Finite Group Databasebeta

← #13053633

Subgroups — #13053633

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
2C3×#6471C211yesmaximal, characteristic, complemented
3#10620926C211yesmaximal, characteristic, complemented
4#10620860C211yesmaximal, characteristic, complemented
5#10620663C211yesmaximal, characteristic, complemented
6C7×#825C211yesmaximal, characteristic, complemented
7#10620512C211yesmaximal, characteristic, complemented
8#89599—31nomaximal, Hall
9C2×Dic42C2211yescharacteristic
10C7×#235C2211yescharacteristic
11C3×#1157C2211yescharacteristic
12C2×C3×D28C2211yescharacteristic
13#66684—22no—
14#66676—21no—
15D3×Dic14—21no—
16#66669—21no—
17D7×Dic6—21no—
18C3×C7×M16C2211yescharacteristic
19#66627C2211yescharacteristic
20C4×C7×Dic3C2211yescharacteristic
21C3×D56—21no—
22C3×#1132—21no—
23(C2×D7)⋊Dic3—22no—
24#66563—21no—
25#66560—21no—
26#6554—31no—
27#6512—31no—
28#6471D311yescharacteristic, complemented
29#6438—31no—
30C7×#104—31no—
31#6398—31no—
32#6379—31no—
33#5201—71nomaximal, Hall
34C22×C3×D7—21no—
35C2×C3×C4×C7C2311yescharacteristic
36C2×Dic21D412yes—
37Dic42—21no—
38Dic42—21no—
39C7×#67—21no—
40C2×C7×Dic3—21no—
41C7×D12D411yescharacteristic
42C4×D3×C7—21no—
43C7×Dic6D411yescharacteristic
44C3×C22⋊D7—21no—
45C3×D28—21no—
46C3×D28—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—
57Q8×D7—61no—
58#1159—61no—
59#1159—62no—
60#1157—61no—
61#1157C2×D311yescharacteristic
62C2×D28C2×D311yescharacteristic
63C2×Dic14—31no—
64C2×Dic14—61no—
65C7×M16C2×D311yescharacteristic
66C42×C7—31no—
67#1143—61no—
68#1141—61no—
69(C2×D7)⋊C4—62no—
70#1136—31no—
71D56—21no—
72#1132—21no—
73#1010—71no—
74C3×AGL(1,ℤ/8)—71no—
75#936—71no—
76#932—71no—
77#837—71no—
78#825D711yescharacteristic, complemented
79#824—71no—
80C22×C3×C7C2×D411yescharacteristic
81C2×D3×C7—21no—
82C2×C3×D7—21no—
83C2×C3×D7—42no—
84C2×C3×D7—41no—
85C3×C4×C7C2×D411yescharacteristic
86C3×C4×C7C2×D411yescharacteristic
87Dic21—22no—
88C3×Dic7—21no—
89C7×Dic3—21no—
90C7×Dic3—41no—
91#454—211noSylow
92C22×D7—21no—
93C7×Q8—31no—
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—32no—
102C2×Dic7—62no—
103D28—21no—
104D28—21no—
105D28#6711yescharacteristic
106C4×D7—21no—
107C4×D7—62no—
108Dic14—62no—
109Dic14—61no—
110Dic14—61no—
111Dic14#6711yescharacteristic
112C7×C8—21no—
113C7⋊C8—61no—
114C3×#41—71no—
115C2×C3×D4—71no—
116D3×Q8—141no—
117#237—141no—
118#237—142no—
119#235—141no—
120#235C2×D711yescharacteristic
121C2×Dic6—71no—
122C2×Dic6—141no—
123C3×SD16—141no—
124C3×D8—141no—
125C3×M16C2×D711yescharacteristic
126#217—142no—
127#214—141no—
128#213—141no—
129C4×Dic3C2×D711yescharacteristic
130#208—71no—
131C2×C3×C7—21no—
132C2×C3×C7#12011yescharacteristic
133C3×D7—41no—
134C3×D7—81no—
135D3×C7—41no—
1362−1+4—211no—
137AGL(1,ℤ/8)—71no—
138AGL(1,ℤ/8)—211no—
139#124—211no—
140#104—31no—
141#104—211no—
142#101—211no—
143C22×C7—61no—
144C22×C7D3×D411yescharacteristic
145C2×D7—21no—
146C2×D7—42no—
147C2×D7—41no—
148C4×C7—61no—
149C4×C7—121no—
150C4×C7D3×D411yescharacteristic
151C4×C7C2×#6711yescharacteristic
152Dic7—21no—
153Dic7—62no—
154C23×C3—141no—
155C3×Q8—71no—
156C3×D4—71no—
157C3×D4—141no—
158C3×D4—141no—
159C3×D4—141no—
160C2×C3×C4—141no—
161C2×C3×C4C22×D711yescharacteristic
162#67—21no—
163#67—142no—
164C2×Dic3—21no—
165C2×Dic3—72no—
166C2×Dic3—142no—
167D12D2811yescharacteristic
168C4×D3—21no—
169C4×D3—142no—
170Dic6—142no—
171Dic6—141no—
172Dic6—141no—
173Dic6D2811yescharacteristic
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—71no—
186SD16—141no—
187SD16—421no—
188D8—141no—
189D8—421no—
190M16—211no—
191M16D3×D711yescharacteristic
192C22⋊C4—422no—
193C42—31no—
194C2×C7—21no—
195C2×C7—121no—
196C2×C7#93711yescharacteristic
197D7—41no—
198D7—81no—
199C22×C3—141no—
200C22×C3—282no—
201C22×C3—281no—
202C22×C3D4×D711yescharacteristic
203C2×D3—21no—
204C3×C4—141no—
205C3×C4C2×D2811yescharacteristic
206C3×C4D4×D711yescharacteristic
207Dic3—21no—
208Dic3—41no—
209Dic3—142no—
210C23—141no—
211Q8—31no—
212Q8—71no—
213Q8—422no—
214Q8—421no—
215Q8—421no—
216D4—31no—
217D4—61no—
218D4—71no—
219D4—141no—
220D4—141no—
221D4—141no—
222D4—422no—
223C2×C4—61no—
224C2×C4—61no—
225C2×C4—141no—
226C2×C4—212no—
227C2×C4—422no—
228C2×C4—422no—
229C2×C4C2×D3×D711yescharacteristic
230C8—21no—
231C8—421no—
232C7#520111yescharacteristic, complemented, Sylow
233C2×C3—21no—
234C2×C3—281no—
235C2×C3—561no—
236C2×C3#644511yescharacteristic
237D3—41no—
238C22—61no—
239C22—141no—
240C22—282no—
241C22—281no—
242C22#6668111yescharacteristic
243C4—61no—
244C4—121no—
245C4—141no—
246C4—422no—
247C4D3×D2811yescharacteristic
248C4D7×#6711yescharacteristic
249C3#8959911yescharacteristic, complemented, Sylow
250C2—21no—
251C2—121no—
252C2—281no—
253C2—561no—
254C2#1062103611yescharacteristic
subgroupG/H
centerC2#10621036
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. C3×#6471C2semidirect
3. #10620926C2semidirect
4. #10620860C2semidirect
5. #10620663C2semidirect
6. C7×#825C2semidirect
7. #10620512C2semidirect
28. #6471D3semidirect
78. #825D7semidirect
176. C3×C7#454semidirect
232. C7#5201semidirect
249. C3#89599semidirect

Maximal subgroups

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

subgroupindex
#106215482
C3×#64712
#106209262
#106208602
#106206632
C7×#8252
#106205122
#895993
#52017