characteristic, complemented
8#89843—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×C7×#67—21no—
11C2×C7×#67—21no—
12C2×C7×D12C2211yescharacteristic
13C4⋊Dic21C2211yescharacteristic
14C2×C3⋊(C7⋊C8)C2211yescharacteristic
15C4×C7×Dic3C2211yescharacteristic
16C3×C4⋊Dic7C2211yescharacteristic
17C2×C3×C7⋊C8C2211yescharacteristic
18#66577—21no—
19Dic7⋊Dic3—21no—
20C7×#127—31no—
21#6433—31no—
22#6406—31no—
23#6406—31no—
24#6406—31no—
25#6406D311yescharacteristic, complemented
26#6377—31no—
27#4764—71nomaximal, Hall
28C23×C3×C7—21no—
29C22×D3×C7—21no—
30C3×C7×D4—21no—
31C3×C7×D4—21no—
32C2×C3×C4×C7C2311yescharacteristic
33C2×Dic21—21no—
34C7×#67—42no—
35C7×#67—42no—
36C2×C7×Dic3D411yescharacteristic
37C2×C7×Dic3D411yescharacteristic
38C7×D12—21no—
39C7×D12—21no—
40C2×C3×Dic7—21no—
41C3⋊(C7⋊C8)—21no—
42C3×C7⋊C8—21no—
43C2×C7×D4—31no—
44C2×C7×D4—61no—
45C2×C7×D4—61no—
46C2×C7×D4C2×D311yescharacteristic
47C42×C7—31no—
48C4⋊Dic7—31no—
49C4⋊Dic7C2×D311yescharacteristic
50Dic7⋊C4—61no—
51Dic7⋊C4—61no—
52C2×C7⋊C8—31no—
53C2×C7⋊C8C2×D311yescharacteristic
54#940D711yescharacteristic
55#890—71no—
56C3×#102—71no—
57#832—71no—
58#821—71no—
59#815—71no—
60#809—71no—
61C22×C3×C7—41no—
62C22×C3×C7—41no—
63C22×C3×C7—41no—
64C22×C3×C7C2×D411yescharacteristic
65C2×D3×C7—41no—
66C2×D3×C7—41no—
67C2×D3×C7—41no—
68C3×C4×C7#4111yescharacteristic
69C3×C4×C7#4111yescharacteristic
70Dic21—41no—
71C3×Dic7—41no—
72C7×Dic3—21no—
73C7×Dic3—21no—
74#490—211noSylow
75C23×C7—21no—
76C23×C7—61no—
77C7×D4—21no—
78C7×D4—21no—
79C7×D4—61no—
80C7×D4—61no—
81C7×D4—122no—
82C7×D4—122no—
83C2×C4×C7—31no—
84C2×C4×C7—31no—
85C2×C4×C7C22×D311yescharacteristic
86C2×Dic7—21no—
87C2×Dic7—61no—
88C7⋊C8—21no—
89C7⋊C8—61no—
90C2×C3×D4C2×D711yescharacteristic
91C2×#67—21no—
92C2×#67—21no—
93C2×D12C2×D711yescharacteristic
94C2×C3×C8—71no—
95C3×C4⋊C4—71no—
96C4⋊Dic3—71no—
97C4⋊Dic3—141no—
98Dic3⋊C4—141no—
99C4×Dic3C2×D711yescharacteristic
100C2×C3⋊C8—71no—
101C2×C3×C7—81no—
102C2×C3×C7#12411yescharacteristic
103C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
104C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
105D3×C7—81no—
106#127—31no—
107#125—211no—
108#102—71no—
109#102—211no—
110#102—211no—
111#102—211no—
112#97—211no—
113C22×C7—41no—
114C22×C7—41no—
115C22×C7—41no—
116C22×C7—121no—
117C22×C7—121no—
118C22×C7—121no—
119C22×C7D3×D411yescharacteristic
120C4×C7—61no—
121C4×C7—61no—
122C4×C7#23511yescharacteristic
123C4×C7#23711yescharacteristic
124Dic7—41no—
125Dic7—121no—
126C23×C3—21no—
127C22×D3—21no—
128C3×D4—21no—
129C3×D4—21no—
130C2×C3×C4—141no—
131C2×C3×C4C22×D711yescharacteristic
132#67—42no—
133#67—42no—
134C2×Dic3—141no—
135C2×Dic3C22⋊D711yescharacteristic
136C2×Dic3C22⋊D711yescharacteristic
137D12—21no—
138D12—21no—
139C3×C8—141no—
140C3⋊C8—141no—
141C3×C7#49011yescharacteristic, complemented, Hall
142C2×D4—31no—
143C2×D4—61no—
144C2×D4—61no—
145C2×D4D3×D711yescharacteristic
146C2×C8—71no—
147C2×C8—211no—
148C4⋊C4—71no—
149C4⋊C4—211no—
150C4⋊C4—421no—
151C4⋊C4—421no—
152C42—31no—
153C2×C7—81no—
154C2×C7—241no—
155C2×C7#88511yescharacteristic
156C2×C7#91111yescharacteristic
157C2×C7#91411yescharacteristic
158C22×C3—41no—
159C22×C3—41no—
160C22×C3—41no—
161C22×C3C2×C22⋊D711yescharacteristic
162C2×D3—41no—
163C2×D3—41no—
164C2×D3—41no—
165C3×C4—281no—
166C3×C4#115911yescharacteristic
167C3×C4#115911yescharacteristic
168Dic3—21no—
169Dic3—21no—
170Dic3—281no—
171C23—21no—
172C23—61no—
173D4—21no—
174D4—21no—
175D4—61no—
176D4—61no—
177D4—122no—
178D4—122no—
179C2×C4—31no—
180C2×C4—31no—
181C2×C4—141no—
182C2×C4—421no—
183C2×C4C2×D3×D711yescharacteristic
184C8—141no—
185C8—421no—
186C7#476411yescharacteristic, complemented, Sylow
187C2×C3—81no—
188C2×C3#649511yescharacteristic
189C2×C3#649511yescharacteristic
190C2×C3#649911yescharacteristic
191D3—81no—
192C22—41no—
193C22—41no—
194C22—41no—
195C22—121no—
196C22—121no—
197C22—121no—
198C22D3×C22⋊D711yescharacteristic
199C4—61no—
200C4—61no—
201C4—281no—
202C4—841no—
203C4#6668411yescharacteristic
204C4#6668611yescharacteristic
205C3#8984311yescharacteristic, complemented, Sylow
206C2—81no—
207C2—241no—
208C2#1062105611yescharacteristic
209C2#1062105811yescharacteristic
210C2#1062108711yescharacteristic
subgroupG/H
centerC22D3×C22⋊D7
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
prime radicalC7×#940C2
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. #10620943C2semidirect
3. #10620671C2semidirect
4. C3×#6406C2semidirect
5. #10620502C2semidirect
6. #10620501C2semidirect
7. #10620488C2semidirect
25. #6406D3semidirect
141. C3×C7#490semidirect
186. C7#4764semidirect
205. C3#89843semidirect

Maximal subgroups

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

subgroupindex
C7×#9402
#106209432
#106206712
C3×#64062
#106205022
#106205012
#106204882
#898433
#47647