Finite Group Databasebeta

← C7×#23480 #13200696

Subgroups — C7×#23480 #13200696

Order 1792.

1902 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#121C211yesmaximal, characteristic
2C2×C7×#428C211yesmaximal, characteristic, complemented
3C2×C7×#418C211yesmaximal, characteristic, complemented
4C2×C7×#418C211yesmaximal, characteristic, complemented
5C2×C7×#407C211yesmaximal, characteristic, complemented
6C7×#2280C211yesmaximal, characteristic, complemented
7C7×#2034C214yesmaximal, complemented
8C7×#2033C214yesmaximal, complemented
9C7×#1736C211yesmaximal, characteristic, complemented
10C23×C7×D4—21no—
11C23×C7×D4C2211yescharacteristic
12C24×C4×C7C2211yescharacteristic
13C22×C7×M16C2211yescharacteristic
14C22×C7×M16C2211yescharacteristic
15C2×C7×#121—24no—
16C2×C7×#121—22no—
17C2×C7×#121—24no—
18C2×C7×#121C2214yes—
19C2×C7×#117C2211yescharacteristic
20C2×C7×#117C2211yescharacteristic
21C22×C7×C4⋊C4C2211yescharacteristic
22C22×C7×C22⋊C4—21no—
23C7×#428—24no—
24C2×C7×#107C2212yescomplemented
25C2×C7×#106C2212yescomplemented
26C7×#418—24no—
27C7×#418—24no—
28C2×C7×#102C2212yescomplemented
29C2×C7×#102C2214yescomplemented
30C2×C7×#102C2212yescomplemented
31C7×#407—24no—
32C2×C7×C22⋊C8C2214yescomplemented
33C7×#381—21no—
34C7×#380C2212yescomplemented
35C7×#380C2212yescomplemented
36C7×#379—21no—
37C4×C7×C22⋊C4—22no—
38C7×#340C2214yescomplemented
39#23480C711yesmaximal, characteristic, direct product factor, Sylow
40C25×C7—21no—
41C25×C7D411yescharacteristic
42C22×C7×D4—28no—
43C22×C7×D4—24no—
44C22×C7×D4—22no—
45C22×C7×D4—21no—
46C22×C7×D4—22no—
47C22×C7×D4—22no—
48C22×C7×D4—22no—
49C22×C7×D4—22no—
50C22×C7×D4—48no—
51C22×C7×D4D414yes—
52C22×C7×D4C2312yes—
53C23×C4×C7—21no—
54C23×C4×C7—22no—
55C23×C4×C7—22no—
56C23×C4×C7—22no—
57C23×C4×C7—22no—
58C23×C4×C7—22no—
59C23×C4×C7—22no—
60C23×C4×C7D411yescharacteristic
61C23×C4×C7D412yes—
62C23×C4×C7C2312yes—
63C23×C4×C7C2311yescharacteristic
64C2×C7×M16—22no—
65C2×C7×M16—22no—
66C2×C7×M16—22no—
67C2×C7×M16—22no—
68C2×C7×M16—22no—
69C2×C7×M16D414yescomplemented
70C22×C7×C8C2312yes—
71C22×C7×C8C2312yes—
72C7×#121—216no—
73C7×#121—28no—
74C7×#121—28no—
75C7×#117—22no—
76C7×#117—22no—
77C7×#117—22no—
78C7×#117—22no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4—22no—
81C2×C7×C4⋊C4D414yes—
82C2×C7×C4⋊C4C2312yes—
83C2×C7×C4⋊C4C2311yescharacteristic
84C2×C7×C4⋊C4C2311yescharacteristic
85C2×C7×C4⋊C4C2312yes—
86C2×C7×C22⋊C4—28no—
87C2×C7×C22⋊C4—24no—
88C2×C7×C22⋊C4—22no—
89C2×C7×C22⋊C4—22no—
90C2×C7×C22⋊C4—21no—
91C2×C7×C22⋊C4—21no—
92C2×C42×C7—21no—
93C2×C42×C7—21no—
94C7×#107—24no—
95C7×#106—24no—
96C7×#102—24no—
97C7×#102—24no—
98C7×#102—28no—
99C7×C22⋊C8—28no—
100C7×#95—22no—
101C7×#95—22no—
102C22×#121C2×C711yescharacteristic
103C2×#428C2×C711yescharacteristic, complemented
104C2×#418C2×C711yescharacteristic, complemented
105C2×#418C2×C711yescharacteristic, complemented
106C2×#407C2×C711yescharacteristic, complemented
107#2280C2×C711yescharacteristic, complemented
108#2034C2×C714yescomplemented
109#2033C2×C714yescomplemented
110#1736C2×C711yescharacteristic, complemented
111C24×C7—22no—
112C24×C7—24no—
113C24×C7—24no—
114C24×C7—22no—
115C24×C7—22no—
116C24×C7—22no—
117C24×C7—42no—
118C24×C7—44no—
119C24×C7—44no—
120C24×C7—42no—
121C24×C7—42no—
122C24×C7C2×D412yes—
123C24×C7C2×D411yescharacteristic
124C2×C7×D4—216no—
125C2×C7×D4—24no—
126C2×C7×D4—28no—
127C2×C7×D4—28no—
128C2×C7×D4—22no—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4—22no—
132C2×C7×D4—22no—
133C2×C7×D4—432no—
134C2×C7×D4—48no—
135C2×C7×D4—48no—
136C2×C7×D4—44no—
137C2×C7×D4—42no—
138C2×C7×D4—42no—
139C2×C7×D4—44no—
140C2×C7×D4C2×D414yes—
141C22×C4×C7—22no—
142C22×C4×C7—24no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—24no—
147C22×C4×C7—24no—
148C22×C4×C7—24no—
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—21no—
156C22×C4×C7—21no—
157C22×C4×C7—22no—
158C22×C4×C7—44no—
159C22×C4×C7—44no—
160C22×C4×C7—44no—
161C22×C4×C7—44no—
162C22×C4×C7—44no—
163C22×C4×C7C2×D412yes—
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D412yes—
166C22×C4×C7C2×D412yes—
167C22×C4×C7C2×D412yes—
168C22×C4×C7#4112yes—
169C22×C4×C7#4112yes—
170C22×C4×C7#4112yes—
171C22×C4×C7C2411yescharacteristic
172C7×M16—28no—
173C7×M16—44no—
174C2×C7×C8—22no—
175C2×C7×C8—22no—
176C2×C7×C8—22no—
177C2×C7×C8—22no—
178C2×C7×C8—22no—
179C2×C7×C8C2×D414yescomplemented
180C7×C4⋊C4—24no—
181C7×C4⋊C4—22no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4C2×D414yes—
189C7×C22⋊C4—216no—
190C7×C22⋊C4—44no—
191C7×C22⋊C4—42no—
192C7×C22⋊C4—42no—
193C7×C22⋊C4—44no—
194C42×C7—42no—
195C42×C7—42no—
196C23×D4—21no—
197C23×D4C22×C711yescharacteristic
198C24×C4C22×C711yescharacteristic
199C22×M16C22×C711yescharacteristic
200C22×M16C22×C711yescharacteristic
201C2×#121—24no—
202C2×#121—22no—
203C2×#121—24no—
204C2×#121C22×C714yes—
205C2×#117C22×C711yescharacteristic
206C2×#117C22×C711yescharacteristic
207C22×C4⋊C4C22×C711yescharacteristic
208C22×C22⋊C4—21no—
209#428—24no—
210C2×#107C22×C712yescomplemented
211C2×#106C22×C712yescomplemented
212#418—24no—
213#418—24no—
214C2×#102C22×C712yescomplemented
215C2×#102C22×C714yescomplemented
216C2×#102C22×C712yescomplemented
217#407—24no—
218C2×C22⋊C8C22×C714yescomplemented
219#381—21no—
220#380C22×C712yescomplemented
221#380C22×C712yescomplemented
222#379—21no—
223C4×C22⋊C4—22no—
224#340C22×C714yescomplemented
225C23×C7—22no—
226C23×C7—22no—
227C23×C7—28no—
228C23×C7—22no—
229C23×C7—22no—
230C23×C7—22no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—22no—
234C23×C7—24no—
235C23×C7—48no—
236C23×C7—48no—
237C23×C7—44no—
238C23×C7—48no—
239C23×C7—48no—
240C23×C7—44no—
241C23×C7—42no—
242C23×C7—42no—
243C23×C7—44no—
244C23×C7—44no—
245C23×C7—48no—
246C23×C7—88no—
247C23×C7—88no—
248C23×C7#12012yes—
249C23×C7#12112yes—
250C23×C7#12312yes—
251C23×C7AGL(1,ℤ/8)14yes—
252C23×C7C22×D411yescharacteristic
253C7×D4—24no—
254C7×D4—24no—
255C7×D4—432no—
256C7×D4—42no—
257C7×D4—42no—
258C7×D4—48no—
259C2×C4×C7—22no—
260C2×C4×C7—24no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—48no—
264C2×C4×C7—44no—
265C2×C4×C7—44no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7—42no—
270C2×C4×C7—48no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7#12012yes—
279C2×C4×C7#12114yes—
280C2×C4×C7#12114yes—
281C2×C4×C7#12112yes—
282C2×C4×C7#12314yes—
283C2×C4×C7#12312yes—
284C2×C4×C7#12414yes—
285C2×C4×C7C22×D411yescharacteristic
286C2×C4×C7C22×D411yescharacteristic
287C2×C4×C7C22×D411yescharacteristic
288C2×C4×C7C2×#4111yescharacteristic
289C2×C4×C7C2×#4111yescharacteristic
290C2×C4×C7C2×#4111yescharacteristic
291C7×C8—24no—
292C7×C8—42no—
293C25—21no—
294C25C7×D411yescharacteristic
295C22×D4—28no—
296C22×D4—24no—
297C22×D4—22no—
298C22×D4—21no—
299C22×D4—22no—
300C22×D4—22no—
301C22×D4—22no—
302C22×D4—22no—
303C22×D4—48no—
304C22×D4C7×D414yes—
305C22×D4C23×C712yes—
306C23×C4—21no—
307C23×C4—22no—
308C23×C4—22no—
309C23×C4—22no—
310C23×C4—22no—
311C23×C4—22no—
312C23×C4—22no—
313C23×C4C7×D411yescharacteristic
314C23×C4C7×D412yes—
315C23×C4C23×C712yes—
316C23×C4C23×C711yescharacteristic
317C2×M16—22no—
318C2×M16—22no—
319C2×M16—22no—
320C2×M16—22no—
321C2×M16—22no—
322C2×M16C7×D414yescomplemented
323C22×C8C23×C712yes—
324C22×C8C23×C712yes—
325#121—216no—
326#121—28no—
327#121—28no—
328#117—22no—
329#117—22no—
330#117—22no—
331#117—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4C7×D414yes—
335C2×C4⋊C4C23×C712yes—
336C2×C4⋊C4C23×C711yescharacteristic
337C2×C4⋊C4C23×C711yescharacteristic
338C2×C4⋊C4C23×C712yes—
339C2×C22⋊C4—28no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C42—21no—
346C2×C42—21no—
347#107—24no—
348#106—24no—
349#102—24no—
350#102—24no—
351#102—28no—
352C22⋊C8—28no—
353#95—22no—
354#95—22no—
355C22×C7—24no—
356C22×C7—22no—
357C22×C7—22no—
358C22×C7—24no—
359C22×C7—24no—
360C22×C7—24no—
361C22×C7—24no—
362C22×C7—416no—
363C22×C7—42no—
364C22×C7—42no—
365C22×C7—42no—
366C22×C7—42no—
367C22×C7—44no—
368C22×C7—44no—
369C22×C7—44no—
370C22×C7—816no—
371C22×C7—84no—
372C22×C7—84no—
373C22×C7#39414yes—
374C22×C7C2×#12011yescharacteristic
375C22×C7C2×#12111yescharacteristic
376C22×C7C2×#12311yescharacteristic
377C22×C7C2×AGL(1,ℤ/8)12yes—
378C22×C7#57512yes—
379C4×C7—42no—
380C4×C7—44no—
381C4×C7—81no—
382C4×C7—81no—
383C4×C7—82no—
384C4×C7#39414yes—
385C4×C7C2×#12111yescharacteristic
386C4×C7C2×#12111yescharacteristic
387C4×C7C2×#12311yescharacteristic
388C4×C7C2×#12411yescharacteristic
389C24—22no—
390C24—24no—
391C24—24no—
392C24—22no—
393C24—22no—
394C24—22no—
395C24—42no—
396C24—44no—
397C24—44no—
398C24—42no—
399C24—42no—
400C24C2×C7×D412yes—
401C24C2×C7×D411yescharacteristic
402C2×D4—216no—
403C2×D4—24no—
404C2×D4—28no—
405C2×D4—28no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—432no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—44no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4C2×C7×D414yes—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4C2×C7×D412yes—
442C22×C4C2×C7×D411yescharacteristic
443C22×C4C2×C7×D412yes—
444C22×C4C2×C7×D412yes—
445C22×C4C2×C7×D412yes—
446C22×C4C7×#4112yes—
447C22×C4C7×#4112yes—
448C22×C4C7×#4112yes—
449C22×C4C24×C711yescharacteristic
450M16—28no—
451M16—44no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8C2×C7×D414yescomplemented
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4C2×C7×D414yes—
467C22⋊C4—216no—
468C22⋊C4—44no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—44no—
472C42—42no—
473C42—42no—
474C2×C7—24no—
475C2×C7—44no—
476C2×C7—84no—
477C2×C7C2×#39411yescharacteristic
478C2×C7#300311yescharacteristic
479C2×C7#300511yescharacteristic
480C2×C7#305512yes—
481C2×C7#309412yes—
482C23—22no—
483C23—22no—
484C23—28no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—48no—
493C23—48no—
494C23—44no—
495C23—48no—
496C23—48no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—48no—
501C23—44no—
502C23—44no—
503C23—88no—
504C23—88no—
505C23C7×#12012yes—
506C23C7×#12112yes—
507C23C7×#12312yes—
508C23C7×AGL(1,ℤ/8)14yes—
509C23C22×C7×D411yescharacteristic
510D4—24no—
511D4—24no—
512D4—432no—
513D4—42no—
514D4—42no—
515D4—48no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—48no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—48no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4C7×#12012yes—
536C2×C4C7×#12114yes—
537C2×C4C7×#12114yes—
538C2×C4C7×#12112yes—
539C2×C4C7×#12314yes—
540C2×C4C7×#12312yes—
541C2×C4C7×#12414yes—
542C2×C4C22×C7×D411yescharacteristic
543C2×C4C22×C7×D411yescharacteristic
544C2×C4C22×C7×D411yescharacteristic
545C2×C4C2×C7×#4111yescharacteristic
546C2×C4C2×C7×#4111yescharacteristic
547C2×C4C2×C7×#4111yescharacteristic
548C8—24no—
549C8—42no—
550C7#2348011yescharacteristic, direct product factor, Sylow
551C22—22no—
552C22—22no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—24no—
557C22—24no—
558C22—416no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—816no—
567C22—84no—
568C22—84no—
569C22C7×#39414yes—
570C22C2×C7×#12011yescharacteristic
571C22C2×C7×#12111yescharacteristic
572C22C2×C7×#12311yescharacteristic
573C22C2×C7×AGL(1,ℤ/8)12yes—
574C22C7×#57512yes—
575C4—44no—
576C4—42no—
577C4—81no—
578C4—81no—
579C4—82no—
580C4C7×#39414yes—
581C4C2×C7×#12111yescharacteristic
582C4C2×C7×#12111yescharacteristic
583C4C2×C7×#12311yescharacteristic
584C4C2×C7×#12411yescharacteristic
585C2—24no—
586C2—44no—
587C2—84no—
588C2C2×C7×#39411yescharacteristic
589C2C7×#300311yescharacteristic
590C2C7×#300511yescharacteristic
591C2C7×#305512yes—
592C2C7×#309412yes—
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC22×C7×#121C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C24×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#428C2semidirect
3. C2×C7×#418C2semidirect
4. C2×C7×#418C2semidirect
5. C2×C7×#407C2semidirect
6. C7×#2280C2semidirect
7. C7×#2034C2semidirect
8. C7×#2033C2semidirect
9. C7×#1736C2semidirect
24. C2×C7×#107C22semidirect
25. C2×C7×#106C22semidirect
28. C2×C7×#102C22semidirect
29. C2×C7×#102C22semidirect
30. C2×C7×#102C22semidirect
32. C2×C7×C22⋊C8C22semidirect
34. C7×#380C22semidirect
35. C7×#380C22semidirect
38. C7×#340C22semidirect
39. #23480C7direct (N × H)
69. C2×C7×M16D4semidirect
103. C2×#428C2×C7semidirect
104. C2×#418C2×C7semidirect
105. C2×#418C2×C7semidirect
106. C2×#407C2×C7semidirect
107. #2280C2×C7semidirect
108. #2034C2×C7semidirect
109. #2033C2×C7semidirect
110. #1736C2×C7semidirect
179. C2×C7×C8C2×D4semidirect
210. C2×#107C22×C7semidirect
211. C2×#106C22×C7semidirect
214. C2×#102C22×C7semidirect
215. C2×#102C22×C7semidirect
216. C2×#102C22×C7semidirect
218. C2×C22⋊C8C22×C7semidirect
220. #380C22×C7semidirect
221. #380C22×C7semidirect
224. #340C22×C7semidirect
322. C2×M16C7×D4semidirect
457. C2×C8C2×C7×D4semidirect
550. C7#23480direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×#1212
C2×C7×#4282
C2×C7×#4182
C2×C7×#4072
C7×#22802
C7×#20342
C7×#20332
C7×#17362
#234807