Finite Group Databasebeta

← C7×#15927 #13193143

Subgroups — C7×#15927 #13193143

Order 1792.

1566 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
1C22×C7×#126C211yesmaximal, characteristic
2C2×C7×#399C211yesmaximal, characteristic
3C2×C7×#398C211yesmaximal, characteristic, complemented
4C2×C7×#382C211yesmaximal, characteristic, complemented
5C7×#1445C211yesmaximal, characteristic, complemented
6C7×#1444C212yesmaximal, complemented
7C7×#1440C212yesmaximal
8C7×#1439C212yesmaximal
9C7×#1437C211yesmaximal, characteristic
10C7×#1436C212yesmaximal, complemented
11C7×#1436C211yesmaximal, characteristic, complemented
12C2×C7×#126—24no—
13C2×C7×#126—24no—
14C2×C7×#126—24no—
15C22×C7×C4⋊C4C2211yescharacteristic
16C22×C7×C4⋊C4C2211yescharacteristic
17C22×C7×C4⋊C4C2212yes—
18C22×C7×C4⋊C4C2211yescharacteristic
19C22×C7×C4⋊C4C2211yescharacteristic
20C22×C7×C4⋊C4C2212yes—
21C22×C7×C22⋊C4C2211yescharacteristic
22C22×C7×C22⋊C4C2211yescharacteristic
23C22×C7×C22⋊C4C2212yes—
24C22×C7×C22⋊C4C2212yes—
25C22×C7×C22⋊C4C2211yescharacteristic
26C22×C42×C7C2211yescharacteristic
27C22×C42×C7C2211yescharacteristic
28C7×#399—24no—
29C7×#398—24no—
30C7×#382—24no—
31C2×C7×#95C2212yes—
32C2×C7×#95C2211yescharacteristic
33C2×C7×#95C2211yescharacteristic
34C2×C7×#95C2212yes—
35C2×C7×#95C2212yes—
36C2×C7×#95C2212yes—
37C2×C7×#95C2212yes—
38C2×C7×#95C2212yes—
39C2×C7×#95C2212yes—
40C2×C7×#95C2212yes—
41#15927C711yesmaximal, characteristic, direct product factor, Sylow
42C25×C7C2311yescharacteristic
43C23×C4×C7C2312yes—
44C23×C4×C7C2311yescharacteristic
45C23×C4×C7C2311yescharacteristic
46C23×C4×C7C2311yescharacteristic
47C23×C4×C7C2311yescharacteristic
48C23×C4×C7C2311yescharacteristic
49C23×C4×C7C2312yes—
50C23×C4×C7C2312yes—
51C23×C4×C7C2312yes—
52C23×C4×C7C2311yescharacteristic
53C7×#126—232no—
54C2×C7×C4⋊C4—22no—
55C2×C7×C4⋊C4—22no—
56C2×C7×C4⋊C4—22no—
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4—22no—
59C2×C7×C4⋊C4—22no—
60C2×C7×C4⋊C4—22no—
61C2×C7×C4⋊C4—22no—
62C2×C7×C4⋊C4—24no—
63C2×C7×C4⋊C4—22no—
64C2×C7×C4⋊C4—24no—
65C2×C7×C4⋊C4—22no—
66C2×C7×C4⋊C4—24no—
67C2×C7×C4⋊C4—42no—
68C2×C7×C4⋊C4—42no—
69C2×C7×C4⋊C4—42no—
70C2×C7×C4⋊C4D414yes—
71C2×C7×C4⋊C4D414yes—
72C2×C7×C22⋊C4—22no—
73C2×C7×C22⋊C4—22no—
74C2×C7×C22⋊C4—22no—
75C2×C7×C22⋊C4—22no—
76C2×C7×C22⋊C4—22no—
77C2×C7×C22⋊C4—22no—
78C2×C7×C22⋊C4—22no—
79C2×C7×C22⋊C4—22no—
80C2×C7×C22⋊C4—24no—
81C2×C7×C22⋊C4—24no—
82C2×C7×C22⋊C4—24no—
83C2×C7×C22⋊C4—24no—
84C2×C7×C22⋊C4—42no—
85C2×C7×C22⋊C4—42no—
86C2×C7×C22⋊C4D414yes—
87C2×C42×C7—22no—
88C2×C42×C7—22no—
89C2×C42×C7—22no—
90C2×C42×C7—22no—
91C2×C42×C7—24no—
92C7×#95—24no—
93C7×#95—24no—
94C7×#95—22no—
95C7×#95—24no—
96C7×#95—22no—
97C7×#95—24no—
98C7×#95—24no—
99C7×#95—24no—
100C7×#95—24no—
101C7×#95—42no—
102C22×#126C2×C711yescharacteristic
103C2×#399C2×C711yescharacteristic
104C2×#398C2×C711yescharacteristic, complemented
105C2×#382C2×C711yescharacteristic, complemented
106#1445C2×C711yescharacteristic, complemented
107#1444C2×C712yescomplemented
108#1440C2×C712yes—
109#1439C2×C712yes—
110#1437C2×C711yescharacteristic
111#1436C2×C712yescomplemented
112#1436C2×C711yescharacteristic, complemented
113C24×C7—21no—
114C24×C7—21no—
115C24×C7—22no—
116C24×C7—22no—
117C24×C7—22no—
118C24×C7—22no—
119C24×C7—22no—
120C24×C7—22no—
121C24×C7C2×D412yes—
122C24×C7C2411yescharacteristic
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—21no—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—21no—
134C22×C4×C7—21no—
135C22×C4×C7—21no—
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
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—22no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—22no—
154C22×C4×C7—22no—
155C22×C4×C7—22no—
156C22×C4×C7—22no—
157C22×C4×C7—22no—
158C22×C4×C7—22no—
159C22×C4×C7—22no—
160C22×C4×C7—22no—
161C22×C4×C7—22no—
162C22×C4×C7—22no—
163C22×C4×C7—22no—
164C22×C4×C7—22no—
165C22×C4×C7—22no—
166C22×C4×C7—22no—
167C22×C4×C7—22no—
168C22×C4×C7—22no—
169C22×C4×C7—22no—
170C22×C4×C7C2×D412yes—
171C22×C4×C7C2×D412yes—
172C22×C4×C7C2×D412yes—
173C22×C4×C7C2×D412yes—
174C22×C4×C7C2×D412yes—
175C22×C4×C7C2×D412yes—
176C22×C4×C7C2×D412yes—
177C22×C4×C7C2×D412yes—
178C22×C4×C7#4114yes—
179C22×C4×C7#4114yes—
180C22×C4×C7#4114yes—
181C7×C4⋊C4—28no—
182C7×C4⋊C4—28no—
183C7×C4⋊C4—28no—
184C7×C4⋊C4—48no—
185C7×C4⋊C4—48no—
186C7×C4⋊C4—44no—
187C7×C22⋊C4—28no—
188C7×C22⋊C4—28no—
189C7×C22⋊C4—28no—
190C7×C22⋊C4—48no—
191C7×C22⋊C4—48no—
192C42×C7—28no—
193C42×C7—28no—
194C2×#126—24no—
195C2×#126—24no—
196C2×#126—24no—
197C22×C4⋊C4C22×C711yescharacteristic
198C22×C4⋊C4C22×C711yescharacteristic
199C22×C4⋊C4C22×C712yes—
200C22×C4⋊C4C22×C711yescharacteristic
201C22×C4⋊C4C22×C711yescharacteristic
202C22×C4⋊C4C22×C712yes—
203C22×C22⋊C4C22×C711yescharacteristic
204C22×C22⋊C4C22×C711yescharacteristic
205C22×C22⋊C4C22×C712yes—
206C22×C22⋊C4C22×C712yes—
207C22×C22⋊C4C22×C711yescharacteristic
208C22×C42C22×C711yescharacteristic
209C22×C42C22×C711yescharacteristic
210#399—24no—
211#398—24no—
212#382—24no—
213C2×#95C22×C712yes—
214C2×#95C22×C711yescharacteristic
215C2×#95C22×C711yescharacteristic
216C2×#95C22×C712yes—
217C2×#95C22×C712yes—
218C2×#95C22×C712yes—
219C2×#95C22×C712yes—
220C2×#95C22×C712yes—
221C2×#95C22×C712yes—
222C2×#95C22×C712yes—
223C23×C7—22no—
224C23×C7—22no—
225C23×C7—22no—
226C23×C7—24no—
227C23×C7—24no—
228C23×C7—41no—
229C23×C7—42no—
230C23×C7—41no—
231C23×C7—41no—
232C23×C7—41no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7—42no—
237C23×C7—42no—
238C23×C7—42no—
239C23×C7—42no—
240C23×C7—42no—
241C23×C7—42no—
242C23×C7—42no—
243C23×C7—42no—
244C23×C7C22×D411yescharacteristic
245C23×C7C22×D411yescharacteristic
246C23×C7C22×D411yescharacteristic
247C23×C7C2×#4112yes—
248C23×C7C2×#4112yes—
249C23×C7C2×#4112yes—
250C23×C7H(5,𝔽2)12yes—
251C23×C72−1+412yes—
252C23×C72−1+412yes—
253C2×C4×C7—22no—
254C2×C4×C7—22no—
255C2×C4×C7—24no—
256C2×C4×C7—24no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—22no—
260C2×C4×C7—22no—
261C2×C4×C7—22no—
262C2×C4×C7—24no—
263C2×C4×C7—24no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—24no—
267C2×C4×C7—24no—
268C2×C4×C7—24no—
269C2×C4×C7—24no—
270C2×C4×C7—24no—
271C2×C4×C7—24no—
272C2×C4×C7—24no—
273C2×C4×C7—24no—
274C2×C4×C7—24no—
275C2×C4×C7—24no—
276C2×C4×C7—24no—
277C2×C4×C7—41no—
278C2×C4×C7—42no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—42no—
283C2×C4×C7—42no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—42no—
289C2×C4×C7—42no—
290C2×C4×C7—42no—
291C2×C4×C7—42no—
292C2×C4×C7—42no—
293C2×C4×C7—42no—
294C2×C4×C7—42no—
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—42no—
306C2×C4×C7—42no—
307C2×C4×C7—42no—
308C2×C4×C7—42no—
309C2×C4×C7—42no—
310C2×C4×C7#12014yes—
311C25C23×C711yescharacteristic
312C23×C4C23×C712yes—
313C23×C4C23×C711yescharacteristic
314C23×C4C23×C711yescharacteristic
315C23×C4C23×C711yescharacteristic
316C23×C4C23×C711yescharacteristic
317C23×C4C23×C711yescharacteristic
318C23×C4C23×C712yes—
319C23×C4C23×C712yes—
320C23×C4C23×C712yes—
321C23×C4C23×C711yescharacteristic
322#126—232no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—42no—
337C2×C4⋊C4—42no—
338C2×C4⋊C4—42no—
339C2×C4⋊C4C7×D414yes—
340C2×C4⋊C4C7×D414yes—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—42no—
354C2×C22⋊C4—42no—
355C2×C22⋊C4C7×D414yes—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—24no—
361#95—24no—
362#95—24no—
363#95—22no—
364#95—24no—
365#95—22no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—42no—
371C22×C7—42no—
372C22×C7—42no—
373C22×C7—42no—
374C22×C7—44no—
375C22×C7—44no—
376C22×C7—82no—
377C22×C7—82no—
378C22×C7—82no—
379C22×C7—82no—
380C22×C7C2×#12011yescharacteristic
381C22×C7#52511yescharacteristic
382C22×C7#52912yes—
383C22×C7#53211yescharacteristic
384C22×C7#53211yescharacteristic
385C22×C7#53511yescharacteristic
386C22×C7#53611yescharacteristic
387C22×C7#53611yescharacteristic
388C22×C7#53912yes—
389C22×C7#54112yes—
390C22×C7#54212yes—
391C22×C7#54612yes—
392C22×C7#54712yes—
393C22×C7#54712yes—
394C22×C7#54712yes—
395C22×C7#54812yes—
396C22×C7#55012yes—
397C22×C7#55112yes—
398C22×C7#55212yes—
399C22×C7#55212yes—
400C22×C7#56312yes—
401C4×C7—24no—
402C4×C7—42no—
403C4×C7—42no—
404C4×C7—42no—
405C4×C7—42no—
406C4×C7—42no—
407C4×C7—44no—
408C4×C7—44no—
409C4×C7—44no—
410C4×C7—82no—
411C24—21no—
412C24—21no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24C2×C7×D412yes—
420C24C24×C711yescharacteristic
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4C2×C7×D412yes—
469C22×C4C2×C7×D412yes—
470C22×C4C2×C7×D412yes—
471C22×C4C2×C7×D412yes—
472C22×C4C2×C7×D412yes—
473C22×C4C2×C7×D412yes—
474C22×C4C2×C7×D412yes—
475C22×C4C2×C7×D412yes—
476C22×C4C7×#4114yes—
477C22×C4C7×#4114yes—
478C22×C4C7×#4114yes—
479C4⋊C4—28no—
480C4⋊C4—28no—
481C4⋊C4—28no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—44no—
485C22⋊C4—28no—
486C22⋊C4—28no—
487C22⋊C4—28no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C42—28no—
491C42—28no—
492C2×C7—82no—
493C2×C7#240611yescharacteristic
494C2×C7#243111yescharacteristic
495C2×C7#243211yescharacteristic
496C2×C7#247912yes—
497C2×C7#251912yes—
498C2×C7#255312yes—
499C2×C7#269812yes—
500C2×C7#272112yes—
501C2×C7#272312yes—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—24no—
506C23—24no—
507C23—41no—
508C23—42no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23C22×C7×D411yescharacteristic
524C23C22×C7×D411yescharacteristic
525C23C22×C7×D411yescharacteristic
526C23C2×C7×#4112yes—
527C23C2×C7×#4112yes—
528C23C2×C7×#4112yes—
529C23C7×H(5,𝔽2)12yes—
530C23C7×2−1+412yes—
531C23C7×2−1+412yes—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4C7×#12014yes—
590C7#1592711yescharacteristic, direct product factor, Sylow
591C22—42no—
592C22—42no—
593C22—42no—
594C22—44no—
595C22—44no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22C2×C7×#12011yescharacteristic
601C22C7×#52511yescharacteristic
602C22C7×#52912yes—
603C22C7×#53211yescharacteristic
604C22C7×#53211yescharacteristic
605C22C7×#53511yescharacteristic
606C22C7×#53611yescharacteristic
607C22C7×#53611yescharacteristic
608C22C7×#53912yes—
609C22C7×#54112yes—
610C22C7×#54212yes—
611C22C7×#54612yes—
612C22C7×#54712yes—
613C22C7×#54712yes—
614C22C7×#54712yes—
615C22C7×#54812yes—
616C22C7×#55012yes—
617C22C7×#55112yes—
618C22C7×#55212yes—
619C22C7×#55212yes—
620C22C7×#56312yes—
621C4—24no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—44no—
628C4—44no—
629C4—44no—
630C4—82no—
631C2—82no—
632C2C7×#240611yescharacteristic
633C2C7×#243111yescharacteristic
634C2C7×#243211yescharacteristic
635C2C7×#247912yes—
636C2C7×#251912yes—
637C2C7×#255312yes—
638C2C7×#269812yes—
639C2C7×#272112yes—
640C2C7×#272312yes—
subgroupG/H
center, socleC24×C7C24
commutator, Frattini, cosocleC24C24×C7
prime radicalC25×C7C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C7 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C7×#398C2semidirect
4. C2×C7×#382C2semidirect
5. C7×#1445C2semidirect
6. C7×#1444C2semidirect
10. C7×#1436C2semidirect
11. C7×#1436C2semidirect
41. #15927C7direct (N × H)
104. C2×#398C2×C7semidirect
105. C2×#382C2×C7semidirect
106. #1445C2×C7semidirect
107. #1444C2×C7semidirect
111. #1436C2×C7semidirect
112. #1436C2×C7semidirect
590. C7#15927direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×#1262
C2×C7×#3992
C2×C7×#3982
C2×C7×#3822
C7×#14452
C7×#14442
C7×#14402
C7×#14392
C7×#14372
C7×#14362
#159277