Finite Group Databasebeta

← C7×#18801 #13196017

Subgroups — C7×#18801 #13196017

Order 1792.

936 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×Q8×#41C211yesmaximal, characteristic
2C7×#3230C211yesmaximal, characteristic, complemented
3C7×#3079C211yesmaximal, characteristic
4C2×C7×#462C211yesmaximal, characteristic, complemented
5C7×#1649C212yesmaximal, complemented
6C7×#1648C212yesmaximal, complemented
7C7×#1645C212yesmaximal, complemented
8C7×#1644C212yesmaximal, complemented
9C7×#1550C211yesmaximal, characteristic, complemented
10C7×#1487C211yesmaximal, characteristic, complemented
11C7×#1477C211yesmaximal, characteristic
12C22×C7×Q16—21no—
13C7×Q82C2211yescharacteristic
14C7×#557—21no—
15C7×#557C2211yescharacteristic
16C7×#554—22no—
17C7×#554—21no—
18C7×#554—21no—
19C7×#554C2211yescharacteristic
20C7×D4×Q8—21no—
21C7×D4×Q8C2211yescharacteristic
22C7×#533—22no—
23C7×#533—21no—
24C7×#533—21no—
25C7×#533C2211yescharacteristic
26C2×C7×C4⋊Q8C2211yescharacteristic
27C4×C7×#41—21no—
28C4×C7×#41C2211yescharacteristic
29C2×C4×C7×Q8—21no—
30C2×C4×C7×Q8C2211yescharacteristic
31C7×#500C2212yes—
32C7×#487C2212yes—
33C7×#484—22no—
34C7×#477—22no—
35C7×#477—21no—
36C7×#476—21no—
37C7×#475C2212yes—
38C7×#474C2212yes—
39C7×#462—21no—
40C7×#462—21no—
41C7×#462C2212yes—
42C7×#462C2212yes—
43C7×#431C2211yescharacteristic
44C7×#428—21no—
45C7×#423C2211yescharacteristic
46C2×C7×C4⋊C8C2211yescharacteristic
47C7×#417—21no—
48C2×C7×#103—21no—
49C7×#333C2212yes—
50C7×#332C2212yes—
51C7×Q8⋊C8C2212yes—
52C7×Q8⋊C8C2212yes—
53C7×Q8⋊C8C2212yes—
54C7×D4⋊C8C2212yes—
55#18801C711yesmaximal, characteristic, direct product factor, Sylow
56C2×C7×#41D411yescharacteristic
57C22×C7×Q8—21no—
58C22×C7×Q8—21no—
59C22×C7×Q8D411yescharacteristic
60C2×C7×Q16—22no—
61C2×C7×Q16—22no—
62C2×C7×Q16—44no—
63C2×C7×M16—21no—
64C22×C7×C8—21no—
65C7×C4⋊Q8—22no—
66C7×C4⋊Q8—21no—
67C7×C4⋊Q8—22no—
68C7×C4⋊Q8—21no—
69C7×C4⋊Q8—21no—
70C7×C4⋊Q8—21no—
71C7×C4⋊Q8—21no—
72C7×C4⋊Q8—21no—
73C7×C4⋊Q8—21no—
74C7×C4⋊Q8C2312yes—
75C7×C4⋊Q8C2311yescharacteristic
76C7×C4⋊Q8C2311yescharacteristic
77C7×#125—22no—
78C7×#125—22no—
79C7×#125—22no—
80C7×#125—21no—
81C7×#125—21no—
82C7×#125—21no—
83C7×C22⋊Q8—22no—
84C7×C22⋊Q8—22no—
85C7×C22⋊Q8—22no—
86C7×C22⋊Q8—22no—
87C7×C22⋊Q8—22no—
88C7×C22⋊Q8—21no—
89C7×C22⋊Q8—22no—
90C7×C22⋊Q8—21no—
91C7×C22⋊Q8—21no—
92C7×C22⋊Q8—22no—
93C7×C22⋊Q8D412yes—
94C4×C7×Q8—22no—
95C4×C7×Q8—22no—
96C4×C7×Q8—22no—
97C4×C7×Q8—21no—
98C4×C7×Q8—21no—
99C4×C7×Q8—21no—
100C4×C7×Q8—21no—
101C4×C7×Q8—21no—
102C4×C7×Q8—21no—
103C4×C7×Q8—21no—
104C4×C7×Q8D411yescharacteristic
105C4×C7×Q8D412yes—
106C4×C7×Q8D411yescharacteristic
107C4×C7×Q8D411yescharacteristic
108C4×C7×Q8C2311yescharacteristic
109C4×C7×Q8C2311yescharacteristic
110C4×C7×Q8C2311yescharacteristic
111C4×C7×D4—22no—
112C4×C7×D4—22no—
113C4×C7×D4—21no—
114C4×C7×D4—21no—
115C4×C7×D4—21no—
116C4×C7×D4—21no—
117C4×C7×D4D411yescharacteristic
118C4×C7×D4C2311yescharacteristic
119C7×#117—22no—
120C7×#117—21no—
121C7×#117—21no—
122C7×#117D411yescharacteristic
123C2×C7×C4⋊C4—22no—
124C2×C7×C4⋊C4—21no—
125C2×C7×C4⋊C4—21no—
126C2×C7×C4⋊C4—21no—
127C2×C7×C4⋊C4—21no—
128C2×C7×C4⋊C4D411yescharacteristic
129C2×C42×C7—22no—
130C2×C42×C7—21no—
131C2×C42×C7—21no—
132C2×C42×C7C2311yescharacteristic
133C7×#107—22no—
134C7×#107—24no—
135C7×#106—22no—
136C7×C4⋊C8—21no—
137C7×C4⋊C8—21no—
138C7×C4⋊C8—21no—
139C7×C4⋊C8C2312yes—
140C7×C4⋊C8C2312yes—
141C7×#103—22no—
142C7×#103—24no—
143C7×#103—22no—
144C7×#103—22no—
145C7×#102—22no—
146C7×C22⋊C8—21no—
147C4×C7×C8C2312yes—
148Q8×#41C2×C711yescharacteristic
149#3230C2×C711yescharacteristic, complemented
150#3079C2×C711yescharacteristic
151C2×#462C2×C711yescharacteristic, complemented
152#1649C2×C712yescomplemented
153#1648C2×C712yescomplemented
154#1645C2×C712yescomplemented
155#1644C2×C712yescomplemented
156#1550C2×C711yescharacteristic, complemented
157#1487C2×C711yescharacteristic, complemented
158#1477C2×C711yescharacteristic
159C7×#41—22no—
160C7×#41Q1614yescomplemented
161C2×C7×Q8—22no—
162C2×C7×Q8—22no—
163C2×C7×Q8—21no—
164C2×C7×Q8—21no—
165C2×C7×Q8—22no—
166C2×C7×Q8—22no—
167C2×C7×Q8—41no—
168C2×C7×Q8—42no—
169C2×C7×Q8—42no—
170C2×C7×Q8—42no—
171C2×C7×Q8—42no—
172C2×C7×Q8—41no—
173C2×C7×Q8C2×D411yescharacteristic
174C2×C7×Q8C2×D411yescharacteristic
175C2×C7×Q8C2×D411yescharacteristic
176C2×C7×Q8C2×D412yes—
177C2×C7×D4—21no—
178C2×C7×D4C2×D411yescharacteristic
179C22×C4×C7—21no—
180C22×C4×C7—22no—
181C22×C4×C7—21no—
182C22×C4×C7—22no—
183C22×C4×C7—21no—
184C22×C4×C7—21no—
185C22×C4×C7—21no—
186C22×C4×C7—41no—
187C22×C4×C7C2×D411yescharacteristic
188C22×C4×C7C2×D411yescharacteristic
189C22×C4×C7C2×D411yescharacteristic
190C7×Q16—44no—
191C7×Q16—44no—
192C7×M16—42no—
193C2×C7×C8—22no—
194C2×C7×C8—22no—
195C2×C7×C8—22no—
196C2×C7×C8—42no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4—22no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—22no—
201C7×C4⋊C4—22no—
202C7×C4⋊C4—22no—
203C7×C4⋊C4—22no—
204C7×C4⋊C4—22no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—22no—
207C7×C4⋊C4—21no—
208C7×C4⋊C4—21no—
209C7×C4⋊C4—21no—
210C7×C4⋊C4—22no—
211C7×C4⋊C4—21no—
212C7×C4⋊C4—21no—
213C7×C4⋊C4—21no—
214C7×C4⋊C4—21no—
215C7×C4⋊C4—21no—
216C7×C4⋊C4—21no—
217C7×C4⋊C4—22no—
218C7×C4⋊C4—22no—
219C7×C4⋊C4—41no—
220C7×C4⋊C4—41no—
221C7×C4⋊C4C2×D411yescharacteristic
222C7×C4⋊C4C2×D411yescharacteristic
223C7×C4⋊C4C2×D411yescharacteristic
224C7×C4⋊C4C2×D411yescharacteristic
225C7×C4⋊C4C2×D412yes—
226C7×C22⋊C4—22no—
227C7×C22⋊C4—22no—
228C7×C22⋊C4—22no—
229C7×C22⋊C4—21no—
230C7×C22⋊C4—21no—
231C7×C22⋊C4—21no—
232C42×C7—22no—
233C42×C7—22no—
234C42×C7—21no—
235C42×C7—22no—
236C42×C7—21no—
237C42×C7—21no—
238C42×C7—21no—
239C42×C7—21no—
240C42×C7—21no—
241C42×C7—21no—
242C42×C7C2×D411yescharacteristic
243C42×C7C2×D411yescharacteristic
244C42×C7C2×D411yescharacteristic
245C42×C7C2411yescharacteristic
246C22×Q16—21no—
247Q82C22×C711yescharacteristic
248#557—21no—
249#557C22×C711yescharacteristic
250#554—22no—
251#554—21no—
252#554—21no—
253#554C22×C711yescharacteristic
254D4×Q8—21no—
255D4×Q8C22×C711yescharacteristic
256#533—22no—
257#533—21no—
258#533—21no—
259#533C22×C711yescharacteristic
260C2×C4⋊Q8C22×C711yescharacteristic
261C4×#41—21no—
262C4×#41C22×C711yescharacteristic
263C2×C4×Q8—21no—
264C2×C4×Q8C22×C711yescharacteristic
265#500C22×C712yes—
266#487C22×C712yes—
267#484—22no—
268#477—21no—
269#477—22no—
270#476—21no—
271#475C22×C712yes—
272#474C22×C712yes—
273#462—21no—
274#462—21no—
275#462C22×C712yes—
276#462C22×C712yes—
277#431C22×C711yescharacteristic
278#428—21no—
279#423C22×C711yescharacteristic
280C2×C4⋊C8C22×C711yescharacteristic
281#417—21no—
282C2×#103—21no—
283#333C22×C712yes—
284#332C22×C712yes—
285Q8⋊C8C22×C712yes—
286Q8⋊C8C22×C712yes—
287Q8⋊C8C22×C712yes—
288D4⋊C8C22×C712yes—
289C23×C7—21no—
290C23×C7#12011yescharacteristic
291C7×Q8—21no—
292C7×Q8—21no—
293C7×Q8—21no—
294C7×Q8—21no—
295C7×Q8—42no—
296C7×Q8—42no—
297C7×Q8—44no—
298C7×Q8—41no—
299C7×Q8—41no—
300C7×Q8—42no—
301C7×Q8—42no—
302C7×Q8—82no—
303C7×Q8—82no—
304C7×Q8C2×Q1612yescomplemented
305C7×Q8AGL(1,ℤ/8)12yes—
306C7×D4—24no—
307C7×D4—21no—
308C7×D4C2×Q1612yescomplemented
309C2×C4×C7—21no—
310C2×C4×C7—21no—
311C2×C4×C7—21no—
312C2×C4×C7—21no—
313C2×C4×C7—24no—
314C2×C4×C7—21no—
315C2×C4×C7—21no—
316C2×C4×C7—22no—
317C2×C4×C7—22no—
318C2×C4×C7—21no—
319C2×C4×C7—21no—
320C2×C4×C7—21no—
321C2×C4×C7—41no—
322C2×C4×C7—42no—
323C2×C4×C7—44no—
324C2×C4×C7—41no—
325C2×C4×C7—41no—
326C2×C4×C7—42no—
327C2×C4×C7—81no—
328C2×C4×C7—81no—
329C2×C4×C7—82no—
330C2×C4×C7—82no—
331C2×C4×C7#12011yescharacteristic
332C2×C4×C7#12011yescharacteristic
333C2×C4×C7#12011yescharacteristic
334C2×C4×C7C2×Q1612yes—
335C2×C4×C7#13712yes—
336C2×C4×C7C22×D411yescharacteristic
337C2×C4×C7C22×D411yescharacteristic
338C2×C4×C7C22×D411yescharacteristic
339C7×C8—42no—
340C7×C8—42no—
341C7×C8—42no—
342C2×#41C7×D411yescharacteristic
343C22×Q8—21no—
344C22×Q8—21no—
345C22×Q8C7×D411yescharacteristic
346C2×Q16—22no—
347C2×Q16—22no—
348C2×Q16—44no—
349C2×M16—21no—
350C22×C8—21no—
351C4⋊Q8—22no—
352C4⋊Q8—21no—
353C4⋊Q8—22no—
354C4⋊Q8—21no—
355C4⋊Q8—21no—
356C4⋊Q8—21no—
357C4⋊Q8—21no—
358C4⋊Q8—21no—
359C4⋊Q8—21no—
360C4⋊Q8C23×C712yes—
361C4⋊Q8C23×C711yescharacteristic
362C4⋊Q8C23×C711yescharacteristic
363#125—22no—
364#125—22no—
365#125—22no—
366#125—21no—
367#125—21no—
368#125—21no—
369C22⋊Q8—22no—
370C22⋊Q8—22no—
371C22⋊Q8—22no—
372C22⋊Q8—22no—
373C22⋊Q8—22no—
374C22⋊Q8—21no—
375C22⋊Q8—22no—
376C22⋊Q8—21no—
377C22⋊Q8—21no—
378C22⋊Q8—22no—
379C22⋊Q8C7×D412yes—
380C4×Q8—22no—
381C4×Q8—22no—
382C4×Q8—22no—
383C4×Q8—21no—
384C4×Q8—21no—
385C4×Q8—21no—
386C4×Q8—21no—
387C4×Q8—21no—
388C4×Q8—21no—
389C4×Q8—21no—
390C4×Q8C7×D411yescharacteristic
391C4×Q8C7×D411yescharacteristic
392C4×Q8C7×D411yescharacteristic
393C4×Q8C7×D412yes—
394C4×Q8C23×C711yescharacteristic
395C4×Q8C23×C711yescharacteristic
396C4×Q8C23×C711yescharacteristic
397C4×D4—22no—
398C4×D4—22no—
399C4×D4—21no—
400C4×D4—21no—
401C4×D4—21no—
402C4×D4—21no—
403C4×D4C7×D411yescharacteristic
404C4×D4C23×C711yescharacteristic
405#117—22no—
406#117—21no—
407#117—21no—
408#117C7×D411yescharacteristic
409C2×C4⋊C4—22no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4C7×D411yescharacteristic
415C2×C42—22no—
416C2×C42—21no—
417C2×C42—21no—
418C2×C42C23×C711yescharacteristic
419#107—22no—
420#107—24no—
421#106—22no—
422C4⋊C8—21no—
423C4⋊C8—21no—
424C4⋊C8—21no—
425C4⋊C8C23×C712yes—
426C4⋊C8C23×C712yes—
427#103—22no—
428#103—24no—
429#103—22no—
430#103—22no—
431#102—22no—
432C22⋊C8—21no—
433C4×C8C23×C712yes—
434C22×C7—21no—
435C22×C7—21no—
436C22×C7—22no—
437C22×C7—41no—
438C22×C7—41no—
439C22×C7#45112yes—
440C22×C7C2×#12011yescharacteristic
441C4×C7—22no—
442C4×C7—21no—
443C4×C7—22no—
444C4×C7—21no—
445C4×C7—21no—
446C4×C7—41no—
447C4×C7—42no—
448C4×C7—41no—
449C4×C7—41no—
450C4×C7—42no—
451C4×C7—41no—
452C4×C7—41no—
453C4×C7—82no—
454C4×C7#45112yes—
455C4×C7C22×Q1611yescharacteristic
456C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
457C4×C7C2×#13711yescharacteristic
458C4×C7#57511yescharacteristic
459#41—22no—
460#41C7×Q1614yescomplemented
461C2×Q8—22no—
462C2×Q8—22no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—22no—
466C2×Q8—22no—
467C2×Q8—41no—
468C2×Q8—42no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×Q8—41no—
473C2×Q8C2×C7×D411yescharacteristic
474C2×Q8C2×C7×D411yescharacteristic
475C2×Q8C2×C7×D411yescharacteristic
476C2×Q8C2×C7×D412yes—
477C2×D4—21no—
478C2×D4C2×C7×D411yescharacteristic
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—41no—
487C22×C4C2×C7×D411yescharacteristic
488C22×C4C2×C7×D411yescharacteristic
489C22×C4C2×C7×D411yescharacteristic
490Q16—44no—
491Q16—44no—
492M16—42no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—42no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—41no—
520C4⋊C4—41no—
521C4⋊C4C2×C7×D411yescharacteristic
522C4⋊C4C2×C7×D411yescharacteristic
523C4⋊C4C2×C7×D411yescharacteristic
524C4⋊C4C2×C7×D411yescharacteristic
525C4⋊C4C2×C7×D412yes—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—21no—
530C22⋊C4—21no—
531C22⋊C4—21no—
532C42—22no—
533C42—22no—
534C42—21no—
535C42—22no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42C2×C7×D411yescharacteristic
543C42C2×C7×D411yescharacteristic
544C42C2×C7×D411yescharacteristic
545C42C24×C711yescharacteristic
546C2×C7—22no—
547C2×C7—42no—
548C2×C7C2×#45111yescharacteristic
549C2×C7#300611yescharacteristic
550C2×C7#302211yescharacteristic
551C23—21no—
552C23C7×#12011yescharacteristic
553Q8—21no—
554Q8—21no—
555Q8—21no—
556Q8—21no—
557Q8—42no—
558Q8—42no—
559Q8—44no—
560Q8—41no—
561Q8—41no—
562Q8—42no—
563Q8—42no—
564Q8—82no—
565Q8—82no—
566Q8C2×C7×Q1612yescomplemented
567Q8C7×AGL(1,ℤ/8)12yes—
568D4—24no—
569D4—21no—
570D4C2×C7×Q1612yescomplemented
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—24no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4C7×#12011yescharacteristic
594C2×C4C7×#12011yescharacteristic
595C2×C4C7×#12011yescharacteristic
596C2×C4C2×C7×Q1612yes—
597C2×C4C7×#13712yes—
598C2×C4C22×C7×D411yescharacteristic
599C2×C4C22×C7×D411yescharacteristic
600C2×C4C22×C7×D411yescharacteristic
601C8—42no—
602C8—42no—
603C8—42no—
604C7#1880111yescharacteristic, direct product factor, Sylow
605C22—21no—
606C22—21no—
607C22—22no—
608C22—41no—
609C22—41no—
610C22C7×#45112yes—
611C22C2×C7×#12011yescharacteristic
612C4—22no—
613C4—21no—
614C4—22no—
615C4—21no—
616C4—21no—
617C4—41no—
618C4—42no—
619C4—41no—
620C4—41no—
621C4—42no—
622C4—41no—
623C4—41no—
624C4—82no—
625C4C7×#45112yes—
626C4C22×C7×Q1611yescharacteristic
627C4C2×C7×AGL(1,ℤ/8)11yescharacteristic
628C4C2×C7×#13711yescharacteristic
629C4C7×#57511yescharacteristic
630C2—22no—
631C2—42no—
632C2C2×C7×#45111yescharacteristic
633C2C7×#300611yescharacteristic
634C2C7×#302211yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC2×C7×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#3230C2semidirect
4. C2×C7×#462C2semidirect
5. C7×#1649C2semidirect
6. C7×#1648C2semidirect
7. C7×#1645C2semidirect
8. C7×#1644C2semidirect
9. C7×#1550C2semidirect
10. C7×#1487C2semidirect
55. #18801C7direct (N × H)
149. #3230C2×C7semidirect
151. C2×#462C2×C7semidirect
152. #1649C2×C7semidirect
153. #1648C2×C7semidirect
154. #1645C2×C7semidirect
155. #1644C2×C7semidirect
156. #1550C2×C7semidirect
157. #1487C2×C7semidirect
160. C7×#41Q16semidirect
304. C7×Q8C2×Q16semidirect
308. C7×D4C2×Q16semidirect
460. #41C7×Q16semidirect
566. Q8C2×C7×Q16semidirect
570. D4C2×C7×Q16semidirect
604. C7#18801direct (N × H)

Maximal subgroups

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

subgroupindex
C7×Q8×#412
C7×#32302
C7×#30792
C2×C7×#4622
C7×#16492
C7×#16482
C7×#16452
C7×#16442
C7×#15502
C7×#14872
C7×#14772
#188017