Finite Group Databasebeta

← C7×#24185 #13201401

Subgroups — C7×#24185 #13201401

Order 1792.

1150 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#556C211yesmaximal, characteristic, complemented
2C2×C7×#461C211yesmaximal, characteristic, complemented
3C2×C4×C7×D8C211yesmaximal, characteristic
4C7×#2538C211yesmaximal, characteristic, complemented
5C7×#2042C212yesmaximal, complemented
6C7×#2012C212yesmaximal
7C7×#1984C211yesmaximal, characteristic, complemented
8C7×#1956C211yesmaximal, characteristic, complemented
9C7×#1880C212yesmaximal, complemented
10C7×#1863C212yesmaximal, complemented
11C7×#1802C211yesmaximal, characteristic, complemented
12C22×C7×SD16C2212yes—
13C22×C7×D8C2211yescharacteristic
14C7×#556—24no—
15C7×#556—24no—
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×#126—22no—
18C2×C7×#124C2212yes—
19C2×C7×C22⋊Q8—21no—
20C2×C7×C22⋊Q8—21no—
21C2×C4×C7×Q8C2212yes—
22C2×C4×C7×Q8C2211yescharacteristic
23C2×C4×C7×D4C2211yescharacteristic
24C2×C4×C7×D4C2211yescharacteristic
25C2×C7×#117—21no—
26C7×#461—24no—
27C4×C7×D8—24no—
28C2×C7×#107C2211yescharacteristic
29C2×C7×C4⋊C8C2211yescharacteristic
30C2×C7×#103C2212yes—
31C2×C7×#103C2212yes—
32C2×C7×#103C2212yes—
33C2×C7×#102C2211yescharacteristic
34C2×C7×#102C2211yescharacteristic
35C2×C7×#102C2212yes—
36C2×C4×C7×C8C2211yescharacteristic
37C7×#396—22no—
38C7×#390C2212yes—
39C7×#389C2212yes—
40C7×#384C2211yescharacteristic
41C7×#380—21no—
42C7×#340C2212yes—
43C7×#336C2212yes—
44C7×#336C2212yes—
45#24185C711yesmaximal, characteristic, direct product factor, Sylow
46C22×C7×Q8D411yescharacteristic
47C22×C7×Q8C2312yes—
48C22×C7×D4C2311yescharacteristic
49C22×C7×D4C2311yescharacteristic
50C23×C4×C7—21no—
51C23×C4×C7—21no—
52C2×C7×SD16—24no—
53C2×C7×SD16—24no—
54C2×C7×SD16—42no—
55C2×C7×D8—22no—
56C2×C7×D8—22no—
57C2×C7×D8—22no—
58C22×C7×C8C2312yes—
59C22×C7×C8C2311yescharacteristic
60C22×C7×C8C2311yescharacteristic
61C7×C4⋊Q8—22no—
62C7×C4⋊Q8D414yes—
63C7×#126—216no—
64C7×#124—24no—
65C7×#124—24no—
66C7×C22⋊Q8—28no—
67C7×C22⋊Q8—44no—
68C4×C7×Q8—24no—
69C4×C7×Q8—24no—
70C4×C7×Q8—22no—
71C4×C7×Q8—22no—
72C4×C7×D4—22no—
73C4×C7×D4—22no—
74C4×C7×D4—22no—
75C4×C7×D4D414yes—
76C7×#117—28no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—22no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—22no—
82C2×C7×C4⋊C4—21no—
83C2×C7×C4⋊C4D411yescharacteristic
84C2×C7×C4⋊C4D412yes—
85C2×C7×C4⋊C4C2312yes—
86C2×C7×C4⋊C4C2311yescharacteristic
87C2×C7×C4⋊C4C2311yescharacteristic
88C2×C7×C22⋊C4—22no—
89C2×C7×C22⋊C4—21no—
90C2×C7×C22⋊C4—22no—
91C2×C7×C22⋊C4—21no—
92C2×C7×C22⋊C4—22no—
93C2×C7×C22⋊C4—22no—
94C2×C42×C7—22no—
95C2×C42×C7C2312yes—
96C2×C42×C7C2311yescharacteristic
97C7×#107—22no—
98C7×C4⋊C8—22no—
99C7×#103—24no—
100C7×#103—24no—
101C7×#103—42no—
102C7×#102—22no—
103C7×#102—22no—
104C7×#102—42no—
105C4×C7×C8—22no—
106C7×#95—22no—
107C2×#556C2×C711yescharacteristic, complemented
108C2×#461C2×C711yescharacteristic, complemented
109C2×C4×D8C2×C711yescharacteristic
110#2538C2×C711yescharacteristic, complemented
111#2042C2×C712yescomplemented
112#2012C2×C712yes—
113#1984C2×C711yescharacteristic, complemented
114#1956C2×C711yescharacteristic, complemented
115#1880C2×C712yescomplemented
116#1863C2×C712yescomplemented
117#1802C2×C711yescharacteristic, complemented
118C24×C7—21no—
119C24×C7—21no—
120C2×C7×Q8—22no—
121C2×C7×Q8—22no—
122C2×C7×Q8—22no—
123C2×C7×Q8—22no—
124C2×C7×Q8—22no—
125C2×C7×Q8—22no—
126C2×C7×Q8—22no—
127C2×C7×Q8—22no—
128C2×C7×Q8C2×D414yes—
129C2×C7×D4—21no—
130C2×C7×D4—21no—
131C2×C7×D4—21no—
132C2×C7×D4—21no—
133C2×C7×D4—21no—
134C2×C7×D4—21no—
135C2×C7×D4—21no—
136C2×C7×D4—21no—
137C2×C7×D4—21no—
138C2×C7×D4—21no—
139C2×C7×D4C2×D412yes—
140C2×C7×D4#4112yes—
141C22×C4×C7—21no—
142C22×C4×C7—22no—
143C22×C4×C7—24no—
144C22×C4×C7—21no—
145C22×C4×C7—22no—
146C22×C4×C7—42no—
147C22×C4×C7—42no—
148C22×C4×C7—42no—
149C22×C4×C7—42no—
150C22×C4×C7—42no—
151C22×C4×C7C2×D412yes—
152C22×C4×C7C2×D412yes—
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2411yescharacteristic
156C7×SD16—48no—
157C7×D8—28no—
158C2×C7×C8—21no—
159C2×C7×C8—21no—
160C2×C7×C8—21no—
161C2×C7×C8—22no—
162C2×C7×C8—22no—
163C2×C7×C8—22no—
164C2×C7×C8—21no—
165C2×C7×C8—21no—
166C2×C7×C8#4112yes—
167C7×C4⋊C4—28no—
168C7×C4⋊C4—24no—
169C7×C4⋊C4—24no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—24no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—44no—
178C7×C4⋊C4—42no—
179C7×C4⋊C4—81no—
180C7×C4⋊C4C2×D412yes—
181C7×C22⋊C4—28no—
182C7×C22⋊C4—28no—
183C7×C22⋊C4—24no—
184C7×C22⋊C4—44no—
185C7×C22⋊C4—42no—
186C7×C22⋊C4—82no—
187C42×C7—28no—
188C42×C7—22no—
189C42×C7—22no—
190C42×C7C2×D412yes—
191C42×C7C2×D412yes—
192C22×SD16C22×C712yes—
193C22×D8C22×C711yescharacteristic
194#556—24no—
195#556—24no—
196C2×C4⋊Q8C22×C711yescharacteristic
197C2×#126—22no—
198C2×#124C22×C712yes—
199C2×C22⋊Q8—21no—
200C2×C22⋊Q8—21no—
201C2×C4×Q8C22×C712yes—
202C2×C4×Q8C22×C711yescharacteristic
203C2×C4×D4C22×C711yescharacteristic
204C2×C4×D4C22×C711yescharacteristic
205C2×#117—21no—
206#461—24no—
207C4×D8—24no—
208C2×#107C22×C711yescharacteristic
209C2×C4⋊C8C22×C711yescharacteristic
210C2×#103C22×C712yes—
211C2×#103C22×C712yes—
212C2×#103C22×C712yes—
213C2×#102C22×C711yescharacteristic
214C2×#102C22×C711yescharacteristic
215C2×#102C22×C712yes—
216C2×C4×C8C22×C711yescharacteristic
217#396—22no—
218#390C22×C712yes—
219#389C22×C712yes—
220#384C22×C711yescharacteristic
221#380—21no—
222#340C22×C712yes—
223#336C22×C712yes—
224#336C22×C712yes—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7—41no—
228C23×C7—41no—
229C23×C7—41no—
230C23×C7—41no—
231C23×C7—41no—
232C23×C7—41no—
233C23×C7—41no—
234C23×C7—41no—
235C23×C7—41no—
236C23×C7—41no—
237C23×C7—41no—
238C23×C7—41no—
239C23×C7C22×D411yescharacteristic
240C7×Q8—24no—
241C7×Q8—24no—
242C7×Q8—42no—
243C7×Q8—42no—
244C7×Q8—44no—
245C7×D4—22no—
246C7×D4—22no—
247C7×D4—22no—
248C7×D4—22no—
249C7×D4—41no—
250C7×D4—41no—
251C7×D4—41no—
252C7×D4—41no—
253C2×C4×C7—21no—
254C2×C4×C7—21no—
255C2×C4×C7—22no—
256C2×C4×C7—22no—
257C2×C4×C7—24no—
258C2×C4×C7—22no—
259C2×C4×C7—22no—
260C2×C4×C7—22no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—41no—
266C2×C4×C7—41no—
267C2×C4×C7—48no—
268C2×C4×C7—41no—
269C2×C4×C7—42no—
270C2×C4×C7—41no—
271C2×C4×C7—42no—
272C2×C4×C7—41no—
273C2×C4×C7—42no—
274C2×C4×C7—84no—
275C2×C4×C7#12012yes—
276C2×C4×C7#12012yes—
277C2×C4×C7#13512yes—
278C2×C4×C7AGL(1,ℤ/8)12yes—
279C2×C4×C7C22×D411yescharacteristic
280C2×C4×C7C22×D411yescharacteristic
281C2×C4×C7C2×#4111yescharacteristic
282C2×C4×C7C2×#4111yescharacteristic
283C2×C4×C7H(5,𝔽2)11yescharacteristic
284C2×C4×C72−1+411yescharacteristic
285C7×C8—22no—
286C7×C8—41no—
287C7×C8—42no—
288C22×Q8C7×D411yescharacteristic
289C22×Q8C23×C712yes—
290C22×D4C23×C711yescharacteristic
291C22×D4C23×C711yescharacteristic
292C23×C4—21no—
293C23×C4—21no—
294C2×SD16—24no—
295C2×SD16—24no—
296C2×SD16—42no—
297C2×D8—22no—
298C2×D8—22no—
299C2×D8—22no—
300C22×C8C23×C712yes—
301C22×C8C23×C711yescharacteristic
302C22×C8C23×C711yescharacteristic
303C4⋊Q8—22no—
304C4⋊Q8C7×D414yes—
305#126—216no—
306#124—24no—
307#124—24no—
308C22⋊Q8—28no—
309C22⋊Q8—44no—
310C4×Q8—24no—
311C4×Q8—24no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×D4—22no—
315C4×D4—22no—
316C4×D4—22no—
317C4×D4C7×D414yes—
318#117—28no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4C7×D411yescharacteristic
326C2×C4⋊C4C7×D412yes—
327C2×C4⋊C4C23×C711yescharacteristic
328C2×C4⋊C4C23×C712yes—
329C2×C4⋊C4C23×C711yescharacteristic
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C42—22no—
337C2×C42C23×C712yes—
338C2×C42C23×C711yescharacteristic
339#107—22no—
340C4⋊C8—22no—
341#103—24no—
342#103—24no—
343#103—42no—
344#102—22no—
345#102—22no—
346#102—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—81no—
356C22×C7—81no—
357C22×C7—81no—
358C22×C7—81no—
359C22×C7—81no—
360C22×C7—81no—
361C22×C7—81no—
362C22×C7—81no—
363C22×C7C2×#12011yescharacteristic
364C22×C7#54811yescharacteristic
365C22×C7#55011yescharacteristic
366C22×C7C2×#13511yescharacteristic
367C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
368C22×C7#57711yescharacteristic
369C22×C7#57811yescharacteristic
370C4×C7—22no—
371C4×C7—44no—
372C4×C7—42no—
373C4×C7—42no—
374C4×C7—42no—
375C4×C7—82no—
376C4×C7—81no—
377C4×C7#44914yes—
378C4×C7#52511yescharacteristic
379C4×C7#53711yescharacteristic
380C4×C7#54611yescharacteristic
381C4×C7#54711yescharacteristic
382C24—21no—
383C24—21no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8C2×C7×D414yes—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4C2×C7×D412yes—
404C2×D4C7×#4112yes—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4C2×C7×D412yes—
416C22×C4C2×C7×D412yes—
417C22×C4C2×C7×D411yescharacteristic
418C22×C4C2×C7×D411yescharacteristic
419C22×C4C24×C711yescharacteristic
420SD16—48no—
421D8—28no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8C7×#4112yes—
431C4⋊C4—28no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—44no—
442C4⋊C4—42no—
443C4⋊C4—81no—
444C4⋊C4C2×C7×D412yes—
445C22⋊C4—28no—
446C22⋊C4—28no—
447C22⋊C4—24no—
448C22⋊C4—44no—
449C22⋊C4—42no—
450C22⋊C4—82no—
451C42—28no—
452C42—22no—
453C42—22no—
454C42C2×C7×D412yes—
455C42C2×C7×D412yes—
456C2×C7—82no—
457C2×C7—82no—
458C2×C7#240711yescharacteristic
459C2×C7C2×#44911yescharacteristic
460C2×C7#301311yescharacteristic
461C2×C7#336011yescharacteristic
462C2×C7#336711yescharacteristic
463C2×C7#339311yescharacteristic
464C2×C7#339711yescharacteristic
465C23—22no—
466C23—22no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23C22×C7×D411yescharacteristic
480Q8—24no—
481Q8—24no—
482Q8—42no—
483Q8—42no—
484Q8—44no—
485D4—22no—
486D4—22no—
487D4—22no—
488D4—22no—
489D4—41no—
490D4—41no—
491D4—41no—
492D4—41no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—48no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—84no—
515C2×C4C7×#12012yes—
516C2×C4C7×#12012yes—
517C2×C4C7×#13512yes—
518C2×C4C7×AGL(1,ℤ/8)12yes—
519C2×C4C22×C7×D411yescharacteristic
520C2×C4C22×C7×D411yescharacteristic
521C2×C4C2×C7×#4111yescharacteristic
522C2×C4C2×C7×#4111yescharacteristic
523C2×C4C7×H(5,𝔽2)11yescharacteristic
524C2×C4C7×2−1+411yescharacteristic
525C8—22no—
526C8—41no—
527C8—42no—
528C7#2418511yescharacteristic, direct product factor, Sylow
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22C2×C7×#12011yescharacteristic
544C22C7×#54811yescharacteristic
545C22C7×#55011yescharacteristic
546C22C2×C7×#13511yescharacteristic
547C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
548C22C7×#57711yescharacteristic
549C22C7×#57811yescharacteristic
550C4—22no—
551C4—44no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—81no—
556C4—82no—
557C4C7×#44914yes—
558C4C7×#52511yescharacteristic
559C4C7×#53711yescharacteristic
560C4C7×#54611yescharacteristic
561C4C7×#54711yescharacteristic
562C2—82no—
563C2—82no—
564C2C7×#240711yescharacteristic
565C2C2×C7×#44911yescharacteristic
566C2C7×#301311yescharacteristic
567C2C7×#336011yescharacteristic
568C2C7×#336711yescharacteristic
569C2C7×#339311yescharacteristic
570C2C7×#339711yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC22×C7×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C4×C7×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#556C2semidirect
2. C2×C7×#461C2semidirect
4. C7×#2538C2semidirect
5. C7×#2042C2semidirect
7. C7×#1984C2semidirect
8. C7×#1956C2semidirect
9. C7×#1880C2semidirect
10. C7×#1863C2semidirect
11. C7×#1802C2semidirect
45. #24185C7direct (N × H)
107. C2×#556C2×C7semidirect
108. C2×#461C2×C7semidirect
110. #2538C2×C7semidirect
111. #2042C2×C7semidirect
113. #1984C2×C7semidirect
114. #1956C2×C7semidirect
115. #1880C2×C7semidirect
116. #1863C2×C7semidirect
117. #1802C2×C7semidirect
528. C7#24185direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5562
C2×C7×#4612
C2×C4×C7×D82
C7×#25382
C7×#20422
C7×#20122
C7×#19842
C7×#19562
C7×#18802
C7×#18632
C7×#18022
#241857