Finite Group Databasebeta

← C7×#18322 #13195538

Subgroups — C7×#18322 #13195538

Order 1792.

1078 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3459C211yesmaximal, characteristic, complemented
2C7×#3194C212yesmaximal, complemented
3C7×#2543C211yesmaximal, characteristic, complemented
4C7×#1609C212yesmaximal, complemented
5C7×#1608C211yesmaximal, characteristic, complemented
6C7×#1607C211yesmaximal, characteristic, complemented
7C7×#1597C212yesmaximal, complemented
8C7×#1596C211yesmaximal, characteristic, complemented
9C7×#1595C211yesmaximal, characteristic, complemented
10C7×#1504C212yesmaximal, complemented
11C7×#1465C211yesmaximal, characteristic, complemented
12C7×#543C2211yescharacteristic
13C7×#539—22no—
14C7×#539C2212yes—
15C7×#538—22no—
16C7×#538—21no—
17C7×#534—21no—
18C7×#534C2211yescharacteristic
19C7×#529—22no—
20C7×#529—21no—
21C7×#525—22no—
22C7×#525C2212yes—
23C2×C7×#121—21no—
24C2×C7×#121C2211yescharacteristic, complemented
25C2×C4×C7×D4—21no—
26C2×C4×C7×D4C2211yescharacteristic
27C7×#483C2212yescomplemented
28C7×#482—22no—
29C7×#480C2212yes—
30C7×#469—22no—
31C7×#468—22no—
32C7×#450C2212yescomplemented
33C7×#449—22no—
34C7×#447C2212yes—
35C7×#407C2212yescomplemented
36C7×#398C2211yescharacteristic, complemented
37C7×#397C2211yescharacteristic
38C7×#394C2211yescharacteristic, complemented
39C7×#392C2211yescharacteristic
40C7×#388—21no—
41C7×#386—21no—
42C7×#386—21no—
43C2×C7×#95C2211yescharacteristic, complemented
44C7×#328C2212yescomplemented
45C7×#328C2212yescomplemented
46C7×#327C2212yes—
47C7×#327C2212yes—
48C7×#324C2212yescomplemented
49C7×#323C2211yescharacteristic
50C7×#323C2211yescharacteristic
51#18322C711yesmaximal, characteristic, direct product factor, Sylow
52C2×C7×#41D412yescomplemented
53C22×C7×D4—21no—
54C22×C7×D4—21no—
55C22×C7×D4—21no—
56C22×C7×D4D411yescharacteristic
57C22×C7×D4C2311yescharacteristic
58C23×C4×C7—21no—
59C23×C4×C7—21no—
60C23×C4×C7D411yescharacteristic
61C23×C4×C7C2311yescharacteristic
62C2×C7×SD16—22no—
63C2×C7×D8—22no—
64C2×C7×M16—22no—
65C7×#126—22no—
66C7×#126—22no—
67C7×#126—22no—
68C7×#125—21no—
69C7×#125—22no—
70C7×#124—22no—
71C7×#124—21no—
72C7×#123—22no—
73C7×#123—22no—
74C7×#123—22no—
75C7×#123—22no—
76C7×#123—22no—
77C7×#123—22no—
78C7×C22⋊Q8—22no—
79C7×C22⋊Q8—22no—
80C7×C22⋊Q8—21no—
81C7×C22⋊Q8—22no—
82C7×C22⋊Q8D412yescomplemented
83C7×C22⋊Q8C2312yes—
84C7×#121—22no—
85C7×#121—22no—
86C7×#121—22no—
87C7×#121—21no—
88C7×#121—22no—
89C7×#121—22no—
90C7×#121—44no—
91C7×#121D412yescomplemented
92C7×#121C2312yes—
93C7×#120—22no—
94C7×#120—22no—
95C7×#120—22no—
96C4×C7×Q8—21no—
97C4×C7×D4—22no—
98C4×C7×D4—22no—
99C4×C7×D4—21no—
100C4×C7×D4—22no—
101C4×C7×D4—22no—
102C4×C7×D4—22no—
103C4×C7×D4—22no—
104C4×C7×D4—22no—
105C7×#117—22no—
106C7×#117D412yescomplemented
107C2×C7×C4⋊C4—21no—
108C2×C7×C4⋊C4—21no—
109C2×C7×C4⋊C4C2311yescharacteristic
110C2×C7×C4⋊C4C2311yescharacteristic
111C2×C7×C22⋊C4—21no—
112C2×C7×C22⋊C4—21no—
113C2×C7×C22⋊C4—21no—
114C2×C7×C22⋊C4—21no—
115C2×C7×C22⋊C4—21no—
116C2×C7×C22⋊C4—21no—
117C2×C7×C22⋊C4—21no—
118C2×C7×C22⋊C4D411yescharacteristic
119C2×C7×C22⋊C4C2311yescharacteristic
120C2×C42×C7—21no—
121C2×C42×C7D411yescharacteristic
122C7×#107—22no—
123C7×#106—22no—
124C7×#103—22no—
125C7×#102—22no—
126C7×#102—22no—
127C7×#102—22no—
128C7×C22⋊C8—22no—
129C7×C22⋊C8C2312yes—
130C7×C22⋊C8C2312yes—
131C7×#95—21no—
132C7×#95—21no—
133C7×#95—21no—
134C7×#95C2311yescharacteristic
135C7×#95C2311yescharacteristic
136#3459C2×C711yescharacteristic, complemented
137#3194C2×C712yescomplemented
138#2543C2×C711yescharacteristic, complemented
139#1609C2×C712yescomplemented
140#1608C2×C711yescharacteristic, complemented
141#1607C2×C711yescharacteristic, complemented
142#1597C2×C712yescomplemented
143#1596C2×C711yescharacteristic, complemented
144#1595C2×C711yescharacteristic, complemented
145#1504C2×C712yescomplemented
146#1465C2×C711yescharacteristic, complemented
147C24×C7—21no—
148C24×C7—21no—
149C24×C7C2×D411yescharacteristic
150C24×C7C2×D411yescharacteristic
151C7×#41—82no—
152C2×C7×Q8—21no—
153C2×C7×Q8C2×D412yes—
154C2×C7×D4—22no—
155C2×C7×D4—22no—
156C2×C7×D4—22no—
157C2×C7×D4—22no—
158C2×C7×D4—21no—
159C2×C7×D4—22no—
160C2×C7×D4—22no—
161C2×C7×D4—22no—
162C2×C7×D4—41no—
163C2×C7×D4—42no—
164C2×C7×D4—42no—
165C2×C7×D4—42no—
166C2×C7×D4—42no—
167C2×C7×D4—42no—
168C2×C7×D4—42no—
169C2×C7×D4—42no—
170C2×C7×D4—81no—
171C2×C7×D4—81no—
172C2×C7×D4C2×D412yes—
173C22×C4×C7—21no—
174C22×C4×C7—21no—
175C22×C4×C7—21no—
176C22×C4×C7—21no—
177C22×C4×C7—22no—
178C22×C4×C7—21no—
179C22×C4×C7—22no—
180C22×C4×C7—21no—
181C22×C4×C7—21no—
182C22×C4×C7—21no—
183C22×C4×C7—42no—
184C22×C4×C7—42no—
185C22×C4×C7—81no—
186C22×C4×C7—81no—
187C22×C4×C7—82no—
188C22×C4×C7—81no—
189C22×C4×C7C2×D411yescharacteristic
190C22×C4×C7C2×D411yescharacteristic
191C22×C4×C7C2×D411yescharacteristic
192C22×C4×C7C2×D411yescharacteristic
193C22×C4×C7C2×D412yes—
194C22×C4×C7C2×D412yes—
195C22×C4×C7C2411yescharacteristic
196C7×SD16—82no—
197C7×D8—82no—
198C7×M16—82no—
199C2×C7×C8—22no—
200C2×C7×C8—22no—
201C7×C4⋊C4—22no—
202C7×C4⋊C4—22no—
203C7×C4⋊C4—22no—
204C7×C4⋊C4—21no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—22no—
207C7×C4⋊C4—22no—
208C7×C4⋊C4—22no—
209C7×C4⋊C4—22no—
210C7×C4⋊C4—22no—
211C7×C4⋊C4—42no—
212C7×C4⋊C4C2×D412yes—
213C7×C4⋊C4C2×D412yes—
214C7×C22⋊C4—22no—
215C7×C22⋊C4—22no—
216C7×C22⋊C4—22no—
217C7×C22⋊C4—22no—
218C7×C22⋊C4—22no—
219C7×C22⋊C4—22no—
220C7×C22⋊C4—22no—
221C7×C22⋊C4—22no—
222C7×C22⋊C4—22no—
223C7×C22⋊C4—22no—
224C7×C22⋊C4—22no—
225C7×C22⋊C4—22no—
226C7×C22⋊C4—42no—
227C7×C22⋊C4—42no—
228C7×C22⋊C4—42no—
229C7×C22⋊C4—41no—
230C7×C22⋊C4—41no—
231C7×C22⋊C4—81no—
232C42×C7—22no—
233C42×C7—22no—
234C42×C7—21no—
235C42×C7—21no—
236#543C22×C711yescharacteristic
237#539—22no—
238#539C22×C712yes—
239#538—22no—
240#538—21no—
241#534—21no—
242#534C22×C711yescharacteristic
243#529—22no—
244#529—21no—
245#525—22no—
246#525C22×C712yes—
247C2×#121—21no—
248C2×#121C22×C711yescharacteristic, complemented
249C2×C4×D4—21no—
250C2×C4×D4C22×C711yescharacteristic
251#483C22×C712yescomplemented
252#482—22no—
253#480C22×C712yes—
254#469—22no—
255#468—22no—
256#450C22×C712yescomplemented
257#449—22no—
258#447C22×C712yes—
259#407C22×C712yescomplemented
260#398C22×C711yescharacteristic, complemented
261#397C22×C711yescharacteristic
262#394C22×C711yescharacteristic, complemented
263#392C22×C711yescharacteristic
264#388—21no—
265#386—21no—
266#386—21no—
267C2×#95C22×C711yescharacteristic, complemented
268#328C22×C712yescomplemented
269#328C22×C712yescomplemented
270#327C22×C712yes—
271#327C22×C712yes—
272#324C22×C712yescomplemented
273#323C22×C711yescharacteristic
274#323C22×C711yescharacteristic
275C23×C7—21no—
276C23×C7—21no—
277C23×C7—21no—
278C23×C7—21no—
279C23×C7—22no—
280C23×C7—41no—
281C23×C7—41no—
282C23×C7—41no—
283C23×C7—41no—
284C23×C7—42no—
285C23×C7—41no—
286C23×C7—41no—
287C23×C7—41no—
288C23×C7—41no—
289C23×C7—41no—
290C23×C7—41no—
291C23×C7—41no—
292C23×C7—81no—
293C23×C7—81no—
294C23×C7#12012yes—
295C23×C7AGL(1,ℤ/8)12yes—
296C23×C7C22×D411yescharacteristic
297C7×Q8—42no—
298C7×Q8—81no—
299C7×D4—42no—
300C7×D4—42no—
301C7×D4—42no—
302C7×D4—44no—
303C7×D4—44no—
304C7×D4—42no—
305C7×D4—42no—
306C7×D4—82no—
307C7×D4—82no—
308C7×D4—81no—
309C7×D4—84no—
310C7×D4—84no—
311C2×C4×C7—22no—
312C2×C4×C7—21no—
313C2×C4×C7—22no—
314C2×C4×C7—21no—
315C2×C4×C7—22no—
316C2×C4×C7—21no—
317C2×C4×C7—22no—
318C2×C4×C7—21no—
319C2×C4×C7—21no—
320C2×C4×C7—42no—
321C2×C4×C7—41no—
322C2×C4×C7—42no—
323C2×C4×C7—41no—
324C2×C4×C7—42no—
325C2×C4×C7—44no—
326C2×C4×C7—41no—
327C2×C4×C7—42no—
328C2×C4×C7—41no—
329C2×C4×C7—41no—
330C2×C4×C7—41no—
331C2×C4×C7—41no—
332C2×C4×C7—41no—
333C2×C4×C7—41no—
334C2×C4×C7—41no—
335C2×C4×C7—41no—
336C2×C4×C7—41no—
337C2×C4×C7—81no—
338C2×C4×C7—81no—
339C2×C4×C7—82no—
340C2×C4×C7—81no—
341C2×C4×C7—84no—
342C2×C4×C7#12011yescharacteristic
343C2×C4×C7#12011yescharacteristic
344C2×C4×C7C22×D412yes—
345C7×C8—42no—
346C7×C8—42no—
347C2×#41C7×D412yescomplemented
348C22×D4—21no—
349C22×D4—21no—
350C22×D4—21no—
351C22×D4C7×D411yescharacteristic
352C22×D4C23×C711yescharacteristic
353C23×C4—21no—
354C23×C4—21no—
355C23×C4C7×D411yescharacteristic
356C23×C4C23×C711yescharacteristic
357C2×SD16—22no—
358C2×D8—22no—
359C2×M16—22no—
360#126—22no—
361#126—22no—
362#126—22no—
363#125—21no—
364#125—22no—
365#124—22no—
366#124—21no—
367#123—22no—
368#123—22no—
369#123—22no—
370#123—22no—
371#123—22no—
372#123—22no—
373C22⋊Q8—22no—
374C22⋊Q8—22no—
375C22⋊Q8—21no—
376C22⋊Q8—22no—
377C22⋊Q8C7×D412yescomplemented
378C22⋊Q8C23×C712yes—
379#121—22no—
380#121—22no—
381#121—22no—
382#121—21no—
383#121—22no—
384#121—22no—
385#121—44no—
386#121C7×D412yescomplemented
387#121C23×C712yes—
388#120—22no—
389#120—22no—
390#120—22no—
391C4×Q8—21no—
392C4×D4—22no—
393C4×D4—22no—
394C4×D4—21no—
395C4×D4—22no—
396C4×D4—22no—
397C4×D4—22no—
398C4×D4—22no—
399C4×D4—22no—
400#117—22no—
401#117C7×D412yescomplemented
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4C23×C711yescharacteristic
405C2×C4⋊C4C23×C711yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4C7×D411yescharacteristic
414C2×C22⋊C4C23×C711yescharacteristic
415C2×C42—21no—
416C2×C42C7×D411yescharacteristic
417#107—22no—
418#106—22no—
419#103—22no—
420#102—22no—
421#102—22no—
422#102—22no—
423C22⋊C8—22no—
424C22⋊C8C23×C712yes—
425C22⋊C8C23×C712yes—
426#95—21no—
427#95—21no—
428#95—21no—
429#95C23×C711yescharacteristic
430#95C23×C711yescharacteristic
431C22×C7—21no—
432C22×C7—21no—
433C22×C7—22no—
434C22×C7—22no—
435C22×C7—22no—
436C22×C7—22no—
437C22×C7—42no—
438C22×C7—42no—
439C22×C7—42no—
440C22×C7—41no—
441C22×C7—41no—
442C22×C7—42no—
443C22×C7—42no—
444C22×C7—42no—
445C22×C7—81no—
446C22×C7—81no—
447C22×C7—81no—
448C22×C7—81no—
449C22×C7—81no—
450C22×C7—84no—
451C22×C7—81no—
452C22×C7—81no—
453C22×C7—81no—
454C22×C7C2×#12011yescharacteristic
455C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
456C22×C7#57511yescharacteristic
457C4×C7—22no—
458C4×C7—21no—
459C4×C7—21no—
460C4×C7—42no—
461C4×C7—41no—
462C4×C7—42no—
463C4×C7—41no—
464C4×C7—42no—
465C4×C7—41no—
466C4×C7—42no—
467C4×C7—81no—
468C4×C7—81no—
469C24—21no—
470C24—21no—
471C24C2×C7×D411yescharacteristic
472C24C2×C7×D411yescharacteristic
473#41—82no—
474C2×Q8—21no—
475C2×Q8C2×C7×D412yes—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—21no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—41no—
485C2×D4—42no—
486C2×D4—42no—
487C2×D4—42no—
488C2×D4—42no—
489C2×D4—42no—
490C2×D4—42no—
491C2×D4—42no—
492C2×D4—81no—
493C2×D4—81no—
494C2×D4C2×C7×D412yes—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—22no—
500C22×C4—21no—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—81no—
508C22×C4—81no—
509C22×C4—82no—
510C22×C4—81no—
511C22×C4C2×C7×D411yescharacteristic
512C22×C4C2×C7×D411yescharacteristic
513C22×C4C2×C7×D411yescharacteristic
514C22×C4C2×C7×D411yescharacteristic
515C22×C4C2×C7×D412yes—
516C22×C4C2×C7×D412yes—
517C22×C4C24×C711yescharacteristic
518SD16—82no—
519D8—82no—
520M16—82no—
521C2×C8—22no—
522C2×C8—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—42no—
534C4⋊C4C2×C7×D412yes—
535C4⋊C4C2×C7×D412yes—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C22⋊C4—41no—
552C22⋊C4—41no—
553C22⋊C4—81no—
554C42—22no—
555C42—22no—
556C42—21no—
557C42—21no—
558C2×C7—21no—
559C2×C7—21no—
560C2×C7—42no—
561C2×C7—42no—
562C2×C7—82no—
563C2×C7—161no—
564C2×C7#300611yescharacteristic
565C2×C7#301811yescharacteristic
566C2×C7#302611yescharacteristic
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—22no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—42no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—81no—
585C23—81no—
586C23C7×#12012yes—
587C23C7×AGL(1,ℤ/8)12yes—
588C23C22×C7×D411yescharacteristic
589Q8—42no—
590Q8—81no—
591D4—42no—
592D4—42no—
593D4—42no—
594D4—44no—
595D4—44no—
596D4—42no—
597D4—42no—
598D4—82no—
599D4—82no—
600D4—81no—
601D4—84no—
602D4—84no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—81no—
630C2×C4—81no—
631C2×C4—82no—
632C2×C4—81no—
633C2×C4—84no—
634C2×C4C7×#12011yescharacteristic
635C2×C4C7×#12011yescharacteristic
636C2×C4C22×C7×D412yes—
637C8—42no—
638C8—42no—
639C7#1832211yescharacteristic, direct product factor, Sylow
640C22—21no—
641C22—21no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—41no—
650C22—41no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—84no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22C2×C7×#12011yescharacteristic
664C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
665C22C7×#57511yescharacteristic
666C4—22no—
667C4—21no—
668C4—21no—
669C4—42no—
670C4—41no—
671C4—42no—
672C4—41no—
673C4—42no—
674C4—41no—
675C4—42no—
676C4—81no—
677C4—81no—
678C2—21no—
679C2—21no—
680C2—42no—
681C2—42no—
682C2—82no—
683C2—161no—
684C2C7×#300611yescharacteristic
685C2C7×#301811yescharacteristic
686C2C7×#302611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3459C2semidirect
2. C7×#3194C2semidirect
3. C7×#2543C2semidirect
4. C7×#1609C2semidirect
5. C7×#1608C2semidirect
6. C7×#1607C2semidirect
7. C7×#1597C2semidirect
8. C7×#1596C2semidirect
9. C7×#1595C2semidirect
10. C7×#1504C2semidirect
11. C7×#1465C2semidirect
24. C2×C7×#121C22semidirect
27. C7×#483C22semidirect
32. C7×#450C22semidirect
35. C7×#407C22semidirect
36. C7×#398C22semidirect
38. C7×#394C22semidirect
43. C2×C7×#95C22semidirect
44. C7×#328C22semidirect
45. C7×#328C22semidirect
48. C7×#324C22semidirect
51. #18322C7direct (N × H)
52. C2×C7×#41D4semidirect
82. C7×C22⋊Q8D4semidirect
91. C7×#121D4semidirect
106. C7×#117D4semidirect
136. #3459C2×C7semidirect
137. #3194C2×C7semidirect
138. #2543C2×C7semidirect
139. #1609C2×C7semidirect
140. #1608C2×C7semidirect
141. #1607C2×C7semidirect
142. #1597C2×C7semidirect
143. #1596C2×C7semidirect
144. #1595C2×C7semidirect
145. #1504C2×C7semidirect
146. #1465C2×C7semidirect
248. C2×#121C22×C7semidirect
251. #483C22×C7semidirect
256. #450C22×C7semidirect
259. #407C22×C7semidirect
260. #398C22×C7semidirect
262. #394C22×C7semidirect
267. C2×#95C22×C7semidirect
268. #328C22×C7semidirect
269. #328C22×C7semidirect
272. #324C22×C7semidirect
347. C2×#41C7×D4semidirect
377. C22⋊Q8C7×D4semidirect
386. #121C7×D4semidirect
401. #117C7×D4semidirect
639. C7#18322direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34592
C7×#31942
C7×#25432
C7×#16092
C7×#16082
C7×#16072
C7×#15972
C7×#15962
C7×#15952
C7×#15042
C7×#14652
#183227