Finite Group Databasebeta

← #13060187

Subgroups — #13060187

Order 1344.

504 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 120 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#10621423C211yesmaximal, characteristic, complemented
2C7×#935C211yesmaximal, characteristic, complemented
3C3×#6500C211yesmaximal, characteristic, complemented
4#10621131C212yesmaximal, complemented
5#10621126C212yesmaximal, complemented
6#10621124C212yesmaximal, complemented
7#10621095C212yesmaximal, complemented
8#10621021C212yesmaximal, complemented
9#10620928C212yesmaximal, complemented
10#90357—31nomaximal, Hall
11C23×C3×D7C2211yescharacteristic
12C2×C3×C7×D4C2211yescharacteristic
13C2×#4268C2212yes—
14C2×#4268C2212yescomplemented
15C2×D84C2211yescharacteristic, complemented
16C22×C7×Dic3C2211yescharacteristic, complemented
17C2×C3×C22⋊D7C2212yes—
18C2×#4246C2212yes—
19C2×#4246C2212yes—
20C2×D21⋊C4C2212yescomplemented
21C4×Dic21C2211yescharacteristic, complemented
22C7×#217C2212yes—
23C7×#217C2211yescharacteristic, complemented
24C7×Dic3⋊C4C2212yescomplemented
25C3×C22⋊Dic7C2211yescharacteristic
26C3×(C2×D7)⋊C4C2212yes—
27Dic3⋊Dic7C2212yescomplemented
28Dic7⋊Dic3C2212yescomplemented
29#66575C2212yescomplemented
30(C2×D7)⋊Dic3C2212yes—
31(C2×D7)⋊Dic3C2212yes—
32C7×#123—31no—
33#6503—31no—
34#6500D311yescharacteristic, complemented
35#6493—32no—
36#6461—32no—
37#6458—32no—
38#6447—32no—
39#6446—32no—
40#6445—32no—
41#5595—71nomaximal, Hall
42C23×C3×C7C2311yescharacteristic
43C23×C3×C7C2311yescharacteristic
44C22×D21C2312yes—
45C22×C3×D7—41no—
46C22×C3×D7—42no—
47C22×C3×D7C2312yes—
48C3×C7×D4—41no—
49C2×C3×C4×C7C2311yescharacteristic
50#4268—42no—
51#4268—42no—
52C2×Dic21C2312yes—
53D84—41no—
54C2×C7×Dic3—41no—
55C2×C7×Dic3C2312yes—
56C2×C7×Dic3C2312yes—
57C3×C22⋊D7—42no—
58C2×C3×Dic7C2312yes—
59#4246—42no—
60#4246—42no—
61D21⋊C4—42no—
62C23×D7C2×D311yescharacteristic
63C2×C7×D4C2×D311yescharacteristic
64C22×C4×C7—31no—
65C2×C22⋊D7—32no—
66C2×C22⋊D7—32no—
67C2×C22⋊D7C2×D312yes—
68C2×D28—32no—
69C2×D28—32no—
70C2×D28—31no—
71C2×C4×D7—32no—
72C7×C4⋊C4—32no—
73C7×C22⋊C4—32no—
74C7×C22⋊C4—31no—
75C22⋊Dic7C2×D311yescharacteristic
76(C2×D7)⋊C4—32no—
77(C2×D7)⋊C4—32no—
78(C2×D7)⋊C4—32no—
79(C2×D7)⋊C4C2×D312yes—
80C4⋊Dic7—32no—
81Dic7⋊C4—32no—
82C4×Dic7—31no—
83C3×#120—71no—
84#953—72no—
85#940—71no—
86#939—72no—
87#938—72no—
88#935D711yescharacteristic, complemented
89#884—72no—
90#883—72no—
91#879—72no—
92C22×C3×C7—22no—
93C22×C3×C7—21no—
94C22×C3×C7—22no—
95C22×C3×C7—21no—
96C22×C3×C7C2411yescharacteristic
97C2×D21—22no—
98C2×D21—22no—
99C2×D21—22no—
100C2×C3×D7—22no—
101C2×C3×D7—22no—
102C2×C3×D7—22no—
103C2×C3×D7—44no—
104C3×C4×C7—21no—
105Dic21—22no—
106C3×Dic7—22no—
107C7×Dic3—22no—
108C7×Dic3—22no—
109#560—211noSylow
110C23×C7C22×D311yescharacteristic
111C23×C7C22×D311yescharacteristic
112C22×D7—32no—
113C22×D7—41no—
114C22×D7—42no—
115C22×D7C22×D312yes—
116C7×D4—41no—
117C2×C4×C7—32no—
118C2×C4×C7—32no—
119C2×C4×C7—121no—
120C2×C4×C7C22×D311yescharacteristic
121C22⋊D7—42no—
122C22⋊D7—122no—
123C22⋊D7—122no—
124C2×Dic7—32no—
125C2×Dic7C22×D312yes—
126D28—121no—
127D28—122no—
128D28—122no—
129C4×D7—122no—
130C24×C3—71no—
131C2×C3×D4—72no—
132C2×C3×D4C2×D711yescharacteristic
133C2×#67—72no—
134C2×#67—72no—
135C2×#67—72no—
136C2×#67—72no—
137C22×Dic3C2×D711yescharacteristic, complemented
138C2×D12—71no—
139C2×C4×D3—72no—
140C3×C22⋊C4—71no—
141C3×C22⋊C4—72no—
142#217—72no—
143#217—72no—
144#217C2×D712yes—
145#217C2×D711yescharacteristic, complemented
146(C2×D3)⋊C4—72no—
147C4⋊Dic3—72no—
148Dic3⋊C4—72no—
149Dic3⋊C4C2×D712yescomplemented
150C4×Dic3—71no—
151C2×C3×C7—41no—
152C2×C3×C7—41no—
153C2×C3×C7H(5,𝔽2)11yescharacteristic
154C2×C3×C7H(5,𝔽2)12yes—
155D21—42no—
156C3×D7—42no—
157#127—211no—
158#126—212no—
159#123—31no—
160#123—212no—
161#121—212no—
162#121—212no—
163#121—212no—
164#120—71no—
165#120—212no—
166C22×C7—22no—
167C22×C7—22no—
168C22×C7—21no—
169C22×C7—21no—
170C22×C7C23×D311yescharacteristic
171C2×D7—22no—
172C2×D7—22no—
173C2×D7—22no—
174C2×D7—44no—
175C2×D7—62no—
176C2×D7—62no—
177C2×D7—62no—
178C4×C7—21no—
179C4×C7—62no—
180C4×C7—62no—
181Dic7—22no—
182Dic7—62no—
183C23×C3—72no—
184C23×C3—281no—
185C23×C3—282no—
186C23×C3C22×D711yescharacteristic
187C23×C3C22×D711yescharacteristic
188C22×D3—72no—
189C3×D4—41no—
190C3×D4—282no—
191C2×C3×C4—72no—
192C2×C3×C4C22×D711yescharacteristic
193#67—282no—
194#67—282no—
195#67—282no—
196#67—282no—
197C2×Dic3—41no—
198C2×Dic3—72no—
199C2×Dic3C22×D712yes—
200C2×Dic3C22×D712yes—
201D12—281no—
202C4×D3—282no—
203C3×C7#56011yescharacteristic, complemented, Hall
204C24—71no—
205C2×D4—72no—
206C2×D4—212no—
207C2×D4—212no—
208C2×D4—212no—
209C2×D4—212no—
210C2×D4—211no—
211C2×D4D3×D711yescharacteristic
212C22×C4—31no—
213C22×C4—212no—
214C4⋊C4—32no—
215C4⋊C4—212no—
216C4⋊C4—212no—
217C22⋊C4—32no—
218C22⋊C4—31no—
219C22⋊C4—71no—
220C22⋊C4—72no—
221C22⋊C4—212no—
222C22⋊C4—212no—
223C22⋊C4—212no—
224C42—211no—
225C2×C7—41no—
226C2×C7—41no—
227C2×C7#100412yes—
228C2×C7#100411yescharacteristic
229D7—42no—
230D7—122no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—22no—
234C22×C3—22no—
235C22×C3—142no—
236C22×C3—142no—
237C22×C3—142no—
238C22×C3—284no—
239C22×C3C23×D711yescharacteristic
240C2×D3—142no—
241C2×D3—142no—
242C2×D3—142no—
243C3×C4—21no—
244C3×C4—142no—
245Dic3—22no—
246Dic3—22no—
247Dic3—142no—
248C23—72no—
249C23—212no—
250C23—282no—
251C23—281no—
252C23C2×D3×D711yescharacteristic
253C23C2×D3×D711yescharacteristic
254D4—41no—
255D4—282no—
256D4—841no—
257D4—842no—
258D4—842no—
259D4—842no—
260D4—842no—
261C2×C4—32no—
262C2×C4—32no—
263C2×C4—72no—
264C2×C4—121no—
265C2×C4—212no—
266C2×C4—842no—
267C2×C4C2×D3×D711yescharacteristic
268C7#559511yescharacteristic, complemented, Sylow
269C2×C3—41no—
270C2×C3—41no—
271C2×C3—282no—
272C2×C3#654811yescharacteristic
273C2×C3#655312yes—
274D3—282no—
275C22—22no—
276C22—22no—
277C22—21no—
278C22—21no—
279C22—142no—
280C22—142no—
281C22—142no—
282C22—284no—
283C22—422no—
284C22—422no—
285C22—422no—
286C22C22×D3×D711yescharacteristic
287C4—21no—
288C4—62no—
289C4—62no—
290C4—142no—
291C4—422no—
292C3#9035711yescharacteristic, complemented, Sylow
293C2—41no—
294C2—41no—
295C2—282no—
296C2—842no—
297C2#1062157312yes—
298C2#1062159611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC2×C3×C7×D4C22
hypercenterC2×D4D3×D7
hypocenterC3×C7#560
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #10621423C2semidirect
2. C7×#935C2semidirect
3. C3×#6500C2semidirect
4. #10621131C2semidirect
5. #10621126C2semidirect
6. #10621124C2semidirect
7. #10621095C2semidirect
8. #10621021C2semidirect
9. #10620928C2semidirect
14. C2×#4268C22semidirect
15. C2×D84C22semidirect
16. C22×C7×Dic3C22semidirect
20. C2×D21⋊C4C22semidirect
21. C4×Dic21C22semidirect
23. C7×#217C22semidirect
24. C7×Dic3⋊C4C22semidirect
27. Dic3⋊Dic7C22semidirect
28. Dic7⋊Dic3C22semidirect
29. #66575C22semidirect
34. #6500D3semidirect
88. #935D7semidirect
137. C22×Dic3C2×D7semidirect
145. #217C2×D7semidirect
149. Dic3⋊C4C2×D7semidirect
203. C3×C7#560semidirect
268. C7#5595semidirect
292. C3#90357semidirect

Maximal subgroups

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

subgroupindex
#106214232
C7×#9352
C3×#65002
#106211312
#106211262
#106211242
#106210952
#106210212
#106209282
#903573
#55957