Finite Group Databasebeta

← C7×#24180 #13201396

Subgroups — C7×#24180 #13201396

Order 1792.

1146 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3506C211yesmaximal, characteristic, complemented
2C7×#3077C211yesmaximal, characteristic, complemented
3C7×#3036C211yesmaximal, characteristic, complemented
4C7×#2328C211yesmaximal, characteristic, complemented
5C7×#2037C212yesmaximal, complemented
6C7×#2002C212yesmaximal, complemented
7C7×#1995C211yesmaximal, characteristic, complemented
8C7×#1963C211yesmaximal, characteristic, complemented
9C7×#1883C212yesmaximal, complemented
10C7×#1860C212yesmaximal, complemented
11C7×#1800C211yesmaximal, characteristic, complemented
12C2×C7×AGL(1,ℤ/8)C2212yes—
13C7×#559—22no—
14C7×#556—21no—
15C7×#555—21no—
16C7×#553—22no—
17C7×#550—21no—
18C7×#547—21no—
19C7×#538—22no—
20C7×#535C2212yes—
21C7×#533C2211yescharacteristic
22C7×#532C2211yescharacteristic
23C2×C7×#127C2211yescharacteristic
24C7×#525—22no—
25C7×#520C2212yes—
26C2×C4×C7×D4C2211yescharacteristic
27C2×C4×C7×D4C2211yescharacteristic
28C7×#478—22no—
29C7×#476—21no—
30C7×#474—21no—
31C7×#463—22no—
32C7×#461—21no—
33C7×#459—21no—
34C7×#428C2212yes—
35C7×#423C2211yescharacteristic
36C7×#423C2211yescharacteristic
37C2×C7×#104C2212yes—
38C2×C7×#104C2212yes—
39C7×#418C2212yes—
40C7×#417C2212yes—
41C2×C7×#100C2212yes—
42C7×#390C2211yescharacteristic
43C7×#390C2211yescharacteristic
44C7×#386—22no—
45C7×#386—22no—
46C7×#382C2211yescharacteristic
47C7×#376C2211yescharacteristic
48C7×#343C2212yes—
49C7×#339C2212yes—
50C7×#339C2212yes—
51#24180C711yesmaximal, characteristic, direct product factor, Sylow
52C2×C7×#41—22no—
53C2×C7×#41C2312yes—
54C22×C7×D4—22no—
55C22×C7×D4C2311yescharacteristic
56C22×C7×D4C2311yescharacteristic
57C23×C4×C7—21no—
58C23×C4×C7D412yes—
59C7×AGL(1,ℤ/8)—24no—
60C7×AGL(1,ℤ/8)—42no—
61C2×C7×SD16—22no—
62C2×C7×D8—22no—
63C2×C7×M16C2312yes—
64C2×C7×M16C2312yes—
65C7×C4⋊Q8D411yescharacteristic, complemented
66C7×C4⋊Q8D411yescharacteristic, complemented
67C7×C4⋊Q8D411yescharacteristic, complemented
68C7×#127—22no—
69C7×#127—22no—
70C7×#127—22no—
71C7×#127D411yescharacteristic, complemented
72C7×#126—24no—
73C7×#125D412yescomplemented
74C7×#124—24no—
75C7×#124—22no—
76C7×#124D412yescomplemented
77C7×#123—24no—
78C7×#123—24no—
79C7×C22⋊Q8—22no—
80C7×C22⋊Q8—21no—
81C7×C22⋊Q8—21no—
82C7×#121—22no—
83C7×#121—24no—
84C7×#121—21no—
85C7×#121—21no—
86C7×#121—22no—
87C7×#121—22no—
88C7×#120—24no—
89C4×C7×Q8—22no—
90C4×C7×Q8—21no—
91C4×C7×Q8—21no—
92C4×C7×D4—24no—
93C4×C7×D4—22no—
94C4×C7×D4—24no—
95C4×C7×D4—21no—
96C4×C7×D4—21no—
97C4×C7×D4—21no—
98C4×C7×D4—21no—
99C4×C7×D4—21no—
100C4×C7×D4—21no—
101C4×C7×D4—21no—
102C4×C7×D4—21no—
103C4×C7×D4—21no—
104C4×C7×D4—21no—
105C7×#117—22no—
106C7×#117—22no—
107C7×#117C2312yes—
108C2×C7×C4⋊C4—22no—
109C2×C7×C4⋊C4C2311yescharacteristic
110C2×C7×C4⋊C4C2311yescharacteristic
111C2×C7×C22⋊C4—21no—
112C2×C7×C22⋊C4—22no—
113C2×C7×C22⋊C4—22no—
114C2×C7×C22⋊C4D412yes—
115C2×C42×C7C2311yescharacteristic
116C2×C42×C7C2311yescharacteristic
117C2×C42×C7C2311yescharacteristic
118C7×#107—22no—
119C7×#106—22no—
120C7×C4⋊C8—21no—
121C7×C4⋊C8—21no—
122C7×C4⋊C8—21no—
123C7×C4⋊C8—21no—
124C7×#104—24no—
125C7×#104—42no—
126C7×#103—22no—
127C7×#102—22no—
128C7×#102—22no—
129C7×#102—22no—
130C7×#100—42no—
131C7×#95—21no—
132C7×#95—21no—
133C7×#95—22no—
134#3506C2×C711yescharacteristic, complemented
135#3077C2×C711yescharacteristic, complemented
136#3036C2×C711yescharacteristic, complemented
137#2328C2×C711yescharacteristic, complemented
138#2037C2×C712yescomplemented
139#2002C2×C712yescomplemented
140#1995C2×C711yescharacteristic, complemented
141#1963C2×C711yescharacteristic, complemented
142#1883C2×C712yescomplemented
143#1860C2×C712yescomplemented
144#1800C2×C711yescharacteristic, complemented
145C24×C7—21no—
146C24×C7C2×D412yes—
147C7×#41—22no—
148C7×#41—22no—
149C7×#41—22no—
150C7×#41—22no—
151C7×#41—84no—
152C2×C7×Q8—21no—
153C2×C7×Q8—21no—
154C2×C7×Q8C2×D412yes—
155C2×C7×D4—22no—
156C2×C7×D4—24no—
157C2×C7×D4—24no—
158C2×C7×D4—28no—
159C2×C7×D4—21no—
160C2×C7×D4—21no—
161C2×C7×D4—21no—
162C2×C7×D4—21no—
163C2×C7×D4—21no—
164C2×C7×D4—21no—
165C2×C7×D4—21no—
166C2×C7×D4—21no—
167C2×C7×D4—21no—
168C2×C7×D4—21no—
169C2×C7×D4—21no—
170C2×C7×D4—21no—
171C2×C7×D4—21no—
172C2×C7×D4—82no—
173C2×C7×D4—82no—
174C2×C7×D4C2×D412yes—
175C2×C7×D4#4112yescomplemented
176C22×C4×C7—21no—
177C22×C4×C7—24no—
178C22×C4×C7—22no—
179C22×C4×C7—22no—
180C22×C4×C7—22no—
181C22×C4×C7—21no—
182C22×C4×C7—21no—
183C22×C4×C7—41no—
184C22×C4×C7—41no—
185C22×C4×C7—82no—
186C22×C4×C7—82no—
187C22×C4×C7C2×D412yes—
188C22×C4×C7C2×D411yescharacteristic
189C22×C4×C7C2×D411yescharacteristic
190C22×C4×C7C2411yescharacteristic
191C7×SD16—44no—
192C7×D8—44no—
193C7×M16—22no—
194C7×M16—22no—
195C7×M16—22no—
196C7×M16—22no—
197C2×C7×C8—22no—
198C2×C7×C8—22no—
199C7×C4⋊C4—24no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—22no—
203C7×C4⋊C4—21no—
204C7×C4⋊C4—22no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—21no—
207C7×C4⋊C4—21no—
208C7×C4⋊C4—21no—
209C7×C4⋊C4—21no—
210C7×C4⋊C4C2×D412yes—
211C7×C4⋊C4C2×D412yes—
212C7×C22⋊C4—24no—
213C7×C22⋊C4—24no—
214C7×C22⋊C4—24no—
215C7×C22⋊C4—22no—
216C7×C22⋊C4—22no—
217C7×C22⋊C4—22no—
218C7×C22⋊C4—22no—
219C7×C22⋊C4—22no—
220C7×C22⋊C4—41no—
221C7×C22⋊C4—41no—
222C7×C22⋊C4—82no—
223C7×C22⋊C4—82no—
224C42×C7—24no—
225C42×C7—22no—
226C42×C7—21no—
227C42×C7—21no—
228C42×C7—21no—
229C42×C7C2×D411yescharacteristic
230C42×C7C2×D411yescharacteristic
231C42×C7C2×D411yescharacteristic
232C42×C7C2×D411yescharacteristic
233C42×C7#4112yescomplemented
234C2×AGL(1,ℤ/8)C22×C712yes—
235#559—22no—
236#556—21no—
237#555—21no—
238#553—22no—
239#550—21no—
240#547—21no—
241#538—22no—
242#535C22×C712yes—
243#533C22×C711yescharacteristic
244#532C22×C711yescharacteristic
245C2×#127C22×C711yescharacteristic
246#525—22no—
247#520C22×C712yes—
248C2×C4×D4C22×C711yescharacteristic
249C2×C4×D4C22×C711yescharacteristic
250#478—22no—
251#476—21no—
252#474—21no—
253#463—22no—
254#461—21no—
255#459—21no—
256#428C22×C712yes—
257#423C22×C711yescharacteristic
258#423C22×C711yescharacteristic
259C2×#104C22×C712yes—
260C2×#104C22×C712yes—
261#418C22×C712yes—
262#417C22×C712yes—
263C2×#100C22×C712yes—
264#390C22×C711yescharacteristic
265#390C22×C711yescharacteristic
266#386—22no—
267#386—22no—
268#382C22×C711yescharacteristic
269#376C22×C711yescharacteristic
270#343C22×C712yes—
271#339C22×C712yes—
272#339C22×C712yes—
273C23×C7—22no—
274C23×C7—22no—
275C23×C7—22no—
276C23×C7—22no—
277C23×C7—22no—
278C23×C7—42no—
279C23×C7—42no—
280C23×C7—42no—
281C23×C7—42no—
282C23×C7—41no—
283C23×C7—41no—
284C23×C7C22×D411yescharacteristic
285C7×Q8—22no—
286C7×Q8—22no—
287C7×Q8—42no—
288C7×Q8—81no—
289C7×D4—22no—
290C7×D4—22no—
291C7×D4—22no—
292C7×D4—22no—
293C7×D4—42no—
294C7×D4—42no—
295C7×D4—42no—
296C7×D4—41no—
297C7×D4—41no—
298C7×D4—41no—
299C7×D4—41no—
300C7×D4—41no—
301C7×D4—41no—
302C7×D4—81no—
303C7×D4—84no—
304C7×D4—88no—
305C7×D4—84no—
306C2×C4×C7—22no—
307C2×C4×C7—22no—
308C2×C4×C7—22no—
309C2×C4×C7—22no—
310C2×C4×C7—22no—
311C2×C4×C7—41no—
312C2×C4×C7—41no—
313C2×C4×C7—44no—
314C2×C4×C7—41no—
315C2×C4×C7—41no—
316C2×C4×C7—42no—
317C2×C4×C7—42no—
318C2×C4×C7—42no—
319C2×C4×C7—41no—
320C2×C4×C7—41no—
321C2×C4×C7—42no—
322C2×C4×C7—41no—
323C2×C4×C7—41no—
324C2×C4×C7—82no—
325C2×C4×C7—82no—
326C2×C4×C7—82no—
327C2×C4×C7—84no—
328C2×C4×C7#12011yescharacteristic
329C2×C4×C7#12011yescharacteristic
330C2×C4×C7#12011yescharacteristic
331C2×C4×C7#12011yescharacteristic
332C2×C4×C7C22×D411yescharacteristic
333C2×C4×C7C22×D411yescharacteristic
334C2×C4×C7C2×#4111yescharacteristic
335C2×C4×C7C2×#4111yescharacteristic
336C2×C4×C7H(5,𝔽2)11yescharacteristic
337C2×C4×C72−1+411yescharacteristic
338C7×C8—42no—
339C7×C8—42no—
340C2×#41—22no—
341C2×#41C23×C712yes—
342C22×D4—22no—
343C22×D4C23×C711yescharacteristic
344C22×D4C23×C711yescharacteristic
345C23×C4—21no—
346C23×C4C7×D412yes—
347AGL(1,ℤ/8)—24no—
348AGL(1,ℤ/8)—42no—
349C2×SD16—22no—
350C2×D8—22no—
351C2×M16C23×C712yes—
352C2×M16C23×C712yes—
353C4⋊Q8C7×D411yescharacteristic, complemented
354C4⋊Q8C7×D411yescharacteristic, complemented
355C4⋊Q8C7×D411yescharacteristic, complemented
356#127—22no—
357#127—22no—
358#127—22no—
359#127C7×D411yescharacteristic, complemented
360#126—24no—
361#125C7×D412yescomplemented
362#124—24no—
363#124—22no—
364#124C7×D412yescomplemented
365#123—24no—
366#123—24no—
367C22⋊Q8—22no—
368C22⋊Q8—21no—
369C22⋊Q8—21no—
370#121—22no—
371#121—24no—
372#121—21no—
373#121—21no—
374#121—22no—
375#121—22no—
376#120—24no—
377C4×Q8—22no—
378C4×Q8—21no—
379C4×Q8—21no—
380C4×D4—24no—
381C4×D4—22no—
382C4×D4—24no—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4—21no—
386C4×D4—21no—
387C4×D4—21no—
388C4×D4—21no—
389C4×D4—21no—
390C4×D4—21no—
391C4×D4—21no—
392C4×D4—21no—
393#117—22no—
394#117—22no—
395#117C23×C712yes—
396C2×C4⋊C4—22no—
397C2×C4⋊C4C23×C711yescharacteristic
398C2×C4⋊C4C23×C711yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4C7×D412yes—
403C2×C42C23×C711yescharacteristic
404C2×C42C23×C711yescharacteristic
405C2×C42C23×C711yescharacteristic
406#107—22no—
407#106—22no—
408C4⋊C8—21no—
409C4⋊C8—21no—
410C4⋊C8—21no—
411C4⋊C8—21no—
412#104—24no—
413#104—42no—
414#103—22no—
415#102—22no—
416#102—22no—
417#102—22no—
418#100—42no—
419#95—21no—
420#95—21no—
421#95—22no—
422C22×C7—21no—
423C22×C7—21no—
424C22×C7—44no—
425C22×C7—44no—
426C22×C7—42no—
427C22×C7—42no—
428C22×C7—42no—
429C22×C7—42no—
430C22×C7—42no—
431C22×C7—42no—
432C22×C7—42no—
433C22×C7—82no—
434C22×C7—82no—
435C22×C7—81no—
436C22×C7—81no—
437C22×C7C2×#12011yescharacteristic
438C22×C7#52511yescharacteristic
439C22×C7#53711yescharacteristic
440C4×C7—21no—
441C4×C7—21no—
442C4×C7—21no—
443C4×C7—21no—
444C4×C7—42no—
445C4×C7—42no—
446C4×C7—42no—
447C4×C7—42no—
448C4×C7—42no—
449C4×C7—81no—
450C4×C7—81no—
451C4×C7#54611yescharacteristic
452C4×C7#54711yescharacteristic
453C4×C7#54811yescharacteristic
454C4×C7#55011yescharacteristic
455C24—21no—
456C24C2×C7×D412yes—
457#41—22no—
458#41—22no—
459#41—22no—
460#41—22no—
461#41—84no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8C2×C7×D412yes—
465C2×D4—22no—
466C2×D4—24no—
467C2×D4—24no—
468C2×D4—28no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—82no—
483C2×D4—82no—
484C2×D4C2×C7×D412yes—
485C2×D4C7×#4112yescomplemented
486C22×C4—21no—
487C22×C4—24no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—41no—
494C22×C4—41no—
495C22×C4—82no—
496C22×C4—82no—
497C22×C4C2×C7×D412yes—
498C22×C4C2×C7×D411yescharacteristic
499C22×C4C2×C7×D411yescharacteristic
500C22×C4C24×C711yescharacteristic
501SD16—44no—
502D8—44no—
503M16—22no—
504M16—22no—
505M16—22no—
506M16—22no—
507C2×C8—22no—
508C2×C8—22no—
509C4⋊C4—24no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4C2×C7×D412yes—
521C4⋊C4C2×C7×D412yes—
522C22⋊C4—24no—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—41no—
531C22⋊C4—41no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C42—24no—
535C42—22no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42C2×C7×D411yescharacteristic
540C42C2×C7×D411yescharacteristic
541C42C2×C7×D411yescharacteristic
542C42C2×C7×D411yescharacteristic
543C42C7×#4112yescomplemented
544C2×C7—21no—
545C2×C7—21no—
546C2×C7—82no—
547C2×C7—82no—
548C2×C7—82no—
549C2×C7#240711yescharacteristic
550C2×C7#301911yescharacteristic
551C2×C7#302111yescharacteristic
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—41no—
562C23—41no—
563C23C22×C7×D411yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—42no—
567Q8—81no—
568D4—22no—
569D4—22no—
570D4—22no—
571D4—22no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—41no—
576D4—41no—
577D4—41no—
578D4—41no—
579D4—41no—
580D4—41no—
581D4—81no—
582D4—84no—
583D4—88no—
584D4—84no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—84no—
607C2×C4C7×#12011yescharacteristic
608C2×C4C7×#12011yescharacteristic
609C2×C4C7×#12011yescharacteristic
610C2×C4C7×#12011yescharacteristic
611C2×C4C22×C7×D411yescharacteristic
612C2×C4C22×C7×D411yescharacteristic
613C2×C4C2×C7×#4111yescharacteristic
614C2×C4C2×C7×#4111yescharacteristic
615C2×C4C7×H(5,𝔽2)11yescharacteristic
616C2×C4C7×2−1+411yescharacteristic
617C8—42no—
618C8—42no—
619C7#2418011yescharacteristic, direct product factor, Sylow
620C22—21no—
621C22—21no—
622C22—44no—
623C22—44no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—82no—
632C22—82no—
633C22—81no—
634C22—81no—
635C22C2×C7×#12011yescharacteristic
636C22C7×#52511yescharacteristic
637C22C7×#53711yescharacteristic
638C4—21no—
639C4—21no—
640C4—21no—
641C4—21no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—81no—
648C4—81no—
649C4C7×#54611yescharacteristic
650C4C7×#54711yescharacteristic
651C4C7×#54811yescharacteristic
652C4C7×#55011yescharacteristic
653C2—21no—
654C2—21no—
655C2—82no—
656C2—82no—
657C2—82no—
658C2C7×#240711yescharacteristic
659C2C7×#301911yescharacteristic
660C2C7×#302111yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3506C2semidirect
2. C7×#3077C2semidirect
3. C7×#3036C2semidirect
4. C7×#2328C2semidirect
5. C7×#2037C2semidirect
6. C7×#2002C2semidirect
7. C7×#1995C2semidirect
8. C7×#1963C2semidirect
9. C7×#1883C2semidirect
10. C7×#1860C2semidirect
11. C7×#1800C2semidirect
51. #24180C7direct (N × H)
65. C7×C4⋊Q8D4semidirect
66. C7×C4⋊Q8D4semidirect
67. C7×C4⋊Q8D4semidirect
71. C7×#127D4semidirect
73. C7×#125D4semidirect
76. C7×#124D4semidirect
134. #3506C2×C7semidirect
135. #3077C2×C7semidirect
136. #3036C2×C7semidirect
137. #2328C2×C7semidirect
138. #2037C2×C7semidirect
139. #2002C2×C7semidirect
140. #1995C2×C7semidirect
141. #1963C2×C7semidirect
142. #1883C2×C7semidirect
143. #1860C2×C7semidirect
144. #1800C2×C7semidirect
175. C2×C7×D4#41semidirect
233. C42×C7#41semidirect
353. C4⋊Q8C7×D4semidirect
354. C4⋊Q8C7×D4semidirect
355. C4⋊Q8C7×D4semidirect
359. #127C7×D4semidirect
361. #125C7×D4semidirect
364. #124C7×D4semidirect
485. C2×D4C7×#41semidirect
543. C42C7×#41semidirect
619. C7#24180direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35062
C7×#30772
C7×#30362
C7×#23282
C7×#20372
C7×#20022
C7×#19952
C7×#19632
C7×#18832
C7×#18602
C7×#18002
#241807