Finite Group Databasebeta

← C7×#5806 #13061522

Subgroups — C7×#5806 #13061522

Order 1344.

668 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C7×D4C211yesmaximal, characteristic, complemented
2C2×C7×#237C211yesmaximal, characteristic, complemented
3C7×#953C212yesmaximal, complemented
4C2×C7×#217C212yesmaximal, complemented
5C7×#939C212yesmaximal, complemented
6C7×#938C211yesmaximal, characteristic, complemented
7C7×#936C211yesmaximal, characteristic, complemented
8C7×#935C212yesmaximal, complemented
9C7×D4×Dic3C211yesmaximal, characteristic, complemented
10C4×C7×#67C211yesmaximal, characteristic, complemented
11C7×#924C211yesmaximal, characteristic, complemented
12C7×#546—31nomaximal, Hall
13C24×C3×C7C2212yes—
14C2×C3×C7×D4—24no—
15C2×C3×C7×D4C2212yes—
16C2×C3×C7×D4C2211yescharacteristic
17C2×C3×C7×D4C2211yescharacteristic
18C22×C3×C4×C7C2211yescharacteristic
19C2×C7×#67C2211yescharacteristic
20C2×C7×#67C2212yescomplemented
21C2×C7×#67C2212yescomplemented
22C22×C7×Dic3C2212yescomplemented
23C22×C7×Dic3C2212yescomplemented
24C7×#237—24no—
25C2×C4×D3×C7C2211yescharacteristic, complemented
26C2×C7×Dic6C2211yescharacteristic, complemented
27C7×#217C2212yes—
28C7×#217C2211yescharacteristic
29C7×#217C2212yescomplemented
30C7×#217C2212yescomplemented
31C7×#217C2212yescomplemented
32C7×#217C2212yescomplemented
33C7×(C2×D3)⋊C4C2211yescharacteristic, complemented
34C7×C4⋊Dic3C2211yescharacteristic, complemented
35C7×Dic3⋊C4C2212yescomplemented
36C7×Dic3⋊C4C2211yescharacteristic, complemented
37C4×C7×Dic3C2211yescharacteristic, complemented
38C2×C7×#41—31no—
39C22×C7×D4D311yescharacteristic, complemented
40C7×#124—31no—
41C7×#123—32no—
42C7×C22⋊Q8—31no—
43C7×#121—31no—
44C7×#121—32no—
45C7×#120—32no—
46C4×C7×D4—31no—
47C4×C7×D4—31no—
48C2×C7×C22⋊C4—32no—
49#5806C711yesmaximal, characteristic, direct product factor, Hall
50C23×C3×C7—24no—
51C23×C3×C7—24no—
52C23×C3×C7—42no—
53C23×C3×C7C2312yes—
54C23×C3×C7C2312yes—
55C23×C3×C7C2311yescharacteristic
56C22×D3×C7C2311yescharacteristic
57C3×C7×D4—24no—
58C3×C7×D4—22no—
59C3×C7×D4D414yescomplemented
60C2×C3×C4×C7—22no—
61C2×C3×C4×C7C2311yescharacteristic
62C2×C3×C4×C7C2311yescharacteristic
63C7×#67—24no—
64C7×#67—22no—
65C7×#67—42no—
66C2×C7×Dic3—24no—
67C2×C7×Dic3—42no—
68C2×C7×Dic3C2311yescharacteristic
69C2×C7×Dic3C2311yescharacteristic
70C2×C7×Dic3C2312yes—
71C2×C7×Dic3C2312yes—
72C2×C7×Dic3C2311yescharacteristic
73C4×D3×C7—22no—
74C7×Dic6—22no—
75C24×C7C2×D312yes—
76C7×#41—64no—
77C2×C7×Q8—31no—
78C2×C7×D4—24no—
79C2×C7×D4—31no—
80C2×C7×D4—32no—
81C2×C7×D4—32no—
82C2×C7×D4C2×D312yes—
83C2×C7×D4C2×D311yescharacteristic
84C2×C7×D4C2×D311yescharacteristic
85C22×C4×C7—32no—
86C22×C4×C7—32no—
87C22×C4×C7—31no—
88C22×C4×C7C2×D311yescharacteristic
89C7×C4⋊C4—31no—
90C7×C4⋊C4—32no—
91C7×C4⋊C4—31no—
92C7×C22⋊C4—31no—
93C7×C22⋊C4—32no—
94C7×C22⋊C4—32no—
95C7×C22⋊C4—32no—
96C7×C22⋊C4—32no—
97C7×C22⋊C4—32no—
98C7×C22⋊C4—31no—
99C42×C7—31no—
100C22×C3×D4C2×C711yescharacteristic, complemented
101C2×#237C2×C711yescharacteristic, complemented
102#953C2×C712yescomplemented
103C2×#217C2×C712yescomplemented
104#939C2×C712yescomplemented
105#938C2×C711yescharacteristic, complemented
106#936C2×C711yescharacteristic, complemented
107#935C2×C712yescomplemented
108D4×Dic3C2×C711yescharacteristic, complemented
109C4×#67C2×C711yescharacteristic, complemented
110#924C2×C711yescharacteristic, complemented
111C22×C3×C7—21no—
112C22×C3×C7—22no—
113C22×C3×C7—22no—
114C22×C3×C7—21no—
115C22×C3×C7—22no—
116C22×C3×C7—22no—
117C22×C3×C7—22no—
118C22×C3×C7—48no—
119C22×C3×C7C2×D414yes—
120C22×C3×C7#4112yescomplemented
121C22×C3×C7C2411yescharacteristic
122C2×D3×C7—21no—
123C2×D3×C7—21no—
124C2×D3×C7—21no—
125C3×C4×C7—21no—
126C3×C4×C7C2×D412yes—
127C7×Dic3—22no—
128C7×Dic3—21no—
129C7×Dic3—21no—
130C7×Dic3—21no—
131C7×Dic3—22no—
132#546—31noSylow
133C23×C7—24no—
134C23×C7—24no—
135C23×C7—31no—
136C23×C7—42no—
137C23×C7C22×D312yes—
138C23×C7C22×D311yescharacteristic
139C23×C7C22×D312yes—
140C7×Q8—62no—
141C7×D4—24no—
142C7×D4—22no—
143C7×D4—62no—
144C7×D4—64no—
145C7×D4—122no—
146C7×D4#6714yescomplemented
147C2×C4×C7—22no—
148C2×C4×C7—31no—
149C2×C4×C7—31no—
150C2×C4×C7—32no—
151C2×C4×C7—31no—
152C2×C4×C7—32no—
153C2×C4×C7—64no—
154C2×C4×C7—62no—
155C2×C4×C7—122no—
156C2×C4×C7C22×D311yescharacteristic
157C2×C4×C7C22×D311yescharacteristic
158C24×C3C22×C712yes—
159C2×C3×D4—24no—
160C2×C3×D4C22×C712yes—
161C2×C3×D4C22×C711yescharacteristic
162C2×C3×D4C22×C711yescharacteristic
163C22×C3×C4C22×C711yescharacteristic
164C2×#67C22×C711yescharacteristic
165C2×#67C22×C712yescomplemented
166C2×#67C22×C712yescomplemented
167C22×Dic3C22×C712yescomplemented
168C22×Dic3C22×C712yescomplemented
169#237—24no—
170C2×C4×D3C22×C711yescharacteristic, complemented
171C2×Dic6C22×C711yescharacteristic, complemented
172#217C22×C712yes—
173#217C22×C711yescharacteristic
174#217C22×C712yescomplemented
175#217C22×C712yescomplemented
176#217C22×C712yescomplemented
177#217C22×C712yescomplemented
178(C2×D3)⋊C4C22×C711yescharacteristic, complemented
179C4⋊Dic3C22×C711yescharacteristic, complemented
180Dic3⋊C4C22×C712yescomplemented
181Dic3⋊C4C22×C711yescharacteristic, complemented
182C4×Dic3C22×C711yescharacteristic, complemented
183C2×C3×C7—22no—
184C2×C3×C7—24no—
185C2×C3×C7—42no—
186C2×C3×C7C22×D411yescharacteristic
187C2×C3×C7C2×#4111yescharacteristic
188C2×C3×C7H(5,𝔽2)11yescharacteristic
189D3×C7—41no—
190C2×#41—31no—
191C22×D4D3×C711yescharacteristic, complemented
192#124—31no—
193#123—32no—
194C22⋊Q8—31no—
195#121—31no—
196#121—32no—
197#120—32no—
198C4×D4—31no—
199C4×D4—31no—
200C2×C22⋊C4—32no—
201C22×C7—22no—
202C22×C7—22no—
203C22×C7—21no—
204C22×C7—21no—
205C22×C7—22no—
206C22×C7—22no—
207C22×C7—22no—
208C22×C7—48no—
209C22×C7—61no—
210C22×C7—61no—
211C22×C7—61no—
212C22×C7#23712yescomplemented
213C22×C7C2×#6714yes—
214C22×C7C23×D311yescharacteristic
215C4×C7—21no—
216C4×C7—61no—
217C4×C7—61no—
218C4×C7—62no—
219C4×C7—61no—
220C4×C7—62no—
221C4×C7C2×#6712yes—
222C23×C3—24no—
223C23×C3—24no—
224C23×C3—42no—
225C23×C3C23×C712yes—
226C23×C3C23×C712yes—
227C23×C3C23×C711yescharacteristic
228C22×D3C23×C711yescharacteristic
229C3×D4—24no—
230C3×D4—22no—
231C3×D4C7×D414yescomplemented
232C2×C3×C4—22no—
233C2×C3×C4C23×C711yescharacteristic
234C2×C3×C4C23×C711yescharacteristic
235#67—24no—
236#67—22no—
237#67—42no—
238C2×Dic3—24no—
239C2×Dic3—42no—
240C2×Dic3C23×C711yescharacteristic
241C2×Dic3C23×C711yescharacteristic
242C2×Dic3C23×C712yes—
243C2×Dic3C23×C712yes—
244C2×Dic3C23×C711yescharacteristic
245C4×D3—22no—
246Dic6—22no—
247C3×C7#54611yescharacteristic, complemented, Hall
248C24C2×D3×C712yes—
249#41—64no—
250C2×Q8—31no—
251C2×D4—24no—
252C2×D4—31no—
253C2×D4—32no—
254C2×D4—32no—
255C2×D4C2×D3×C712yes—
256C2×D4C2×D3×C711yescharacteristic
257C2×D4C2×D3×C711yescharacteristic
258C22×C4—32no—
259C22×C4—32no—
260C22×C4—31no—
261C22×C4C2×D3×C711yescharacteristic
262C4⋊C4—31no—
263C4⋊C4—31no—
264C4⋊C4—32no—
265C22⋊C4—31no—
266C22⋊C4—32no—
267C22⋊C4—32no—
268C22⋊C4—32no—
269C22⋊C4—32no—
270C22⋊C4—32no—
271C22⋊C4—31no—
272C42—31no—
273C2×C7—22no—
274C2×C7—24no—
275C2×C7—42no—
276C2×C7—121no—
277C2×C7C2×#23711yescharacteristic
278C2×C7#100411yescharacteristic
279C2×C7C22×#6711yescharacteristic
280C22×C3—21no—
281C22×C3—22no—
282C22×C3—22no—
283C22×C3—21no—
284C22×C3—22no—
285C22×C3—22no—
286C22×C3—22no—
287C22×C3—48no—
288C22×C3C2×C7×D414yes—
289C22×C3C7×#4112yescomplemented
290C22×C3C24×C711yescharacteristic
291C2×D3—21no—
292C2×D3—21no—
293C2×D3—21no—
294C3×C4—21no—
295C3×C4C2×C7×D412yes—
296Dic3—22no—
297Dic3—21no—
298Dic3—21no—
299Dic3—21no—
300Dic3—22no—
301C23—24no—
302C23—24no—
303C23—31no—
304C23—42no—
305C23C22×D3×C712yes—
306C23C22×D3×C711yescharacteristic
307C23C22×D3×C712yes—
308Q8—62no—
309D4—24no—
310D4—22no—
311D4—62no—
312D4—64no—
313D4—122no—
314D4C7×#6714yescomplemented
315C2×C4—22no—
316C2×C4—31no—
317C2×C4—31no—
318C2×C4—32no—
319C2×C4—31no—
320C2×C4—32no—
321C2×C4—64no—
322C2×C4—62no—
323C2×C4—122no—
324C2×C4C22×D3×C711yescharacteristic
325C2×C4C22×D3×C711yescharacteristic
326C7#580611yescharacteristic, direct product factor, Sylow
327C2×C3—22no—
328C2×C3—24no—
329C2×C3—42no—
330C2×C3C22×C7×D411yescharacteristic
331C2×C3C2×C7×#4111yescharacteristic
332C2×C3C7×H(5,𝔽2)11yescharacteristic
333D3—41no—
334C22—22no—
335C22—22no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—48no—
342C22—61no—
343C22—61no—
344C22—61no—
345C22C7×#23712yescomplemented
346C22C2×C7×#6714yes—
347C22C23×D3×C711yescharacteristic
348C4—21no—
349C4—61no—
350C4—61no—
351C4—62no—
352C4—61no—
353C4—62no—
354C4C2×C7×#6712yes—
355C3C7×#54611yescharacteristic, complemented, Sylow
356C2—22no—
357C2—24no—
358C2—42no—
359C2—121no—
360C2C2×C7×#23711yescharacteristic
361C2C7×#100411yescharacteristic
362C2C22×C7×#6711yescharacteristic
subgroupG/H
centerC22×C7C23×D3
commutator, cosocleC22×C3C24×C7
FrattiniC22C23×D3×C7
FittingC22×C3×C7×D4C2
socleC22×C3×C7C24
hypercenterC22×C7×D4D3
hypocenterC3C7×#546
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C7 ◁ C22×C7×D4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C7×D4C2semidirect
2. C2×C7×#237C2semidirect
3. C7×#953C2semidirect
4. C2×C7×#217C2semidirect
5. C7×#939C2semidirect
6. C7×#938C2semidirect
7. C7×#936C2semidirect
8. C7×#935C2semidirect
9. C7×D4×Dic3C2semidirect
10. C4×C7×#67C2semidirect
11. C7×#924C2semidirect
20. C2×C7×#67C22semidirect
21. C2×C7×#67C22semidirect
22. C22×C7×Dic3C22semidirect
23. C22×C7×Dic3C22semidirect
25. C2×C4×D3×C7C22semidirect
26. C2×C7×Dic6C22semidirect
29. C7×#217C22semidirect
30. C7×#217C22semidirect
31. C7×#217C22semidirect
32. C7×#217C22semidirect
33. C7×(C2×D3)⋊C4C22semidirect
34. C7×C4⋊Dic3C22semidirect
35. C7×Dic3⋊C4C22semidirect
36. C7×Dic3⋊C4C22semidirect
37. C4×C7×Dic3C22semidirect
39. C22×C7×D4D3semidirect
49. #5806C7direct (N × H)
59. C3×C7×D4D4semidirect
100. C22×C3×D4C2×C7semidirect
101. C2×#237C2×C7semidirect
102. #953C2×C7semidirect
103. C2×#217C2×C7semidirect
104. #939C2×C7semidirect
105. #938C2×C7semidirect
106. #936C2×C7semidirect
107. #935C2×C7semidirect
108. D4×Dic3C2×C7semidirect
109. C4×#67C2×C7semidirect
110. #924C2×C7semidirect
120. C22×C3×C7#41semidirect
146. C7×D4#67semidirect
165. C2×#67C22×C7semidirect
166. C2×#67C22×C7semidirect
167. C22×Dic3C22×C7semidirect
168. C22×Dic3C22×C7semidirect
170. C2×C4×D3C22×C7semidirect
171. C2×Dic6C22×C7semidirect
174. #217C22×C7semidirect
175. #217C22×C7semidirect
176. #217C22×C7semidirect
177. #217C22×C7semidirect
178. (C2×D3)⋊C4C22×C7semidirect
179. C4⋊Dic3C22×C7semidirect
180. Dic3⋊C4C22×C7semidirect
181. Dic3⋊C4C22×C7semidirect
182. C4×Dic3C22×C7semidirect
191. C22×D4D3×C7semidirect
212. C22×C7#237semidirect
231. C3×D4C7×D4semidirect
247. C3×C7#546semidirect
289. C22×C3C7×#41semidirect
314. D4C7×#67semidirect
326. C7#5806direct (N × H)
345. C22C7×#237semidirect
355. C3C7×#546semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C7×D42
C2×C7×#2372
C7×#9532
C2×C7×#2172
C7×#9392
C7×#9382
C7×#9362
C7×#9352
C7×D4×Dic32
C4×C7×#672
C7×#9242
C7×#5463
#58067