Finite Group Databasebeta

← C7×#18957 #13196173

Subgroups — C7×#18957 #13196173

Order 1792.

874 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3547C211yesmaximal, characteristic
2C7×#3238C211yesmaximal, characteristic, complemented
3C7×#3227C211yesmaximal, characteristic, complemented
4C7×#3209C211yesmaximal, characteristic, complemented
5C7×#1637C212yesmaximal, complemented
6C7×#1635C212yesmaximal, complemented
7C7×#1634C212yesmaximal, complemented
8C7×#1632C212yesmaximal, complemented
9C7×#1554C211yesmaximal, characteristic, complemented
10C7×#1496C211yesmaximal, characteristic, complemented
11C7×#1482C211yesmaximal, characteristic, complemented
12C7×#562—22no—
13C7×#555—21no—
14C7×#555C2211yescharacteristic
15C7×#554C2211yescharacteristic
16C7×#553—21no—
17C7×#552—21no—
18C7×D4×Q8C2211yescharacteristic
19C7×#548—21no—
20C7×#547C2211yescharacteristic
21C7×#544—21no—
22C7×#542—22no—
23C7×#540—21no—
24C7×#537—21no—
25C7×#536—21no—
26C7×#533C2211yescharacteristic
27C7×#532C2211yescharacteristic
28C2×C7×C4⋊Q8C2211yescharacteristic
29C7×#529C2211yescharacteristic
30C7×#501—21no—
31C7×#498—21no—
32C7×#494C2212yes—
33C7×#489—21no—
34C7×#487C2212yes—
35C7×#486—21no—
36C7×#483—21no—
37C7×#481—21no—
38C7×#481—21no—
39C7×#480—21no—
40C7×#477C2212yes—
41C7×#476C2212yes—
42C7×#476C2212yes—
43C7×#476C2212yes—
44C7×#462—22no—
45C7×#451—22no—
46C7×#432C2211yescharacteristic
47C7×#431C2211yescharacteristic
48C7×#431C2211yescharacteristic
49C7×#333C2212yes—
50C7×#332C2212yes—
51C7×Q8⋊C8C2212yes—
52C7×D4⋊C8C2212yes—
53C7×D4⋊C8C2212yes—
54C7×D4⋊C8C2212yes—
55#18957C711yesmaximal, characteristic, direct product factor, Sylow
56C2×C7×#41—21no—
57C2×C7×#41—21no—
58C22×C7×Q8—21no—
59C22×C7×Q8—21no—
60C2×C7×Q16—24no—
61C7×C4⋊Q8—22no—
62C7×C4⋊Q8—21no—
63C7×C4⋊Q8—21no—
64C7×C4⋊Q8—21no—
65C7×C4⋊Q8—21no—
66C7×C4⋊Q8D411yescharacteristic
67C7×C4⋊Q8C2312yes—
68C7×C4⋊Q8C2311yescharacteristic
69C7×C4⋊Q8C2311yescharacteristic
70C7×#127D411yescharacteristic
71C7×#126—22no—
72C7×#126—21no—
73C7×#125—22no—
74C7×#125—21no—
75C7×#125—21no—
76C7×#125D411yescharacteristic
77C7×#124—22no—
78C7×#124—22no—
79C7×#124—22no—
80C7×#124—21no—
81C7×#124D411yescharacteristic
82C7×#123—22no—
83C7×#123—22no—
84C7×#123—22no—
85C7×#123—21no—
86C7×#123—22no—
87C7×C22⋊Q8—22no—
88C7×C22⋊Q8—22no—
89C7×C22⋊Q8—21no—
90C7×C22⋊Q8—22no—
91C7×C22⋊Q8—21no—
92C7×C22⋊Q8—21no—
93C7×C22⋊Q8—21no—
94C7×C22⋊Q8—22no—
95C7×C22⋊Q8D411yescharacteristic
96C7×C22⋊Q8D412yes—
97C7×#121—22no—
98C7×#121—22no—
99C7×#121—22no—
100C7×#121—22no—
101C7×#121—21no—
102C7×#121—21no—
103C7×#121—21no—
104C7×#121—21no—
105C7×#121D411yescharacteristic
106C7×#121D411yescharacteristic
107C7×#121D411yescharacteristic
108C4×C7×Q8—21no—
109C4×C7×Q8D412yes—
110C4×C7×Q8C2311yescharacteristic
111C4×C7×D4—22no—
112C4×C7×D4—21no—
113C4×C7×D4—21no—
114C4×C7×D4—21no—
115C4×C7×D4—21no—
116C4×C7×D4C2311yescharacteristic
117C4×C7×D4C2311yescharacteristic
118C4×C7×D4C2311yescharacteristic
119C7×#117—22no—
120C7×#117—21no—
121C7×#117—21no—
122C7×#117—21no—
123C7×#117—21no—
124C2×C7×C4⋊C4—21no—
125C2×C7×C4⋊C4—21no—
126C2×C7×C4⋊C4—21no—
127C2×C7×C4⋊C4—21no—
128C2×C42×C7C2311yescharacteristic
129C7×#107—22no—
130C7×#106—22no—
131C7×#106—22no—
132C7×#106—22no—
133C7×C4⋊C8—21no—
134C7×C4⋊C8C2312yes—
135C7×C4⋊C8C2312yes—
136C7×#103—22no—
137C7×#103—24no—
138C7×#102—22no—
139C7×#102—22no—
140C7×#102—22no—
141C7×C22⋊C8—21no—
142C7×C22⋊C8—21no—
143C7×C22⋊C8—21no—
144C7×#97—21no—
145C4×C7×C8—21no—
146C4×C7×C8C2312yes—
147#3547C2×C711yescharacteristic
148#3238C2×C711yescharacteristic, complemented
149#3227C2×C711yescharacteristic, complemented
150#3209C2×C711yescharacteristic, complemented
151#1637C2×C712yescomplemented
152#1635C2×C712yescomplemented
153#1634C2×C712yescomplemented
154#1632C2×C712yescomplemented
155#1554C2×C711yescharacteristic, complemented
156#1496C2×C711yescharacteristic, complemented
157#1482C2×C711yescharacteristic, complemented
158C7×#41—82no—
159C7×#41—82no—
160C2×C7×Q8—22no—
161C2×C7×Q8—21no—
162C2×C7×Q8—21no—
163C2×C7×Q8—22no—
164C2×C7×Q8—41no—
165C2×C7×Q8—82no—
166C2×C7×Q8—82no—
167C2×C7×Q8C2×D411yescharacteristic
168C2×C7×Q8C2×D412yes—
169C2×C7×D4—22no—
170C2×C7×D4—22no—
171C2×C7×D4—21no—
172C2×C7×D4—21no—
173C2×C7×D4—21no—
174C2×C7×D4—21no—
175C2×C7×D4—21no—
176C2×C7×D4C2×D411yescharacteristic
177C2×C7×D4C2×D411yescharacteristic
178C2×C7×D4C2×D411yescharacteristic
179C22×C4×C7—22no—
180C22×C4×C7—22no—
181C22×C4×C7—22no—
182C22×C4×C7—21no—
183C22×C4×C7—21no—
184C22×C4×C7—21no—
185C22×C4×C7—41no—
186C22×C4×C7C2×D411yescharacteristic
187C22×C4×C7C2×D411yescharacteristic
188C22×C4×C7C2×D411yescharacteristic
189C7×Q16—84no—
190C2×C7×C8—22no—
191C2×C7×C8—22no—
192C2×C7×C8—22no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—21no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—22no—
202C7×C4⋊C4—22no—
203C7×C4⋊C4—21no—
204C7×C4⋊C4—21no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—21no—
207C7×C4⋊C4—22no—
208C7×C4⋊C4—22no—
209C7×C4⋊C4—41no—
210C7×C4⋊C4—41no—
211C7×C4⋊C4C2×D412yes—
212C7×C4⋊C4C2×D411yescharacteristic
213C7×C4⋊C4C2×D411yescharacteristic
214C7×C4⋊C4C2×D411yescharacteristic
215C7×C4⋊C4C2×D411yescharacteristic
216C7×C22⋊C4—22no—
217C7×C22⋊C4—22no—
218C7×C22⋊C4—22no—
219C7×C22⋊C4—22no—
220C7×C22⋊C4—22no—
221C7×C22⋊C4—22no—
222C7×C22⋊C4—21no—
223C7×C22⋊C4—21no—
224C7×C22⋊C4—21no—
225C7×C22⋊C4—21no—
226C7×C22⋊C4—21no—
227C7×C22⋊C4—21no—
228C7×C22⋊C4—21no—
229C7×C22⋊C4—21no—
230C7×C22⋊C4—21no—
231C42×C7—22no—
232C42×C7—22no—
233C42×C7—21no—
234C42×C7C2×D411yescharacteristic
235C42×C7C2×D411yescharacteristic
236C42×C7C2×D411yescharacteristic
237C42×C7C2411yescharacteristic
238#562—22no—
239#555—21no—
240#555C22×C711yescharacteristic
241#554C22×C711yescharacteristic
242#553—21no—
243#552—21no—
244D4×Q8C22×C711yescharacteristic
245#548—21no—
246#547C22×C711yescharacteristic
247#544—21no—
248#542—22no—
249#540—21no—
250#537—21no—
251#536—21no—
252#533C22×C711yescharacteristic
253#532C22×C711yescharacteristic
254C2×C4⋊Q8C22×C711yescharacteristic
255#529C22×C711yescharacteristic
256#501—21no—
257#498—21no—
258#494C22×C712yes—
259#489—21no—
260#487C22×C712yes—
261#486—21no—
262#483—21no—
263#481—21no—
264#481—21no—
265#480—21no—
266#477C22×C712yes—
267#476C22×C712yes—
268#476C22×C712yes—
269#476C22×C712yes—
270#462—22no—
271#451—22no—
272#432C22×C711yescharacteristic
273#431C22×C711yescharacteristic
274#431C22×C711yescharacteristic
275#333C22×C712yes—
276#332C22×C712yes—
277Q8⋊C8C22×C712yes—
278D4⋊C8C22×C712yes—
279D4⋊C8C22×C712yes—
280D4⋊C8C22×C712yes—
281C23×C7—21no—
282C23×C7—21no—
283C23×C7—21no—
284C23×C7#12011yescharacteristic
285C7×Q8—22no—
286C7×Q8—42no—
287C7×Q8—44no—
288C7×Q8—42no—
289C7×Q8—81no—
290C7×Q8—82no—
291C7×Q8—82no—
292C7×D4—22no—
293C7×D4—22no—
294C7×D4—22no—
295C7×D4—81no—
296C7×D4—81no—
297C7×D4—81no—
298C7×D4—81no—
299C7×D4—81no—
300C7×D4—82no—
301C7×D4—82no—
302C2×C4×C7—22no—
303C2×C4×C7—22no—
304C2×C4×C7—21no—
305C2×C4×C7—22no—
306C2×C4×C7—21no—
307C2×C4×C7—21no—
308C2×C4×C7—21no—
309C2×C4×C7—22no—
310C2×C4×C7—21no—
311C2×C4×C7—41no—
312C2×C4×C7—42no—
313C2×C4×C7—42no—
314C2×C4×C7—42no—
315C2×C4×C7—42no—
316C2×C4×C7—82no—
317C2×C4×C7—82no—
318C2×C4×C7—82no—
319C2×C4×C7—82no—
320C2×C4×C7#12011yescharacteristic
321C2×C4×C7#12011yescharacteristic
322C2×C4×C7#12011yescharacteristic
323C2×C4×C7C22×D411yescharacteristic
324C2×C4×C7C22×D411yescharacteristic
325C2×C4×C7C22×D411yescharacteristic
326C7×C8—42no—
327C7×C8—42no—
328C7×C8—42no—
329C2×#41—21no—
330C2×#41—21no—
331C22×Q8—21no—
332C22×Q8—21no—
333C2×Q16—24no—
334C4⋊Q8—22no—
335C4⋊Q8—21no—
336C4⋊Q8—21no—
337C4⋊Q8—21no—
338C4⋊Q8—21no—
339C4⋊Q8C7×D411yescharacteristic
340C4⋊Q8C23×C712yes—
341C4⋊Q8C23×C711yescharacteristic
342C4⋊Q8C23×C711yescharacteristic
343#127C7×D411yescharacteristic
344#126—22no—
345#126—21no—
346#125—22no—
347#125—21no—
348#125—21no—
349#125C7×D411yescharacteristic
350#124—22no—
351#124—22no—
352#124—22no—
353#124—21no—
354#124C7×D411yescharacteristic
355#123—22no—
356#123—22no—
357#123—22no—
358#123—21no—
359#123—22no—
360C22⋊Q8—22no—
361C22⋊Q8—22no—
362C22⋊Q8—21no—
363C22⋊Q8—22no—
364C22⋊Q8—21no—
365C22⋊Q8—21no—
366C22⋊Q8—21no—
367C22⋊Q8—22no—
368C22⋊Q8C7×D411yescharacteristic
369C22⋊Q8C7×D412yes—
370#121—22no—
371#121—22no—
372#121—22no—
373#121—22no—
374#121—21no—
375#121—21no—
376#121—21no—
377#121—21no—
378#121C7×D411yescharacteristic
379#121C7×D411yescharacteristic
380#121C7×D411yescharacteristic
381C4×Q8—21no—
382C4×Q8C7×D412yes—
383C4×Q8C23×C711yescharacteristic
384C4×D4—22no—
385C4×D4—21no—
386C4×D4—21no—
387C4×D4—21no—
388C4×D4—21no—
389C4×D4C23×C711yescharacteristic
390C4×D4C23×C711yescharacteristic
391C4×D4C23×C711yescharacteristic
392#117—22no—
393#117—21no—
394#117—21no—
395#117—21no—
396#117—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C42C23×C711yescharacteristic
402#107—22no—
403#106—22no—
404#106—22no—
405#106—22no—
406C4⋊C8—21no—
407C4⋊C8C23×C712yes—
408C4⋊C8C23×C712yes—
409#103—22no—
410#103—24no—
411#102—22no—
412#102—22no—
413#102—22no—
414C22⋊C8—21no—
415C22⋊C8—21no—
416C22⋊C8—21no—
417#97—21no—
418C4×C8—21no—
419C4×C8C23×C712yes—
420C22×C7—21no—
421C22×C7—21no—
422C22×C7—21no—
423C22×C7—41no—
424C22×C7—41no—
425C22×C7—41no—
426C22×C7—41no—
427C22×C7—41no—
428C22×C7—41no—
429C22×C7—41no—
430C22×C7—41no—
431C22×C7—41no—
432C22×C7C2×#12011yescharacteristic
433C4×C7—21no—
434C4×C7—21no—
435C4×C7—21no—
436C4×C7—21no—
437C4×C7—41no—
438C4×C7—41no—
439C4×C7—42no—
440C4×C7—41no—
441C4×C7—41no—
442C4×C7—41no—
443C4×C7—42no—
444C4×C7—82no—
445C4×C7#57712yes—
446C4×C7#57812yes—
447#41—82no—
448#41—82no—
449C2×Q8—22no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—22no—
453C2×Q8—41no—
454C2×Q8—82no—
455C2×Q8—82no—
456C2×Q8C2×C7×D411yescharacteristic
457C2×Q8C2×C7×D412yes—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4C2×C7×D411yescharacteristic
466C2×D4C2×C7×D411yescharacteristic
467C2×D4C2×C7×D411yescharacteristic
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—41no—
475C22×C4C2×C7×D411yescharacteristic
476C22×C4C2×C7×D411yescharacteristic
477C22×C4C2×C7×D411yescharacteristic
478Q16—84no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4C2×C7×D412yes—
501C4⋊C4C2×C7×D411yescharacteristic
502C4⋊C4C2×C7×D411yescharacteristic
503C4⋊C4C2×C7×D411yescharacteristic
504C4⋊C4C2×C7×D411yescharacteristic
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—21no—
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C22⋊C4—21no—
520C42—22no—
521C42—22no—
522C42—21no—
523C42C2×C7×D411yescharacteristic
524C42C2×C7×D411yescharacteristic
525C42C2×C7×D411yescharacteristic
526C42C24×C711yescharacteristic
527C2×C7—41no—
528C2×C7—81no—
529C2×C7—81no—
530C2×C7—81no—
531C2×C7#300911yescharacteristic
532C2×C7#301111yescharacteristic
533C2×C7#302111yescharacteristic
534C23—21no—
535C23—21no—
536C23—21no—
537C23C7×#12011yescharacteristic
538Q8—22no—
539Q8—42no—
540Q8—44no—
541Q8—42no—
542Q8—81no—
543Q8—82no—
544Q8—82no—
545D4—22no—
546D4—22no—
547D4—22no—
548D4—81no—
549D4—81no—
550D4—81no—
551D4—81no—
552D4—81no—
553D4—82no—
554D4—82no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4C7×#12011yescharacteristic
574C2×C4C7×#12011yescharacteristic
575C2×C4C7×#12011yescharacteristic
576C2×C4C22×C7×D411yescharacteristic
577C2×C4C22×C7×D411yescharacteristic
578C2×C4C22×C7×D411yescharacteristic
579C8—42no—
580C8—42no—
581C8—42no—
582C7#1895711yescharacteristic, direct product factor, Sylow
583C22—21no—
584C22—21no—
585C22—21no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C2×C7×#12011yescharacteristic
596C4—21no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—41no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—42no—
607C4—82no—
608C4C7×#57712yes—
609C4C7×#57812yes—
610C2—41no—
611C2—81no—
612C2—81no—
613C2—81no—
614C2C7×#300911yescharacteristic
615C2C7×#301111yescharacteristic
616C2C7×#302111yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3547C2
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×#3238C2semidirect
3. C7×#3227C2semidirect
4. C7×#3209C2semidirect
5. C7×#1637C2semidirect
6. C7×#1635C2semidirect
7. C7×#1634C2semidirect
8. C7×#1632C2semidirect
9. C7×#1554C2semidirect
10. C7×#1496C2semidirect
11. C7×#1482C2semidirect
55. #18957C7direct (N × H)
148. #3238C2×C7semidirect
149. #3227C2×C7semidirect
150. #3209C2×C7semidirect
151. #1637C2×C7semidirect
152. #1635C2×C7semidirect
153. #1634C2×C7semidirect
154. #1632C2×C7semidirect
155. #1554C2×C7semidirect
156. #1496C2×C7semidirect
157. #1482C2×C7semidirect
582. C7#18957direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35472
C7×#32382
C7×#32272
C7×#32092
C7×#16372
C7×#16352
C7×#16342
C7×#16322
C7×#15542
C7×#14962
C7×#14822
#189577