Finite Group Databasebeta

← C7×#24172 #13201388

Subgroups — C7×#24172 #13201388

Order 1792.

1058 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3511C211yesmaximal, characteristic
2C7×#3037C211yesmaximal, characteristic, complemented
3C4×C7×#137C211yesmaximal, characteristic, complemented
4C7×#2568C211yesmaximal, characteristic, complemented
5C7×#2040C212yesmaximal, complemented
6C7×#2007C212yesmaximal, complemented
7C7×#1997C211yesmaximal, characteristic, complemented
8C7×#1980C211yesmaximal, characteristic, complemented
9C7×#1882C212yesmaximal, complemented
10C7×#1860C212yesmaximal, complemented
11C7×#1806C211yesmaximal, characteristic, complemented
12C2×C7×#137C2211yescharacteristic
13C2×C7×#137C2212yes—
14C7×#556—21no—
15C7×#555—21no—
16C7×#554—21no—
17C7×#553—22no—
18C7×#548—22no—
19C7×#547—21no—
20C7×#542—22no—
21C7×#537C2212yes—
22C7×#533—21no—
23C7×#532—21no—
24C7×#532C2211yescharacteristic
25C2×C7×C4⋊Q8C2211yescharacteristic
26C7×#520C2212yes—
27C4×C7×#41C2211yescharacteristic
28C2×C4×C7×Q8C2211yescharacteristic
29C7×#464—22no—
30C7×#462—21no—
31C7×#460—21no—
32C7×#441—21no—
33C7×#440—21no—
34C4×C7×Q16—21no—
35C4×C7×SD16—21no—
36C7×#428C2211yescharacteristic
37C7×#423C2211yescharacteristic
38C2×C7×#104C2212yes—
39C2×C7×#104C2212yes—
40C2×C7×#104C2212yes—
41C7×#419C2211yescharacteristic
42C7×#417C2211yescharacteristic
43C7×#417C2212yes—
44C4×C7×M16C2211yescharacteristic
45C7×#400—22no—
46C7×#391—22no—
47C7×#391C2212yes—
48C7×#382C2212yes—
49C4×C7×C4⋊C4C2211yescharacteristic
50C7×#343C2212yes—
51C7×#343C2212yes—
52C7×#339C2212yes—
53#24172C711yesmaximal, characteristic, direct product factor, Sylow
54C2×C7×#41—22no—
55C2×C7×#41C2311yescharacteristic
56C2×C7×#41C2312yes—
57C22×C7×Q8D412yes—
58C22×C7×Q8C2311yescharacteristic
59C7×#137—22no—
60C7×#137—24no—
61C7×#137—22no—
62C7×#137—42no—
63C2×C7×Q16—21no—
64C2×C7×Q16—22no—
65C2×C7×SD16—21no—
66C2×C7×SD16—22no—
67C2×C7×M16C2312yes—
68C2×C7×M16C2311yescharacteristic
69C2×C7×M16C2311yescharacteristic
70C7×C4⋊Q8—21no—
71C7×C4⋊Q8—21no—
72C7×C4⋊Q8—21no—
73C7×C4⋊Q8—21no—
74C7×C4⋊Q8D411yescharacteristic
75C7×#127D411yescharacteristic
76C7×#126—24no—
77C7×#125—22no—
78C7×#124—22no—
79C7×#124—24no—
80C7×#124D412yes—
81C7×#123—24no—
82C7×C22⋊Q8—22no—
83C7×C22⋊Q8—24no—
84C7×C22⋊Q8—22no—
85C7×C22⋊Q8—22no—
86C7×#121—22no—
87C7×#121—22no—
88C7×#121—22no—
89C7×#121—21no—
90C7×#121—21no—
91C7×#121—24no—
92C4×C7×Q8—22no—
93C4×C7×Q8—22no—
94C4×C7×Q8—21no—
95C4×C7×Q8—21no—
96C4×C7×Q8—21no—
97C4×C7×Q8—21no—
98C4×C7×Q8D411yescharacteristic
99C4×C7×Q8D411yescharacteristic
100C4×C7×D4—22no—
101C4×C7×D4—22no—
102C4×C7×D4—21no—
103C4×C7×D4—21no—
104C4×C7×D4—24no—
105C4×C7×D4—21no—
106C4×C7×D4—21no—
107C4×C7×D4D411yescharacteristic
108C4×C7×D4D411yescharacteristic
109C7×#117—22no—
110C7×#117—22no—
111C7×#117—21no—
112C7×#117C2311yescharacteristic
113C2×C7×C4⋊C4—22no—
114C2×C7×C4⋊C4—21no—
115C2×C7×C4⋊C4—21no—
116C2×C7×C4⋊C4D412yes—
117C2×C7×C4⋊C4C2311yescharacteristic
118C2×C7×C4⋊C4C2312yes—
119C2×C42×C7—21no—
120C2×C42×C7—21no—
121C2×C42×C7C2312yes—
122C2×C42×C7C2311yescharacteristic
123C7×#107—21no—
124C7×#106—21no—
125C7×C4⋊C8—21no—
126C7×C4⋊C8—21no—
127C7×#104—24no—
128C7×#104—42no—
129C7×#104—42no—
130C7×#103—21no—
131C7×#103—22no—
132C7×#103—21no—
133C7×#103—21no—
134C7×#102—21no—
135C7×#102—22no—
136C7×#97—21no—
137C4×C7×C8—21no—
138C7×#95—21no—
139C7×#95—24no—
140C7×#95—24no—
141#3511C2×C711yescharacteristic
142#3037C2×C711yescharacteristic, complemented
143C4×#137C2×C711yescharacteristic, complemented
144#2568C2×C711yescharacteristic, complemented
145#2040C2×C712yescomplemented
146#2007C2×C712yescomplemented
147#1997C2×C711yescharacteristic, complemented
148#1980C2×C711yescharacteristic, complemented
149#1882C2×C712yescomplemented
150#1860C2×C712yescomplemented
151#1806C2×C711yescharacteristic, complemented
152C7×#41—22no—
153C7×#41—22no—
154C7×#41—21no—
155C7×#41—21no—
156C7×#41—21no—
157C7×#41—22no—
158C7×#41—22no—
159C7×#41—21no—
160C7×#41—84no—
161C2×C7×Q8—22no—
162C2×C7×Q8—22no—
163C2×C7×Q8—21no—
164C2×C7×Q8—21no—
165C2×C7×Q8—21no—
166C2×C7×Q8—21no—
167C2×C7×Q8—21no—
168C2×C7×Q8—44no—
169C2×C7×Q8C2×D412yes—
170C2×C7×Q8C2×D411yescharacteristic
171C2×C7×Q8#4112yescomplemented
172C2×C7×D4—21no—
173C2×C7×D4—22no—
174C2×C7×D4—24no—
175C2×C7×D4—21no—
176C2×C7×D4—21no—
177C2×C7×D4C2×D412yes—
178C2×C7×D4C2×D411yescharacteristic
179C22×C4×C7—21no—
180C22×C4×C7—24no—
181C22×C4×C7—21no—
182C22×C4×C7—21no—
183C22×C4×C7—21no—
184C22×C4×C7—22no—
185C22×C4×C7—22no—
186C22×C4×C7—21no—
187C22×C4×C7C2×D411yescharacteristic
188C22×C4×C7C2×D414yes—
189C22×C4×C7C2×D411yescharacteristic
190C22×C4×C7C2411yescharacteristic
191C7×Q16—24no—
192C7×Q16—44no—
193C7×SD16—24no—
194C7×SD16—44no—
195C7×M16—21no—
196C7×M16—22no—
197C7×M16—21no—
198C7×M16—22no—
199C7×M16—21no—
200C7×M16#4112yescomplemented
201C2×C7×C8—21no—
202C2×C7×C8—21no—
203C2×C7×C8—22no—
204C7×C4⋊C4—24no—
205C7×C4⋊C4—24no—
206C7×C4⋊C4—22no—
207C7×C4⋊C4—22no—
208C7×C4⋊C4—21no—
209C7×C4⋊C4—21no—
210C7×C4⋊C4—21no—
211C7×C4⋊C4—21no—
212C7×C4⋊C4—22no—
213C7×C4⋊C4—22no—
214C7×C4⋊C4—21no—
215C7×C4⋊C4—21no—
216C7×C4⋊C4—42no—
217C7×C4⋊C4—41no—
218C7×C4⋊C4—41no—
219C7×C4⋊C4C2×D411yescharacteristic
220C7×C4⋊C4C2×D411yescharacteristic
221C7×C22⋊C4—24no—
222C7×C22⋊C4—24no—
223C7×C22⋊C4—24no—
224C7×C22⋊C4—21no—
225C7×C22⋊C4—21no—
226C7×C22⋊C4—21no—
227C7×C22⋊C4—22no—
228C7×C22⋊C4—22no—
229C42×C7—24no—
230C42×C7—21no—
231C42×C7—21no—
232C42×C7—21no—
233C42×C7—22no—
234C42×C7—22no—
235C42×C7—41no—
236C42×C7—41no—
237C42×C7C2×D411yescharacteristic
238C42×C7C2×D411yescharacteristic
239C42×C7C2×D411yescharacteristic
240C42×C7C2×D411yescharacteristic
241C2×#137C22×C711yescharacteristic
242C2×#137C22×C712yes—
243#556—21no—
244#555—21no—
245#554—21no—
246#553—22no—
247#548—22no—
248#547—21no—
249#542—22no—
250#537C22×C712yes—
251#533—21no—
252#532—21no—
253#532C22×C711yescharacteristic
254C2×C4⋊Q8C22×C711yescharacteristic
255#520C22×C712yes—
256C4×#41C22×C711yescharacteristic
257C2×C4×Q8C22×C711yescharacteristic
258#464—22no—
259#462—21no—
260#460—21no—
261#441—21no—
262#440—21no—
263C4×Q16—21no—
264C4×SD16—21no—
265#428C22×C711yescharacteristic
266#423C22×C711yescharacteristic
267C2×#104C22×C712yes—
268C2×#104C22×C712yes—
269C2×#104C22×C712yes—
270#419C22×C711yescharacteristic
271#417C22×C712yes—
272#417C22×C711yescharacteristic
273C4×M16C22×C711yescharacteristic
274#400—22no—
275#391—22no—
276#391C22×C712yes—
277#382C22×C712yes—
278C4×C4⋊C4C22×C711yescharacteristic
279#343C22×C712yes—
280#343C22×C712yes—
281#339C22×C712yes—
282C23×C7—21no—
283C23×C7—22no—
284C23×C7C22×D411yescharacteristic
285C7×Q8—22no—
286C7×Q8—21no—
287C7×Q8—21no—
288C7×Q8—22no—
289C7×Q8—22no—
290C7×Q8—22no—
291C7×Q8—41no—
292C7×Q8—41no—
293C7×Q8—44no—
294C7×Q8—82no—
295C7×D4—21no—
296C7×D4—21no—
297C7×D4—22no—
298C7×D4—22no—
299C7×D4—42no—
300C7×D4—42no—
301C7×D4—41no—
302C7×D4—42no—
303C7×D4—41no—
304C7×D4—42no—
305C7×D4—84no—
306C2×C4×C7—22no—
307C2×C4×C7—21no—
308C2×C4×C7—22no—
309C2×C4×C7—21no—
310C2×C4×C7—21no—
311C2×C4×C7—24no—
312C2×C4×C7—22no—
313C2×C4×C7—41no—
314C2×C4×C7—42no—
315C2×C4×C7—41no—
316C2×C4×C7—41no—
317C2×C4×C7—42no—
318C2×C4×C7—42no—
319C2×C4×C7—41no—
320C2×C4×C7—41no—
321C2×C4×C7—42no—
322C2×C4×C7—44no—
323C2×C4×C7—42no—
324C2×C4×C7—42no—
325C2×C4×C7—42no—
326C2×C4×C7—41no—
327C2×C4×C7—42no—
328C2×C4×C7—84no—
329C2×C4×C7#12011yescharacteristic
330C2×C4×C7#12011yescharacteristic
331C2×C4×C7#12011yescharacteristic
332C2×C4×C7#12011yescharacteristic
333C2×C4×C7C22×D411yescharacteristic
334C2×C4×C7C22×D411yescharacteristic
335C2×C4×C7C2×#4111yescharacteristic
336C2×C4×C7C2×#4111yescharacteristic
337C2×C4×C7H(5,𝔽2)11yescharacteristic
338C2×C4×C72−1+411yescharacteristic
339C7×C8—22no—
340C7×C8—41no—
341C7×C8—42no—
342C2×#41—22no—
343C2×#41C23×C711yescharacteristic
344C2×#41C23×C712yes—
345C22×Q8C7×D412yes—
346C22×Q8C23×C711yescharacteristic
347#137—22no—
348#137—24no—
349#137—22no—
350#137—42no—
351C2×Q16—21no—
352C2×Q16—22no—
353C2×SD16—21no—
354C2×SD16—22no—
355C2×M16C23×C712yes—
356C2×M16C23×C711yescharacteristic
357C2×M16C23×C711yescharacteristic
358C4⋊Q8—21no—
359C4⋊Q8—21no—
360C4⋊Q8—21no—
361C4⋊Q8—21no—
362C4⋊Q8C7×D411yescharacteristic
363#127C7×D411yescharacteristic
364#126—24no—
365#125—22no—
366#124—22no—
367#124—24no—
368#124C7×D412yes—
369#123—24no—
370C22⋊Q8—22no—
371C22⋊Q8—24no—
372C22⋊Q8—22no—
373C22⋊Q8—22no—
374#121—22no—
375#121—22no—
376#121—22no—
377#121—21no—
378#121—21no—
379#121—24no—
380C4×Q8—22no—
381C4×Q8—22no—
382C4×Q8—21no—
383C4×Q8—21no—
384C4×Q8—21no—
385C4×Q8—21no—
386C4×Q8C7×D411yescharacteristic
387C4×Q8C7×D411yescharacteristic
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—21no—
391C4×D4—21no—
392C4×D4—24no—
393C4×D4—21no—
394C4×D4—21no—
395C4×D4C7×D411yescharacteristic
396C4×D4C7×D411yescharacteristic
397#117—22no—
398#117—22no—
399#117—21no—
400#117C23×C711yescharacteristic
401C2×C4⋊C4—22no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4C7×D412yes—
405C2×C4⋊C4C23×C711yescharacteristic
406C2×C4⋊C4C23×C712yes—
407C2×C42—21no—
408C2×C42—21no—
409C2×C42C23×C712yes—
410C2×C42C23×C711yescharacteristic
411#107—21no—
412#106—21no—
413C4⋊C8—21no—
414C4⋊C8—21no—
415#104—24no—
416#104—42no—
417#104—42no—
418#103—21no—
419#103—22no—
420#103—21no—
421#103—21no—
422#102—21no—
423#102—22no—
424#97—21no—
425C4×C8—21no—
426#95—21no—
427#95—24no—
428#95—24no—
429C22×C7—21no—
430C22×C7—21no—
431C22×C7—42no—
432C22×C7—41no—
433C22×C7—41no—
434C22×C7—42no—
435C22×C7—41no—
436C22×C7—42no—
437C22×C7C2×#12011yescharacteristic
438C22×C7#54811yescharacteristic
439C22×C7#55011yescharacteristic
440C4×C7—21no—
441C4×C7—21no—
442C4×C7—21no—
443C4×C7—42no—
444C4×C7—41no—
445C4×C7—42no—
446C4×C7—41no—
447C4×C7—44no—
448C4×C7—41no—
449C4×C7—82no—
450C4×C7—81no—
451C4×C7#45512yes—
452C4×C7#52511yescharacteristic
453C4×C7#53711yescharacteristic
454C4×C7#54611yescharacteristic
455C4×C7#54711yescharacteristic
456#41—22no—
457#41—22no—
458#41—21no—
459#41—21no—
460#41—21no—
461#41—22no—
462#41—22no—
463#41—21no—
464#41—84no—
465C2×Q8—22no—
466C2×Q8—22no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—44no—
473C2×Q8C2×C7×D412yes—
474C2×Q8C2×C7×D411yescharacteristic
475C2×Q8C7×#4112yescomplemented
476C2×D4—21no—
477C2×D4—22no—
478C2×D4—24no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4C2×C7×D412yes—
482C2×D4C2×C7×D411yescharacteristic
483C22×C4—21no—
484C22×C4—24no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4C2×C7×D411yescharacteristic
492C22×C4C2×C7×D414yes—
493C22×C4C2×C7×D411yescharacteristic
494C22×C4C24×C711yescharacteristic
495Q16—24no—
496Q16—44no—
497SD16—24no—
498SD16—44no—
499M16—21no—
500M16—22no—
501M16—21no—
502M16—22no—
503M16—21no—
504M16C7×#4112yescomplemented
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—22no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—42no—
521C4⋊C4—41no—
522C4⋊C4—41no—
523C4⋊C4C2×C7×D411yescharacteristic
524C4⋊C4C2×C7×D411yescharacteristic
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—21no—
529C22⋊C4—21no—
530C22⋊C4—21no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C42—24no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—22no—
538C42—22no—
539C42—41no—
540C42—41no—
541C42C2×C7×D411yescharacteristic
542C42C2×C7×D411yescharacteristic
543C42C2×C7×D411yescharacteristic
544C42C2×C7×D411yescharacteristic
545C2×C7—21no—
546C2×C7—21no—
547C2×C7—81no—
548C2×C7—82no—
549C2×C7#240711yescharacteristic
550C2×C7C2×#45511yescharacteristic
551C2×C7#302011yescharacteristic
552C23—21no—
553C23—22no—
554C23C22×C7×D411yescharacteristic
555Q8—22no—
556Q8—21no—
557Q8—21no—
558Q8—22no—
559Q8—22no—
560Q8—22no—
561Q8—41no—
562Q8—41no—
563Q8—44no—
564Q8—82no—
565D4—21no—
566D4—21no—
567D4—22no—
568D4—22no—
569D4—42no—
570D4—42no—
571D4—41no—
572D4—42no—
573D4—41no—
574D4—42no—
575D4—84no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—84no—
599C2×C4C7×#12011yescharacteristic
600C2×C4C7×#12011yescharacteristic
601C2×C4C7×#12011yescharacteristic
602C2×C4C7×#12011yescharacteristic
603C2×C4C22×C7×D411yescharacteristic
604C2×C4C22×C7×D411yescharacteristic
605C2×C4C2×C7×#4111yescharacteristic
606C2×C4C2×C7×#4111yescharacteristic
607C2×C4C7×H(5,𝔽2)11yescharacteristic
608C2×C4C7×2−1+411yescharacteristic
609C8—22no—
610C8—41no—
611C8—42no—
612C7#2417211yescharacteristic, direct product factor, Sylow
613C22—21no—
614C22—21no—
615C22—42no—
616C22—41no—
617C22—41no—
618C22—42no—
619C22—41no—
620C22—42no—
621C22C2×C7×#12011yescharacteristic
622C22C7×#54811yescharacteristic
623C22C7×#55011yescharacteristic
624C4—21no—
625C4—21no—
626C4—21no—
627C4—42no—
628C4—41no—
629C4—42no—
630C4—41no—
631C4—44no—
632C4—41no—
633C4—82no—
634C4—81no—
635C4C7×#45512yes—
636C4C7×#52511yescharacteristic
637C4C7×#53711yescharacteristic
638C4C7×#54611yescharacteristic
639C4C7×#54711yescharacteristic
640C2—21no—
641C2—21no—
642C2—81no—
643C2—82no—
644C2C7×#240711yescharacteristic
645C2C2×C7×#45511yescharacteristic
646C2C7×#302011yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3511C2
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×#3037C2semidirect
3. C4×C7×#137C2semidirect
4. C7×#2568C2semidirect
5. C7×#2040C2semidirect
6. C7×#2007C2semidirect
7. C7×#1997C2semidirect
8. C7×#1980C2semidirect
9. C7×#1882C2semidirect
10. C7×#1860C2semidirect
11. C7×#1806C2semidirect
53. #24172C7direct (N × H)
142. #3037C2×C7semidirect
143. C4×#137C2×C7semidirect
144. #2568C2×C7semidirect
145. #2040C2×C7semidirect
146. #2007C2×C7semidirect
147. #1997C2×C7semidirect
148. #1980C2×C7semidirect
149. #1882C2×C7semidirect
150. #1860C2×C7semidirect
151. #1806C2×C7semidirect
171. C2×C7×Q8#41semidirect
200. C7×M16#41semidirect
475. C2×Q8C7×#41semidirect
504. M16C7×#41semidirect
612. C7#24172direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35112
C7×#30372
C4×C7×#1372
C7×#25682
C7×#20402
C7×#20072
C7×#19972
C7×#19802
C7×#18822
C7×#18602
C7×#18062
#241727