Finite Group Databasebeta

← C7×#24182 #13201398

Subgroups — C7×#24182 #13201398

Order 1792.

1082 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3506C211yesmaximal, characteristic
2C7×#3078C211yesmaximal, characteristic, complemented
3C7×#3037C211yesmaximal, characteristic, complemented
4C7×#2326C211yesmaximal, characteristic, complemented
5C7×#2037C212yesmaximal, complemented
6C7×#2004C212yesmaximal, complemented
7C7×#1995C211yesmaximal, characteristic, complemented
8C7×#1963C211yesmaximal, characteristic, complemented
9C7×#1882C212yesmaximal, complemented
10C7×#1859C212yesmaximal, complemented
11C7×#1800C211yesmaximal, characteristic, complemented
12C2×C7×#137C2212yes—
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×#124C2211yescharacteristic
24C7×#525—22no—
25C7×#520C2212yes—
26C2×C4×C7×Q8C2211yescharacteristic
27C2×C4×C7×D4C2211yescharacteristic
28C7×#479—22no—
29C7×#477—21no—
30C7×#475—21no—
31C7×#464—22no—
32C7×#462—21no—
33C7×#460—21no—
34C7×#428C2212yes—
35C7×#423C2211yescharacteristic
36C7×#423C2211yescharacteristic
37C2×C7×#104C2212yes—
38C2×C7×#104C2212yes—
39C7×#419C2212yes—
40C7×#417C2212yes—
41C2×C7×#100C2212yes—
42C7×#391C2211yescharacteristic
43C7×#390C2211yescharacteristic
44C7×#388—22no—
45C7×#385—22no—
46C7×#384C2211yescharacteristic
47C7×#376C2211yescharacteristic
48C7×#343C2212yes—
49C7×#339C2212yes—
50C7×#339C2212yes—
51#24182C711yesmaximal, characteristic, direct product factor, Sylow
52C2×C7×#41—22no—
53C2×C7×#41C2312yes—
54C22×C7×Q8C2311yescharacteristic
55C22×C7×D4C2311yescharacteristic
56C23×C4×C7D412yes—
57C7×#137—24no—
58C7×#137—42no—
59C2×C7×Q16—22no—
60C2×C7×SD16—22no—
61C2×C7×M16C2312yes—
62C2×C7×M16C2312yes—
63C7×C4⋊Q8D411yescharacteristic
64C7×C4⋊Q8D411yescharacteristic
65C7×C4⋊Q8D411yescharacteristic
66C7×#127—22no—
67C7×#127D411yescharacteristic
68C7×#126—24no—
69C7×#125D412yes—
70C7×#124—24no—
71C7×#124—22no—
72C7×#124—22no—
73C7×#124—22no—
74C7×#124D412yes—
75C7×#123—24no—
76C7×#123—24no—
77C7×C22⋊Q8—22no—
78C7×C22⋊Q8—21no—
79C7×C22⋊Q8—21no—
80C7×#121—22no—
81C7×#121—24no—
82C7×#121—21no—
83C7×#121—21no—
84C7×#121—22no—
85C7×#121—22no—
86C7×#120—24no—
87C4×C7×Q8—22no—
88C4×C7×Q8—21no—
89C4×C7×Q8—21no—
90C4×C7×Q8—21no—
91C4×C7×Q8—21no—
92C4×C7×Q8—21no—
93C4×C7×Q8—21no—
94C4×C7×D4—24no—
95C4×C7×D4—22no—
96C4×C7×D4—24no—
97C4×C7×D4—21no—
98C4×C7×D4—21no—
99C4×C7×D4—21no—
100C4×C7×D4—21no—
101C4×C7×D4—21no—
102C4×C7×D4—21no—
103C7×#117—22no—
104C7×#117—22no—
105C7×#117C2312yes—
106C2×C7×C4⋊C4—22no—
107C2×C7×C4⋊C4—21no—
108C2×C7×C4⋊C4—22no—
109C2×C7×C4⋊C4C2311yescharacteristic
110C2×C7×C4⋊C4C2311yescharacteristic
111C2×C7×C22⋊C4—22no—
112C2×C7×C22⋊C4—22no—
113C2×C7×C22⋊C4D412yes—
114C2×C42×C7—21no—
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×#103—22no—
128C7×#103—22no—
129C7×#102—22no—
130C7×#100—42no—
131C7×#95—21no—
132C7×#95—21no—
133C7×#95—22no—
134#3506C2×C711yescharacteristic
135#3078C2×C711yescharacteristic, complemented
136#3037C2×C711yescharacteristic, complemented
137#2326C2×C711yescharacteristic, complemented
138#2037C2×C712yescomplemented
139#2004C2×C712yescomplemented
140#1995C2×C711yescharacteristic, complemented
141#1963C2×C711yescharacteristic, complemented
142#1882C2×C712yescomplemented
143#1859C2×C712yescomplemented
144#1800C2×C711yescharacteristic, complemented
145C24×C7C2×D412yes—
146C7×#41—22no—
147C7×#41—22no—
148C7×#41—22no—
149C7×#41—22no—
150C7×#41—84no—
151C2×C7×Q8—21no—
152C2×C7×Q8—21no—
153C2×C7×Q8—21no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8—21no—
158C2×C7×Q8C2×D412yes—
159C2×C7×Q8#4112yescomplemented
160C2×C7×D4—22no—
161C2×C7×D4—24no—
162C2×C7×D4—24no—
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×D4C2×D412yes—
172C22×C4×C7—21no—
173C22×C4×C7—21no—
174C22×C4×C7—24no—
175C22×C4×C7—22no—
176C22×C4×C7—22no—
177C22×C4×C7—22no—
178C22×C4×C7—21no—
179C22×C4×C7—21no—
180C22×C4×C7—82no—
181C22×C4×C7C2×D412yes—
182C22×C4×C7C2×D411yescharacteristic
183C22×C4×C7C2×D411yescharacteristic
184C22×C4×C7C2411yescharacteristic
185C7×Q16—44no—
186C7×SD16—44no—
187C7×M16—22no—
188C7×M16—22no—
189C7×M16—22no—
190C7×M16—22no—
191C2×C7×C8—22no—
192C2×C7×C8—22no—
193C7×C4⋊C4—24no—
194C7×C4⋊C4—21no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—22no—
199C7×C4⋊C4—21no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—21no—
204C7×C4⋊C4—41no—
205C7×C4⋊C4—41no—
206C7×C4⋊C4—82no—
207C7×C4⋊C4C2×D412yes—
208C7×C4⋊C4C2×D412yes—
209C7×C22⋊C4—28no—
210C7×C22⋊C4—24no—
211C7×C22⋊C4—24no—
212C7×C22⋊C4—24no—
213C7×C22⋊C4—22no—
214C7×C22⋊C4—22no—
215C7×C22⋊C4—22no—
216C7×C22⋊C4—22no—
217C7×C22⋊C4—22no—
218C7×C22⋊C4—82no—
219C42×C7—24no—
220C42×C7—22no—
221C42×C7—21no—
222C42×C7—21no—
223C42×C7—21no—
224C42×C7—41no—
225C42×C7—41no—
226C42×C7C2×D411yescharacteristic
227C42×C7C2×D411yescharacteristic
228C42×C7C2×D411yescharacteristic
229C42×C7C2×D411yescharacteristic
230C42×C7#4112yescomplemented
231C2×#137C22×C712yes—
232#559—22no—
233#556—21no—
234#555—21no—
235#553—22no—
236#550—21no—
237#547—21no—
238#538—22no—
239#535C22×C712yes—
240#533C22×C711yescharacteristic
241#532C22×C711yescharacteristic
242C2×#124C22×C711yescharacteristic
243#525—22no—
244#520C22×C712yes—
245C2×C4×Q8C22×C711yescharacteristic
246C2×C4×D4C22×C711yescharacteristic
247#479—22no—
248#477—21no—
249#475—21no—
250#464—22no—
251#462—21no—
252#460—21no—
253#428C22×C712yes—
254#423C22×C711yescharacteristic
255#423C22×C711yescharacteristic
256C2×#104C22×C712yes—
257C2×#104C22×C712yes—
258#419C22×C712yes—
259#417C22×C712yes—
260C2×#100C22×C712yes—
261#391C22×C711yescharacteristic
262#390C22×C711yescharacteristic
263#388—22no—
264#385—22no—
265#384C22×C711yescharacteristic
266#376C22×C711yescharacteristic
267#343C22×C712yes—
268#339C22×C712yes—
269#339C22×C712yes—
270C23×C7—22no—
271C23×C7—22no—
272C23×C7—22no—
273C23×C7—22no—
274C23×C7—42no—
275C23×C7—42no—
276C23×C7C22×D411yescharacteristic
277C7×Q8—22no—
278C7×Q8—22no—
279C7×Q8—22no—
280C7×Q8—22no—
281C7×Q8—42no—
282C7×Q8—41no—
283C7×Q8—41no—
284C7×Q8—81no—
285C7×D4—22no—
286C7×D4—22no—
287C7×D4—42no—
288C7×D4—42no—
289C7×D4—42no—
290C7×D4—41no—
291C7×D4—41no—
292C7×D4—41no—
293C7×D4—41no—
294C7×D4—81no—
295C7×D4—84no—
296C7×D4—84no—
297C2×C4×C7—22no—
298C2×C4×C7—22no—
299C2×C4×C7—22no—
300C2×C4×C7—22no—
301C2×C4×C7—22no—
302C2×C4×C7—22no—
303C2×C4×C7—41no—
304C2×C4×C7—41no—
305C2×C4×C7—44no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
308C2×C4×C7—41no—
309C2×C4×C7—41no—
310C2×C4×C7—42no—
311C2×C4×C7—42no—
312C2×C4×C7—42no—
313C2×C4×C7—41no—
314C2×C4×C7—42no—
315C2×C4×C7—41no—
316C2×C4×C7—41no—
317C2×C4×C7—41no—
318C2×C4×C7—82no—
319C2×C4×C7—84no—
320C2×C4×C7#12011yescharacteristic
321C2×C4×C7#12011yescharacteristic
322C2×C4×C7#12011yescharacteristic
323C2×C4×C7#12011yescharacteristic
324C2×C4×C7C22×D411yescharacteristic
325C2×C4×C7C22×D411yescharacteristic
326C2×C4×C7C2×#4111yescharacteristic
327C2×C4×C7C2×#4111yescharacteristic
328C2×C4×C7H(5,𝔽2)11yescharacteristic
329C2×C4×C72−1+411yescharacteristic
330C7×C8—42no—
331C7×C8—42no—
332C2×#41—22no—
333C2×#41C23×C712yes—
334C22×Q8C23×C711yescharacteristic
335C22×D4C23×C711yescharacteristic
336C23×C4C7×D412yes—
337#137—24no—
338#137—42no—
339C2×Q16—22no—
340C2×SD16—22no—
341C2×M16C23×C712yes—
342C2×M16C23×C712yes—
343C4⋊Q8C7×D411yescharacteristic
344C4⋊Q8C7×D411yescharacteristic
345C4⋊Q8C7×D411yescharacteristic
346#127—22no—
347#127C7×D411yescharacteristic
348#126—24no—
349#125C7×D412yes—
350#124—24no—
351#124—22no—
352#124—22no—
353#124—22no—
354#124C7×D412yes—
355#123—24no—
356#123—24no—
357C22⋊Q8—22no—
358C22⋊Q8—21no—
359C22⋊Q8—21no—
360#121—22no—
361#121—24no—
362#121—21no—
363#121—21no—
364#121—22no—
365#121—22no—
366#120—24no—
367C4×Q8—22no—
368C4×Q8—21no—
369C4×Q8—21no—
370C4×Q8—21no—
371C4×Q8—21no—
372C4×Q8—21no—
373C4×Q8—21no—
374C4×D4—24no—
375C4×D4—22no—
376C4×D4—24no—
377C4×D4—21no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4—21no—
383#117—22no—
384#117—22no—
385#117C23×C712yes—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4C23×C711yescharacteristic
390C2×C4⋊C4C23×C711yescharacteristic
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4C7×D412yes—
394C2×C42—21no—
395C2×C42C23×C711yescharacteristic
396C2×C42C23×C711yescharacteristic
397C2×C42C23×C711yescharacteristic
398#107—22no—
399#106—22no—
400C4⋊C8—21no—
401C4⋊C8—21no—
402C4⋊C8—21no—
403C4⋊C8—21no—
404#104—24no—
405#104—42no—
406#103—22no—
407#103—22no—
408#103—22no—
409#102—22no—
410#100—42no—
411#95—21no—
412#95—21no—
413#95—22no—
414C22×C7—21no—
415C22×C7—21no—
416C22×C7—44no—
417C22×C7—42no—
418C22×C7—42no—
419C22×C7—42no—
420C22×C7—42no—
421C22×C7—42no—
422C22×C7—42no—
423C22×C7—82no—
424C22×C7C2×#12011yescharacteristic
425C22×C7#52511yescharacteristic
426C22×C7#53711yescharacteristic
427C4×C7—21no—
428C4×C7—21no—
429C4×C7—21no—
430C4×C7—21no—
431C4×C7—42no—
432C4×C7—42no—
433C4×C7—42no—
434C4×C7—42no—
435C4×C7—42no—
436C4×C7—42no—
437C4×C7—81no—
438C4×C7—81no—
439C4×C7#54611yescharacteristic
440C4×C7#54711yescharacteristic
441C4×C7#54811yescharacteristic
442C4×C7#55011yescharacteristic
443C24C2×C7×D412yes—
444#41—22no—
445#41—22no—
446#41—22no—
447#41—22no—
448#41—84no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8C2×C7×D412yes—
457C2×Q8C7×#4112yescomplemented
458C2×D4—22no—
459C2×D4—24no—
460C2×D4—24no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4C2×C7×D412yes—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—82no—
479C22×C4C2×C7×D412yes—
480C22×C4C2×C7×D411yescharacteristic
481C22×C4C2×C7×D411yescharacteristic
482C22×C4C24×C711yescharacteristic
483Q16—44no—
484SD16—44no—
485M16—22no—
486M16—22no—
487M16—22no—
488M16—22no—
489C2×C8—22no—
490C2×C8—22no—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—82no—
505C4⋊C4C2×C7×D412yes—
506C4⋊C4C2×C7×D412yes—
507C22⋊C4—28no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—82no—
517C42—24no—
518C42—22no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—41no—
523C42—41no—
524C42C2×C7×D411yescharacteristic
525C42C2×C7×D411yescharacteristic
526C42C2×C7×D411yescharacteristic
527C42C2×C7×D411yescharacteristic
528C42C7×#4112yescomplemented
529C2×C7—21no—
530C2×C7—21no—
531C2×C7—82no—
532C2×C7—82no—
533C2×C7#240711yescharacteristic
534C2×C7#301911yescharacteristic
535C2×C7#302211yescharacteristic
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—42no—
541C23—42no—
542C23C22×C7×D411yescharacteristic
543Q8—22no—
544Q8—22no—
545Q8—22no—
546Q8—22no—
547Q8—42no—
548Q8—41no—
549Q8—41no—
550Q8—81no—
551D4—22no—
552D4—22no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—41no—
557D4—41no—
558D4—41no—
559D4—41no—
560D4—81no—
561D4—84no—
562D4—84no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—82no—
585C2×C4—84no—
586C2×C4C7×#12011yescharacteristic
587C2×C4C7×#12011yescharacteristic
588C2×C4C7×#12011yescharacteristic
589C2×C4C7×#12011yescharacteristic
590C2×C4C22×C7×D411yescharacteristic
591C2×C4C22×C7×D411yescharacteristic
592C2×C4C2×C7×#4111yescharacteristic
593C2×C4C2×C7×#4111yescharacteristic
594C2×C4C7×H(5,𝔽2)11yescharacteristic
595C2×C4C7×2−1+411yescharacteristic
596C8—42no—
597C8—42no—
598C7#2418211yescharacteristic, direct product factor, Sylow
599C22—21no—
600C22—21no—
601C22—44no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—82no—
609C22C2×C7×#12011yescharacteristic
610C22C7×#52511yescharacteristic
611C22C7×#53711yescharacteristic
612C4—21no—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—42no—
620C4—42no—
621C4—42no—
622C4—81no—
623C4—81no—
624C4C7×#54611yescharacteristic
625C4C7×#54711yescharacteristic
626C4C7×#54811yescharacteristic
627C4C7×#55011yescharacteristic
628C2—21no—
629C2—21no—
630C2—82no—
631C2—82no—
632C2C7×#240711yescharacteristic
633C2C7×#301911yescharacteristic
634C2C7×#302211yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3506C2
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
2. C7×#3078C2semidirect
3. C7×#3037C2semidirect
4. C7×#2326C2semidirect
5. C7×#2037C2semidirect
6. C7×#2004C2semidirect
7. C7×#1995C2semidirect
8. C7×#1963C2semidirect
9. C7×#1882C2semidirect
10. C7×#1859C2semidirect
11. C7×#1800C2semidirect
51. #24182C7direct (N × H)
135. #3078C2×C7semidirect
136. #3037C2×C7semidirect
137. #2326C2×C7semidirect
138. #2037C2×C7semidirect
139. #2004C2×C7semidirect
140. #1995C2×C7semidirect
141. #1963C2×C7semidirect
142. #1882C2×C7semidirect
143. #1859C2×C7semidirect
144. #1800C2×C7semidirect
159. C2×C7×Q8#41semidirect
230. C42×C7#41semidirect
457. C2×Q8C7×#41semidirect
528. C42C7×#41semidirect
598. C7#24182direct (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×#30782
C7×#30372
C7×#23262
C7×#20372
C7×#20042
C7×#19952
C7×#19632
C7×#18822
C7×#18592
C7×#18002
#241827