Finite Group Databasebeta

← #13053641

Subgroups — #13053641

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#10621392C211yesmaximal, characteristic, complemented
3#10620951C211yesmaximal, characteristic, complemented
4#10620884C211yesmaximal, characteristic, complemented
5C7×#825C211yesmaximal, characteristic, complemented
6C3×#6398C211yesmaximal, characteristic, complemented
7#10620515C211yesmaximal, characteristic, complemented
8#89598—31nomaximal, Hall
9#66728C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11C7×#235C2211yescharacteristic
12C2×C3×Dic14C2211yescharacteristic
13#66686—22no—
14D21⋊Q8—21no—
15#66673—21no—
16#66672—21no—
17D3×Dic14—21no—
18C3×C7×M16C2211yescharacteristic
19Dic84—21no—
20#66623—21no—
21C4×C7×Dic3C2211yescharacteristic
22C3×#1136C2211yescharacteristic
23Dic3⋊Dic7—22no—
24#66571—21no—
25#66567—21no—
26#6554—31no—
27#6514—31no—
28#6472—31no—
29#6432—31no—
30C7×#104—31no—
31#6398D311yescharacteristic, complemented
32#6379—31no—
33#4752—71nomaximal, Hall
34C2×C3×C4×C7C2311yescharacteristic
35#4268—21no—
36C2×Dic21—21no—
37D84D411yescharacteristic
38C4×D21—21no—
39Dic42—21no—
40Dic42—21no—
41Dic42D411yescharacteristic
42C7×#67—21no—
43C2×C7×Dic3—21no—
44C7×D12D411yescharacteristic
45C4×D3×C7—21no—
46C7×Dic6D411yescharacteristic
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—
55C3×C7⋊C8—21no—
56C7×#41—31no—
57Q8×D7—61no—
58#1159—61no—
59#1159—62no—
60#1157—31no—
61#1157—61no—
62C2×Dic14—31no—
63C2×Dic14—61no—
64C2×Dic14C2×D311yescharacteristic
65C7×M16C2×D311yescharacteristic
66C42×C7—31no—
67#1144—61no—
68#1142—61no—
69C4⋊Dic7—62no—
70#1136C2×D311yescharacteristic
71Dic28—61no—
72#1132—61no—
73#1007—71no—
74#944—71no—
75#909—71no—
76#909—71no—
77C3×#101—71no—
78#825—71no—
79#825D711yescharacteristic, complemented
80C22×C3×C7C2×D411yescharacteristic
81C2×D21—21no—
82C2×D3×C7—21no—
83C3×C4×C7C2×D411yescharacteristic
84C3×C4×C7C2×D411yescharacteristic
85Dic21—21no—
86Dic21—41no—
87C3×Dic7—22no—
88C7×Dic3—22no—
89C7×Dic3—21no—
90#456—211noSylow
91C7×Q8—31no—
92C7×D4—31no—
93C7×D4—61no—
94C2×C4×C7—61no—
95C2×C4×C7—61no—
96C2×C4×C7C22×D311yescharacteristic
97C22⋊D7—62no—
98C22⋊D7—61no—
99C2×Dic7—62no—
100C2×Dic7—61no—
101C2×Dic7D1212yes—
102D28—31no—
103C4×D7—62no—
104C4×D7—61no—
105Dic14—21no—
106Dic14—21no—
107Dic14—31no—
108Dic14—62no—
109Dic14—61no—
110Dic14—61no—
111C7×C8—21no—
112C7⋊C8—21no—
113C2×C3×Q8—71no—
114#239—141no—
115#239—141no—
116D3×Q8—141no—
117D3×Q8—141no—
118#235—71no—
119#235—142no—
120#235C2×D711yescharacteristic
121C2×Dic6—71no—
122C3×M16—71no—
123C3×M16C2×D711yescharacteristic
124Dic3⋊C4—142no—
125C4×Dic3C2×D711yescharacteristic
126Dic12—141no—
127Dic12—141no—
128#204—141no—
129#204—141no—
130C2×C3×C7—21no—
131C2×C3×C7#12011yescharacteristic
132D21—41no—
133D3×C7—41no—
1342−1+4—211no—
135#137—211no—
136#137—211no—
137C4⋊Q8—211no—
138#104—31no—
139#104—211no—
140#101—71no—
141C22×C7—61no—
142C22×C7C2×D1211yescharacteristic
143C2×D7—61no—
144C4×C7—62no—
145C4×C7—61no—
146C4×C7D3×D411yescharacteristic
147C4×C7D3×D411yescharacteristic
148Dic7—22no—
149Dic7—61no—
150Dic7—121no—
151C3×Q8—141no—
152C3×Q8—141no—
153C2×C3×C4—72no—
154C2×C3×C4C22×D711yescharacteristic
155#67—21no—
156#67—141no—
157C2×Dic3—21no—
158C2×Dic3—141no—
159D12—71no—
160D12—142no—
161D12D2811yescharacteristic
162C4×D3—21no—
163C4×D3—142no—
164C4×D3—142no—
165C4×D3—141no—
166Dic6—71no—
167Dic6—142no—
168Dic6—141no—
169Dic6—141no—
170Dic6D2811yescharacteristic
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—211no—
181C2×Q8—421no—
182C2×Q8—421no—
183Q16—421no—
184Q16—421no—
185SD16—421no—
186SD16—421no—
187M16—71no—
188M16D3×D711yescharacteristic
189C4⋊C4—422no—
190C42—31no—
191C2×C7—21no—
192C2×C7—121no—
193C2×C7#88211yescharacteristic
194D7—121no—
195C22×C3D4×D711yescharacteristic
196C2×D3—21no—
197C2×D3—141no—
198C3×C4—142no—
199C3×C4C2×D2811yescharacteristic
200C3×C4D4×D711yescharacteristic
201Dic3—22no—
202Dic3—21no—
203Dic3—141no—
204Dic3—281no—
205Q8—31no—
206Q8—141no—
207Q8—141no—
208Q8—211no—
209Q8—422no—
210Q8—421no—
211Q8—421no—
212D4—31no—
213D4—61no—
214D4—211no—
215D4—422no—
216D4—421no—
217C2×C4—61no—
218C2×C4—61no—
219C2×C4—72no—
220C2×C4—422no—
221C2×C4—422no—
222C2×C4—421no—
223C2×C4—421no—
224C2×C4C2×D3×D711yescharacteristic
225C8—21no—
226C8—141no—
227C7#475211yescharacteristic, complemented, Sylow
228C2×C3—21no—
229C2×C3#644511yescharacteristic
230D3—41no—
231D3—281no—
232C22—61no—
233C22—421no—
234C22D7×D1211yescharacteristic
235C4—62no—
236C4—61no—
237C4—142no—
238C4—421no—
239C4—841no—
240C4D3×D2811yescharacteristic
241C4#6669411yescharacteristic
242C3#8959811yescharacteristic, complemented, Sylow
243C2—21no—
244C2—121no—
245C2—841no—
246C2#1062103811yescharacteristic
subgroupG/H
centerC2#10621038
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×M16C22
socleC2×C3×C7#120
prime radical#10621551C2
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. #10621392C2semidirect
3. #10620951C2semidirect
4. #10620884C2semidirect
5. C7×#825C2semidirect
6. C3×#6398C2semidirect
7. #10620515C2semidirect
31. #6398D3semidirect
79. #825D7semidirect
173. C3×C7#456semidirect
227. C7#4752semidirect
242. C3#89598semidirect

Maximal subgroups

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

subgroupindex
#106215512
#106213922
#106209512
#106208842
C7×#8252
C3×#63982
#106205152
#895983
#47527