Finite Group Databasebeta

← C2×C7×#3228 #13238059

Subgroups — C2×C7×#3228 #13238059

Order 1792.

2058 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#533C211yesmaximal, characteristic, complemented
2C2×C7×#532C211yesmaximal, characteristic
3C7×#3228C2116yesmaximal, direct product factor
4C2×C7×#500C211yesmaximal, characteristic, complemented
5C2×C7×#498C211yesmaximal, characteristic, complemented
6C2×C7×#486C211yesmaximal, characteristic, complemented
7C2×C7×#486C211yesmaximal, characteristic, complemented
8C2×C7×#482C214yesmaximal, complemented
9C2×C7×#478C214yesmaximal, complemented
10C2×C7×#431C211yesmaximal, characteristic, complemented
11C22×C7×#41—21no—
12C7×#533—24no—
13C7×#533C2218yescomplemented
14C7×#532—24no—
15C7×#532C2218yes—
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×C4⋊Q8C2211yescharacteristic
18C2×C7×C4⋊Q8C2211yescharacteristic
19C2×C7×#127C2211yescharacteristic
20C2×C7×#125C2212yes—
21C2×C7×#124—21no—
22C2×C7×C22⋊Q8—21no—
23C2×C7×C22⋊Q8—21no—
24C2×C7×#121—21no—
25C2×C7×#121C2212yes—
26C2×C4×C7×Q8—21no—
27C2×C4×C7×Q8—21no—
28C2×C4×C7×D4—21no—
29C2×C4×C7×D4C2212yes—
30C2×C7×#117C2212yes—
31C22×C42×C7C2211yescharacteristic
32C7×#500C2218yescomplemented
33C7×#498C2218yescomplemented
34C7×#486C2218yescomplemented
35C7×#486C2218yescomplemented
36C7×#482C22132yescomplemented
37C7×#478C22132yescomplemented
38C7×#431C2218yescomplemented
39C2×C7×#107C2214yes—
40C2×C7×#106C2214yes—
41C2×C7×C4⋊C8C2212yes—
42C2×C7×#102C2214yes—
43C2×C7×#102C2214yes—
44C2×C7×C22⋊C8C2212yes—
45C2×C4×C7×C8C2211yescharacteristic
46C2×C4×C7×C8C2211yescharacteristic
47C2×#3228C711yesmaximal, characteristic, direct product factor, Sylow
48C2×C7×#41—28no—
49C2×C7×#41—48no—
50C22×C7×Q8—21no—
51C22×C7×Q8—21no—
52C22×C7×Q8—21no—
53C22×C7×D4—22no—
54C22×C7×D4—21no—
55C22×C7×D4—21no—
56C22×C7×D4C2312yes—
57C23×C4×C7—22no—
58C23×C4×C7D411yescharacteristic
59C23×C4×C7D411yescharacteristic
60C23×C4×C7C2311yescharacteristic
61C22×C7×C8C2312yes—
62C22×C7×C8C2312yes—
63C7×C4⋊Q8—22no—
64C7×C4⋊Q8—22no—
65C7×C4⋊Q8—22no—
66C7×C4⋊Q8C2314yes—
67C7×C4⋊Q8C2314yes—
68C7×C4⋊Q8C2314yes—
69C7×#127—22no—
70C7×#127C2314yes—
71C7×#125—24no—
72C7×#125C2318yes—
73C7×#124—28no—
74C7×C22⋊Q8—28no—
75C7×C22⋊Q8—28no—
76C7×#121—28no—
77C7×#121—24no—
78C7×#121C2318yes—
79C4×C7×Q8—28no—
80C4×C7×Q8—28no—
81C4×C7×D4—28no—
82C4×C7×D4—24no—
83C4×C7×D4C2318yes—
84C7×#117—24no—
85C7×#117C2318yes—
86C2×C7×C4⋊C4—21no—
87C2×C7×C4⋊C4—22no—
88C2×C7×C4⋊C4—22no—
89C2×C7×C4⋊C4—21no—
90C2×C7×C4⋊C4—21no—
91C2×C7×C4⋊C4C2312yes—
92C2×C7×C4⋊C4C2312yes—
93C2×C7×C4⋊C4C2312yes—
94C2×C7×C22⋊C4—22no—
95C2×C7×C22⋊C4—22no—
96C2×C7×C22⋊C4—22no—
97C2×C42×C7—22no—
98C2×C42×C7D411yescharacteristic
99C2×C42×C7D411yescharacteristic
100C2×C42×C7D414yes—
101C2×C42×C7C2311yescharacteristic
102C2×C42×C7C2311yescharacteristic
103C2×C42×C7C2314yes—
104C7×#107C23116yes—
105C7×#106C23116yes—
106C7×C4⋊C8C2318yes—
107C7×#102C23116yes—
108C7×#102C23116yes—
109C7×C22⋊C8C2318yes—
110C4×C7×C8C2314yes—
111C4×C7×C8C2314yes—
112C2×#533C2×C711yescharacteristic, complemented
113C2×#532C2×C711yescharacteristic
114#3228C2×C7116yesdirect product factor
115C2×#500C2×C711yescharacteristic, complemented
116C2×#498C2×C711yescharacteristic, complemented
117C2×#486C2×C711yescharacteristic, complemented
118C2×#486C2×C711yescharacteristic, complemented
119C2×#482C2×C714yescomplemented
120C2×#478C2×C714yescomplemented
121C2×#431C2×C711yescharacteristic, complemented
122C24×C7—22no—
123C24×C7C2×D411yescharacteristic
124C7×#41—432no—
125C2×C7×Q8—24no—
126C2×C7×Q8—24no—
127C2×C7×Q8—24no—
128C2×C7×Q8—44no—
129C2×C7×Q8—44no—
130C2×C7×Q8—44no—
131C2×C7×D4—28no—
132C2×C7×D4—22no—
133C2×C7×D4—24no—
134C2×C7×D4—24no—
135C2×C7×D4—28no—
136C2×C7×D4—48no—
137C2×C7×D4—44no—
138C2×C7×D4—44no—
139C2×C7×D4C2414yes—
140C22×C4×C7—24no—
141C22×C4×C7—24no—
142C22×C4×C7—24no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—28no—
147C22×C4×C7—48no—
148C22×C4×C7C2×D412yes—
149C22×C4×C7C2×D414yes—
150C22×C4×C7C2×D414yes—
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2412yes—
157C22×C4×C7C2411yescharacteristic
158C2×C7×C8—24no—
159C2×C7×C8—24no—
160C2×C7×C8C2414yes—
161C2×C7×C8C2414yes—
162C7×C4⋊C4—24no—
163C7×C4⋊C4—28no—
164C7×C4⋊C4—28no—
165C7×C4⋊C4—24no—
166C7×C4⋊C4—24no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4C2414yes—
171C7×C4⋊C4C2414yes—
172C7×C4⋊C4C2414yes—
173C7×C22⋊C4—28no—
174C7×C22⋊C4—28no—
175C7×C22⋊C4—28no—
176C42×C7—28no—
177C42×C7C2×D414yes—
178C42×C7C2×D412yes—
179C42×C7C2×D414yes—
180C42×C7C2×D412yes—
181C42×C7C2412yes—
182C42×C7C2412yes—
183C22×#41—21no—
184#533—24no—
185#533C22×C718yescomplemented
186#532—24no—
187#532C22×C718yes—
188C2×C4⋊Q8C22×C711yescharacteristic
189C2×C4⋊Q8C22×C711yescharacteristic
190C2×C4⋊Q8C22×C711yescharacteristic
191C2×#127C22×C711yescharacteristic
192C2×#125C22×C712yes—
193C2×#124—21no—
194C2×C22⋊Q8—21no—
195C2×C22⋊Q8—21no—
196C2×#121—21no—
197C2×#121C22×C712yes—
198C2×C4×Q8—21no—
199C2×C4×Q8—21no—
200C2×C4×D4—21no—
201C2×C4×D4C22×C712yes—
202C2×#117C22×C712yes—
203C22×C42C22×C711yescharacteristic
204#500C22×C718yescomplemented
205#498C22×C718yescomplemented
206#486C22×C718yescomplemented
207#486C22×C718yescomplemented
208#482C22×C7132yescomplemented
209#478C22×C7132yescomplemented
210#431C22×C718yescomplemented
211C2×#107C22×C714yes—
212C2×#106C22×C714yes—
213C2×C4⋊C8C22×C712yes—
214C2×#102C22×C714yes—
215C2×#102C22×C714yes—
216C2×C22⋊C8C22×C712yes—
217C2×C4×C8C22×C711yescharacteristic
218C2×C4×C8C22×C711yescharacteristic
219C23×C7—22no—
220C23×C7—22no—
221C23×C7—22no—
222C23×C7—24no—
223C23×C7—44no—
224C23×C7—44no—
225C23×C7—44no—
226C23×C7C22×D412yes—
227C23×C7C22×D411yescharacteristic
228C7×Q8—48no—
229C7×Q8—48no—
230C7×Q8—48no—
231C7×D4—28no—
232C7×D4—416no—
233C7×D4—44no—
234C7×D4—48no—
235C7×D4—48no—
236C2×C4×C7—22no—
237C2×C4×C7—22no—
238C2×C4×C7—22no—
239C2×C4×C7—24no—
240C2×C4×C7—28no—
241C2×C4×C7—28no—
242C2×C4×C7—24no—
243C2×C4×C7—24no—
244C2×C4×C7—24no—
245C2×C4×C7—24no—
246C2×C4×C7—416no—
247C2×C4×C7—44no—
248C2×C4×C7—44no—
249C2×C4×C7—44no—
250C2×C4×C7#13514yes—
251C2×C4×C7AGL(1,ℤ/8)14yes—
252C2×C4×C7C22×D412yes—
253C2×C4×C7C22×D412yes—
254C2×C4×C7C22×D412yes—
255C2×C4×C7C22×D411yescharacteristic
256C2×C4×C7C22×D412yes—
257C2×C4×C7C22×D412yes—
258C2×C4×C72−1+414yes—
259C2×C4×C7C2511yescharacteristic
260C7×C8—24no—
261C7×C8—24no—
262C2×#41—28no—
263C2×#41—48no—
264C22×Q8—21no—
265C22×Q8—21no—
266C22×Q8—21no—
267C22×D4—22no—
268C22×D4—21no—
269C22×D4—21no—
270C22×D4C23×C712yes—
271C23×C4—22no—
272C23×C4C7×D411yescharacteristic
273C23×C4C7×D411yescharacteristic
274C23×C4C23×C711yescharacteristic
275C22×C8C23×C712yes—
276C22×C8C23×C712yes—
277C4⋊Q8—22no—
278C4⋊Q8—22no—
279C4⋊Q8—22no—
280C4⋊Q8C23×C714yes—
281C4⋊Q8C23×C714yes—
282C4⋊Q8C23×C714yes—
283#127—22no—
284#127C23×C714yes—
285#125—24no—
286#125C23×C718yes—
287#124—28no—
288C22⋊Q8—28no—
289C22⋊Q8—28no—
290#121—28no—
291#121—24no—
292#121C23×C718yes—
293C4×Q8—28no—
294C4×Q8—28no—
295C4×D4—28no—
296C4×D4—24no—
297C4×D4C23×C718yes—
298#117—24no—
299#117C23×C718yes—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4C23×C712yes—
306C2×C4⋊C4C23×C712yes—
307C2×C4⋊C4C23×C712yes—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C42—22no—
312C2×C42C7×D411yescharacteristic
313C2×C42C7×D411yescharacteristic
314C2×C42C7×D414yes—
315C2×C42C23×C711yescharacteristic
316C2×C42C23×C711yescharacteristic
317C2×C42C23×C714yes—
318#107C23×C7116yes—
319#106C23×C7116yes—
320C4⋊C8C23×C718yes—
321#102C23×C7116yes—
322#102C23×C7116yes—
323C22⋊C8C23×C718yes—
324C4×C8C23×C714yes—
325C4×C8C23×C714yes—
326C22×C7—22no—
327C22×C7—22no—
328C22×C7—22no—
329C22×C7—44no—
330C22×C7—44no—
331C22×C7—44no—
332C22×C7—44no—
333C22×C7—88no—
334C22×C7#53612yes—
335C22×C7C2×#13512yes—
336C22×C7C2×AGL(1,ℤ/8)12yes—
337C22×C7C23×D411yescharacteristic
338C4×C7—22no—
339C4×C7—22no—
340C4×C7—22no—
341C4×C7—44no—
342C4×C7—44no—
343C4×C7—44no—
344C4×C7#53612yes—
345C4×C7C2×#13514yes—
346C4×C7C2×AGL(1,ℤ/8)14yes—
347C4×C7C2×2−1+412yes—
348C24—22no—
349C24C2×C7×D411yescharacteristic
350#41—432no—
351C2×Q8—24no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×D4—28no—
358C2×D4—22no—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—28no—
362C2×D4—48no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4C24×C714yes—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—28no—
373C22×C4—48no—
374C22×C4C2×C7×D411yescharacteristic
375C22×C4C2×C7×D412yes—
376C22×C4C2×C7×D411yescharacteristic
377C22×C4C2×C7×D411yescharacteristic
378C22×C4C2×C7×D414yes—
379C22×C4C2×C7×D414yes—
380C22×C4C2×C7×D411yescharacteristic
381C22×C4C2×C7×D411yescharacteristic
382C22×C4C24×C712yes—
383C22×C4C24×C711yescharacteristic
384C2×C8—24no—
385C2×C8—24no—
386C2×C8C24×C714yes—
387C2×C8C24×C714yes—
388C4⋊C4—24no—
389C4⋊C4—28no—
390C4⋊C4—28no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4C24×C714yes—
397C4⋊C4C24×C714yes—
398C4⋊C4C24×C714yes—
399C22⋊C4—28no—
400C22⋊C4—28no—
401C22⋊C4—28no—
402C42—28no—
403C42C2×C7×D414yes—
404C42C2×C7×D412yes—
405C42C2×C7×D414yes—
406C42C2×C7×D412yes—
407C42C24×C712yes—
408C42C24×C712yes—
409C2×C7—42no—
410C2×C7—84no—
411C2×C7#322814yesdirect product factor
412C2×C7C2×#53611yescharacteristic
413C2×C7C22×#13511yescharacteristic
414C2×C7C22×AGL(1,ℤ/8)11yescharacteristic
415C23—22no—
416C23—22no—
417C23—22no—
418C23—24no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23C22×C7×D412yes—
423C23C22×C7×D411yescharacteristic
424Q8—48no—
425Q8—48no—
426Q8—48no—
427D4—28no—
428D4—416no—
429D4—44no—
430D4—48no—
431D4—48no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—28no—
438C2×C4—28no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—416no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4C7×#13514yes—
447C2×C4C7×AGL(1,ℤ/8)14yes—
448C2×C4C22×C7×D412yes—
449C2×C4C22×C7×D412yes—
450C2×C4C22×C7×D412yes—
451C2×C4C22×C7×D411yescharacteristic
452C2×C4C22×C7×D412yes—
453C2×C4C22×C7×D412yes—
454C2×C4C7×2−1+414yes—
455C2×C4C25×C711yescharacteristic
456C8—24no—
457C8—24no—
458C7C2×#322811yescharacteristic, direct product factor, Sylow
459C22—22no—
460C22—22no—
461C22—22no—
462C22—44no—
463C22—44no—
464C22—44no—
465C22—44no—
466C22—88no—
467C22C7×#53612yes—
468C22C2×C7×#13512yes—
469C22C2×C7×AGL(1,ℤ/8)12yes—
470C22C23×C7×D411yescharacteristic
471C4—22no—
472C4—22no—
473C4—22no—
474C4—44no—
475C4—44no—
476C4—44no—
477C4C7×#53612yes—
478C4C2×C7×#13514yes—
479C4C2×C7×AGL(1,ℤ/8)14yes—
480C4C2×C7×2−1+412yes—
481C2—42no—
482C2—84no—
483C2C7×#322814yesdirect product factor
484C2C2×C7×#53611yescharacteristic
485C2C22×C7×#13511yescharacteristic
486C2C22×C7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC2×C4C25×C7
prime radicalC2×C7×#532C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C22×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#533C2semidirect
3. C7×#3228C2direct (N × H)
4. C2×C7×#500C2semidirect
5. C2×C7×#498C2semidirect
6. C2×C7×#486C2semidirect
7. C2×C7×#486C2semidirect
8. C2×C7×#482C2semidirect
9. C2×C7×#478C2semidirect
10. C2×C7×#431C2semidirect
13. C7×#533C22semidirect
32. C7×#500C22semidirect
33. C7×#498C22semidirect
34. C7×#486C22semidirect
35. C7×#486C22semidirect
36. C7×#482C22semidirect
37. C7×#478C22semidirect
38. C7×#431C22semidirect
47. C2×#3228C7direct (N × H)
112. C2×#533C2×C7semidirect
114. #3228C2×C7direct (N × H)
115. C2×#500C2×C7semidirect
116. C2×#498C2×C7semidirect
117. C2×#486C2×C7semidirect
118. C2×#486C2×C7semidirect
119. C2×#482C2×C7semidirect
120. C2×#478C2×C7semidirect
121. C2×#431C2×C7semidirect
185. #533C22×C7semidirect
204. #500C22×C7semidirect
205. #498C22×C7semidirect
206. #486C22×C7semidirect
207. #486C22×C7semidirect
208. #482C22×C7semidirect
209. #478C22×C7semidirect
210. #431C22×C7semidirect
411. C2×C7#3228direct (N × H)
458. C7C2×#3228direct (N × H)
483. C2C7×#3228direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5332
C2×C7×#5322
C7×#32282
C2×C7×#5002
C2×C7×#4982
C2×C7×#4862
C2×C7×#4822
C2×C7×#4782
C2×C7×#4312
C2×#32287