Finite Group Databasebeta

← #13054791

Subgroups — #13054791

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
2#10621427C211yesmaximal, characteristic, complemented
3C7×#942C211yesmaximal, characteristic, complemented
4C3×#6505C211yesmaximal, characteristic, complemented
5#10620513C211yesmaximal, characteristic, complemented
6#10620511C211yesmaximal, characteristic, complemented
7#10620498C211yesmaximal, characteristic, complemented
8#89977—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11C7×#235C2211yescharacteristic
12C3×#1157C2211yescharacteristic
13#66684—22no—
14#66676—21no—
15D3×Dic14—21no—
16#66669—21no—
17D7×Dic6—21no—
18#66629—22no—
19C4×Dic21C2211yescharacteristic
20C7×#216—21no—
21C7×#215—21no—
22C7×#208C2211yescharacteristic
23C3×#1144—21no—
24C3×#1143—21no—
25C3×#1136C2211yescharacteristic
26#6554—31no—
27C7×#137—31no—
28#6507—31no—
29#6505D311yescharacteristic, complemented
30#6411—31no—
31#6399—31no—
32#6398—31no—
33#5205—71nomaximal, Hall
34C3×C7×Q8—21no—
35C3×C7×Q8—21no—
36C2×C3×C4×C7—21no—
37C2×C3×C4×C7C2311yescharacteristic
38C2×Dic21—21no—
39C2×Dic21D412yes—
40Dic42—21no—
41Dic42—21no—
42C7×#67—21no—
43C7×D12D411yescharacteristic
44C4×D3×C7—21no—
45C7×Dic6D411yescharacteristic
46C3×C22⋊D7—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—
55C7×C3⋊C8—21no—
56C7×#41—31no—
57C2×C7×Q8C2×D311yescharacteristic
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×M16—31no—
68#1144—21no—
69#1143—21no—
70Dic7⋊C4—62no—
71C4×Dic7—31no—
72#1136C2×D311yescharacteristic
73#1010—71no—
74C3×#137—71no—
75#944—71no—
76#942D711yescharacteristic, complemented
77#837—71no—
78#825—71no—
79#824—71no—
80C22×C3×C7C2×D411yescharacteristic
81C2×D3×C7—21no—
82C2×C3×D7—21no—
83C3×C4×C7—41no—
84C3×C4×C7C2×D411yescharacteristic
85C3×C4×C7C2×D411yescharacteristic
86Dic21—22no—
87Dic21—22no—
88C3×Dic7—21no—
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—21no—
100C22⋊D7—62no—
101C2×Dic7—32no—
102C2×Dic7—62no—
103C2×Dic7—61no—
104D28#6711yescharacteristic
105C4×D7—21no—
106C4×D7—62no—
107Dic14—62no—
108Dic14—61no—
109Dic14—61no—
110Dic14#6711yescharacteristic
111C7×C8—61no—
112C7⋊C8—21no—
113C3×#41—71no—
114C2×C3×Q8C2×D711yescharacteristic
115D3×Q8—141no—
116#237—141no—
117#237—142no—
118#235—141no—
119#235C2×D711yescharacteristic
120C2×Dic6—71no—
121C2×Dic6—141no—
122C3×Q16—141no—
123C3×SD16—141no—
124C3×M16—71no—
125#216—21no—
126#215—21no—
127Dic3⋊C4—142no—
128C4×Dic3—71no—
129#208C2×D711yescharacteristic
130C2×C3×C7—21no—
131C2×C3×C7#12011yescharacteristic
132C3×D7—41no—
133D3×C7—41no—
1342−1+4—211no—
135#137—31no—
136#137—71no—
137C4⋊Q8—211no—
138#104—211no—
139#104—211no—
140#101—211no—
141C22×C7—61no—
142C22×C7D3×D411yescharacteristic
143C2×D7—21no—
144C4×C7—41no—
145C4×C7—61no—
146C4×C7D3×D411yescharacteristic
147C4×C7C2×#6711yescharacteristic
148Dic7—21no—
149Dic7—62no—
150Dic7—62no—
151C3×Q8—21no—
152C3×Q8—21no—
153C3×Q8—71no—
154C3×D4—71no—
155C3×D4—141no—
156C2×C3×C4—21no—
157C2×C3×C4—141no—
158C2×C3×C4C22×D711yescharacteristic
159#67—21no—
160#67—142no—
161C2×Dic3—72no—
162C2×Dic3—142no—
163C2×Dic3—141no—
164D12C22⋊D711yescharacteristic
165C4×D3—21no—
166C4×D3—142no—
167Dic6—142no—
168Dic6—141no—
169Dic6—141no—
170Dic6C22⋊D711yescharacteristic
171C3×C8—141no—
172C3⋊C8—21no—
173C3×C7#45611yescharacteristic, complemented, Hall
174#41—31no—
175#41—71no—
176#41—421no—
177#41—421no—
178#41—422no—
179C2×Q8—211no—
180C2×Q8—421no—
181C2×Q8—421no—
182C2×Q8D3×D711yescharacteristic
183Q16—61no—
184Q16—141no—
185SD16—61no—
186SD16—141no—
187M16—31no—
188M16—71no—
189C4⋊C4—422no—
190C42—211no—
191C2×C7—21no—
192C2×C7—121no—
193C2×C7#93711yescharacteristic
194D7—41no—
195C22×C3—141no—
196C22×C3D4×D711yescharacteristic
197C2×D3—21no—
198C3×C4—41no—
199C3×C4—141no—
200C3×C4D4×D711yescharacteristic
201C3×C4C2×C22⋊D711yescharacteristic
202Dic3—21no—
203Dic3—142no—
204Dic3—142no—
205Q8—21no—
206Q8—21no—
207Q8—31no—
208Q8—71no—
209Q8—422no—
210Q8—421no—
211Q8—421no—
212D4—31no—
213D4—61no—
214D4—71no—
215D4—141no—
216D4—422no—
217C2×C4—21no—
218C2×C4—61no—
219C2×C4—141no—
220C2×C4—212no—
221C2×C4—422no—
222C2×C4—422no—
223C2×C4—421no—
224C2×C4C2×D3×D711yescharacteristic
225C8—61no—
226C8—141no—
227C7#520511yescharacteristic, complemented, Sylow
228C2×C3—21no—
229C2×C3—281no—
230C2×C3#650011yescharacteristic
231D3—41no—
232C22—61no—
233C22—141no—
234C22#6669411yescharacteristic
235C4—41no—
236C4—61no—
237C4—141no—
238C4—422no—
239C4—422no—
240C4D7×#6711yescharacteristic
241C4D3×C22⋊D711yescharacteristic
242C3#8997711yescharacteristic, complemented, Sylow
243C2—21no—
244C2—121no—
245C2—281no—
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#10621548C2
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. #10621427C2semidirect
3. C7×#942C2semidirect
4. C3×#6505C2semidirect
5. #10620513C2semidirect
6. #10620511C2semidirect
7. #10620498C2semidirect
29. #6505D3semidirect
76. #942D7semidirect
173. C3×C7#456semidirect
227. C7#5205semidirect
242. C3#89977semidirect

Maximal subgroups

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

subgroupindex
#106215482
#106214272
C7×#9422
C3×#65052
#106205132
#106205112
#106204982
#899773
#52057