Finite Group Databasebeta

← #13054778

Subgroups — #13054778

Order 1344.

376 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C2×#66681C211yesmaximal, characteristic
2#10621429C211yesmaximal, characteristic, complemented
3C2×C7×#215C211yesmaximal, characteristic, complemented
4C2×C3×#1143C211yesmaximal, characteristic, complemented
5#10620502C211yesmaximal, characteristic, complemented
6#10620500C211yesmaximal, characteristic, complemented
7#10620492C211yesmaximal, characteristic, complemented
8#89941—31nomaximal, Hall
9C22×D3×D7—21no—
10C2×C3×C7×Q8C2211yescharacteristic, complemented
11C2×C4×D21C2211yescharacteristic
12C2×C7×D12C2211yescharacteristic
13C2×C3×D28C2211yescharacteristic
14C2×#4245—21no—
15#66681—22no—
16#66681—22no—
17#66631—21no—
18C4⋊Dic21C2211yescharacteristic, complemented
19#66629—21no—
20C7×#215—22no—
21C2×C7×C3⋊C8C2211yescharacteristic, complemented
22C3×#1143—22no—
23C2×C3×C7⋊C8C2211yescharacteristic, complemented
24C2×D4×D7—31no—
25C2×C7×SD16—31no—
26#6509—31no—
27C2×#1143D311yescharacteristic, complemented
28#6406—31no—
29#6395—31no—
30(C2×D7)⋊C8—31no—
31#5169—71nomaximal, Hall
32C22×D21D411yescharacteristic
33C22×D3×C7—21no—
34C22×C3×D7—21no—
35C2×D3×D7—24no—
36C2×D3×D7—42no—
37C2×D3×D7—42no—
38C3×C7×Q8—21no—
39C3×C7×Q8—21no—
40C2×C3×C4×C7—21no—
41C2×C3×C4×C7C2311yescharacteristic
42C2×Dic21—21no—
43C2×Dic21D411yescharacteristic
44C4×D21—21no—
45C4×D21—21no—
46C7×D12—21no—
47C7×D12D412yes—
48C3×D28—21no—
49C3×D28D412yes—
50#4245—24no—
51C3×C7⋊C8—21no—
52C7×C3⋊C8—21no—
53C23×D7—61no—
54C2×C7×Q8C2×D311yescharacteristic, complemented
55C2×C7×D4—31no—
56C2×C22⋊D7—61no—
57D4×D7—62no—
58D4×D7—62no—
59C2×D28C2×D311yescharacteristic
60C2×C4×D7—31no—
61C7×SD16—62no—
62C2×C7×C8—31no—
63#1143—22no—
64(C2×D7)⋊C4—61no—
65C4⋊Dic7—31no—
66Dic7⋊C4—61no—
67C2×C7⋊C8C2×D311yescharacteristic, complemented
68C2×D3×D4—71no—
69C2×C3×SD16—71no—
70#946—71no—
71C2×#215D711yescharacteristic, complemented
72#832—71no—
73#821—71no—
74(C2×D3)⋊C8—71no—
75C22×C3×C7C2×D411yescharacteristic
76C2×D21—21no—
77C2×D21—21no—
78C2×D21SD1612yescomplemented
79C2×D3×C7—22no—
80C2×D3×C7—41no—
81C2×D3×C7—41no—
82C2×C3×D7—22no—
83C2×C3×D7—41no—
84C2×C3×D7—41no—
85D3×D7—48no—
86C3×C4×C7—41no—
87C3×C4×C7C2×D411yescharacteristic
88C3×C4×C7C2×D411yescharacteristic
89Dic21—21no—
90Dic21—41no—
91#450—211noSylow
92C23×C7—61no—
93C22×D7—21no—
94C22×D7—31no—
95C22×D7—64no—
96C22×D7—122no—
97C22×D7—122no—
98C7×Q8—21no—
99C7×Q8—21no—
100C7×D4—32no—
101C7×D4—61no—
102C2×C4×C7—21no—
103C2×C4×C7C22×D311yescharacteristic
104C22⋊D7—64no—
105C2×Dic7—31no—
106C2×Dic7—61no—
107D28—21no—
108D28#6712yes—
109C4×D7—61no—
110C4×D7—61no—
111C7×C8—61no—
112C7⋊C8—21no—
113C23×D3—141no—
114C2×C3×Q8C2×D711yescharacteristic, complemented
115C2×C3×D4—71no—
116C2×#67—141no—
117D3×D4—142no—
118D3×D4—142no—
119C2×D12C2×D711yescharacteristic
120C2×C4×D3—71no—
121C3×SD16—142no—
122C2×C3×C8—71no—
123#215—22no—
124(C2×D3)⋊C4—141no—
125C4⋊Dic3—71no—
126Dic3⋊C4—141no—
127C2×C3⋊C8C2×D711yescharacteristic, complemented
128C2×C3×C7#12011yescharacteristic
129C2×C3×C7C2×SD1611yescharacteristic
130C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
131D21—22no—
132C3×D7—42no—
133D3×C7—42no—
134C22×D4—211no—
135C2×SD16—31no—
136C2×SD16—71no—
137C22⋊Q8—211no—
138#102—211no—
139#102—211no—
140C22⋊C8—211no—
141C22×C7—62no—
142C22×C7—121no—
143C22×C7—121no—
144C22×C7D3×D411yescharacteristic
145C2×D7—22no—
146C2×D7—32no—
147C2×D7—41no—
148C2×D7—41no—
149C2×D7—61no—
150C2×D7—61no—
151C2×D7—128no—
152C4×C7—41no—
153C4×C7D3×D411yescharacteristic
154C4×C7C2×#6711yescharacteristic
155Dic7—61no—
156Dic7—121no—
157C23×C3—141no—
158C22×D3—21no—
159C22×D3—71no—
160C22×D3—144no—
161C22×D3—282no—
162C22×D3—282no—
163C3×Q8—21no—
164C3×Q8—21no—
165C3×D4—72no—
166C3×D4—141no—
167C2×C3×C4—21no—
168C2×C3×C4C22×D711yescharacteristic
169#67—144no—
170C2×Dic3—71no—
171C2×Dic3—141no—
172D12—21no—
173D12C22⋊D712yes—
174C4×D3—141no—
175C4×D3—141no—
176C3×C8—141no—
177C3⋊C8—21no—
178C3×C7#45011yescharacteristic, complemented, Hall
179C24—421no—
180C2×Q8D3×D711yescharacteristic, complemented
181C2×D4—31no—
182C2×D4—71no—
183C2×D4—421no—
184C2×D4—422no—
185C2×D4—422no—
186C22×C4—211no—
187SD16—62no—
188SD16—142no—
189C2×C8—31no—
190C2×C8—71no—
191C4⋊C4—211no—
192C4⋊C4—421no—
193C22⋊C4—421no—
194C2×C7—122no—
195C2×C7D3×SD1611yescharacteristic
196C2×C7#91411yescharacteristic
197C2×C7#93711yescharacteristic
198D7—42no—
199D7—62no—
200C22×C3—142no—
201C22×C3—281no—
202C22×C3—281no—
203C22×C3D4×D711yescharacteristic
204C2×D3—22no—
205C2×D3—41no—
206C2×D3—41no—
207C2×D3—72no—
208C2×D3—141no—
209C2×D3—141no—
210C2×D3—288no—
211C3×C4—41no—
212C3×C4D4×D711yescharacteristic
213C3×C4C2×C22⋊D711yescharacteristic
214Dic3—141no—
215Dic3—281no—
216C23—61no—
217C23—141no—
218C23—211no—
219C23—424no—
220C23—842no—
221C23—842no—
222Q8—21no—
223Q8—21no—
224D4—32no—
225D4—61no—
226D4—72no—
227D4—141no—
228D4—424no—
229C2×C4—21no—
230C2×C4—211no—
231C2×C4—421no—
232C2×C4—421no—
233C2×C4—421no—
234C2×C4C2×D3×D711yescharacteristic
235C8—61no—
236C8—141no—
237C7#516911yescharacteristic, complemented, Sylow
238C2×C3—282no—
239C2×C3D7×SD1611yescharacteristic
240C2×C3#647711yescharacteristic
241C2×C3#650011yescharacteristic
242D3—42no—
243D3—142no—
244C22—62no—
245C22—121no—
246C22—121no—
247C22—142no—
248C22—212no—
249C22—281no—
250C22—281no—
251C22—421no—
252C22—421no—
253C22—848no—
254C22#6669411yescharacteristic
255C4—41no—
256C4—421no—
257C4—841no—
258C4D7×#6711yescharacteristic
259C4D3×C22⋊D711yescharacteristic
260C3#8994111yescharacteristic, complemented, Sylow
261C2—122no—
262C2—282no—
263C2—422no—
264C2#1062106711yescharacteristic
265C2#1062106811yescharacteristic
266C2#1062113911yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radicalC2×#66681C2
hypercenterC2×Q8D3×D7
hypocenterC3×C7#450
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #10621429C2semidirect
3. C2×C7×#215C2semidirect
4. C2×C3×#1143C2semidirect
5. #10620502C2semidirect
6. #10620500C2semidirect
7. #10620492C2semidirect
10. C2×C3×C7×Q8C22semidirect
18. C4⋊Dic21C22semidirect
21. C2×C7×C3⋊C8C22semidirect
23. C2×C3×C7⋊C8C22semidirect
27. C2×#1143D3semidirect
54. C2×C7×Q8C2×D3semidirect
67. C2×C7⋊C8C2×D3semidirect
71. C2×#215D7semidirect
78. C2×D21SD16semidirect
114. C2×C3×Q8C2×D7semidirect
127. C2×C3⋊C8C2×D7semidirect
178. C3×C7#450semidirect
180. C2×Q8D3×D7semidirect
237. C7#5169semidirect
260. C3#89941semidirect

Maximal subgroups

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

subgroupindex
C2×#666812
#106214292
C2×C7×#2152
C2×C3×#11432
#106205022
#106205002
#106204922
#899413
#51697