Finite Group Databasebeta

← C7×#24300 #13201516

Subgroups — C7×#24300 #13201516

Order 1792.

1150 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#533C211yesmaximal, characteristic, complemented
2C2×C7×#476C211yesmaximal, characteristic, complemented
3C2×C7×#474C211yesmaximal, characteristic, complemented
4C4×C7×#121C211yesmaximal, characteristic
5C7×#2059C212yesmaximal, complemented
6C7×#2025C212yesmaximal, complemented
7C7×#1988C211yesmaximal, characteristic, complemented
8C7×#1987C211yesmaximal, characteristic, complemented
9C7×#1875C212yesmaximal
10C7×#1854C212yesmaximal, complemented
11C7×#1797C211yesmaximal, characteristic, complemented
12C7×#533—24no—
13C7×#533—24no—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#125C2212yes—
17C2×C7×C22⋊Q8—21no—
18C2×C7×C22⋊Q8—21no—
19C2×C7×#121C2211yescharacteristic
20C2×C4×C7×Q8—21no—
21C2×C4×C7×Q8—21no—
22C2×C4×C7×D4—21no—
23C2×C4×C7×D4C2211yescharacteristic
24C2×C7×#117C2212yes—
25C22×C42×C7C2211yescharacteristic
26C7×#476—24no—
27C7×#474—24no—
28C2×C7×#107C2212yes—
29C2×C7×#106C2212yes—
30C2×C7×C4⋊C8C2211yescharacteristic
31C2×C7×C4⋊C8C2211yescharacteristic
32C2×C7×#102C2212yes—
33C2×C7×#102C2212yes—
34C2×C7×C22⋊C8C2212yes—
35C7×#390C2211yescharacteristic
36C7×#389C2211yescharacteristic
37C7×#388—21no—
38C7×#386—21no—
39C7×#380C2211yescharacteristic
40C4×C7×C4⋊C4C2211yescharacteristic
41C4×C7×C22⋊C4—21no—
42C7×#340C2212yes—
43C7×#340C2212yes—
44C7×#340C2212yes—
45C7×#340C2212yes—
46C7×#336C2212yes—
47#24300C711yesmaximal, characteristic, direct product factor, Sylow
48C22×C7×Q8—21no—
49C22×C7×Q8—21no—
50C22×C7×D4—21no—
51C22×C7×D4C2311yescharacteristic
52C23×C4×C7—21no—
53C23×C4×C7D411yescharacteristic
54C23×C4×C7D411yescharacteristic
55C23×C4×C7C2311yescharacteristic
56C22×C7×C8C2312yes—
57C22×C7×C8C2312yes—
58C7×C4⋊Q8—22no—
59C7×C4⋊Q8—22no—
60C7×C4⋊Q8D414yescomplemented
61C7×C4⋊Q8D414yescomplemented
62C7×#125—24no—
63C7×#125—24no—
64C7×C22⋊Q8—28no—
65C7×C22⋊Q8—28no—
66C7×#121—22no—
67C7×#121Q814yescomplemented
68C4×C7×Q8—28no—
69C4×C7×Q8—28no—
70C4×C7×D4—22no—
71C4×C7×D4—22no—
72C4×C7×D4—44no—
73C7×#117—24no—
74C7×#117—24no—
75C2×C7×C4⋊C4—22no—
76C2×C7×C4⋊C4—22no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C4⋊C4C2311yescharacteristic
82C2×C7×C4⋊C4C2312yes—
83C2×C7×C4⋊C4C2312yes—
84C2×C7×C4⋊C4C2311yescharacteristic
85C2×C7×C22⋊C4—22no—
86C2×C7×C22⋊C4—21no—
87C2×C7×C22⋊C4—21no—
88C2×C7×C22⋊C4—21no—
89C2×C7×C22⋊C4—21no—
90C2×C7×C22⋊C4—21no—
91C2×C42×C7—22no—
92C2×C42×C7—22no—
93C2×C42×C7—22no—
94C2×C42×C7—21no—
95C2×C42×C7D411yescharacteristic
96C2×C42×C7D411yescharacteristic
97C2×C42×C7C2311yescharacteristic
98C2×C42×C7C2311yescharacteristic
99C7×#107—24no—
100C7×#106—24no—
101C7×C4⋊C8—22no—
102C7×C4⋊C8—22no—
103C7×#102—24no—
104C7×#102—24no—
105C7×C22⋊C8—24no—
106C7×#95—21no—
107C7×#95—21no—
108C7×#95—21no—
109C2×#533C2×C711yescharacteristic, complemented
110C2×#476C2×C711yescharacteristic, complemented
111C2×#474C2×C711yescharacteristic, complemented
112C4×#121C2×C711yescharacteristic
113#2059C2×C712yescomplemented
114#2025C2×C712yescomplemented
115#1988C2×C711yescharacteristic, complemented
116#1987C2×C711yescharacteristic, complemented
117#1875C2×C712yes—
118#1854C2×C712yescomplemented
119#1797C2×C711yescharacteristic, complemented
120C24×C7—21no—
121C24×C7C2×D411yescharacteristic
122C2×C7×Q8—24no—
123C2×C7×Q8—24no—
124C2×C7×Q8—42no—
125C2×C7×Q8—42no—
126C2×C7×Q8—42no—
127C2×C7×Q8—42no—
128C2×C7×D4—24no—
129C2×C7×D4—21no—
130C2×C7×D4—21no—
131C2×C7×D4—21no—
132C2×C7×D4—21no—
133C2×C7×D4—21no—
134C2×C7×D4—42no—
135C2×C7×D4—42no—
136C2×C7×D4C2×Q812yes—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—41no—
151C22×C4×C7—41no—
152C22×C4×C7—42no—
153C22×C4×C7—42no—
154C22×C4×C7—42no—
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×Q812yes—
161C22×C4×C7#4112yes—
162C22×C4×C7C2411yescharacteristic
163C2×C7×C8—22no—
164C2×C7×C8—22no—
165C2×C7×C8—22no—
166C2×C7×C8—22no—
167C2×C7×C8—22no—
168C2×C7×C8—22no—
169C7×C4⋊C4—28no—
170C7×C4⋊C4—28no—
171C7×C4⋊C4—24no—
172C7×C4⋊C4—24no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—42no—
181C7×C4⋊C4C2×D414yes—
182C7×C4⋊C4C2×D414yes—
183C7×C4⋊C4C2×Q812yes—
184C7×C22⋊C4—28no—
185C7×C22⋊C4—24no—
186C7×C22⋊C4—42no—
187C7×C22⋊C4—42no—
188C7×C22⋊C4—42no—
189C7×C22⋊C4—42no—
190C42×C7—28no—
191C42×C7—22no—
192C42×C7—22no—
193C42×C7—21no—
194C42×C7—21no—
195C42×C7—42no—
196C42×C7C2×D412yes—
197C42×C7C2×D412yes—
198#533—24no—
199#533—24no—
200C2×C4⋊Q8C22×C711yescharacteristic
201C2×C4⋊Q8C22×C711yescharacteristic
202C2×#125C22×C712yes—
203C2×C22⋊Q8—21no—
204C2×C22⋊Q8—21no—
205C2×#121C22×C711yescharacteristic
206C2×C4×Q8—21no—
207C2×C4×Q8—21no—
208C2×C4×D4—21no—
209C2×C4×D4C22×C711yescharacteristic
210C2×#117C22×C712yes—
211C22×C42C22×C711yescharacteristic
212#476—24no—
213#474—24no—
214C2×#107C22×C712yes—
215C2×#106C22×C712yes—
216C2×C4⋊C8C22×C711yescharacteristic
217C2×C4⋊C8C22×C711yescharacteristic
218C2×#102C22×C712yes—
219C2×#102C22×C712yes—
220C2×C22⋊C8C22×C712yes—
221#390C22×C711yescharacteristic
222#389C22×C711yescharacteristic
223#388—21no—
224#386—21no—
225#380C22×C711yescharacteristic
226C4×C4⋊C4C22×C711yescharacteristic
227C4×C22⋊C4—21no—
228#340C22×C712yes—
229#340C22×C712yes—
230#340C22×C712yes—
231#340C22×C712yes—
232#336C22×C712yes—
233C23×C7—21no—
234C23×C7—21no—
235C23×C7—21no—
236C23×C7—21no—
237C23×C7—21no—
238C23×C7—21no—
239C23×C7—22no—
240C23×C7—41no—
241C23×C7—41no—
242C23×C7—41no—
243C23×C7—41no—
244C23×C7—41no—
245C23×C7—41no—
246C23×C7C22⋊Q812yes—
247C23×C7C22×D411yescharacteristic
248C7×Q8—48no—
249C7×Q8—48no—
250C7×D4—22no—
251C7×D4—22no—
252C7×D4—48no—
253C7×D4—41no—
254C7×D4—41no—
255C2×C4×C7—21no—
256C2×C4×C7—21no—
257C2×C4×C7—21no—
258C2×C4×C7—21no—
259C2×C4×C7—22no—
260C2×C4×C7—22no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—24no—
266C2×C4×C7—24no—
267C2×C4×C7—22no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—44no—
273C2×C4×C7—42no—
274C2×C4×C7—41no—
275C2×C4×C7—42no—
276C2×C4×C7—41no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—44no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—84no—
287C2×C4×C7#12012yes—
288C2×C4×C7#12012yes—
289C2×C4×C7C22⋊Q812yes—
290C2×C4×C7#13512yes—
291C2×C4×C7AGL(1,ℤ/8)12yes—
292C2×C4×C7C22×D411yescharacteristic
293C2×C4×C7C22×D411yescharacteristic
294C2×C4×C7C22×Q811yescharacteristic
295C2×C4×C7C2×#4111yescharacteristic
296C2×C4×C72−1+411yescharacteristic
297C2×C4×C72−1+411yescharacteristic
298C7×C8—42no—
299C7×C8—42no—
300C22×Q8—21no—
301C22×Q8—21no—
302C22×D4—21no—
303C22×D4C23×C711yescharacteristic
304C23×C4—21no—
305C23×C4C7×D411yescharacteristic
306C23×C4C7×D411yescharacteristic
307C23×C4C23×C711yescharacteristic
308C22×C8C23×C712yes—
309C22×C8C23×C712yes—
310C4⋊Q8—22no—
311C4⋊Q8—22no—
312C4⋊Q8C7×D414yescomplemented
313C4⋊Q8C7×D414yescomplemented
314#125—24no—
315#125—24no—
316C22⋊Q8—28no—
317C22⋊Q8—28no—
318#121—22no—
319#121C7×Q814yescomplemented
320C4×Q8—28no—
321C4×Q8—28no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—44no—
325#117—24no—
326#117—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4C23×C711yescharacteristic
333C2×C4⋊C4C23×C711yescharacteristic
334C2×C4⋊C4C23×C712yes—
335C2×C4⋊C4C23×C712yes—
336C2×C4⋊C4C23×C711yescharacteristic
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42C7×D411yescharacteristic
348C2×C42C7×D411yescharacteristic
349C2×C42C23×C711yescharacteristic
350C2×C42C23×C711yescharacteristic
351#107—24no—
352#106—24no—
353C4⋊C8—22no—
354C4⋊C8—22no—
355#102—24no—
356#102—24no—
357C22⋊C8—24no—
358#95—21no—
359#95—21no—
360#95—21no—
361C22×C7—22no—
362C22×C7—22no—
363C22×C7—22no—
364C22×C7—41no—
365C22×C7—41no—
366C22×C7—41no—
367C22×C7—41no—
368C22×C7—42no—
369C22×C7—42no—
370C22×C7—42no—
371C22×C7—81no—
372C22×C7—81no—
373C22×C7—81no—
374C22×C7—81no—
375C22×C7C2×#12011yescharacteristic
376C22×C7C2×C22⋊Q811yescharacteristic
377C22×C7#53611yescharacteristic
378C22×C7C2×#13511yescharacteristic
379C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
380C22×C7#57511yescharacteristic
381C22×C7#57511yescharacteristic
382C4×C7—22no—
383C4×C7—22no—
384C4×C7—42no—
385C4×C7—41no—
386C4×C7—44no—
387C4×C7—41no—
388C4×C7—44no—
389C4×C7—81no—
390C4×C7—81no—
391C4×C7#44914yes—
392C4×C7#54811yescharacteristic
393C4×C7#54811yescharacteristic
394C4×C7D4×Q811yescharacteristic
395C4×C7D4×Q811yescharacteristic
396C24—21no—
397C24C2×C7×D411yescharacteristic
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×D4—24no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4C2×C7×Q812yes—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4C2×C7×D411yescharacteristic
432C22×C4C2×C7×D411yescharacteristic
433C22×C4C2×C7×D411yescharacteristic
434C22×C4C2×C7×D411yescharacteristic
435C22×C4C2×C7×D411yescharacteristic
436C22×C4C2×C7×Q812yes—
437C22×C4C7×#4112yes—
438C22×C4C24×C711yescharacteristic
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C4⋊C4—28no—
446C4⋊C4—28no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—42no—
457C4⋊C4C2×C7×D414yes—
458C4⋊C4C2×C7×D414yes—
459C4⋊C4C2×C7×Q812yes—
460C22⋊C4—28no—
461C22⋊C4—24no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C42—28no—
467C42—22no—
468C42—22no—
469C42—21no—
470C42—21no—
471C42—42no—
472C42C2×C7×D412yes—
473C42C2×C7×D412yes—
474C2×C7—42no—
475C2×C7—82no—
476C2×C7#240911yescharacteristic
477C2×C7C2×#44911yescharacteristic
478C2×C7#300511yescharacteristic
479C2×C7#307611yescharacteristic
480C2×C7#307711yescharacteristic
481C2×C7#323311yescharacteristic
482C2×C7#323511yescharacteristic
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—22no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23C7×C22⋊Q812yes—
497C23C22×C7×D411yescharacteristic
498Q8—48no—
499Q8—48no—
500D4—22no—
501D4—22no—
502D4—48no—
503D4—41no—
504D4—41no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—84no—
537C2×C4C7×#12012yes—
538C2×C4C7×#12012yes—
539C2×C4C7×C22⋊Q812yes—
540C2×C4C7×#13512yes—
541C2×C4C7×AGL(1,ℤ/8)12yes—
542C2×C4C22×C7×D411yescharacteristic
543C2×C4C22×C7×D411yescharacteristic
544C2×C4C22×C7×Q811yescharacteristic
545C2×C4C2×C7×#4111yescharacteristic
546C2×C4C7×2−1+411yescharacteristic
547C2×C4C7×2−1+411yescharacteristic
548C8—42no—
549C8—42no—
550C7#2430011yescharacteristic, direct product factor, Sylow
551C22—22no—
552C22—22no—
553C22—22no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22C2×C7×#12011yescharacteristic
566C22C2×C7×C22⋊Q811yescharacteristic
567C22C7×#53611yescharacteristic
568C22C2×C7×#13511yescharacteristic
569C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
570C22C7×#57511yescharacteristic
571C22C7×#57511yescharacteristic
572C4—22no—
573C4—22no—
574C4—41no—
575C4—44no—
576C4—41no—
577C4—44no—
578C4—42no—
579C4—81no—
580C4—81no—
581C4C7×#44914yes—
582C4C7×#54811yescharacteristic
583C4C7×#54811yescharacteristic
584C4C7×D4×Q811yescharacteristic
585C4C7×D4×Q811yescharacteristic
586C2—42no—
587C2—82no—
588C2C7×#240911yescharacteristic
589C2C2×C7×#44911yescharacteristic
590C2C7×#300511yescharacteristic
591C2C7×#307611yescharacteristic
592C2C7×#307711yescharacteristic
593C2C7×#323311yescharacteristic
594C2C7×#323511yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C22×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#533C2semidirect
2. C2×C7×#476C2semidirect
3. C2×C7×#474C2semidirect
5. C7×#2059C2semidirect
6. C7×#2025C2semidirect
7. C7×#1988C2semidirect
8. C7×#1987C2semidirect
10. C7×#1854C2semidirect
11. C7×#1797C2semidirect
47. #24300C7direct (N × H)
60. C7×C4⋊Q8D4semidirect
61. C7×C4⋊Q8D4semidirect
67. C7×#121Q8semidirect
109. C2×#533C2×C7semidirect
110. C2×#476C2×C7semidirect
111. C2×#474C2×C7semidirect
113. #2059C2×C7semidirect
114. #2025C2×C7semidirect
115. #1988C2×C7semidirect
116. #1987C2×C7semidirect
118. #1854C2×C7semidirect
119. #1797C2×C7semidirect
312. C4⋊Q8C7×D4semidirect
313. C4⋊Q8C7×D4semidirect
319. #121C7×Q8semidirect
550. C7#24300direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5332
C2×C7×#4762
C2×C7×#4742
C4×C7×#1212
C7×#20592
C7×#20252
C7×#19882
C7×#19872
C7×#18752
C7×#18542
C7×#17972
#243007