Finite Group Databasebeta

← C7×#20849 #13198065

Subgroups — C7×#20849 #13198065

Order 1792.

1462 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#520C211yesmaximal, characteristic
2C2×C7×#428C211yesmaximal, characteristic, complemented
3C2×C7×#427C211yesmaximal, characteristic, complemented
4C2×C7×#408C211yesmaximal, characteristic
5C7×#2311C211yesmaximal, characteristic
6C7×#1854C212yesmaximal, complemented
7C7×#1854C212yesmaximal, complemented
8C7×#1853C212yesmaximal, complemented
9C7×#1852C212yesmaximal, complemented
10C7×#1736C211yesmaximal, characteristic, complemented
11C2×C7×#340C211yesmaximal, characteristic, complemented
12C22×C7×#41C2211yescharacteristic
13C22×C7×M16C2211yescharacteristic
14C23×C7×C8C2211yescharacteristic
15C7×#520—24no—
16C7×#520—24no—
17C2×C4×C7×Q8—21no—
18C2×C4×C7×D4—22no—
19C2×C4×C7×D4—21no—
20C2×C7×#117C2211yescharacteristic
21C2×C7×#117C2212yes—
22C2×C7×#117C2211yescharacteristic
23C22×C7×C4⋊C4C2211yescharacteristic
24C22×C7×C4⋊C4C2212yes—
25C22×C7×C4⋊C4C2211yescharacteristic
26C7×#428—24no—
27C7×#427—24no—
28C2×C7×#107C2211yescharacteristic
29C2×C7×#107C2211yescharacteristic
30C2×C7×#107C2212yes—
31C2×C7×#106C2211yescharacteristic
32C2×C7×#106C2211yescharacteristic
33C2×C7×#106C2212yes—
34C7×#408C418yescomplemented
35C2×C7×C22⋊C8C2212yes—
36C2×C7×C22⋊C8C2212yes—
37C7×#391—21no—
38C7×#390—21no—
39C7×#386—22no—
40C7×#380C2212yes—
41C7×#380C2212yes—
42C7×#340C2212yes—
43C7×#340C2212yes—
44C7×#340C2212yes—
45C7×#340C2212yes—
46#20849C711yesmaximal, characteristic, direct product factor, Sylow
47C2×C7×#41—22no—
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41—22no—
51C2×C7×#41C2×C414yes—
52C2×C7×#41Q814yes—
53C22×C7×Q8D411yescharacteristic
54C22×C7×D4D412yes—
55C22×C7×D4D411yescharacteristic
56C23×C4×C7—22no—
57C23×C4×C7—21no—
58C23×C4×C7—21no—
59C23×C4×C7C2312yes—
60C23×C4×C7C2311yescharacteristic
61C2×C7×M16—22no—
62C2×C7×M16—22no—
63C2×C7×M16C2×C414yescomplemented
64C22×C7×C8—22no—
65C22×C7×C8—22no—
66C22×C7×C8C2×C414yescomplemented
67C22×C7×C8C2311yescharacteristic
68C22×C7×C8C2312yes—
69C22×C7×C8C2311yescharacteristic
70C4×C7×Q8—28no—
71C4×C7×D4—216no—
72C4×C7×D4—28no—
73C7×#117—22no—
74C7×#117—24no—
75C7×#117—24no—
76C7×#117—22no—
77C7×#117—22no—
78C7×#117D414yes—
79C2×C7×C4⋊C4—22no—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—22no—
82C2×C7×C4⋊C4—24no—
83C2×C7×C4⋊C4—24no—
84C2×C7×C4⋊C4—22no—
85C2×C7×C4⋊C4—22no—
86C2×C7×C4⋊C4—21no—
87C2×C7×C4⋊C4D414yes—
88C2×C7×C4⋊C4C2312yes—
89C2×C7×C4⋊C4C2311yescharacteristic
90C2×C7×C4⋊C4C2311yescharacteristic
91C2×C7×C4⋊C4C2311yescharacteristic
92C2×C7×C4⋊C4C2312yes—
93C2×C7×C4⋊C4C2311yescharacteristic
94C2×C7×C22⋊C4—22no—
95C2×C7×C22⋊C4—21no—
96C2×C7×C22⋊C4—21no—
97C2×C7×C22⋊C4—22no—
98C2×C7×C22⋊C4—22no—
99C2×C42×C7—22no—
100C2×C42×C7—21no—
101C2×C42×C7—21no—
102C7×#107—22no—
103C7×#107—22no—
104C7×#107—24no—
105C7×#106—22no—
106C7×#106—22no—
107C7×#106—24no—
108C7×C22⋊C8C2×C418yescomplemented
109C7×C22⋊C8C2×C418yescomplemented
110C7×#95—22no—
111C7×#95—22no—
112C2×#520C2×C711yescharacteristic
113C2×#428C2×C711yescharacteristic, complemented
114C2×#427C2×C711yescharacteristic, complemented
115C2×#408C2×C711yescharacteristic
116#2311C2×C711yescharacteristic
117#1854C2×C712yescomplemented
118#1854C2×C712yescomplemented
119#1853C2×C712yescomplemented
120#1852C2×C712yescomplemented
121#1736C2×C711yescharacteristic, complemented
122C2×#340C2×C711yescharacteristic, complemented
123C24×C7C2×D412yes—
124C24×C7C2×D411yescharacteristic
125C7×#41—28no—
126C7×#41—28no—
127C7×#41—28no—
128C7×#41—28no—
129C2×C7×Q8—22no—
130C2×C7×Q8—22no—
131C2×C7×Q8C4⋊C414yes—
132C2×C7×D4—24no—
133C2×C7×D4—24no—
134C2×C7×D4—22no—
135C2×C7×D4—22no—
136C2×C7×D4C4⋊C418yes—
137C2×C7×D4C4⋊C414yes—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—21no—
143C22×C4×C7—22no—
144C22×C4×C7—21no—
145C22×C4×C7—28no—
146C22×C4×C7—24no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—22no—
150C22×C4×C7—42no—
151C22×C4×C7—81no—
152C22×C4×C7—82no—
153C22×C4×C7—81no—
154C22×C4×C7—82no—
155C22×C4×C7—81no—
156C22×C4×C7—81no—
157C22×C4×C7C22×C414yes—
158C22×C4×C7C22×C412yes—
159C22×C4×C7C2×D412yes—
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D412yes—
162C22×C4×C7C2×D412yes—
163C22×C4×C7C2×Q814yes—
164C22×C4×C7C2×Q812yes—
165C22×C4×C7#4112yes—
166C22×C4×C7#4112yes—
167C22×C4×C7C2411yescharacteristic
168C7×M16—28no—
169C2×C7×C8—28no—
170C2×C7×C8—21no—
171C2×C7×C8—22no—
172C2×C7×C8—21no—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C2×C7×C8—22no—
176C2×C7×C8C22×C412yes—
177C2×C7×C8C22×C412yes—
178C2×C7×C8C22×C414yes—
179C7×C4⋊C4—28no—
180C7×C4⋊C4—24no—
181C7×C4⋊C4—22no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—42no—
191C7×C4⋊C4C2×D412yes—
192C7×C4⋊C4C2×D414yes—
193C7×C4⋊C4C2×D412yes—
194C7×C22⋊C4—28no—
195C7×C22⋊C4—24no—
196C7×C22⋊C4—42no—
197C7×C22⋊C4—82no—
198C7×C22⋊C4—82no—
199C42×C7—28no—
200C42×C7—24no—
201C42×C7—42no—
202C22×#41C22×C711yescharacteristic
203C22×M16C22×C711yescharacteristic
204C23×C8C22×C711yescharacteristic
205#520—24no—
206#520—24no—
207C2×C4×Q8—21no—
208C2×C4×D4—22no—
209C2×C4×D4—21no—
210C2×#117C22×C711yescharacteristic
211C2×#117C22×C712yes—
212C2×#117C22×C711yescharacteristic
213C22×C4⋊C4C22×C711yescharacteristic
214C22×C4⋊C4C22×C712yes—
215C22×C4⋊C4C22×C711yescharacteristic
216#428—24no—
217#427—24no—
218C2×#107C22×C711yescharacteristic
219C2×#107C22×C712yes—
220C2×#107C22×C711yescharacteristic
221C2×#106C22×C711yescharacteristic
222C2×#106C22×C712yes—
223C2×#106C22×C711yescharacteristic
224#408C4×C718yescomplemented
225C2×C22⋊C8C22×C712yes—
226C2×C22⋊C8C22×C712yes—
227#391—21no—
228#390—21no—
229#386—22no—
230#380C22×C712yes—
231#380C22×C712yes—
232#340C22×C712yes—
233#340C22×C712yes—
234#340C22×C712yes—
235#340C22×C712yes—
236C23×C7—21no—
237C23×C7—21no—
238C23×C7—21no—
239C23×C7—21no—
240C23×C7—22no—
241C23×C7—22no—
242C23×C7—22no—
243C23×C7—22no—
244C23×C7—22no—
245C23×C7—22no—
246C23×C7C2×C4⋊C414yes—
247C23×C7C2×C4⋊C412yes—
248C23×C7#12012yes—
249C23×C7#12312yes—
250C23×C7C22×D411yescharacteristic
251C7×Q8—28no—
252C7×D4—216no—
253C7×D4—28no—
254C2×C4×C7—22no—
255C2×C4×C7—22no—
256C2×C4×C7—24no—
257C2×C4×C7—24no—
258C2×C4×C7—24no—
259C2×C4×C7—24no—
260C2×C4×C7—22no—
261C2×C4×C7—24no—
262C2×C4×C7—22no—
263C2×C4×C7—42no—
264C2×C4×C7—41no—
265C2×C4×C7—41no—
266C2×C4×C7—42no—
267C2×C4×C7—41no—
268C2×C4×C7—41no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—42no—
272C2×C4×C7—41no—
273C2×C4×C7—42no—
274C2×C4×C7—41no—
275C2×C4×C7—42no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—84no—
279C2×C4×C7—88no—
280C2×C4×C7—84no—
281C2×C4×C7C2×C4⋊C414yes—
282C2×C4×C7C2×C4⋊C412yes—
283C2×C4×C7C4×D414yes—
284C2×C4×C7C4×D414yes—
285C2×C4×C7#12012yes—
286C2×C4×C7C22⋊Q814yes—
287C2×C4×C7C22⋊Q814yes—
288C2×C4×C7#12312yes—
289C2×C4×C7C23×C411yescharacteristic
290C2×C4×C7C22×D411yescharacteristic
291C2×C4×C7C22×D411yescharacteristic
292C2×C4×C7C22×Q811yescharacteristic
293C2×C4×C7C2×#4111yescharacteristic
294C2×C4×C7C2×#4111yescharacteristic
295C7×C8—22no—
296C7×C8—22no—
297C7×C8—24no—
298C2×#41—22no—
299C2×#41—22no—
300C2×#41—22no—
301C2×#41—22no—
302C2×#41C2×C4×C714yes—
303C2×#41C7×Q814yes—
304C22×Q8C7×D411yescharacteristic
305C22×D4C7×D412yes—
306C22×D4C7×D411yescharacteristic
307C23×C4—22no—
308C23×C4—21no—
309C23×C4—21no—
310C23×C4C23×C712yes—
311C23×C4C23×C711yescharacteristic
312C2×M16—22no—
313C2×M16—22no—
314C2×M16C2×C4×C714yescomplemented
315C22×C8—22no—
316C22×C8—22no—
317C22×C8C2×C4×C714yescomplemented
318C22×C8C23×C711yescharacteristic
319C22×C8C23×C711yescharacteristic
320C22×C8C23×C712yes—
321C4×Q8—28no—
322C4×D4—216no—
323C4×D4—28no—
324#117—22no—
325#117—24no—
326#117—24no—
327#117—22no—
328#117—22no—
329#117C7×D414yes—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4C7×D414yes—
339C2×C4⋊C4C23×C712yes—
340C2×C4⋊C4C23×C711yescharacteristic
341C2×C4⋊C4C23×C711yescharacteristic
342C2×C4⋊C4C23×C711yescharacteristic
343C2×C4⋊C4C23×C712yes—
344C2×C4⋊C4C23×C711yescharacteristic
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C42—22no—
351C2×C42—21no—
352C2×C42—21no—
353#107—22no—
354#107—22no—
355#107—24no—
356#106—22no—
357#106—22no—
358#106—24no—
359C22⋊C8C2×C4×C718yescomplemented
360C22⋊C8C2×C4×C718yescomplemented
361#95—22no—
362#95—22no—
363C22×C7—22no—
364C22×C7—22no—
365C22×C7—22no—
366C22×C7—22no—
367C22×C7—22no—
368C22×C7—22no—
369C22×C7—24no—
370C22×C7—24no—
371C22×C7—24no—
372C22×C7—42no—
373C22×C7—42no—
374C22×C7—42no—
375C22×C7—42no—
376C22×C7#38514yes—
377C22×C7C22×C4⋊C411yescharacteristic
378C22×C7C2×#12011yescharacteristic
379C22×C7C2×#12311yescharacteristic
380C22×C7#57611yescharacteristic
381C22×C7#57711yescharacteristic
382C22×C7#57711yescharacteristic
383C22×C7#57811yescharacteristic
384C4×C7—24no—
385C4×C7—44no—
386C4×C7—42no—
387C4×C7—42no—
388C4×C7—82no—
389C4×C7—81no—
390C4×C7—81no—
391C4×C7#38514yes—
392C4×C7C2×C4×D411yescharacteristic
393C4×C7C2×C4×D411yescharacteristic
394C4×C7C2×C22⋊Q811yescharacteristic
395C4×C7C2×C22⋊Q811yescharacteristic
396C24C2×C7×D412yes—
397C24C2×C7×D411yescharacteristic
398#41—28no—
399#41—28no—
400#41—28no—
401#41—28no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8C7×C4⋊C414yes—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4C7×C4⋊C418yes—
410C2×D4C7×C4⋊C414yes—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—28no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—42no—
424C22×C4—81no—
425C22×C4—82no—
426C22×C4—81no—
427C22×C4—82no—
428C22×C4—81no—
429C22×C4—81no—
430C22×C4C22×C4×C714yes—
431C22×C4C22×C4×C712yes—
432C22×C4C2×C7×D412yes—
433C22×C4C2×C7×D411yescharacteristic
434C22×C4C2×C7×D412yes—
435C22×C4C2×C7×D412yes—
436C22×C4C2×C7×Q814yes—
437C22×C4C2×C7×Q812yes—
438C22×C4C7×#4112yes—
439C22×C4C7×#4112yes—
440C22×C4C24×C711yescharacteristic
441M16—28no—
442C2×C8—28no—
443C2×C8—21no—
444C2×C8—22no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—22no—
449C2×C8C22×C4×C712yes—
450C2×C8C22×C4×C712yes—
451C2×C8C22×C4×C714yes—
452C4⋊C4—28no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—42no—
464C4⋊C4C2×C7×D412yes—
465C4⋊C4C2×C7×D414yes—
466C4⋊C4C2×C7×D412yes—
467C22⋊C4—28no—
468C22⋊C4—24no—
469C22⋊C4—42no—
470C22⋊C4—82no—
471C22⋊C4—82no—
472C42—28no—
473C42—24no—
474C42—42no—
475C2×C7—24no—
476C2×C7—44no—
477C2×C7C2×#38511yescharacteristic
478C2×C7#291211yescharacteristic
479C2×C7#291311yescharacteristic
480C2×C7#301211yescharacteristic
481C2×C7#301311yescharacteristic
482C2×C7#309611yescharacteristic
483C2×C7#309711yescharacteristic
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23C2×C7×C4⋊C414yes—
495C23C2×C7×C4⋊C412yes—
496C23C7×#12012yes—
497C23C7×#12312yes—
498C23C22×C7×D411yescharacteristic
499Q8—28no—
500D4—216no—
501D4—28no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—84no—
527C2×C4—88no—
528C2×C4—84no—
529C2×C4C2×C7×C4⋊C414yes—
530C2×C4C2×C7×C4⋊C412yes—
531C2×C4C4×C7×D414yes—
532C2×C4C4×C7×D414yes—
533C2×C4C7×#12012yes—
534C2×C4C7×C22⋊Q814yes—
535C2×C4C7×C22⋊Q814yes—
536C2×C4C7×#12312yes—
537C2×C4C23×C4×C711yescharacteristic
538C2×C4C22×C7×D411yescharacteristic
539C2×C4C22×C7×D411yescharacteristic
540C2×C4C22×C7×Q811yescharacteristic
541C2×C4C2×C7×#4111yescharacteristic
542C2×C4C2×C7×#4111yescharacteristic
543C8—22no—
544C8—22no—
545C8—24no—
546C7#2084911yescharacteristic, direct product factor, Sylow
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22C7×#38514yes—
561C22C22×C7×C4⋊C411yescharacteristic
562C22C2×C7×#12011yescharacteristic
563C22C2×C7×#12311yescharacteristic
564C22C7×#57611yescharacteristic
565C22C7×#57711yescharacteristic
566C22C7×#57711yescharacteristic
567C22C7×#57811yescharacteristic
568C4—24no—
569C4—44no—
570C4—42no—
571C4—42no—
572C4—81no—
573C4—81no—
574C4—82no—
575C4C7×#38514yes—
576C4C2×C4×C7×D411yescharacteristic
577C4C2×C4×C7×D411yescharacteristic
578C4C2×C7×C22⋊Q811yescharacteristic
579C4C2×C7×C22⋊Q811yescharacteristic
580C2—24no—
581C2—44no—
582C2C2×C7×#38511yescharacteristic
583C2C7×#291211yescharacteristic
584C2C7×#291311yescharacteristic
585C2C7×#301211yescharacteristic
586C2C7×#301311yescharacteristic
587C2C7×#309611yescharacteristic
588C2C7×#309711yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutatorC2×C4C23×C4×C7
Frattini, cosocleC22×C4C24×C7
prime radicalC22×C7×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C22×C7×#41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#428C2semidirect
3. C2×C7×#427C2semidirect
6. C7×#1854C2semidirect
7. C7×#1854C2semidirect
8. C7×#1853C2semidirect
9. C7×#1852C2semidirect
10. C7×#1736C2semidirect
11. C2×C7×#340C2semidirect
34. C7×#408C4semidirect
46. #20849C7direct (N × H)
63. C2×C7×M16C2×C4semidirect
66. C22×C7×C8C2×C4semidirect
108. C7×C22⋊C8C2×C4semidirect
109. C7×C22⋊C8C2×C4semidirect
113. C2×#428C2×C7semidirect
114. C2×#427C2×C7semidirect
117. #1854C2×C7semidirect
118. #1854C2×C7semidirect
119. #1853C2×C7semidirect
120. #1852C2×C7semidirect
121. #1736C2×C7semidirect
122. C2×#340C2×C7semidirect
224. #408C4×C7semidirect
314. C2×M16C2×C4×C7semidirect
317. C22×C8C2×C4×C7semidirect
359. C22⋊C8C2×C4×C7semidirect
360. C22⋊C8C2×C4×C7semidirect
546. C7#20849direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5202
C2×C7×#4282
C2×C7×#4272
C2×C7×#4082
C7×#23112
C7×#18542
C7×#18532
C7×#18522
C7×#17362
C2×C7×#3402
#208497