Finite Group Databasebeta

← C7×#15941 #13193157

Subgroups — C7×#15941 #13193157

Order 1792.

1538 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#394C211yesmaximal, characteristic
2C2×C7×#382C211yesmaximal, characteristic, complemented
3C7×#1443C212yesmaximal, complemented
4C7×#1440C212yesmaximal, complemented
5C7×#1439C212yesmaximal, complemented
6C7×#1438C212yesmaximal
7C7×#1437C212yesmaximal, complemented
8C7×#1437C212yesmaximal, complemented
9C7×#1436C211yesmaximal, characteristic, complemented
10C23×C7×D4C2211yescharacteristic
11C22×C7×C4⋊C4C2211yescharacteristic
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C7×C22⋊C4C2212yes—
14C22×C7×C22⋊C4C2212yes—
15C22×C7×C22⋊C4C2212yes—
16C22×C7×C22⋊C4C2212yes—
17C22×C7×C22⋊C4C2212yes—
18C22×C7×C22⋊C4C2212yes—
19C22×C42×C7C2211yescharacteristic
20C22×C42×C7C2211yescharacteristic
21C7×#394—24no—
22C7×#382—24no—
23C2×C7×#95C2212yes—
24C2×C7×#95C2212yes—
25C2×C7×#95C2212yes—
26C2×C7×#95C2212yes—
27C2×C7×#95C2211yescharacteristic
28C2×C7×#95C2212yes—
29C2×C7×#95C2211yescharacteristic
30C2×C7×#95C2212yes—
31C2×C7×#95C2212yes—
32C2×C7×#95C2212yes—
33#15941C711yesmaximal, characteristic, direct product factor, Sylow
34C25×C7C2312yes—
35C22×C7×D4—24no—
36C22×C7×D4—22no—
37C22×C7×D4—42no—
38C22×C7×D4—41no—
39C22×C7×D4—41no—
40C23×C4×C7C2312yes—
41C23×C4×C7C2312yes—
42C23×C4×C7C2312yes—
43C23×C4×C7C2311yescharacteristic
44C23×C4×C7C2311yescharacteristic
45C23×C4×C7C2312yes—
46C23×C4×C7C2312yes—
47C23×C4×C7C2311yescharacteristic
48C2×C7×C4⋊C4—22no—
49C2×C7×C4⋊C4—41no—
50C2×C7×C4⋊C4—41no—
51C2×C7×C4⋊C4—41no—
52C2×C7×C4⋊C4—41no—
53C2×C7×C4⋊C4D414yes—
54C2×C7×C22⋊C4—24no—
55C2×C7×C22⋊C4—24no—
56C2×C7×C22⋊C4—24no—
57C2×C7×C22⋊C4—24no—
58C2×C7×C22⋊C4—24no—
59C2×C7×C22⋊C4—24no—
60C2×C7×C22⋊C4—24no—
61C2×C7×C22⋊C4—24no—
62C2×C7×C22⋊C4—24no—
63C2×C7×C22⋊C4—24no—
64C2×C7×C22⋊C4—42no—
65C2×C7×C22⋊C4—42no—
66C2×C7×C22⋊C4—42no—
67C2×C7×C22⋊C4—42no—
68C2×C7×C22⋊C4—42no—
69C2×C7×C22⋊C4—42no—
70C2×C7×C22⋊C4—42no—
71C2×C7×C22⋊C4—42no—
72C2×C42×C7—22no—
73C2×C42×C7—24no—
74C2×C42×C7—42no—
75C2×C42×C7—41no—
76C7×#95—24no—
77C7×#95—24no—
78C7×#95—22no—
79C7×#95—24no—
80C7×#95—24no—
81C7×#95—42no—
82C7×#95—42no—
83C7×#95—42no—
84C7×#95—41no—
85C7×#95—42no—
86C2×#394C2×C711yescharacteristic
87C2×#382C2×C711yescharacteristic, complemented
88#1443C2×C712yescomplemented
89#1440C2×C712yescomplemented
90#1439C2×C712yescomplemented
91#1438C2×C712yes—
92#1437C2×C712yescomplemented
93#1437C2×C712yescomplemented
94#1436C2×C711yescharacteristic, complemented
95C24×C7—22no—
96C24×C7—22no—
97C24×C7—22no—
98C24×C7—22no—
99C24×C7—22no—
100C24×C7—22no—
101C24×C7—22no—
102C24×C7—22no—
103C24×C7—22no—
104C24×C7—22no—
105C24×C7—22no—
106C24×C7—22no—
107C24×C7—22no—
108C24×C7—22no—
109C24×C7#4114yes—
110C24×C7C2411yescharacteristic
111C2×C7×D4—48no—
112C2×C7×D4—44no—
113C2×C7×D4—44no—
114C2×C7×D4—48no—
115C2×C7×D4—44no—
116C22×C4×C7—22no—
117C22×C4×C7—21no—
118C22×C4×C7—21no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—21no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
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×C7C2×D412yes—
161C22×C4×C7C2×D412yes—
162C22×C4×C7C2×D412yes—
163C22×C4×C7#4114yes—
164C22×C4×C7#4114yes—
165C22×C4×C7#4114yes—
166C7×C4⋊C4—44no—
167C7×C4⋊C4—44no—
168C7×C22⋊C4—48no—
169C7×C22⋊C4—48no—
170C7×C22⋊C4—48no—
171C7×C22⋊C4—48no—
172C7×C22⋊C4—48no—
173C7×C22⋊C4—48no—
174C42×C7—28no—
175C42×C7—44no—
176C23×D4C22×C711yescharacteristic
177C22×C4⋊C4C22×C711yescharacteristic
178C22×C4⋊C4C22×C711yescharacteristic
179C22×C22⋊C4C22×C712yes—
180C22×C22⋊C4C22×C712yes—
181C22×C22⋊C4C22×C712yes—
182C22×C22⋊C4C22×C712yes—
183C22×C22⋊C4C22×C712yes—
184C22×C22⋊C4C22×C712yes—
185C22×C42C22×C711yescharacteristic
186C22×C42C22×C711yescharacteristic
187#394—24no—
188#382—24no—
189C2×#95C22×C711yescharacteristic
190C2×#95C22×C712yes—
191C2×#95C22×C712yes—
192C2×#95C22×C712yes—
193C2×#95C22×C712yes—
194C2×#95C22×C712yes—
195C2×#95C22×C711yescharacteristic
196C2×#95C22×C712yes—
197C2×#95C22×C712yes—
198C2×#95C22×C712yes—
199C23×C7—24no—
200C23×C7—24no—
201C23×C7—24no—
202C23×C7—24no—
203C23×C7—24no—
204C23×C7—24no—
205C23×C7—24no—
206C23×C7—42no—
207C23×C7—42no—
208C23×C7—42no—
209C23×C7—42no—
210C23×C7—42no—
211C23×C7—42no—
212C23×C7—42no—
213C23×C7—42no—
214C23×C7—42no—
215C23×C7—42no—
216C23×C7—42no—
217C23×C7—42no—
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—42no—
227C23×C7—42no—
228C23×C7—42no—
229C23×C7—42no—
230C23×C7—42no—
231C23×C7—42no—
232C23×C7—42no—
233C23×C7—42no—
234C23×C7C22×D411yescharacteristic
235C23×C7C2×#4112yes—
236C23×C7C2×#4112yes—
237C23×C7C2×#4112yes—
238C23×C7C2×#4112yes—
239C23×C7H(5,𝔽2)11yescharacteristic
240C23×C7H(5,𝔽2)11yescharacteristic
241C23×C7H(5,𝔽2)12yes—
242C23×C7H(5,𝔽2)12yes—
243C7×D4—416no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—24no—
247C2×C4×C7—24no—
248C2×C4×C7—24no—
249C2×C4×C7—22no—
250C2×C4×C7—22no—
251C2×C4×C7—24no—
252C2×C4×C7—24no—
253C2×C4×C7—24no—
254C2×C4×C7—24no—
255C2×C4×C7—24no—
256C2×C4×C7—24no—
257C2×C4×C7—24no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—42no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—41no—
267C2×C4×C7—41no—
268C2×C4×C7—42no—
269C2×C4×C7—41no—
270C2×C4×C7—42no—
271C2×C4×C7—41no—
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#12414yes—
294C25C23×C712yes—
295C22×D4—24no—
296C22×D4—22no—
297C22×D4—42no—
298C22×D4—41no—
299C22×D4—41no—
300C23×C4C23×C712yes—
301C23×C4C23×C712yes—
302C23×C4C23×C712yes—
303C23×C4C23×C711yescharacteristic
304C23×C4C23×C711yescharacteristic
305C23×C4C23×C712yes—
306C23×C4C23×C712yes—
307C23×C4C23×C711yescharacteristic
308C2×C4⋊C4—22no—
309C2×C4⋊C4—41no—
310C2×C4⋊C4—41no—
311C2×C4⋊C4—41no—
312C2×C4⋊C4—41no—
313C2×C4⋊C4C7×D414yes—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—42no—
325C2×C22⋊C4—42no—
326C2×C22⋊C4—42no—
327C2×C22⋊C4—42no—
328C2×C22⋊C4—42no—
329C2×C22⋊C4—42no—
330C2×C22⋊C4—42no—
331C2×C22⋊C4—42no—
332C2×C42—22no—
333C2×C42—24no—
334C2×C42—42no—
335C2×C42—41no—
336#95—24no—
337#95—24no—
338#95—22no—
339#95—24no—
340#95—24no—
341#95—42no—
342#95—42no—
343#95—42no—
344#95—41no—
345#95—42no—
346C22×C7—44no—
347C22×C7—44no—
348C22×C7—44no—
349C22×C7—44no—
350C22×C7—44no—
351C22×C7—44no—
352C22×C7—44no—
353C22×C7—82no—
354C22×C7—82no—
355C22×C7—82no—
356C22×C7—82no—
357C22×C7—82no—
358C22×C7—82no—
359C22×C7—82no—
360C22×C7—82no—
361C22×C7#52511yescharacteristic
362C22×C7C2×#12411yescharacteristic
363C22×C7#52912yes—
364C22×C7#52912yes—
365C22×C7#53411yescharacteristic
366C22×C7#53812yes—
367C22×C7#53812yes—
368C22×C7#54012yes—
369C22×C7#54012yes—
370C22×C7#54612yes—
371C22×C7#55012yes—
372C22×C7#55111yescharacteristic
373C22×C7#55112yes—
374C22×C7#55112yes—
375C22×C7#55312yes—
376C22×C7#55312yes—
377C22×C7#55312yes—
378C22×C7#55312yes—
379C22×C7#55511yescharacteristic
380C22×C7H(3,𝔽4)11yescharacteristic
381C22×C7H(3,𝔽4)11yescharacteristic
382C4×C7—24no—
383C4×C7—42no—
384C4×C7—42no—
385C4×C7—44no—
386C4×C7—44no—
387C4×C7—82no—
388C4×C7—82no—
389C4×C7—82no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—22no—
394C24—22no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—22no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24C7×#4114yes—
405C24C24×C711yescharacteristic
406C2×D4—48no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—48no—
410C2×D4—44no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
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×C4C2×C7×D412yes—
456C22×C4C2×C7×D412yes—
457C22×C4C2×C7×D412yes—
458C22×C4C7×#4114yes—
459C22×C4C7×#4114yes—
460C22×C4C7×#4114yes—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C42—28no—
470C42—44no—
471C2×C7—84no—
472C2×C7#247112yes—
473C2×C7#249012yes—
474C2×C7#251812yes—
475C2×C7#257612yes—
476C2×C7#259111yescharacteristic
477C2×C7#268411yescharacteristic
478C2×C7#273311yescharacteristic
479C2×C7#275312yes—
480C2×C7#275312yes—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23C22×C7×D411yescharacteristic
517C23C2×C7×#4112yes—
518C23C2×C7×#4112yes—
519C23C2×C7×#4112yes—
520C23C2×C7×#4112yes—
521C23C7×H(5,𝔽2)11yescharacteristic
522C23C7×H(5,𝔽2)11yescharacteristic
523C23C7×H(5,𝔽2)12yes—
524C23C7×H(5,𝔽2)12yes—
525D4—416no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—42no—
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—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×C4C7×#12414yes—
576C7#1594111yescharacteristic, direct product factor, Sylow
577C22—44no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22C7×#52511yescharacteristic
593C22C2×C7×#12411yescharacteristic
594C22C7×#52912yes—
595C22C7×#52912yes—
596C22C7×#53411yescharacteristic
597C22C7×#53812yes—
598C22C7×#53812yes—
599C22C7×#54012yes—
600C22C7×#54012yes—
601C22C7×#54612yes—
602C22C7×#55012yes—
603C22C7×#55111yescharacteristic
604C22C7×#55112yes—
605C22C7×#55112yes—
606C22C7×#55312yes—
607C22C7×#55312yes—
608C22C7×#55312yes—
609C22C7×#55312yes—
610C22C7×#55511yescharacteristic
611C22C7×H(3,𝔽4)11yescharacteristic
612C22C7×H(3,𝔽4)11yescharacteristic
613C4—24no—
614C4—42no—
615C4—42no—
616C4—44no—
617C4—44no—
618C4—82no—
619C4—82no—
620C4—82no—
621C2—84no—
622C2C7×#247112yes—
623C2C7×#249012yes—
624C2C7×#251812yes—
625C2C7×#257612yes—
626C2C7×#259111yescharacteristic
627C2C7×#268411yescharacteristic
628C2C7×#273311yescharacteristic
629C2C7×#275312yes—
630C2C7×#275312yes—
subgroupG/H
center, socleC24×C7C24
commutator, Frattini, cosocleC24C24×C7
prime radicalC23×C7×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#382C2semidirect
3. C7×#1443C2semidirect
4. C7×#1440C2semidirect
5. C7×#1439C2semidirect
7. C7×#1437C2semidirect
8. C7×#1437C2semidirect
9. C7×#1436C2semidirect
33. #15941C7direct (N × H)
87. C2×#382C2×C7semidirect
88. #1443C2×C7semidirect
89. #1440C2×C7semidirect
90. #1439C2×C7semidirect
92. #1437C2×C7semidirect
93. #1437C2×C7semidirect
94. #1436C2×C7semidirect
576. C7#15941direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#3942
C2×C7×#3822
C7×#14432
C7×#14402
C7×#14392
C7×#14382
C7×#14372
C7×#14362
#159417