Finite Group Databasebeta

← C2×C7×#3369 #13238526

Subgroups — C2×C7×#3369 #13238526

Order 1792.

1842 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×Q82C211yesmaximal, characteristic, complemented
2C2×C7×#556C211yesmaximal, characteristic
3C7×#3369C2116yesmaximal, direct product factor
4C2×C7×#489C212yesmaximal
5C2×C7×#487C211yesmaximal, characteristic, complemented
6C2×C7×#475C212yesmaximal, complemented
7C2×C7×#476C211yesmaximal, characteristic
8C2×C7×#464C212yesmaximal
9C2×C7×#462C211yesmaximal, characteristic, complemented
10C2×C7×#446C211yesmaximal, characteristic, complemented
11C2×C7×#441C212yesmaximal, complemented
12C2×C4×C7×SD16C211yesmaximal, characteristic
13C22×C7×Q16C2212yes—
14C22×C7×SD16C2211yescharacteristic
15C7×Q82—24no—
16C7×Q82C2218yescomplemented
17C7×#556—24no—
18C7×#556C2218yes—
19C2×C7×C4⋊Q8—21no—
20C2×C7×C4⋊Q8—22no—
21C2×C7×C4⋊Q8C2211yescharacteristic
22C2×C7×C4⋊Q8C2211yescharacteristic
23C2×C7×C4⋊Q8C2212yes—
24C2×C7×#126—22no—
25C2×C7×#124C2212yes—
26C2×C7×C22⋊Q8—21no—
27C2×C4×C7×Q8—21no—
28C2×C4×C7×Q8C2211yescharacteristic
29C2×C4×C7×Q8C2212yes—
30C2×C4×C7×Q8C2212yes—
31C2×C4×C7×Q8C2211yescharacteristic
32C2×C4×C7×D4C2211yescharacteristic
33C2×C7×#117—21no—
34C7×#489C22116yes—
35C7×#487C2218yescomplemented
36C7×#476C2218yes—
37C7×#475C22116yescomplemented
38C7×#464C22116yes—
39C7×#462C2218yescomplemented
40C7×#446C2218yescomplemented
41C7×#441C22116yescomplemented
42C4×C7×SD16C2218yes—
43C2×C7×#106C2211yescharacteristic
44C2×C7×#106C2212yes—
45C2×C7×C4⋊C8C2211yescharacteristic
46C2×C7×C4⋊C8C2212yes—
47C2×C7×#103C2211yescharacteristic
48C2×C7×#103C2212yes—
49C2×C7×#103C2212yes—
50C2×C7×#103C2212yes—
51C2×C7×#102C2211yescharacteristic
52C2×C7×#102C2212yes—
53C2×C7×#97C2212yes—
54C2×C4×C7×C8C2211yescharacteristic
55C2×#3369C711yesmaximal, characteristic, direct product factor, Sylow
56C22×C7×Q8—22no—
57C22×C7×Q8—21no—
58C22×C7×Q8D411yescharacteristic
59C22×C7×Q8C2312yes—
60C22×C7×Q8C2311yescharacteristic
61C22×C7×D4C2311yescharacteristic
62C23×C4×C7—21no—
63C2×C7×Q16—28no—
64C2×C7×Q16C2318yes—
65C2×C7×SD16—24no—
66C2×C7×SD16C2314yes—
67C22×C7×C8C2311yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C22×C7×C8C2312yes—
70C7×C4⋊Q8—28no—
71C7×C4⋊Q8—216no—
72C7×C4⋊Q8—22no—
73C7×C4⋊Q8—22no—
74C7×C4⋊Q8—24no—
75C7×C4⋊Q8C2314yes—
76C7×C4⋊Q8C2318yes—
77C7×C4⋊Q8C2314yes—
78C7×#126—216no—
79C7×#124—24no—
80C7×#124C2318yes—
81C7×C22⋊Q8—28no—
82C4×C7×Q8—28no—
83C4×C7×Q8—24no—
84C4×C7×Q8—22no—
85C4×C7×Q8—24no—
86C4×C7×Q8D414yes—
87C4×C7×Q8C2318yes—
88C4×C7×Q8C2314yes—
89C4×C7×Q8C2318yes—
90C4×C7×Q8C2314yes—
91C4×C7×D4—22no—
92C4×C7×D4C2314yes—
93C7×#117—28no—
94C2×C7×C4⋊C4—21no—
95C2×C7×C4⋊C4—22no—
96C2×C7×C4⋊C4—21no—
97C2×C7×C4⋊C4—22no—
98C2×C7×C4⋊C4—22no—
99C2×C7×C4⋊C4—21no—
100C2×C7×C4⋊C4—21no—
101C2×C7×C4⋊C4D411yescharacteristic
102C2×C7×C4⋊C4D412yes—
103C2×C7×C4⋊C4C2311yescharacteristic
104C2×C7×C4⋊C4C2311yescharacteristic
105C2×C7×C4⋊C4C2312yes—
106C2×C7×C22⋊C4—22no—
107C2×C7×C22⋊C4—21no—
108C2×C7×C22⋊C4—22no—
109C2×C42×C7—22no—
110C2×C42×C7—22no—
111C2×C42×C7—21no—
112C2×C42×C7C2311yescharacteristic
113C2×C42×C7C2312yes—
114C7×#106C2314yes—
115C7×#106C2318yes—
116C7×C4⋊C8C2318yes—
117C7×C4⋊C8C2314yes—
118C7×#103C2318yes—
119C7×#103C2318yes—
120C7×#103C2318yes—
121C7×#103C2314yes—
122C7×#102C2318yes—
123C7×#102C2314yes—
124C7×#97C2318yes—
125C4×C7×C8C2314yes—
126C2×Q82C2×C711yescharacteristic, complemented
127C2×#556C2×C711yescharacteristic
128#3369C2×C7116yesdirect product factor
129C2×#489C2×C712yes—
130C2×#487C2×C711yescharacteristic, complemented
131C2×#475C2×C712yescomplemented
132C2×#476C2×C711yescharacteristic
133C2×#464C2×C712yes—
134C2×#462C2×C711yescharacteristic, complemented
135C2×#446C2×C711yescharacteristic, complemented
136C2×#441C2×C712yescomplemented
137C2×C4×SD16C2×C711yescharacteristic
138C24×C7—21no—
139C2×C7×Q8—24no—
140C2×C7×Q8—21no—
141C2×C7×Q8—22no—
142C2×C7×Q8—28no—
143C2×C7×Q8—24no—
144C2×C7×Q8—22no—
145C2×C7×Q8—24no—
146C2×C7×Q8—24no—
147C2×C7×Q8—84no—
148C2×C7×Q8—82no—
149C2×C7×Q8C2×D414yes—
150C2×C7×Q8#4114yes—
151C2×C7×Q8C2414yes—
152C2×C7×Q8C2412yes—
153C2×C7×D4—21no—
154C2×C7×D4—22no—
155C2×C7×D4—22no—
156C2×C7×D4C2412yes—
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7—22no—
160C22×C4×C7—22no—
161C22×C4×C7—21no—
162C22×C4×C7—24no—
163C22×C4×C7—44no—
164C22×C4×C7C2×D412yes—
165C22×C4×C7C2×D412yes—
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2411yescharacteristic
169C7×Q16—216no—
170C7×SD16—28no—
171C2×C7×C8—22no—
172C2×C7×C8—24no—
173C2×C7×C8—22no—
174C2×C7×C8C2412yes—
175C2×C7×C8C2412yes—
176C2×C7×C8C2414yes—
177C7×C4⋊C4—24no—
178C7×C4⋊C4—28no—
179C7×C4⋊C4—24no—
180C7×C4⋊C4—28no—
181C7×C4⋊C4—28no—
182C7×C4⋊C4—24no—
183C7×C4⋊C4—24no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4C2×D414yes—
188C7×C4⋊C4C2×D418yes—
189C7×C4⋊C4C2414yes—
190C7×C4⋊C4C2412yes—
191C7×C4⋊C4C2412yes—
192C7×C22⋊C4—28no—
193C7×C22⋊C4—24no—
194C7×C22⋊C4—28no—
195C42×C7—28no—
196C42×C7—28no—
197C42×C7—24no—
198C42×C7C2×D412yes—
199C42×C7C2×D414yes—
200C42×C7C2412yes—
201C42×C7C2414yes—
202C22×Q16C22×C712yes—
203C22×SD16C22×C711yescharacteristic
204Q82—24no—
205Q82C22×C718yescomplemented
206#556—24no—
207#556C22×C718yes—
208C2×C4⋊Q8—21no—
209C2×C4⋊Q8—22no—
210C2×C4⋊Q8C22×C711yescharacteristic
211C2×C4⋊Q8C22×C712yes—
212C2×C4⋊Q8C22×C711yescharacteristic
213C2×#126—22no—
214C2×#124C22×C712yes—
215C2×C22⋊Q8—21no—
216C2×C4×Q8—21no—
217C2×C4×Q8C22×C712yes—
218C2×C4×Q8C22×C711yescharacteristic
219C2×C4×Q8C22×C712yes—
220C2×C4×Q8C22×C711yescharacteristic
221C2×C4×D4C22×C711yescharacteristic
222C2×#117—21no—
223#489C22×C7116yes—
224#487C22×C718yescomplemented
225#476C22×C718yes—
226#475C22×C7116yescomplemented
227#464C22×C7116yes—
228#462C22×C718yescomplemented
229#446C22×C718yescomplemented
230#441C22×C7116yescomplemented
231C4×SD16C22×C718yes—
232C2×#106C22×C711yescharacteristic
233C2×#106C22×C712yes—
234C2×C4⋊C8C22×C712yes—
235C2×C4⋊C8C22×C711yescharacteristic
236C2×#103C22×C712yes—
237C2×#103C22×C712yes—
238C2×#103C22×C712yes—
239C2×#103C22×C711yescharacteristic
240C2×#102C22×C711yescharacteristic
241C2×#102C22×C712yes—
242C2×#97C22×C712yes—
243C2×C4×C8C22×C711yescharacteristic
244C23×C7—22no—
245C23×C7—42no—
246C23×C7—42no—
247C23×C7—42no—
248C23×C7C22×D411yescharacteristic
249C7×Q8—24no—
250C7×Q8—28no—
251C7×Q8—22no—
252C7×Q8—24no—
253C7×Q8—24no—
254C7×Q8—44no—
255C7×Q8—88no—
256C7×Q8—84no—
257C7×Q8C2×#4114yes—
258C7×D4—22no—
259C7×D4—22no—
260C7×D4—42no—
261C2×C4×C7—24no—
262C2×C4×C7—24no—
263C2×C4×C7—24no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—24no—
268C2×C4×C7—24no—
269C2×C4×C7—22no—
270C2×C4×C7—44no—
271C2×C4×C7—42no—
272C2×C4×C7—44no—
273C2×C4×C7—42no—
274C2×C4×C7—48no—
275C2×C4×C7#13714yes—
276C2×C4×C7C22×D414yes—
277C2×C4×C7C22×D414yes—
278C2×C4×C7C22×D412yes—
279C2×C4×C7C22×D411yescharacteristic
280C2×C4×C7C22×D412yes—
281C2×C4×C7C2×#4112yes—
282C2×C4×C72−1+412yes—
283C2×C4×C7C2511yescharacteristic
284C7×C8—22no—
285C7×C8—24no—
286C7×C8—22no—
287C22×Q8—22no—
288C22×Q8—21no—
289C22×Q8C7×D411yescharacteristic
290C22×Q8C23×C712yes—
291C22×Q8C23×C711yescharacteristic
292C22×D4C23×C711yescharacteristic
293C23×C4—21no—
294C2×Q16—28no—
295C2×Q16C23×C718yes—
296C2×SD16—24no—
297C2×SD16C23×C714yes—
298C22×C8C23×C711yescharacteristic
299C22×C8C23×C712yes—
300C22×C8C23×C711yescharacteristic
301C4⋊Q8—28no—
302C4⋊Q8—216no—
303C4⋊Q8—22no—
304C4⋊Q8—22no—
305C4⋊Q8—24no—
306C4⋊Q8C23×C714yes—
307C4⋊Q8C23×C718yes—
308C4⋊Q8C23×C714yes—
309#126—216no—
310#124—24no—
311#124C23×C718yes—
312C22⋊Q8—28no—
313C4×Q8—28no—
314C4×Q8—24no—
315C4×Q8—22no—
316C4×Q8—24no—
317C4×Q8C7×D414yes—
318C4×Q8C23×C718yes—
319C4×Q8C23×C714yes—
320C4×Q8C23×C718yes—
321C4×Q8C23×C714yes—
322C4×D4—22no—
323C4×D4C23×C714yes—
324#117—28no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4C7×D411yescharacteristic
333C2×C4⋊C4C7×D412yes—
334C2×C4⋊C4C23×C711yescharacteristic
335C2×C4⋊C4C23×C711yescharacteristic
336C2×C4⋊C4C23×C712yes—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42C23×C711yescharacteristic
344C2×C42C23×C712yes—
345#106C23×C714yes—
346#106C23×C718yes—
347C4⋊C8C23×C714yes—
348C4⋊C8C23×C718yes—
349#103C23×C714yes—
350#103C23×C718yes—
351#103C23×C718yes—
352#103C23×C718yes—
353#102C23×C714yes—
354#102C23×C718yes—
355#97C23×C718yes—
356C4×C8C23×C714yes—
357C22×C7—42no—
358C22×C7—42no—
359C22×C7—42no—
360C22×C7—84no—
361C22×C7#54812yes—
362C22×C7C2×#13712yes—
363C22×C7#57712yes—
364C22×C7C23×D411yescharacteristic
365C4×C7—24no—
366C4×C7—24no—
367C4×C7—24no—
368C4×C7—22no—
369C4×C7—44no—
370C4×C7—42no—
371C4×C7—44no—
372C4×C7—42no—
373C4×C7#54812yes—
374C4×C7C2×#13714yes—
375C4×C7C22×#4111yescharacteristic
376C4×C7C2×2−1+411yescharacteristic
377C24—21no—
378C2×Q8—24no—
379C2×Q8—21no—
380C2×Q8—22no—
381C2×Q8—28no—
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8—84no—
387C2×Q8—82no—
388C2×Q8C2×C7×D414yes—
389C2×Q8C7×#4114yes—
390C2×Q8C24×C714yes—
391C2×Q8C24×C712yes—
392C2×D4—21no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4C24×C712yes—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—44no—
403C22×C4C2×C7×D412yes—
404C22×C4C2×C7×D412yes—
405C22×C4C2×C7×D411yescharacteristic
406C22×C4C2×C7×D411yescharacteristic
407C22×C4C24×C711yescharacteristic
408Q16—216no—
409SD16—28no—
410C2×C8—22no—
411C2×C8—24no—
412C2×C8—22no—
413C2×C8C24×C712yes—
414C2×C8C24×C712yes—
415C2×C8C24×C714yes—
416C4⋊C4—24no—
417C4⋊C4—28no—
418C4⋊C4—24no—
419C4⋊C4—28no—
420C4⋊C4—28no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—22no—
426C4⋊C4C2×C7×D414yes—
427C4⋊C4C2×C7×D418yes—
428C4⋊C4C24×C712yes—
429C4⋊C4C24×C714yes—
430C4⋊C4C24×C712yes—
431C22⋊C4—28no—
432C22⋊C4—24no—
433C22⋊C4—28no—
434C42—28no—
435C42—28no—
436C42—24no—
437C42C2×C7×D412yes—
438C42C2×C7×D414yes—
439C42C24×C712yes—
440C42C24×C714yes—
441C2×C7—82no—
442C2×C7#336914yesdirect product factor
443C2×C7C2×#54811yescharacteristic
444C2×C7C22×#13711yescharacteristic
445C2×C7C2×#57711yescharacteristic
446C23—22no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23C22×C7×D411yescharacteristic
451Q8—22no—
452Q8—24no—
453Q8—28no—
454Q8—24no—
455Q8—24no—
456Q8—44no—
457Q8—88no—
458Q8—84no—
459Q8C2×C7×#4114yes—
460D4—22no—
461D4—22no—
462D4—42no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—48no—
477C2×C4C7×#13714yes—
478C2×C4C22×C7×D414yes—
479C2×C4C22×C7×D414yes—
480C2×C4C22×C7×D412yes—
481C2×C4C22×C7×D411yescharacteristic
482C2×C4C22×C7×D412yes—
483C2×C4C2×C7×#4112yes—
484C2×C4C7×2−1+412yes—
485C2×C4C25×C711yescharacteristic
486C8—22no—
487C8—24no—
488C8—22no—
489C7C2×#336911yescharacteristic, direct product factor, Sylow
490C22—42no—
491C22—42no—
492C22—42no—
493C22—84no—
494C22C7×#54812yes—
495C22C2×C7×#13712yes—
496C22C7×#57712yes—
497C22C23×C7×D411yescharacteristic
498C4—24no—
499C4—24no—
500C4—22no—
501C4—24no—
502C4—42no—
503C4—44no—
504C4—42no—
505C4—44no—
506C4C7×#54812yes—
507C4C2×C7×#13714yes—
508C4C22×C7×#4111yescharacteristic
509C4C2×C7×2−1+411yescharacteristic
510C2—82no—
511C2C7×#336914yesdirect product factor
512C2C2×C7×#54811yescharacteristic
513C2C22×C7×#13711yescharacteristic
514C2C2×C7×#57711yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC2×C4C25×C7
prime radicalC22×C7×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C4×C7×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×Q82C2semidirect
3. C7×#3369C2direct (N × H)
5. C2×C7×#487C2semidirect
6. C2×C7×#475C2semidirect
9. C2×C7×#462C2semidirect
10. C2×C7×#446C2semidirect
11. C2×C7×#441C2semidirect
16. C7×Q82C22semidirect
35. C7×#487C22semidirect
37. C7×#475C22semidirect
39. C7×#462C22semidirect
40. C7×#446C22semidirect
41. C7×#441C22semidirect
55. C2×#3369C7direct (N × H)
126. C2×Q82C2×C7semidirect
128. #3369C2×C7direct (N × H)
130. C2×#487C2×C7semidirect
131. C2×#475C2×C7semidirect
134. C2×#462C2×C7semidirect
135. C2×#446C2×C7semidirect
136. C2×#441C2×C7semidirect
205. Q82C22×C7semidirect
224. #487C22×C7semidirect
226. #475C22×C7semidirect
228. #462C22×C7semidirect
229. #446C22×C7semidirect
230. #441C22×C7semidirect
442. C2×C7#3369direct (N × H)
489. C7C2×#3369direct (N × H)
511. C2C7×#3369direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×Q822
C2×C7×#5562
C7×#33692
C2×C7×#4892
C2×C7×#4872
C2×C7×#4752
C2×C7×#4762
C2×C7×#4642
C2×C7×#4622
C2×C7×#4462
C2×C7×#4412
C2×C4×C7×SD162
C2×#33697