Finite Group Databasebeta

← #13059902

Subgroups — #13059902

Order 1344.

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

normal subgroups only

This group has 692 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×#66694C212yesmaximal, complemented
2C2×#66631C211yesmaximal, characteristic, complemented
3C7×#930C211yesmaximal, characteristic, complemented
4C3×#6493C211yesmaximal, characteristic, complemented
5#10621133C212yesmaximal, complemented
6#10621037C212yesmaximal, complemented
7#10621036C212yesmaximal, complemented
8#10621000C212yesmaximal, complemented
9#10620918C212yesmaximal, complemented
10#90296—31nomaximal, Hall
11C23×D21C2211yescharacteristic
12C22×D3×D7C2212yescomplemented
13C22×C3×C4×C7C2211yescharacteristic, complemented
14C22×Dic21C2211yescharacteristic, complemented
15C2×C7×#67C2212yescomplemented
16C2×C7×D12C2211yescharacteristic, complemented
17C2×C3×C22⋊D7C2212yescomplemented
18C2×C3×D28C2211yescharacteristic, complemented
19#66694—28no—
20C2×#4247C2212yes—
21C2×#4246C2212yes—
22C2×#4245C2212yescomplemented
23C2×D21⋊C4C2212yescomplemented
24#66631C2212yescomplemented
25#66631C2212yescomplemented
26C7×(C2×D3)⋊C4C2212yescomplemented
27C7×C4⋊Dic3C2211yescharacteristic, complemented
28C3×(C2×D7)⋊C4C2212yescomplemented
29C3×C4⋊Dic7C2211yescharacteristic, complemented
30#66577C2212yescomplemented
31(C2×D3)⋊Dic7C2212yescomplemented
32(C2×D7)⋊Dic3C2212yescomplemented
33C2×D4×D7—32no—
34C7×#121—31no—
35#6502—32no—
36#6500—32no—
37#6493D311yescharacteristic, complemented
38C2×(C2×D7)⋊C4—31no—
39#6457—32no—
40#6446—32no—
41#6445—32no—
42#5605—71nomaximal, Hall
43C23×C3×C7C2311yescharacteristic
44C22×D21—41no—
45C22×D21—41no—
46C22×D21D414yescomplemented
47C22×D21C2312yes—
48C22×D3×C7C2312yes—
49C22×C3×D7C2312yes—
50C2×D3×D7—24no—
51C2×D3×D7—42no—
52C2×D3×D7—42no—
53C2×C3×C4×C7—41no—
54C2×C3×C4×C7C2311yescharacteristic
55C2×C3×C4×C7C2311yescharacteristic
56C2×Dic21—41no—
57C2×Dic21C2312yes—
58C7×#67—24no—
59C2×C7×Dic3C2312yes—
60C7×D12—41no—
61C3×C22⋊D7—24no—
62C2×C3×Dic7C2312yes—
63C3×D28—41no—
64#4247—24no—
65#4246—24no—
66#4245—42no—
67D21⋊C4—24no—
68C23×D7—31no—
69C23×D7—32no—
70C2×C7×D4—32no—
71C2×C7×D4—31no—
72C22×C4×C7C2×D311yescharacteristic, complemented
73C2×C22⋊D7—32no—
74C2×C22⋊D7—32no—
75C2×C22⋊D7C2×D312yescomplemented
76C22×Dic7—31no—
77D4×D7—68no—
78C2×D28—32no—
79C2×D28C2×D311yescharacteristic, complemented
80C2×C4×D7—32no—
81C7×C4⋊C4—31no—
82C7×C22⋊C4—32no—
83C22⋊Dic7—32no—
84(C2×D7)⋊C4—32no—
85(C2×D7)⋊C4—32no—
86(C2×D7)⋊C4—32no—
87(C2×D7)⋊C4C2×D312yescomplemented
88C4⋊Dic7C2×D311yescharacteristic, complemented
89Dic7⋊C4—32no—
90C2×D3×D4—72no—
91C3×#121—71no—
92#939—72no—
93#937—72no—
94#930D711yescharacteristic, complemented
95C2×(C2×D3)⋊C4—71no—
96#894—72no—
97#883—72no—
98#882—72no—
99C22×C3×C7—21no—
100C22×C3×C7—21no—
101C22×C3×C7C2×D412yes—
102C22×C3×C7C2411yescharacteristic
103C2×D21—22no—
104C2×D21—22no—
105C2×D21—44no—
106C2×D21C2×D414yes—
107C2×D3×C7—22no—
108C2×D3×C7—22no—
109C2×D3×C7—22no—
110C2×C3×D7—22no—
111C2×C3×D7—22no—
112C2×C3×D7—22no—
113D3×D7—48no—
114C3×C4×C7—21no—
115C3×C4×C7—21no—
116Dic21—22no—
117C3×Dic7—22no—
118C7×Dic3—22no—
119#534—211noSylow
120C23×C7—32no—
121C23×C7C22×D311yescharacteristic
122C22×D7—32no—
123C22×D7—34no—
124C22×D7—64no—
125C22×D7—122no—
126C22×D7—122no—
127C22×D7—121no—
128C22×D7—121no—
129C22×D7C22×D312yes—
130C7×D4—64no—
131C7×D4—121no—
132C2×C4×C7—32no—
133C2×C4×C7—41no—
134C2×C4×C7C22×D311yescharacteristic
135C2×C4×C7C22×D311yescharacteristic
136C22⋊D7—24no—
137C22⋊D7—64no—
138C22⋊D7—122no—
139C2×Dic7—32no—
140C2×Dic7—121no—
141C2×Dic7C22×D312yes—
142D28—41no—
143D28—64no—
144C4×D7—64no—
145C23×D3—71no—
146C23×D3—72no—
147C2×C3×D4—72no—
148C2×C3×D4—71no—
149C22×C3×C4C2×D711yescharacteristic, complemented
150C2×#67—72no—
151C2×#67—72no—
152C2×#67C2×D712yescomplemented
153C22×Dic3—71no—
154D3×D4—148no—
155C2×D12—72no—
156C2×D12C2×D711yescharacteristic, complemented
157C2×C4×D3—72no—
158C3×C4⋊C4—71no—
159C3×C22⋊C4—72no—
160#217—72no—
161(C2×D3)⋊C4—72no—
162(C2×D3)⋊C4—72no—
163(C2×D3)⋊C4—72no—
164(C2×D3)⋊C4C2×D712yescomplemented
165C4⋊Dic3C2×D711yescharacteristic, complemented
166Dic3⋊C4—72no—
167C2×C3×C7—22no—
168C2×C3×C7C22×D411yescharacteristic
169C2×C3×C7H(5,𝔽2)11yescharacteristic
170C2×C3×C7H(5,𝔽2)11yescharacteristic
171D21—24no—
172C3×D7—42no—
173D3×C7—42no—
174C22×D4—212no—
175#123—212no—
176#123—212no—
177#121—31no—
178#121—71no—
179#121—212no—
180#120—212no—
181#120—212no—
182C2×C22⋊C4—211no—
183C22×C7—21no—
184C22×C7—21no—
185C22×C7—62no—
186C22×C7—62no—
187C22×C7—62no—
188C22×C7D3×D412yes—
189C22×C7C23×D311yescharacteristic
190C2×D7—22no—
191C2×D7—22no—
192C2×D7—22no—
193C2×D7—34no—
194C2×D7—62no—
195C2×D7—62no—
196C2×D7—128no—
197C2×D7—124no—
198C4×C7—21no—
199C4×C7—21no—
200C4×C7—62no—
201Dic7—22no—
202Dic7—62no—
203C23×C3—72no—
204C23×C3C22×D711yescharacteristic
205C22×D3—72no—
206C22×D3—74no—
207C22×D3—144no—
208C22×D3—282no—
209C22×D3—282no—
210C22×D3—281no—
211C22×D3—281no—
212C22×D3C22×D712yes—
213C3×D4—144no—
214C3×D4—281no—
215C2×C3×C4—41no—
216C2×C3×C4—72no—
217C2×C3×C4C22×D711yescharacteristic
218C2×C3×C4C22×D711yescharacteristic
219#67—24no—
220#67—144no—
221#67—282no—
222C2×Dic3—72no—
223C2×Dic3—281no—
224C2×Dic3C22×D712yes—
225D12—41no—
226D12—144no—
227C4×D3—144no—
228C3×C7#53411yescharacteristic, complemented, Hall
229C24—212no—
230C24—211no—
231C2×D4—32no—
232C2×D4—31no—
233C2×D4—72no—
234C2×D4—71no—
235C2×D4—212no—
236C2×D4—212no—
237C2×D4—212no—
238C2×D4—428no—
239C22×C4—212no—
240C22×C4—211no—
241C22×C4D3×D711yescharacteristic, complemented
242C4⋊C4—31no—
243C4⋊C4—71no—
244C4⋊C4—212no—
245C22⋊C4—32no—
246C22⋊C4—72no—
247C22⋊C4—212no—
248C22⋊C4—212no—
249C22⋊C4—212no—
250C22⋊C4—212no—
251C2×C7—22no—
252C2×C7—122no—
253C2×C7C2×D3×D411yescharacteristic
254C2×C7#100411yescharacteristic
255C2×C7#100911yescharacteristic
256D7—42no—
257D7—64no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—142no—
261C22×C3—142no—
262C22×C3—142no—
263C22×C3D4×D712yes—
264C22×C3C23×D711yescharacteristic
265C2×D3—22no—
266C2×D3—22no—
267C2×D3—22no—
268C2×D3—74no—
269C2×D3—142no—
270C2×D3—142no—
271C2×D3—288no—
272C2×D3—284no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—142no—
276Dic3—22no—
277Dic3—142no—
278C23—32no—
279C23—72no—
280C23—212no—
281C23—214no—
282C23—424no—
283C23—841no—
284C23—841no—
285C23—842no—
286C23—842no—
287C23C2×D3×D711yescharacteristic
288D4—64no—
289D4—121no—
290D4—144no—
291D4—281no—
292D4—424no—
293D4—424no—
294D4—842no—
295C2×C4—32no—
296C2×C4—41no—
297C2×C4—72no—
298C2×C4—212no—
299C2×C4—424no—
300C2×C4—841no—
301C2×C4C2×D3×D711yescharacteristic
302C2×C4C2×D3×D711yescharacteristic
303C7#560511yescharacteristic, complemented, Sylow
304C2×C3—22no—
305C2×C3—282no—
306C2×C3C2×D4×D711yescharacteristic
307C2×C3#654811yescharacteristic
308C2×C3#655311yescharacteristic
309D3—42no—
310D3—144no—
311C22—21no—
312C22—21no—
313C22—62no—
314C22—62no—
315C22—62no—
316C22—142no—
317C22—142no—
318C22—142no—
319C22—214no—
320C22—422no—
321C22—422no—
322C22—844no—
323C22—848no—
324C22#6668112yes—
325C22C22×D3×D711yescharacteristic
326C4—21no—
327C4—21no—
328C4—62no—
329C4—142no—
330C4—422no—
331C3#9029611yescharacteristic, complemented, Sylow
332C2—22no—
333C2—122no—
334C2—282no—
335C2—424no—
336C2C2×#6668111yescharacteristic
337C2#1062156611yescharacteristic
338C2#1062159611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC22×C3×C4×C7C22
hypercenterC22×C4D3×D7
hypocenterC3×C7#534
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#66694C2semidirect
2. C2×#66631C2semidirect
3. C7×#930C2semidirect
4. C3×#6493C2semidirect
5. #10621133C2semidirect
6. #10621037C2semidirect
7. #10621036C2semidirect
8. #10621000C2semidirect
9. #10620918C2semidirect
12. C22×D3×D7C22semidirect
13. C22×C3×C4×C7C22semidirect
14. C22×Dic21C22semidirect
15. C2×C7×#67C22semidirect
16. C2×C7×D12C22semidirect
17. C2×C3×C22⋊D7C22semidirect
18. C2×C3×D28C22semidirect
22. C2×#4245C22semidirect
23. C2×D21⋊C4C22semidirect
24. #66631C22semidirect
25. #66631C22semidirect
26. C7×(C2×D3)⋊C4C22semidirect
27. C7×C4⋊Dic3C22semidirect
28. C3×(C2×D7)⋊C4C22semidirect
29. C3×C4⋊Dic7C22semidirect
30. #66577C22semidirect
31. (C2×D3)⋊Dic7C22semidirect
32. (C2×D7)⋊Dic3C22semidirect
37. #6493D3semidirect
46. C22×D21D4semidirect
72. C22×C4×C7C2×D3semidirect
75. C2×C22⋊D7C2×D3semidirect
79. C2×D28C2×D3semidirect
87. (C2×D7)⋊C4C2×D3semidirect
88. C4⋊Dic7C2×D3semidirect
94. #930D7semidirect
149. C22×C3×C4C2×D7semidirect
152. C2×#67C2×D7semidirect
156. C2×D12C2×D7semidirect
164. (C2×D3)⋊C4C2×D7semidirect
165. C4⋊Dic3C2×D7semidirect
228. C3×C7#534semidirect
241. C22×C4D3×D7semidirect
303. C7#5605semidirect
331. C3#90296semidirect

Maximal subgroups

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

subgroupindex
C2×#666942
C2×#666312
C7×#9302
C3×#64932
#106211332
#106210372
#106210362
#106210002
#106209182
#902963
#56057