Finite Group Databasebeta

← C7×#24506 #13201722

Subgroups — C7×#24506 #13201722

Order 1792.

1250 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#447C211yesmaximal, characteristic
2C2×C7×#432C211yesmaximal, characteristic, complemented
3C7×#2854C211yesmaximal, characteristic, complemented
4C2×C7×#390C211yesmaximal, characteristic, complemented
5C7×#2065C212yesmaximal, complemented
6C7×#2054C212yesmaximal, complemented
7C7×#1972C212yesmaximal, complemented
8C7×#1936C212yesmaximal, complemented
9C7×#1874C211yesmaximal, characteristic, complemented
10C7×#1853C211yesmaximal, characteristic, complemented
11C7×#1786C211yesmaximal, characteristic, complemented
12C23×C7×D4C2211yescharacteristic
13C22×C7×D8C2212yes—
14C2×C7×#123—21no—
15C2×C7×#121C2211yescharacteristic
16C22×C7×C4⋊C4C2211yescharacteristic
17C22×C7×C22⋊C4—21no—
18C22×C7×C22⋊C4—21no—
19C22×C42×C7C2211yescharacteristic
20C7×#447—24no—
21C7×#432—24no—
22C2×C7×#106C2212yescomplemented
23C2×C7×C4⋊C8C2212yescomplemented
24C2×C7×#102C2212yes—
25C2×C7×#102C2212yes—
26C2×C7×#102C2212yes—
27C2×C7×#102C2212yescomplemented
28C2×C7×C22⋊C8C2211yescharacteristic
29C2×C7×C22⋊C8C2211yescharacteristic
30C2×C7×#97C2212yescomplemented
31C7×#390—21no—
32C7×#390—21no—
33C7×#390C2212yes—
34C7×#390C2212yes—
35C7×#390C2211yescharacteristic, complemented
36C7×#390C2211yescharacteristic, complemented
37C7×#389C2211yescharacteristic, complemented
38C7×#389C2211yescharacteristic, complemented
39C7×#388—21no—
40C7×#388—21no—
41C7×#386—21no—
42C7×#380C2211yescharacteristic
43C7×#340C2212yescomplemented
44C7×#340C2212yescomplemented
45#24506C711yesmaximal, characteristic, direct product factor, Sylow
46C25×C7—21no—
47C22×C7×D4—24no—
48C22×C7×D4—21no—
49C22×C7×D4—21no—
50C22×C7×D4—22no—
51C22×C7×D4—22no—
52C22×C7×D4—42no—
53C22×C7×D4C2311yescharacteristic
54C22×C7×D4C2312yes—
55C23×C4×C7—21no—
56C23×C4×C7D411yescharacteristic
57C23×C4×C7D411yescharacteristic
58C23×C4×C7C2311yescharacteristic
59C2×C7×D8—24no—
60C2×C7×D8—24no—
61C2×C7×D8—42no—
62C22×C7×C8C2312yes—
63C22×C7×C8C2312yes—
64C7×#123—44no—
65C7×#121—22no—
66C7×#121—22no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—42no—
71C2×C7×C4⋊C4C2312yes—
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
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—21no—
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4—21no—
83C2×C7×C22⋊C4—21no—
84C2×C7×C22⋊C4—21no—
85C2×C7×C22⋊C4—44no—
86C2×C7×C22⋊C4—44no—
87C2×C42×C7—22no—
88C2×C42×C7—22no—
89C2×C42×C7D411yescharacteristic
90C2×C42×C7D411yescharacteristic
91C2×C42×C7C2311yescharacteristic
92C2×C42×C7C2311yescharacteristic
93C7×#106—42no—
94C7×C4⋊C8—24no—
95C7×#102—24no—
96C7×#102—24no—
97C7×#102—42no—
98C7×#102—42no—
99C7×C22⋊C8—22no—
100C7×C22⋊C8D414yes—
101C7×#97—24no—
102C7×#95—21no—
103C7×#95—21no—
104C7×#95—21no—
105C2×#447C2×C711yescharacteristic
106C2×#432C2×C711yescharacteristic, complemented
107#2854C2×C711yescharacteristic, complemented
108C2×#390C2×C711yescharacteristic, complemented
109#2065C2×C712yescomplemented
110#2054C2×C712yescomplemented
111#1972C2×C712yescomplemented
112#1936C2×C712yescomplemented
113#1874C2×C711yescharacteristic, complemented
114#1853C2×C711yescharacteristic, complemented
115#1786C2×C711yescharacteristic, complemented
116C24×C7—24no—
117C24×C7—22no—
118C24×C7—21no—
119C24×C7—42no—
120C24×C7—42no—
121C24×C7—42no—
122C24×C7—42no—
123C24×C7—42no—
124C24×C7—42no—
125C24×C7C2×D411yescharacteristic
126C2×C7×D4—24no—
127C2×C7×D4—22no—
128C2×C7×D4—21no—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4—21no—
132C2×C7×D4—22no—
133C2×C7×D4—22no—
134C2×C7×D4—22no—
135C2×C7×D4—22no—
136C2×C7×D4—48no—
137C2×C7×D4—48no—
138C2×C7×D4—42no—
139C2×C7×D4—42no—
140C2×C7×D4—42no—
141C2×C7×D4—42no—
142C2×C7×D4—42no—
143C2×C7×D4#4114yes—
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—22no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—22no—
155C22×C4×C7—41no—
156C22×C4×C7—42no—
157C22×C4×C7—42no—
158C22×C4×C7—42no—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D412yes—
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7#4112yes—
166C22×C4×C7C2411yescharacteristic
167C7×D8—48no—
168C2×C7×C8—22no—
169C2×C7×C8—22no—
170C2×C7×C8—22no—
171C2×C7×C8—22no—
172C2×C7×C8—22no—
173C2×C7×C8C2×D414yes—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—42no—
181C7×C4⋊C4—42no—
182C7×C4⋊C4—42no—
183C7×C22⋊C4—42no—
184C7×C22⋊C4—44no—
185C7×C22⋊C4—44no—
186C7×C22⋊C4—44no—
187C7×C22⋊C4—42no—
188C7×C22⋊C4—42no—
189C7×C22⋊C4—42no—
190C7×C22⋊C4—42no—
191C7×C22⋊C4—42no—
192C7×C22⋊C4—44no—
193C42×C7—22no—
194C42×C7—22no—
195C42×C7—21no—
196C42×C7—21no—
197C42×C7—21no—
198C42×C7—21no—
199C23×D4C22×C711yescharacteristic
200C22×D8C22×C712yes—
201C2×#123—21no—
202C2×#121C22×C711yescharacteristic
203C22×C4⋊C4C22×C711yescharacteristic
204C22×C22⋊C4—21no—
205C22×C22⋊C4—21no—
206C22×C42C22×C711yescharacteristic
207#447—24no—
208#432—24no—
209C2×#106C22×C712yescomplemented
210C2×C4⋊C8C22×C712yescomplemented
211C2×#102C22×C712yes—
212C2×#102C22×C712yes—
213C2×#102C22×C712yes—
214C2×#102C22×C712yescomplemented
215C2×C22⋊C8C22×C711yescharacteristic
216C2×C22⋊C8C22×C711yescharacteristic
217C2×#97C22×C712yescomplemented
218#390—21no—
219#390—21no—
220#390C22×C712yes—
221#390C22×C712yes—
222#390C22×C711yescharacteristic, complemented
223#390C22×C711yescharacteristic, complemented
224#389C22×C711yescharacteristic, complemented
225#389C22×C711yescharacteristic, complemented
226#388—21no—
227#388—21no—
228#386—21no—
229#380C22×C711yescharacteristic
230#340C22×C712yescomplemented
231#340C22×C712yescomplemented
232C23×C7—21no—
233C23×C7—21no—
234C23×C7—21no—
235C23×C7—21no—
236C23×C7—24no—
237C23×C7—48no—
238C23×C7—48no—
239C23×C7—48no—
240C23×C7—42no—
241C23×C7—42no—
242C23×C7—42no—
243C23×C7—42no—
244C23×C7—42no—
245C23×C7—42no—
246C23×C7—41no—
247C23×C7—41no—
248C23×C7—41no—
249C23×C7—41no—
250C23×C7—41no—
251C23×C7—41no—
252C23×C7—41no—
253C23×C7—84no—
254C23×C7—84no—
255C23×C7—84no—
256C23×C7—84no—
257C23×C7#12112yes—
258C23×C7AGL(1,ℤ/8)12yes—
259C23×C7AGL(1,ℤ/8)12yes—
260C23×C7C22×D411yescharacteristic
261C7×D4—24no—
262C7×D4—24no—
263C7×D4—48no—
264C7×D4—42no—
265C7×D4—42no—
266C7×D4—42no—
267C7×D4—42no—
268C7×D4—84no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—24no—
274C2×C4×C7—24no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—44no—
285C2×C4×C7—44no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—42no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—42no—
293C2×C4×C7—42no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—84no—
299C2×C4×C7#12112yes—
300C2×C4×C7AGL(1,ℤ/8)12yes—
301C2×C4×C7C22×D411yescharacteristic
302C2×C4×C7C2×#4112yes—
303C2×C4×C7C2×#4111yescharacteristic
304C2×C4×C7H(5,𝔽2)12yes—
305C7×C8—24no—
306C7×C8—42no—
307C25—21no—
308C22×D4—24no—
309C22×D4—21no—
310C22×D4—21no—
311C22×D4—22no—
312C22×D4—22no—
313C22×D4—42no—
314C22×D4C23×C711yescharacteristic
315C22×D4C23×C712yes—
316C23×C4—21no—
317C23×C4C7×D411yescharacteristic
318C23×C4C7×D411yescharacteristic
319C23×C4C23×C711yescharacteristic
320C2×D8—24no—
321C2×D8—24no—
322C2×D8—42no—
323C22×C8C23×C712yes—
324C22×C8C23×C712yes—
325#123—44no—
326#121—22no—
327#121—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—42no—
332C2×C4⋊C4C23×C712yes—
333C2×C4⋊C4C23×C711yescharacteristic
334C2×C4⋊C4C23×C711yescharacteristic
335C2×C4⋊C4C23×C711yescharacteristic
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—44no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42C7×D411yescharacteristic
351C2×C42C7×D411yescharacteristic
352C2×C42C23×C711yescharacteristic
353C2×C42C23×C711yescharacteristic
354#106—42no—
355C4⋊C8—24no—
356#102—24no—
357#102—24no—
358#102—42no—
359#102—42no—
360C22⋊C8—22no—
361C22⋊C8C7×D414yes—
362#97—24no—
363#95—21no—
364#95—21no—
365#95—21no—
366C22×C7—22no—
367C22×C7—22no—
368C22×C7—22no—
369C22×C7—22no—
370C22×C7—22no—
371C22×C7—22no—
372C22×C7—44no—
373C22×C7—44no—
374C22×C7—44no—
375C22×C7—816no—
376C22×C7—82no—
377C22×C7—82no—
378C22×C7—82no—
379C22×C7—82no—
380C22×C7—81no—
381C22×C7—81no—
382C22×C7—81no—
383C22×C7—81no—
384C22×C7—81no—
385C22×C7—81no—
386C22×C7—81no—
387C22×C7#46914yes—
388C22×C7C2×#12111yescharacteristic
389C22×C7#52511yescharacteristic
390C22×C7#53511yescharacteristic
391C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
392C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
393C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
394C22×C7#57511yescharacteristic
395C4×C7—22no—
396C4×C7—22no—
397C4×C7—41no—
398C4×C7—41no—
399C4×C7—41no—
400C4×C7—81no—
401C4×C7—81no—
402C4×C7—82no—
403C4×C7—81no—
404C4×C7#54612yes—
405C4×C7#55112yes—
406C24—24no—
407C24—22no—
408C24—21no—
409C24—42no—
410C24—42no—
411C24—42no—
412C24—42no—
413C24—42no—
414C24—42no—
415C24C2×C7×D411yescharacteristic
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—48no—
427C2×D4—48no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4C7×#4114yes—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—41no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4C2×C7×D411yescharacteristic
450C22×C4C2×C7×D411yescharacteristic
451C22×C4C2×C7×D411yescharacteristic
452C22×C4C2×C7×D411yescharacteristic
453C22×C4C2×C7×D412yes—
454C22×C4C2×C7×D411yescharacteristic
455C22×C4C7×#4112yes—
456C22×C4C24×C711yescharacteristic
457D8—48no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8C2×C7×D414yes—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—44no—
482C22⋊C4—42no—
483C42—22no—
484C42—22no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C2×C7—24no—
490C2×C7—84no—
491C2×C7—161no—
492C2×C7#241111yescharacteristic
493C2×C7C2×#46911yescharacteristic
494C2×C7#305511yescharacteristic
495C2×C7#310911yescharacteristic
496C2×C7#311411yescharacteristic
497C2×C7#321311yescharacteristic
498C2×C7#321711yescharacteristic
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—24no—
504C23—48no—
505C23—48no—
506C23—48no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—84no—
521C23—84no—
522C23—84no—
523C23—84no—
524C23C7×#12112yes—
525C23C7×AGL(1,ℤ/8)12yes—
526C23C7×AGL(1,ℤ/8)12yes—
527C23C22×C7×D411yescharacteristic
528D4—24no—
529D4—24no—
530D4—48no—
531D4—42no—
532D4—42no—
533D4—42no—
534D4—42no—
535D4—84no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—84no—
566C2×C4C7×#12112yes—
567C2×C4C7×AGL(1,ℤ/8)12yes—
568C2×C4C22×C7×D411yescharacteristic
569C2×C4C2×C7×#4112yes—
570C2×C4C2×C7×#4111yescharacteristic
571C2×C4C7×H(5,𝔽2)12yes—
572C8—24no—
573C8—42no—
574C7#2450611yescharacteristic, direct product factor, Sylow
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—816no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22C7×#46914yes—
597C22C2×C7×#12111yescharacteristic
598C22C7×#52511yescharacteristic
599C22C7×#53511yescharacteristic
600C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
601C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
602C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
603C22C7×#57511yescharacteristic
604C4—22no—
605C4—22no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—81no—
610C4—82no—
611C4—81no—
612C4—81no—
613C4C7×#54612yes—
614C4C7×#55112yes—
615C2—24no—
616C2—84no—
617C2—161no—
618C2C7×#241111yescharacteristic
619C2C2×C7×#46911yescharacteristic
620C2C7×#305511yescharacteristic
621C2C7×#310911yescharacteristic
622C2C7×#311411yescharacteristic
623C2C7×#321311yescharacteristic
624C2C7×#321711yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#447C2
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×#432C2semidirect
3. C7×#2854C2semidirect
4. C2×C7×#390C2semidirect
5. C7×#2065C2semidirect
6. C7×#2054C2semidirect
7. C7×#1972C2semidirect
8. C7×#1936C2semidirect
9. C7×#1874C2semidirect
10. C7×#1853C2semidirect
11. C7×#1786C2semidirect
22. C2×C7×#106C22semidirect
23. C2×C7×C4⋊C8C22semidirect
27. C2×C7×#102C22semidirect
30. C2×C7×#97C22semidirect
35. C7×#390C22semidirect
36. C7×#390C22semidirect
37. C7×#389C22semidirect
38. C7×#389C22semidirect
43. C7×#340C22semidirect
44. C7×#340C22semidirect
45. #24506C7direct (N × H)
106. C2×#432C2×C7semidirect
107. #2854C2×C7semidirect
108. C2×#390C2×C7semidirect
109. #2065C2×C7semidirect
110. #2054C2×C7semidirect
111. #1972C2×C7semidirect
112. #1936C2×C7semidirect
113. #1874C2×C7semidirect
114. #1853C2×C7semidirect
115. #1786C2×C7semidirect
209. C2×#106C22×C7semidirect
210. C2×C4⋊C8C22×C7semidirect
214. C2×#102C22×C7semidirect
217. C2×#97C22×C7semidirect
222. #390C22×C7semidirect
223. #390C22×C7semidirect
224. #389C22×C7semidirect
225. #389C22×C7semidirect
230. #340C22×C7semidirect
231. #340C22×C7semidirect
574. C7#24506direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4472
C2×C7×#4322
C7×#28542
C2×C7×#3902
C7×#20652
C7×#20542
C7×#19722
C7×#19362
C7×#18742
C7×#18532
C7×#17862
#245067