Finite Group Databasebeta

← C7×#13947 #13191163

Subgroups — C7×#13947 #13191163

Order 1792.

1486 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C7×C22⋊C4C211yesmaximal, characteristic
2C7×#1443C212yesmaximal, complemented
3C7×#1440C212yesmaximal
4C7×#1437C212yesmaximal
5C7×#1434C212yesmaximal, complemented
6C7×#1434C211yesmaximal, characteristic, complemented
7C7×#1434C211yesmaximal, characteristic, complemented
8C7×#1434C211yesmaximal, characteristic, complemented
9C7×C22⋊#95C211yesmaximal, characteristic
10C7×C22⋊#95C211yesmaximal, characteristic
11C4×C7×#95C211yesmaximal, characteristic, complemented
12C24×C4×C7C2211yescharacteristic
13C22×C7×C22⋊C4C2212yes—
14C22×C7×C22⋊C4C2212yes—
15C22×C7×C22⋊C4C2211yescharacteristic
16C22×C7×C22⋊C4C2211yescharacteristic
17C22×C42×C7C2212yes—
18C22×C42×C7C2211yescharacteristic
19C22×C42×C7C2211yescharacteristic
20C4×C7×C22⋊C4C418yescomplemented
21C2×C7×#95C2211yescharacteristic
22C2×C7×#95C2211yescharacteristic
23C2×C7×#95C2212yes—
24C2×C7×#95C2211yescharacteristic
25C2×C7×#95C2211yescharacteristic
26C2×C7×#95C2212yes—
27C2×C7×#95C2212yes—
28C2×C7×#95C2212yes—
29C2×C7×#95C2212yes—
30C2×C7×#95C2212yes—
31C2×C7×#95C2212yes—
32C2×C7×#95C2212yes—
33C2×C7×#95C2212yes—
34C2×C7×#95C2211yescharacteristic
35C2×C7×#95C2211yescharacteristic
36#13947C711yesmaximal, characteristic, direct product factor, Sylow
37C25×C7C2311yescharacteristic
38C23×C4×C7—22no—
39C23×C4×C7—22no—
40C23×C4×C7—42no—
41C23×C4×C7—42no—
42C23×C4×C7C2×C414yes—
43C23×C4×C7C2312yes—
44C23×C4×C7C2312yes—
45C23×C4×C7C2311yescharacteristic
46C23×C4×C7C2311yescharacteristic
47C23×C4×C7C2311yescharacteristic
48C23×C4×C7C2311yescharacteristic
49C23×C4×C7C2312yes—
50C23×C4×C7C2312yes—
51C23×C4×C7C2311yescharacteristic
52C23×C4×C7C2311yescharacteristic
53C2×C7×C22⋊C4—22no—
54C2×C7×C22⋊C4—22no—
55C2×C7×C22⋊C4—22no—
56C2×C7×C22⋊C4—22no—
57C2×C7×C22⋊C4—24no—
58C2×C7×C22⋊C4—24no—
59C2×C7×C22⋊C4—24no—
60C2×C7×C22⋊C4—24no—
61C2×C7×C22⋊C4—42no—
62C2×C7×C22⋊C4—42no—
63C2×C7×C22⋊C4C2×C414yes—
64C2×C7×C22⋊C4C2×C414yes—
65C2×C42×C7—24no—
66C2×C42×C7—24no—
67C2×C42×C7—42no—
68C2×C42×C7—42no—
69C2×C42×C7—42no—
70C2×C42×C7C2×C414yes—
71C2×C42×C7C2×C414yes—
72C7×#95—24no—
73C7×#95—22no—
74C7×#95—22no—
75C7×#95—24no—
76C7×#95—24no—
77C7×#95—24no—
78C7×#95—22no—
79C7×#95—22no—
80C7×#95—42no—
81C7×#95—42no—
82C7×#95—42no—
83C7×#95—42no—
84C7×#95—42no—
85C7×#95C2×C414yes—
86C7×#95C2×C414yes—
87C2×C4×C22⋊C4C2×C711yescharacteristic
88#1443C2×C712yescomplemented
89#1440C2×C712yes—
90#1437C2×C712yes—
91#1434C2×C711yescharacteristic, complemented
92#1434C2×C711yescharacteristic, complemented
93#1434C2×C711yescharacteristic, complemented
94#1434C2×C712yescomplemented
95C22⋊#95C2×C711yescharacteristic
96C22⋊#95C2×C711yescharacteristic
97C4×#95C2×C711yescharacteristic, complemented
98C24×C7—21no—
99C24×C7—21no—
100C24×C7—22no—
101C24×C7—22no—
102C24×C7—21no—
103C24×C7—21no—
104C24×C7—22no—
105C24×C7—22no—
106C24×C7C22×C412yes—
107C24×C7#4112yes—
108C24×C7#4112yes—
109C24×C7C2411yescharacteristic
110C22×C4×C7—22no—
111C22×C4×C7—22no—
112C22×C4×C7—22no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—21no—
124C22×C4×C7—21no—
125C22×C4×C7—21no—
126C22×C4×C7—21no—
127C22×C4×C7—21no—
128C22×C4×C7—21no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—28no—
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—44no—
158C22×C4×C7—44no—
159C22×C4×C7—48no—
160C22×C4×C7—48no—
161C22×C4×C7C22×C412yes—
162C22×C4×C7C22×C412yes—
163C22×C4×C7C22×C412yes—
164C22×C4×C7C22×C412yes—
165C22×C4×C7C22×C412yes—
166C22×C4×C7C22×C412yes—
167C22×C4×C7#4114yes—
168C22×C4×C7#4114yes—
169C22×C4×C7#4112yes—
170C22×C4×C7#4112yes—
171C22×C4×C7#4112yes—
172C22×C4×C7#4112yes—
173C7×C22⋊C4—28no—
174C7×C22⋊C4—28no—
175C7×C22⋊C4—48no—
176C7×C22⋊C4—48no—
177C42×C7—44no—
178C42×C7—44no—
179C42×C7—48no—
180C24×C4C22×C711yescharacteristic
181C22×C22⋊C4C22×C712yes—
182C22×C22⋊C4C22×C712yes—
183C22×C22⋊C4C22×C711yescharacteristic
184C22×C22⋊C4C22×C711yescharacteristic
185C22×C42C22×C712yes—
186C22×C42C22×C711yescharacteristic
187C22×C42C22×C711yescharacteristic
188C4×C22⋊C4C4×C718yescomplemented
189C2×#95C22×C711yescharacteristic
190C2×#95C22×C711yescharacteristic
191C2×#95C22×C712yes—
192C2×#95C22×C711yescharacteristic
193C2×#95C22×C711yescharacteristic
194C2×#95C22×C712yes—
195C2×#95C22×C712yes—
196C2×#95C22×C712yes—
197C2×#95C22×C712yes—
198C2×#95C22×C712yes—
199C2×#95C22×C712yes—
200C2×#95C22×C712yes—
201C2×#95C22×C712yes—
202C2×#95C22×C711yescharacteristic
203C2×#95C22×C711yescharacteristic
204C23×C7—22no—
205C23×C7—22no—
206C23×C7—22no—
207C23×C7—22no—
208C23×C7—22no—
209C23×C7—22no—
210C23×C7—22no—
211C23×C7—22no—
212C23×C7—22no—
213C23×C7—22no—
214C23×C7—24no—
215C23×C7—24no—
216C23×C7—24no—
217C23×C7—24no—
218C23×C7—42no—
219C23×C7—42no—
220C23×C7—42no—
221C23×C7—42no—
222C23×C7—42no—
223C23×C7—42no—
224C23×C7—42no—
225C23×C7—42no—
226C23×C7#11714yes—
227C23×C7C23×C411yescharacteristic
228C23×C7C2×#4111yescharacteristic
229C23×C7C2×#4111yescharacteristic
230C23×C7C2×#4111yescharacteristic
231C23×C7C2×#4111yescharacteristic
232C23×C7C2×#4111yescharacteristic
233C23×C7C2×#4111yescharacteristic
234C23×C7C2×#4112yes—
235C23×C7C2×#4112yes—
236C23×C7H(5,𝔽2)12yes—
237C23×C7H(5,𝔽2)12yes—
238C2×C4×C7—22no—
239C2×C4×C7—22no—
240C2×C4×C7—22no—
241C2×C4×C7—22no—
242C2×C4×C7—24no—
243C2×C4×C7—24no—
244C2×C4×C7—24no—
245C2×C4×C7—24no—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—22no—
250C2×C4×C7—22no—
251C2×C4×C7—22no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—22no—
255C2×C4×C7—22no—
256C2×C4×C7—22no—
257C2×C4×C7—22no—
258C2×C4×C7—24no—
259C2×C4×C7—24no—
260C2×C4×C7—24no—
261C2×C4×C7—24no—
262C2×C4×C7—24no—
263C2×C4×C7—24no—
264C2×C4×C7—416no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7—42no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—42no—
280C2×C4×C7—42no—
281C2×C4×C7—42no—
282C2×C4×C7—42no—
283C2×C4×C7—42no—
284C2×C4×C7—42no—
285C2×C4×C7—42no—
286C2×C4×C7—42no—
287C2×C4×C7—42no—
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#11714yes—
296C2×C4×C7#11714yes—
297C25C23×C711yescharacteristic
298C23×C4—22no—
299C23×C4—22no—
300C23×C4—42no—
301C23×C4—42no—
302C23×C4C2×C4×C714yes—
303C23×C4C23×C712yes—
304C23×C4C23×C712yes—
305C23×C4C23×C711yescharacteristic
306C23×C4C23×C711yescharacteristic
307C23×C4C23×C711yescharacteristic
308C23×C4C23×C711yescharacteristic
309C23×C4C23×C712yes—
310C23×C4C23×C712yes—
311C23×C4C23×C711yescharacteristic
312C23×C4C23×C711yescharacteristic
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—42no—
322C2×C22⋊C4—42no—
323C2×C22⋊C4C2×C4×C714yes—
324C2×C22⋊C4C2×C4×C714yes—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—42no—
328C2×C42—42no—
329C2×C42—42no—
330C2×C42C2×C4×C714yes—
331C2×C42C2×C4×C714yes—
332#95—24no—
333#95—22no—
334#95—22no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—22no—
339#95—22no—
340#95—42no—
341#95—42no—
342#95—42no—
343#95—42no—
344#95—42no—
345#95C2×C4×C714yes—
346#95C2×C4×C714yes—
347C22×C7—24no—
348C22×C7—24no—
349C22×C7—24no—
350C22×C7—42no—
351C22×C7—44no—
352C22×C7—44no—
353C22×C7—44no—
354C22×C7—44no—
355C22×C7—42no—
356C22×C7—42no—
357C22×C7—42no—
358C22×C7C2×#11711yescharacteristic
359C22×C7C2×#11711yescharacteristic
360C22×C7C2×#11711yescharacteristic
361C22×C7C4×#4111yescharacteristic
362C22×C7C4×#4111yescharacteristic
363C22×C7#51811yescharacteristic
364C22×C7#51811yescharacteristic
365C22×C7#52911yescharacteristic
366C22×C7#52912yes—
367C22×C7#52911yescharacteristic
368C22×C7#52912yes—
369C22×C7#52911yescharacteristic
370C22×C7#52911yescharacteristic
371C22×C7#53812yes—
372C22×C7#53812yes—
373C22×C7#55112yes—
374C22×C7#55112yes—
375C22×C7#55112yes—
376C22×C7#55312yes—
377C22×C7#55312yes—
378C22×C7#55312yes—
379C22×C7#55312yes—
380C22×C7#55912yes—
381C4×C7—24no—
382C4×C7—24no—
383C4×C7—42no—
384C4×C7—44no—
385C4×C7—44no—
386C4×C7—42no—
387C4×C7—42no—
388C4×C7—42no—
389C4×C7—82no—
390C4×C7—82no—
391C24—21no—
392C24—21no—
393C24—22no—
394C24—22no—
395C24—21no—
396C24—21no—
397C24—22no—
398C24—22no—
399C24C22×C4×C712yes—
400C24C7×#4112yes—
401C24C7×#4112yes—
402C24C24×C711yescharacteristic
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—28no—
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—44no—
451C22×C4—44no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4C22×C4×C712yes—
455C22×C4C22×C4×C712yes—
456C22×C4C22×C4×C712yes—
457C22×C4C22×C4×C712yes—
458C22×C4C22×C4×C712yes—
459C22×C4C22×C4×C712yes—
460C22×C4C7×#4114yes—
461C22×C4C7×#4114yes—
462C22×C4C7×#4112yes—
463C22×C4C7×#4112yes—
464C22×C4C7×#4112yes—
465C22×C4C7×#4112yes—
466C22⋊C4—28no—
467C22⋊C4—28no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C42—44no—
471C42—44no—
472C42—48no—
473C2×C7—44no—
474C2×C7#228311yescharacteristic
475C2×C7#229811yescharacteristic
476C2×C7#231211yescharacteristic
477C2×C7#235311yescharacteristic
478C2×C7#235311yescharacteristic
479C2×C7#235311yescharacteristic
480C2×C7#235711yescharacteristic
481C2×C7#257612yes—
482C2×C7#257812yes—
483C2×C7#259512yes—
484C2×C7#259712yes—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23C7×#11714yes—
508C23C23×C4×C711yescharacteristic
509C23C2×C7×#4111yescharacteristic
510C23C2×C7×#4111yescharacteristic
511C23C2×C7×#4111yescharacteristic
512C23C2×C7×#4111yescharacteristic
513C23C2×C7×#4111yescharacteristic
514C23C2×C7×#4111yescharacteristic
515C23C2×C7×#4112yes—
516C23C2×C7×#4112yes—
517C23C7×H(5,𝔽2)12yes—
518C23C7×H(5,𝔽2)12yes—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—416no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
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—42no—
565C2×C4—42no—
566C2×C4—42no—
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×C4C7×#11714yes—
577C2×C4C7×#11714yes—
578C7#1394711yescharacteristic, direct product factor, Sylow
579C22—24no—
580C22—24no—
581C22—24no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—42no—
589C22—42no—
590C22C2×C7×#11711yescharacteristic
591C22C2×C7×#11711yescharacteristic
592C22C2×C7×#11711yescharacteristic
593C22C4×C7×#4111yescharacteristic
594C22C4×C7×#4111yescharacteristic
595C22C7×#51811yescharacteristic
596C22C7×#51811yescharacteristic
597C22C7×#52911yescharacteristic
598C22C7×#52911yescharacteristic
599C22C7×#52911yescharacteristic
600C22C7×#52911yescharacteristic
601C22C7×#52912yes—
602C22C7×#52912yes—
603C22C7×#53812yes—
604C22C7×#53812yes—
605C22C7×#55112yes—
606C22C7×#55112yes—
607C22C7×#55112yes—
608C22C7×#55312yes—
609C22C7×#55312yes—
610C22C7×#55312yes—
611C22C7×#55312yes—
612C22C7×#55912yes—
613C4—24no—
614C4—24no—
615C4—42no—
616C4—44no—
617C4—44no—
618C4—42no—
619C4—42no—
620C4—42no—
621C4—82no—
622C4—82no—
623C2—44no—
624C2C7×#228311yescharacteristic
625C2C7×#229811yescharacteristic
626C2C7×#231211yescharacteristic
627C2C7×#235311yescharacteristic
628C2C7×#235311yescharacteristic
629C2C7×#235311yescharacteristic
630C2C7×#235711yescharacteristic
631C2C7×#257612yes—
632C2C7×#257812yes—
633C2C7×#259512yes—
634C2C7×#259712yes—
subgroupG/H
center, socleC24×C7C24
commutatorC23C23×C4×C7
Frattini, cosocleC24C24×C7
prime radicalC25×C7C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#1443C2semidirect
5. C7×#1434C2semidirect
6. C7×#1434C2semidirect
7. C7×#1434C2semidirect
8. C7×#1434C2semidirect
11. C4×C7×#95C2semidirect
20. C4×C7×C22⋊C4C4semidirect
36. #13947C7direct (N × H)
88. #1443C2×C7semidirect
91. #1434C2×C7semidirect
92. #1434C2×C7semidirect
93. #1434C2×C7semidirect
94. #1434C2×C7semidirect
97. C4×#95C2×C7semidirect
188. C4×C22⋊C4C4×C7semidirect
578. C7#13947direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×C7×C22⋊C42
C7×#14432
C7×#14402
C7×#14372
C7×#14342
C7×C22⋊#952
C4×C7×#952
#139477