Finite Group Databasebeta

← #13053625

Subgroups — #13053625

Order 1344.

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

normal subgroups only

This group has 336 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#10621566C211yesmaximal, characteristic, complemented
2C7×#908C211yesmaximal, characteristic, complemented
3#10621013C211yesmaximal, characteristic, complemented
4#10620861C211yesmaximal, characteristic, complemented
5#10620662C211yesmaximal, characteristic, complemented
6C3×#6399C211yesmaximal, characteristic, complemented
7#10620514C211yesmaximal, characteristic, complemented
8#89970—31nomaximal, Hall
9C2×D84C2211yescharacteristic
10C7×#235C2211yescharacteristic
11C2×C7×D12C2211yescharacteristic, complemented
12C3×#1157C2211yescharacteristic
13#66694—22no—
14C2×#4247—22no—
15D3×D28—21no—
16D7×D12—21no—
17#66677—21no—
18#66673—21no—
19C3×C7×M16C2211yescharacteristic, complemented
20#66627C2211yescharacteristic, complemented
21C7×D24—21no—
22C7×#204—21no—
23C3×C4×Dic7C2211yescharacteristic, complemented
24#66564—21no—
25#66560—21no—
26#6553—31no—
27C7×AGL(1,ℤ/8)—31no—
28#6503—31no—
29#6495—31no—
30#6411—31no—
31#6399D311yescharacteristic, complemented
32#6397—31no—
33#4822—71nomaximal, Hall
34C22×D21D412yes—
35C22×D3×C7—21no—
36C2×D3×D7—22no—
37C2×C3×C4×C7C2311yescharacteristic
38D84—21no—
39D84—21no—
40C7×#67—21no—
41C7×D12—21no—
42C7×D12—21no—
43C7×D12D411yescharacteristic, complemented
44C4×D3×C7—21no—
45C7×Dic6D411yescharacteristic, complemented
46C3×C22⋊D7—21no—
47C2×C3×Dic7—21no—
48C3×D28D411yescharacteristic, complemented
49C3×C4×D7—21no—
50C3×Dic14D411yescharacteristic, complemented
51#4247—22no—
52#4247—44no—
53#4246—22no—
54D21⋊C4—22no—
55C3×C7×C8—21no—
56C3⋊(C7⋊C8)—21no—
57C7×#41—31no—
58C2×C7×D4—31no—
59C2×C22⋊D7—62no—
60#1161—61no—
61D4×D7—61no—
62D4×D7—62no—
63#1157—61no—
64#1157C2×D311yescharacteristic
65C2×D28—31no—
66C2×D28—61no—
67C7×SD16—61no—
68C7×D8—61no—
69C7×M16C2×D311yescharacteristic, complemented
70#1142—61no—
71#1141—61no—
72C4×Dic7C2×D311yescharacteristic, complemented
73#1136—31no—
74#1009—71no—
75#949—71no—
76#908D711yescharacteristic, complemented
77#874—71no—
78C3×#104—71no—
79#823—71no—
80#805—71no—
81C22×C3×C7C2×D411yescharacteristic
82C2×D21—22no—
83C2×D21—42no—
84C2×D3×C7—21no—
85C2×D3×C7—42no—
86C2×D3×C7—41no—
87C2×C3×D7—21no—
88D3×D7—82no—
89C3×C4×C7C2×D411yescharacteristic
90C3×C4×C7C2×D411yescharacteristic
91C3×Dic7—22no—
92C3×Dic7—21no—
93C7×Dic3—21no—
94#453—211noSylow
95C23×C7—61no—
96C22×D7—32no—
97C22×D7—62no—
98C7×Q8—31no—
99C7×D4—31no—
100C7×D4—61no—
101C7×D4—61no—
102C7×D4—61no—
103C2×C4×C7—61no—
104C2×C4×C7C22×D311yescharacteristic
105C22⋊D7—21no—
106C22⋊D7—62no—
107C22⋊D7—124no—
108C2×Dic7—21no—
109D28—62no—
110D28—61no—
111D28—61no—
112D28D1211yescharacteristic, complemented
113C4×D7—21no—
114C4×D7—62no—
115Dic14D1211yescharacteristic, complemented
116C7×C8—21no—
117C7⋊C8—61no—
118C3×#41—71no—
119#239—141no—
120D3×D4—141no—
121D3×D4—142no—
122#235—141no—
123#235C2×D711yescharacteristic
124C2×D12—71no—
125C2×D12—141no—
126C2×D12—142no—
127C2×D12C2×D711yescharacteristic, complemented
128C3×M16C2×D711yescharacteristic, complemented
129C3×C42—71no—
130#215—141no—
131#213—141no—
132#208—71no—
133D24—21no—
134#204—21no—
135C2×C3×C7—21no—
136C2×C3×C7#12011yescharacteristic
137D21—42no—
138C3×D7—41no—
139D3×C7—41no—
140D3×C7—81no—
141H(5,𝔽2)—211no—
142AGL(1,ℤ/8)—31no—
143AGL(1,ℤ/8)—211no—
144#127—211no—
145#104—71no—
146#104—211no—
147#100—211no—
148C22×C7—61no—
149C22×C7—122no—
150C22×C7—121no—
151C22×C7D3×D411yescharacteristic
152C2×D7—21no—
153C2×D7—62no—
154C2×D7—122no—
155C2×D7—242no—
156C4×C7—61no—
157C4×C7C2×D1211yescharacteristic
158C4×C7D3×D411yescharacteristic
159Dic7—22no—
160Dic7—21no—
161C22×D3—21no—
162C22×D3—72no—
163C22×D3—142no—
164C3×Q8—71no—
165C3×D4—71no—
166C3×D4—141no—
167C2×C3×C4—141no—
168C2×C3×C4—141no—
169C2×C3×C4C22×D711yescharacteristic
170#67—21no—
171#67—142no—
172D12—21no—
173D12—21no—
174D12—142no—
175D12—141no—
176D12—141no—
177D12—284no—
178D12C22⋊D711yescharacteristic, complemented
179C4×D3—21no—
180C4×D3—142no—
181Dic6C22⋊D711yescharacteristic, complemented
182C3×C8—21no—
183C3⋊C8—141no—
184C3×C7#45311yescharacteristic, complemented, Hall
185#41—31no—
186#41—71no—
187#41—421no—
188#41—421no—
189C2×D4—31no—
190C2×D4—211no—
191C2×D4—421no—
192C2×D4—421no—
193C2×D4—422no—
194C2×D4—422no—
195SD16—61no—
196SD16—421no—
197D8—61no—
198D8—421no—
199M16—211no—
200M16D3×D711yescharacteristic, complemented
201C42—71no—
202C2×C7—21no—
203C2×C7—121no—
204C2×C7—241no—
205C2×C7#88211yescharacteristic
206D7—41no—
207D7—122no—
208C22×C3—141no—
209C22×C3D4×D711yescharacteristic
210C2×D3—21no—
211C2×D3—42no—
212C2×D3—41no—
213C2×D3—142no—
214C2×D3—282no—
215C2×D3—562no—
216C3×C4—142no—
217C3×C4—141no—
218C3×C4D4×D711yescharacteristic
219C3×C4C2×C22⋊D711yescharacteristic
220Dic3—21no—
221C23—61no—
222C23—212no—
223C23—422no—
224Q8—31no—
225Q8—71no—
226D4—31no—
227D4—61no—
228D4—61no—
229D4—61no—
230D4—71no—
231D4—141no—
232D4—422no—
233D4—422no—
234D4—421no—
235D4—421no—
236D4—844no—
237C2×C4—61no—
238C2×C4—141no—
239C2×C4—141no—
240C2×C4—422no—
241C2×C4C2×D3×D711yescharacteristic
242C8—21no—
243C8—421no—
244C7#482211yescharacteristic, complemented, Sylow
245C2×C3—21no—
246C2×C3—281no—
247C2×C3#650011yescharacteristic
248D3—41no—
249D3—81no—
250D3—282no—
251C22—61no—
252C22—122no—
253C22—121no—
254C22—141no—
255C22—422no—
256C22—842no—
257C22—1682no—
258C22#6668111yescharacteristic
259C4—61no—
260C4—142no—
261C4—141no—
262C4D7×D1211yescharacteristic
263C4D3×C22⋊D711yescharacteristic
264C3#8997011yescharacteristic, complemented, Sylow
265C2—21no—
266C2—121no—
267C2—241no—
268C2—281no—
269C2—842no—
270C2#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#453
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. #10621566C2semidirect
2. C7×#908C2semidirect
3. #10621013C2semidirect
4. #10620861C2semidirect
5. #10620662C2semidirect
6. C3×#6399C2semidirect
7. #10620514C2semidirect
11. C2×C7×D12C22semidirect
19. C3×C7×M16C22semidirect
20. #66627C22semidirect
23. C3×C4×Dic7C22semidirect
31. #6399D3semidirect
43. C7×D12D4semidirect
45. C7×Dic6D4semidirect
48. C3×D28D4semidirect
50. C3×Dic14D4semidirect
69. C7×M16C2×D3semidirect
72. C4×Dic7C2×D3semidirect
76. #908D7semidirect
112. D28D12semidirect
115. Dic14D12semidirect
127. C2×D12C2×D7semidirect
128. C3×M16C2×D7semidirect
178. D12C22⋊D7semidirect
181. Dic6C22⋊D7semidirect
184. C3×C7#453semidirect
200. M16D3×D7semidirect
244. C7#4822semidirect
264. C3#89970semidirect

Maximal subgroups

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

subgroupindex
#106215662
C7×#9082
#106210132
#106208612
#106206622
C3×#63992
#106205142
#899703
#48227