Finite Group Databasebeta

← #13061765

Subgroups — #13061765

Order 1344.

668 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 147 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×#66732C211yesmaximal, characteristic, complemented
2C2×D4×D21C211yesmaximal, characteristic, complemented
3C3×C7×#124C211yesmaximal, characteristic, complemented
4#10621430C211yesmaximal, characteristic, complemented
5#10621423C211yesmaximal, characteristic, complemented
6#10621367C214yesmaximal, complemented
7#10621365C214yesmaximal, complemented
8#10621357C211yesmaximal, characteristic, complemented
9#10621355C211yesmaximal, characteristic, complemented
10#90365—31nomaximal, Hall
11C23×D21C2212yes—
12C2×C3×C7×Q8C2211yescharacteristic, complemented
13C2×C3×C7×D4C2211yescharacteristic
14C2×#4268C2212yes—
15C2×#4268C2214yescomplemented
16#66732—24no—
17D4×D21—24no—
18C2×D84C2211yescharacteristic
19C2×D84C2212yes—
20C2×D84C2211yescharacteristic, complemented
21C2×D84C2214yescomplemented
22C2×C4×D21C2211yescharacteristic
23C2×C4×D21C2212yescomplemented
24C3×C7×C22⋊C4C2214yescomplemented
25C3×C42×C7C2211yescharacteristic, complemented
26#66631C2212yes—
27#66631C2214yescomplemented
28#66629C2212yescomplemented
29C4×Dic21C2211yescharacteristic, complemented
30C2×#1161—31no—
31C2×D4×D7—31no—
32C7×#124D311yescharacteristic, complemented
33#6510—31no—
34#6503—31no—
35#6447—34no—
36#6445—34no—
37#6437—31no—
38#6435—31no—
39#5674—71nomaximal, Hall
40C23×C3×C7C2312yes—
41C22×D21—24no—
42C22×D21—44no—
43C22×D21C2312yes—
44C22×D21C2312yes—
45C22×D21C2311yescharacteristic
46C3×C7×Q8—22no—
47C3×C7×D4—22no—
48C2×C3×C4×C7C2312yes—
49C2×C3×C4×C7C2312yes—
50C2×C3×C4×C7C2311yescharacteristic
51#4268—24no—
52#4268—44no—
53C2×Dic21C2312yes—
54C2×Dic21C2311yescharacteristic
55D84—24no—
56D84—22no—
57D84—22no—
58D84—44no—
59C4×D21—24no—
60C4×D21D414yescomplemented
61C23×D7—32no—
62C2×C7×Q8C2×D311yescharacteristic, complemented
63C2×C7×D4C2×D311yescharacteristic
64C2×C22⋊D7—32no—
65C2×C22⋊D7—34no—
66#1161—64no—
67D4×D7—64no—
68C2×D28—31no—
69C2×D28—31no—
70C2×D28—32no—
71C2×D28—34no—
72C2×C4×D7—31no—
73C2×C4×D7—32no—
74C7×C22⋊C4C2×D314yescomplemented
75C42×C7C2×D311yescharacteristic, complemented
76(C2×D7)⋊C4—32no—
77(C2×D7)⋊C4—34no—
78Dic7⋊C4—32no—
79C4×Dic7—31no—
80C2×#239—71no—
81C2×D3×D4—71no—
82C3×#124D711yescharacteristic, complemented
83#947—71no—
84#940—71no—
85#884—74no—
86#882—74no—
87#874—71no—
88#872—71no—
89C22×C3×C7—22no—
90C22×C3×C7—24no—
91C22×C3×C7C2411yescharacteristic
92C2×D21—22no—
93C2×D21—22no—
94C2×D21—22no—
95C2×D21—24no—
96C2×D21—24no—
97C2×D21—48no—
98C2×D21C2×D412yes—
99C3×C4×C7—22no—
100C3×C4×C7—22no—
101C3×C4×C7C2×D412yes—
102Dic21—22no—
103Dic21C2×D412yes—
104#535—211noSylow
105C23×C7C22×D312yes—
106C22×D7—31no—
107C22×D7—32no—
108C22×D7—32no—
109C22×D7—64no—
110C22×D7—124no—
111C7×Q8—22no—
112C7×D4—22no—
113C2×C4×C7C22×D312yes—
114C2×C4×C7C22×D312yes—
115C2×C4×C7C22×D311yescharacteristic
116C22⋊D7—64no—
117C22⋊D7—124no—
118C2×Dic7—31no—
119C2×Dic7—32no—
120D28—64no—
121D28—62no—
122D28—62no—
123D28—124no—
124C4×D7—34no—
125C4×D7—64no—
126C23×D3—72no—
127C2×C3×Q8C2×D711yescharacteristic, complemented
128C2×C3×D4C2×D711yescharacteristic
129C2×#67—72no—
130C2×#67—74no—
131#239—144no—
132D3×D4—144no—
133C2×D12—71no—
134C2×D12—71no—
135C2×D12—72no—
136C2×D12—74no—
137C2×C4×D3—71no—
138C2×C4×D3—72no—
139C3×C22⋊C4C2×D714yescomplemented
140C3×C42C2×D711yescharacteristic, complemented
141(C2×D3)⋊C4—72no—
142(C2×D3)⋊C4—74no—
143Dic3⋊C4—72no—
144C4×Dic3—71no—
145C2×C3×C7—42no—
146C2×C3×C7C22×D411yescharacteristic
147C2×C3×C7H(5,𝔽2)12yes—
148D21—22no—
149D21—42no—
150D21—42no—
151C2×#41—211no—
152C22×D4—211no—
153#127—211no—
154#127—211no—
155#124—211no—
156#124D2111yescharacteristic, complemented
157#121—214no—
158#120—214no—
159#117—211no—
160C22×C7—22no—
161C22×C7—24no—
162C22×C7C23×D311yescharacteristic
163C2×D7—32no—
164C2×D7—62no—
165C2×D7—62no—
166C2×D7—62no—
167C2×D7—64no—
168C2×D7—64no—
169C2×D7—128no—
170C4×C7—22no—
171C4×C7—22no—
172C4×C7D3×D412yes—
173Dic7—32no—
174Dic7—62no—
175C23×C3C22×D712yes—
176C22×D3—71no—
177C22×D3—72no—
178C22×D3—72no—
179C22×D3—144no—
180C22×D3—284no—
181C3×Q8—22no—
182C3×D4—22no—
183C2×C3×C4C22×D712yes—
184C2×C3×C4C22×D712yes—
185C2×C3×C4C22×D711yescharacteristic
186#67—144no—
187#67—284no—
188C2×Dic3—71no—
189C2×Dic3—72no—
190D12—144no—
191D12—142no—
192D12—142no—
193D12—284no—
194C4×D3—74no—
195C4×D3—144no—
196C3×C7#53511yescharacteristic, complemented, Hall
197C24—212no—
198#41—424no—
199C2×Q8C2×D2111yescharacteristic, complemented
200C2×D4—212no—
201C2×D4—214no—
202C2×D4—212no—
203C2×D4—211no—
204C2×D4—211no—
205C2×D4—214no—
206C2×D4—424no—
207C2×D4C2×D2111yescharacteristic
208C22×C4—211no—
209C22×C4—212no—
210C4⋊C4—212no—
211C22⋊C4—212no—
212C22⋊C4—214no—
213C22⋊C4C2×D2114yescomplemented
214C42—211no—
215C42C2×D2111yescharacteristic, complemented
216C2×C7—42no—
217C2×C7C2×D3×D411yescharacteristic
218C2×C7#100912yes—
219D7—62no—
220D7—122no—
221D7—122no—
222C22×C3—22no—
223C22×C3—24no—
224C22×C3C23×D711yescharacteristic
225C2×D3—72no—
226C2×D3—142no—
227C2×D3—142no—
228C2×D3—142no—
229C2×D3—144no—
230C2×D3—144no—
231C2×D3—288no—
232C3×C4—22no—
233C3×C4—22no—
234C3×C4D4×D712yes—
235Dic3—72no—
236Dic3—142no—
237C23—211no—
238C23—212no—
239C23—212no—
240C23—424no—
241C23—844no—
242C23C22×D2112yes—
243Q8—22no—
244D4—22no—
245D4—424no—
246D4—424no—
247D4—422no—
248D4—422no—
249D4—844no—
250D4—844no—
251C2×C4—211no—
252C2×C4—214no—
253C2×C4—212no—
254C2×C4—424no—
255C2×C4C22×D2112yes—
256C2×C4C22×D2112yes—
257C2×C4C22×D2111yescharacteristic
258C7#567411yescharacteristic, complemented, Sylow
259C2×C3—42no—
260C2×C3C2×D4×D711yescharacteristic
261C2×C3#655312yes—
262D3—142no—
263D3—282no—
264D3—282no—
265C22—22no—
266C22—24no—
267C22—212no—
268C22—422no—
269C22—422no—
270C22—422no—
271C22—424no—
272C22—424no—
273C22—848no—
274C22C23×D2111yescharacteristic
275C4—22no—
276C4—22no—
277C4—212no—
278C4—422no—
279C4D4×D2112yes—
280C3#9036511yescharacteristic, complemented, Sylow
281C2—42no—
282C2—422no—
283C2—842no—
284C2—842no—
285C2C2×D4×D2111yescharacteristic
286C2#1062166212yes—
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×#124C2
hypercenter#124D21
hypocenterC3×C7#535
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C2×#66732C2semidirect
2. C2×D4×D21C2semidirect
3. C3×C7×#124C2semidirect
4. #10621430C2semidirect
5. #10621423C2semidirect
6. #10621367C2semidirect
7. #10621365C2semidirect
8. #10621357C2semidirect
9. #10621355C2semidirect
12. C2×C3×C7×Q8C22semidirect
15. C2×#4268C22semidirect
20. C2×D84C22semidirect
21. C2×D84C22semidirect
23. C2×C4×D21C22semidirect
24. C3×C7×C22⋊C4C22semidirect
25. C3×C42×C7C22semidirect
27. #66631C22semidirect
28. #66629C22semidirect
29. C4×Dic21C22semidirect
32. C7×#124D3semidirect
60. C4×D21D4semidirect
62. C2×C7×Q8C2×D3semidirect
74. C7×C22⋊C4C2×D3semidirect
75. C42×C7C2×D3semidirect
82. C3×#124D7semidirect
127. C2×C3×Q8C2×D7semidirect
139. C3×C22⋊C4C2×D7semidirect
140. C3×C42C2×D7semidirect
156. #124D21semidirect
196. C3×C7#535semidirect
199. C2×Q8C2×D21semidirect
213. C22⋊C4C2×D21semidirect
215. C42C2×D21semidirect
258. C7#5674semidirect
280. C3#90365semidirect

Maximal subgroups

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

subgroupindex
C2×#667322
C2×D4×D212
C3×C7×#1242
#106214302
#106214232
#106213672
#106213652
#106213572
#106213552
#903653
#56747