Finite Group Databasebeta

← C7×#24173 #13201389

Subgroups — C7×#24173 #13201389

Order 1792.

1074 conjugacy classes of subgroups.

The subgroup lattice has 650 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×#2567C211yesmaximal, characteristic, complemented
5C7×#2039C212yesmaximal, complemented
6C7×#2007C212yesmaximal, complemented
7C7×#1997C211yesmaximal, characteristic, complemented
8C7×#1980C211yesmaximal, characteristic, complemented
9C7×#1882C212yesmaximal, complemented
10C7×#1859C212yesmaximal, 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×#395—22no—
46C7×#391C2212yes—
47C7×#389—22no—
48C7×#384C2212yes—
49C4×C7×C4⋊C4C2211yescharacteristic
50C7×#343C2212yes—
51C7×#343C2212yes—
52C7×#339C2212yes—
53#24173C711yesmaximal, 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—21no—
114C2×C7×C4⋊C4—22no—
115C2×C7×C4⋊C4—21no—
116C2×C7×C4⋊C4—24no—
117C2×C7×C4⋊C4D412yes—
118C2×C7×C4⋊C4C2311yescharacteristic
119C2×C7×C4⋊C4C2312yes—
120C2×C42×C7—21no—
121C2×C42×C7—21no—
122C2×C42×C7C2312yes—
123C2×C42×C7C2311yescharacteristic
124C7×#107—21no—
125C7×#106—21no—
126C7×C4⋊C8—21no—
127C7×C4⋊C8—21no—
128C7×#104—24no—
129C7×#104—42no—
130C7×#104—42no—
131C7×#103—21no—
132C7×#103—22no—
133C7×#103—21no—
134C7×#103—21no—
135C7×#102—21no—
136C7×#102—22no—
137C7×#97—21no—
138C4×C7×C8—21no—
139C7×#95—21no—
140C7×#95—24no—
141#3511C2×C711yescharacteristic
142#3037C2×C711yescharacteristic, complemented
143C4×#137C2×C711yescharacteristic, complemented
144#2567C2×C711yescharacteristic, complemented
145#2039C2×C712yescomplemented
146#2007C2×C712yescomplemented
147#1997C2×C711yescharacteristic, complemented
148#1980C2×C711yescharacteristic, complemented
149#1882C2×C712yescomplemented
150#1859C2×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—82no—
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—22no—
181C22×C4×C7—24no—
182C22×C4×C7—21no—
183C22×C4×C7—21no—
184C22×C4×C7—21no—
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—24no—
207C7×C4⋊C4—22no—
208C7×C4⋊C4—22no—
209C7×C4⋊C4—21no—
210C7×C4⋊C4—21no—
211C7×C4⋊C4—21no—
212C7×C4⋊C4—21no—
213C7×C4⋊C4—22no—
214C7×C4⋊C4—22no—
215C7×C4⋊C4—21no—
216C7×C4⋊C4—21no—
217C7×C4⋊C4—41no—
218C7×C4⋊C4—41no—
219C7×C4⋊C4—84no—
220C7×C4⋊C4C2×D411yescharacteristic
221C7×C4⋊C4C2×D411yescharacteristic
222C7×C22⋊C4—24no—
223C7×C22⋊C4—24no—
224C7×C22⋊C4—24no—
225C7×C22⋊C4—21no—
226C7×C22⋊C4—21no—
227C7×C22⋊C4—21no—
228C7×C22⋊C4—22no—
229C7×C22⋊C4—22no—
230C42×C7—24no—
231C42×C7—21no—
232C42×C7—21no—
233C42×C7—21no—
234C42×C7—22no—
235C42×C7—22no—
236C42×C7—41no—
237C42×C7—41no—
238C42×C7C2×D411yescharacteristic
239C42×C7C2×D411yescharacteristic
240C42×C7C2×D411yescharacteristic
241C42×C7C2×D411yescharacteristic
242C2×#137C22×C711yescharacteristic
243C2×#137C22×C712yes—
244#556—21no—
245#555—21no—
246#554—21no—
247#553—22no—
248#548—22no—
249#547—21no—
250#542—22no—
251#537C22×C712yes—
252#533—21no—
253#532—21no—
254#532C22×C711yescharacteristic
255C2×C4⋊Q8C22×C711yescharacteristic
256#520C22×C712yes—
257C4×#41C22×C711yescharacteristic
258C2×C4×Q8C22×C711yescharacteristic
259#464—22no—
260#462—21no—
261#460—21no—
262#441—21no—
263#440—21no—
264C4×Q16—21no—
265C4×SD16—21no—
266#428C22×C711yescharacteristic
267#423C22×C711yescharacteristic
268C2×#104C22×C712yes—
269C2×#104C22×C712yes—
270C2×#104C22×C712yes—
271#419C22×C711yescharacteristic
272#417C22×C712yes—
273#417C22×C711yescharacteristic
274C4×M16C22×C711yescharacteristic
275#395—22no—
276#391C22×C712yes—
277#389—22no—
278#384C22×C712yes—
279C4×C4⋊C4C22×C711yescharacteristic
280#343C22×C712yes—
281#343C22×C712yes—
282#339C22×C712yes—
283C23×C7—21no—
284C23×C7—22no—
285C23×C7C22×D411yescharacteristic
286C7×Q8—22no—
287C7×Q8—21no—
288C7×Q8—21no—
289C7×Q8—22no—
290C7×Q8—22no—
291C7×Q8—22no—
292C7×Q8—44no—
293C7×Q8—41no—
294C7×Q8—41no—
295C7×Q8—82no—
296C7×D4—21no—
297C7×D4—21no—
298C7×D4—22no—
299C7×D4—22no—
300C7×D4—42no—
301C7×D4—42no—
302C7×D4—41no—
303C7×D4—42no—
304C7×D4—41no—
305C7×D4—42no—
306C7×D4—84no—
307C2×C4×C7—22no—
308C2×C4×C7—24no—
309C2×C4×C7—21no—
310C2×C4×C7—22no—
311C2×C4×C7—21no—
312C2×C4×C7—21no—
313C2×C4×C7—24no—
314C2×C4×C7—22no—
315C2×C4×C7—41no—
316C2×C4×C7—42no—
317C2×C4×C7—42no—
318C2×C4×C7—41no—
319C2×C4×C7—42no—
320C2×C4×C7—41no—
321C2×C4×C7—42no—
322C2×C4×C7—42no—
323C2×C4×C7—41no—
324C2×C4×C7—41no—
325C2×C4×C7—42no—
326C2×C4×C7—44no—
327C2×C4×C7—42no—
328C2×C4×C7—41no—
329C2×C4×C7—84no—
330C2×C4×C7#12011yescharacteristic
331C2×C4×C7#12011yescharacteristic
332C2×C4×C7#12011yescharacteristic
333C2×C4×C7#12011yescharacteristic
334C2×C4×C7C22×D411yescharacteristic
335C2×C4×C7C22×D411yescharacteristic
336C2×C4×C7C2×#4111yescharacteristic
337C2×C4×C7C2×#4111yescharacteristic
338C2×C4×C7H(5,𝔽2)11yescharacteristic
339C2×C4×C72−1+411yescharacteristic
340C7×C8—22no—
341C7×C8—41no—
342C7×C8—42no—
343C2×#41—22no—
344C2×#41C23×C711yescharacteristic
345C2×#41C23×C712yes—
346C22×Q8C7×D412yes—
347C22×Q8C23×C711yescharacteristic
348#137—22no—
349#137—24no—
350#137—22no—
351#137—42no—
352C2×Q16—21no—
353C2×Q16—22no—
354C2×SD16—21no—
355C2×SD16—22no—
356C2×M16C23×C712yes—
357C2×M16C23×C711yescharacteristic
358C2×M16C23×C711yescharacteristic
359C4⋊Q8—21no—
360C4⋊Q8—21no—
361C4⋊Q8—21no—
362C4⋊Q8—21no—
363C4⋊Q8C7×D411yescharacteristic
364#127C7×D411yescharacteristic
365#126—24no—
366#125—22no—
367#124—22no—
368#124—24no—
369#124C7×D412yes—
370#123—24no—
371C22⋊Q8—22no—
372C22⋊Q8—24no—
373C22⋊Q8—22no—
374C22⋊Q8—22no—
375#121—22no—
376#121—22no—
377#121—22no—
378#121—21no—
379#121—21no—
380#121—24no—
381C4×Q8—22no—
382C4×Q8—22no—
383C4×Q8—21no—
384C4×Q8—21no—
385C4×Q8—21no—
386C4×Q8—21no—
387C4×Q8C7×D411yescharacteristic
388C4×Q8C7×D411yescharacteristic
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—21no—
392C4×D4—21no—
393C4×D4—24no—
394C4×D4—21no—
395C4×D4—21no—
396C4×D4C7×D411yescharacteristic
397C4×D4C7×D411yescharacteristic
398#117—22no—
399#117—22no—
400#117—21no—
401#117C23×C711yescharacteristic
402C2×C4⋊C4—21no—
403C2×C4⋊C4—22no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—24no—
406C2×C4⋊C4C7×D412yes—
407C2×C4⋊C4C23×C711yescharacteristic
408C2×C4⋊C4C23×C712yes—
409C2×C42—21no—
410C2×C42—21no—
411C2×C42C23×C712yes—
412C2×C42C23×C711yescharacteristic
413#107—21no—
414#106—21no—
415C4⋊C8—21no—
416C4⋊C8—21no—
417#104—24no—
418#104—42no—
419#104—42no—
420#103—21no—
421#103—22no—
422#103—21no—
423#103—21no—
424#102—21no—
425#102—22no—
426#97—21no—
427C4×C8—21no—
428#95—21no—
429#95—24no—
430C22×C7—21no—
431C22×C7—21no—
432C22×C7—42no—
433C22×C7—41no—
434C22×C7—41no—
435C22×C7—42no—
436C22×C7—41no—
437C22×C7—42no—
438C22×C7C2×#12011yescharacteristic
439C22×C7#54811yescharacteristic
440C22×C7#55011yescharacteristic
441C4×C7—21no—
442C4×C7—21no—
443C4×C7—21no—
444C4×C7—44no—
445C4×C7—42no—
446C4×C7—41no—
447C4×C7—42no—
448C4×C7—41no—
449C4×C7—41no—
450C4×C7—44no—
451C4×C7—81no—
452C4×C7#45612yes—
453C4×C7#52511yescharacteristic
454C4×C7#53711yescharacteristic
455C4×C7#54611yescharacteristic
456C4×C7#54711yescharacteristic
457#41—22no—
458#41—22no—
459#41—21no—
460#41—21no—
461#41—21no—
462#41—22no—
463#41—22no—
464#41—21no—
465#41—84no—
466C2×Q8—22no—
467C2×Q8—22no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—82no—
474C2×Q8C2×C7×D412yes—
475C2×Q8C2×C7×D411yescharacteristic
476C2×Q8C7×#4112yescomplemented
477C2×D4—21no—
478C2×D4—22no—
479C2×D4—24no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4C2×C7×D412yes—
483C2×D4C2×C7×D411yescharacteristic
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—24no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—21no—
492C22×C4C2×C7×D411yescharacteristic
493C22×C4C2×C7×D414yes—
494C22×C4C2×C7×D411yescharacteristic
495C22×C4C24×C711yescharacteristic
496Q16—24no—
497Q16—44no—
498SD16—24no—
499SD16—44no—
500M16—21no—
501M16—22no—
502M16—21no—
503M16—22no—
504M16—21no—
505M16C7×#4112yescomplemented
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—22no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—41no—
523C4⋊C4—41no—
524C4⋊C4—84no—
525C4⋊C4C2×C7×D411yescharacteristic
526C4⋊C4C2×C7×D411yescharacteristic
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—21no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C42—24no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—22no—
540C42—22no—
541C42—41no—
542C42—41no—
543C42C2×C7×D411yescharacteristic
544C42C2×C7×D411yescharacteristic
545C42C2×C7×D411yescharacteristic
546C42C2×C7×D411yescharacteristic
547C2×C7—21no—
548C2×C7—21no—
549C2×C7—81no—
550C2×C7—82no—
551C2×C7#240711yescharacteristic
552C2×C7C2×#45611yescharacteristic
553C2×C7#301911yescharacteristic
554C23—21no—
555C23—22no—
556C23C22×C7×D411yescharacteristic
557Q8—22no—
558Q8—21no—
559Q8—21no—
560Q8—22no—
561Q8—22no—
562Q8—22no—
563Q8—44no—
564Q8—41no—
565Q8—41no—
566Q8—82no—
567D4—21no—
568D4—21no—
569D4—22no—
570D4—22no—
571D4—42no—
572D4—42no—
573D4—41no—
574D4—42no—
575D4—41no—
576D4—42no—
577D4—84no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—84no—
601C2×C4C7×#12011yescharacteristic
602C2×C4C7×#12011yescharacteristic
603C2×C4C7×#12011yescharacteristic
604C2×C4C7×#12011yescharacteristic
605C2×C4C22×C7×D411yescharacteristic
606C2×C4C22×C7×D411yescharacteristic
607C2×C4C2×C7×#4111yescharacteristic
608C2×C4C2×C7×#4111yescharacteristic
609C2×C4C7×H(5,𝔽2)11yescharacteristic
610C2×C4C7×2−1+411yescharacteristic
611C8—22no—
612C8—41no—
613C8—42no—
614C7#2417311yescharacteristic, direct product factor, Sylow
615C22—21no—
616C22—21no—
617C22—42no—
618C22—41no—
619C22—41no—
620C22—42no—
621C22—41no—
622C22—42no—
623C22C2×C7×#12011yescharacteristic
624C22C7×#54811yescharacteristic
625C22C7×#55011yescharacteristic
626C4—21no—
627C4—21no—
628C4—21no—
629C4—44no—
630C4—42no—
631C4—41no—
632C4—42no—
633C4—41no—
634C4—41no—
635C4—44no—
636C4—81no—
637C4C7×#45612yes—
638C4C7×#52511yescharacteristic
639C4C7×#53711yescharacteristic
640C4C7×#54611yescharacteristic
641C4C7×#54711yescharacteristic
642C2—21no—
643C2—21no—
644C2—81no—
645C2—82no—
646C2C7×#240711yescharacteristic
647C2C2×C7×#45611yescharacteristic
648C2C7×#301911yescharacteristic
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×#2567C2semidirect
5. C7×#2039C2semidirect
6. C7×#2007C2semidirect
7. C7×#1997C2semidirect
8. C7×#1980C2semidirect
9. C7×#1882C2semidirect
10. C7×#1859C2semidirect
11. C7×#1806C2semidirect
53. #24173C7direct (N × H)
142. #3037C2×C7semidirect
143. C4×#137C2×C7semidirect
144. #2567C2×C7semidirect
145. #2039C2×C7semidirect
146. #2007C2×C7semidirect
147. #1997C2×C7semidirect
148. #1980C2×C7semidirect
149. #1882C2×C7semidirect
150. #1859C2×C7semidirect
151. #1806C2×C7semidirect
171. C2×C7×Q8#41semidirect
200. C7×M16#41semidirect
476. C2×Q8C7×#41semidirect
505. M16C7×#41semidirect
614. C7#24173direct (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×#25672
C7×#20392
C7×#20072
C7×#19972
C7×#19802
C7×#18822
C7×#18592
C7×#18062
#241737