Finite Group Databasebeta

← C7×#18328 #13195544

Subgroups — C7×#18328 #13195544

Order 1792.

1192 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#538C211yesmaximal, characteristic, complemented
2C7×#3193C211yesmaximal, characteristic, complemented
3C7×#3192C211yesmaximal, characteristic, complemented
4C7×#2539C211yesmaximal, characteristic, complemented
5C7×#1618C214yesmaximal, complemented
6C7×#1600C214yesmaximal, complemented
7C7×#1508C211yesmaximal, characteristic, complemented
8C7×#1507C211yesmaximal, characteristic, complemented
9C7×#1462C211yesmaximal, characteristic, complemented
10C7×#538—22no—
11C7×#538—22no—
12C7×#538—22no—
13C7×#538C2214yes—
14C2×C7×#126—21no—
15C2×C7×#124—21no—
16C2×C7×#123—21no—
17C2×C7×#123—21no—
18C2×C7×C22⋊Q8C2211yescharacteristic
19C2×C7×#121—21no—
20C2×C7×#121C2211yescharacteristic
21C2×C7×#121C2211yescharacteristic
22C2×C7×#120—21no—
23C2×C7×#120—21no—
24C2×C4×C7×D4—21no—
25C22×C7×C22⋊C4C2211yescharacteristic
26C7×#484C2212yescomplemented
27C7×#481—21no—
28C7×#481C2212yescomplemented
29C7×#480—21no—
30C7×#469—22no—
31C7×#468—22no—
32C7×#450C2212yescomplemented
33C7×#448—21no—
34C7×#447—21no—
35C7×#447C2212yescomplemented
36C7×#407C2211yescharacteristic
37C7×#407C2211yescharacteristic
38C7×#399C2212yescomplemented
39C7×#392C2212yescomplemented
40C7×#390—22no—
41C4×C7×C22⋊C4C2211yescharacteristic
42C7×#328C2214yescomplemented
43C7×#327C2214yescomplemented
44C7×#323C2214yescomplemented
45#18328C711yesmaximal, characteristic, direct product factor, Sylow
46C25×C7D411yescharacteristic
47C22×C7×Q8D411yescharacteristic, complemented
48C22×C7×D4—21no—
49C22×C7×D4—21no—
50C22×C7×D4—21no—
51C22×C7×D4—21no—
52C22×C7×D4—22no—
53C22×C7×D4D411yescharacteristic, complemented
54C22×C7×D4C2311yescharacteristic
55C23×C4×C7—21no—
56C23×C4×C7D411yescharacteristic
57C23×C4×C7C2311yescharacteristic
58C2×C7×SD16—22no—
59C2×C7×D8—22no—
60C2×C7×M16—21no—
61C2×C7×M16—21no—
62C7×#126—24no—
63C7×#126—24no—
64C7×#124—24no—
65C7×#124—24no—
66C7×#123—24no—
67C7×#123—24no—
68C7×#123—44no—
69C7×C22⋊Q8—21no—
70C7×C22⋊Q8—21no—
71C7×C22⋊Q8D412yescomplemented
72C7×C22⋊Q8C2312yes—
73C7×#121—24no—
74C7×#121—24no—
75C7×#121—21no—
76C7×#121—21no—
77C7×#121—21no—
78C7×#121—21no—
79C7×#121—21no—
80C7×#121—21no—
81C7×#121D412yescomplemented
82C7×#121C2312yes—
83C7×#120—24no—
84C7×#120—24no—
85C7×#120—44no—
86C4×C7×D4—24no—
87C4×C7×D4—24no—
88C2×C7×C4⋊C4—21no—
89C2×C7×C4⋊C4—21no—
90C2×C7×C4⋊C4—22no—
91C2×C7×C4⋊C4D411yescharacteristic, complemented
92C2×C7×C4⋊C4D411yescharacteristic, complemented
93C2×C7×C4⋊C4C2311yescharacteristic
94C2×C7×C22⋊C4—21no—
95C2×C7×C22⋊C4—21no—
96C2×C7×C22⋊C4—21no—
97C2×C7×C22⋊C4—21no—
98C2×C7×C22⋊C4—21no—
99C2×C7×C22⋊C4—21no—
100C2×C7×C22⋊C4—21no—
101C2×C7×C22⋊C4—21no—
102C2×C7×C22⋊C4—21no—
103C2×C7×C22⋊C4—21no—
104C2×C7×C22⋊C4—22no—
105C2×C7×C22⋊C4—21no—
106C2×C7×C22⋊C4—22no—
107C2×C7×C22⋊C4D412yes—
108C2×C7×C22⋊C4C2312yes—
109C2×C42×C7—21no—
110C2×C42×C7—21no—
111C7×#107—22no—
112C7×#106—22no—
113C7×#103—22no—
114C7×#102—22no—
115C7×#102—22no—
116C7×#102—22no—
117C7×C22⋊C8—21no—
118C7×C22⋊C8—21no—
119C7×C22⋊C8C2312yes—
120C7×C22⋊C8C2312yes—
121C7×#95C2312yes—
122C2×#538C2×C711yescharacteristic, complemented
123#3193C2×C711yescharacteristic, complemented
124#3192C2×C711yescharacteristic, complemented
125#2539C2×C711yescharacteristic, complemented
126#1618C2×C714yescomplemented
127#1600C2×C714yescomplemented
128#1508C2×C711yescharacteristic, complemented
129#1507C2×C711yescharacteristic, complemented
130#1462C2×C711yescharacteristic, complemented
131C24×C7—21no—
132C24×C7—21no—
133C24×C7—21no—
134C24×C7—21no—
135C24×C7—42no—
136C24×C7—42no—
137C24×C7—81no—
138C24×C7C2×D411yescharacteristic
139C24×C7C2×D412yes—
140C2×C7×Q8—21no—
141C2×C7×Q8—81no—
142C2×C7×Q8C2×D412yes—
143C2×C7×D4—22no—
144C2×C7×D4—22no—
145C2×C7×D4—24no—
146C2×C7×D4—24no—
147C2×C7×D4—21no—
148C2×C7×D4—21no—
149C2×C7×D4—21no—
150C2×C7×D4—44no—
151C2×C7×D4—44no—
152C2×C7×D4—42no—
153C2×C7×D4—42no—
154C2×C7×D4—44no—
155C2×C7×D4—41no—
156C2×C7×D4—41no—
157C2×C7×D4—81no—
158C2×C7×D4—81no—
159C2×C7×D4—82no—
160C2×C7×D4—82no—
161C2×C7×D4—82no—
162C2×C7×D4C2×D412yes—
163C22×C4×C7—21no—
164C22×C4×C7—21no—
165C22×C4×C7—21no—
166C22×C4×C7—21no—
167C22×C4×C7—22no—
168C22×C4×C7—24no—
169C22×C4×C7—21no—
170C22×C4×C7—21no—
171C22×C4×C7—42no—
172C22×C4×C7—81no—
173C22×C4×C7—82no—
174C22×C4×C7C2×D412yes—
175C22×C4×C7C2×D411yescharacteristic
176C22×C4×C7C2×D411yescharacteristic
177C22×C4×C7C2×D411yescharacteristic
178C22×C4×C7C2×D411yescharacteristic
179C22×C4×C7C2×D411yescharacteristic
180C22×C4×C7C2411yescharacteristic
181C7×SD16—82no—
182C7×D8—82no—
183C7×M16—81no—
184C7×M16—81no—
185C2×C7×C8—22no—
186C2×C7×C8—22no—
187C7×C4⋊C4—24no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—22no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4—21no—
194C7×C4⋊C4—44no—
195C7×C4⋊C4C2×D412yes—
196C7×C4⋊C4C2×D412yes—
197C7×C22⋊C4—24no—
198C7×C22⋊C4—24no—
199C7×C22⋊C4—24no—
200C7×C22⋊C4—24no—
201C7×C22⋊C4—22no—
202C7×C22⋊C4—22no—
203C7×C22⋊C4—22no—
204C7×C22⋊C4—22no—
205C7×C22⋊C4—22no—
206C7×C22⋊C4—22no—
207C7×C22⋊C4—22no—
208C7×C22⋊C4—22no—
209C7×C22⋊C4—44no—
210C7×C22⋊C4—44no—
211C7×C22⋊C4—41no—
212C7×C22⋊C4—41no—
213C42×C7—24no—
214C42×C7—81no—
215#538—22no—
216#538—22no—
217#538—22no—
218#538C22×C714yes—
219C2×#126—21no—
220C2×#124—21no—
221C2×#123—21no—
222C2×#123—21no—
223C2×C22⋊Q8C22×C711yescharacteristic
224C2×#121—21no—
225C2×#121C22×C711yescharacteristic
226C2×#121C22×C711yescharacteristic
227C2×#120—21no—
228C2×#120—21no—
229C2×C4×D4—21no—
230C22×C22⋊C4C22×C711yescharacteristic
231#484C22×C712yescomplemented
232#481—21no—
233#481C22×C712yescomplemented
234#480—21no—
235#469—22no—
236#468—22no—
237#450C22×C712yescomplemented
238#448—21no—
239#447—21no—
240#447C22×C712yescomplemented
241#407C22×C711yescharacteristic
242#407C22×C711yescharacteristic
243#399C22×C712yescomplemented
244#392C22×C712yescomplemented
245#390—22no—
246C4×C22⋊C4C22×C711yescharacteristic
247#328C22×C714yescomplemented
248#327C22×C714yescomplemented
249#323C22×C714yescomplemented
250C23×C7—22no—
251C23×C7—22no—
252C23×C7—22no—
253C23×C7—22no—
254C23×C7—22no—
255C23×C7—22no—
256C23×C7—41no—
257C23×C7—42no—
258C23×C7—42no—
259C23×C7—42no—
260C23×C7—42no—
261C23×C7—42no—
262C23×C7—42no—
263C23×C7—42no—
264C23×C7—41no—
265C23×C7—41no—
266C23×C7—42no—
267C23×C7—42no—
268C23×C7—42no—
269C23×C7—41no—
270C23×C7—42no—
271C23×C7—81no—
272C23×C7—81no—
273C23×C7—88no—
274C23×C7#12011yescharacteristic
275C23×C7#12011yescharacteristic
276C23×C7C22×D411yescharacteristic
277C7×Q8—42no—
278C7×Q8—81no—
279C7×D4—42no—
280C7×D4—42no—
281C7×D4—42no—
282C7×D4—44no—
283C7×D4—44no—
284C7×D4—81no—
285C7×D4—84no—
286C7×D4—84no—
287C7×D4—88no—
288C7×D4—82no—
289C7×D4—82no—
290C2×C4×C7—24no—
291C2×C4×C7—22no—
292C2×C4×C7—22no—
293C2×C4×C7—22no—
294C2×C4×C7—22no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7—42no—
302C2×C4×C7—42no—
303C2×C4×C7—42no—
304C2×C4×C7—42no—
305C2×C4×C7—42no—
306C2×C4×C7—42no—
307C2×C4×C7—42no—
308C2×C4×C7—41no—
309C2×C4×C7—41no—
310C2×C4×C7—41no—
311C2×C4×C7—81no—
312C2×C4×C7—84no—
313C2×C4×C7#12011yescharacteristic
314C2×C4×C7#12011yescharacteristic
315C2×C4×C7C22×D411yescharacteristic
316C2×C4×C7C22×D411yescharacteristic
317C7×C8—42no—
318C7×C8—42no—
319C25C7×D411yescharacteristic
320C22×Q8C7×D411yescharacteristic, complemented
321C22×D4—21no—
322C22×D4—21no—
323C22×D4—21no—
324C22×D4—21no—
325C22×D4—22no—
326C22×D4C7×D411yescharacteristic, complemented
327C22×D4C23×C711yescharacteristic
328C23×C4—21no—
329C23×C4C7×D411yescharacteristic
330C23×C4C23×C711yescharacteristic
331C2×SD16—22no—
332C2×D8—22no—
333C2×M16—21no—
334C2×M16—21no—
335#126—24no—
336#126—24no—
337#124—24no—
338#124—24no—
339#123—24no—
340#123—24no—
341#123—44no—
342C22⋊Q8—21no—
343C22⋊Q8—21no—
344C22⋊Q8C7×D412yescomplemented
345C22⋊Q8C23×C712yes—
346#121—24no—
347#121—24no—
348#121—21no—
349#121—21no—
350#121—21no—
351#121—21no—
352#121—21no—
353#121—21no—
354#121C7×D412yescomplemented
355#121C23×C712yes—
356#120—24no—
357#120—24no—
358#120—44no—
359C4×D4—24no—
360C4×D4—24no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4C7×D411yescharacteristic, complemented
365C2×C4⋊C4C7×D411yescharacteristic, complemented
366C2×C4⋊C4C23×C711yescharacteristic
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4C7×D412yes—
381C2×C22⋊C4C23×C712yes—
382C2×C42—21no—
383C2×C42—21no—
384#107—22no—
385#106—22no—
386#103—22no—
387#102—22no—
388#102—22no—
389#102—22no—
390C22⋊C8—21no—
391C22⋊C8—21no—
392C22⋊C8C23×C712yes—
393C22⋊C8C23×C712yes—
394#95C23×C712yes—
395C22×C7—21no—
396C22×C7—21no—
397C22×C7—21no—
398C22×C7—21no—
399C22×C7—21no—
400C22×C7—21no—
401C22×C7—21no—
402C22×C7—44no—
403C22×C7—44no—
404C22×C7—42no—
405C22×C7—42no—
406C22×C7—42no—
407C22×C7—42no—
408C22×C7—42no—
409C22×C7—42no—
410C22×C7—82no—
411C22×C7—84no—
412C22×C7—84no—
413C22×C7—82no—
414C22×C7—82no—
415C22×C7—84no—
416C22×C7—81no—
417C22×C7—81no—
418C22×C7—81no—
419C22×C7#45312yes—
420C22×C7C2×#12011yescharacteristic
421C22×C7#57512yes—
422C4×C7—21no—
423C4×C7—21no—
424C4×C7—21no—
425C4×C7—21no—
426C4×C7—44no—
427C4×C7—42no—
428C4×C7—42no—
429C4×C7—42no—
430C4×C7—42no—
431C4×C7—81no—
432C24—21no—
433C24—21no—
434C24—21no—
435C24—21no—
436C24—42no—
437C24—42no—
438C24—81no—
439C24C2×C7×D411yescharacteristic
440C24C2×C7×D412yes—
441C2×Q8—21no—
442C2×Q8—81no—
443C2×Q8C2×C7×D412yes—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—44no—
452C2×D4—44no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—44no—
456C2×D4—41no—
457C2×D4—41no—
458C2×D4—81no—
459C2×D4—81no—
460C2×D4—82no—
461C2×D4—82no—
462C2×D4—82no—
463C2×D4C2×C7×D412yes—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—42no—
473C22×C4—81no—
474C22×C4—82no—
475C22×C4C2×C7×D412yes—
476C22×C4C2×C7×D411yescharacteristic
477C22×C4C2×C7×D411yescharacteristic
478C22×C4C2×C7×D411yescharacteristic
479C22×C4C2×C7×D411yescharacteristic
480C22×C4C2×C7×D411yescharacteristic
481C22×C4C24×C711yescharacteristic
482SD16—82no—
483D8—82no—
484M16—81no—
485M16—81no—
486C2×C8—22no—
487C2×C8—22no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—44no—
496C4⋊C4C2×C7×D412yes—
497C4⋊C4C2×C7×D412yes—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C42—24no—
515C42—81no—
516C2×C7—22no—
517C2×C7—22no—
518C2×C7—41no—
519C2×C7—82no—
520C2×C7—82no—
521C2×C7—161no—
522C2×C7#300511yescharacteristic
523C2×C7C2×#45311yescharacteristic
524C2×C7#302611yescharacteristic
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—41no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—41no—
545C23—42no—
546C23—81no—
547C23—81no—
548C23—88no—
549C23C7×#12011yescharacteristic
550C23C7×#12011yescharacteristic
551C23C22×C7×D411yescharacteristic
552Q8—42no—
553Q8—81no—
554D4—42no—
555D4—42no—
556D4—42no—
557D4—44no—
558D4—44no—
559D4—81no—
560D4—84no—
561D4—84no—
562D4—88no—
563D4—82no—
564D4—82no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—81no—
587C2×C4—84no—
588C2×C4C7×#12011yescharacteristic
589C2×C4C7×#12011yescharacteristic
590C2×C4C22×C7×D411yescharacteristic
591C2×C4C22×C7×D411yescharacteristic
592C8—42no—
593C8—42no—
594C7#1832811yescharacteristic, direct product factor, Sylow
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—44no—
603C22—44no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—82no—
611C22—84no—
612C22—84no—
613C22—82no—
614C22—82no—
615C22—84no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22C7×#45312yes—
620C22C2×C7×#12011yescharacteristic
621C22C7×#57512yes—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—44no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—81no—
632C2—22no—
633C2—22no—
634C2—41no—
635C2—82no—
636C2—82no—
637C2—161no—
638C2C7×#300511yescharacteristic
639C2C2×C7×#45311yescharacteristic
640C2C7×#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. C2×C7×#538C2semidirect
2. C7×#3193C2semidirect
3. C7×#3192C2semidirect
4. C7×#2539C2semidirect
5. C7×#1618C2semidirect
6. C7×#1600C2semidirect
7. C7×#1508C2semidirect
8. C7×#1507C2semidirect
9. C7×#1462C2semidirect
26. C7×#484C22semidirect
28. C7×#481C22semidirect
32. C7×#450C22semidirect
35. C7×#447C22semidirect
38. C7×#399C22semidirect
39. C7×#392C22semidirect
42. C7×#328C22semidirect
43. C7×#327C22semidirect
44. C7×#323C22semidirect
45. #18328C7direct (N × H)
47. C22×C7×Q8D4semidirect
53. C22×C7×D4D4semidirect
71. C7×C22⋊Q8D4semidirect
81. C7×#121D4semidirect
91. C2×C7×C4⋊C4D4semidirect
92. C2×C7×C4⋊C4D4semidirect
122. C2×#538C2×C7semidirect
123. #3193C2×C7semidirect
124. #3192C2×C7semidirect
125. #2539C2×C7semidirect
126. #1618C2×C7semidirect
127. #1600C2×C7semidirect
128. #1508C2×C7semidirect
129. #1507C2×C7semidirect
130. #1462C2×C7semidirect
231. #484C22×C7semidirect
233. #481C22×C7semidirect
237. #450C22×C7semidirect
240. #447C22×C7semidirect
243. #399C22×C7semidirect
244. #392C22×C7semidirect
247. #328C22×C7semidirect
248. #327C22×C7semidirect
249. #323C22×C7semidirect
320. C22×Q8C7×D4semidirect
326. C22×D4C7×D4semidirect
344. C22⋊Q8C7×D4semidirect
354. #121C7×D4semidirect
364. C2×C4⋊C4C7×D4semidirect
365. C2×C4⋊C4C7×D4semidirect
594. C7#18328direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5382
C7×#31932
C7×#31922
C7×#25392
C7×#16182
C7×#16002
C7×#15082
C7×#15072
C7×#14622
#183287