Finite Group Databasebeta

← C7×#24323 #13201539

Subgroups — C7×#24323 #13201539

Order 1792.

1004 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3503C211yesmaximal, characteristic
2C7×#3078C211yesmaximal, characteristic, complemented
3C4×C7×#137C211yesmaximal, characteristic
4C7×#2567C211yesmaximal, characteristic, complemented
5C7×#2060C212yesmaximal, complemented
6C7×#2028C212yesmaximal, complemented
7C7×#1997C211yesmaximal, characteristic, complemented
8C7×#1986C211yesmaximal, characteristic, complemented
9C7×#1882C212yesmaximal
10C7×#1859C212yesmaximal
11C7×#1806C211yesmaximal, characteristic
12C2×C7×#137C2211yescharacteristic
13C7×Q82—21no—
14C7×#557—21no—
15C7×#554—21no—
16C7×#554—22no—
17C7×D4×Q8—22no—
18C7×D4×Q8—21no—
19C7×#544—22no—
20C7×#544C2212yes—
21C7×#533—21no—
22C7×#533C2211yescharacteristic
23C2×C7×C4⋊Q8—21no—
24C2×C7×C4⋊Q8C2211yescharacteristic
25C7×#520C2212yes—
26C4×C7×#41C2211yescharacteristic
27C2×C4×C7×Q8C2211yescharacteristic
28C7×#479—22no—
29C7×#477—21no—
30C7×#475—21no—
31C7×#441—21no—
32C7×#440—21no—
33C4×C7×Q16—21no—
34C4×C7×SD16—21no—
35C7×#428C2211yescharacteristic
36C7×#428C2212yes—
37C7×#423C2211yescharacteristic
38C2×C7×#104C2212yes—
39C7×#419C2211yescharacteristic
40C7×#419C2212yes—
41C7×#417C2211yescharacteristic
42C4×C7×M16C2211yescharacteristic
43C7×#395—22no—
44C7×#391C2212yes—
45C7×#389—22no—
46C7×#384C2212yes—
47C4×C7×C4⋊C4C2211yescharacteristic
48C7×#343C2212yes—
49C7×#343C2212yes—
50C7×#339C2212yes—
51C7×#339C2212yes—
52C7×#339C2212yes—
53#24323C711yesmaximal, characteristic, direct product factor, Sylow
54C2×C7×#41C2311yescharacteristic
55C22×C7×Q8—22no—
56C22×C7×Q8D412yes—
57C22×C7×Q8C2311yescharacteristic
58C7×#137—22no—
59C7×#137Q814yescomplemented
60C2×C7×Q16—21no—
61C2×C7×SD16—21no—
62C2×C7×M16C2311yescharacteristic
63C2×C7×M16C2312yes—
64C2×C7×M16C2311yescharacteristic
65C7×C4⋊Q8—22no—
66C7×C4⋊Q8—22no—
67C7×C4⋊Q8—24no—
68C7×C4⋊Q8—22no—
69C7×C4⋊Q8—22no—
70C7×C4⋊Q8—21no—
71C7×C4⋊Q8—21no—
72C7×C4⋊Q8—21no—
73C7×C4⋊Q8—21no—
74C7×C4⋊Q8D411yescharacteristic
75C7×C4⋊Q8D411yescharacteristic
76C7×#125—24no—
77C7×#125—22no—
78C7×#125—22no—
79C7×#125D412yes—
80C7×C22⋊Q8—22no—
81C7×C22⋊Q8—22no—
82C7×C22⋊Q8—24no—
83C7×C22⋊Q8—24no—
84C7×C22⋊Q8—21no—
85C7×C22⋊Q8—21no—
86C7×C22⋊Q8—24no—
87C4×C7×Q8—22no—
88C4×C7×Q8—22no—
89C4×C7×Q8—22no—
90C4×C7×Q8—21no—
91C4×C7×Q8—21no—
92C4×C7×Q8—21no—
93C4×C7×Q8—21no—
94C4×C7×Q8—21no—
95C4×C7×Q8—21no—
96C4×C7×Q8D411yescharacteristic
97C4×C7×Q8D411yescharacteristic
98C4×C7×D4—22no—
99C4×C7×D4—21no—
100C4×C7×D4—21no—
101C4×C7×D4—24no—
102C4×C7×D4D411yescharacteristic
103C4×C7×D4D411yescharacteristic
104C7×#117—22no—
105C7×#117—21no—
106C7×#117C2311yescharacteristic
107C7×#117C2312yes—
108C2×C7×C4⋊C4—22no—
109C2×C7×C4⋊C4—21no—
110C2×C7×C4⋊C4—22no—
111C2×C7×C4⋊C4—21no—
112C2×C7×C4⋊C4—24no—
113C2×C7×C4⋊C4D412yes—
114C2×C7×C4⋊C4C2311yescharacteristic
115C2×C7×C4⋊C4C2312yes—
116C2×C42×C7—21no—
117C2×C42×C7—21no—
118C2×C42×C7C2312yes—
119C2×C42×C7C2311yescharacteristic
120C7×#107—22no—
121C7×#107—21no—
122C7×#106—22no—
123C7×#106—21no—
124C7×C4⋊C8—21no—
125C7×C4⋊C8—21no—
126C7×#104—24no—
127C7×#103—21no—
128C7×#103—22no—
129C7×#103—22no—
130C7×#103—21no—
131C7×#103—21no—
132C7×#102—21no—
133C7×#97—21no—
134C4×C7×C8—21no—
135C7×#95—21no—
136C7×#95—24no—
137#3503C2×C711yescharacteristic
138#3078C2×C711yescharacteristic, complemented
139C4×#137C2×C711yescharacteristic
140#2567C2×C711yescharacteristic, complemented
141#2060C2×C712yescomplemented
142#2028C2×C712yescomplemented
143#1997C2×C711yescharacteristic, complemented
144#1986C2×C711yescharacteristic, complemented
145#1882C2×C712yes—
146#1859C2×C712yes—
147#1806C2×C711yescharacteristic
148C7×#41—21no—
149C7×#41—21no—
150C7×#41C2×Q812yes—
151C7×#41#4112yes—
152C2×C7×Q8—24no—
153C2×C7×Q8—21no—
154C2×C7×Q8—22no—
155C2×C7×Q8—21no—
156C2×C7×Q8—22no—
157C2×C7×Q8—21no—
158C2×C7×Q8—21no—
159C2×C7×Q8—21no—
160C2×C7×Q8—21no—
161C2×C7×Q8—21no—
162C2×C7×Q8—82no—
163C2×C7×Q8—84no—
164C2×C7×Q8C2×D411yescharacteristic
165C2×C7×Q8C2×Q812yescomplemented
166C2×C7×D4—21no—
167C2×C7×D4C2×D411yescharacteristic
168C22×C4×C7—21no—
169C22×C4×C7—21no—
170C22×C4×C7—22no—
171C22×C4×C7—24no—
172C22×C4×C7—21no—
173C22×C4×C7—21no—
174C22×C4×C7—21no—
175C22×C4×C7C2×D411yescharacteristic
176C22×C4×C7C2×D414yes—
177C22×C4×C7C2×D411yescharacteristic
178C22×C4×C7C2411yescharacteristic
179C7×Q16—24no—
180C7×SD16—24no—
181C7×M16—21no—
182C7×M16—22no—
183C7×M16—21no—
184C7×M16—21no—
185C7×M16—22no—
186C7×M16C2×Q812yescomplemented
187C2×C7×C8—21no—
188C2×C7×C8—21no—
189C2×C7×C8—22no—
190C7×C4⋊C4—24no—
191C7×C4⋊C4—24no—
192C7×C4⋊C4—24no—
193C7×C4⋊C4—24no—
194C7×C4⋊C4—24no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—21no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—22no—
202C7×C4⋊C4—22no—
203C7×C4⋊C4—22no—
204C7×C4⋊C4—22no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—21no—
207C7×C4⋊C4—41no—
208C7×C4⋊C4—41no—
209C7×C4⋊C4—84no—
210C7×C4⋊C4C2×D412yes—
211C7×C4⋊C4C2×D411yescharacteristic
212C7×C4⋊C4C2×D412yes—
213C7×C4⋊C4C2×D411yescharacteristic
214C7×C22⋊C4—24no—
215C7×C22⋊C4—22no—
216C7×C22⋊C4—21no—
217C7×C22⋊C4—21no—
218C7×C22⋊C4—21no—
219C42×C7—24no—
220C42×C7—21no—
221C42×C7—21no—
222C42×C7—21no—
223C42×C7—22no—
224C42×C7—22no—
225C42×C7—22no—
226C42×C7—41no—
227C42×C7—41no—
228C42×C7C2×D411yescharacteristic
229C42×C7C2×D411yescharacteristic
230C42×C7C2×D411yescharacteristic
231C42×C7C2×D411yescharacteristic
232C2×#137C22×C711yescharacteristic
233Q82—21no—
234#557—21no—
235#554—21no—
236#554—22no—
237D4×Q8—22no—
238D4×Q8—21no—
239#544—22no—
240#544C22×C712yes—
241#533—21no—
242#533C22×C711yescharacteristic
243C2×C4⋊Q8—21no—
244C2×C4⋊Q8C22×C711yescharacteristic
245#520C22×C712yes—
246C4×#41C22×C711yescharacteristic
247C2×C4×Q8C22×C711yescharacteristic
248#479—22no—
249#477—21no—
250#475—21no—
251#441—21no—
252#440—21no—
253C4×Q16—21no—
254C4×SD16—21no—
255#428C22×C712yes—
256#428C22×C711yescharacteristic
257#423C22×C711yescharacteristic
258C2×#104C22×C712yes—
259#419C22×C712yes—
260#419C22×C711yescharacteristic
261#417C22×C711yescharacteristic
262C4×M16C22×C711yescharacteristic
263#395—22no—
264#391C22×C712yes—
265#389—22no—
266#384C22×C712yes—
267C4×C4⋊C4C22×C711yescharacteristic
268#343C22×C712yes—
269#343C22×C712yes—
270#339C22×C712yes—
271#339C22×C712yes—
272#339C22×C712yes—
273C23×C7—21no—
274C23×C7C22×D411yescharacteristic
275C7×Q8—21no—
276C7×Q8—22no—
277C7×Q8—22no—
278C7×Q8—44no—
279C7×Q8—42no—
280C7×Q8—41no—
281C7×Q8—41no—
282C7×Q8—42no—
283C7×Q8—82no—
284C7×Q8—84no—
285C7×Q8C22⋊Q812yes—
286C7×D4—22no—
287C7×D4—22no—
288C7×D4—21no—
289C7×D4C22⋊Q812yes—
290C2×C4×C7—21no—
291C2×C4×C7—22no—
292C2×C4×C7—22no—
293C2×C4×C7—22no—
294C2×C4×C7—24no—
295C2×C4×C7—22no—
296C2×C4×C7—21no—
297C2×C4×C7—21no—
298C2×C4×C7—24no—
299C2×C4×C7—22no—
300C2×C4×C7—22no—
301C2×C4×C7—41no—
302C2×C4×C7—42no—
303C2×C4×C7—42no—
304C2×C4×C7—41no—
305C2×C4×C7—42no—
306C2×C4×C7—41no—
307C2×C4×C7—42no—
308C2×C4×C7—44no—
309C2×C4×C7—42no—
310C2×C4×C7—41no—
311C2×C4×C7—84no—
312C2×C4×C7#12011yescharacteristic
313C2×C4×C7#12011yescharacteristic
314C2×C4×C7#12011yescharacteristic
315C2×C4×C7#12011yescharacteristic
316C2×C4×C7C22×D411yescharacteristic
317C2×C4×C7C22×D411yescharacteristic
318C2×C4×C7C22×Q811yescharacteristic
319C2×C4×C7C2×#4111yescharacteristic
320C2×C4×C72−1+411yescharacteristic
321C2×C4×C72−1+411yescharacteristic
322C7×C8—22no—
323C7×C8—41no—
324C7×C8—42no—
325C2×#41C23×C711yescharacteristic
326C22×Q8—22no—
327C22×Q8C7×D412yes—
328C22×Q8C23×C711yescharacteristic
329#137—22no—
330#137C7×Q814yescomplemented
331C2×Q16—21no—
332C2×SD16—21no—
333C2×M16C23×C711yescharacteristic
334C2×M16C23×C712yes—
335C2×M16C23×C711yescharacteristic
336C4⋊Q8—22no—
337C4⋊Q8—22no—
338C4⋊Q8—24no—
339C4⋊Q8—22no—
340C4⋊Q8—22no—
341C4⋊Q8—21no—
342C4⋊Q8—21no—
343C4⋊Q8—21no—
344C4⋊Q8—21no—
345C4⋊Q8C7×D411yescharacteristic
346C4⋊Q8C7×D411yescharacteristic
347#125—24no—
348#125—22no—
349#125—22no—
350#125C7×D412yes—
351C22⋊Q8—22no—
352C22⋊Q8—22no—
353C22⋊Q8—24no—
354C22⋊Q8—24no—
355C22⋊Q8—21no—
356C22⋊Q8—21no—
357C22⋊Q8—24no—
358C4×Q8—22no—
359C4×Q8—22no—
360C4×Q8—22no—
361C4×Q8—21no—
362C4×Q8—21no—
363C4×Q8—21no—
364C4×Q8—21no—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8C7×D411yescharacteristic
368C4×Q8C7×D411yescharacteristic
369C4×D4—22no—
370C4×D4—21no—
371C4×D4—21no—
372C4×D4—24no—
373C4×D4C7×D411yescharacteristic
374C4×D4C7×D411yescharacteristic
375#117—22no—
376#117—21no—
377#117C23×C711yescharacteristic
378#117C23×C712yes—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—24no—
384C2×C4⋊C4C7×D412yes—
385C2×C4⋊C4C23×C711yescharacteristic
386C2×C4⋊C4C23×C712yes—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42C23×C712yes—
390C2×C42C23×C711yescharacteristic
391#107—21no—
392#107—22no—
393#106—21no—
394#106—22no—
395C4⋊C8—21no—
396C4⋊C8—21no—
397#104—24no—
398#103—21no—
399#103—21no—
400#103—21no—
401#103—22no—
402#103—22no—
403#102—21no—
404#97—21no—
405C4×C8—21no—
406#95—21no—
407#95—24no—
408C22×C7—21no—
409C22×C7—21no—
410C22×C7—22no—
411C22×C7—41no—
412C22×C7—41no—
413C22×C7C2×#12011yescharacteristic
414C22×C7#54811yescharacteristic
415C22×C7D4×Q811yescharacteristic
416C4×C7—22no—
417C4×C7—21no—
418C4×C7—21no—
419C4×C7—21no—
420C4×C7—44no—
421C4×C7—42no—
422C4×C7—41no—
423C4×C7—42no—
424C4×C7—41no—
425C4×C7—44no—
426C4×C7—42no—
427C4×C7—81no—
428C4×C7#45612yes—
429C4×C7C2×C22⋊Q811yescharacteristic
430C4×C7#53611yescharacteristic
431C4×C7#54811yescharacteristic
432C4×C7D4×Q811yescharacteristic
433#41—21no—
434#41—21no—
435#41C2×C7×Q812yes—
436#41C7×#4112yes—
437C2×Q8—24no—
438C2×Q8—21no—
439C2×Q8—22no—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—82no—
448C2×Q8—84no—
449C2×Q8C2×C7×D411yescharacteristic
450C2×Q8C2×C7×Q812yescomplemented
451C2×D4—21no—
452C2×D4C2×C7×D411yescharacteristic
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4C2×C7×D411yescharacteristic
461C22×C4C2×C7×D414yes—
462C22×C4C2×C7×D411yescharacteristic
463C22×C4C24×C711yescharacteristic
464Q16—24no—
465SD16—24no—
466M16—21no—
467M16—22no—
468M16—21no—
469M16—21no—
470M16—22no—
471M16C2×C7×Q812yescomplemented
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—22no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—84no—
495C4⋊C4C2×C7×D412yes—
496C4⋊C4C2×C7×D411yescharacteristic
497C4⋊C4C2×C7×D412yes—
498C4⋊C4C2×C7×D411yescharacteristic
499C22⋊C4—24no—
500C22⋊C4—22no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C42—24no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—41no—
512C42—41no—
513C42C2×C7×D411yescharacteristic
514C42C2×C7×D411yescharacteristic
515C42C2×C7×D411yescharacteristic
516C42C2×C7×D411yescharacteristic
517C2×C7—21no—
518C2×C7—21no—
519C2×C7—42no—
520C2×C7#240911yescharacteristic
521C2×C7C2×#45611yescharacteristic
522C2×C7#301911yescharacteristic
523C23—21no—
524C23C22×C7×D411yescharacteristic
525Q8—21no—
526Q8—22no—
527Q8—22no—
528Q8—44no—
529Q8—42no—
530Q8—41no—
531Q8—41no—
532Q8—42no—
533Q8—82no—
534Q8—84no—
535Q8C7×C22⋊Q812yes—
536D4—22no—
537D4—22no—
538D4—21no—
539D4C7×C22⋊Q812yes—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—84no—
562C2×C4C7×#12011yescharacteristic
563C2×C4C7×#12011yescharacteristic
564C2×C4C7×#12011yescharacteristic
565C2×C4C7×#12011yescharacteristic
566C2×C4C22×C7×D411yescharacteristic
567C2×C4C22×C7×D411yescharacteristic
568C2×C4C22×C7×Q811yescharacteristic
569C2×C4C2×C7×#4111yescharacteristic
570C2×C4C7×2−1+411yescharacteristic
571C2×C4C7×2−1+411yescharacteristic
572C8—22no—
573C8—41no—
574C8—42no—
575C7#2432311yescharacteristic, direct product factor, Sylow
576C22—21no—
577C22—21no—
578C22—22no—
579C22—41no—
580C22—41no—
581C22C2×C7×#12011yescharacteristic
582C22C7×#54811yescharacteristic
583C22C7×D4×Q811yescharacteristic
584C4—22no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—44no—
589C4—42no—
590C4—41no—
591C4—42no—
592C4—41no—
593C4—44no—
594C4—42no—
595C4—81no—
596C4C7×#45612yes—
597C4C2×C7×C22⋊Q811yescharacteristic
598C4C7×#53611yescharacteristic
599C4C7×#54811yescharacteristic
600C4C7×D4×Q811yescharacteristic
601C2—21no—
602C2—21no—
603C2—42no—
604C2C7×#240911yescharacteristic
605C2C2×C7×#45611yescharacteristic
606C2C7×#301911yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#3078C2semidirect
4. C7×#2567C2semidirect
5. C7×#2060C2semidirect
6. C7×#2028C2semidirect
7. C7×#1997C2semidirect
8. C7×#1986C2semidirect
53. #24323C7direct (N × H)
59. C7×#137Q8semidirect
138. #3078C2×C7semidirect
140. #2567C2×C7semidirect
141. #2060C2×C7semidirect
142. #2028C2×C7semidirect
143. #1997C2×C7semidirect
144. #1986C2×C7semidirect
165. C2×C7×Q8C2×Q8semidirect
186. C7×M16C2×Q8semidirect
330. #137C7×Q8semidirect
450. C2×Q8C2×C7×Q8semidirect
471. M16C2×C7×Q8semidirect
575. C7#24323direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35032
C7×#30782
C4×C7×#1372
C7×#25672
C7×#20602
C7×#20282
C7×#19972
C7×#19862
C7×#18822
C7×#18592
C7×#18062
#243237