Finite Group Databasebeta

← C7×#24147 #13201363

Subgroups — C7×#24147 #13201363

Order 1792.

1006 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3509C211yesmaximal, characteristic
2C7×#3078C211yesmaximal, characteristic, complemented
3C4×C7×#137C211yesmaximal, characteristic
4C7×#2546C211yesmaximal, characteristic, complemented
5C7×#2038C212yesmaximal
6C7×#2005C212yesmaximal
7C7×#1986C211yesmaximal, characteristic, complemented
8C7×#1966C211yesmaximal, characteristic, complemented
9C7×#1886C212yesmaximal, complemented
10C7×#1857C212yesmaximal, complemented
11C7×#1806C211yesmaximal, characteristic
12C2×C7×#137C2211yescharacteristic
13C7×#557—21no—
14C7×#557—21no—
15C7×#556—21no—
16C7×#556—21no—
17C7×#552—22no—
18C7×#552—22no—
19C7×#541—22no—
20C7×#536C2212yes—
21C7×#533—21no—
22C7×#533—21no—
23C7×#533C2211yescharacteristic
24C2×C7×C4⋊Q8C2212yes—
25C2×C7×C4⋊Q8C2211yescharacteristic
26C7×#519C2212yes—
27C4×C7×#41C2211yescharacteristic
28C2×C4×C7×Q8C2211yescharacteristic
29C7×#479—22no—
30C7×#477—21no—
31C7×#475—21no—
32C7×#441—21no—
33C7×#440—21no—
34C4×C7×Q16—21no—
35C4×C7×SD16—21no—
36C7×#428C2211yescharacteristic
37C7×#428C2212yes—
38C7×#423C2211yescharacteristic
39C2×C7×#104C2212yes—
40C7×#419C2212yes—
41C7×#419C2211yescharacteristic
42C7×#417C2211yescharacteristic
43C2×C7×#101C2212yes—
44C2×C7×#101C2212yes—
45C4×C7×M16C2211yescharacteristic
46C7×#398—22no—
47C7×#391C2212yes—
48C7×#389—22no—
49C4×C7×C4⋊C4C2211yescharacteristic
50C7×#339C2212yes—
51C7×#339C2212yes—
52C7×#339C2212yes—
53#24147C711yesmaximal, characteristic, direct product factor, Sylow
54C2×C7×#41C2311yescharacteristic
55C22×C7×Q8C2311yescharacteristic
56C22×C7×Q8C2312yes—
57C7×#137—22no—
58C7×#137—22no—
59C2×C7×Q16—21no—
60C2×C7×SD16—21no—
61C2×C7×M16C2311yescharacteristic
62C2×C7×M16C2312yes—
63C2×C7×M16C2311yescharacteristic
64C7×C4⋊Q8—22no—
65C7×C4⋊Q8—24no—
66C7×C4⋊Q8—21no—
67C7×C4⋊Q8—21no—
68C7×C4⋊Q8—21no—
69C7×C4⋊Q8—21no—
70C7×C4⋊Q8—42no—
71C7×C4⋊Q8D411yescharacteristic
72C7×C4⋊Q8D411yescharacteristic
73C7×#126—24no—
74C7×#126—24no—
75C7×#125—22no—
76C7×#125—22no—
77C7×#125—24no—
78C7×#125—24no—
79C7×#125D412yes—
80C7×#124—22no—
81C7×#123—24no—
82C7×C22⋊Q8—22no—
83C7×C22⋊Q8—22no—
84C7×C22⋊Q8—21no—
85C7×C22⋊Q8—21no—
86C7×C22⋊Q8—22no—
87C7×C22⋊Q8—22no—
88C4×C7×Q8—22no—
89C4×C7×Q8—22no—
90C4×C7×Q8—22no—
91C4×C7×Q8—22no—
92C4×C7×Q8—22no—
93C4×C7×Q8—22no—
94C4×C7×Q8—21no—
95C4×C7×Q8—21no—
96C4×C7×Q8—21no—
97C4×C7×Q8—21no—
98C4×C7×Q8—21no—
99C4×C7×Q8—21no—
100C4×C7×Q8D411yescharacteristic
101C4×C7×Q8D411yescharacteristic
102C4×C7×D4—21no—
103C4×C7×D4—21no—
104C4×C7×D4D411yescharacteristic
105C4×C7×D4D411yescharacteristic
106C7×#117—22no—
107C7×#117—22no—
108C7×#117—24no—
109C7×#117—21no—
110C7×#117C2311yescharacteristic
111C7×#117C2312yes—
112C2×C7×C4⋊C4—21no—
113C2×C7×C4⋊C4—21no—
114C2×C7×C4⋊C4—24no—
115C2×C7×C4⋊C4D412yes—
116C2×C7×C4⋊C4D412yes—
117C2×C7×C4⋊C4C2311yescharacteristic
118C2×C42×C7—21no—
119C2×C42×C7—21no—
120C2×C42×C7C2312yes—
121C2×C42×C7C2311yescharacteristic
122C7×#107—22no—
123C7×#107—21no—
124C7×#106—22no—
125C7×#106—21no—
126C7×C4⋊C8—21no—
127C7×C4⋊C8—21no—
128C7×#104—42no—
129C7×#103—21no—
130C7×#103—22no—
131C7×#103—22no—
132C7×#103—21no—
133C7×#103—21no—
134C7×#102—21no—
135C7×#101—24no—
136C7×#101—42no—
137C7×#97—21no—
138C4×C7×C8—21no—
139C7×#95—21no—
140C7×#95—24no—
141#3509C2×C711yescharacteristic
142#3078C2×C711yescharacteristic, complemented
143C4×#137C2×C711yescharacteristic
144#2546C2×C711yescharacteristic, complemented
145#2038C2×C712yes—
146#2005C2×C712yes—
147#1986C2×C711yescharacteristic, complemented
148#1966C2×C711yescharacteristic, complemented
149#1886C2×C712yescomplemented
150#1857C2×C712yescomplemented
151#1806C2×C711yescharacteristic
152C7×#41—21no—
153C7×#41—21no—
154C7×#41—21no—
155C7×#41—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8—21no—
158C2×C7×Q8—22no—
159C2×C7×Q8—22no—
160C2×C7×Q8—22no—
161C2×C7×Q8—22no—
162C2×C7×Q8—22no—
163C2×C7×Q8—21no—
164C2×C7×Q8—22no—
165C2×C7×Q8—21no—
166C2×C7×Q8—21no—
167C2×C7×Q8—21no—
168C2×C7×Q8—21no—
169C2×C7×Q8C2×D411yescharacteristic
170C2×C7×Q8#4112yes—
171C2×C7×D4—21no—
172C2×C7×D4C2×D411yescharacteristic
173C22×C4×C7—22no—
174C22×C4×C7—21no—
175C22×C4×C7—21no—
176C22×C4×C7—21no—
177C22×C4×C7—21no—
178C22×C4×C7—21no—
179C22×C4×C7C2×D414yes—
180C22×C4×C7C2×D411yescharacteristic
181C22×C4×C7C2×D411yescharacteristic
182C22×C4×C7C2411yescharacteristic
183C7×Q16—24no—
184C7×SD16—24no—
185C7×M16—21no—
186C7×M16—22no—
187C7×M16—21no—
188C7×M16—22no—
189C7×M16—21no—
190C7×M16#4112yes—
191C2×C7×C8—21no—
192C2×C7×C8—21no—
193C2×C7×C8—22no—
194C7×C4⋊C4—24no—
195C7×C4⋊C4—24no—
196C7×C4⋊C4—24no—
197C7×C4⋊C4—24no—
198C7×C4⋊C4—24no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—22no—
201C7×C4⋊C4—22no—
202C7×C4⋊C4—22no—
203C7×C4⋊C4—21no—
204C7×C4⋊C4—21no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—21no—
207C7×C4⋊C4—22no—
208C7×C4⋊C4—22no—
209C7×C4⋊C4—21no—
210C7×C4⋊C4—21no—
211C7×C4⋊C4—48no—
212C7×C4⋊C4—41no—
213C7×C4⋊C4—41no—
214C7×C4⋊C4C2×D411yescharacteristic
215C7×C4⋊C4C2×D412yes—
216C7×C4⋊C4C2×D411yescharacteristic
217C7×C4⋊C4C2×D412yes—
218C7×C22⋊C4—24no—
219C7×C22⋊C4—24no—
220C7×C22⋊C4—21no—
221C7×C22⋊C4—22no—
222C7×C22⋊C4—21no—
223C7×C22⋊C4—21no—
224C42×C7—24no—
225C42×C7—24no—
226C42×C7—21no—
227C42×C7—21no—
228C42×C7—21no—
229C42×C7—22no—
230C42×C7—22no—
231C42×C7—22no—
232C42×C7—41no—
233C42×C7—41no—
234C42×C7C2×D411yescharacteristic
235C42×C7C2×D411yescharacteristic
236C42×C7C2×D411yescharacteristic
237C42×C7C2×D411yescharacteristic
238C2×#137C22×C711yescharacteristic
239#557—21no—
240#557—21no—
241#556—21no—
242#556—21no—
243#552—22no—
244#552—22no—
245#541—22no—
246#536C22×C712yes—
247#533—21no—
248#533—21no—
249#533C22×C711yescharacteristic
250C2×C4⋊Q8C22×C712yes—
251C2×C4⋊Q8C22×C711yescharacteristic
252#519C22×C712yes—
253C4×#41C22×C711yescharacteristic
254C2×C4×Q8C22×C711yescharacteristic
255#479—22no—
256#477—21no—
257#475—21no—
258#441—21no—
259#440—21no—
260C4×Q16—21no—
261C4×SD16—21no—
262#428C22×C712yes—
263#428C22×C711yescharacteristic
264#423C22×C711yescharacteristic
265C2×#104C22×C712yes—
266#419C22×C712yes—
267#419C22×C711yescharacteristic
268#417C22×C711yescharacteristic
269C2×#101C22×C712yes—
270C2×#101C22×C712yes—
271C4×M16C22×C711yescharacteristic
272#398—22no—
273#391C22×C712yes—
274#389—22no—
275C4×C4⋊C4C22×C711yescharacteristic
276#339C22×C712yes—
277#339C22×C712yes—
278#339C22×C712yes—
279C23×C7—21no—
280C23×C7C22×D411yescharacteristic
281C7×Q8—22no—
282C7×Q8—21no—
283C7×Q8—21no—
284C7×Q8—22no—
285C7×Q8—42no—
286C7×Q8—42no—
287C7×Q8—42no—
288C7×Q8—41no—
289C7×Q8—42no—
290C7×Q8—42no—
291C7×Q8—41no—
292C7×Q8—42no—
293C7×D4—21no—
294C7×D4—21no—
295C7×D4—41no—
296C7×D4—41no—
297C2×C4×C7—22no—
298C2×C4×C7—24no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—22no—
302C2×C4×C7—22no—
303C2×C4×C7—24no—
304C2×C4×C7—21no—
305C2×C4×C7—22no—
306C2×C4×C7—24no—
307C2×C4×C7—24no—
308C2×C4×C7—41no—
309C2×C4×C7—42no—
310C2×C4×C7—42no—
311C2×C4×C7—41no—
312C2×C4×C7—42no—
313C2×C4×C7—41no—
314C2×C4×C7—41no—
315C2×C4×C7—41no—
316C2×C4×C7—42no—
317C2×C4×C7—42no—
318C2×C4×C7—41no—
319C2×C4×C7#12011yescharacteristic
320C2×C4×C7#12011yescharacteristic
321C2×C4×C7#12011yescharacteristic
322C2×C4×C7#12011yescharacteristic
323C2×C4×C7C22×D411yescharacteristic
324C2×C4×C7C22×D411yescharacteristic
325C2×C4×C7C2×#4111yescharacteristic
326C2×C4×C7C2×#4111yescharacteristic
327C2×C4×C7H(5,𝔽2)11yescharacteristic
328C2×C4×C72−1+411yescharacteristic
329C7×C8—22no—
330C7×C8—41no—
331C7×C8—42no—
332C2×#41C23×C711yescharacteristic
333C22×Q8C23×C711yescharacteristic
334C22×Q8C23×C712yes—
335#137—22no—
336#137—22no—
337C2×Q16—21no—
338C2×SD16—21no—
339C2×M16C23×C711yescharacteristic
340C2×M16C23×C712yes—
341C2×M16C23×C711yescharacteristic
342C4⋊Q8—22no—
343C4⋊Q8—24no—
344C4⋊Q8—21no—
345C4⋊Q8—21no—
346C4⋊Q8—21no—
347C4⋊Q8—21no—
348C4⋊Q8—42no—
349C4⋊Q8C7×D411yescharacteristic
350C4⋊Q8C7×D411yescharacteristic
351#126—24no—
352#126—24no—
353#125—22no—
354#125—22no—
355#125—24no—
356#125—24no—
357#125C7×D412yes—
358#124—22no—
359#123—24no—
360C22⋊Q8—22no—
361C22⋊Q8—22no—
362C22⋊Q8—21no—
363C22⋊Q8—21no—
364C22⋊Q8—22no—
365C22⋊Q8—22no—
366C4×Q8—22no—
367C4×Q8—22no—
368C4×Q8—22no—
369C4×Q8—22no—
370C4×Q8—22no—
371C4×Q8—22no—
372C4×Q8—21no—
373C4×Q8—21no—
374C4×Q8—21no—
375C4×Q8—21no—
376C4×Q8—21no—
377C4×Q8—21no—
378C4×Q8C7×D411yescharacteristic
379C4×Q8C7×D411yescharacteristic
380C4×D4—21no—
381C4×D4—21no—
382C4×D4C7×D411yescharacteristic
383C4×D4C7×D411yescharacteristic
384#117—22no—
385#117—22no—
386#117—24no—
387#117—21no—
388#117C23×C711yescharacteristic
389#117C23×C712yes—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—24no—
393C2×C4⋊C4C7×D412yes—
394C2×C4⋊C4C7×D412yes—
395C2×C4⋊C4C23×C711yescharacteristic
396C2×C42—21no—
397C2×C42—21no—
398C2×C42C23×C712yes—
399C2×C42C23×C711yescharacteristic
400#107—21no—
401#107—22no—
402#106—21no—
403#106—22no—
404C4⋊C8—21no—
405C4⋊C8—21no—
406#104—42no—
407#103—21no—
408#103—21no—
409#103—21no—
410#103—22no—
411#103—22no—
412#102—21no—
413#101—24no—
414#101—42no—
415#97—21no—
416C4×C8—21no—
417#95—21no—
418#95—24no—
419C22×C7—21no—
420C22×C7—21no—
421C22×C7—41no—
422C22×C7—41no—
423C22×C7—41no—
424C22×C7C2×#12011yescharacteristic
425C22×C7#54611yescharacteristic
426C22×C7#54711yescharacteristic
427C4×C7—21no—
428C4×C7—21no—
429C4×C7—21no—
430C4×C7—44no—
431C4×C7—44no—
432C4×C7—41no—
433C4×C7—42no—
434C4×C7—41no—
435C4×C7—42no—
436C4×C7—41no—
437C4×C7—42no—
438C4×C7—81no—
439C4×C7#45612yes—
440C4×C7#52511yescharacteristic
441C4×C7#53711yescharacteristic
442C4×C7#54811yescharacteristic
443C4×C7#55011yescharacteristic
444#41—21no—
445#41—21no—
446#41—21no—
447#41—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—21no—
456C2×Q8—22no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8C2×C7×D411yescharacteristic
462C2×Q8C7×#4112yes—
463C2×D4—21no—
464C2×D4C2×C7×D411yescharacteristic
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4C2×C7×D414yes—
472C22×C4C2×C7×D411yescharacteristic
473C22×C4C2×C7×D411yescharacteristic
474C22×C4C24×C711yescharacteristic
475Q16—24no—
476SD16—24no—
477M16—21no—
478M16—22no—
479M16—21no—
480M16—22no—
481M16—21no—
482M16C7×#4112yes—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—22no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—48no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4C2×C7×D411yescharacteristic
507C4⋊C4C2×C7×D412yes—
508C4⋊C4C2×C7×D411yescharacteristic
509C4⋊C4C2×C7×D412yes—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—21no—
513C22⋊C4—22no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C42—24no—
517C42—24no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—41no—
525C42—41no—
526C42C2×C7×D411yescharacteristic
527C42C2×C7×D411yescharacteristic
528C42C2×C7×D411yescharacteristic
529C42C2×C7×D411yescharacteristic
530C2×C7—21no—
531C2×C7—21no—
532C2×C7—81no—
533C2×C7#240711yescharacteristic
534C2×C7C2×#45611yescharacteristic
535C2×C7#302011yescharacteristic
536C23—21no—
537C23C22×C7×D411yescharacteristic
538Q8—22no—
539Q8—21no—
540Q8—21no—
541Q8—22no—
542Q8—42no—
543Q8—42no—
544Q8—42no—
545Q8—41no—
546Q8—42no—
547Q8—42no—
548Q8—41no—
549Q8—42no—
550D4—21no—
551D4—21no—
552D4—41no—
553D4—41no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4C7×#12011yescharacteristic
577C2×C4C7×#12011yescharacteristic
578C2×C4C7×#12011yescharacteristic
579C2×C4C7×#12011yescharacteristic
580C2×C4C22×C7×D411yescharacteristic
581C2×C4C22×C7×D411yescharacteristic
582C2×C4C2×C7×#4111yescharacteristic
583C2×C4C2×C7×#4111yescharacteristic
584C2×C4C7×H(5,𝔽2)11yescharacteristic
585C2×C4C7×2−1+411yescharacteristic
586C8—22no—
587C8—41no—
588C8—42no—
589C7#2414711yescharacteristic, direct product factor, Sylow
590C22—21no—
591C22—21no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C2×C7×#12011yescharacteristic
596C22C7×#54611yescharacteristic
597C22C7×#54711yescharacteristic
598C4—21no—
599C4—21no—
600C4—21no—
601C4—44no—
602C4—44no—
603C4—41no—
604C4—42no—
605C4—41no—
606C4—42no—
607C4—41no—
608C4—42no—
609C4—81no—
610C4C7×#45612yes—
611C4C7×#52511yescharacteristic
612C4C7×#53711yescharacteristic
613C4C7×#54811yescharacteristic
614C4C7×#55011yescharacteristic
615C2—21no—
616C2—21no—
617C2—81no—
618C2C7×#240711yescharacteristic
619C2C2×C7×#45611yescharacteristic
620C2C7×#302011yescharacteristic
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×#2546C2semidirect
7. C7×#1986C2semidirect
8. C7×#1966C2semidirect
9. C7×#1886C2semidirect
10. C7×#1857C2semidirect
53. #24147C7direct (N × H)
142. #3078C2×C7semidirect
144. #2546C2×C7semidirect
147. #1986C2×C7semidirect
148. #1966C2×C7semidirect
149. #1886C2×C7semidirect
150. #1857C2×C7semidirect
589. C7#24147direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35092
C7×#30782
C4×C7×#1372
C7×#25462
C7×#20382
C7×#20052
C7×#19862
C7×#19662
C7×#18862
C7×#18572
C7×#18062
#241477