Finite Group Databasebeta

← C2×C7×#3336 #13238443

Subgroups — C2×C7×#3336 #13238443

Order 1792.

1866 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#552C212yesmaximal
2C7×#3336C2116yesmaximal, direct product factor
3C2×C7×#499C211yesmaximal, characteristic, complemented
4C2×C7×#485C212yesmaximal, complemented
5C2×C7×#483C212yesmaximal, complemented
6C2×C7×#479C212yesmaximal, complemented
7C2×C7×#467C211yesmaximal, characteristic, complemented
8C2×C7×#467C211yesmaximal, characteristic, complemented
9C2×C4×C7×Q16C211yesmaximal, characteristic, complemented
10C2×C7×C8×D4C211yesmaximal, characteristic
11C2×C7×#427C211yesmaximal, characteristic, complemented
12C22×C7×#107C211yesmaximal, characteristic, complemented
13C22×C7×Q16C2211yescharacteristic
14C23×C7×C8C2211yescharacteristic
15C23×C7×C8C2211yescharacteristic
16C7×#552—28no—
17C7×#552C22116yes—
18C2×C7×#126—22no—
19C2×C7×#125—22no—
20C2×C7×#125C2212yes—
21C2×C7×#123—22no—
22C2×C7×C22⋊Q8C2212yes—
23C2×C7×C22⋊Q8C2212yes—
24C2×C4×C7×Q8C2212yes—
25C2×C4×C7×D4C2211yescharacteristic
26C2×C7×#117—22no—
27C2×C7×#117C2212yes—
28C22×C7×C4⋊C4C2212yes—
29C7×#499C2218yescomplemented
30C7×#485C22116yescomplemented
31C7×#483C22116yescomplemented
32C7×#479C22116yescomplemented
33C7×#467C2218yescomplemented
34C7×#467C2218yescomplemented
35C4×C7×Q16C2218yescomplemented
36C7×C8×D4C2218yes—
37C7×#427C2218yescomplemented
38C2×C7×#107C2211yescharacteristic
39C2×C7×#107C2211yescharacteristic
40C2×C7×#107C2211yescharacteristic
41C2×C7×#107C2212yes—
42C2×C7×#107C2218yescomplemented
43C2×C7×#106C2212yes—
44C2×C7×#106C2212yes—
45C2×C7×C4⋊C8C2211yescharacteristic
46C2×C7×#103C2212yes—
47C2×C7×#103C2212yes—
48C2×C7×#103C2212yes—
49C2×C7×C22⋊C8C2211yescharacteristic
50C2×C7×C22⋊C8C2211yescharacteristic
51C2×C4×C7×C8C2211yescharacteristic
52C2×#3336C711yesmaximal, characteristic, direct product factor, Sylow
53C22×C7×Q8C2312yes—
54C22×C7×D4D411yescharacteristic
55C23×C4×C7—22no—
56C23×C4×C7C2311yescharacteristic
57C23×C4×C7C2311yescharacteristic
58C2×C7×Q16—24no—
59C2×C7×Q16C2314yes—
60C22×C7×C8—24no—
61C22×C7×C8—24no—
62C22×C7×C8C2314yes—
63C22×C7×C8C2314yes—
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C7×#126—216no—
69C7×#125—216no—
70C7×#125—24no—
71C7×#125C2318yes—
72C7×#123—216no—
73C7×C22⋊Q8—24no—
74C7×C22⋊Q8—24no—
75C7×C22⋊Q8C2318yes—
76C7×C22⋊Q8C2318yes—
77C4×C7×Q8—24no—
78C4×C7×Q8C2318yes—
79C4×C7×D4D414yes—
80C4×C7×D4C2314yes—
81C7×#117—24no—
82C7×#117—216no—
83C7×#117C2318yes—
84C2×C7×C4⋊C4—22no—
85C2×C7×C4⋊C4—24no—
86C2×C7×C4⋊C4—22no—
87C2×C7×C4⋊C4—22no—
88C2×C7×C4⋊C4—22no—
89C2×C7×C4⋊C4—22no—
90C2×C7×C4⋊C4—22no—
91C2×C7×C4⋊C4D411yescharacteristic
92C2×C7×C4⋊C4C2318yes—
93C2×C7×C4⋊C4C2312yes—
94C2×C7×C4⋊C4C2312yes—
95C2×C7×C4⋊C4C2312yes—
96C2×C7×C22⋊C4—22no—
97C2×C7×C22⋊C4—22no—
98C2×C7×C22⋊C4—22no—
99C2×C7×C22⋊C4D411yescharacteristic
100C2×C7×C22⋊C4D411yescharacteristic
101C2×C42×C7—22no—
102C2×C42×C7—22no—
103C2×C42×C7C2311yescharacteristic
104C7×#107C2314yes—
105C7×#107C2314yes—
106C7×#107C2318yes—
107C7×#107C2314yes—
108C7×#106C2318yes—
109C7×#106C2318yes—
110C7×C4⋊C8C2314yes—
111C7×#103C2318yes—
112C7×#103C2318yes—
113C7×#103C2318yes—
114C7×C22⋊C8C2314yes—
115C7×C22⋊C8C2314yes—
116C4×C7×C8C2314yes—
117C2×#552C2×C712yes—
118#3336C2×C7116yesdirect product factor
119C2×#499C2×C711yescharacteristic, complemented
120C2×#485C2×C712yescomplemented
121C2×#483C2×C712yescomplemented
122C2×#479C2×C712yescomplemented
123C2×#467C2×C711yescharacteristic, complemented
124C2×#467C2×C711yescharacteristic, complemented
125C2×C4×Q16C2×C711yescharacteristic, complemented
126C2×C8×D4C2×C711yescharacteristic
127C2×#427C2×C711yescharacteristic, complemented
128C22×#107C2×C711yescharacteristic, complemented
129C24×C7C2×D411yescharacteristic
130C24×C7C2×D411yescharacteristic
131C2×C7×Q8—22no—
132C2×C7×Q8—24no—
133C2×C7×Q8—24no—
134C2×C7×Q8C2414yes—
135C2×C7×D4—24no—
136C2×C7×D4C2×D414yes—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—28no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—48no—
147C22×C4×C7C2×D412yes—
148C22×C4×C7C2×D412yes—
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2412yes—
154C22×C4×C7C2412yes—
155C22×C4×C7C2411yescharacteristic
156C7×Q16—28no—
157C2×C7×C8—22no—
158C2×C7×C8—22no—
159C2×C7×C8—22no—
160C2×C7×C8—28no—
161C2×C7×C8—28no—
162C2×C7×C8#4114yes—
163C2×C7×C8C2412yes—
164C2×C7×C8C2412yes—
165C2×C7×C8C2412yes—
166C2×C7×C8C2412yes—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—28no—
170C7×C4⋊C4—28no—
171C7×C4⋊C4—28no—
172C7×C4⋊C4—28no—
173C7×C4⋊C4—28no—
174C7×C4⋊C4—28no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4C2×D414yes—
177C7×C4⋊C4C2414yes—
178C7×C4⋊C4C2414yes—
179C7×C4⋊C4C2414yes—
180C7×C22⋊C4—28no—
181C7×C22⋊C4—28no—
182C7×C22⋊C4—28no—
183C7×C22⋊C4C2×D414yes—
184C7×C22⋊C4C2×D414yes—
185C42×C7—28no—
186C42×C7—28no—
187C42×C7C2×D412yes—
188C42×C7C2412yes—
189C22×Q16C22×C711yescharacteristic
190C23×C8C22×C711yescharacteristic
191C23×C8C22×C711yescharacteristic
192#552—28no—
193#552C22×C7116yes—
194C2×#126—22no—
195C2×#125—22no—
196C2×#125C22×C712yes—
197C2×#123—22no—
198C2×C22⋊Q8C22×C712yes—
199C2×C22⋊Q8C22×C712yes—
200C2×C4×Q8C22×C712yes—
201C2×C4×D4C22×C711yescharacteristic
202C2×#117—22no—
203C2×#117C22×C712yes—
204C22×C4⋊C4C22×C712yes—
205#499C22×C718yescomplemented
206#485C22×C7116yescomplemented
207#483C22×C7116yescomplemented
208#479C22×C7116yescomplemented
209#467C22×C718yescomplemented
210#467C22×C718yescomplemented
211C4×Q16C22×C718yescomplemented
212C8×D4C22×C718yes—
213#427C22×C718yescomplemented
214C2×#107C22×C712yes—
215C2×#107C22×C711yescharacteristic
216C2×#107C22×C711yescharacteristic
217C2×#107C22×C711yescharacteristic
218C2×#107C22×C718yescomplemented
219C2×#106C22×C712yes—
220C2×#106C22×C712yes—
221C2×C4⋊C8C22×C711yescharacteristic
222C2×#103C22×C712yes—
223C2×#103C22×C712yes—
224C2×#103C22×C712yes—
225C2×C22⋊C8C22×C711yescharacteristic
226C2×C22⋊C8C22×C711yescharacteristic
227C2×C4×C8C22×C711yescharacteristic
228C23×C7—22no—
229C23×C7—22no—
230C23×C7—22no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7#13514yes—
234C23×C7C22×D412yes—
235C23×C7C22×D412yes—
236C23×C7C22×D411yescharacteristic
237C7×Q8—24no—
238C7×Q8—24no—
239C7×Q8—44no—
240C7×D4—28no—
241C2×C4×C7—22no—
242C2×C4×C7—22no—
243C2×C4×C7—22no—
244C2×C4×C7—22no—
245C2×C4×C7—24no—
246C2×C4×C7—24no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—22no—
250C2×C4×C7—22no—
251C2×C4×C7—24no—
252C2×C4×C7—24no—
253C2×C4×C7—24no—
254C2×C4×C7—24no—
255C2×C4×C7—44no—
256C2×C4×C7—416no—
257C2×C4×C7—44no—
258C2×C4×C7—44no—
259C2×C4×C7—44no—
260C2×C4×C7C22×D412yes—
261C2×C4×C7C22×D412yes—
262C2×C4×C7C22×D412yes—
263C2×C4×C7C22×D412yes—
264C2×C4×C7C22×D411yescharacteristic
265C2×C4×C7C2×#4112yes—
266C2×C4×C72−1+412yes—
267C2×C4×C7C2511yescharacteristic
268C7×C8—22no—
269C7×C8—22no—
270C7×C8—22no—
271C7×C8C2×#4114yes—
272C22×Q8C23×C712yes—
273C22×D4C7×D411yescharacteristic
274C23×C4—22no—
275C23×C4C23×C711yescharacteristic
276C23×C4C23×C711yescharacteristic
277C2×Q16—24no—
278C2×Q16C23×C714yes—
279C22×C8—24no—
280C22×C8—24no—
281C22×C8C23×C711yescharacteristic
282C22×C8C23×C711yescharacteristic
283C22×C8C23×C714yes—
284C22×C8C23×C714yes—
285C22×C8C23×C711yescharacteristic
286C22×C8C23×C711yescharacteristic
287#126—216no—
288#125—216no—
289#125—24no—
290#125C23×C718yes—
291#123—216no—
292C22⋊Q8—24no—
293C22⋊Q8—24no—
294C22⋊Q8C23×C718yes—
295C22⋊Q8C23×C718yes—
296C4×Q8—24no—
297C4×Q8C23×C718yes—
298C4×D4C7×D414yes—
299C4×D4C23×C714yes—
300#117—24no—
301#117—216no—
302#117C23×C718yes—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4C7×D411yescharacteristic
311C2×C4⋊C4C23×C712yes—
312C2×C4⋊C4C23×C712yes—
313C2×C4⋊C4C23×C718yes—
314C2×C4⋊C4C23×C712yes—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4C7×D411yescharacteristic
319C2×C22⋊C4C7×D411yescharacteristic
320C2×C42—22no—
321C2×C42—22no—
322C2×C42C23×C711yescharacteristic
323#107C23×C714yes—
324#107C23×C714yes—
325#107C23×C718yes—
326#107C23×C714yes—
327#106C23×C718yes—
328#106C23×C718yes—
329C4⋊C8C23×C714yes—
330#103C23×C718yes—
331#103C23×C718yes—
332#103C23×C718yes—
333C22⋊C8C23×C714yes—
334C22⋊C8C23×C714yes—
335C4×C8C23×C714yes—
336C22×C7—28no—
337C22×C7—22no—
338C22×C7—22no—
339C22×C7—22no—
340C22×C7—22no—
341C22×C7—22no—
342C22×C7—44no—
343C22×C7#54712yes—
344C22×C7C2×#13514yes—
345C22×C7C2×#13512yes—
346C22×C7#57812yes—
347C22×C7C23×D411yescharacteristic
348C4×C7—22no—
349C4×C7—22no—
350C4×C7—22no—
351C4×C7—22no—
352C4×C7—44no—
353C4×C7—44no—
354C4×C7—44no—
355C4×C7—44no—
356C4×C7#54712yes—
357C4×C7C22×#4111yescharacteristic
358C4×C7C2×2−1+411yescharacteristic
359C24C2×C7×D411yescharacteristic
360C24C2×C7×D411yescharacteristic
361C2×Q8—22no—
362C2×Q8—24no—
363C2×Q8—24no—
364C2×Q8C24×C714yes—
365C2×D4—24no—
366C2×D4C2×C7×D414yes—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—28no—
376C22×C4—48no—
377C22×C4C2×C7×D411yescharacteristic
378C22×C4C2×C7×D411yescharacteristic
379C22×C4C2×C7×D411yescharacteristic
380C22×C4C2×C7×D411yescharacteristic
381C22×C4C2×C7×D412yes—
382C22×C4C2×C7×D412yes—
383C22×C4C24×C712yes—
384C22×C4C24×C712yes—
385C22×C4C24×C711yescharacteristic
386Q16—28no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—28no—
391C2×C8—28no—
392C2×C8C7×#4114yes—
393C2×C8C24×C712yes—
394C2×C8C24×C712yes—
395C2×C8C24×C712yes—
396C2×C8C24×C712yes—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—28no—
400C4⋊C4—28no—
401C4⋊C4—28no—
402C4⋊C4—28no—
403C4⋊C4—28no—
404C4⋊C4—28no—
405C4⋊C4—22no—
406C4⋊C4C2×C7×D414yes—
407C4⋊C4C24×C714yes—
408C4⋊C4C24×C714yes—
409C4⋊C4C24×C714yes—
410C22⋊C4—28no—
411C22⋊C4—28no—
412C22⋊C4—28no—
413C22⋊C4C2×C7×D414yes—
414C22⋊C4C2×C7×D414yes—
415C42—28no—
416C42—28no—
417C42C2×C7×D412yes—
418C42C24×C712yes—
419C2×C7—24no—
420C2×C7—42no—
421C2×C7#333614yesdirect product factor
422C2×C7C2×#54711yescharacteristic
423C2×C7C22×#13511yescharacteristic
424C2×C7C2×#57811yescharacteristic
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23C7×#13514yes—
431C23C22×C7×D412yes—
432C23C22×C7×D412yes—
433C23C22×C7×D411yescharacteristic
434Q8—24no—
435Q8—24no—
436Q8—44no—
437D4—28no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—44no—
453C2×C4—416no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4C22×C7×D412yes—
458C2×C4C22×C7×D412yes—
459C2×C4C22×C7×D412yes—
460C2×C4C22×C7×D412yes—
461C2×C4C22×C7×D411yescharacteristic
462C2×C4C2×C7×#4112yes—
463C2×C4C7×2−1+412yes—
464C2×C4C25×C711yescharacteristic
465C8—22no—
466C8—22no—
467C8—22no—
468C8C2×C7×#4114yes—
469C7C2×#333611yescharacteristic, direct product factor, Sylow
470C22—22no—
471C22—22no—
472C22—28no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—44no—
477C22C7×#54712yes—
478C22C2×C7×#13514yes—
479C22C2×C7×#13512yes—
480C22C7×#57812yes—
481C22C23×C7×D411yescharacteristic
482C4—22no—
483C4—22no—
484C4—22no—
485C4—22no—
486C4—44no—
487C4—44no—
488C4—44no—
489C4—44no—
490C4C7×#54712yes—
491C4C22×C7×#4111yescharacteristic
492C4C2×C7×2−1+411yescharacteristic
493C2—24no—
494C2—42no—
495C2C7×#333614yesdirect product factor
496C2C2×C7×#54711yescharacteristic
497C2C22×C7×#13511yescharacteristic
498C2C2×C7×#57811yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC2×C4C25×C7
prime radicalC22×C7×D4D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#3336C2direct (N × H)
3. C2×C7×#499C2semidirect
4. C2×C7×#485C2semidirect
5. C2×C7×#483C2semidirect
6. C2×C7×#479C2semidirect
7. C2×C7×#467C2semidirect
8. C2×C7×#467C2semidirect
9. C2×C4×C7×Q16C2semidirect
11. C2×C7×#427C2semidirect
12. C22×C7×#107C2semidirect
29. C7×#499C22semidirect
30. C7×#485C22semidirect
31. C7×#483C22semidirect
32. C7×#479C22semidirect
33. C7×#467C22semidirect
34. C7×#467C22semidirect
35. C4×C7×Q16C22semidirect
37. C7×#427C22semidirect
42. C2×C7×#107C22semidirect
52. C2×#3336C7direct (N × H)
118. #3336C2×C7direct (N × H)
119. C2×#499C2×C7semidirect
120. C2×#485C2×C7semidirect
121. C2×#483C2×C7semidirect
122. C2×#479C2×C7semidirect
123. C2×#467C2×C7semidirect
124. C2×#467C2×C7semidirect
125. C2×C4×Q16C2×C7semidirect
127. C2×#427C2×C7semidirect
128. C22×#107C2×C7semidirect
205. #499C22×C7semidirect
206. #485C22×C7semidirect
207. #483C22×C7semidirect
208. #479C22×C7semidirect
209. #467C22×C7semidirect
210. #467C22×C7semidirect
211. C4×Q16C22×C7semidirect
213. #427C22×C7semidirect
218. C2×#107C22×C7semidirect
421. C2×C7#3336direct (N × H)
469. C7C2×#3336direct (N × H)
495. C2C7×#3336direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5522
C7×#33362
C2×C7×#4992
C2×C7×#4852
C2×C7×#4832
C2×C7×#4792
C2×C7×#4672
C2×C4×C7×Q162
C2×C7×C8×D42
C2×C7×#4272
C22×C7×#1072
C2×#33367