Finite Group Databasebeta

← C7×#24121 #13201337

Subgroups — C7×#24121 #13201337

Order 1792.

1154 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#555C211yesmaximal, characteristic
2C2×C7×#477C211yesmaximal, characteristic, complemented
3C2×C4×C7×SD16C211yesmaximal, characteristic, complemented
4C7×#2538C211yesmaximal, characteristic, complemented
5C7×#2033C212yesmaximal, complemented
6C7×#2000C212yesmaximal, complemented
7C7×#1990C211yesmaximal, characteristic, complemented
8C7×#1954C211yesmaximal, characteristic, complemented
9C7×#1873C212yesmaximal, complemented
10C7×#1854C212yesmaximal, complemented
11C7×#1802C211yesmaximal, characteristic, complemented
12C22×C7×SD16C2211yescharacteristic
13C7×#555—24no—
14C7×#555—24no—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#124—22no—
17C2×C7×C22⋊Q8C2212yes—
18C2×C7×#121—21no—
19C2×C7×#121C2212yes—
20C2×C4×C7×Q8C2211yescharacteristic
21C2×C4×C7×D4C2211yescharacteristic
22C2×C4×C7×D4C2211yescharacteristic
23C2×C7×#117—21no—
24C2×C7×#117C2212yes—
25C7×#477—24no—
26C4×C7×SD16—24no—
27C2×C7×#107C2212yes—
28C2×C7×#106C2211yescharacteristic
29C2×C7×C4⋊C8C2211yescharacteristic
30C2×C7×#103C2212yes—
31C2×C7×#103C2211yescharacteristic
32C2×C7×#102C2211yescharacteristic
33C2×C7×#102C2212yes—
34C2×C7×C22⋊C8C2212yes—
35C2×C7×C22⋊C8C2212yes—
36C2×C4×C7×C8C2211yescharacteristic
37C7×#396—22no—
38C7×#390—21no—
39C7×#389—21no—
40C7×#384C2211yescharacteristic
41C7×#380C2212yes—
42C7×#340C2212yes—
43C7×#340C2212yes—
44C7×#340C2212yes—
45#24121C711yesmaximal, characteristic, direct product factor, Sylow
46C22×C7×Q8—21no—
47C22×C7×Q8C2311yescharacteristic
48C22×C7×D4—22no—
49C22×C7×D4D411yescharacteristic
50C22×C7×D4C2311yescharacteristic
51C23×C4×C7—21no—
52C23×C4×C7C2312yes—
53C2×C7×SD16—22no—
54C2×C7×SD16—22no—
55C2×C7×SD16—22no—
56C22×C7×C8C2311yescharacteristic
57C22×C7×C8C2312yes—
58C22×C7×C8C2311yescharacteristic
59C7×C4⋊Q8—22no—
60C7×C4⋊Q8D414yes—
61C7×#124—216no—
62C7×C22⋊Q8—24no—
63C7×C22⋊Q8—42no—
64C7×#121—28no—
65C7×#121—24no—
66C7×#121—24no—
67C4×C7×Q8—22no—
68C4×C7×Q8—22no—
69C4×C7×D4—22no—
70C4×C7×D4—22no—
71C4×C7×D4—22no—
72C4×C7×D4D414yes—
73C7×#117—28no—
74C7×#117—24no—
75C7×#117—24no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4—22no—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4D411yescharacteristic
82C2×C7×C4⋊C4C2312yes—
83C2×C7×C4⋊C4C2311yescharacteristic
84C2×C7×C4⋊C4C2311yescharacteristic
85C2×C7×C4⋊C4C2312yes—
86C2×C7×C22⋊C4—22no—
87C2×C7×C22⋊C4—22no—
88C2×C7×C22⋊C4—21no—
89C2×C7×C22⋊C4—22no—
90C2×C7×C22⋊C4—22no—
91C2×C7×C22⋊C4D412yes—
92C2×C42×C7—22no—
93C2×C42×C7—21no—
94C2×C42×C7C2311yescharacteristic
95C7×#107—24no—
96C7×#106—22no—
97C7×C4⋊C8—22no—
98C7×#103—24no—
99C7×#103—22no—
100C7×#102—22no—
101C7×#102—42no—
102C7×C22⋊C8—24no—
103C7×C22⋊C8—42no—
104C4×C7×C8—22no—
105C7×#95—22no—
106C2×#555C2×C711yescharacteristic
107C2×#477C2×C711yescharacteristic, complemented
108C2×C4×SD16C2×C711yescharacteristic, complemented
109#2538C2×C711yescharacteristic, complemented
110#2033C2×C712yescomplemented
111#2000C2×C712yescomplemented
112#1990C2×C711yescharacteristic, complemented
113#1954C2×C711yescharacteristic, complemented
114#1873C2×C712yescomplemented
115#1854C2×C712yescomplemented
116#1802C2×C711yescharacteristic, complemented
117C24×C7—21no—
118C24×C7C2×D412yes—
119C2×C7×Q8—24no—
120C2×C7×Q8—21no—
121C2×C7×Q8—21no—
122C2×C7×Q8—21no—
123C2×C7×Q8—21no—
124C2×C7×Q8—21no—
125C2×C7×Q8—81no—
126C2×C7×Q8—81no—
127C2×C7×Q8#4112yes—
128C2×C7×D4—28no—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4—22no—
132C2×C7×D4—21no—
133C2×C7×D4—21no—
134C2×C7×D4—21no—
135C2×C7×D4—21no—
136C2×C7×D4—21no—
137C2×C7×D4—82no—
138C2×C7×D4—82no—
139C2×C7×D4C2×D412yes—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—22no—
143C22×C4×C7—24no—
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—21no—
151C22×C4×C7—22no—
152C22×C4×C7—42no—
153C22×C4×C7—42no—
154C22×C4×C7C2×D412yes—
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2411yescharacteristic
158C7×SD16—28no—
159C2×C7×C8—21no—
160C2×C7×C8—21no—
161C2×C7×C8—21no—
162C2×C7×C8—22no—
163C2×C7×C8—22no—
164C2×C7×C8—21no—
165C2×C7×C8—22no—
166C2×C7×C8—21no—
167C2×C7×C8#4112yes—
168C7×C4⋊C4—24no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—44no—
177C7×C4⋊C4—42no—
178C7×C4⋊C4—42no—
179C7×C4⋊C4—81no—
180C7×C4⋊C4C2×D412yes—
181C7×C4⋊C4C2×D414yes—
182C7×C22⋊C4—28no—
183C7×C22⋊C4—28no—
184C7×C22⋊C4—24no—
185C7×C22⋊C4—24no—
186C7×C22⋊C4—44no—
187C7×C22⋊C4—82no—
188C42×C7—28no—
189C42×C7—42no—
190C42×C7C2×D412yes—
191C42×C7C2×D412yes—
192C22×SD16C22×C711yescharacteristic
193#555—24no—
194#555—24no—
195C2×C4⋊Q8C22×C711yescharacteristic
196C2×#124—22no—
197C2×C22⋊Q8C22×C712yes—
198C2×#121—21no—
199C2×#121C22×C712yes—
200C2×C4×Q8C22×C711yescharacteristic
201C2×C4×D4C22×C711yescharacteristic
202C2×C4×D4C22×C711yescharacteristic
203C2×#117—21no—
204C2×#117C22×C712yes—
205#477—24no—
206C4×SD16—24no—
207C2×#107C22×C712yes—
208C2×#106C22×C711yescharacteristic
209C2×C4⋊C8C22×C711yescharacteristic
210C2×#103C22×C712yes—
211C2×#103C22×C711yescharacteristic
212C2×#102C22×C711yescharacteristic
213C2×#102C22×C712yes—
214C2×C22⋊C8C22×C712yes—
215C2×C22⋊C8C22×C712yes—
216C2×C4×C8C22×C711yescharacteristic
217#396—22no—
218#390—21no—
219#389—21no—
220#384C22×C711yescharacteristic
221#380C22×C712yes—
222#340C22×C712yes—
223#340C22×C712yes—
224#340C22×C712yes—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7—22no—
228C23×C7—22no—
229C23×C7—22no—
230C23×C7—22no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—41no—
234C23×C7—41no—
235C23×C7—41no—
236C23×C7—41no—
237C23×C7—41no—
238C23×C7—41no—
239C23×C7C22×D411yescharacteristic
240C7×Q8—22no—
241C7×Q8—22no—
242C7×Q8—41no—
243C7×Q8—41no—
244C7×Q8—84no—
245C7×D4—22no—
246C7×D4—22no—
247C7×D4—41no—
248C7×D4—41no—
249C7×D4—44no—
250C7×D4—88no—
251C2×C4×C7—21no—
252C2×C4×C7—21no—
253C2×C4×C7—22no—
254C2×C4×C7—22no—
255C2×C4×C7—24no—
256C2×C4×C7—22no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—41no—
260C2×C4×C7—41no—
261C2×C4×C7—41no—
262C2×C4×C7—41no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7—48no—
270C2×C4×C7—41no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7—41no—
274C2×C4×C7—42no—
275C2×C4×C7—41no—
276C2×C4×C7#12012yes—
277C2×C4×C7#12012yes—
278C2×C4×C7#13512yes—
279C2×C4×C7#13712yes—
280C2×C4×C7C22×D411yescharacteristic
281C2×C4×C7C22×D411yescharacteristic
282C2×C4×C7C2×#4111yescharacteristic
283C2×C4×C7C2×#4111yescharacteristic
284C2×C4×C7H(5,𝔽2)11yescharacteristic
285C2×C4×C72−1+411yescharacteristic
286C7×C8—22no—
287C7×C8—41no—
288C7×C8—42no—
289C22×Q8—21no—
290C22×Q8C23×C711yescharacteristic
291C22×D4—22no—
292C22×D4C7×D411yescharacteristic
293C22×D4C23×C711yescharacteristic
294C23×C4—21no—
295C23×C4C23×C712yes—
296C2×SD16—22no—
297C2×SD16—22no—
298C2×SD16—22no—
299C22×C8C23×C711yescharacteristic
300C22×C8C23×C712yes—
301C22×C8C23×C711yescharacteristic
302C4⋊Q8—22no—
303C4⋊Q8C7×D414yes—
304#124—216no—
305C22⋊Q8—24no—
306C22⋊Q8—42no—
307#121—28no—
308#121—24no—
309#121—24no—
310C4×Q8—22no—
311C4×Q8—22no—
312C4×D4—22no—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4C7×D414yes—
316#117—28no—
317#117—24no—
318#117—24no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4C7×D411yescharacteristic
325C2×C4⋊C4C23×C711yescharacteristic
326C2×C4⋊C4C23×C712yes—
327C2×C4⋊C4C23×C711yescharacteristic
328C2×C4⋊C4C23×C712yes—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4C7×D412yes—
335C2×C42—22no—
336C2×C42—21no—
337C2×C42C23×C711yescharacteristic
338#107—24no—
339#106—22no—
340C4⋊C8—22no—
341#103—22no—
342#103—24no—
343#102—22no—
344#102—42no—
345C22⋊C8—24no—
346C22⋊C8—42no—
347C4×C8—22no—
348#95—22no—
349C22×C7—42no—
350C22×C7—42no—
351C22×C7—42no—
352C22×C7—42no—
353C22×C7—42no—
354C22×C7—42no—
355C22×C7—42no—
356C22×C7—42no—
357C22×C7—42no—
358C22×C7—42no—
359C22×C7—81no—
360C22×C7—81no—
361C22×C7—81no—
362C22×C7—81no—
363C22×C7C2×#12011yescharacteristic
364C22×C7#54611yescharacteristic
365C22×C7#54711yescharacteristic
366C22×C7C2×#13511yescharacteristic
367C22×C7C2×#13711yescharacteristic
368C22×C7#57611yescharacteristic
369C22×C7#57711yescharacteristic
370C4×C7—22no—
371C4×C7—42no—
372C4×C7—42no—
373C4×C7—42no—
374C4×C7—44no—
375C4×C7—81no—
376C4×C7—82no—
377C4×C7#45214yes—
378C4×C7#52511yescharacteristic
379C4×C7#53711yescharacteristic
380C4×C7#54811yescharacteristic
381C4×C7#55011yescharacteristic
382C24—21no—
383C24C2×C7×D412yes—
384C2×Q8—24no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—81no—
391C2×Q8—81no—
392C2×Q8C7×#4112yes—
393C2×D4—28no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—82no—
403C2×D4—82no—
404C2×D4C2×C7×D412yes—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4C2×C7×D412yes—
420C22×C4C2×C7×D411yescharacteristic
421C22×C4C2×C7×D411yescharacteristic
422C22×C4C24×C711yescharacteristic
423SD16—28no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—21no—
430C2×C8—22no—
431C2×C8—21no—
432C2×C8C7×#4112yes—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—44no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—81no—
445C4⋊C4C2×C7×D412yes—
446C4⋊C4C2×C7×D414yes—
447C22⋊C4—28no—
448C22⋊C4—28no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—44no—
452C22⋊C4—82no—
453C42—28no—
454C42—42no—
455C42C2×C7×D412yes—
456C42C2×C7×D412yes—
457C2×C7—82no—
458C2×C7—82no—
459C2×C7#240711yescharacteristic
460C2×C7C2×#45211yescharacteristic
461C2×C7#301211yescharacteristic
462C2×C7#330511yescharacteristic
463C2×C7#332911yescharacteristic
464C2×C7#334011yescharacteristic
465C2×C7#335711yescharacteristic
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23C22×C7×D411yescharacteristic
481Q8—22no—
482Q8—22no—
483Q8—41no—
484Q8—41no—
485Q8—84no—
486D4—22no—
487D4—22no—
488D4—41no—
489D4—41no—
490D4—44no—
491D4—88no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—48no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4C7×#12012yes—
518C2×C4C7×#12012yes—
519C2×C4C7×#13512yes—
520C2×C4C7×#13712yes—
521C2×C4C22×C7×D411yescharacteristic
522C2×C4C22×C7×D411yescharacteristic
523C2×C4C2×C7×#4111yescharacteristic
524C2×C4C2×C7×#4111yescharacteristic
525C2×C4C7×H(5,𝔽2)11yescharacteristic
526C2×C4C7×2−1+411yescharacteristic
527C8—22no—
528C8—41no—
529C8—42no—
530C7#2412111yescharacteristic, direct product factor, Sylow
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22C2×C7×#12011yescharacteristic
546C22C7×#54611yescharacteristic
547C22C7×#54711yescharacteristic
548C22C2×C7×#13511yescharacteristic
549C22C2×C7×#13711yescharacteristic
550C22C7×#57611yescharacteristic
551C22C7×#57711yescharacteristic
552C4—22no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—44no—
557C4—81no—
558C4—82no—
559C4C7×#45214yes—
560C4C7×#52511yescharacteristic
561C4C7×#53711yescharacteristic
562C4C7×#54811yescharacteristic
563C4C7×#55011yescharacteristic
564C2—82no—
565C2—82no—
566C2C7×#240711yescharacteristic
567C2C2×C7×#45211yescharacteristic
568C2C7×#301211yescharacteristic
569C2C7×#330511yescharacteristic
570C2C7×#332911yescharacteristic
571C2C7×#334011yescharacteristic
572C2C7×#335711yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#555C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#477C2semidirect
3. C2×C4×C7×SD16C2semidirect
4. C7×#2538C2semidirect
5. C7×#2033C2semidirect
6. C7×#2000C2semidirect
7. C7×#1990C2semidirect
8. C7×#1954C2semidirect
9. C7×#1873C2semidirect
10. C7×#1854C2semidirect
11. C7×#1802C2semidirect
45. #24121C7direct (N × H)
107. C2×#477C2×C7semidirect
108. C2×C4×SD16C2×C7semidirect
109. #2538C2×C7semidirect
110. #2033C2×C7semidirect
111. #2000C2×C7semidirect
112. #1990C2×C7semidirect
113. #1954C2×C7semidirect
114. #1873C2×C7semidirect
115. #1854C2×C7semidirect
116. #1802C2×C7semidirect
530. C7#24121direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5552
C2×C7×#4772
C2×C4×C7×SD162
C7×#25382
C7×#20332
C7×#20002
C7×#19902
C7×#19542
C7×#18732
C7×#18542
C7×#18022
#241217