Finite Group Databasebeta

← #13057979

Subgroups — #13057979

Order 1344.

668 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 136 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, complemented
2C2×#66670C211yesmaximal, characteristic, complemented
3#10621355C211yesmaximal, characteristic, complemented
4C7×#875C211yesmaximal, characteristic, complemented
5C3×#6437C211yesmaximal, characteristic, complemented
6#10621036C214yesmaximal, complemented
7#10621025C211yesmaximal, characteristic, complemented
8#10620970C214yesmaximal, complemented
9#10620930C211yesmaximal, characteristic, complemented
10#90365—31nomaximal, Hall
11C22×D3×D7C2212yes—
12C2×C4×D21C2211yescharacteristic
13C2×C7×D12C2211yescharacteristic
14C2×C7×Dic6C2211yescharacteristic, complemented
15C2×C3×D28C2211yescharacteristic
16C2×C3×D28C2211yescharacteristic, complemented
17C2×C3×D28C2214yescomplemented
18C2×#4246C2212yes—
19C2×#4245C2212yes—
20C2×#4245C2214yescomplemented
21C2×Dic3×D7C2212yescomplemented
22#66681—24no—
23#66670—24no—
24C3×C42×C7C2211yescharacteristic, complemented
25#66631C2212yes—
26#66629C2212yescomplemented
27C4×Dic21C2211yescharacteristic, complemented
28C7×(C2×D3)⋊C4C2214yescomplemented
29(C2×D7)⋊Dic3C2214yescomplemented
30C2×#1161—31no—
31C2×D4×D7—31no—
32C7×#124—31no—
33#6510—31no—
34#6503—31no—
35#6447—34no—
36#6445—34no—
37#6437D311yescharacteristic, complemented
38#6435—31no—
39#5715—71nomaximal, Hall
40C22×D21C2311yescharacteristic
41C22×D3×C7C2312yes—
42C22×C3×D7C2312yes—
43C22×C3×D7C2312yes—
44C2×D3×D7—24no—
45C2×D3×D7—44no—
46C2×C3×C4×C7C2312yes—
47C2×C3×C4×C7C2311yescharacteristic
48C2×Dic21C2312yes—
49C2×Dic21C2311yescharacteristic
50C4×D21D414yescomplemented
51C2×C7×Dic3C2312yes—
52C7×D12—22no—
53C7×Dic6—22no—
54C3×D28—22no—
55C3×D28—22no—
56C3×D28—44no—
57#4246—24no—
58#4245—24no—
59#4245—44no—
60Dic3×D7—24no—
61C23×D7—32no—
62C2×C7×Q8—31no—
63C2×C7×D4—31no—
64C2×C22⋊D7—32no—
65C2×C22⋊D7—34no—
66#1161—64no—
67D4×D7—64no—
68C2×D28—32no—
69C2×D28C2×D311yescharacteristic
70C2×D28C2×D311yescharacteristic, complemented
71C2×D28C2×D314yescomplemented
72C2×C4×D7—31no—
73C2×C4×D7—32no—
74C7×C22⋊C4—34no—
75C42×C7C2×D311yescharacteristic, complemented
76(C2×D7)⋊C4—34no—
77(C2×D7)⋊C4—32no—
78Dic7⋊C4—32no—
79C4×Dic7—31no—
80C2×#237—71no—
81C2×D3×D4—71no—
82C3×#127—71no—
83#940—71no—
84#939—74no—
85#937—74no—
86#936—71no—
87#875D711yescharacteristic, complemented
88#872—71no—
89C22×C3×C7C2411yescharacteristic
90C2×D21—22no—
91C2×D21C2×D412yes—
92C2×D3×C7—22no—
93C2×D3×C7—24no—
94C2×C3×D7—22no—
95C2×C3×D7—22no—
96C2×C3×D7—24no—
97C2×C3×D7—24no—
98D3×D7—48no—
99C3×C4×C7—22no—
100C3×C4×C7C2×D412yes—
101Dic21—22no—
102Dic21C2×D412yes—
103C7×Dic3—22no—
104#535—211noSylow
105C23×C7—32no—
106C22×D7—31no—
107C22×D7—64no—
108C22×D7—124no—
109C22×D7C22×D312yes—
110C22×D7C22×D312yes—
111C7×Q8—62no—
112C7×D4—62no—
113C2×C4×C7—32no—
114C2×C4×C7C22×D312yes—
115C2×C4×C7C22×D311yescharacteristic
116C22⋊D7—64no—
117C22⋊D7—124no—
118C2×Dic7—31no—
119C2×Dic7—32no—
120D28—22no—
121D28—22no—
122D28—44no—
123D28—64no—
124C4×D7—34no—
125C4×D7—64no—
126C23×D3—72no—
127C2×C3×D4—71no—
128C2×C3×D4—71no—
129C2×C3×D4—74no—
130C2×#67—72no—
131C2×#67—74no—
132C2×#67—72no—
133C22×Dic3—72no—
134#237—144no—
135D3×D4—144no—
136C2×D12C2×D711yescharacteristic
137C2×C4×D3—71no—
138C2×Dic6C2×D711yescharacteristic, complemented
139C3×C42C2×D711yescharacteristic, complemented
140#217—74no—
141(C2×D3)⋊C4—72no—
142(C2×D3)⋊C4C2×D714yescomplemented
143Dic3⋊C4—72no—
144C4×Dic3—71no—
145C2×C3×C7C22×D411yescharacteristic
146C2×C3×C7H(5,𝔽2)12yes—
147D21—22no—
148C3×D7—42no—
149C3×D7—42no—
150D3×C7—42no—
151C2×#41—211no—
152C22×D4—211no—
153#127—71no—
154#127—211no—
155#124—31no—
156#124—211no—
157#121—214no—
158#120—214no—
159#117—211no—
160C22×C7—62no—
161C22×C7—64no—
162C22×C7C23×D311yescharacteristic
163C2×D7—22no—
164C2×D7—22no—
165C2×D7—24no—
166C2×D7—24no—
167C2×D7—32no—
168C2×D7—62no—
169C2×D7—128no—
170C4×C7—22no—
171C4×C7—62no—
172C4×C7D3×D412yes—
173Dic7—32no—
174Dic7—62no—
175C23×C3—72no—
176C23×C3—72no—
177C22×D3—71no—
178C22×D3—144no—
179C22×D3—284no—
180C22×D3C22×D712yes—
181C3×D4—142no—
182C3×D4—142no—
183C3×D4—284no—
184C2×C3×C4C22×D712yes—
185C2×C3×C4C22×D711yescharacteristic
186#67—144no—
187#67—144no—
188#67—284no—
189C2×Dic3—71no—
190C2×Dic3—72no—
191C2×Dic3—144no—
192C2×Dic3C22×D712yes—
193D12—22no—
194C4×D3—74no—
195Dic6—22no—
196C3×C7#53511yescharacteristic, complemented, Hall
197C24—212no—
198#41—424no—
199C2×Q8—31no—
200C2×D4—31no—
201C2×D4—71no—
202C2×D4—71no—
203C2×D4—74no—
204C2×D4—212no—
205C2×D4—212no—
206C2×D4—214no—
207C2×D4—424no—
208C22×C4—212no—
209C22×C4—211no—
210C4⋊C4—212no—
211C22⋊C4—34no—
212C22⋊C4—214no—
213C22⋊C4—212no—
214C42—211no—
215C42D3×D711yescharacteristic, complemented
216C2×C7—122no—
217C2×C7C2×D3×D411yescharacteristic
218C2×C7#100412yes—
219D7—42no—
220D7—42no—
221D7—62no—
222C22×C3—142no—
223C22×C3—142no—
224C22×C3—144no—
225C22×C3—144no—
226C22×C3C23×D711yescharacteristic
227C2×D3—22no—
228C2×D3—24no—
229C2×D3—72no—
230C2×D3—142no—
231C2×D3—288no—
232C3×C4—22no—
233C3×C4D4×D712yes—
234Dic3—22no—
235Dic3—72no—
236Dic3—142no—
237C23—32no—
238C23—72no—
239C23—72no—
240C23—211no—
241C23—424no—
242C23—844no—
243Q8—62no—
244D4—62no—
245D4—142no—
246D4—142no—
247D4—284no—
248D4—424no—
249D4—424no—
250D4—844no—
251C2×C4—32no—
252C2×C4—211no—
253C2×C4—214no—
254C2×C4—212no—
255C2×C4—424no—
256C2×C4C2×D3×D712yes—
257C2×C4C2×D3×D711yescharacteristic
258C7#571511yescharacteristic, complemented, Sylow
259C2×C3—282no—
260C2×C3—282no—
261C2×C3C2×D4×D711yescharacteristic
262C2×C3#655312yes—
263D3—42no—
264D3—142no—
265C22—62no—
266C22—64no—
267C22—142no—
268C22—142no—
269C22—144no—
270C22—144no—
271C22—212no—
272C22—422no—
273C22—848no—
274C22C22×D3×D711yescharacteristic
275C4—22no—
276C4—62no—
277C4—212no—
278C4—422no—
279C4#6668112yes—
280C3#9036511yescharacteristic, complemented, Sylow
281C2—122no—
282C2—282no—
283C2—282no—
284C2—422no—
285C2C2×#6668111yescharacteristic
286C2#1062156412yes—
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C42×C7C22
hypercenterC42D3×D7
hypocenterC3×C7#535
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#66681C2semidirect
2. C2×#66670C2semidirect
3. #10621355C2semidirect
4. C7×#875C2semidirect
5. C3×#6437C2semidirect
6. #10621036C2semidirect
7. #10621025C2semidirect
8. #10620970C2semidirect
9. #10620930C2semidirect
14. C2×C7×Dic6C22semidirect
16. C2×C3×D28C22semidirect
17. C2×C3×D28C22semidirect
20. C2×#4245C22semidirect
21. C2×Dic3×D7C22semidirect
24. C3×C42×C7C22semidirect
26. #66629C22semidirect
27. C4×Dic21C22semidirect
28. C7×(C2×D3)⋊C4C22semidirect
29. (C2×D7)⋊Dic3C22semidirect
37. #6437D3semidirect
50. C4×D21D4semidirect
70. C2×D28C2×D3semidirect
71. C2×D28C2×D3semidirect
75. C42×C7C2×D3semidirect
87. #875D7semidirect
138. C2×Dic6C2×D7semidirect
139. C3×C42C2×D7semidirect
142. (C2×D3)⋊C4C2×D7semidirect
196. C3×C7#535semidirect
215. C42D3×D7semidirect
258. C7#5715semidirect
280. C3#90365semidirect

Maximal subgroups

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

subgroupindex
C2×#666812
C2×#666702
#106213552
C7×#8752
C3×#64372
#106210362
#106210252
#106209702
#106209302
#903653
#57157