Finite Group Databasebeta

← #13053636

Subgroups — #13053636

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#10621548C211yesmaximal, characteristic
2C7×#909C211yesmaximal, characteristic, complemented
3#10620938C211yesmaximal, characteristic, complemented
4#10620869C211yesmaximal, characteristic, complemented
5#10620663C211yesmaximal, characteristic, complemented
6C3×#6399C211yesmaximal, characteristic, complemented
7#10620514C211yesmaximal, characteristic, complemented
8#89977—31nomaximal, Hall
9C2×Dic42C2211yescharacteristic
10C7×#235C2211yescharacteristic
11C2×C7×Dic6C2211yescharacteristic
12C3×#1157C2211yescharacteristic
13#66684—22no—
14#66676—21no—
15D3×Dic14—21no—
16#66669—21no—
17D7×Dic6—21no—
18C3×C7×M16C2211yescharacteristic
19#66627C2211yescharacteristic
20C7×Dic12—21no—
21C7×#204—21no—
22C3×C4×Dic7C2211yescharacteristic
23Dic7⋊Dic3—22no—
24#66569—21no—
25#66563—21no—
26#6554—31no—
27C7×#137—31no—
28#6507—31no—
29#6505—31no—
30#6411—31no—
31#6399D311yescharacteristic, complemented
32#6398—31no—
33#4826—71nomaximal, Hall
34C2×C3×C4×C7C2311yescharacteristic
35C2×Dic21D412yes—
36Dic42—21no—
37Dic42—21no—
38C7×#67—21no—
39C2×C7×Dic3—21no—
40C7×D12D411yescharacteristic
41C4×D3×C7—21no—
42C7×Dic6—21no—
43C7×Dic6—21no—
44C7×Dic6D411yescharacteristic
45C3×C22⋊D7—21no—
46C2×C3×Dic7—21no—
47C3×D28D411yescharacteristic
48C3×C4×D7—21no—
49C3×Dic14D411yescharacteristic
50#4248—22no—
51#4245—22no—
52D3×Dic7—22no—
53Dic3×D7—22no—
54C3×C7×C8—21no—
55C3⋊(C7⋊C8)—21no—
56C7×#41—31no—
57C2×C7×Q8—31no—
58Q8×D7—61no—
59#1159—61no—
60#1159—62no—
61#1157—61no—
62#1157C2×D311yescharacteristic
63C2×Dic14—31no—
64C2×Dic14—61no—
65C7×Q16—61no—
66C7×SD16—61no—
67C7×M16C2×D311yescharacteristic
68#1144—61no—
69#1143—61no—
70Dic7⋊C4—62no—
71C4×Dic7C2×D311yescharacteristic
72#1136—31no—
73#1010—71no—
74#951—71no—
75#909D711yescharacteristic, complemented
76#869—71no—
77C3×#104—71no—
78#824—71no—
79#805—71no—
80C22×C3×C7C2×D411yescharacteristic
81C2×D3×C7—21no—
82C2×C3×D7—21no—
83C3×C4×C7C2×D411yescharacteristic
84C3×C4×C7C2×D411yescharacteristic
85Dic21—22no—
86C3×Dic7—22no—
87C3×Dic7—21no—
88C7×Dic3—21no—
89C7×Dic3—41no—
90#456—211noSylow
91C7×Q8—31no—
92C7×Q8—61no—
93C7×Q8—61no—
94C7×D4—31no—
95C7×D4—61no—
96C2×C4×C7—61no—
97C2×C4×C7—61no—
98C2×C4×C7C22×D311yescharacteristic
99C22⋊D7—21no—
100C22⋊D7—62no—
101C2×Dic7—21no—
102C2×Dic7—32no—
103C2×Dic7—62no—
104D28D1211yescharacteristic
105C4×D7—21no—
106C4×D7—62no—
107Dic14—62no—
108Dic14—61no—
109Dic14—61no—
110Dic14D1211yescharacteristic
111C7×C8—21no—
112C7⋊C8—61no—
113C3×#41—71no—
114D3×Q8—141no—
115#237—141no—
116#237—142no—
117#235—141no—
118#235C2×D711yescharacteristic
119C2×Dic6—71no—
120C2×Dic6—141no—
121C2×Dic6C2×D711yescharacteristic
122C3×M16C2×D711yescharacteristic
123C3×C42—71no—
124#216—141no—
125#214—141no—
126C4⋊Dic3—142no—
127#208—71no—
128Dic12—21no—
129#204—21no—
130C2×C3×C7—21no—
131C2×C3×C7#12011yescharacteristic
132C3×D7—41no—
133D3×C7—41no—
1342−1+4—211no—
135#137—31no—
136#137—211no—
137C4⋊Q8—211no—
138#104—71no—
139#104—211no—
140#101—211no—
141C22×C7—61no—
142C22×C7D3×D411yescharacteristic
143C2×D7—21no—
144C4×C7—61no—
145C4×C7—121no—
146C4×C7C2×D1211yescharacteristic
147C4×C7D3×D411yescharacteristic
148Dic7—22no—
149Dic7—21no—
150Dic7—62no—
151C3×Q8—71no—
152C3×D4—71no—
153C3×D4—141no—
154C2×C3×C4—141no—
155C2×C3×C4—141no—
156C2×C3×C4C22×D711yescharacteristic
157#67—21no—
158#67—142no—
159C2×Dic3—21no—
160C2×Dic3—72no—
161C2×Dic3—142no—
162D12C22⋊D711yescharacteristic
163C4×D3—21no—
164C4×D3—142no—
165Dic6—21no—
166Dic6—21no—
167Dic6—142no—
168Dic6—141no—
169Dic6—141no—
170Dic6C22⋊D711yescharacteristic
171C3×C8—21no—
172C3⋊C8—141no—
173C3×C7#45611yescharacteristic, complemented, Hall
174#41—31no—
175#41—71no—
176#41—421no—
177#41—421no—
178#41—422no—
179C2×Q8—31no—
180C2×Q8—211no—
181C2×Q8—421no—
182C2×Q8—421no—
183Q16—61no—
184Q16—421no—
185SD16—61no—
186SD16—421no—
187M16—211no—
188M16D3×D711yescharacteristic
189C4⋊C4—422no—
190C42—71no—
191C2×C7—21no—
192C2×C7—121no—
193C2×C7#88211yescharacteristic
194D7—41no—
195C22×C3—141no—
196C22×C3D4×D711yescharacteristic
197C2×D3—21no—
198C3×C4—142no—
199C3×C4—141no—
200C3×C4D4×D711yescharacteristic
201C3×C4C2×C22⋊D711yescharacteristic
202Dic3—21no—
203Dic3—41no—
204Dic3—142no—
205Q8—31no—
206Q8—61no—
207Q8—61no—
208Q8—71no—
209Q8—422no—
210Q8—421no—
211Q8—421no—
212D4—31no—
213D4—61no—
214D4—71no—
215D4—141no—
216D4—422no—
217C2×C4—61no—
218C2×C4—61no—
219C2×C4—141no—
220C2×C4—141no—
221C2×C4—212no—
222C2×C4—422no—
223C2×C4—422no—
224C2×C4C2×D3×D711yescharacteristic
225C8—21no—
226C8—421no—
227C7#482611yescharacteristic, complemented, Sylow
228C2×C3—21no—
229C2×C3—281no—
230C2×C3#650011yescharacteristic
231D3—41no—
232C22—61no—
233C22—141no—
234C22#6668111yescharacteristic
235C4—61no—
236C4—121no—
237C4—142no—
238C4—141no—
239C4—422no—
240C4D7×D1211yescharacteristic
241C4D3×C22⋊D711yescharacteristic
242C3#8997711yescharacteristic, complemented, Sylow
243C2—21no—
244C2—121no—
245C2—281no—
246C2#1062103711yescharacteristic
subgroupG/H
centerC2#10621037
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×M16C22
socleC2×C3×C7#120
prime radical#10621548C2
hypercenterM16D3×D7
hypocenterC3×C7#456
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. C7×#909C2semidirect
3. #10620938C2semidirect
4. #10620869C2semidirect
5. #10620663C2semidirect
6. C3×#6399C2semidirect
7. #10620514C2semidirect
31. #6399D3semidirect
75. #909D7semidirect
173. C3×C7#456semidirect
227. C7#4826semidirect
242. C3#89977semidirect

Maximal subgroups

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

subgroupindex
#106215482
C7×#9092
#106209382
#106208692
#106206632
C3×#63992
#106205142
#899773
#48267