Finite Group Databasebeta

← C7×#24045 #13201261

Subgroups — C7×#24045 #13201261

Order 1792.

1410 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#550C211yesmaximal, characteristic
2C2×C7×#481C211yesmaximal, characteristic, complemented
3C2×C7×#417C211yesmaximal, characteristic, complemented
4C7×#2486C211yesmaximal, characteristic, complemented
5C7×#2041C212yesmaximal, complemented
6C7×#2034C212yesmaximal, complemented
7C7×#1895C211yesmaximal, characteristic, complemented
8C7×#1870C211yesmaximal, characteristic, complemented
9C7×#1865C212yesmaximal, complemented
10C7×#1788C212yesmaximal, complemented
11C7×#1788C211yesmaximal, characteristic, complemented
12C22×C7×#41C2211yescharacteristic
13C22×C7×#41C2211yescharacteristic
14C22×C7×M16C2211yescharacteristic
15C7×#550—24no—
16C7×#550—24no—
17C2×C7×#127—21no—
18C2×C7×#127C2211yescharacteristic
19C2×C7×#123—21no—
20C2×C7×#121—21no—
21C2×C7×#121—21no—
22C2×C7×#121C2211yescharacteristic
23C2×C7×#121C2211yescharacteristic
24C2×C4×C7×Q8C2211yescharacteristic
25C2×C4×C7×D4—21no—
26C2×C4×C7×D4C2211yescharacteristic
27C22×C7×C4⋊C4C2211yescharacteristic
28C7×#481—24no—
29C2×C7×#106C2212yes—
30C7×#417—24no—
31C2×C7×#103C2212yes—
32C2×C7×#103C2212yes—
33C2×C7×#102C2212yes—
34C2×C7×#102C2212yes—
35C2×C7×#102C2212yes—
36C2×C7×#102C2212yes—
37C2×C7×C22⋊C8C2211yescharacteristic
38C2×C7×C22⋊C8C2212yes—
39C7×#398—21no—
40C7×#397—21no—
41C7×#389C2212yes—
42C7×#381—21no—
43C7×#380C2212yes—
44C7×#380C2211yescharacteristic
45C7×#340C2212yes—
46C7×#336C2212yes—
47#24045C711yesmaximal, characteristic, direct product factor, Sylow
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41—22no—
51C2×C7×#41—24no—
52C2×C7×#41—24no—
53C2×C7×#41—42no—
54C2×C7×#41—42no—
55C2×C7×#41D414yes—
56C22×C7×Q8C2311yescharacteristic
57C22×C7×D4—21no—
58C22×C7×D4—21no—
59C22×C7×D4—21no—
60C22×C7×D4—21no—
61C22×C7×D4—21no—
62C22×C7×D4—21no—
63C22×C7×D4D411yescharacteristic
64C22×C7×D4C2311yescharacteristic
65C22×C7×D4C2311yescharacteristic
66C23×C4×C7—21no—
67C23×C4×C7—21no—
68C23×C4×C7—21no—
69C23×C4×C7C2311yescharacteristic
70C23×C4×C7C2311yescharacteristic
71C2×C7×M16—22no—
72C2×C7×M16—42no—
73C22×C7×C8C2312yes—
74C22×C7×C8C2312yes—
75C7×#127—28no—
76C7×#127—22no—
77C7×#127—22no—
78C7×#123—44no—
79C7×#121—28no—
80C7×#121—28no—
81C7×#121—22no—
82C7×#121—22no—
83C7×#121—22no—
84C7×#121D414yes—
85C4×C7×Q8—22no—
86C4×C7×Q8—22no—
87C4×C7×D4—28no—
88C4×C7×D4—22no—
89C4×C7×D4—22no—
90C2×C7×C4⋊C4—21no—
91C2×C7×C4⋊C4—22no—
92C2×C7×C4⋊C4—22no—
93C2×C7×C4⋊C4—21no—
94C2×C7×C4⋊C4—22no—
95C2×C7×C4⋊C4D411yescharacteristic
96C2×C7×C4⋊C4C2312yes—
97C2×C7×C4⋊C4C2312yes—
98C2×C7×C4⋊C4C2311yescharacteristic
99C2×C7×C22⋊C4—21no—
100C2×C7×C22⋊C4—21no—
101C2×C7×C22⋊C4—21no—
102C2×C7×C22⋊C4—22no—
103C2×C7×C22⋊C4D411yescharacteristic
104C2×C7×C22⋊C4D411yescharacteristic
105C2×C42×C7—21no—
106C2×C42×C7C2311yescharacteristic
107C7×#106—24no—
108C7×#103—24no—
109C7×#103—24no—
110C7×#102—24no—
111C7×#102—24no—
112C7×#102—42no—
113C7×#102—42no—
114C7×C22⋊C8—22no—
115C7×C22⋊C8—42no—
116C7×#95—22no—
117C7×#95—21no—
118C7×#95—22no—
119C2×#550C2×C711yescharacteristic
120C2×#481C2×C711yescharacteristic, complemented
121C2×#417C2×C711yescharacteristic, complemented
122#2486C2×C711yescharacteristic, complemented
123#2041C2×C712yescomplemented
124#2034C2×C712yescomplemented
125#1895C2×C711yescharacteristic, complemented
126#1870C2×C711yescharacteristic, complemented
127#1865C2×C712yescomplemented
128#1788C2×C712yescomplemented
129#1788C2×C711yescharacteristic, complemented
130C24×C7—21no—
131C24×C7—21no—
132C24×C7C2×D411yescharacteristic
133C24×C7C2×D411yescharacteristic
134C7×#41—28no—
135C7×#41—44no—
136C7×#41—48no—
137C7×#41—48no—
138C7×#41—44no—
139C7×#41—44no—
140C2×C7×Q8—21no—
141C2×C7×Q8—22no—
142C2×C7×Q8C2×D412yes—
143C2×C7×Q8#4114yes—
144C2×C7×D4—24no—
145C2×C7×D4—24no—
146C2×C7×D4—22no—
147C2×C7×D4—24no—
148C2×C7×D4—24no—
149C2×C7×D4—24no—
150C2×C7×D4—24no—
151C2×C7×D4—22no—
152C2×C7×D4—21no—
153C2×C7×D4—21no—
154C2×C7×D4—22no—
155C2×C7×D4—22no—
156C2×C7×D4—22no—
157C2×C7×D4—22no—
158C2×C7×D4—41no—
159C2×C7×D4—44no—
160C2×C7×D4—44no—
161C2×C7×D4—42no—
162C2×C7×D4—42no—
163C2×C7×D4—82no—
164C2×C7×D4—82no—
165C2×C7×D4—82no—
166C2×C7×D4C2×D412yes—
167C2×C7×D4C2×D412yes—
168C22×C4×C7—21no—
169C22×C4×C7—22no—
170C22×C4×C7—22no—
171C22×C4×C7—24no—
172C22×C4×C7—21no—
173C22×C4×C7—24no—
174C22×C4×C7—21no—
175C22×C4×C7—21no—
176C22×C4×C7—22no—
177C22×C4×C7—22no—
178C22×C4×C7—22no—
179C22×C4×C7—22no—
180C22×C4×C7—44no—
181C22×C4×C7—44no—
182C22×C4×C7—42no—
183C22×C4×C7—42no—
184C22×C4×C7—42no—
185C22×C4×C7C2×D411yescharacteristic
186C22×C4×C7C2×D411yescharacteristic
187C22×C4×C7C2×D411yescharacteristic
188C22×C4×C7C2×D412yes—
189C22×C4×C7C2×D412yes—
190C22×C4×C7C2×D411yescharacteristic
191C22×C4×C7C2411yescharacteristic
192C7×M16—44no—
193C2×C7×C8—22no—
194C2×C7×C8—22no—
195C2×C7×C8—22no—
196C2×C7×C8—22no—
197C2×C7×C8—22no—
198C2×C7×C8—22no—
199C7×C4⋊C4—24no—
200C7×C4⋊C4—22no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—22no—
203C7×C4⋊C4—22no—
204C7×C4⋊C4—22no—
205C7×C4⋊C4—22no—
206C7×C4⋊C4—42no—
207C7×C4⋊C4—44no—
208C7×C4⋊C4C2×D412yes—
209C7×C22⋊C4—24no—
210C7×C22⋊C4—22no—
211C7×C22⋊C4—24no—
212C7×C22⋊C4—22no—
213C7×C22⋊C4—42no—
214C7×C22⋊C4—44no—
215C42×C7—24no—
216C42×C7—21no—
217C42×C7—21no—
218C22×#41C22×C711yescharacteristic
219C22×#41C22×C711yescharacteristic
220C22×M16C22×C711yescharacteristic
221#550—24no—
222#550—24no—
223C2×#127—21no—
224C2×#127C22×C711yescharacteristic
225C2×#123—21no—
226C2×#121—21no—
227C2×#121—21no—
228C2×#121C22×C711yescharacteristic
229C2×#121C22×C711yescharacteristic
230C2×C4×Q8C22×C711yescharacteristic
231C2×C4×D4—21no—
232C2×C4×D4C22×C711yescharacteristic
233C22×C4⋊C4C22×C711yescharacteristic
234#481—24no—
235C2×#106C22×C712yes—
236#417—24no—
237C2×#103C22×C712yes—
238C2×#103C22×C712yes—
239C2×#102C22×C712yes—
240C2×#102C22×C712yes—
241C2×#102C22×C712yes—
242C2×#102C22×C712yes—
243C2×C22⋊C8C22×C711yescharacteristic
244C2×C22⋊C8C22×C712yes—
245#398—21no—
246#397—21no—
247#389C22×C712yes—
248#381—21no—
249#380C22×C711yescharacteristic
250#380C22×C712yes—
251#340C22×C712yes—
252#336C22×C712yes—
253C23×C7—22no—
254C23×C7—21no—
255C23×C7—21no—
256C23×C7—22no—
257C23×C7—21no—
258C23×C7—22no—
259C23×C7—22no—
260C23×C7—22no—
261C23×C7—21no—
262C23×C7—21no—
263C23×C7—22no—
264C23×C7—42no—
265C23×C7—42no—
266C23×C7—41no—
267C23×C7—41no—
268C23×C7—41no—
269C23×C7—41no—
270C23×C7—42no—
271C23×C7—42no—
272C23×C7#12012yes—
273C23×C7C22×D411yescharacteristic
274C7×Q8—22no—
275C7×Q8—24no—
276C7×Q8#12314yescomplemented
277C7×D4—22no—
278C7×D4—22no—
279C7×D4—24no—
280C7×D4—44no—
281C7×D4—48no—
282C7×D4—48no—
283C7×D4—48no—
284C7×D4—41no—
285C7×D4—41no—
286C7×D4—42no—
287C7×D4—84no—
288C7×D4—84no—
289C7×D4—84no—
290C2×C4×C7—21no—
291C2×C4×C7—22no—
292C2×C4×C7—22no—
293C2×C4×C7—24no—
294C2×C4×C7—21no—
295C2×C4×C7—22no—
296C2×C4×C7—22no—
297C2×C4×C7—24no—
298C2×C4×C7—21no—
299C2×C4×C7—22no—
300C2×C4×C7—21no—
301C2×C4×C7—22no—
302C2×C4×C7—22no—
303C2×C4×C7—48no—
304C2×C4×C7—48no—
305C2×C4×C7—42no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
308C2×C4×C7—42no—
309C2×C4×C7—42no—
310C2×C4×C7—42no—
311C2×C4×C7—42no—
312C2×C4×C7—42no—
313C2×C4×C7—42no—
314C2×C4×C7—42no—
315C2×C4×C7—84no—
316C2×C4×C7#12012yes—
317C2×C4×C7AGL(1,ℤ/8)12yes—
318C2×C4×C7C22×D411yescharacteristic
319C2×C4×C7C22×D411yescharacteristic
320C2×C4×C7C2×#4112yes—
321C2×C4×C7H(5,𝔽2)12yes—
322C7×C8—42no—
323C7×C8—42no—
324C2×#41—22no—
325C2×#41—22no—
326C2×#41—22no—
327C2×#41—24no—
328C2×#41—24no—
329C2×#41—42no—
330C2×#41—42no—
331C2×#41C7×D414yes—
332C22×Q8C23×C711yescharacteristic
333C22×D4—21no—
334C22×D4—21no—
335C22×D4—21no—
336C22×D4—21no—
337C22×D4—21no—
338C22×D4—21no—
339C22×D4C7×D411yescharacteristic
340C22×D4C23×C711yescharacteristic
341C22×D4C23×C711yescharacteristic
342C23×C4—21no—
343C23×C4—21no—
344C23×C4—21no—
345C23×C4C23×C711yescharacteristic
346C23×C4C23×C711yescharacteristic
347C2×M16—22no—
348C2×M16—42no—
349C22×C8C23×C712yes—
350C22×C8C23×C712yes—
351#127—28no—
352#127—22no—
353#127—22no—
354#123—44no—
355#121—28no—
356#121—28no—
357#121—22no—
358#121—22no—
359#121—22no—
360#121C7×D414yes—
361C4×Q8—22no—
362C4×Q8—22no—
363C4×D4—28no—
364C4×D4—22no—
365C4×D4—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4C7×D411yescharacteristic
372C2×C4⋊C4C23×C712yes—
373C2×C4⋊C4C23×C712yes—
374C2×C4⋊C4C23×C711yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4C7×D411yescharacteristic
380C2×C22⋊C4C7×D411yescharacteristic
381C2×C42—21no—
382C2×C42C23×C711yescharacteristic
383#106—24no—
384#103—24no—
385#103—24no—
386#102—24no—
387#102—24no—
388#102—42no—
389#102—42no—
390C22⋊C8—22no—
391C22⋊C8—42no—
392#95—22no—
393#95—21no—
394#95—22no—
395C22×C7—22no—
396C22×C7—22no—
397C22×C7—22no—
398C22×C7—42no—
399C22×C7—42no—
400C22×C7—42no—
401C22×C7—42no—
402C22×C7—42no—
403C22×C7—41no—
404C22×C7—41no—
405C22×C7—42no—
406C22×C7—42no—
407C22×C7—41no—
408C22×C7—42no—
409C22×C7—42no—
410C22×C7—42no—
411C22×C7—82no—
412C22×C7—82no—
413C22×C7—82no—
414C22×C7—82no—
415C22×C7C2×#12011yescharacteristic
416C22×C7#54612yes—
417C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
418C22×C7#57511yescharacteristic
419C22×C7#57712yes—
420C4×C7—22no—
421C4×C7—24no—
422C4×C7—22no—
423C4×C7—41no—
424C4×C7—41no—
425C4×C7—42no—
426C4×C7—82no—
427C4×C7—82no—
428C4×C7C2×#12311yescharacteristic
429C4×C7#53411yescharacteristic
430C4×C7#54612yes—
431C24—21no—
432C24—21no—
433C24C2×C7×D411yescharacteristic
434C24C2×C7×D411yescharacteristic
435#41—28no—
436#41—44no—
437#41—48no—
438#41—48no—
439#41—44no—
440#41—44no—
441C2×Q8—21no—
442C2×Q8—22no—
443C2×Q8C2×C7×D412yes—
444C2×Q8C7×#4114yes—
445C2×D4—24no—
446C2×D4—24no—
447C2×D4—22no—
448C2×D4—24no—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—24no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—41no—
460C2×D4—44no—
461C2×D4—44no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—82no—
465C2×D4—82no—
466C2×D4—82no—
467C2×D4C2×C7×D412yes—
468C2×D4C2×C7×D412yes—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—21no—
474C22×C4—24no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4C2×C7×D411yescharacteristic
487C22×C4C2×C7×D411yescharacteristic
488C22×C4C2×C7×D411yescharacteristic
489C22×C4C2×C7×D412yes—
490C22×C4C2×C7×D412yes—
491C22×C4C2×C7×D411yescharacteristic
492C22×C4C24×C711yescharacteristic
493M16—44no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4C2×C7×D412yes—
510C22⋊C4—24no—
511C22⋊C4—22no—
512C22⋊C4—24no—
513C22⋊C4—22no—
514C22⋊C4—42no—
515C22⋊C4—44no—
516C42—24no—
517C42—21no—
518C42—21no—
519C2×C7—42no—
520C2×C7—81no—
521C2×C7—82no—
522C2×C7—82no—
523C2×C7#240511yescharacteristic
524C2×C7#300611yescharacteristic
525C2×C7#300911yescharacteristic
526C2×C7#331712yes—
527C2×C7#333212yes—
528C23—22no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—21no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—21no—
537C23—21no—
538C23—22no—
539C23—42no—
540C23—42no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—42no—
546C23—42no—
547C23C7×#12012yes—
548C23C22×C7×D411yescharacteristic
549Q8—22no—
550Q8—24no—
551Q8C7×#12314yescomplemented
552D4—22no—
553D4—22no—
554D4—24no—
555D4—44no—
556D4—48no—
557D4—48no—
558D4—48no—
559D4—41no—
560D4—41no—
561D4—42no—
562D4—84no—
563D4—84no—
564D4—84no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—48no—
579C2×C4—48no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—84no—
591C2×C4C7×#12012yes—
592C2×C4C7×AGL(1,ℤ/8)12yes—
593C2×C4C22×C7×D411yescharacteristic
594C2×C4C22×C7×D411yescharacteristic
595C2×C4C2×C7×#4112yes—
596C2×C4C7×H(5,𝔽2)12yes—
597C8—42no—
598C8—42no—
599C7#2404511yescharacteristic, direct product factor, Sylow
600C22—22no—
601C22—22no—
602C22—22no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—41no—
608C22—41no—
609C22—42no—
610C22—42no—
611C22—41no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22C2×C7×#12011yescharacteristic
621C22C7×#54612yes—
622C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
623C22C7×#57511yescharacteristic
624C22C7×#57712yes—
625C4—22no—
626C4—24no—
627C4—22no—
628C4—41no—
629C4—41no—
630C4—42no—
631C4—82no—
632C4—82no—
633C4C2×C7×#12311yescharacteristic
634C4C7×#53411yescharacteristic
635C4C7×#54612yes—
636C2—42no—
637C2—81no—
638C2—82no—
639C2—82no—
640C2C7×#240511yescharacteristic
641C2C7×#300611yescharacteristic
642C2C7×#300911yescharacteristic
643C2C7×#331712yes—
644C2C7×#333212yes—
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#550C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#481C2semidirect
3. C2×C7×#417C2semidirect
4. C7×#2486C2semidirect
5. C7×#2041C2semidirect
6. C7×#2034C2semidirect
7. C7×#1895C2semidirect
8. C7×#1870C2semidirect
9. C7×#1865C2semidirect
10. C7×#1788C2semidirect
11. C7×#1788C2semidirect
47. #24045C7direct (N × H)
120. C2×#481C2×C7semidirect
121. C2×#417C2×C7semidirect
122. #2486C2×C7semidirect
123. #2041C2×C7semidirect
124. #2034C2×C7semidirect
125. #1895C2×C7semidirect
126. #1870C2×C7semidirect
127. #1865C2×C7semidirect
128. #1788C2×C7semidirect
129. #1788C2×C7semidirect
276. C7×Q8#123semidirect
551. Q8C7×#123semidirect
599. C7#24045direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5502
C2×C7×#4812
C2×C7×#4172
C7×#24862
C7×#20412
C7×#20342
C7×#18952
C7×#18702
C7×#18652
C7×#17882
#240457