Finite Group Databasebeta

← #13059897

Subgroups — #13059897

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
1C22×D84C211yesmaximal, characteristic, complemented
2C2×#66694C211yesmaximal, characteristic, complemented
3C7×#929C211yesmaximal, characteristic, complemented
4C3×#6492C211yesmaximal, characteristic, complemented
5#10621135C211yesmaximal, characteristic, complemented
6#10621134C211yesmaximal, characteristic, complemented
7#10621123C211yesmaximal, characteristic, complemented
8#10621038C212yesmaximal, complemented
9#10621021C212yesmaximal, complemented
10#10620995C212yesmaximal, complemented
11#10620982C212yesmaximal, complemented
12#90344—31nomaximal, Hall
13C23×D21C2211yescharacteristic
14C23×D21C2211yescharacteristic, complemented
15C22×D3×D7C2211yescharacteristic, complemented
16C22×C3×C4×C7C2211yescharacteristic, complemented
17C2×D84—24no—
18C2×D84C2212yes—
19C2×D84C2212yescomplemented
20C2×C7×#67C2211yescharacteristic
21C2×C3×C22⋊D7C2211yescharacteristic
22#66694—24no—
23C2×#4247C2211yescharacteristic, complemented
24C2×#4247C2212yescomplemented
25C2×#4246C2211yescharacteristic, complemented
26C2×#4246C2212yescomplemented
27C2×D21⋊C4C2211yescharacteristic, complemented
28C2×D21⋊C4C2212yescomplemented
29C7×#217C2211yescharacteristic, complemented
30C7×(C2×D3)⋊C4C2212yescomplemented
31C7×Dic3⋊C4C2212yescomplemented
32C3×C22⋊Dic7C2211yescharacteristic, complemented
33C3×(C2×D7)⋊C4C2212yescomplemented
34C3×Dic7⋊C4C2212yescomplemented
35#66575C2212yescomplemented
36#66575C2212yescomplemented
37#66575C2212yescomplemented
38C2×D4×D7—31no—
39C22×D28—31no—
40C7×#123—31no—
41#6500—31no—
42#6492D311yescharacteristic, complemented
43#6458—32no—
44#6457—32no—
45#6447—32no—
46#6445—31no—
47#6445—32no—
48D7×C22⋊C4—31no—
49#5653—71nomaximal, Hall
50C23×C3×C7C2311yescharacteristic
51C22×D21—22no—
52C22×D21—42no—
53C22×D21—42no—
54C22×D21D414yescomplemented
55C22×D21C2312yes—
56C22×D21C2311yescharacteristic
57C22×D21C2311yescharacteristic
58C22×D3×C7C2311yescharacteristic
59C22×C3×D7C2311yescharacteristic
60C2×D3×D7—22no—
61C2×D3×D7—42no—
62C2×C3×C4×C7—22no—
63C2×C3×C4×C7C2312yes—
64D84—24no—
65D84—24no—
66C7×#67—22no—
67C2×C7×Dic3C2311yescharacteristic
68C2×C7×Dic3C2312yes—
69C3×C22⋊D7—22no—
70C2×C3×Dic7C2311yescharacteristic
71C2×C3×Dic7C2312yes—
72#4247—22no—
73#4247—42no—
74#4246—22no—
75#4246—42no—
76D21⋊C4—22no—
77D21⋊C4—42no—
78C23×D7—31no—
79C23×D7—31no—
80C23×D7—31no—
81C2×C7×D4—31no—
82C22×C4×C7C2×D311yescharacteristic, complemented
83C2×C22⋊D7—31no—
84C2×C22⋊D7—32no—
85C2×C22⋊D7C2×D311yescharacteristic
86D4×D7—64no—
87C2×D28—31no—
88C2×D28—32no—
89C2×D28—32no—
90C2×D28—32no—
91C2×D28—64no—
92C2×C4×D7—31no—
93C2×C4×D7—32no—
94C7×C4⋊C4—32no—
95C7×C22⋊C4—31no—
96C7×C22⋊C4—32no—
97C22⋊Dic7C2×D311yescharacteristic, complemented
98(C2×D7)⋊C4—32no—
99(C2×D7)⋊C4—32no—
100(C2×D7)⋊C4—32no—
101(C2×D7)⋊C4C2×D312yescomplemented
102Dic7⋊C4C2×D312yescomplemented
103C2×D3×D4—71no—
104C22×D12—71no—
105C3×#123—71no—
106#937—71no—
107#929D711yescharacteristic, complemented
108#895—72no—
109#894—72no—
110#884—72no—
111#882—71no—
112#882—72no—
113D3×C22⋊C4—71no—
114C22×C3×C7—22no—
115C22×C3×C7C2×D412yes—
116C22×C3×C7C2411yescharacteristic
117C2×D21—22no—
118C2×D21—22no—
119C2×D21—22no—
120C2×D21—22no—
121C2×D21—22no—
122C2×D21—44no—
123C2×D21—44no—
124C2×D21C2×D412yes—
125C2×D21C2×D412yes—
126C2×D3×C7—21no—
127C2×D3×C7—22no—
128C2×C3×D7—21no—
129C2×C3×D7—22no—
130D3×D7—44no—
131C3×C4×C7—22no—
132C3×Dic7—21no—
133C3×Dic7—22no—
134C7×Dic3—21no—
135C7×Dic3—22no—
136#534—211noSylow
137C23×C7—31no—
138C23×C7C22×D311yescharacteristic
139C22×D7—31no—
140C22×D7—31no—
141C22×D7—34no—
142C22×D7—32no—
143C22×D7—62no—
144C22×D7—62no—
145C22×D7—122no—
146C22×D7—122no—
147C22×D7—122no—
148C22×D7C22×D311yescharacteristic
149C7×D4—62no—
150C2×C4×C7—22no—
151C2×C4×C7—32no—
152C2×C4×C7—31no—
153C2×C4×C7C22×D312yes—
154C22⋊D7—22no—
155C22⋊D7—62no—
156C22⋊D7—122no—
157C2×Dic7C22×D311yescharacteristic
158C2×Dic7C22×D312yes—
159D28—64no—
160D28—64no—
161D28—62no—
162D28—122no—
163C4×D7—62no—
164C4×D7—122no—
165C23×D3—71no—
166C23×D3—71no—
167C23×D3—71no—
168C2×C3×D4—71no—
169C22×C3×C4C2×D711yescharacteristic, complemented
170C2×#67—71no—
171C2×#67—72no—
172C2×#67C2×D711yescharacteristic
173D3×D4—144no—
174C2×D12—71no—
175C2×D12—72no—
176C2×D12—72no—
177C2×D12—72no—
178C2×D12—144no—
179C2×C4×D3—71no—
180C2×C4×D3—72no—
181C3×C4⋊C4—72no—
182C3×C22⋊C4—71no—
183C3×C22⋊C4—72no—
184#217C2×D711yescharacteristic, complemented
185(C2×D3)⋊C4—72no—
186(C2×D3)⋊C4—72no—
187(C2×D3)⋊C4—72no—
188(C2×D3)⋊C4C2×D712yescomplemented
189Dic3⋊C4C2×D712yescomplemented
190C2×C3×C7—22no—
191C2×C3×C7C22×D411yescharacteristic
192C2×C3×C7H(5,𝔽2)12yes—
193D21—22no—
194D21—22no—
195D21—42no—
196C3×D7—41no—
197D3×C7—41no—
198C22×D4—211no—
199C22×D4—211no—
200#123—31no—
201#123—71no—
202#123—212no—
203#121—212no—
204#121—212no—
205#120—212no—
206#120—211no—
207#120—211no—
208C2×C22⋊C4—211no—
209C22×C7—22no—
210C22×C7—61no—
211C22×C7—62no—
212C22×C7D3×D412yes—
213C22×C7C23×D311yescharacteristic
214C2×D7—21no—
215C2×D7—22no—
216C2×D7—32no—
217C2×D7—32no—
218C2×D7—62no—
219C2×D7—62no—
220C2×D7—62no—
221C2×D7—62no—
222C2×D7—62no—
223C2×D7—124no—
224C2×D7—124no—
225C2×D7—124no—
226C4×C7—22no—
227C4×C7—61no—
228C4×C7—62no—
229Dic7—21no—
230Dic7—22no—
231C23×C3—71no—
232C23×C3C22×D711yescharacteristic
233C22×D3—71no—
234C22×D3—71no—
235C22×D3—74no—
236C22×D3—72no—
237C22×D3—142no—
238C22×D3—142no—
239C22×D3—282no—
240C22×D3—282no—
241C22×D3—282no—
242C22×D3C22×D711yescharacteristic
243C3×D4—142no—
244C2×C3×C4—22no—
245C2×C3×C4—72no—
246C2×C3×C4—71no—
247C2×C3×C4C22×D712yes—
248#67—22no—
249#67—142no—
250#67—282no—
251C2×Dic3C22×D711yescharacteristic
252C2×Dic3C22×D712yes—
253D12—144no—
254D12—144no—
255D12—142no—
256D12—282no—
257C4×D3—142no—
258C4×D3—282no—
259C3×C7#53411yescharacteristic, complemented, Hall
260C24—211no—
261C24—211no—
262C24—211no—
263C2×D4—31no—
264C2×D4—71no—
265C2×D4—211no—
266C2×D4—212no—
267C2×D4—211no—
268C2×D4—212no—
269C2×D4—212no—
270C2×D4—212no—
271C2×D4—424no—
272C2×D4—424no—
273C22×C4—211no—
274C22×C4—212no—
275C22×C4D3×D711yescharacteristic, complemented
276C4⋊C4—32no—
277C4⋊C4—72no—
278C22⋊C4—31no—
279C22⋊C4—32no—
280C22⋊C4—71no—
281C22⋊C4—72no—
282C22⋊C4—212no—
283C22⋊C4—212no—
284C22⋊C4—212no—
285C2×C7—22no—
286C2×C7—121no—
287C2×C7C2×D3×D411yescharacteristic
288C2×C7#100912yes—
289D7—41no—
290D7—62no—
291D7—62no—
292D7—122no—
293C22×C3—22no—
294C22×C3—141no—
295C22×C3—142no—
296C22×C3D4×D712yes—
297C22×C3C23×D711yescharacteristic
298C2×D3—21no—
299C2×D3—22no—
300C2×D3—72no—
301C2×D3—72no—
302C2×D3—142no—
303C2×D3—142no—
304C2×D3—142no—
305C2×D3—142no—
306C2×D3—142no—
307C2×D3—284no—
308C2×D3—284no—
309C2×D3—284no—
310C3×C4—22no—
311C3×C4—141no—
312C3×C4—142no—
313Dic3—21no—
314Dic3—22no—
315C23—31no—
316C23—71no—
317C23—211no—
318C23—211no—
319C23—214no—
320C23—212no—
321C23—422no—
322C23—422no—
323C23—842no—
324C23—842no—
325C23—842no—
326C23C2×D3×D711yescharacteristic
327D4—62no—
328D4—142no—
329D4—424no—
330D4—424no—
331D4—422no—
332D4—422no—
333D4—842no—
334D4—842no—
335C2×C4—22no—
336C2×C4—32no—
337C2×C4—31no—
338C2×C4—72no—
339C2×C4—71no—
340C2×C4—422no—
341C2×C4—842no—
342C2×C4C2×D3×D712yes—
343C7#565311yescharacteristic, complemented, Sylow
344C2×C3—22no—
345C2×C3—281no—
346C2×C3C2×D4×D711yescharacteristic
347C2×C3#655312yes—
348D3—41no—
349D3—142no—
350D3—142no—
351D3—282no—
352C22—22no—
353C22—62no—
354C22—61no—
355C22—142no—
356C22—141no—
357C22—212no—
358C22—212no—
359C22—422no—
360C22—422no—
361C22—422no—
362C22—422no—
363C22—422no—
364C22—844no—
365C22—844no—
366C22—844no—
367C22#6669412yes—
368C22C22×D3×D711yescharacteristic
369C4—22no—
370C4—62no—
371C4—61no—
372C4—142no—
373C4—141no—
374C3#9034411yescharacteristic, complemented, Sylow
375C2—22no—
376C2—121no—
377C2—281no—
378C2—422no—
379C2—422no—
380C2—842no—
381C2#1062156612yes—
382C2C2×#6669411yescharacteristic
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. C22×D84C2semidirect
2. C2×#66694C2semidirect
3. C7×#929C2semidirect
4. C3×#6492C2semidirect
5. #10621135C2semidirect
6. #10621134C2semidirect
7. #10621123C2semidirect
8. #10621038C2semidirect
9. #10621021C2semidirect
10. #10620995C2semidirect
11. #10620982C2semidirect
14. C23×D21C22semidirect
15. C22×D3×D7C22semidirect
16. C22×C3×C4×C7C22semidirect
19. C2×D84C22semidirect
23. C2×#4247C22semidirect
24. C2×#4247C22semidirect
25. C2×#4246C22semidirect
26. C2×#4246C22semidirect
27. C2×D21⋊C4C22semidirect
28. C2×D21⋊C4C22semidirect
29. C7×#217C22semidirect
30. C7×(C2×D3)⋊C4C22semidirect
31. C7×Dic3⋊C4C22semidirect
32. C3×C22⋊Dic7C22semidirect
33. C3×(C2×D7)⋊C4C22semidirect
34. C3×Dic7⋊C4C22semidirect
35. #66575C22semidirect
36. #66575C22semidirect
37. #66575C22semidirect
42. #6492D3semidirect
54. C22×D21D4semidirect
82. C22×C4×C7C2×D3semidirect
97. C22⋊Dic7C2×D3semidirect
101. (C2×D7)⋊C4C2×D3semidirect
102. Dic7⋊C4C2×D3semidirect
107. #929D7semidirect
169. C22×C3×C4C2×D7semidirect
184. #217C2×D7semidirect
188. (C2×D3)⋊C4C2×D7semidirect
189. Dic3⋊C4C2×D7semidirect
259. C3×C7#534semidirect
275. C22×C4D3×D7semidirect
343. C7#5653semidirect
374. C3#90344semidirect

Maximal subgroups

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

subgroupindex
C22×D842
C2×#666942
C7×#9292
C3×#64922
#106211352
#106211342
#106211232
#106210382
#106210212
#106209952
#106209822
#903443
#56537