Finite Group Databasebeta

← #13053622

Subgroups — #13053622

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#10621565C211yesmaximal, characteristic, complemented
2C3×#6471C211yesmaximal, characteristic, complemented
3#10621025C211yesmaximal, characteristic, complemented
4#10620874C211yesmaximal, characteristic, complemented
5#10620664C211yesmaximal, characteristic, complemented
6C7×#823C211yesmaximal, characteristic, complemented
7#10620511C211yesmaximal, characteristic, complemented
8#89519—31nomaximal, Hall
9#66728C2211yescharacteristic
10C2×C7×D12C2211yescharacteristic
11C3×#1157C2211yescharacteristic
12C2×C3×D28C2211yescharacteristic, complemented
13D3×C22⋊D7—22no—
14C2×#4245—22no—
15#66681—21no—
16D7×D12—21no—
17#66676—21no—
18#66672—21no—
19C3×C7×M16C2211yescharacteristic, complemented
20C4×Dic21C2211yescharacteristic, complemented
21C7×#208C2211yescharacteristic, complemented
22C3×D56—21no—
23C3×#1132—21no—
24#66565—21no—
25#66561—21no—
26#6548—31no—
27#6503—31no—
28#6471—31no—
29#6471D311yescharacteristic, complemented
30C7×#100—31no—
31#6399—31no—
32#6399—31no—
33#5198—71nomaximal, Hall
34C22×D3×C7D412yes—
35C22×C3×D7—21no—
36C2×D3×D7—22no—
37C2×C3×C4×C7C2311yescharacteristic
38#4268—21no—
39C2×Dic21—21no—
40D84D411yescharacteristic, complemented
41C4×D21—21no—
42Dic42D411yescharacteristic, complemented
43C7×D12—21no—
44C7×D12—21no—
45C3×C22⋊D7—21no—
46C3×D28—21no—
47C3×D28—21no—
48C3×D28D411yescharacteristic, complemented
49C3×C4×D7—21no—
50C3×Dic14D411yescharacteristic, complemented
51#4247—22no—
52#4245—22no—
53#4245—44no—
54D3×Dic7—22no—
55C3×C7×C8—21no—
56C7×C3⋊C8—21no—
57C2×C7×D4—31no—
58C2×C22⋊D7—62no—
59C2×C22⋊D7—62no—
60#1159—61no—
61#1159—61no—
62D4×D7—61no—
63D4×D7—61no—
64#1157—31no—
65#1157C2×D311yescharacteristic
66C2×D28C2×D311yescharacteristic, complemented
67C7×M16—31no—
68C7×M16C2×D311yescharacteristic, complemented
69C4×Dic7—31no—
70D56—21no—
71D56—61no—
72#1132—21no—
73#1132—61no—
74#1009—71no—
75C3×AGL(1,ℤ/8)—71no—
76#940—71no—
77#932—71no—
78#837—71no—
79#825—71no—
80#823D711yescharacteristic, complemented
81C22×C3×C7C2×D411yescharacteristic
82C2×D21—21no—
83C2×D3×C7—22no—
84C2×D3×C7—42no—
85C2×C3×D7—21no—
86C2×C3×D7—42no—
87C2×C3×D7—41no—
88D3×D7—82no—
89C3×C4×C7C2×D411yescharacteristic
90C3×C4×C7C2×D411yescharacteristic
91Dic21—22no—
92Dic21—21no—
93C3×Dic7—21no—
94#453—211noSylow
95C23×C7—32no—
96C22×D7—21no—
97C22×D7—62no—
98C7×D4—61no—
99C7×D4—61no—
100C2×C4×C7C22×D311yescharacteristic
101C22⋊D7—21no—
102C22⋊D7—62no—
103C22⋊D7—62no—
104C22⋊D7—61no—
105C22⋊D7—124no—
106C2×Dic7—62no—
107C2×Dic7—61no—
108D28—21no—
109D28—21no—
110D28—31no—
111D28#6711yescharacteristic, complemented
112C4×D7—21no—
113C4×D7—61no—
114Dic14—31no—
115Dic14#6711yescharacteristic, complemented
116C7×C8—21no—
117C7×C8—61no—
118C3×#41—71no—
119C2×C3×D4—71no—
120C2×#67—142no—
121#239—141no—
122D3×D4—141no—
123D3×D4—142no—
124#235—71no—
125#235—141no—
126C2×D12—141no—
127C2×D12C2×D711yescharacteristic
128C3×SD16—141no—
129C3×D8—141no—
130C3×M16C2×D711yescharacteristic, complemented
131#214—141no—
132#213—141no—
133C4×Dic3—71no—
134#208C2×D711yescharacteristic, complemented
135C2×C3×C7—21no—
136C2×C3×C7#12011yescharacteristic
137D21—41no—
138C3×D7—41no—
139C3×D7—81no—
140D3×C7—42no—
141H(5,𝔽2)—211no—
142AGL(1,ℤ/8)—71no—
143AGL(1,ℤ/8)—211no—
144#127—211no—
145#104—211no—
146#104—211no—
147#100—31no—
148C22×C7—62no—
149C22×C7—122no—
150C22×C7D3×D411yescharacteristic
151C2×D7—21no—
152C2×D7—42no—
153C2×D7—41no—
154C2×D7—61no—
155C2×D7—242no—
156C4×C7D3×D411yescharacteristic
157C4×C7C2×#6711yescharacteristic
158Dic7—21no—
159Dic7—62no—
160Dic7—61no—
161C23×C3—141no—
162C22×D3—142no—
163C22×D3D2812yes—
164C3×Q8—71no—
165C3×D4—71no—
166C3×D4—141no—
167C3×D4—141no—
168C3×D4—141no—
169C2×C3×C4—141no—
170C2×C3×C4C22×D711yescharacteristic
171#67—142no—
172#67—141no—
173#67—284no—
174C2×Dic3—141no—
175D12—21no—
176D12—21no—
177D12—71no—
178D12—142no—
179C4×D3—142no—
180C4×D3—141no—
181Dic6—71no—
182C3×C8—21no—
183C3⋊C8—21no—
184C3×C7#45311yescharacteristic, complemented, Hall
185#41—71no—
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—141no—
196SD16—421no—
197D8—141no—
198D8—421no—
199M16—31no—
200M16D3×D711yescharacteristic, complemented
201C42—211no—
202C2×C7—21no—
203C2×C7—122no—
204C2×C7#93711yescharacteristic
205D7—41no—
206D7—81no—
207D7—121no—
208C22×C3—141no—
209C22×C3—282no—
210C22×C3—281no—
211C22×C3C2×D2811yescharacteristic
212C2×D3—22no—
213C2×D3—42no—
214C2×D3—141no—
215C2×D3—562no—
216C3×C4—141no—
217C3×C4D4×D711yescharacteristic
218C3×C4D4×D711yescharacteristic
219Dic3—142no—
220Dic3—141no—
221C23—32no—
222C23—141no—
223C23—422no—
224Q8—71no—
225Q8—211no—
226D4—61no—
227D4—61no—
228D4—71no—
229D4—141no—
230D4—141no—
231D4—141no—
232D4—211no—
233D4—422no—
234D4—422no—
235D4—421no—
236D4—844no—
237C2×C4—141no—
238C2×C4—422no—
239C2×C4—421no—
240C2×C4—421no—
241C2×C4C2×D3×D711yescharacteristic
242C8—21no—
243C8—61no—
244C7#519811yescharacteristic, complemented, Sylow
245C2×C3—21no—
246C2×C3—281no—
247C2×C3—561no—
248C2×C3#644511yescharacteristic
249D3—42no—
250D3—281no—
251C22—62no—
252C22—122no—
253C22—141no—
254C22—282no—
255C22—281no—
256C22—421no—
257C22—1682no—
258C22D3×D2811yescharacteristic
259C4—141no—
260C4—422no—
261C4—421no—
262C4#6668111yescharacteristic
263C4D7×#6711yescharacteristic
264C3#8951911yescharacteristic, complemented, Sylow
265C2—21no—
266C2—122no—
267C2—281no—
268C2—561no—
269C2—841no—
270C2#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#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. #10621565C2semidirect
2. C3×#6471C2semidirect
3. #10621025C2semidirect
4. #10620874C2semidirect
5. #10620664C2semidirect
6. C7×#823C2semidirect
7. #10620511C2semidirect
12. C2×C3×D28C22semidirect
19. C3×C7×M16C22semidirect
20. C4×Dic21C22semidirect
21. C7×#208C22semidirect
29. #6471D3semidirect
40. D84D4semidirect
42. Dic42D4semidirect
48. C3×D28D4semidirect
50. C3×Dic14D4semidirect
66. C2×D28C2×D3semidirect
68. C7×M16C2×D3semidirect
80. #823D7semidirect
111. D28#67semidirect
115. Dic14#67semidirect
130. C3×M16C2×D7semidirect
134. #208C2×D7semidirect
184. C3×C7#453semidirect
200. M16D3×D7semidirect
244. C7#5198semidirect
264. C3#89519semidirect

Maximal subgroups

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

subgroupindex
#106215652
C3×#64712
#106210252
#106208742
#106206642
C7×#8232
#106205112
#895193
#51987