Finite Group Databasebeta

← D3×C7×#137 #13061492

Subgroups — D3×C7×#137 #13061492

Order 1344.

516 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×C7×#41C211yesmaximal, characteristic
2C2×D3×C7×Q8C211yesmaximal, characteristic, complemented
3C3×C7×#137C211yesmaximal, characteristic, complemented
4C7×#951C211yesmaximal, characteristic, complemented
5C7×#942C211yesmaximal, characteristic, complemented
6C7×#918C212yesmaximal, complemented
7D3×C7×Q16C212yesmaximal, complemented
8C7×#915C212yesmaximal, complemented
9D3×C7×SD16C212yesmaximal, complemented
10C7×#909C211yesmaximal, characteristic, complemented
11D3×C7×M16C211yesmaximal, characteristic, complemented
12C2×C7×#137—31nomaximal, Hall
13C3×C7×#41C2211yescharacteristic
14C2×C3×C7×Q8C2211yescharacteristic, complemented
15C7×#239C2211yescharacteristic
16D3×C7×Q8—21no—
17D3×C7×Q8—21no—
18D3×C7×Q8C2212yes—
19D3×C7×Q8C2211yescharacteristic
20D3×C7×Q8C2212yescomplemented
21C7×#237—21no—
22C7×#237C2211yescharacteristic
23D3×C7×D4—21no—
24D3×C7×D4C2211yescharacteristic
25C7×#235—21no—
26C7×#235C2211yescharacteristic
27C2×C4×D3×C7—21no—
28C2×C4×D3×C7—21no—
29C2×C4×D3×C7C2211yescharacteristic
30C2×C7×Dic6—21no—
31C2×C7×Dic6C2211yescharacteristic, complemented
32C3×C7×Q16C2212yescomplemented
33C3×C7×SD16C2212yescomplemented
34C3×C7×M16C2211yescharacteristic, complemented
35C7×#216C2212yescomplemented
36C7×#216C2212yescomplemented
37C7×#215C2212yescomplemented
38C7×#214C2212yescomplemented
39C7×#208C2211yescharacteristic, complemented
40C7×Dic12C2212yescomplemented
41C7×#204C2212yescomplemented
42C7×ΓL(1,𝔽25)C2212yescomplemented
43D3×C7×C8C2212yes—
44C2×C7×#41—31no—
45C22×C7×Q8—31no—
46C7×#137—31no—
47C7×#137—31no—
48C7×#137—32no—
49C7×#137—32no—
50C7×#137—31no—
51C7×#137D311yescharacteristic, direct product factor
52C2×C7×Q16—32no—
53C2×C7×SD16—32no—
54C2×C7×M16—31no—
55D3×#137C711yesmaximal, characteristic, direct product factor, Hall
56C22×D3×C7—21no—
57C22×D3×C7D411yescharacteristic
58C3×C7×Q8—21no—
59C3×C7×Q8C2312yes—
60C3×C7×Q8C2311yescharacteristic
61C3×C7×D4—21no—
62C3×C7×D4C2311yescharacteristic
63C2×C3×C4×C7—21no—
64C2×C3×C4×C7—21no—
65C2×C3×C4×C7C2311yescharacteristic
66C7×#67—21no—
67C7×#67—21no—
68C7×#67—21no—
69C2×C7×Dic3—21no—
70C2×C7×Dic3—21no—
71C2×C7×Dic3D411yescharacteristic
72C7×D12—21no—
73C7×D12C2311yescharacteristic
74C4×D3×C7—21no—
75C4×D3×C7—22no—
76C4×D3×C7—22no—
77C4×D3×C7—21no—
78C4×D3×C7—21no—
79C4×D3×C7D411yescharacteristic
80C4×D3×C7D411yescharacteristic
81C4×D3×C7C2311yescharacteristic
82C4×D3×C7C2311yescharacteristic
83C7×Dic6—21no—
84C7×Dic6—22no—
85C7×Dic6—22no—
86C7×Dic6—21no—
87C7×Dic6C2312yes—
88C7×Dic6C2311yescharacteristic
89C3×C7×C8C2312yes—
90C7×C3⋊C8C2312yes—
91C7×#41—31no—
92C7×#41—31no—
93C7×#41—31no—
94C7×#41—61no—
95C7×#41—61no—
96C7×#41C2×D311yescharacteristic
97C2×C7×Q8—32no—
98C2×C7×Q8—31no—
99C2×C7×Q8—32no—
100C2×C7×Q8—31no—
101C2×C7×Q8—61no—
102C2×C7×Q8—61no—
103C2×C7×Q8—61no—
104C2×C7×Q8C2×D311yescharacteristic, complemented
105C2×C7×D4—31no—
106C2×C7×D4—61no—
107C22×C4×C7—31no—
108C22×C4×C7—61no—
109C22×C4×C7—61no—
110C7×Q16—32no—
111C7×Q16—32no—
112C7×Q16—32no—
113C7×Q16C2×D312yescomplemented
114C7×SD16—32no—
115C7×SD16—32no—
116C7×SD16—32no—
117C7×SD16C2×D312yescomplemented
118C7×M16—31no—
119C7×M16—32no—
120C7×M16C2×D311yescharacteristic, complemented
121C2×C7×C8—32no—
122D3×#41C2×C711yescharacteristic
123C2×D3×Q8C2×C711yescharacteristic, complemented
124C3×#137C2×C711yescharacteristic, complemented
125#951C2×C711yescharacteristic, complemented
126#942C2×C711yescharacteristic, complemented
127#918C2×C712yescomplemented
128D3×Q16C2×C712yescomplemented
129#915C2×C712yescomplemented
130D3×SD16C2×C712yescomplemented
131#909C2×C711yescharacteristic, complemented
132D3×M16C2×C711yescharacteristic, complemented
133C22×C3×C7—21no—
134C22×C3×C7C2×D411yescharacteristic
135C2×D3×C7—22no—
136C2×D3×C7—21no—
137C2×D3×C7—42no—
138C2×D3×C7C2×D411yescharacteristic
139C2×D3×C7C2×D411yescharacteristic
140C3×C4×C7—22no—
141C3×C4×C7—21no—
142C3×C4×C7C2×D411yescharacteristic
143C3×C4×C7C2411yescharacteristic
144C7×Dic3—22no—
145C7×Dic3—21no—
146C7×Dic3C2×D411yescharacteristic
147C7×Dic3C2×D411yescharacteristic
148C2×#137—31noSylow
149C23×C7—31no—
150C23×C7—61no—
151C7×Q8—21no—
152C7×Q8—32no—
153C7×Q8—31no—
154C7×Q8—62no—
155C7×Q8—62no—
156C7×Q8—61no—
157C7×Q8—61no—
158C7×Q8C22×D312yes—
159C7×Q8C22×D311yescharacteristic
160C7×D4—21no—
161C7×D4—31no—
162C7×D4—61no—
163C7×D4—61no—
164C7×D4—61no—
165C7×D4—61no—
166C7×D4C22×D311yescharacteristic
167C2×C4×C7—21no—
168C2×C4×C7—21no—
169C2×C4×C7—31no—
170C2×C4×C7—31no—
171C2×C4×C7—31no—
172C2×C4×C7—31no—
173C2×C4×C7—31no—
174C2×C4×C7—61no—
175C2×C4×C7—61no—
176C2×C4×C7—62no—
177C2×C4×C7—62no—
178C2×C4×C7—61no—
179C2×C4×C7—61no—
180C2×C4×C7—61no—
181C2×C4×C7C22×D311yescharacteristic
182C7×C8—32no—
183C7×C8C22×D312yes—
184C3×#41C22×C711yescharacteristic
185C2×C3×Q8C22×C711yescharacteristic, complemented
186#239C22×C711yescharacteristic
187D3×Q8—21no—
188D3×Q8—21no—
189D3×Q8C22×C712yes—
190D3×Q8C22×C711yescharacteristic
191D3×Q8C22×C712yescomplemented
192#237—21no—
193#237C22×C711yescharacteristic
194D3×D4—21no—
195D3×D4C22×C711yescharacteristic
196#235—21no—
197#235C22×C711yescharacteristic
198C2×C4×D3—21no—
199C2×C4×D3—21no—
200C2×C4×D3C22×C711yescharacteristic
201C2×Dic6—21no—
202C2×Dic6C22×C711yescharacteristic, complemented
203C3×Q16C22×C712yescomplemented
204C3×SD16C22×C712yescomplemented
205C3×M16C22×C711yescharacteristic, complemented
206#216C22×C712yescomplemented
207#216C22×C712yescomplemented
208#215C22×C712yescomplemented
209#214C22×C712yescomplemented
210#208C22×C711yescharacteristic, complemented
211Dic12C22×C712yescomplemented
212#204C22×C712yescomplemented
213ΓL(1,𝔽25)C22×C712yescomplemented
214D3×C8C22×C712yes—
215C2×C3×C7—21no—
216C2×C3×C7—41no—
217C2×C3×C7C22×D411yescharacteristic
218D3×C7—21no—
219D3×C7—41no—
220D3×C7#13712yesdirect product factor
221C2×#41—31no—
222C22×Q8—31no—
223#137—31no—
224#137—31no—
225#137—32no—
226#137—32no—
227#137—31no—
228#137D3×C711yescharacteristic, direct product factor
229C2×Q16—32no—
230C2×SD16—32no—
231C2×M16—31no—
232C22×C7—21no—
233C22×C7—31no—
234C22×C7—31no—
235C22×C7—62no—
236C22×C7—61no—
237C22×C7—122no—
238C22×C7D3×D411yescharacteristic
239C4×C7—21no—
240C4×C7—22no—
241C4×C7—31no—
242C4×C7—31no—
243C4×C7—61no—
244C4×C7—62no—
245C4×C7D3×D411yescharacteristic
246C4×C7C23×D311yescharacteristic
247C22×D3—21no—
248C22×D3C7×D411yescharacteristic
249C3×Q8—21no—
250C3×Q8C23×C712yes—
251C3×Q8C23×C711yescharacteristic
252C3×D4—21no—
253C3×D4C23×C711yescharacteristic
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4C23×C711yescharacteristic
257#67—21no—
258#67—21no—
259#67—21no—
260C2×Dic3—21no—
261C2×Dic3—21no—
262C2×Dic3C7×D411yescharacteristic
263D12—21no—
264D12C23×C711yescharacteristic
265C4×D3—21no—
266C4×D3—22no—
267C4×D3—22no—
268C4×D3—21no—
269C4×D3—21no—
270C4×D3C7×D411yescharacteristic
271C4×D3C7×D411yescharacteristic
272C4×D3C23×C711yescharacteristic
273C4×D3C23×C711yescharacteristic
274Dic6—21no—
275Dic6—22no—
276Dic6—22no—
277Dic6—21no—
278Dic6C23×C712yes—
279Dic6C23×C711yescharacteristic
280C3×C8C23×C712yes—
281C3⋊C8C23×C712yes—
282C3×C7C2×#13711yescharacteristic, complemented, Hall
283#41—31no—
284#41—31no—
285#41—31no—
286#41—61no—
287#41—61no—
288#41C2×D3×C711yescharacteristic
289C2×Q8—32no—
290C2×Q8—31no—
291C2×Q8—32no—
292C2×Q8—31no—
293C2×Q8—61no—
294C2×Q8—61no—
295C2×Q8—61no—
296C2×Q8C2×D3×C711yescharacteristic, complemented
297C2×D4—31no—
298C2×D4—61no—
299C22×C4—31no—
300C22×C4—61no—
301C22×C4—61no—
302Q16—32no—
303Q16—32no—
304Q16—32no—
305Q16C2×D3×C712yescomplemented
306SD16—32no—
307SD16—32no—
308SD16—32no—
309SD16C2×D3×C712yescomplemented
310M16—31no—
311M16—32no—
312M16C2×D3×C711yescharacteristic, complemented
313C2×C8—32no—
314C2×C7—21no—
315C2×C7—32no—
316C2×C7—41no—
317C2×C7—61no—
318C2×C7—121no—
319C2×C7C2×D3×D411yescharacteristic
320C22×C3—21no—
321C22×C3C2×C7×D411yescharacteristic
322C2×D3—22no—
323C2×D3—21no—
324C2×D3—42no—
325C2×D3C2×C7×D411yescharacteristic
326C2×D3C2×C7×D411yescharacteristic
327C3×C4—22no—
328C3×C4—21no—
329C3×C4C2×C7×D411yescharacteristic
330C3×C4C24×C711yescharacteristic
331Dic3—22no—
332Dic3—21no—
333Dic3C2×C7×D411yescharacteristic
334Dic3C2×C7×D411yescharacteristic
335C23—31no—
336C23—61no—
337Q8—21no—
338Q8—32no—
339Q8—31no—
340Q8—62no—
341Q8—62no—
342Q8—61no—
343Q8—61no—
344Q8C22×D3×C712yes—
345Q8C22×D3×C711yescharacteristic
346D4—21no—
347D4—31no—
348D4—61no—
349D4—61no—
350D4—61no—
351D4—61no—
352D4C22×D3×C711yescharacteristic
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—31no—
356C2×C4—31no—
357C2×C4—31no—
358C2×C4—31no—
359C2×C4—31no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—62no—
363C2×C4—62no—
364C2×C4—61no—
365C2×C4—61no—
366C2×C4—61no—
367C2×C4C22×D3×C711yescharacteristic
368C8—32no—
369C8C22×D3×C712yes—
370C7D3×#13711yescharacteristic, direct product factor, Sylow
371C2×C3—21no—
372C2×C3—41no—
373C2×C3C22×C7×D411yescharacteristic
374D3—21no—
375D3—41no—
376D3C7×#13712yesdirect product factor
377C22—21no—
378C22—31no—
379C22—31no—
380C22—62no—
381C22—61no—
382C22—122no—
383C22D3×C7×D411yescharacteristic
384C4—21no—
385C4—22no—
386C4—31no—
387C4—31no—
388C4—61no—
389C4—62no—
390C4D3×C7×D411yescharacteristic
391C4C23×D3×C711yescharacteristic
392C3C2×C7×#13711yescharacteristic, complemented, Sylow
393C2—21no—
394C2—32no—
395C2—41no—
396C2—61no—
397C2—121no—
398C2C2×D3×C7×D411yescharacteristic
subgroupG/H
centerC2×C7C2×D3×D4
commutator, cosocleC3×C4C24×C7
FrattiniC4C23×D3×C7
FittingC3×C7×#137C2
socleC2×C3×C7C22×D4
prime radicalD3×C7×#41C2
hypercenterC7×#137D3
hypocenterC3C2×C7×#137
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C7×#137

Complements

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

NH ≅ G/Ntype
2. C2×D3×C7×Q8C2semidirect
3. C3×C7×#137C2semidirect
4. C7×#951C2semidirect
5. C7×#942C2semidirect
6. C7×#918C2semidirect
7. D3×C7×Q16C2semidirect
8. C7×#915C2semidirect
9. D3×C7×SD16C2semidirect
10. C7×#909C2semidirect
11. D3×C7×M16C2semidirect
14. C2×C3×C7×Q8C22semidirect
20. D3×C7×Q8C22semidirect
31. C2×C7×Dic6C22semidirect
32. C3×C7×Q16C22semidirect
33. C3×C7×SD16C22semidirect
34. C3×C7×M16C22semidirect
35. C7×#216C22semidirect
36. C7×#216C22semidirect
37. C7×#215C22semidirect
38. C7×#214C22semidirect
39. C7×#208C22semidirect
40. C7×Dic12C22semidirect
41. C7×#204C22semidirect
42. C7×ΓL(1,𝔽25)C22semidirect
51. C7×#137D3direct (N × H)
55. D3×#137C7direct (N × H)
104. C2×C7×Q8C2×D3semidirect
113. C7×Q16C2×D3semidirect
117. C7×SD16C2×D3semidirect
120. C7×M16C2×D3semidirect
123. C2×D3×Q8C2×C7semidirect
124. C3×#137C2×C7semidirect
125. #951C2×C7semidirect
126. #942C2×C7semidirect
127. #918C2×C7semidirect
128. D3×Q16C2×C7semidirect
129. #915C2×C7semidirect
130. D3×SD16C2×C7semidirect
131. #909C2×C7semidirect
132. D3×M16C2×C7semidirect
185. C2×C3×Q8C22×C7semidirect
191. D3×Q8C22×C7semidirect
202. C2×Dic6C22×C7semidirect
203. C3×Q16C22×C7semidirect
204. C3×SD16C22×C7semidirect
205. C3×M16C22×C7semidirect
206. #216C22×C7semidirect
207. #216C22×C7semidirect
208. #215C22×C7semidirect
209. #214C22×C7semidirect
210. #208C22×C7semidirect
211. Dic12C22×C7semidirect
212. #204C22×C7semidirect
213. ΓL(1,𝔽25)C22×C7semidirect
220. D3×C7#137direct (N × H)
228. #137D3×C7direct (N × H)
282. C3×C7C2×#137semidirect
296. C2×Q8C2×D3×C7semidirect
305. Q16C2×D3×C7semidirect
309. SD16C2×D3×C7semidirect
312. M16C2×D3×C7semidirect
370. C7D3×#137direct (N × H)
376. D3C7×#137direct (N × H)
392. C3C2×C7×#137semidirect

Maximal subgroups

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

subgroupindex
D3×C7×#412
C2×D3×C7×Q82
C3×C7×#1372
C7×#9512
C7×#9422
C7×#9182
D3×C7×Q162
C7×#9152
D3×C7×SD162
C7×#9092
D3×C7×M162
C2×C7×#1373
D3×#1377