Finite Group Databasebeta

← #13054793

Subgroups — #13054793

Order 1344.

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

normal subgroups only

This group has 284 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#10621551C211yesmaximal, characteristic
2#10621425C211yesmaximal, characteristic, complemented
3C7×#942C211yesmaximal, characteristic, complemented
4C3×#6507C211yesmaximal, characteristic, complemented
5#10620516C211yesmaximal, characteristic, complemented
6#10620514C211yesmaximal, characteristic, complemented
7#10620496C211yesmaximal, characteristic, complemented
8#89977—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10#66728C2211yescharacteristic
11C7×#235C2211yescharacteristic
12C2×C3×Dic14C2211yescharacteristic
13#66686—22no—
14D21⋊Q8—21no—
15#66673—21no—
16#66672—21no—
17D3×Dic14—21no—
18#66635—21no—
19#66634—21no—
20#66627C2211yescharacteristic
21C7×#216—21no—
22C7×#215—21no—
23C7×#208C2211yescharacteristic
24C3×Dic7⋊C4—22no—
25C3×C4×Dic7C2211yescharacteristic
26#6554—31no—
27C7×#137—31no—
28#6507D311yescharacteristic, complemented
29#6505—31no—
30#6411—31no—
31#6399—31no—
32#6398—31no—
33#5095—71nomaximal, Hall
34C3×C7×Q8—21no—
35C3×C7×Q8—21no—
36C2×C3×C4×C7—21no—
37C2×C3×C4×C7C2311yescharacteristic
38#4268—21no—
39D84D411yescharacteristic
40C4×D21—21no—
41Dic42D411yescharacteristic
42C7×#67—21no—
43C7×D12D411yescharacteristic
44C4×D3×C7—21no—
45C7×Dic6D411yescharacteristic
46C2×C3×Dic7—21no—
47C2×C3×Dic7D412yes—
48C3×Dic14—21no—
49C3×Dic14—21no—
50#4248—22no—
51#4247—22no—
52D21⋊C4—22no—
53D3×Dic7—22no—
54C3⋊(C7⋊C8)—21no—
55C7×C3⋊C8—21no—
56C7×#41—31no—
57C2×C7×Q8C2×D311yescharacteristic
58Q8×D7—61no—
59#1159—61no—
60#1159—62no—
61#1157—31no—
62#1157—61no—
63C2×Dic14—61no—
64C2×Dic14C2×D311yescharacteristic
65C7×Q16—61no—
66C7×SD16—61no—
67C7×M16—31no—
68#1144—61no—
69#1143—61no—
70Dic7⋊C4—22no—
71C4×Dic7C2×D311yescharacteristic
72#1136—31no—
73#1007—71no—
74C3×C4⋊Q8—71no—
75#942—71no—
76#942D711yescharacteristic, complemented
77#836—71no—
78#805—71no—
79#805—71no—
80C22×C3×C7C2×D411yescharacteristic
81C2×D21—21no—
82C2×D3×C7—21no—
83C3×C4×C7—41no—
84C3×C4×C7C2×D411yescharacteristic
85C3×C4×C7C2×D411yescharacteristic
86Dic21—21no—
87C3×Dic7—22no—
88C3×Dic7—22no—
89C7×Dic3—21no—
90#456—211noSylow
91C7×Q8—21no—
92C7×Q8—21no—
93C7×Q8—31no—
94C7×D4—31no—
95C7×D4—61no—
96C2×C4×C7—21no—
97C2×C4×C7—61no—
98C2×C4×C7C22×D311yescharacteristic
99C22⋊D7—62no—
100C22⋊D7—61no—
101C2×Dic7—21no—
102C2×Dic7—62no—
103C2×Dic7#6712yes—
104D28—31no—
105C4×D7—62no—
106C4×D7—61no—
107Dic14—21no—
108Dic14—21no—
109Dic14—31no—
110Dic14—62no—
111C7×C8—61no—
112C7⋊C8—61no—
113C2×C3×Q8—71no—
114C2×C3×Q8C2×D711yescharacteristic
115#239—141no—
116#239—141no—
117D3×Q8—141no—
118D3×Q8—141no—
119#235—71no—
120#235—142no—
121#235C2×D711yescharacteristic
122C3×C4⋊C4—142no—
123C3×C42—71no—
124#216—21no—
125#216—141no—
126#215—21no—
127#215—141no—
128#208—71no—
129#208C2×D711yescharacteristic
130C2×C3×C7—21no—
131C2×C3×C7#12011yescharacteristic
132D21—41no—
133D3×C7—41no—
1342−1+4—211no—
135#137—31no—
136#137—211no—
137C4⋊Q8—71no—
138#104—211no—
139#104—211no—
140#101—211no—
141C22×C7—61no—
142C22×C7C2×#6711yescharacteristic
143C2×D7—61no—
144C4×C7—41no—
145C4×C7—61no—
146C4×C7D3×D411yescharacteristic
147C4×C7D3×D411yescharacteristic
148Dic7—22no—
149Dic7—22no—
150Dic7—61no—
151C3×Q8—21no—
152C3×Q8—21no—
153C3×Q8—141no—
154C3×Q8—141no—
155C2×C3×C4—21no—
156C2×C3×C4—72no—
157C2×C3×C4—141no—
158C2×C3×C4C22×D711yescharacteristic
159#67—21no—
160#67—141no—
161D12—71no—
162D12—142no—
163D12C22⋊D711yescharacteristic
164C4×D3—21no—
165C4×D3—142no—
166C4×D3—142no—
167C4×D3—141no—
168Dic6—71no—
169Dic6—142no—
170Dic6C22⋊D711yescharacteristic
171C3⋊C8—21no—
172C3⋊C8—141no—
173C3×C7#45611yescharacteristic, complemented, Hall
174#41—31no—
175#41—211no—
176#41—421no—
177#41—422no—
178#41—421no—
179C2×Q8—71no—
180C2×Q8—421no—
181C2×Q8—421no—
182C2×Q8D3×D711yescharacteristic
183Q16—61no—
184Q16—421no—
185SD16—61no—
186SD16—421no—
187M16—31no—
188M16—211no—
189C4⋊C4—142no—
190C42—71no—
191C2×C7—21no—
192C2×C7—121no—
193C2×C7#93711yescharacteristic
194D7—121no—
195C22×C3D4×D711yescharacteristic
196C2×D3—21no—
197C2×D3—141no—
198C3×C4—41no—
199C3×C4—142no—
200C3×C4—142no—
201C3×C4D4×D711yescharacteristic
202C3×C4C2×C22⋊D711yescharacteristic
203Dic3—21no—
204Dic3—141no—
205Q8—21no—
206Q8—21no—
207Q8—31no—
208Q8—141no—
209Q8—141no—
210Q8—211no—
211Q8—422no—
212D4—31no—
213D4—61no—
214D4—211no—
215D4—422no—
216D4—421no—
217C2×C4—21no—
218C2×C4—61no—
219C2×C4—72no—
220C2×C4—141no—
221C2×C4—422no—
222C2×C4—422no—
223C2×C4—421no—
224C2×C4C2×D3×D711yescharacteristic
225C8—61no—
226C8—421no—
227C7#509511yescharacteristic, complemented, Sylow
228C2×C3—21no—
229C2×C3#650011yescharacteristic
230D3—41no—
231D3—281no—
232C22—61no—
233C22—421no—
234C22D7×#6711yescharacteristic
235C4—41no—
236C4—61no—
237C4—142no—
238C4—142no—
239C4—421no—
240C4D3×C22⋊D711yescharacteristic
241C4#6669411yescharacteristic
242C3#8997711yescharacteristic, complemented, Sylow
243C2—21no—
244C2—121no—
245C2—841no—
246C2#1062113911yescharacteristic
subgroupG/H
centerC2#10621139
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC2×C3×C7#120
prime radical#10621551C2
hypercenterC2×Q8D3×D7
hypocenterC3×C7#456
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #10621425C2semidirect
3. C7×#942C2semidirect
4. C3×#6507C2semidirect
5. #10620516C2semidirect
6. #10620514C2semidirect
7. #10620496C2semidirect
28. #6507D3semidirect
76. #942D7semidirect
173. C3×C7#456semidirect
227. C7#5095semidirect
242. C3#89977semidirect

Maximal subgroups

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

subgroupindex
#106215512
#106214252
C7×#9422
C3×#65072
#106205162
#106205142
#106204962
#899773
#50957