Finite Group Databasebeta

← C7×#24320 #13201536

Subgroups — C7×#24320 #13201536

Order 1792.

1092 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3503C211yesmaximal, characteristic, complemented
2C7×#3077C211yesmaximal, characteristic, complemented
3C4×C7×AGL(1,ℤ/8)C211yesmaximal, characteristic
4C7×#2542C211yesmaximal, characteristic, complemented
5C7×#2060C212yesmaximal, complemented
6C7×#2027C212yesmaximal, complemented
7C7×#1997C211yesmaximal, characteristic, complemented
8C7×#1986C211yesmaximal, characteristic, complemented
9C7×#1883C212yesmaximal
10C7×#1859C212yesmaximal, complemented
11C7×#1806C211yesmaximal, characteristic, complemented
12C2×C7×AGL(1,ℤ/8)C2211yescharacteristic
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
25C2×C7×C22⋊Q8—22no—
26C7×#520C2212yes—
27C4×C7×#41C2211yescharacteristic
28C2×C4×C7×D4C2211yescharacteristic
29C7×#478—22no—
30C7×#476—21no—
31C7×#474—21no—
32C7×#442—21no—
33C7×#440—21no—
34C4×C7×SD16—21no—
35C4×C7×D8—21no—
36C7×#428C2211yescharacteristic
37C7×#428C2212yes—
38C7×#423C2211yescharacteristic
39C2×C7×#104C2212yes—
40C7×#418C2211yescharacteristic
41C7×#418C2212yes—
42C7×#417C2211yescharacteristic
43C4×C7×M16C2211yescharacteristic
44C7×#399—22no—
45C7×#390C2212yes—
46C7×#384C2212yes—
47C4×C7×C4⋊C4C2211yescharacteristic
48C7×#343C2212yes—
49C7×#343C2212yes—
50C7×#339C2212yes—
51C7×#339C2212yes—
52C7×#339C2212yes—
53#24320C711yesmaximal, characteristic, direct product factor, Sylow
54C2×C7×#41C2311yescharacteristic
55C22×C7×Q8—22no—
56C22×C7×Q8D412yes—
57C22×C7×D4C2311yescharacteristic
58C23×C4×C7—21no—
59C7×AGL(1,ℤ/8)—22no—
60C7×AGL(1,ℤ/8)Q814yescomplemented
61C2×C7×SD16—21no—
62C2×C7×D8—21no—
63C2×C7×M16C2311yescharacteristic
64C2×C7×M16C2312yes—
65C2×C7×M16C2311yescharacteristic
66C7×C4⋊Q8—22no—
67C7×C4⋊Q8—22no—
68C7×C4⋊Q8—24no—
69C7×C4⋊Q8—22no—
70C7×C4⋊Q8—22no—
71C7×C4⋊Q8—21no—
72C7×C4⋊Q8—21no—
73C7×C4⋊Q8—21no—
74C7×C4⋊Q8—21no—
75C7×C4⋊Q8D411yescharacteristic
76C7×C4⋊Q8D411yescharacteristic
77C7×#125—24no—
78C7×#125—22no—
79C7×#125—22no—
80C7×#125D412yes—
81C7×C22⋊Q8—22no—
82C7×C22⋊Q8—22no—
83C7×C22⋊Q8—24no—
84C7×C22⋊Q8—24no—
85C7×C22⋊Q8—21no—
86C7×C22⋊Q8—21no—
87C7×C22⋊Q8—24no—
88C7×C22⋊Q8—48no—
89C4×C7×Q8—22no—
90C4×C7×Q8—22no—
91C4×C7×Q8—22no—
92C4×C7×Q8—21no—
93C4×C7×Q8—21no—
94C4×C7×Q8D411yescharacteristic
95C4×C7×Q8D411yescharacteristic
96C4×C7×D4—22no—
97C4×C7×D4—21no—
98C4×C7×D4—21no—
99C4×C7×D4—24no—
100C4×C7×D4—21no—
101C4×C7×D4—21no—
102C4×C7×D4—21no—
103C4×C7×D4—21no—
104C4×C7×D4D411yescharacteristic
105C4×C7×D4D411yescharacteristic
106C7×#117—22no—
107C7×#117—21no—
108C7×#117C2311yescharacteristic
109C7×#117C2312yes—
110C2×C7×C4⋊C4—22no—
111C2×C7×C4⋊C4—21no—
112C2×C7×C4⋊C4—22no—
113C2×C7×C4⋊C4—24no—
114C2×C7×C4⋊C4D412yes—
115C2×C7×C4⋊C4C2311yescharacteristic
116C2×C7×C4⋊C4C2312yes—
117C2×C7×C22⋊C4—21no—
118C2×C7×C22⋊C4—24no—
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—24no—
129C7×#103—21no—
130C7×#102—21no—
131C7×#102—22no—
132C7×#102—22no—
133C7×#102—21no—
134C7×#102—21no—
135C7×#97—21no—
136C4×C7×C8—21no—
137C7×#95—21no—
138#3503C2×C711yescharacteristic, complemented
139#3077C2×C711yescharacteristic, complemented
140C4×AGL(1,ℤ/8)C2×C711yescharacteristic
141#2542C2×C711yescharacteristic, complemented
142#2060C2×C712yescomplemented
143#2027C2×C712yescomplemented
144#1997C2×C711yescharacteristic, complemented
145#1986C2×C711yescharacteristic, complemented
146#1883C2×C712yes—
147#1859C2×C712yescomplemented
148#1806C2×C711yescharacteristic, complemented
149C24×C7—21no—
150C7×#41—21no—
151C7×#41—21no—
152C7×#41C2×Q812yes—
153C7×#41#4112yes—
154C2×C7×Q8—24no—
155C2×C7×Q8—21no—
156C2×C7×Q8—22no—
157C2×C7×Q8—21no—
158C2×C7×Q8—22no—
159C2×C7×Q8—44no—
160C2×C7×Q8—84no—
161C2×C7×Q8C2×D411yescharacteristic
162C2×C7×D4—21no—
163C2×C7×D4—21no—
164C2×C7×D4—21no—
165C2×C7×D4—21no—
166C2×C7×D4—21no—
167C2×C7×D4—21no—
168C2×C7×D4C2×D411yescharacteristic
169C2×C7×D4C2×Q812yescomplemented
170C22×C4×C7—21no—
171C22×C4×C7—22no—
172C22×C4×C7—24no—
173C22×C4×C7—21no—
174C22×C4×C7—21no—
175C22×C4×C7—21no—
176C22×C4×C7—44no—
177C22×C4×C7—41no—
178C22×C4×C7—41no—
179C22×C4×C7C2×D411yescharacteristic
180C22×C4×C7C2×D414yes—
181C22×C4×C7C2×D411yescharacteristic
182C22×C4×C7C2411yescharacteristic
183C7×SD16—24no—
184C7×D8—24no—
185C7×M16—21no—
186C7×M16—22no—
187C7×M16—21no—
188C7×M16—21no—
189C7×M16—22no—
190C7×M16C2×Q812yescomplemented
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—21no—
200C7×C4⋊C4—22no—
201C7×C4⋊C4—22no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—21no—
204C7×C4⋊C4—21no—
205C7×C4⋊C4—22no—
206C7×C4⋊C4—22no—
207C7×C4⋊C4—22no—
208C7×C4⋊C4—22no—
209C7×C4⋊C4—21no—
210C7×C4⋊C4—21no—
211C7×C4⋊C4—48no—
212C7×C4⋊C4C2×D412yes—
213C7×C4⋊C4C2×D411yescharacteristic
214C7×C4⋊C4C2×D412yes—
215C7×C4⋊C4C2×D411yescharacteristic
216C7×C22⋊C4—24no—
217C7×C22⋊C4—22no—
218C7×C22⋊C4—21no—
219C7×C22⋊C4—21no—
220C7×C22⋊C4—21no—
221C7×C22⋊C4—48no—
222C7×C22⋊C4—41no—
223C7×C22⋊C4—41no—
224C42×C7—24no—
225C42×C7—21no—
226C42×C7—21no—
227C42×C7—21no—
228C42×C7—22no—
229C42×C7—22no—
230C42×C7—22no—
231C42×C7C2×D411yescharacteristic
232C42×C7C2×D411yescharacteristic
233C42×C7C2×D411yescharacteristic
234C42×C7C2×D411yescharacteristic
235C2×AGL(1,ℤ/8)C22×C711yescharacteristic
236Q82—21no—
237#557—21no—
238#554—21no—
239#554—22no—
240D4×Q8—22no—
241D4×Q8—21no—
242#544—22no—
243#544C22×C712yes—
244#533—21no—
245#533C22×C711yescharacteristic
246C2×C4⋊Q8—21no—
247C2×C4⋊Q8C22×C711yescharacteristic
248C2×C22⋊Q8—22no—
249#520C22×C712yes—
250C4×#41C22×C711yescharacteristic
251C2×C4×D4C22×C711yescharacteristic
252#478—22no—
253#476—21no—
254#474—21no—
255#442—21no—
256#440—21no—
257C4×SD16—21no—
258C4×D8—21no—
259#428C22×C712yes—
260#428C22×C711yescharacteristic
261#423C22×C711yescharacteristic
262C2×#104C22×C712yes—
263#418C22×C712yes—
264#418C22×C711yescharacteristic
265#417C22×C711yescharacteristic
266C4×M16C22×C711yescharacteristic
267#399—22no—
268#390C22×C712yes—
269#384C22×C712yes—
270C4×C4⋊C4C22×C711yescharacteristic
271#343C22×C712yes—
272#343C22×C712yes—
273#339C22×C712yes—
274#339C22×C712yes—
275#339C22×C712yes—
276C23×C7—21no—
277C23×C7—22no—
278C23×C7—42no—
279C23×C7—42no—
280C23×C7—41no—
281C23×C7—41no—
282C23×C7C22×D411yescharacteristic
283C7×Q8—21no—
284C7×Q8—42no—
285C7×Q8—44no—
286C7×Q8—42no—
287C7×Q8—82no—
288C7×Q8—84no—
289C7×Q8C22⋊Q812yescomplemented
290C7×D4—22no—
291C7×D4—22no—
292C7×D4—22no—
293C7×D4—21no—
294C7×D4—22no—
295C7×D4—41no—
296C7×D4—41no—
297C7×D4C22⋊Q812yescomplemented
298C2×C4×C7—21no—
299C2×C4×C7—22no—
300C2×C4×C7—22no—
301C2×C4×C7—22no—
302C2×C4×C7—24no—
303C2×C4×C7—22no—
304C2×C4×C7—21no—
305C2×C4×C7—21no—
306C2×C4×C7—24no—
307C2×C4×C7—22no—
308C2×C4×C7—41no—
309C2×C4×C7—42no—
310C2×C4×C7—42no—
311C2×C4×C7—42no—
312C2×C4×C7—42no—
313C2×C4×C7—44no—
314C2×C4×C7—42no—
315C2×C4×C7—41no—
316C2×C4×C7—82no—
317C2×C4×C7—84no—
318C2×C4×C7—82no—
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×C7C22×Q811yescharacteristic
326C2×C4×C7C2×#4111yescharacteristic
327C2×C4×C72−1+411yescharacteristic
328C2×C4×C72−1+411yescharacteristic
329C7×C8—22no—
330C7×C8—41no—
331C7×C8—42no—
332C2×#41C23×C711yescharacteristic
333C22×Q8—22no—
334C22×Q8C7×D412yes—
335C22×D4C23×C711yescharacteristic
336C23×C4—21no—
337AGL(1,ℤ/8)—22no—
338AGL(1,ℤ/8)C7×Q814yescomplemented
339C2×SD16—21no—
340C2×D8—21no—
341C2×M16C23×C711yescharacteristic
342C2×M16C23×C712yes—
343C2×M16C23×C711yescharacteristic
344C4⋊Q8—22no—
345C4⋊Q8—22no—
346C4⋊Q8—24no—
347C4⋊Q8—22no—
348C4⋊Q8—22no—
349C4⋊Q8—21no—
350C4⋊Q8—21no—
351C4⋊Q8—21no—
352C4⋊Q8—21no—
353C4⋊Q8C7×D411yescharacteristic
354C4⋊Q8C7×D411yescharacteristic
355#125—24no—
356#125—22no—
357#125—22no—
358#125C7×D412yes—
359C22⋊Q8—22no—
360C22⋊Q8—22no—
361C22⋊Q8—24no—
362C22⋊Q8—24no—
363C22⋊Q8—21no—
364C22⋊Q8—21no—
365C22⋊Q8—24no—
366C22⋊Q8—48no—
367C4×Q8—22no—
368C4×Q8—22no—
369C4×Q8—22no—
370C4×Q8—21no—
371C4×Q8—21no—
372C4×Q8C7×D411yescharacteristic
373C4×Q8C7×D411yescharacteristic
374C4×D4—22no—
375C4×D4—21no—
376C4×D4—21no—
377C4×D4—24no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4C7×D411yescharacteristic
383C4×D4C7×D411yescharacteristic
384#117—22no—
385#117—21no—
386#117C23×C711yescharacteristic
387#117C23×C712yes—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4C7×D412yes—
393C2×C4⋊C4C23×C711yescharacteristic
394C2×C4⋊C4C23×C712yes—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—24no—
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—24no—
407#103—21no—
408#102—21no—
409#102—21no—
410#102—21no—
411#102—22no—
412#102—22no—
413#97—21no—
414C4×C8—21no—
415#95—21no—
416C22×C7—21no—
417C22×C7—21no—
418C22×C7—22no—
419C22×C7—44no—
420C22×C7—41no—
421C22×C7—41no—
422C22×C7—42no—
423C22×C7—82no—
424C22×C7—81no—
425C22×C7—81no—
426C22×C7C2×#12011yescharacteristic
427C22×C7#54811yescharacteristic
428C22×C7D4×Q811yescharacteristic
429C4×C7—22no—
430C4×C7—21no—
431C4×C7—21no—
432C4×C7—21no—
433C4×C7—44no—
434C4×C7—41no—
435C4×C7—44no—
436C4×C7—42no—
437C4×C7—41no—
438C4×C7—42no—
439C4×C7—81no—
440C4×C7#45312yes—
441C4×C7C2×C22⋊Q811yescharacteristic
442C4×C7#53611yescharacteristic
443C4×C7#54811yescharacteristic
444C4×C7D4×Q811yescharacteristic
445C24—21no—
446#41—21no—
447#41—21no—
448#41C2×C7×Q812yes—
449#41C7×#4112yes—
450C2×Q8—24no—
451C2×Q8—21no—
452C2×Q8—22no—
453C2×Q8—21no—
454C2×Q8—22no—
455C2×Q8—44no—
456C2×Q8—84no—
457C2×Q8C2×C7×D411yescharacteristic
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4C2×C7×D411yescharacteristic
465C2×D4C2×C7×Q812yescomplemented
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—44no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4C2×C7×D411yescharacteristic
476C22×C4C2×C7×D414yes—
477C22×C4C2×C7×D411yescharacteristic
478C22×C4C24×C711yescharacteristic
479SD16—24no—
480D8—24no—
481M16—21no—
482M16—22no—
483M16—21no—
484M16—21no—
485M16—22no—
486M16C2×C7×Q812yescomplemented
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—22no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—48no—
508C4⋊C4C2×C7×D412yes—
509C4⋊C4C2×C7×D411yescharacteristic
510C4⋊C4C2×C7×D412yes—
511C4⋊C4C2×C7×D411yescharacteristic
512C22⋊C4—24no—
513C22⋊C4—22no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—48no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C42—24no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42C2×C7×D411yescharacteristic
528C42C2×C7×D411yescharacteristic
529C42C2×C7×D411yescharacteristic
530C42C2×C7×D411yescharacteristic
531C2×C7—21no—
532C2×C7—21no—
533C2×C7—42no—
534C2×C7—82no—
535C2×C7#240911yescharacteristic
536C2×C7C2×#45311yescharacteristic
537C2×C7#302011yescharacteristic
538C23—21no—
539C23—22no—
540C23—42no—
541C23—42no—
542C23—41no—
543C23—41no—
544C23C22×C7×D411yescharacteristic
545Q8—21no—
546Q8—42no—
547Q8—44no—
548Q8—42no—
549Q8—82no—
550Q8—84no—
551Q8C7×C22⋊Q812yescomplemented
552D4—22no—
553D4—22no—
554D4—22no—
555D4—21no—
556D4—22no—
557D4—41no—
558D4—41no—
559D4C7×C22⋊Q812yescomplemented
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—82no—
579C2×C4—84no—
580C2×C4—82no—
581C2×C4C7×#12011yescharacteristic
582C2×C4C7×#12011yescharacteristic
583C2×C4C7×#12011yescharacteristic
584C2×C4C7×#12011yescharacteristic
585C2×C4C22×C7×D411yescharacteristic
586C2×C4C22×C7×D411yescharacteristic
587C2×C4C22×C7×Q811yescharacteristic
588C2×C4C2×C7×#4111yescharacteristic
589C2×C4C7×2−1+411yescharacteristic
590C2×C4C7×2−1+411yescharacteristic
591C8—22no—
592C8—41no—
593C8—42no—
594C7#2432011yescharacteristic, direct product factor, Sylow
595C22—21no—
596C22—21no—
597C22—22no—
598C22—44no—
599C22—41no—
600C22—41no—
601C22—42no—
602C22—82no—
603C22—81no—
604C22—81no—
605C22C2×C7×#12011yescharacteristic
606C22C7×#54811yescharacteristic
607C22C7×D4×Q811yescharacteristic
608C4—22no—
609C4—21no—
610C4—21no—
611C4—21no—
612C4—44no—
613C4—41no—
614C4—44no—
615C4—42no—
616C4—41no—
617C4—42no—
618C4—81no—
619C4C7×#45312yes—
620C4C2×C7×C22⋊Q811yescharacteristic
621C4C7×#53611yescharacteristic
622C4C7×#54811yescharacteristic
623C4C7×D4×Q811yescharacteristic
624C2—21no—
625C2—21no—
626C2—42no—
627C2—82no—
628C2C7×#240911yescharacteristic
629C2C2×C7×#45311yescharacteristic
630C2C7×#302011yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×AGL(1,ℤ/8)C22
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
1. C7×#3503C2semidirect
2. C7×#3077C2semidirect
4. C7×#2542C2semidirect
5. C7×#2060C2semidirect
6. C7×#2027C2semidirect
7. C7×#1997C2semidirect
8. C7×#1986C2semidirect
10. C7×#1859C2semidirect
11. C7×#1806C2semidirect
53. #24320C7direct (N × H)
60. C7×AGL(1,ℤ/8)Q8semidirect
138. #3503C2×C7semidirect
139. #3077C2×C7semidirect
141. #2542C2×C7semidirect
142. #2060C2×C7semidirect
143. #2027C2×C7semidirect
144. #1997C2×C7semidirect
145. #1986C2×C7semidirect
147. #1859C2×C7semidirect
148. #1806C2×C7semidirect
169. C2×C7×D4C2×Q8semidirect
190. C7×M16C2×Q8semidirect
289. C7×Q8C22⋊Q8semidirect
297. C7×D4C22⋊Q8semidirect
338. AGL(1,ℤ/8)C7×Q8semidirect
465. C2×D4C2×C7×Q8semidirect
486. M16C2×C7×Q8semidirect
551. Q8C7×C22⋊Q8semidirect
559. D4C7×C22⋊Q8semidirect
594. C7#24320direct (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×#30772
C4×C7×AGL(1,ℤ/8)2
C7×#25422
C7×#20602
C7×#20272
C7×#19972
C7×#19862
C7×#18832
C7×#18592
C7×#18062
#243207