Finite Group Databasebeta

← C7×#24178 #13201394

Subgroups — C7×#24178 #13201394

Order 1792.

1102 conjugacy classes of subgroups.

The subgroup lattice has 638 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
4C4×C7×#124C211yesmaximal, characteristic, complemented
5C7×#2037C212yesmaximal, complemented
6C7×#2004C212yesmaximal, complemented
7C7×#1994C211yesmaximal, characteristic, complemented
8C7×#1962C211yesmaximal, characteristic, complemented
9C7×#1882C212yesmaximal, complemented
10C7×#1860C212yesmaximal, 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×#382C2211yescharacteristic
46C4×C7×C22⋊C4—22no—
47C43×C7C2211yescharacteristic
48C7×#343C2212yes—
49C7×#339C2212yes—
50C7×#339C2212yes—
51#24178C711yesmaximal, 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⋊C4C2311yescharacteristic
109C2×C7×C4⋊C4C2311yescharacteristic
110C2×C7×C22⋊C4—22no—
111C2×C7×C22⋊C4—22no—
112C2×C7×C22⋊C4D412yes—
113C2×C42×C7—21no—
114C2×C42×C7—21no—
115C2×C42×C7—21no—
116C2×C42×C7C2311yescharacteristic
117C2×C42×C7C2311yescharacteristic
118C2×C42×C7C2311yescharacteristic
119C7×#107—22no—
120C7×#106—22no—
121C7×C4⋊C8—21no—
122C7×C4⋊C8—21no—
123C7×C4⋊C8—21no—
124C7×C4⋊C8—21no—
125C7×#104—24no—
126C7×#104—42no—
127C7×#103—22no—
128C7×#103—22no—
129C7×#103—22no—
130C7×#102—22no—
131C7×#100—42no—
132C7×#95—22no—
133C7×#95—22no—
134#3506C2×C711yescharacteristic
135#3078C2×C711yescharacteristic, complemented
136#3037C2×C711yescharacteristic, complemented
137C4×#124C2×C711yescharacteristic, complemented
138#2037C2×C712yescomplemented
139#2004C2×C712yescomplemented
140#1994C2×C711yescharacteristic, complemented
141#1962C2×C711yescharacteristic, complemented
142#1882C2×C712yescomplemented
143#1860C2×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—21no—
175C22×C4×C7—24no—
176C22×C4×C7—22no—
177C22×C4×C7—22no—
178C22×C4×C7—22no—
179C22×C4×C7—21no—
180C22×C4×C7—44no—
181C22×C4×C7C2×D411yescharacteristic
182C22×C4×C7C2×D412yes—
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⋊C4C2×D412yes—
207C7×C4⋊C4C2×D412yes—
208C7×C22⋊C4—28no—
209C7×C22⋊C4—24no—
210C7×C22⋊C4—24no—
211C7×C22⋊C4—24no—
212C7×C22⋊C4—22no—
213C7×C22⋊C4—22no—
214C7×C22⋊C4—22no—
215C7×C22⋊C4—22no—
216C7×C22⋊C4—22no—
217C7×C22⋊C4—44no—
218C42×C7—24no—
219C42×C7—22no—
220C42×C7—21no—
221C42×C7—21no—
222C42×C7—21no—
223C42×C7—42no—
224C42×C7—42no—
225C42×C7—41no—
226C42×C7—41no—
227C42×C7C2×D411yescharacteristic
228C42×C7C2×D411yescharacteristic
229C42×C7C2×D411yescharacteristic
230C42×C7C2×D411yescharacteristic
231C42×C7#4112yescomplemented
232C2×#137C22×C712yes—
233#559—22no—
234#556—21no—
235#555—21no—
236#553—22no—
237#550—21no—
238#547—21no—
239#538—22no—
240#535C22×C712yes—
241#533C22×C711yescharacteristic
242#532C22×C711yescharacteristic
243C2×#124C22×C711yescharacteristic
244#525—22no—
245#520C22×C712yes—
246C2×C4×Q8C22×C711yescharacteristic
247C2×C4×D4C22×C711yescharacteristic
248#479—22no—
249#477—21no—
250#475—21no—
251#464—22no—
252#462—21no—
253#460—21no—
254#428C22×C712yes—
255#423C22×C711yescharacteristic
256#423C22×C711yescharacteristic
257C2×#104C22×C712yes—
258C2×#104C22×C712yes—
259#419C22×C712yes—
260#417C22×C712yes—
261C2×#100C22×C712yes—
262#391C22×C711yescharacteristic
263#390C22×C711yescharacteristic
264#388—22no—
265#382C22×C711yescharacteristic
266C4×C22⋊C4—22no—
267C43C22×C711yescharacteristic
268#343C22×C712yes—
269#339C22×C712yes—
270#339C22×C712yes—
271C23×C7—22no—
272C23×C7—22no—
273C23×C7—22no—
274C23×C7—22no—
275C23×C7—42no—
276C23×C7—42no—
277C23×C7C22×D411yescharacteristic
278C7×Q8—22no—
279C7×Q8—22no—
280C7×Q8—22no—
281C7×Q8—22no—
282C7×Q8—41no—
283C7×Q8—41no—
284C7×Q8—42no—
285C7×Q8—81no—
286C7×D4—22no—
287C7×D4—22no—
288C7×D4—42no—
289C7×D4—42no—
290C7×D4—41no—
291C7×D4—41no—
292C7×D4—41no—
293C7×D4—41no—
294C7×D4—42no—
295C7×D4—81no—
296C7×D4—84no—
297C7×D4—84no—
298C2×C4×C7—22no—
299C2×C4×C7—22no—
300C2×C4×C7—22no—
301C2×C4×C7—22no—
302C2×C4×C7—22no—
303C2×C4×C7—22no—
304C2×C4×C7—22no—
305C2×C4×C7—22no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
308C2×C4×C7—41no—
309C2×C4×C7—41no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7—41no—
313C2×C4×C7—42no—
314C2×C4×C7—42no—
315C2×C4×C7—44no—
316C2×C4×C7—42no—
317C2×C4×C7—42no—
318C2×C4×C7—41no—
319C2×C4×C7—82no—
320C2×C4×C7—84no—
321C2×C4×C7#12011yescharacteristic
322C2×C4×C7#12011yescharacteristic
323C2×C4×C7#12011yescharacteristic
324C2×C4×C7#12011yescharacteristic
325C2×C4×C7C22×D411yescharacteristic
326C2×C4×C7C22×D411yescharacteristic
327C2×C4×C7C2×#4111yescharacteristic
328C2×C4×C7C2×#4111yescharacteristic
329C2×C4×C7H(5,𝔽2)11yescharacteristic
330C2×C4×C72−1+411yescharacteristic
331C7×C8—42no—
332C7×C8—42no—
333C2×#41—22no—
334C2×#41C23×C712yes—
335C22×Q8C23×C711yescharacteristic
336C22×D4C23×C711yescharacteristic
337C23×C4C7×D412yes—
338#137—24no—
339#137—42no—
340C2×Q16—22no—
341C2×SD16—22no—
342C2×M16C23×C712yes—
343C2×M16C23×C712yes—
344C4⋊Q8C7×D411yescharacteristic
345C4⋊Q8C7×D411yescharacteristic
346C4⋊Q8C7×D411yescharacteristic
347#127—22no—
348#127C7×D411yescharacteristic
349#126—24no—
350#125C7×D412yes—
351#124—24no—
352#124—22no—
353#124—22no—
354#124—22no—
355#124C7×D412yes—
356#123—24no—
357#123—24no—
358C22⋊Q8—22no—
359C22⋊Q8—21no—
360C22⋊Q8—21no—
361#121—22no—
362#121—24no—
363#121—21no—
364#121—21no—
365#121—22no—
366#121—22no—
367#120—24no—
368C4×Q8—22no—
369C4×Q8—21no—
370C4×Q8—21no—
371C4×Q8—21no—
372C4×Q8—21no—
373C4×Q8—21no—
374C4×Q8—21no—
375C4×D4—24no—
376C4×D4—22no—
377C4×D4—24no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4—21no—
383C4×D4—21no—
384#117—22no—
385#117—22no—
386#117C23×C712yes—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4C23×C711yescharacteristic
390C2×C4⋊C4C23×C711yescharacteristic
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4C7×D412yes—
394C2×C42—21no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42C23×C711yescharacteristic
398C2×C42C23×C711yescharacteristic
399C2×C42C23×C711yescharacteristic
400#107—22no—
401#106—22no—
402C4⋊C8—21no—
403C4⋊C8—21no—
404C4⋊C8—21no—
405C4⋊C8—21no—
406#104—24no—
407#104—42no—
408#103—22no—
409#103—22no—
410#103—22no—
411#102—22no—
412#100—42no—
413#95—22no—
414#95—22no—
415C22×C7—21no—
416C22×C7—21no—
417C22×C7—44no—
418C22×C7—42no—
419C22×C7—42no—
420C22×C7—42no—
421C22×C7—42no—
422C22×C7—42no—
423C22×C7—42no—
424C22×C7—82no—
425C22×C7C2×#12011yescharacteristic
426C22×C7#52511yescharacteristic
427C22×C7#53711yescharacteristic
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—42no—
438C4×C7—81no—
439C4×C7#45512yes—
440C4×C7#54611yescharacteristic
441C4×C7#54711yescharacteristic
442C4×C7#54811yescharacteristic
443C4×C7#55011yescharacteristic
444C24C2×C7×D412yes—
445#41—22no—
446#41—22no—
447#41—22no—
448#41—22no—
449#41—84no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8C2×C7×D412yes—
458C2×Q8C7×#4112yescomplemented
459C2×D4—22no—
460C2×D4—24no—
461C2×D4—24no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4C2×C7×D412yes—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—44no—
480C22×C4C2×C7×D411yescharacteristic
481C22×C4C2×C7×D412yes—
482C22×C4C2×C7×D411yescharacteristic
483C22×C4C24×C711yescharacteristic
484Q16—44no—
485SD16—44no—
486M16—22no—
487M16—22no—
488M16—22no—
489M16—22no—
490C2×C8—22no—
491C2×C8—22no—
492C4⋊C4—24no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—41no—
504C4⋊C4—41no—
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—44no—
517C42—24no—
518C42—22no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—42no—
523C42—42no—
524C42—41no—
525C42—41no—
526C42C2×C7×D411yescharacteristic
527C42C2×C7×D411yescharacteristic
528C42C2×C7×D411yescharacteristic
529C42C2×C7×D411yescharacteristic
530C42C7×#4112yescomplemented
531C2×C7—21no—
532C2×C7—21no—
533C2×C7—82no—
534C2×C7—82no—
535C2×C7#240711yescharacteristic
536C2×C7C2×#45511yescharacteristic
537C2×C7#301811yescharacteristic
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—42no—
543C23—42no—
544C23C22×C7×D411yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—22no—
548Q8—22no—
549Q8—41no—
550Q8—41no—
551Q8—42no—
552Q8—81no—
553D4—22no—
554D4—22no—
555D4—42no—
556D4—42no—
557D4—41no—
558D4—41no—
559D4—41no—
560D4—41no—
561D4—42no—
562D4—81no—
563D4—84no—
564D4—84no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—82no—
587C2×C4—84no—
588C2×C4C7×#12011yescharacteristic
589C2×C4C7×#12011yescharacteristic
590C2×C4C7×#12011yescharacteristic
591C2×C4C7×#12011yescharacteristic
592C2×C4C22×C7×D411yescharacteristic
593C2×C4C22×C7×D411yescharacteristic
594C2×C4C2×C7×#4111yescharacteristic
595C2×C4C2×C7×#4111yescharacteristic
596C2×C4C7×H(5,𝔽2)11yescharacteristic
597C2×C4C7×2−1+411yescharacteristic
598C8—42no—
599C8—42no—
600C7#2417811yescharacteristic, direct product factor, Sylow
601C22—21no—
602C22—21no—
603C22—44no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—82no—
611C22C2×C7×#12011yescharacteristic
612C22C7×#52511yescharacteristic
613C22C7×#53711yescharacteristic
614C4—21no—
615C4—21no—
616C4—21no—
617C4—42no—
618C4—42no—
619C4—42no—
620C4—42no—
621C4—42no—
622C4—42no—
623C4—42no—
624C4—81no—
625C4C7×#45512yes—
626C4C7×#54611yescharacteristic
627C4C7×#54711yescharacteristic
628C4C7×#54811yescharacteristic
629C4C7×#55011yescharacteristic
630C2—21no—
631C2—21no—
632C2—82no—
633C2—82no—
634C2C7×#240711yescharacteristic
635C2C2×C7×#45511yescharacteristic
636C2C7×#301811yescharacteristic
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. C4×C7×#124C2semidirect
5. C7×#2037C2semidirect
6. C7×#2004C2semidirect
7. C7×#1994C2semidirect
8. C7×#1962C2semidirect
9. C7×#1882C2semidirect
10. C7×#1860C2semidirect
11. C7×#1800C2semidirect
51. #24178C7direct (N × H)
135. #3078C2×C7semidirect
136. #3037C2×C7semidirect
137. C4×#124C2×C7semidirect
138. #2037C2×C7semidirect
139. #2004C2×C7semidirect
140. #1994C2×C7semidirect
141. #1962C2×C7semidirect
142. #1882C2×C7semidirect
143. #1860C2×C7semidirect
144. #1800C2×C7semidirect
159. C2×C7×Q8#41semidirect
231. C42×C7#41semidirect
458. C2×Q8C7×#41semidirect
530. C42C7×#41semidirect
600. C7#24178direct (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
C4×C7×#1242
C7×#20372
C7×#20042
C7×#19942
C7×#19622
C7×#18822
C7×#18602
C7×#18002
#241787