Finite Group Databasebeta

← C7×#18959 #13196175

Subgroups — C7×#18959 #13196175

Order 1792.

874 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3554C211yesmaximal, characteristic
2C7×#3238C211yesmaximal, characteristic, complemented
3C7×#3230C211yesmaximal, characteristic, complemented
4C7×#3206C211yesmaximal, characteristic, complemented
5C7×#1637C212yesmaximal, complemented
6C7×#1636C212yesmaximal, complemented
7C7×#1631C212yesmaximal, complemented
8C7×#1630C212yesmaximal, complemented
9C7×#1553C211yesmaximal, characteristic, complemented
10C7×#1497C211yesmaximal, characteristic, complemented
11C7×#1482C211yesmaximal, characteristic, complemented
12C7×#562—22no—
13C7×#559—21no—
14C7×#559C2211yescharacteristic
15C7×#557C2211yescharacteristic
16C7×#554—21no—
17C7×#550C2211yescharacteristic
18C7×D4×Q8C2211yescharacteristic
19C7×#548—21no—
20C7×#547—21no—
21C7×#542—21no—
22C7×#542—21no—
23C7×#540—21no—
24C7×#539—22no—
25C7×#536—21no—
26C7×#533C2211yescharacteristic
27C7×#532C2211yescharacteristic
28C2×C7×C4⋊Q8C2211yescharacteristic
29C7×#529C2211yescharacteristic
30C7×#501—21no—
31C7×#500—21no—
32C7×#492C2212yes—
33C7×#489—21no—
34C7×#487—21no—
35C7×#487C2212yes—
36C7×#484—21no—
37C7×#484—21no—
38C7×#483—21no—
39C7×#480—21no—
40C7×#477C2212yes—
41C7×#477C2212yes—
42C7×#477C2212yes—
43C7×#476C2212yes—
44C7×#461—22no—
45C7×#448—22no—
46C7×#432C2211yescharacteristic
47C7×#431C2211yescharacteristic
48C7×#431C2211yescharacteristic
49C7×#332C2212yes—
50C7×#331C2212yes—
51C7×Q8⋊C8C2212yes—
52C7×Q8⋊C8C2212yes—
53C7×Q8⋊C8C2212yes—
54C7×D4⋊C8C2212yes—
55#18959C711yesmaximal, characteristic, direct product factor, Sylow
56C2×C7×#41—21no—
57C2×C7×#41—21no—
58C22×C7×Q8—21no—
59C22×C7×Q8—21no—
60C2×C7×SD16—24no—
61C7×C4⋊Q8—21no—
62C7×C4⋊Q8—21no—
63C7×C4⋊Q8—21no—
64C7×C4⋊Q8—21no—
65C7×C4⋊Q8—21no—
66C7×C4⋊Q8D411yescharacteristic
67C7×C4⋊Q8D411yescharacteristic
68C7×C4⋊Q8C2312yes—
69C7×C4⋊Q8C2311yescharacteristic
70C7×#127—21no—
71C7×#127C2311yescharacteristic
72C7×#126—22no—
73C7×#126—21no—
74C7×#125—22no—
75C7×#125—21no—
76C7×#125—21no—
77C7×#125D411yescharacteristic
78C7×#124—22no—
79C7×#124—22no—
80C7×#124—21no—
81C7×#124—21no—
82C7×#124—21no—
83C7×#124D411yescharacteristic
84C7×#123—22no—
85C7×#123—22no—
86C7×#123—22no—
87C7×#123—22no—
88C7×#123—22no—
89C7×#123—22no—
90C7×#123—21no—
91C7×C22⋊Q8—22no—
92C7×C22⋊Q8—22no—
93C7×C22⋊Q8—22no—
94C7×C22⋊Q8—22no—
95C7×C22⋊Q8—21no—
96C7×C22⋊Q8—22no—
97C7×C22⋊Q8—21no—
98C7×C22⋊Q8D411yescharacteristic
99C7×C22⋊Q8D411yescharacteristic
100C7×C22⋊Q8D411yescharacteristic
101C7×#121—22no—
102C7×#121—22no—
103C7×#121—22no—
104C7×#121—21no—
105C7×#121—21no—
106C7×#121D411yescharacteristic
107C7×#121D412yes—
108C4×C7×Q8—21no—
109C4×C7×Q8—21no—
110C4×C7×Q8—21no—
111C4×C7×Q8C2311yescharacteristic
112C4×C7×Q8C2311yescharacteristic
113C4×C7×Q8C2311yescharacteristic
114C4×C7×D4—22no—
115C4×C7×D4—21no—
116C4×C7×D4—21no—
117C4×C7×D4—21no—
118C4×C7×D4—21no—
119C4×C7×D4—21no—
120C4×C7×D4—21no—
121C4×C7×D4D412yes—
122C4×C7×D4C2311yescharacteristic
123C7×#117—21no—
124C7×#117—21no—
125C2×C7×C4⋊C4—22no—
126C2×C7×C4⋊C4—21no—
127C2×C7×C4⋊C4—21no—
128C2×C42×C7C2311yescharacteristic
129C7×#107—22no—
130C7×#107—22no—
131C7×#107—22no—
132C7×#106—22no—
133C7×C4⋊C8—21no—
134C7×C4⋊C8C2312yes—
135C7×C4⋊C8C2312yes—
136C7×#103—22no—
137C7×#103—22no—
138C7×#103—22no—
139C7×#103—24no—
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#3554C2×C711yescharacteristic
148#3238C2×C711yescharacteristic, complemented
149#3230C2×C711yescharacteristic, complemented
150#3206C2×C711yescharacteristic, complemented
151#1637C2×C712yescomplemented
152#1636C2×C712yescomplemented
153#1631C2×C712yescomplemented
154#1630C2×C712yescomplemented
155#1553C2×C711yescharacteristic, complemented
156#1497C2×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×D411yescharacteristic
169C2×C7×Q8C2×D411yescharacteristic
170C2×C7×D4—22no—
171C2×C7×D4—22no—
172C2×C7×D4—21no—
173C2×C7×D4—22no—
174C2×C7×D4—21no—
175C2×C7×D4—21no—
176C2×C7×D4C2×D411yescharacteristic
177C2×C7×D4C2×D412yes—
178C22×C4×C7—22no—
179C22×C4×C7—22no—
180C22×C4×C7—22no—
181C22×C4×C7—21no—
182C22×C4×C7—22no—
183C22×C4×C7—41no—
184C22×C4×C7C2×D411yescharacteristic
185C22×C4×C7C2×D411yescharacteristic
186C22×C4×C7C2×D411yescharacteristic
187C7×SD16—84no—
188C2×C7×C8—22no—
189C2×C7×C8—22no—
190C2×C7×C8—22no—
191C7×C4⋊C4—22no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—21no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—21no—
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—21no—
222C7×C22⋊C4—21no—
223C7×C22⋊C4—21no—
224C7×C22⋊C4—21no—
225C7×C22⋊C4—21no—
226C7×C22⋊C4—21no—
227C7×C22⋊C4—22no—
228C7×C22⋊C4—22no—
229C7×C22⋊C4—21no—
230C42×C7—22no—
231C42×C7—21no—
232C42×C7—21no—
233C42×C7—21no—
234C42×C7C2×D411yescharacteristic
235C42×C7C2×D411yescharacteristic
236C42×C7C2×D411yescharacteristic
237C42×C7C2411yescharacteristic
238#562—22no—
239#559—21no—
240#559C22×C711yescharacteristic
241#557C22×C711yescharacteristic
242#554—21no—
243#550C22×C711yescharacteristic
244D4×Q8C22×C711yescharacteristic
245#548—21no—
246#547—21no—
247#542—21no—
248#542—21no—
249#540—21no—
250#539—22no—
251#536—21no—
252#533C22×C711yescharacteristic
253#532C22×C711yescharacteristic
254C2×C4⋊Q8C22×C711yescharacteristic
255#529C22×C711yescharacteristic
256#501—21no—
257#500—21no—
258#492C22×C712yes—
259#489—21no—
260#487—21no—
261#487C22×C712yes—
262#484—21no—
263#484—21no—
264#483—21no—
265#480—21no—
266#477C22×C712yes—
267#477C22×C712yes—
268#477C22×C712yes—
269#476C22×C712yes—
270#461—22no—
271#448—22no—
272#432C22×C711yescharacteristic
273#431C22×C711yescharacteristic
274#431C22×C711yescharacteristic
275#332C22×C712yes—
276#331C22×C712yes—
277Q8⋊C8C22×C712yes—
278Q8⋊C8C22×C712yes—
279Q8⋊C8C22×C712yes—
280D4⋊C8C22×C712yes—
281C23×C7—21no—
282C23×C7—22no—
283C23×C7#12011yescharacteristic
284C7×Q8—22no—
285C7×Q8—22no—
286C7×Q8—22no—
287C7×Q8—44no—
288C7×Q8—42no—
289C7×Q8—81no—
290C7×Q8—82no—
291C7×Q8—82no—
292C7×D4—22no—
293C7×D4—42no—
294C7×D4—42no—
295C7×D4—42no—
296C7×D4—81no—
297C7×D4—82no—
298C7×D4—82no—
299C7×D4—82no—
300C2×C4×C7—22no—
301C2×C4×C7—22no—
302C2×C4×C7—21no—
303C2×C4×C7—21no—
304C2×C4×C7—21no—
305C2×C4×C7—21no—
306C2×C4×C7—21no—
307C2×C4×C7—22no—
308C2×C4×C7—21no—
309C2×C4×C7—21no—
310C2×C4×C7—41no—
311C2×C4×C7—42no—
312C2×C4×C7—42no—
313C2×C4×C7—82no—
314C2×C4×C7—82no—
315C2×C4×C7—82no—
316C2×C4×C7—82no—
317C2×C4×C7—82no—
318C2×C4×C7#12011yescharacteristic
319C2×C4×C7#12011yescharacteristic
320C2×C4×C7#12011yescharacteristic
321C2×C4×C7C22×D411yescharacteristic
322C2×C4×C7C22×D411yescharacteristic
323C2×C4×C7C22×D411yescharacteristic
324C7×C8—42no—
325C7×C8—42no—
326C7×C8—42no—
327C2×#41—21no—
328C2×#41—21no—
329C22×Q8—21no—
330C22×Q8—21no—
331C2×SD16—24no—
332C4⋊Q8—21no—
333C4⋊Q8—21no—
334C4⋊Q8—21no—
335C4⋊Q8—21no—
336C4⋊Q8—21no—
337C4⋊Q8C7×D411yescharacteristic
338C4⋊Q8C7×D411yescharacteristic
339C4⋊Q8C23×C712yes—
340C4⋊Q8C23×C711yescharacteristic
341#127—21no—
342#127C23×C711yescharacteristic
343#126—22no—
344#126—21no—
345#125—22no—
346#125—21no—
347#125—21no—
348#125C7×D411yescharacteristic
349#124—22no—
350#124—22no—
351#124—21no—
352#124—21no—
353#124—21no—
354#124C7×D411yescharacteristic
355#123—22no—
356#123—22no—
357#123—22no—
358#123—22no—
359#123—22no—
360#123—22no—
361#123—21no—
362C22⋊Q8—22no—
363C22⋊Q8—22no—
364C22⋊Q8—22no—
365C22⋊Q8—22no—
366C22⋊Q8—21no—
367C22⋊Q8—22no—
368C22⋊Q8—21no—
369C22⋊Q8C7×D411yescharacteristic
370C22⋊Q8C7×D411yescharacteristic
371C22⋊Q8C7×D411yescharacteristic
372#121—22no—
373#121—22no—
374#121—22no—
375#121—21no—
376#121—21no—
377#121C7×D411yescharacteristic
378#121C7×D412yes—
379C4×Q8—21no—
380C4×Q8—21no—
381C4×Q8—21no—
382C4×Q8C23×C711yescharacteristic
383C4×Q8C23×C711yescharacteristic
384C4×Q8C23×C711yescharacteristic
385C4×D4—22no—
386C4×D4—21no—
387C4×D4—21no—
388C4×D4—21no—
389C4×D4—21no—
390C4×D4—21no—
391C4×D4—21no—
392C4×D4C7×D412yes—
393C4×D4C23×C711yescharacteristic
394#117—21no—
395#117—21no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C42C23×C711yescharacteristic
400#107—22no—
401#107—22no—
402#107—22no—
403#106—22no—
404C4⋊C8—21no—
405C4⋊C8C23×C712yes—
406C4⋊C8C23×C712yes—
407#103—22no—
408#103—22no—
409#103—22no—
410#103—24no—
411#102—22no—
412C22⋊C8—21no—
413C22⋊C8—21no—
414C22⋊C8—21no—
415#97—21no—
416C4×C8—21no—
417C4×C8C23×C712yes—
418C22×C7—21no—
419C22×C7—21no—
420C22×C7—21no—
421C22×C7—41no—
422C22×C7—42no—
423C22×C7—41no—
424C22×C7—42no—
425C22×C7—42no—
426C22×C7—41no—
427C22×C7C2×#12011yescharacteristic
428C4×C7—21no—
429C4×C7—21no—
430C4×C7—21no—
431C4×C7—21no—
432C4×C7—41no—
433C4×C7—41no—
434C4×C7—42no—
435C4×C7—41no—
436C4×C7—41no—
437C4×C7—41no—
438C4×C7—41no—
439C4×C7—41no—
440C4×C7—82no—
441C4×C7#57712yes—
442C4×C7#57812yes—
443#41—82no—
444#41—82no—
445C2×Q8—22no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—22no—
449C2×Q8—41no—
450C2×Q8—82no—
451C2×Q8—82no—
452C2×Q8C2×C7×D411yescharacteristic
453C2×Q8C2×C7×D411yescharacteristic
454C2×Q8C2×C7×D411yescharacteristic
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—21no—
458C2×D4—22no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4C2×C7×D411yescharacteristic
462C2×D4C2×C7×D412yes—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—41no—
469C22×C4C2×C7×D411yescharacteristic
470C22×C4C2×C7×D411yescharacteristic
471C22×C4C2×C7×D411yescharacteristic
472SD16—84no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4C2×C7×D412yes—
497C4⋊C4C2×C7×D411yescharacteristic
498C4⋊C4C2×C7×D411yescharacteristic
499C4⋊C4C2×C7×D411yescharacteristic
500C4⋊C4C2×C7×D411yescharacteristic
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—21no—
515C42—22no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42C2×C7×D411yescharacteristic
520C42C2×C7×D411yescharacteristic
521C42C2×C7×D411yescharacteristic
522C42C24×C711yescharacteristic
523C2×C7—41no—
524C2×C7—81no—
525C2×C7—82no—
526C2×C7#300911yescharacteristic
527C2×C7#301111yescharacteristic
528C2×C7#302111yescharacteristic
529C23—21no—
530C23—22no—
531C23C7×#12011yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—22no—
535Q8—44no—
536Q8—42no—
537Q8—81no—
538Q8—82no—
539Q8—82no—
540D4—22no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—81no—
545D4—82no—
546D4—82no—
547D4—82no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4C7×#12011yescharacteristic
567C2×C4C7×#12011yescharacteristic
568C2×C4C7×#12011yescharacteristic
569C2×C4C22×C7×D411yescharacteristic
570C2×C4C22×C7×D411yescharacteristic
571C2×C4C22×C7×D411yescharacteristic
572C8—42no—
573C8—42no—
574C8—42no—
575C7#1895911yescharacteristic, direct product factor, Sylow
576C22—21no—
577C22—21no—
578C22—21no—
579C22—41no—
580C22—42no—
581C22—41no—
582C22—42no—
583C22—42no—
584C22—41no—
585C22C2×C7×#12011yescharacteristic
586C4—21no—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—41no—
591C4—41no—
592C4—42no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—82no—
599C4C7×#57712yes—
600C4C7×#57812yes—
601C2—41no—
602C2—81no—
603C2—82no—
604C2C7×#300911yescharacteristic
605C2C7×#301111yescharacteristic
606C2C7×#302111yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3554C2
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×#3230C2semidirect
4. C7×#3206C2semidirect
5. C7×#1637C2semidirect
6. C7×#1636C2semidirect
7. C7×#1631C2semidirect
8. C7×#1630C2semidirect
9. C7×#1553C2semidirect
10. C7×#1497C2semidirect
11. C7×#1482C2semidirect
55. #18959C7direct (N × H)
148. #3238C2×C7semidirect
149. #3230C2×C7semidirect
150. #3206C2×C7semidirect
151. #1637C2×C7semidirect
152. #1636C2×C7semidirect
153. #1631C2×C7semidirect
154. #1630C2×C7semidirect
155. #1553C2×C7semidirect
156. #1497C2×C7semidirect
157. #1482C2×C7semidirect
575. C7#18959direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35542
C7×#32382
C7×#32302
C7×#32062
C7×#16372
C7×#16362
C7×#16312
C7×#16302
C7×#15532
C7×#14972
C7×#14822
#189597