Finite Group Databasebeta

← C7×#15948 #13193164

Subgroups — C7×#15948 #13193164

Order 1792.

3730 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#123C211yesmaximal, characteristic, complemented
2C22×C7×#120C211yesmaximal, characteristic
3C2×C7×#386C212yesmaximal, complemented
4C2×C7×#379C212yesmaximal, complemented
5C7×#1440C212yesmaximal, complemented
6C7×#1439C214yesmaximal, complemented
7C7×#1438C211yesmaximal, characteristic, complemented
8C7×#1438C212yesmaximal, complemented
9C26×C7C2211yescharacteristic
10C23×C7×D4C2211yescharacteristic
11C23×C7×D4C2212yes—
12C24×C4×C7C2211yescharacteristic, complemented
13C2×C7×#123—24no—
14C2×C7×#123—28no—
15C2×C7×#120—28no—
16C2×C7×#120—24no—
17C22×C7×C4⋊C4C2212yescomplemented
18C22×C7×C22⋊C4C2212yes—
19C22×C7×C22⋊C4C2211yescharacteristic
20C22×C7×C22⋊C4C2214yes—
21C22×C7×C22⋊C4C2214yes—
22C22×C7×C22⋊C4C2212yes—
23C22×C7×C22⋊C4C2212yes—
24C22×C7×C22⋊C4C2211yescharacteristic, complemented
25C22×C7×C22⋊C4C2212yescomplemented
26C7×#386—28no—
27C7×#379—28no—
28C2×C7×#95C2214yescomplemented
29C2×C7×#95C2212yescomplemented
30C2×C7×#95C2214yescomplemented
31#15948C711yesmaximal, characteristic, direct product factor, Sylow
32C25×C7—24no—
33C25×C7—22no—
34C25×C7—24no—
35C25×C7—28no—
36C25×C7—44no—
37C25×C7—42no—
38C25×C7C2311yescharacteristic
39C25×C7C2311yescharacteristic
40C25×C7C2312yes—
41C22×C7×D4—24no—
42C22×C7×D4—24no—
43C22×C7×D4—24no—
44C22×C7×D4—24no—
45C22×C7×D4—28no—
46C22×C7×D4—44no—
47C22×C7×D4—42no—
48C22×C7×D4D418yes—
49C22×C7×D4D414yes—
50C23×C4×C7—22no—
51C23×C4×C7—24no—
52C23×C4×C7—24no—
53C23×C4×C7—42no—
54C23×C4×C7C2314yes—
55C23×C4×C7C2312yes—
56C23×C4×C7C2311yescharacteristic
57C23×C4×C7C2312yes—
58C23×C4×C7C2312yes—
59C7×#123—232no—
60C7×#120—232no—
61C2×C7×C4⋊C4—24no—
62C2×C7×C4⋊C4—24no—
63C2×C7×C4⋊C4—24no—
64C2×C7×C22⋊C4—22no—
65C2×C7×C22⋊C4—24no—
66C2×C7×C22⋊C4—24no—
67C2×C7×C22⋊C4—24no—
68C2×C7×C22⋊C4—22no—
69C2×C7×C22⋊C4—24no—
70C2×C7×C22⋊C4—24no—
71C2×C7×C22⋊C4—24no—
72C2×C7×C22⋊C4—24no—
73C2×C7×C22⋊C4—24no—
74C2×C7×C22⋊C4—28no—
75C2×C7×C22⋊C4—24no—
76C2×C7×C22⋊C4—28no—
77C2×C7×C22⋊C4—28no—
78C2×C7×C22⋊C4—28no—
79C2×C7×C22⋊C4—44no—
80C2×C7×C22⋊C4—44no—
81C2×C7×C22⋊C4—44no—
82C2×C7×C22⋊C4—44no—
83C2×C7×C22⋊C4D418yescomplemented
84C7×#95—28no—
85C7×#95—44no—
86C7×#95—42no—
87C22×#123C2×C711yescharacteristic, complemented
88C22×#120C2×C711yescharacteristic
89C2×#386C2×C712yescomplemented
90C2×#379C2×C712yescomplemented
91#1440C2×C712yescomplemented
92#1439C2×C714yescomplemented
93#1438C2×C712yescomplemented
94#1438C2×C711yescharacteristic, complemented
95C24×C7—24no—
96C24×C7—22no—
97C24×C7—22no—
98C24×C7—22no—
99C24×C7—22no—
100C24×C7—24no—
101C24×C7—24no—
102C24×C7—24no—
103C24×C7—24no—
104C24×C7—24no—
105C24×C7—24no—
106C24×C7—24no—
107C24×C7—22no—
108C24×C7—22no—
109C24×C7—24no—
110C24×C7—28no—
111C24×C7—216no—
112C24×C7—416no—
113C24×C7—416no—
114C24×C7—416no—
115C24×C7—416no—
116C24×C7—48no—
117C24×C7—44no—
118C24×C7—48no—
119C24×C7—48no—
120C24×C7—416no—
121C24×C7—48no—
122C24×C7—44no—
123C24×C7—48no—
124C24×C7C2×D412yes—
125C24×C7C2×D414yes—
126C24×C7C2×D412yes—
127C24×C7C2×D414yes—
128C24×C7C2×D414yes—
129C24×C7#4118yes—
130C24×C7C2411yescharacteristic
131C2×C7×D4—216no—
132C2×C7×D4—28no—
133C2×C7×D4—28no—
134C2×C7×D4—216no—
135C2×C7×D4—48no—
136C2×C7×D4—44no—
137C2×C7×D4—416no—
138C2×C7×D4—416no—
139C2×C7×D4—416no—
140C22×C4×C7—28no—
141C22×C4×C7—24no—
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—24no—
151C22×C4×C7—24no—
152C22×C4×C7—24no—
153C22×C4×C7—22no—
154C22×C4×C7—24no—
155C22×C4×C7—24no—
156C22×C4×C7—24no—
157C22×C4×C7—24no—
158C22×C4×C7—24no—
159C22×C4×C7—24no—
160C22×C4×C7—24no—
161C22×C4×C7—24no—
162C22×C4×C7—24no—
163C22×C4×C7—416no—
164C22×C4×C7—44no—
165C22×C4×C7—44no—
166C22×C4×C7C2×D414yes—
167C22×C4×C7C2×D414yes—
168C22×C4×C7C2×D414yes—
169C22×C4×C7C2×D412yes—
170C7×C4⋊C4—216no—
171C7×C22⋊C4—216no—
172C7×C22⋊C4—28no—
173C7×C22⋊C4—28no—
174C7×C22⋊C4—216no—
175C7×C22⋊C4—416no—
176C7×C22⋊C4—416no—
177C7×C22⋊C4—48no—
178C7×C22⋊C4—48no—
179C26C22×C711yescharacteristic
180C23×D4C22×C711yescharacteristic
181C23×D4C22×C712yes—
182C24×C4C22×C711yescharacteristic, complemented
183C2×#123—24no—
184C2×#123—28no—
185C2×#120—24no—
186C2×#120—28no—
187C22×C4⋊C4C22×C712yescomplemented
188C22×C22⋊C4C22×C712yes—
189C22×C22⋊C4C22×C711yescharacteristic
190C22×C22⋊C4C22×C714yes—
191C22×C22⋊C4C22×C714yes—
192C22×C22⋊C4C22×C712yes—
193C22×C22⋊C4C22×C712yes—
194C22×C22⋊C4C22×C711yescharacteristic, complemented
195C22×C22⋊C4C22×C712yescomplemented
196#386—28no—
197#379—28no—
198C2×#95C22×C714yescomplemented
199C2×#95C22×C712yescomplemented
200C2×#95C22×C714yescomplemented
201C23×C7—22no—
202C23×C7—22no—
203C23×C7—24no—
204C23×C7—24no—
205C23×C7—24no—
206C23×C7—28no—
207C23×C7—24no—
208C23×C7—22no—
209C23×C7—22no—
210C23×C7—24no—
211C23×C7—24no—
212C23×C7—28no—
213C23×C7—28no—
214C23×C7—28no—
215C23×C7—28no—
216C23×C7—28no—
217C23×C7—28no—
218C23×C7—28no—
219C23×C7—28no—
220C23×C7—28no—
221C23×C7—28no—
222C23×C7—28no—
223C23×C7—28no—
224C23×C7—28no—
225C23×C7—416no—
226C23×C7—432no—
227C23×C7—432no—
228C23×C7—464no—
229C23×C7—464no—
230C23×C7—432no—
231C23×C7—42no—
232C23×C7—42no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7—42no—
237C23×C7—42no—
238C23×C7—42no—
239C23×C7—44no—
240C23×C7—44no—
241C23×C7—44no—
242C23×C7—44no—
243C23×C7—44no—
244C23×C7—44no—
245C23×C7—44no—
246C23×C7—44no—
247C23×C7—44no—
248C23×C7—42no—
249C23×C7—42no—
250C23×C7—42no—
251C23×C7—44no—
252C23×C7—42no—
253C23×C7#12014yes—
254C23×C7#12014yes—
255C23×C7#12318yescomplemented
256C23×C7C22×D411yescharacteristic
257C23×C7C22×D412yes—
258C23×C7C22×D412yes—
259C23×C7C2×#4114yes—
260C23×C7H(5,𝔽2)14yes—
261C23×C7H(5,𝔽2)12yes—
262C7×D4—416no—
263C7×D4—432no—
264C2×C4×C7—24no—
265C2×C4×C7—22no—
266C2×C4×C7—24no—
267C2×C4×C7—24no—
268C2×C4×C7—22no—
269C2×C4×C7—24no—
270C2×C4×C7—22no—
271C2×C4×C7—24no—
272C2×C4×C7—28no—
273C2×C4×C7—28no—
274C2×C4×C7—416no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—44no—
280C2×C4×C7—44no—
281C2×C4×C7—44no—
282C2×C4×C7—44no—
283C2×C4×C7—44no—
284C2×C4×C7—44no—
285C2×C4×C7—44no—
286C2×C4×C7—44no—
287C2×C4×C7—44no—
288C2×C4×C7—44no—
289C2×C4×C7—44no—
290C2×C4×C7—44no—
291C25—22no—
292C25—24no—
293C25—24no—
294C25—28no—
295C25—44no—
296C25—42no—
297C25C23×C711yescharacteristic
298C25C23×C711yescharacteristic
299C25C23×C712yes—
300C22×D4—24no—
301C22×D4—24no—
302C22×D4—24no—
303C22×D4—24no—
304C22×D4—28no—
305C22×D4—44no—
306C22×D4—42no—
307C22×D4C7×D414yes—
308C22×D4C7×D418yes—
309C23×C4—22no—
310C23×C4—24no—
311C23×C4—24no—
312C23×C4—42no—
313C23×C4C23×C714yes—
314C23×C4C23×C712yes—
315C23×C4C23×C712yes—
316C23×C4C23×C711yescharacteristic
317C23×C4C23×C712yes—
318#123—232no—
319#120—232no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—28no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—28no—
336C2×C22⋊C4—28no—
337C2×C22⋊C4—28no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4C7×D418yescomplemented
343#95—28no—
344#95—44no—
345#95—42no—
346C22×C7—24no—
347C22×C7—24no—
348C22×C7—28no—
349C22×C7—28no—
350C22×C7—28no—
351C22×C7—216no—
352C22×C7—464no—
353C22×C7—44no—
354C22×C7—44no—
355C22×C7—48no—
356C22×C7—44no—
357C22×C7—44no—
358C22×C7—48no—
359C22×C7—48no—
360C22×C7—48no—
361C22×C7—48no—
362C22×C7—48no—
363C22×C7—48no—
364C22×C7—48no—
365C22×C7—44no—
366C22×C7—44no—
367C22×C7—48no—
368C22×C7C2×#12011yescharacteristic
369C22×C7C2×#12011yescharacteristic
370C22×C7C2×#12312yes—
371C22×C7#52512yes—
372C22×C7#53411yescharacteristic
373C22×C7#53412yes—
374C22×C7#53512yes—
375C22×C7#53814yes—
376C22×C7D4214yes—
377C22×C7#54614yes—
378C22×C7#54614yes—
379C22×C7#54614yes—
380C22×C7#55112yes—
381C22×C7#56012yes—
382C4×C7—44no—
383C4×C7—42no—
384C4×C7—44no—
385C4×C7—44no—
386C4×C7—84no—
387C24—24no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—24no—
393C24—24no—
394C24—24no—
395C24—24no—
396C24—24no—
397C24—24no—
398C24—24no—
399C24—22no—
400C24—22no—
401C24—24no—
402C24—28no—
403C24—216no—
404C24—416no—
405C24—416no—
406C24—416no—
407C24—416no—
408C24—48no—
409C24—44no—
410C24—48no—
411C24—48no—
412C24—416no—
413C24—48no—
414C24—44no—
415C24—48no—
416C24C2×C7×D412yes—
417C24C2×C7×D414yes—
418C24C2×C7×D412yes—
419C24C2×C7×D414yes—
420C24C2×C7×D414yes—
421C24C7×#4118yes—
422C24C24×C711yescharacteristic
423C2×D4—28no—
424C2×D4—216no—
425C2×D4—28no—
426C2×D4—216no—
427C2×D4—48no—
428C2×D4—44no—
429C2×D4—416no—
430C2×D4—416no—
431C2×D4—416no—
432C22×C4—28no—
433C22×C4—24no—
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—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—416no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4C2×C7×D414yes—
459C22×C4C2×C7×D414yes—
460C22×C4C2×C7×D414yes—
461C22×C4C2×C7×D412yes—
462C4⋊C4—216no—
463C22⋊C4—216no—
464C22⋊C4—28no—
465C22⋊C4—28no—
466C22⋊C4—216no—
467C22⋊C4—416no—
468C22⋊C4—416no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C2×C7—44no—
472C2×C7—44no—
473C2×C7—48no—
474C2×C7#240311yescharacteristic
475C2×C7#240412yes—
476C2×C7#240512yes—
477C2×C7#241814yes—
478C2×C7#267814yes—
479C2×C7#269712yes—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—28no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—28no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—416no—
505C23—432no—
506C23—432no—
507C23—464no—
508C23—464no—
509C23—432no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—42no—
532C23C7×#12014yes—
533C23C7×#12014yes—
534C23C7×#12318yescomplemented
535C23C22×C7×D411yescharacteristic
536C23C22×C7×D412yes—
537C23C22×C7×D412yes—
538C23C2×C7×#4114yes—
539C23C7×H(5,𝔽2)14yes—
540C23C7×H(5,𝔽2)12yes—
541D4—416no—
542D4—432no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—416no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C7#1594811yescharacteristic, direct product factor, Sylow
571C22—24no—
572C22—24no—
573C22—28no—
574C22—28no—
575C22—28no—
576C22—216no—
577C22—464no—
578C22—44no—
579C22—48no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—48no—
584C22—48no—
585C22—48no—
586C22—48no—
587C22—48no—
588C22—48no—
589C22—48no—
590C22—44no—
591C22—48no—
592C22—44no—
593C22C2×C7×#12011yescharacteristic
594C22C2×C7×#12011yescharacteristic
595C22C2×C7×#12312yes—
596C22C7×#52512yes—
597C22C7×#53411yescharacteristic
598C22C7×#53412yes—
599C22C7×#53512yes—
600C22C7×#53814yes—
601C22C7×D4214yes—
602C22C7×#54614yes—
603C22C7×#54614yes—
604C22C7×#54614yes—
605C22C7×#55112yes—
606C22C7×#56012yes—
607C4—44no—
608C4—42no—
609C4—44no—
610C4—44no—
611C4—84no—
612C2—44no—
613C2—48no—
614C2—44no—
615C2C7×#240311yescharacteristic
616C2C7×#240412yes—
617C2C7×#240512yes—
618C2C7×#241814yes—
619C2C7×#267814yes—
620C2C7×#269712yes—
subgroupG/H
center, socleC24×C7C24
commutator, Frattini, cosocleC24C24×C7
prime radicalC22×C7×#120C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C7×#123C2semidirect
3. C2×C7×#386C2semidirect
4. C2×C7×#379C2semidirect
5. C7×#1440C2semidirect
6. C7×#1439C2semidirect
7. C7×#1438C2semidirect
8. C7×#1438C2semidirect
12. C24×C4×C7C22semidirect
17. C22×C7×C4⋊C4C22semidirect
24. C22×C7×C22⋊C4C22semidirect
25. C22×C7×C22⋊C4C22semidirect
28. C2×C7×#95C22semidirect
29. C2×C7×#95C22semidirect
30. C2×C7×#95C22semidirect
31. #15948C7direct (N × H)
83. C2×C7×C22⋊C4D4semidirect
87. C22×#123C2×C7semidirect
89. C2×#386C2×C7semidirect
90. C2×#379C2×C7semidirect
91. #1440C2×C7semidirect
92. #1439C2×C7semidirect
93. #1438C2×C7semidirect
94. #1438C2×C7semidirect
182. C24×C4C22×C7semidirect
187. C22×C4⋊C4C22×C7semidirect
194. C22×C22⋊C4C22×C7semidirect
195. C22×C22⋊C4C22×C7semidirect
198. C2×#95C22×C7semidirect
199. C2×#95C22×C7semidirect
200. C2×#95C22×C7semidirect
255. C23×C7#123semidirect
342. C2×C22⋊C4C7×D4semidirect
534. C23C7×#123semidirect
570. C7#15948direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×#1232
C22×C7×#1202
C2×C7×#3862
C2×C7×#3792
C7×#14402
C7×#14392
C7×#14382
#159487