Finite Group Databasebeta

← #13053635

Subgroups — #13053635

Order 1344.

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

normal subgroups only

This group has 292 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, complemented
2C7×#908C211yesmaximal, characteristic, complemented
3#10620935C211yesmaximal, characteristic, complemented
4#10620861C211yesmaximal, characteristic, complemented
5#10620663C211yesmaximal, characteristic, complemented
6C3×#6399C211yesmaximal, characteristic, complemented
7#10620514C211yesmaximal, characteristic, complemented
8#89973—31nomaximal, Hall
9C2×Dic42C2211yescharacteristic
10C7×#235C2211yescharacteristic
11C2×C7×D12C2211yescharacteristic
12C3×#1157C2211yescharacteristic
13#66684—22no—
14#66676—21no—
15D3×Dic14—21no—
16#66669—21no—
17D7×Dic6—21no—
18C3×C7×M16C2211yescharacteristic
19#66627C2211yescharacteristic
20C7×D24—21no—
21C7×#204—21no—
22C3×C4×Dic7C2211yescharacteristic
23(C2×D3)⋊Dic7—22no—
24#66564—21no—
25#66560—21no—
26#6554—31no—
27C7×AGL(1,ℤ/8)—31no—
28#6499—31no—
29#6495—31no—
30#6411—31no—
31#6399D311yescharacteristic, complemented
32#6398—31no—
33#4827—71nomaximal, Hall
34C22×D3×C7—21no—
35C2×C3×C4×C7C2311yescharacteristic
36C2×Dic21D412yes—
37Dic42—21no—
38Dic42—21no—
39C7×#67—21no—
40C7×D12—21no—
41C7×D12—21no—
42C7×D12D411yescharacteristic
43C4×D3×C7—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×D4—31no—
58Q8×D7—61no—
59#1159—61no—
60#1159—62no—
61#1157—61no—
62#1157C2×D311yescharacteristic
63C2×Dic14—31no—
64C2×Dic14—61no—
65C7×SD16—61no—
66C7×D8—61no—
67C7×M16C2×D311yescharacteristic
68C22⋊Dic7—62no—
69#1142—61no—
70#1141—61no—
71C4×Dic7C2×D311yescharacteristic
72#1136—31no—
73#1010—71no—
74#949—71no—
75#908D711yescharacteristic, complemented
76#875—71no—
77C3×#104—71no—
78#824—71no—
79#805—71no—
80C22×C3×C7C2×D411yescharacteristic
81C2×D3×C7—21no—
82C2×D3×C7—42no—
83C2×D3×C7—41no—
84C2×C3×D7—21no—
85C3×C4×C7C2×D411yescharacteristic
86C3×C4×C7C2×D411yescharacteristic
87Dic21—22no—
88C3×Dic7—21no—
89C3×Dic7—41no—
90C7×Dic3—21no—
91#454—211noSylow
92C23×C7—61no—
93C7×Q8—31no—
94C7×D4—31no—
95C7×D4—61no—
96C7×D4—61no—
97C7×D4—61no—
98C2×C4×C7—61no—
99C2×C4×C7C22×D311yescharacteristic
100C22⋊D7—21no—
101C22⋊D7—62no—
102C2×Dic7—21no—
103C2×Dic7—32no—
104C2×Dic7—62no—
105D28D1211yescharacteristic
106C4×D7—21no—
107C4×D7—62no—
108Dic14—62no—
109Dic14—61no—
110Dic14—61no—
111Dic14D1211yescharacteristic
112C7×C8—21no—
113C7⋊C8—61no—
114C3×#41—71no—
115D3×Q8—141no—
116#237—141no—
117#237—142no—
118#235—141no—
119#235C2×D711yescharacteristic
120C2×D12C2×D711yescharacteristic
121C2×Dic6—71no—
122C2×Dic6—141no—
123C3×M16C2×D711yescharacteristic
124C3×C42—71no—
125#215—141no—
126#213—141no—
127(C2×D3)⋊C4—142no—
128#208—71no—
129D24—21no—
130#204—21no—
131C2×C3×C7—21no—
132C2×C3×C7#12011yescharacteristic
133C3×D7—41no—
134D3×C7—41no—
135D3×C7—81no—
1362−1+4—211no—
137AGL(1,ℤ/8)—31no—
138AGL(1,ℤ/8)—211no—
139#124—211no—
140#104—71no—
141#104—211no—
142#101—211no—
143C22×C7—61no—
144C22×C7—122no—
145C22×C7—121no—
146C22×C7D3×D411yescharacteristic
147C2×D7—21no—
148C4×C7—61no—
149C4×C7C2×D1211yescharacteristic
150C4×C7D3×D411yescharacteristic
151Dic7—21no—
152Dic7—41no—
153Dic7—62no—
154C22×D3—21no—
155C3×Q8—71no—
156C3×D4—71no—
157C3×D4—141no—
158C2×C3×C4—141no—
159C2×C3×C4—141no—
160C2×C3×C4C22×D711yescharacteristic
161#67—21no—
162#67—142no—
163C2×Dic3—72no—
164C2×Dic3—142no—
165D12—21no—
166D12—21no—
167D12C22⋊D711yescharacteristic
168C4×D3—21no—
169C4×D3—142no—
170Dic6—142no—
171Dic6—141no—
172Dic6—141no—
173Dic6C22⋊D711yescharacteristic
174C3×C8—21no—
175C3⋊C8—141no—
176C3×C7#45411yescharacteristic, complemented, Hall
177#41—31no—
178#41—71no—
179#41—421no—
180#41—421no—
181#41—422no—
182C2×Q8—211no—
183C2×Q8—421no—
184C2×Q8—421no—
185C2×D4—31no—
186SD16—61no—
187SD16—421no—
188D8—61no—
189D8—421no—
190M16—211no—
191M16D3×D711yescharacteristic
192C22⋊C4—422no—
193C42—71no—
194C2×C7—21no—
195C2×C7—121no—
196C2×C7—241no—
197C2×C7#88211yescharacteristic
198D7—41no—
199C22×C3—141no—
200C22×C3D4×D711yescharacteristic
201C2×D3—21no—
202C2×D3—42no—
203C2×D3—41no—
204C3×C4—141no—
205C3×C4—281no—
206C3×C4D4×D711yescharacteristic
207C3×C4C2×C22⋊D711yescharacteristic
208Dic3—21no—
209Dic3—142no—
210C23—61no—
211Q8—31no—
212Q8—71no—
213Q8—422no—
214Q8—421no—
215Q8—421no—
216D4—31no—
217D4—61no—
218D4—61no—
219D4—61no—
220D4—71no—
221D4—141no—
222D4—422no—
223C2×C4—61no—
224C2×C4—141no—
225C2×C4—141no—
226C2×C4—212no—
227C2×C4—422no—
228C2×C4—422no—
229C2×C4C2×D3×D711yescharacteristic
230C8—21no—
231C8—421no—
232C7#482711yescharacteristic, complemented, Sylow
233C2×C3—21no—
234C2×C3—281no—
235C2×C3#650011yescharacteristic
236D3—41no—
237D3—81no—
238C22—61no—
239C22—122no—
240C22—121no—
241C22—141no—
242C22#6668111yescharacteristic
243C4—61no—
244C4—141no—
245C4—281no—
246C4—422no—
247C4D7×D1211yescharacteristic
248C4D3×C22⋊D711yescharacteristic
249C3#8997311yescharacteristic, complemented, Sylow
250C2—21no—
251C2—121no—
252C2—241no—
253C2—281no—
254C2#1062103711yescharacteristic
subgroupG/H
centerC2#10621037
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×M16C22
socleC2×C3×C7#120
hypercenterM16D3×D7
hypocenterC3×C7#454
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
1. #10621548C2semidirect
2. C7×#908C2semidirect
3. #10620935C2semidirect
4. #10620861C2semidirect
5. #10620663C2semidirect
6. C3×#6399C2semidirect
7. #10620514C2semidirect
31. #6399D3semidirect
75. #908D7semidirect
176. C3×C7#454semidirect
232. C7#4827semidirect
249. C3#89973semidirect

Maximal subgroups

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

subgroupindex
#106215482
C7×#9082
#106209352
#106208612
#106206632
C3×#63992
#106205142
#899733
#48277