Finite Group Databasebeta

← #13053626

Subgroups — #13053626

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#10621564C211yesmaximal, characteristic, complemented
2C7×#908C211yesmaximal, characteristic, complemented
3#10621025C211yesmaximal, characteristic, complemented
4#10620866C211yesmaximal, characteristic, complemented
5#10620664C211yesmaximal, characteristic, complemented
6C3×#6397C211yesmaximal, characteristic, complemented
7#10620513C211yesmaximal, characteristic, complemented
8#89970—31nomaximal, Hall
9#66728C2211yescharacteristic
10C7×#235C2211yescharacteristic
11C2×C7×D12C2211yescharacteristic, complemented
12C2×C3×D28C2211yescharacteristic
13D7×#67—22no—
14C2×#4245—22no—
15#66681—21no—
16D3×D28—21no—
17#66670—21no—
18#66669—21no—
19C3×C7×M16C2211yescharacteristic, complemented
20C4×Dic21C2211yescharacteristic, complemented
21C7×D24—21no—
22C7×#204—21no—
23C3×#1136C2211yescharacteristic, complemented
24#66567—21no—
25#66562—21no—
26#6553—31no—
27C7×AGL(1,ℤ/8)—31no—
28#6503—31no—
29#6495—31no—
30#6411—31no—
31#6399—31no—
32#6397D311yescharacteristic, complemented
33#4747—71nomaximal, Hall
34C22×D3×C7—21no—
35C22×C3×D7D412yes—
36C2×D3×D7—22no—
37C2×C3×C4×C7C2311yescharacteristic
38#4268—21no—
39C2×Dic21—21no—
40D84D411yescharacteristic, complemented
41C4×D21—21no—
42Dic42D411yescharacteristic, complemented
43C7×#67—21no—
44C7×D12—21no—
45C7×D12—21no—
46C7×D12D411yescharacteristic, complemented
47C4×D3×C7—21no—
48C7×Dic6D411yescharacteristic, complemented
49C3×D28—21no—
50C3×D28—21no—
51#4246—22no—
52#4245—22no—
53#4245—44no—
54Dic3×D7—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—31no—
64#1157—61no—
65C2×D28—61no—
66C2×D28C2×D311yescharacteristic
67C7×SD16—61no—
68C7×D8—61no—
69C7×M16C2×D311yescharacteristic, complemented
70#1142—61no—
71#1141—61no—
72C4×Dic7—31no—
73#1136C2×D311yescharacteristic, complemented
74#1004—71no—
75#940—71no—
76#908—71no—
77#908D711yescharacteristic, complemented
78C3×#100—71no—
79#825—71no—
80#825—71no—
81C22×C3×C7C2×D411yescharacteristic
82C2×D21—21no—
83C2×D3×C7—21no—
84C2×D3×C7—42no—
85C2×D3×C7—41no—
86C2×C3×D7—22no—
87C2×C3×D7—42no—
88D3×D7—82no—
89C3×C4×C7C2×D411yescharacteristic
90C3×C4×C7C2×D411yescharacteristic
91Dic21—22no—
92Dic21—21no—
93C7×Dic3—21no—
94#453—211noSylow
95C23×C7—61no—
96C22×D7—62no—
97C22×D7D1212yes—
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—62no—
106C22⋊D7—61no—
107C22⋊D7—124no—
108C2×Dic7—61no—
109D28—21no—
110D28—21no—
111D28—31no—
112D28—62no—
113C4×D7—62no—
114C4×D7—61no—
115Dic14—31no—
116C7×C8—21no—
117C7⋊C8—21no—
118C2×C3×D4—71no—
119C2×#67—142no—
120C2×#67—142no—
121#237—141no—
122#237—141no—
123D3×D4—141no—
124D3×D4—141no—
125#235—71no—
126#235C2×D711yescharacteristic
127C2×D12C2×D711yescharacteristic, complemented
128C3×M16—71no—
129C3×M16C2×D711yescharacteristic, complemented
130C4×Dic3—71no—
131D24—21no—
132D24—141no—
133#204—21no—
134#204—141no—
135C2×C3×C7—21no—
136C2×C3×C7#12011yescharacteristic
137D21—41no—
138C3×D7—42no—
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—211no—
146#104—211no—
147#100—71no—
148C22×C7—61no—
149C22×C7—122no—
150C22×C7—121no—
151C22×C7C2×D1211yescharacteristic
152C2×D7—22no—
153C2×D7—42no—
154C2×D7—61no—
155C2×D7—242no—
156C4×C7—61no—
157C4×C7D3×D411yescharacteristic
158C4×C7D3×D411yescharacteristic
159Dic7—62no—
160Dic7—61no—
161C23×C3—72no—
162C22×D3—21no—
163C22×D3—142no—
164C3×D4—141no—
165C3×D4—141no—
166C2×C3×C4C22×D711yescharacteristic
167#67—21no—
168#67—142no—
169#67—142no—
170#67—141no—
171#67—284no—
172C2×Dic3—142no—
173C2×Dic3—141no—
174D12—21no—
175D12—21no—
176D12—71no—
177D12C22⋊D711yescharacteristic, complemented
178C4×D3—21no—
179C4×D3—141no—
180Dic6—71no—
181Dic6C22⋊D711yescharacteristic, complemented
182C3×C8—21no—
183C3×C8—141no—
184C3×C7#45311yescharacteristic, complemented, Hall
185#41—31no—
186#41—211no—
187#41—421no—
188#41—421no—
189C2×D4—31no—
190C2×D4—71no—
191C2×D4—421no—
192C2×D4—421no—
193C2×D4—422no—
194C2×D4—422no—
195SD16—61no—
196SD16—421no—
197D8—61no—
198D8—421no—
199M16—71no—
200M16D3×D711yescharacteristic, complemented
201C42—211no—
202C2×C7—21no—
203C2×C7—121no—
204C2×C7—241no—
205C2×C7#88211yescharacteristic
206D7—42no—
207D7—121no—
208C22×C3—142no—
209C22×C3—282no—
210C22×C3D4×D711yescharacteristic
211C2×D3—21no—
212C2×D3—42no—
213C2×D3—41no—
214C2×D3—141no—
215C2×D3—562no—
216C3×C4D4×D711yescharacteristic
217C3×C4C2×C22⋊D711yescharacteristic
218Dic3—21no—
219Dic3—142no—
220Dic3—141no—
221C23—61no—
222C23—72no—
223C23—422no—
224Q8—31no—
225Q8—211no—
226D4—31no—
227D4—61no—
228D4—61no—
229D4—61no—
230D4—141no—
231D4—141no—
232D4—211no—
233D4—422no—
234D4—422no—
235D4—421no—
236D4—844no—
237C2×C4—61no—
238C2×C4—422no—
239C2×C4—421no—
240C2×C4—421no—
241C2×C4C2×D3×D711yescharacteristic
242C8—21no—
243C8—141no—
244C7#474711yescharacteristic, complemented, Sylow
245C2×C3—21no—
246C2×C3—282no—
247C2×C3#650011yescharacteristic
248D3—41no—
249D3—81no—
250D3—281no—
251C22—61no—
252C22—122no—
253C22—121no—
254C22—142no—
255C22—282no—
256C22—421no—
257C22—1682no—
258C22D7×D1211yescharacteristic
259C4—61no—
260C4—422no—
261C4—421no—
262C4#6668111yescharacteristic
263C4D3×C22⋊D711yescharacteristic
264C3#8997011yescharacteristic, complemented, Sylow
265C2—21no—
266C2—121no—
267C2—241no—
268C2—282no—
269C2—841no—
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. #10621564C2semidirect
2. C7×#908C2semidirect
3. #10621025C2semidirect
4. #10620866C2semidirect
5. #10620664C2semidirect
6. C3×#6397C2semidirect
7. #10620513C2semidirect
11. C2×C7×D12C22semidirect
19. C3×C7×M16C22semidirect
20. C4×Dic21C22semidirect
23. C3×#1136C22semidirect
32. #6397D3semidirect
40. D84D4semidirect
42. Dic42D4semidirect
46. C7×D12D4semidirect
48. C7×Dic6D4semidirect
69. C7×M16C2×D3semidirect
73. #1136C2×D3semidirect
77. #908D7semidirect
127. C2×D12C2×D7semidirect
129. C3×M16C2×D7semidirect
177. D12C22⋊D7semidirect
181. Dic6C22⋊D7semidirect
184. C3×C7#453semidirect
200. M16D3×D7semidirect
244. C7#4747semidirect
264. C3#89970semidirect

Maximal subgroups

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

subgroupindex
#106215642
C7×#9082
#106210252
#106208662
#106206642
C3×#63972
#106205132
#899703
#47477