Finite Group Databasebeta

← C7×#24014 #13201230

Subgroups — C7×#24014 #13201230

Order 1792.

1276 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#536C211yesmaximal, characteristic
2C2×C7×#484C211yesmaximal, characteristic, complemented
3C2×C7×#483C211yesmaximal, characteristic, complemented
4C7×#2311C211yesmaximal, characteristic
5C7×#2036C212yesmaximal, complemented
6C7×#2035C212yesmaximal, complemented
7C7×#1873C211yesmaximal, characteristic, complemented
8C7×#1873C211yesmaximal, characteristic, complemented
9C7×#1854C212yesmaximal, complemented
10C7×#1787C212yesmaximal, complemented
11C7×#1780C211yesmaximal, characteristic
12C22×C7×#41C2211yescharacteristic
13C23×C7×Q8C2211yescharacteristic
14C7×#536—24no—
15C7×#536—24no—
16C2×C7×C4⋊Q8—21no—
17C2×C7×#124—21no—
18C2×C7×#123—21no—
19C2×C7×#123—21no—
20C2×C7×C22⋊Q8C2211yescharacteristic
21C2×C7×C22⋊Q8C2211yescharacteristic
22C2×C7×C22⋊Q8C2211yescharacteristic
23C2×C7×C22⋊Q8C2211yescharacteristic
24C2×C7×#117C2211yescharacteristic
25C2×C7×#117C2211yescharacteristic
26C22×C7×C4⋊C4C2211yescharacteristic
27C7×#484—24no—
28C7×#483—24no—
29C2×C7×#107C2212yes—
30C2×C7×#106C2212yes—
31C2×C7×#103C2212yes—
32C2×C7×#103C2212yes—
33C2×C7×#103C2212yes—
34C2×C7×#103C2212yes—
35C2×C7×C22⋊C8C2212yes—
36C2×C7×C22⋊C8C2211yescharacteristic
37C2×C7×C22⋊C8C2211yescharacteristic
38C2×C7×C22⋊C8C2212yes—
39C7×#391—21no—
40C7×#390—21no—
41C7×#386—21no—
42C7×#386—21no—
43C7×#380C2212yes—
44C7×#380C2212yes—
45C7×#340C2212yes—
46C7×#340C2212yes—
47#24014C711yesmaximal, characteristic, direct product factor, Sylow
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41—24no—
51C2×C7×#41—42no—
52C22×C7×Q8—22no—
53C22×C7×Q8—22no—
54C22×C7×Q8—21no—
55C22×C7×Q8—24no—
56C22×C7×Q8—42no—
57C22×C7×Q8D411yescharacteristic
58C22×C7×Q8C2311yescharacteristic
59C22×C7×Q8C2311yescharacteristic
60C22×C7×D4D411yescharacteristic
61C22×C7×D4D411yescharacteristic
62C22×C7×D4D411yescharacteristic
63C23×C4×C7—21no—
64C23×C4×C7—21no—
65C23×C4×C7C2311yescharacteristic
66C23×C4×C7C2311yescharacteristic
67C23×C4×C7C2311yescharacteristic
68C22×C7×C8C2312yes—
69C22×C7×C8C2312yes—
70C7×C4⋊Q8—28no—
71C7×#124—28no—
72C7×#123—28no—
73C7×#123—28no—
74C7×C22⋊Q8—22no—
75C7×C22⋊Q8—22no—
76C7×C22⋊Q8—22no—
77C7×C22⋊Q8—22no—
78C7×C22⋊Q8—22no—
79C7×C22⋊Q8—22no—
80C7×C22⋊Q8D414yes—
81C7×C22⋊Q8D414yes—
82C7×#117—22no—
83C7×#117—22no—
84C7×#117—42no—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4—21no—
87C2×C7×C4⋊C4—21no—
88C2×C7×C4⋊C4—21no—
89C2×C7×C4⋊C4—22no—
90C2×C7×C4⋊C4—22no—
91C2×C7×C4⋊C4—21no—
92C2×C7×C4⋊C4C2312yes—
93C2×C7×C4⋊C4C2312yes—
94C2×C7×C4⋊C4C2311yescharacteristic
95C2×C7×C4⋊C4C2311yescharacteristic
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—22no—
103C2×C7×C22⋊C4—22no—
104C2×C42×C7—21no—
105C2×C42×C7—21no—
106C7×#107—24no—
107C7×#106—24no—
108C7×#103—24no—
109C7×#103—24no—
110C7×#103—42no—
111C7×#103—42no—
112C7×C22⋊C8—22no—
113C7×C22⋊C8—22no—
114C7×C22⋊C8—24no—
115C7×C22⋊C8—42no—
116C7×#95—22no—
117C7×#95—22no—
118C2×#536C2×C711yescharacteristic
119C2×#484C2×C711yescharacteristic, complemented
120C2×#483C2×C711yescharacteristic, complemented
121#2311C2×C711yescharacteristic
122#2036C2×C712yescomplemented
123#2035C2×C712yescomplemented
124#1873C2×C711yescharacteristic, complemented
125#1873C2×C711yescharacteristic, complemented
126#1854C2×C712yescomplemented
127#1787C2×C712yescomplemented
128#1780C2×C711yescharacteristic
129C24×C7C2×D411yescharacteristic
130C24×C7C2×D411yescharacteristic
131C24×C7C2×D411yescharacteristic
132C7×#41—48no—
133C7×#41—48no—
134C2×C7×Q8—24no—
135C2×C7×Q8—22no—
136C2×C7×Q8—22no—
137C2×C7×Q8—21no—
138C2×C7×Q8—21no—
139C2×C7×Q8—22no—
140C2×C7×Q8—22no—
141C2×C7×Q8—22no—
142C2×C7×Q8—22no—
143C2×C7×Q8—41no—
144C2×C7×Q8—48no—
145C2×C7×Q8—48no—
146C2×C7×Q8—42no—
147C2×C7×Q8—42no—
148C2×C7×Q8C2×D412yes—
149C2×C7×Q8C2×D412yes—
150C2×C7×D4—22no—
151C2×C7×D4—22no—
152C2×C7×D4—22no—
153C2×C7×D4—22no—
154C2×C7×D4—22no—
155C2×C7×D4—22no—
156C2×C7×D4—41no—
157C2×C7×D4—41no—
158C2×C7×D4—41no—
159C22×C4×C7—21no—
160C22×C4×C7—21no—
161C22×C4×C7—22no—
162C22×C4×C7—22no—
163C22×C4×C7—22no—
164C22×C4×C7—24no—
165C22×C4×C7—21no—
166C22×C4×C7—21no—
167C22×C4×C7—21no—
168C22×C4×C7—21no—
169C22×C4×C7—21no—
170C22×C4×C7—21no—
171C22×C4×C7—22no—
172C22×C4×C7—22no—
173C22×C4×C7—22no—
174C22×C4×C7—22no—
175C22×C4×C7—42no—
176C22×C4×C7—42no—
177C22×C4×C7—42no—
178C22×C4×C7—82no—
179C22×C4×C7C2×D411yescharacteristic
180C22×C4×C7C2×D411yescharacteristic
181C22×C4×C7C2×D412yes—
182C22×C4×C7C2×D412yes—
183C22×C4×C7C2×D411yescharacteristic
184C22×C4×C7#4114yes—
185C22×C4×C7C2411yescharacteristic
186C2×C7×C8—22no—
187C2×C7×C8—22no—
188C2×C7×C8—22no—
189C2×C7×C8—22no—
190C2×C7×C8—22no—
191C2×C7×C8—22no—
192C7×C4⋊C4—24no—
193C7×C4⋊C4—24no—
194C7×C4⋊C4—24no—
195C7×C4⋊C4—24no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—22no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—22no—
201C7×C4⋊C4—22no—
202C7×C4⋊C4—42no—
203C7×C4⋊C4C2×D412yes—
204C7×C4⋊C4C2×D412yes—
205C7×C22⋊C4—24no—
206C7×C22⋊C4—24no—
207C7×C22⋊C4—24no—
208C7×C22⋊C4—24no—
209C7×C22⋊C4—24no—
210C7×C22⋊C4—42no—
211C7×C22⋊C4—82no—
212C7×C22⋊C4—82no—
213C42×C7—24no—
214C42×C7—42no—
215C22×#41C22×C711yescharacteristic
216C23×Q8C22×C711yescharacteristic
217#536—24no—
218#536—24no—
219C2×C4⋊Q8—21no—
220C2×#124—21no—
221C2×#123—21no—
222C2×#123—21no—
223C2×C22⋊Q8C22×C711yescharacteristic
224C2×C22⋊Q8C22×C711yescharacteristic
225C2×C22⋊Q8C22×C711yescharacteristic
226C2×C22⋊Q8C22×C711yescharacteristic
227C2×#117C22×C711yescharacteristic
228C2×#117C22×C711yescharacteristic
229C22×C4⋊C4C22×C711yescharacteristic
230#484—24no—
231#483—24no—
232C2×#107C22×C712yes—
233C2×#106C22×C712yes—
234C2×#103C22×C712yes—
235C2×#103C22×C712yes—
236C2×#103C22×C712yes—
237C2×#103C22×C712yes—
238C2×C22⋊C8C22×C711yescharacteristic
239C2×C22⋊C8C22×C711yescharacteristic
240C2×C22⋊C8C22×C712yes—
241C2×C22⋊C8C22×C712yes—
242#391—21no—
243#390—21no—
244#386—21no—
245#386—21no—
246#380C22×C712yes—
247#380C22×C712yes—
248#340C22×C712yes—
249#340C22×C712yes—
250C23×C7—21no—
251C23×C7—21no—
252C23×C7—21no—
253C23×C7—21no—
254C23×C7—22no—
255C23×C7—22no—
256C23×C7—22no—
257C23×C7—22no—
258C23×C7—21no—
259C23×C7—21no—
260C23×C7—21no—
261C23×C7—22no—
262C23×C7—22no—
263C23×C7#12012yes—
264C23×C7#12312yes—
265C23×C7C22×D411yescharacteristic
266C7×Q8—24no—
267C7×Q8—24no—
268C7×Q8—44no—
269C7×Q8—48no—
270C7×Q8—42no—
271C7×Q8—42no—
272C7×D4—44no—
273C7×D4—44no—
274C7×D4—44no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—24no—
280C2×C4×C7—24no—
281C2×C4×C7—24no—
282C2×C4×C7—24no—
283C2×C4×C7—22no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—48no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—42no—
296C2×C4×C7—42no—
297C2×C4×C7—42no—
298C2×C4×C7—42no—
299C2×C4×C7—42no—
300C2×C4×C7—42no—
301C2×C4×C7—42no—
302C2×C4×C7—42no—
303C2×C4×C7—42no—
304C2×C4×C7—42no—
305C2×C4×C7—84no—
306C2×C4×C7#12012yes—
307C2×C4×C7#12312yes—
308C2×C4×C7C22×D411yescharacteristic
309C2×C4×C7C22×D411yescharacteristic
310C2×C4×C7C2×#4112yes—
311C2×C4×C7H(5,𝔽2)12yes—
312C7×C8—42no—
313C7×C8—42no—
314C2×#41—22no—
315C2×#41—22no—
316C2×#41—24no—
317C2×#41—42no—
318C22×Q8—22no—
319C22×Q8—22no—
320C22×Q8—21no—
321C22×Q8—24no—
322C22×Q8—42no—
323C22×Q8C7×D411yescharacteristic
324C22×Q8C23×C711yescharacteristic
325C22×Q8C23×C711yescharacteristic
326C22×D4C7×D411yescharacteristic
327C22×D4C7×D411yescharacteristic
328C22×D4C7×D411yescharacteristic
329C23×C4—21no—
330C23×C4—21no—
331C23×C4C23×C711yescharacteristic
332C23×C4C23×C711yescharacteristic
333C23×C4C23×C711yescharacteristic
334C22×C8C23×C712yes—
335C22×C8C23×C712yes—
336C4⋊Q8—28no—
337#124—28no—
338#123—28no—
339#123—28no—
340C22⋊Q8—22no—
341C22⋊Q8—22no—
342C22⋊Q8—22no—
343C22⋊Q8—22no—
344C22⋊Q8—22no—
345C22⋊Q8—22no—
346C22⋊Q8C7×D414yes—
347C22⋊Q8C7×D414yes—
348#117—22no—
349#117—22no—
350#117—42no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4C23×C712yes—
359C2×C4⋊C4C23×C712yes—
360C2×C4⋊C4C23×C711yescharacteristic
361C2×C4⋊C4C23×C711yescharacteristic
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C42—21no—
371C2×C42—21no—
372#107—24no—
373#106—24no—
374#103—24no—
375#103—24no—
376#103—42no—
377#103—42no—
378C22⋊C8—22no—
379C22⋊C8—22no—
380C22⋊C8—24no—
381C22⋊C8—42no—
382#95—22no—
383#95—22no—
384C22×C7—24no—
385C22×C7—22no—
386C22×C7—22no—
387C22×C7—22no—
388C22×C7—22no—
389C22×C7—42no—
390C22×C7—42no—
391C22×C7—42no—
392C22×C7—41no—
393C22×C7—41no—
394C22×C7—42no—
395C22×C7—41no—
396C22×C7—42no—
397C22×C7—41no—
398C22×C7—41no—
399C22×C7C2×#12011yescharacteristic
400C22×C7C2×#12311yescharacteristic
401C22×C7#53411yescharacteristic
402C22×C7#57712yes—
403C22×C7#57812yes—
404C4×C7—22no—
405C4×C7—22no—
406C4×C7—41no—
407C4×C7—42no—
408C4×C7—42no—
409C4×C7—42no—
410C4×C7—42no—
411C4×C7—82no—
412C4×C7—82no—
413C4×C7#54612yes—
414C4×C7#54612yes—
415C24C2×C7×D411yescharacteristic
416C24C2×C7×D411yescharacteristic
417C24C2×C7×D411yescharacteristic
418#41—48no—
419#41—48no—
420C2×Q8—24no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—41no—
430C2×Q8—48no—
431C2×Q8—48no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8C2×C7×D412yes—
435C2×Q8C2×C7×D412yes—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—41no—
443C2×D4—41no—
444C2×D4—41no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—82no—
465C22×C4C2×C7×D411yescharacteristic
466C22×C4C2×C7×D411yescharacteristic
467C22×C4C2×C7×D412yes—
468C22×C4C2×C7×D412yes—
469C22×C4C2×C7×D411yescharacteristic
470C22×C4C7×#4114yes—
471C22×C4C24×C711yescharacteristic
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—42no—
489C4⋊C4C2×C7×D412yes—
490C4⋊C4C2×C7×D412yes—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—42no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C42—24no—
500C42—42no—
501C2×C7—42no—
502C2×C7—42no—
503C2×C7—81no—
504C2×C7#240511yescharacteristic
505C2×C7#301011yescharacteristic
506C2×C7#301111yescharacteristic
507C2×C7#309712yes—
508C2×C7#320112yes—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—22no—
521C23—22no—
522C23C7×#12012yes—
523C23C7×#12312yes—
524C23C22×C7×D411yescharacteristic
525Q8—24no—
526Q8—24no—
527Q8—44no—
528Q8—48no—
529Q8—42no—
530Q8—42no—
531D4—44no—
532D4—44no—
533D4—44no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—48no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—84no—
565C2×C4C7×#12012yes—
566C2×C4C7×#12312yes—
567C2×C4C22×C7×D411yescharacteristic
568C2×C4C22×C7×D411yescharacteristic
569C2×C4C2×C7×#4112yes—
570C2×C4C7×H(5,𝔽2)12yes—
571C8—42no—
572C8—42no—
573C7#2401411yescharacteristic, direct product factor, Sylow
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—24no—
579C22—42no—
580C22—42no—
581C22—41no—
582C22—41no—
583C22—42no—
584C22—41no—
585C22—42no—
586C22—41no—
587C22—41no—
588C22—42no—
589C22C2×C7×#12011yescharacteristic
590C22C2×C7×#12311yescharacteristic
591C22C7×#53411yescharacteristic
592C22C7×#57712yes—
593C22C7×#57812yes—
594C4—22no—
595C4—22no—
596C4—41no—
597C4—42no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—82no—
602C4—82no—
603C4C7×#54612yes—
604C4C7×#54612yes—
605C2—42no—
606C2—42no—
607C2—81no—
608C2C7×#240511yescharacteristic
609C2C7×#301011yescharacteristic
610C2C7×#301111yescharacteristic
611C2C7×#309712yes—
612C2C7×#320112yes—
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC22×C7×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C22×C7×#41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#484C2semidirect
3. C2×C7×#483C2semidirect
5. C7×#2036C2semidirect
6. C7×#2035C2semidirect
7. C7×#1873C2semidirect
8. C7×#1873C2semidirect
9. C7×#1854C2semidirect
10. C7×#1787C2semidirect
47. #24014C7direct (N × H)
119. C2×#484C2×C7semidirect
120. C2×#483C2×C7semidirect
122. #2036C2×C7semidirect
123. #2035C2×C7semidirect
124. #1873C2×C7semidirect
125. #1873C2×C7semidirect
126. #1854C2×C7semidirect
127. #1787C2×C7semidirect
573. C7#24014direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5362
C2×C7×#4842
C2×C7×#4832
C7×#23112
C7×#20362
C7×#20352
C7×#18732
C7×#18542
C7×#17872
C7×#17802
#240147