Finite Group Databasebeta

← C7×#24501 #13201717

Subgroups — C7×#24501 #13201717

Order 1792.

1208 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#450C211yesmaximal, characteristic
2C2×C7×#431C211yesmaximal, characteristic, complemented
3C7×#2855C211yesmaximal, characteristic, complemented
4C2×C7×#390C211yesmaximal, characteristic
5C7×#2065C212yesmaximal, complemented
6C7×#2055C212yesmaximal, complemented
7C7×#1968C212yesmaximal, complemented
8C7×#1934C212yesmaximal, complemented
9C7×#1872C211yesmaximal, characteristic, complemented
10C7×#1853C211yesmaximal, characteristic, complemented
11C7×#1786C211yesmaximal, characteristic
12C23×C7×D4C2211yescharacteristic
13C22×C7×SD16C2212yes—
14C2×C7×C22⋊Q8C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16C22×C7×C22⋊C4—21no—
17C22×C7×C22⋊C4—21no—
18C22×C42×C7C2211yescharacteristic
19C7×#450—24no—
20C7×#431—24no—
21C2×C7×#106C2212yescomplemented
22C2×C7×C4⋊C8C2212yescomplemented
23C2×C7×#103C2212yescomplemented
24C2×C7×#102C2212yes—
25C2×C7×#102C2212yes—
26C2×C7×#102C2212yes—
27C2×C7×C22⋊C8C2211yescharacteristic
28C2×C7×C22⋊C8C2211yescharacteristic
29C2×C4×C7×C8C2212yescomplemented
30C7×#391C2211yescharacteristic, complemented
31C7×#391C2211yescharacteristic, complemented
32C7×#390—21no—
33C7×#390—21no—
34C7×#390C2212yes—
35C7×#390C2212yes—
36C7×#389C2211yescharacteristic, complemented
37C7×#389C2211yescharacteristic, complemented
38C7×#387—21no—
39C7×#387—21no—
40C7×#385—21no—
41C7×#381—21no—
42C7×#380C2211yescharacteristic
43C7×#340C2212yescomplemented
44C7×#340C2212yescomplemented
45#24501C711yesmaximal, characteristic, direct product factor, Sylow
46C25×C7—21no—
47C22×C7×Q8C2311yescharacteristic
48C22×C7×D4—24no—
49C22×C7×D4—21no—
50C22×C7×D4—22no—
51C22×C7×D4—22no—
52C22×C7×D4—42no—
53C22×C7×D4C2312yes—
54C23×C4×C7—21no—
55C23×C4×C7D411yescharacteristic
56C23×C4×C7D411yescharacteristic
57C23×C4×C7C2311yescharacteristic
58C2×C7×SD16—24no—
59C2×C7×SD16—24no—
60C2×C7×SD16—42no—
61C22×C7×C8C2312yes—
62C22×C7×C8C2312yes—
63C7×C22⋊Q8—22no—
64C7×C22⋊Q8—22no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—42no—
70C2×C7×C4⋊C4C2312yes—
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C22⋊C4—21no—
75C2×C7×C22⋊C4—21no—
76C2×C7×C22⋊C4—22no—
77C2×C7×C22⋊C4—22no—
78C2×C7×C22⋊C4—22no—
79C2×C7×C22⋊C4—22no—
80C2×C7×C22⋊C4—44no—
81C2×C7×C22⋊C4—44no—
82C2×C42×C7—22no—
83C2×C42×C7—22no—
84C2×C42×C7D411yescharacteristic
85C2×C42×C7D411yescharacteristic
86C2×C42×C7C2311yescharacteristic
87C2×C42×C7C2311yescharacteristic
88C7×#106—42no—
89C7×C4⋊C8—24no—
90C7×#103—42no—
91C7×#102—24no—
92C7×#102—24no—
93C7×#102—42no—
94C7×C22⋊C8—22no—
95C7×C22⋊C8D414yes—
96C4×C7×C8—24no—
97C7×#95—21no—
98C7×#95—21no—
99C7×#95—21no—
100C7×#95—21no—
101C7×#95—21no—
102C7×#95—21no—
103C7×#95—21no—
104C2×#450C2×C711yescharacteristic
105C2×#431C2×C711yescharacteristic, complemented
106#2855C2×C711yescharacteristic, complemented
107C2×#390C2×C711yescharacteristic
108#2065C2×C712yescomplemented
109#2055C2×C712yescomplemented
110#1968C2×C712yescomplemented
111#1934C2×C712yescomplemented
112#1872C2×C711yescharacteristic, complemented
113#1853C2×C711yescharacteristic, complemented
114#1786C2×C711yescharacteristic
115C24×C7—24no—
116C24×C7—22no—
117C24×C7—42no—
118C24×C7—42no—
119C24×C7—42no—
120C24×C7—42no—
121C24×C7—42no—
122C24×C7—42no—
123C24×C7C2×D411yescharacteristic
124C2×C7×Q8—21no—
125C2×C7×Q8—21no—
126C2×C7×Q8—22no—
127C2×C7×Q8—22no—
128C2×C7×D4—24no—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4—22no—
132C2×C7×D4—22no—
133C2×C7×D4—22no—
134C2×C7×D4—48no—
135C2×C7×D4—48no—
136C2×C7×D4—42no—
137C2×C7×D4—42no—
138C2×C7×D4#4114yes—
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—21no—
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—22no—
150C22×C4×C7—41no—
151C22×C4×C7—42no—
152C22×C4×C7—42no—
153C22×C4×C7—42no—
154C22×C4×C7SD1614yescomplemented
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D412yes—
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7#4112yes—
162C22×C4×C7C2411yescharacteristic
163C7×SD16—48no—
164C2×C7×C8—22no—
165C2×C7×C8—22no—
166C2×C7×C8—22no—
167C2×C7×C8—22no—
168C2×C7×C8—22no—
169C2×C7×C8C2×D414yes—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—42no—
177C7×C4⋊C4—42no—
178C7×C4⋊C4—42no—
179C7×C4⋊C4—42no—
180C7×C22⋊C4—44no—
181C7×C22⋊C4—42no—
182C7×C22⋊C4—42no—
183C7×C22⋊C4—44no—
184C7×C22⋊C4—44no—
185C7×C22⋊C4—44no—
186C42×C7—22no—
187C42×C7—22no—
188C42×C7—21no—
189C42×C7—21no—
190C42×C7—21no—
191C42×C7—21no—
192C23×D4C22×C711yescharacteristic
193C22×SD16C22×C712yes—
194C2×C22⋊Q8C22×C711yescharacteristic
195C22×C4⋊C4C22×C711yescharacteristic
196C22×C22⋊C4—21no—
197C22×C22⋊C4—21no—
198C22×C42C22×C711yescharacteristic
199#450—24no—
200#431—24no—
201C2×#106C22×C712yescomplemented
202C2×C4⋊C8C22×C712yescomplemented
203C2×#103C22×C712yescomplemented
204C2×#102C22×C712yes—
205C2×#102C22×C712yes—
206C2×#102C22×C712yes—
207C2×C22⋊C8C22×C711yescharacteristic
208C2×C22⋊C8C22×C711yescharacteristic
209C2×C4×C8C22×C712yescomplemented
210#391C22×C711yescharacteristic, complemented
211#391C22×C711yescharacteristic, complemented
212#390—21no—
213#390—21no—
214#390C22×C712yes—
215#390C22×C712yes—
216#389C22×C711yescharacteristic, complemented
217#389C22×C711yescharacteristic, complemented
218#387—21no—
219#387—21no—
220#385—21no—
221#381—21no—
222#380C22×C711yescharacteristic
223#340C22×C712yescomplemented
224#340C22×C712yescomplemented
225C23×C7—21no—
226C23×C7—21no—
227C23×C7—21no—
228C23×C7—21no—
229C23×C7—24no—
230C23×C7—48no—
231C23×C7—48no—
232C23×C7—48no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7—42no—
237C23×C7—42no—
238C23×C7—42no—
239C23×C7—84no—
240C23×C7—84no—
241C23×C7—84no—
242C23×C7—84no—
243C23×C7#12112yes—
244C23×C7C2×SD1612yes—
245C23×C7#13512yes—
246C23×C7C22×D411yescharacteristic
247C7×Q8—42no—
248C7×Q8—42no—
249C7×D4—24no—
250C7×D4—24no—
251C7×D4—48no—
252C7×D4—42no—
253C7×D4—42no—
254C2×C4×C7—21no—
255C2×C4×C7—21no—
256C2×C4×C7—21no—
257C2×C4×C7—21no—
258C2×C4×C7—21no—
259C2×C4×C7—21no—
260C2×C4×C7—21no—
261C2×C4×C7—21no—
262C2×C4×C7—28no—
263C2×C4×C7—24no—
264C2×C4×C7—24no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—44no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7—42no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—42no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7—42no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—84no—
285C2×C4×C7#12112yes—
286C2×C4×C7C2×SD1612yes—
287C2×C4×C7C2×SD1612yes—
288C2×C4×C7AGL(1,ℤ/8)12yes—
289C2×C4×C7C22×D411yescharacteristic
290C2×C4×C7C2×#4112yes—
291C2×C4×C7C2×#4111yescharacteristic
292C2×C4×C7H(5,𝔽2)12yes—
293C7×C8—24no—
294C7×C8—42no—
295C25—21no—
296C22×Q8C23×C711yescharacteristic
297C22×D4—24no—
298C22×D4—21no—
299C22×D4—22no—
300C22×D4—22no—
301C22×D4—42no—
302C22×D4C23×C712yes—
303C23×C4—21no—
304C23×C4C7×D411yescharacteristic
305C23×C4C7×D411yescharacteristic
306C23×C4C23×C711yescharacteristic
307C2×SD16—24no—
308C2×SD16—24no—
309C2×SD16—42no—
310C22×C8C23×C712yes—
311C22×C8C23×C712yes—
312C22⋊Q8—22no—
313C22⋊Q8—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4C23×C712yes—
320C2×C4⋊C4C23×C711yescharacteristic
321C2×C4⋊C4C23×C711yescharacteristic
322C2×C4⋊C4C23×C711yescharacteristic
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42C7×D411yescharacteristic
334C2×C42C7×D411yescharacteristic
335C2×C42C23×C711yescharacteristic
336C2×C42C23×C711yescharacteristic
337#106—42no—
338C4⋊C8—24no—
339#103—42no—
340#102—24no—
341#102—24no—
342#102—42no—
343C22⋊C8—22no—
344C22⋊C8C7×D414yes—
345C4×C8—24no—
346#95—21no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
353C22×C7—22no—
354C22×C7—22no—
355C22×C7—22no—
356C22×C7—22no—
357C22×C7—22no—
358C22×C7—22no—
359C22×C7—44no—
360C22×C7—44no—
361C22×C7—44no—
362C22×C7—816no—
363C22×C7—82no—
364C22×C7—82no—
365C22×C7—82no—
366C22×C7—82no—
367C22×C7#46514yes—
368C22×C7C2×#12111yescharacteristic
369C22×C7#52511yescharacteristic
370C22×C7#53511yescharacteristic
371C22×C7C22×SD1611yescharacteristic
372C22×C7C2×#13511yescharacteristic
373C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
374C22×C7#57511yescharacteristic
375C4×C7—22no—
376C4×C7—22no—
377C4×C7—22no—
378C4×C7—22no—
379C4×C7—41no—
380C4×C7—81no—
381C4×C7—81no—
382C4×C7—82no—
383C4×C7—81no—
384C4×C7—81no—
385C4×C7#54612yes—
386C4×C7#55112yes—
387C24—24no—
388C24—22no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24C2×C7×D411yescharacteristic
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4C7×#4114yes—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—41no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4C7×SD1614yescomplemented
427C22×C4C2×C7×D411yescharacteristic
428C22×C4C2×C7×D411yescharacteristic
429C22×C4C2×C7×D412yes—
430C22×C4C2×C7×D411yescharacteristic
431C22×C4C2×C7×D411yescharacteristic
432C22×C4C2×C7×D411yescharacteristic
433C22×C4C7×#4112yes—
434C22×C4C24×C711yescharacteristic
435SD16—48no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8C2×C7×D414yes—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C22⋊C4—44no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C42—22no—
459C42—22no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C2×C7—24no—
465C2×C7—84no—
466C2×C7#241111yescharacteristic
467C2×C7C2×#46511yescharacteristic
468C2×C7#305511yescharacteristic
469C2×C7#310111yescharacteristic
470C2×C7#310511yescharacteristic
471C2×C7#320811yescharacteristic
472C2×C7#321111yescharacteristic
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—24no—
478C23—48no—
479C23—48no—
480C23—48no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—84no—
488C23—84no—
489C23—84no—
490C23—84no—
491C23C7×#12112yes—
492C23C2×C7×SD1612yes—
493C23C7×#13512yes—
494C23C22×C7×D411yescharacteristic
495Q8—42no—
496Q8—42no—
497D4—24no—
498D4—24no—
499D4—48no—
500D4—42no—
501D4—42no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—44no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—84no—
533C2×C4C7×#12112yes—
534C2×C4C2×C7×SD1612yes—
535C2×C4C2×C7×SD1612yes—
536C2×C4C7×AGL(1,ℤ/8)12yes—
537C2×C4C22×C7×D411yescharacteristic
538C2×C4C2×C7×#4112yes—
539C2×C4C2×C7×#4111yescharacteristic
540C2×C4C7×H(5,𝔽2)12yes—
541C8—24no—
542C8—42no—
543C7#2450111yescharacteristic, direct product factor, Sylow
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—816no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22C7×#46514yes—
559C22C2×C7×#12111yescharacteristic
560C22C7×#52511yescharacteristic
561C22C7×#53511yescharacteristic
562C22C22×C7×SD1611yescharacteristic
563C22C2×C7×#13511yescharacteristic
564C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
565C22C7×#57511yescharacteristic
566C4—22no—
567C4—22no—
568C4—22no—
569C4—22no—
570C4—41no—
571C4—81no—
572C4—82no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4C7×#54612yes—
577C4C7×#55112yes—
578C2—24no—
579C2—84no—
580C2C7×#241111yescharacteristic
581C2C2×C7×#46511yescharacteristic
582C2C7×#305511yescharacteristic
583C2C7×#310111yescharacteristic
584C2C7×#310511yescharacteristic
585C2C7×#320811yescharacteristic
586C2C7×#321111yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC23×C7×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C22×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#431C2semidirect
3. C7×#2855C2semidirect
5. C7×#2065C2semidirect
6. C7×#2055C2semidirect
7. C7×#1968C2semidirect
8. C7×#1934C2semidirect
9. C7×#1872C2semidirect
10. C7×#1853C2semidirect
21. C2×C7×#106C22semidirect
22. C2×C7×C4⋊C8C22semidirect
23. C2×C7×#103C22semidirect
29. C2×C4×C7×C8C22semidirect
30. C7×#391C22semidirect
31. C7×#391C22semidirect
36. C7×#389C22semidirect
37. C7×#389C22semidirect
43. C7×#340C22semidirect
44. C7×#340C22semidirect
45. #24501C7direct (N × H)
105. C2×#431C2×C7semidirect
106. #2855C2×C7semidirect
108. #2065C2×C7semidirect
109. #2055C2×C7semidirect
110. #1968C2×C7semidirect
111. #1934C2×C7semidirect
112. #1872C2×C7semidirect
113. #1853C2×C7semidirect
154. C22×C4×C7SD16semidirect
201. C2×#106C22×C7semidirect
202. C2×C4⋊C8C22×C7semidirect
203. C2×#103C22×C7semidirect
209. C2×C4×C8C22×C7semidirect
210. #391C22×C7semidirect
211. #391C22×C7semidirect
216. #389C22×C7semidirect
217. #389C22×C7semidirect
223. #340C22×C7semidirect
224. #340C22×C7semidirect
426. C22×C4C7×SD16semidirect
543. C7#24501direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4502
C2×C7×#4312
C7×#28552
C2×C7×#3902
C7×#20652
C7×#20552
C7×#19682
C7×#19342
C7×#18722
C7×#18532
C7×#17862
#245017