Finite Group Databasebeta

← #13057981

Subgroups — #13057981

Order 1344.

660 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 144 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
2C2×D21⋊Q8C211yesmaximal, characteristic, complemented
3C42×D21C211yesmaximal, characteristic, complemented
4C7×#875C211yesmaximal, characteristic, complemented
5C3×#6438C211yesmaximal, characteristic, complemented
6#10621035C214yesmaximal, complemented
7#10621026C214yesmaximal, complemented
8#10620932C211yesmaximal, characteristic, complemented
9#10620930C211yesmaximal, characteristic, complemented
10D7×#124—31nomaximal, Hall
11C22×D3×D7C2212yes—
12C2×C4×D21C2212yes—
13C2×C4×D21C2211yescharacteristic
14C2×C7×D12C2211yescharacteristic
15C2×C7×Dic6C2211yescharacteristic, complemented
16C2×C3×D28C2211yescharacteristic
17C2×C3×Dic14C2211yescharacteristic, complemented
18C2×#4248C2212yescomplemented
19C2×#4245C2212yes—
20C2×D21⋊C4C2212yes—
21#66681—24no—
22D21⋊Q8—24no—
23C3×C42×C7C2211yescharacteristic, complemented
24C4×Dic21C2211yescharacteristic, complemented
25C4×Dic21C2212yescomplemented
26C7×(C2×D3)⋊C4C2214yescomplemented
27C3×(C2×D7)⋊C4C2214yescomplemented
28(C2×D3)⋊Dic7C2214yescomplemented
29(C2×D7)⋊Dic3C2214yescomplemented
30C2×Q8×D7—31no—
31C2×D4×D7—31no—
32C7×#124—31no—
33#6510—31no—
34#6499—31no—
35#6448—34no—
36D7×C22⋊C4—34no—
37#6438D311yescharacteristic, complemented
38C42×D7—31no—
39D3×#124—71nomaximal, Hall
40C22×D21C2311yescharacteristic
41C22×D3×C7C2312yes—
42C22×C3×D7C2312yes—
43C2×D3×D7—24no—
44C2×D3×D7—28no—
45C2×C3×C4×C7C2312yes—
46C2×C3×C4×C7C2311yescharacteristic
47C2×Dic21C2312yes—
48C2×Dic21C2311yescharacteristic
49C4×D21—24no—
50C4×D21D414yes—
51C2×C7×Dic3C2312yes—
52C7×D12—22no—
53C7×Dic6—22no—
54C2×C3×Dic7C2312yes—
55C3×D28—22no—
56C3×Dic14—22no—
57#4248—24no—
58#4245—24no—
59D21⋊C4—24no—
60C23×D7—32no—
61C2×C7×Q8—31no—
62C2×C7×D4—31no—
63C2×C22⋊D7—32no—
64Q8×D7—64no—
65D4×D7—64no—
66C2×D28C2×D311yescharacteristic
67C2×C4×D7—31no—
68C2×C4×D7—32no—
69C2×C4×D7—32no—
70C2×Dic14—32no—
71C2×Dic14C2×D311yescharacteristic, complemented
72C7×C22⋊C4—34no—
73C42×C7C2×D311yescharacteristic, complemented
74C22⋊Dic7—34no—
75(C2×D7)⋊C4—34no—
76(C2×D7)⋊C4C2×D314yescomplemented
77C4×Dic7—32no—
78C4×Dic7—31no—
79C2×D3×Q8—71no—
80C2×D3×D4—71no—
81C3×#124—71no—
82#947—71no—
83#936—71no—
84#885—74no—
85D3×C22⋊C4—74no—
86#875D711yescharacteristic, complemented
87C42×D3—71no—
88C22×C3×C7C2411yescharacteristic
89C2×D21C2×D412yes—
90C2×D21#4114yes—
91C2×D3×C7—22no—
92C2×D3×C7—24no—
93C2×C3×D7—22no—
94C2×C3×D7—24no—
95D3×D7—48no—
96C3×C4×C7—22no—
97C3×C4×C7C2×D412yes—
98Dic21—22no—
99Dic21C2×D412yes—
100C3×Dic7—22no—
101C7×Dic3—22no—
102C2×#124—211noSylow
103C23×C7—32no—
104C22×D7—31no—
105C22×D7—64no—
106C22×D7—68no—
107C22×D7C22×D312yes—
108C7×Q8—62no—
109C7×D4—62no—
110C2×C4×C7—32no—
111C2×C4×C7C22×D312yes—
112C2×C4×C7C22×D311yescharacteristic
113C22⋊D7—64no—
114C2×Dic7—31no—
115C2×Dic7—32no—
116C2×Dic7C22×D312yes—
117D28—22no—
118C4×D7—34no—
119C4×D7—64no—
120C4×D7—64no—
121Dic14—22no—
122Dic14—64no—
123C23×D3—72no—
124C2×C3×Q8—71no—
125C2×C3×D4—71no—
126C2×#67—72no—
127D3×Q8—144no—
128D3×D4—144no—
129C2×D12C2×D711yescharacteristic
130C2×C4×D3—71no—
131C2×C4×D3—72no—
132C2×C4×D3—72no—
133C2×Dic6—72no—
134C2×Dic6C2×D711yescharacteristic, complemented
135C3×C22⋊C4—74no—
136C3×C42C2×D711yescharacteristic, complemented
137#217—74no—
138(C2×D3)⋊C4—74no—
139(C2×D3)⋊C4C2×D714yescomplemented
140C4×Dic3—72no—
141C4×Dic3—71no—
142C2×C3×C7C22×D411yescharacteristic
143C2×C3×C7C2×#4112yes—
144D21#12414yescomplemented
145C3×D7—42no—
146D3×C7—42no—
147C22×Q8—211no—
148C22×D4—211no—
149#124—31no—
150#124—71no—
151#124—214no—
152#124—211no—
153#124—211no—
154C2×C22⋊C4—214no—
155C2×C42—211no—
156C22×C7—62no—
157C22×C7—64no—
158C22×C7C23×D311yescharacteristic
159C2×D7—22no—
160C2×D7—24no—
161C2×D7—32no—
162C2×D7—34no—
163C2×D7—128no—
164C4×C7—22no—
165C4×C7—62no—
166C4×C7D3×D412yes—
167Dic7—22no—
168Dic7—32no—
169Dic7—62no—
170C23×C3—72no—
171C22×D3—71no—
172C22×D3—144no—
173C22×D3—148no—
174C22×D3C22×D712yes—
175C3×Q8—142no—
176C3×D4—142no—
177C2×C3×C4—72no—
178C2×C3×C4C22×D712yes—
179C2×C3×C4C22×D711yescharacteristic
180#67—144no—
181C2×Dic3—71no—
182C2×Dic3—72no—
183C2×Dic3C22×D712yes—
184D12—22no—
185C4×D3—74no—
186C4×D3—144no—
187C4×D3—144no—
188Dic6—22no—
189Dic6—144no—
190C3×C7C2×#12411yescharacteristic, complemented, Hall
191C24—212no—
192C2×Q8—31no—
193C2×Q8—71no—
194C2×Q8—212no—
195C2×Q8—424no—
196C2×D4—31no—
197C2×D4—71no—
198C2×D4—212no—
199C2×D4—424no—
200C22×C4—212no—
201C22×C4—211no—
202C22×C4—212no—
203C22⋊C4—34no—
204C22⋊C4—74no—
205C22⋊C4—214no—
206C22⋊C4—214no—
207C42—212no—
208C42—211no—
209C42D3×D711yescharacteristic, complemented
210C2×C7—122no—
211C2×C7C2×D3×D411yescharacteristic
212C2×C7D3×#4112yes—
213D7—34no—
214D7—42no—
215C22×C3—142no—
216C22×C3—144no—
217C22×C3C23×D711yescharacteristic
218C2×D3—22no—
219C2×D3—24no—
220C2×D3—72no—
221C2×D3—74no—
222C2×D3—288no—
223C3×C4—22no—
224C3×C4—142no—
225C3×C4D4×D712yes—
226Dic3—22no—
227Dic3—72no—
228Dic3—142no—
229C23—32no—
230C23—72no—
231C23—211no—
232C23—424no—
233C23—428no—
234Q8—62no—
235Q8—142no—
236Q8—424no—
237D4—62no—
238D4—142no—
239D4—424no—
240C2×C4—32no—
241C2×C4—72no—
242C2×C4—211no—
243C2×C4—214no—
244C2×C4—212no—
245C2×C4—424no—
246C2×C4—424no—
247C2×C4C2×D3×D712yes—
248C2×C4C2×D3×D711yescharacteristic
249C7D3×#12411yescharacteristic, complemented, Sylow
250C2×C3—282no—
251C2×C3C2×D4×D711yescharacteristic
252C2×C3D7×#4112yes—
253D3—42no—
254D3—74no—
255C22—62no—
256C22—64no—
257C22—142no—
258C22—144no—
259C22—212no—
260C22—214no—
261C22—848no—
262C22C22×D3×D711yescharacteristic
263C4—22no—
264C4—62no—
265C4—142no—
266C4—212no—
267C4—422no—
268C4#6668112yes—
269C3D7×#12411yescharacteristic, complemented, Sylow
270C2—122no—
271C2—214no—
272C2—282no—
273C2C2×#6668111yescharacteristic
274C2#1062156312yes—
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C42×C7C22
prime radicalC2×#66681C2
hypercenterC42D3×D7
hypocenterC3×C7C2×#124
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C2×D21⋊Q8C2semidirect
3. C42×D21C2semidirect
4. C7×#875C2semidirect
5. C3×#6438C2semidirect
6. #10621035C2semidirect
7. #10621026C2semidirect
8. #10620932C2semidirect
9. #10620930C2semidirect
15. C2×C7×Dic6C22semidirect
17. C2×C3×Dic14C22semidirect
18. C2×#4248C22semidirect
23. C3×C42×C7C22semidirect
24. C4×Dic21C22semidirect
25. C4×Dic21C22semidirect
26. C7×(C2×D3)⋊C4C22semidirect
27. C3×(C2×D7)⋊C4C22semidirect
28. (C2×D3)⋊Dic7C22semidirect
29. (C2×D7)⋊Dic3C22semidirect
37. #6438D3semidirect
71. C2×Dic14C2×D3semidirect
73. C42×C7C2×D3semidirect
76. (C2×D7)⋊C4C2×D3semidirect
86. #875D7semidirect
134. C2×Dic6C2×D7semidirect
136. C3×C42C2×D7semidirect
139. (C2×D3)⋊C4C2×D7semidirect
144. D21#124semidirect
190. C3×C7C2×#124semidirect
209. C42D3×D7semidirect
249. C7D3×#124semidirect
269. C3D7×#124semidirect

Maximal subgroups

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

subgroupindex
C2×#666812
C2×D21⋊Q82
C42×D212
C7×#8752
C3×#64382
#106210352
#106210262
#106209322
#106209302
D7×#1243
D3×#1247