Finite Group Databasebeta

← C7×#18296 #13195512

Subgroups — C7×#18296 #13195512

Order 1792.

1162 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#538C211yesmaximal, characteristic, complemented
2C7×#3187C211yesmaximal, characteristic, complemented
3C7×#3186C211yesmaximal, characteristic, complemented
4C7×#2617C211yesmaximal, characteristic, complemented
5C7×#1612C214yesmaximal, complemented
6C7×#1600C214yesmaximal, complemented
7C7×#1520C211yesmaximal, characteristic, complemented
8C7×#1519C211yesmaximal, characteristic, complemented
9C2×C7×#323C211yesmaximal, 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×C22⋊Q8C2211yescharacteristic
18C2×C7×#121—21no—
19C2×C7×#121C2211yescharacteristic
20C2×C7×#121C2211yescharacteristic
21C2×C7×#120—21no—
22C2×C7×#120—21no—
23C2×C4×C7×D4—21no—
24C22×C7×C22⋊C4C2211yescharacteristic
25C7×#483C2212yescomplemented
26C7×#481—21no—
27C7×#480—21no—
28C7×#480C2212yescomplemented
29C7×#466—22no—
30C7×#465—22no—
31C7×#450C2212yescomplemented
32C7×#448—21no—
33C7×#447—21no—
34C7×#447C2212yescomplemented
35C2×C7×C22⋊C8C2211yescharacteristic
36C2×C7×C22⋊C8C2211yescharacteristic
37C7×#397C2212yescomplemented
38C7×#394—22no—
39C7×#392C2212yescomplemented
40C7×#386—21no—
41C7×#380C2211yescharacteristic
42C7×#328C2214yescomplemented
43C7×#327C2214yescomplemented
44C7×#323C2214yescomplemented
45#18296C711yesmaximal, 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—
60C22×C7×C8—21no—
61C22×C7×C8—21no—
62C7×#126—24no—
63C7×#126—24no—
64C7×#124—24no—
65C7×#124—24no—
66C7×#123—24no—
67C7×#123—24no—
68C7×C22⋊Q8—21no—
69C7×C22⋊Q8—21no—
70C7×C22⋊Q8D412yescomplemented
71C7×C22⋊Q8C2312yes—
72C7×#121—24no—
73C7×#121—24no—
74C7×#121—21no—
75C7×#121—21no—
76C7×#121—21no—
77C7×#121—21no—
78C7×#121—21no—
79C7×#121—21no—
80C7×#121D412yescomplemented
81C7×#121C2312yes—
82C7×#120—24no—
83C7×#120—24no—
84C7×#120—44no—
85C4×C7×D4—24no—
86C4×C7×D4—24no—
87C2×C7×C4⋊C4—21no—
88C2×C7×C4⋊C4—21no—
89C2×C7×C4⋊C4—21no—
90C2×C7×C4⋊C4D411yescharacteristic, complemented
91C2×C7×C4⋊C4D411yescharacteristic, complemented
92C2×C7×C4⋊C4C2311yescharacteristic
93C2×C7×C22⋊C4—21no—
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—22no—
104C2×C7×C22⋊C4—21no—
105C2×C7×C22⋊C4—22no—
106C2×C7×C22⋊C4D412yes—
107C2×C7×C22⋊C4C2312yes—
108C2×C42×C7—21no—
109C7×#107—22no—
110C7×#106—22no—
111C7×#103—22no—
112C7×#102—22no—
113C7×#102—22no—
114C7×#102—22no—
115C7×C22⋊C8—21no—
116C7×C22⋊C8—21no—
117C7×C22⋊C8C2312yes—
118C7×C22⋊C8C2312yes—
119C7×#95—22no—
120C7×#95C2312yes—
121C2×#538C2×C711yescharacteristic, complemented
122#3187C2×C711yescharacteristic, complemented
123#3186C2×C711yescharacteristic, complemented
124#2617C2×C711yescharacteristic, complemented
125#1612C2×C714yescomplemented
126#1600C2×C714yescomplemented
127#1520C2×C711yescharacteristic, complemented
128#1519C2×C711yescharacteristic, complemented
129C2×#323C2×C711yescharacteristic, complemented
130C24×C7—21no—
131C24×C7—21no—
132C24×C7—21no—
133C24×C7—22no—
134C24×C7—22no—
135C24×C7—21no—
136C24×C7—81no—
137C24×C7—81no—
138C24×C7C2×D412yes—
139C24×C7C2×D411yescharacteristic
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—42no—
151C2×C7×D4—42no—
152C2×C7×D4—44no—
153C2×C7×D4—41no—
154C2×C7×D4—41no—
155C2×C7×D4—42no—
156C2×C7×D4—42no—
157C2×C7×D4—81no—
158C2×C7×D4—81no—
159C2×C7×D4—82no—
160C2×C7×D4—82no—
161C2×C7×D4—82no—
162C2×C7×D4—82no—
163C2×C7×D4C2×D412yes—
164C22×C4×C7—21no—
165C22×C4×C7—21no—
166C22×C4×C7—21no—
167C22×C4×C7—21no—
168C22×C4×C7—24no—
169C22×C4×C7—21no—
170C22×C4×C7—21no—
171C22×C4×C7—22no—
172C22×C4×C7—81no—
173C22×C4×C7—81no—
174C22×C4×C7—82no—
175C22×C4×C7C2×D412yes—
176C22×C4×C7C2×D411yescharacteristic
177C22×C4×C7C2×D411yescharacteristic
178C22×C4×C7C2×D411yescharacteristic
179C22×C4×C7C2×D411yescharacteristic
180C22×C4×C7C2×D411yescharacteristic
181C22×C4×C7C2411yescharacteristic
182C7×SD16—82no—
183C7×D8—82no—
184C2×C7×C8—22no—
185C2×C7×C8—22no—
186C2×C7×C8—81no—
187C2×C7×C8—81no—
188C7×C4⋊C4—24no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—21no—
194C7×C4⋊C4—21no—
195C7×C4⋊C4—81no—
196C7×C4⋊C4C2×D412yes—
197C7×C4⋊C4C2×D412yes—
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—42no—
211C7×C22⋊C4—41no—
212C7×C22⋊C4—41no—
213C7×C22⋊C4—82no—
214C42×C7—24no—
215#538—22no—
216#538—22no—
217#538—22no—
218#538C22×C714yes—
219C2×#126—21no—
220C2×#124—21no—
221C2×#123—21no—
222C2×C22⋊Q8C22×C711yescharacteristic
223C2×#121—21no—
224C2×#121C22×C711yescharacteristic
225C2×#121C22×C711yescharacteristic
226C2×#120—21no—
227C2×#120—21no—
228C2×C4×D4—21no—
229C22×C22⋊C4C22×C711yescharacteristic
230#483C22×C712yescomplemented
231#481—21no—
232#480—21no—
233#480C22×C712yescomplemented
234#466—22no—
235#465—22no—
236#450C22×C712yescomplemented
237#448—21no—
238#447—21no—
239#447C22×C712yescomplemented
240C2×C22⋊C8C22×C711yescharacteristic
241C2×C22⋊C8C22×C711yescharacteristic
242#397C22×C712yescomplemented
243#394—22no—
244#392C22×C712yescomplemented
245#386—21no—
246#380C22×C711yescharacteristic
247#328C22×C714yescomplemented
248#327C22×C714yescomplemented
249#323C22×C714yescomplemented
250C23×C7—22no—
251C23×C7—22no—
252C23×C7—22no—
253C23×C7—22no—
254C23×C7—41no—
255C23×C7—41no—
256C23×C7—42no—
257C23×C7—42no—
258C23×C7—42no—
259C23×C7—42no—
260C23×C7—42no—
261C23×C7—42no—
262C23×C7—42no—
263C23×C7—42no—
264C23×C7—42no—
265C23×C7—42no—
266C23×C7—42no—
267C23×C7—41no—
268C23×C7—41no—
269C23×C7—41no—
270C23×C7—81no—
271C23×C7—81no—
272C23×C7—84no—
273C23×C7—84no—
274C23×C7#12011yescharacteristic
275C23×C7#12011yescharacteristic
276C23×C7#13512yes—
277C23×C7AGL(1,ℤ/8)12yes—
278C23×C7C22×D411yescharacteristic
279C7×Q8—42no—
280C7×Q8—81no—
281C7×D4—42no—
282C7×D4—42no—
283C7×D4—42no—
284C7×D4—42no—
285C7×D4—42no—
286C7×D4—44no—
287C7×D4—81no—
288C7×D4—84no—
289C7×D4—84no—
290C7×D4—88no—
291C7×D4—82no—
292C7×D4—82no—
293C2×C4×C7—22no—
294C2×C4×C7—22no—
295C2×C4×C7—22no—
296C2×C4×C7—22no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—42no—
304C2×C4×C7—42no—
305C2×C4×C7—42no—
306C2×C4×C7—42no—
307C2×C4×C7—42no—
308C2×C4×C7—42no—
309C2×C4×C7—42no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7—42no—
313C2×C4×C7—41no—
314C2×C4×C7—81no—
315C2×C4×C7—84no—
316C2×C4×C7#12011yescharacteristic
317C2×C4×C7#12011yescharacteristic
318C2×C4×C7C22×D411yescharacteristic
319C2×C4×C7C22×D411yescharacteristic
320C7×C8—42no—
321C7×C8—42no—
322C25C7×D411yescharacteristic
323C22×Q8C7×D411yescharacteristic, complemented
324C22×D4—21no—
325C22×D4—21no—
326C22×D4—21no—
327C22×D4—21no—
328C22×D4—22no—
329C22×D4C7×D411yescharacteristic, complemented
330C22×D4C23×C711yescharacteristic
331C23×C4—21no—
332C23×C4C7×D411yescharacteristic
333C23×C4C23×C711yescharacteristic
334C2×SD16—22no—
335C2×D8—22no—
336C22×C8—21no—
337C22×C8—21no—
338#126—24no—
339#126—24no—
340#124—24no—
341#124—24no—
342#123—24no—
343#123—24no—
344C22⋊Q8—21no—
345C22⋊Q8—21no—
346C22⋊Q8C7×D412yescomplemented
347C22⋊Q8C23×C712yes—
348#121—24no—
349#121—24no—
350#121—21no—
351#121—21no—
352#121—21no—
353#121—21no—
354#121—21no—
355#121—21no—
356#121C7×D412yescomplemented
357#121C23×C712yes—
358#120—24no—
359#120—24no—
360#120—44no—
361C4×D4—24no—
362C4×D4—24no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4C7×D411yescharacteristic, complemented
367C2×C4⋊C4C7×D411yescharacteristic, complemented
368C2×C4⋊C4C23×C711yescharacteristic
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—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4C7×D412yes—
383C2×C22⋊C4C23×C712yes—
384C2×C42—21no—
385#107—22no—
386#106—22no—
387#103—22no—
388#102—22no—
389#102—22no—
390#102—22no—
391C22⋊C8—21no—
392C22⋊C8—21no—
393C22⋊C8C23×C712yes—
394C22⋊C8C23×C712yes—
395#95—22no—
396#95C23×C712yes—
397C22×C7—21no—
398C22×C7—21no—
399C22×C7—21no—
400C22×C7—21no—
401C22×C7—21no—
402C22×C7—21no—
403C22×C7—42no—
404C22×C7—42no—
405C22×C7—44no—
406C22×C7—42no—
407C22×C7—42no—
408C22×C7—42no—
409C22×C7—42no—
410C22×C7—42no—
411C22×C7—42no—
412C22×C7—82no—
413C22×C7—84no—
414C22×C7—84no—
415C22×C7—84no—
416C22×C7—82no—
417C22×C7—82no—
418C22×C7—81no—
419C22×C7—81no—
420C22×C7—81no—
421C22×C7#44912yes—
422C22×C7#44912yes—
423C22×C7C2×#12011yescharacteristic
424C22×C7C2×#13511yescharacteristic
425C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
426C4×C7—21no—
427C4×C7—21no—
428C4×C7—21no—
429C4×C7—21no—
430C4×C7—42no—
431C4×C7—42no—
432C4×C7—42no—
433C4×C7—42no—
434C4×C7—82no—
435C4×C7—81no—
436C24—21no—
437C24—21no—
438C24—21no—
439C24—22no—
440C24—22no—
441C24—21no—
442C24—81no—
443C24—81no—
444C24C2×C7×D412yes—
445C24C2×C7×D411yescharacteristic
446C2×Q8—21no—
447C2×Q8—81no—
448C2×Q8C2×C7×D412yes—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—24no—
452C2×D4—24no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—44no—
459C2×D4—41no—
460C2×D4—41no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—81no—
464C2×D4—81no—
465C2×D4—82no—
466C2×D4—82no—
467C2×D4—82no—
468C2×D4—82no—
469C2×D4C2×C7×D412yes—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—24no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—81no—
479C22×C4—81no—
480C22×C4—82no—
481C22×C4C2×C7×D412yes—
482C22×C4C2×C7×D411yescharacteristic
483C22×C4C2×C7×D411yescharacteristic
484C22×C4C2×C7×D411yescharacteristic
485C22×C4C2×C7×D411yescharacteristic
486C22×C4C2×C7×D411yescharacteristic
487C22×C4C24×C711yescharacteristic
488SD16—82no—
489D8—82no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—81no—
493C2×C8—81no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—81no—
502C4⋊C4C2×C7×D412yes—
503C4⋊C4C2×C7×D412yes—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—44no—
516C22⋊C4—42no—
517C22⋊C4—41no—
518C22⋊C4—41no—
519C22⋊C4—82no—
520C42—24no—
521C2×C7—21no—
522C2×C7—21no—
523C2×C7—22no—
524C2×C7—22no—
525C2×C7—82no—
526C2×C7—82no—
527C2×C7—161no—
528C2×C7C2×#44911yescharacteristic
529C2×C7#301911yescharacteristic
530C2×C7#302611yescharacteristic
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—41no—
536C23—41no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—81no—
552C23—81no—
553C23—84no—
554C23—84no—
555C23C7×#12011yescharacteristic
556C23C7×#12011yescharacteristic
557C23C7×#13512yes—
558C23C7×AGL(1,ℤ/8)12yes—
559C23C22×C7×D411yescharacteristic
560Q8—42no—
561Q8—81no—
562D4—42no—
563D4—42no—
564D4—42no—
565D4—42no—
566D4—42no—
567D4—44no—
568D4—81no—
569D4—84no—
570D4—84no—
571D4—88no—
572D4—82no—
573D4—82no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—81no—
596C2×C4—84no—
597C2×C4C7×#12011yescharacteristic
598C2×C4C7×#12011yescharacteristic
599C2×C4C22×C7×D411yescharacteristic
600C2×C4C22×C7×D411yescharacteristic
601C8—42no—
602C8—42no—
603C7#1829611yescharacteristic, direct product factor, Sylow
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—82no—
620C22—84no—
621C22—84no—
622C22—84no—
623C22—82no—
624C22—82no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22C7×#44912yes—
629C22C7×#44912yes—
630C22C2×C7×#12011yescharacteristic
631C22C2×C7×#13511yescharacteristic
632C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
633C4—21no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—82no—
642C4—81no—
643C2—21no—
644C2—21no—
645C2—22no—
646C2—22no—
647C2—82no—
648C2—82no—
649C2—161no—
650C2C2×C7×#44911yescharacteristic
651C2C7×#301911yescharacteristic
652C2C7×#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×#3187C2semidirect
3. C7×#3186C2semidirect
4. C7×#2617C2semidirect
5. C7×#1612C2semidirect
6. C7×#1600C2semidirect
7. C7×#1520C2semidirect
8. C7×#1519C2semidirect
9. C2×C7×#323C2semidirect
25. C7×#483C22semidirect
28. C7×#480C22semidirect
31. C7×#450C22semidirect
34. C7×#447C22semidirect
37. C7×#397C22semidirect
39. C7×#392C22semidirect
42. C7×#328C22semidirect
43. C7×#327C22semidirect
44. C7×#323C22semidirect
45. #18296C7direct (N × H)
47. C22×C7×Q8D4semidirect
53. C22×C7×D4D4semidirect
70. C7×C22⋊Q8D4semidirect
80. C7×#121D4semidirect
90. C2×C7×C4⋊C4D4semidirect
91. C2×C7×C4⋊C4D4semidirect
121. C2×#538C2×C7semidirect
122. #3187C2×C7semidirect
123. #3186C2×C7semidirect
124. #2617C2×C7semidirect
125. #1612C2×C7semidirect
126. #1600C2×C7semidirect
127. #1520C2×C7semidirect
128. #1519C2×C7semidirect
129. C2×#323C2×C7semidirect
230. #483C22×C7semidirect
233. #480C22×C7semidirect
236. #450C22×C7semidirect
239. #447C22×C7semidirect
242. #397C22×C7semidirect
244. #392C22×C7semidirect
247. #328C22×C7semidirect
248. #327C22×C7semidirect
249. #323C22×C7semidirect
323. C22×Q8C7×D4semidirect
329. C22×D4C7×D4semidirect
346. C22⋊Q8C7×D4semidirect
356. #121C7×D4semidirect
366. C2×C4⋊C4C7×D4semidirect
367. C2×C4⋊C4C7×D4semidirect
603. C7#18296direct (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×#31872
C7×#31862
C7×#26172
C7×#16122
C7×#16002
C7×#15202
C7×#15192
C2×C7×#3232
#182967