Finite Group Databasebeta

← #13054749

Subgroups — #13054749

Order 1344.

224 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 62 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#10621430C211yesmaximal, characteristic
2C2×C7×#216C211yesmaximal, characteristic, complemented
3#10620906C211yesmaximal, characteristic, complemented
4C2×#66568C211yesmaximal, characteristic
5C3×#6409C211yesmaximal, characteristic, complemented
6#10620518C211yesmaximal, characteristic, complemented
7#10620503C211yesmaximal, characteristic
8#89957—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×D84C2211yescharacteristic
11C2×C7×Dic6C2211yescharacteristic
12#66631—21no—
13#66631—21no—
14C4×Dic21C2211yescharacteristic
15C7×#216—22no—
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×C4⋊Dic7C2211yescharacteristic
18C2×C3×C7⋊C8C2211yescharacteristic
19Dic3⋊Dic7—21no—
20Dic3×Dic7—21no—
21#66568—22no—
22C2×C7×Q16—31no—
23#6510—31no—
24Q8×Dic7—31no—
25C2×#1143—31no—
26#6409D311yescharacteristic, complemented
27#6395—31no—
28Dic7⋊C8—31no—
29#4798—71nomaximal, Hall
30C22×D21—21no—
31C3×C7×Q8—21no—
32C3×C7×Q8—21no—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7C2311yescharacteristic
35C2×Dic21D411yescharacteristic
36C2×Dic21D411yescharacteristic
37D84—21no—
38D84—21no—
39C2×C7×Dic3—21no—
40C7×Dic6—21no—
41C7×Dic6D412yes—
42C2×C3×Dic7—21no—
43C3×C7⋊C8—21no—
44C7×C3⋊C8—21no—
45C2×C7×Q8—31no—
46C2×C7×Q8C2×D311yescharacteristic
47C2×D28—31no—
48C7×Q16—62no—
49C2×C7×C8—31no—
50#1143—62no—
51(C2×D7)⋊C4—61no—
52(C2×D7)⋊C4—61no—
53C4⋊Dic7—61no—
54C4⋊Dic7C2×D311yescharacteristic
55C4×Dic7—31no—
56C4×Dic7—61no—
57C2×C7⋊C8C2×D311yescharacteristic
58#947—71no—
59C2×#216D711yescharacteristic, complemented
60C2×#204—71no—
61#887—71no—
62C3×#103—71no—
63Dic3⋊C8—71no—
64#809—71no—
65C22×C3×C7C2×D411yescharacteristic
66C2×D21—41no—
67C2×D21—41no—
68C2×D21—41no—
69C3×C4×C7—41no—
70C3×C4×C7C2×D411yescharacteristic
71C3×C4×C7#4111yescharacteristic
72Dic21—21no—
73Dic21—21no—
74C3×Dic7—41no—
75C7×Dic3—22no—
76#464—211noSylow
77C22×D7—61no—
78C7×Q8—21no—
79C7×Q8—21no—
80C7×Q8—32no—
81C7×Q8—61no—
82C2×C4×C7—21no—
83C2×C4×C7—61no—
84C2×C4×C7C22×D311yescharacteristic
85C2×Dic7—21no—
86C2×Dic7—31no—
87C2×Dic7—31no—
88D28—61no—
89D28—61no—
90C7×C8—61no—
91C7⋊C8—21no—
92C2×C3×Q8C2×D711yescharacteristic
93C2×D12—71no—
94C2×Dic6C2×D711yescharacteristic
95C2×C3×C8—71no—
96C3×C4⋊C4—71no—
97#216—22no—
98(C2×D3)⋊C4—141no—
99(C2×D3)⋊C4—141no—
100Dic3⋊C4—141no—
101C4×Dic3—71no—
102C4×Dic3—141no—
103C2×C3⋊C8C2×D711yescharacteristic
104#204—142no—
105C2×C3×C7#12111yescharacteristic
106C2×C3×C7#13511yescharacteristic
107C2×C3×C7#13711yescharacteristic
108D21—81no—
109C2×Q16—31no—
110C2×SD16—211no—
111#124—211no—
112C4×Q8—211no—
113C4⋊C8—211no—
114#103—71no—
115#102—211no—
116C22×C7D3×D411yescharacteristic
117C2×D7—121no—
118C2×D7—121no—
119C2×D7—121no—
120C4×C7—41no—
121C4×C7—62no—
122C4×C7#23511yescharacteristic
123C4×C7D3×D411yescharacteristic
124Dic7—41no—
125Dic7—61no—
126Dic7—61no—
127C22×D3—141no—
128C3×Q8—21no—
129C3×Q8—21no—
130C2×C3×C4—21no—
131C2×C3×C4—141no—
132C2×C3×C4C22×D711yescharacteristic
133C2×Dic3—21no—
134C2×Dic3—71no—
135C2×Dic3—71no—
136D12—141no—
137D12—141no—
138Dic6—21no—
139Dic6C22⋊D712yes—
140C3×C8—141no—
141C3⋊C8—21no—
142C3×C7#46411yescharacteristic, complemented, Hall
143C2×Q8—31no—
144C2×Q8D3×D711yescharacteristic
145C2×D4—211no—
146Q16—62no—
147SD16—422no—
148C2×C8—31no—
149C2×C8—71no—
150C4⋊C4—71no—
151C4⋊C4—421no—
152C22⋊C4—421no—
153C22⋊C4—421no—
154C42—211no—
155C42—421no—
156C2×C7#88411yescharacteristic
157C2×C7#91611yescharacteristic
158C2×C7#91811yescharacteristic
159D7—241no—
160C22×C3D4×D711yescharacteristic
161C2×D3—281no—
162C2×D3—281no—
163C2×D3—281no—
164C3×C4—41no—
165C3×C4—281no—
166C3×C4#115911yescharacteristic
167C3×C4C2×C22⋊D711yescharacteristic
168Dic3—22no—
169Dic3—141no—
170Dic3—141no—
171C23—421no—
172Q8—21no—
173Q8—21no—
174Q8—32no—
175Q8—61no—
176D4—421no—
177D4—421no—
178C2×C4—21no—
179C2×C4—61no—
180C2×C4—141no—
181C2×C4—211no—
182C2×C4—211no—
183C2×C4C2×D3×D711yescharacteristic
184C8—61no—
185C8—141no—
186C7#479811yescharacteristic, complemented, Sylow
187C2×C3#648111yescharacteristic
188C2×C3#648211yescharacteristic
189C2×C3#650211yescharacteristic
190D3—561no—
191C22—841no—
192C22—841no—
193C22—841no—
194C22#6669411yescharacteristic
195C4—41no—
196C4—62no—
197C4—281no—
198C4—421no—
199C4—421no—
200C4#6668611yescharacteristic
201C4D3×C22⋊D711yescharacteristic
202C3#8995711yescharacteristic, complemented, Sylow
203C2—1681no—
204C2#1062107411yescharacteristic
205C2#1062108311yescharacteristic
206C2#1062112011yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radicalC2×D84C22
hypercenterC2×Q8D3×D7
hypocenterC3×C7#464
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C7×#216C2semidirect
3. #10620906C2semidirect
5. C3×#6409C2semidirect
6. #10620518C2semidirect
26. #6409D3semidirect
59. C2×#216D7semidirect
142. C3×C7#464semidirect
186. C7#4798semidirect
202. C3#89957semidirect

Maximal subgroups

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

subgroupindex
#106214302
C2×C7×#2162
#106209062
C2×#665682
C3×#64092
#106205182
#106205032
#899573
#47987