Finite Group Databasebeta

← C7×#18958 #13196174

Subgroups — C7×#18958 #13196174

Order 1792.

950 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3554C211yesmaximal, characteristic, complemented
2C7×#3217C211yesmaximal, characteristic, complemented
3C7×#3208C211yesmaximal, characteristic, complemented
4C7×#3205C211yesmaximal, characteristic, complemented
5C7×#1627C212yesmaximal, complemented
6C7×#1626C212yesmaximal, complemented
7C7×#1621C212yesmaximal, complemented
8C7×#1620C212yesmaximal, complemented
9C7×#1553C211yesmaximal, characteristic, complemented
10C7×#1497C211yesmaximal, characteristic, complemented
11C7×#1482C211yesmaximal, characteristic, complemented
12C7×#562—22no—
13C7×#559—21no—
14C7×#559C2211yescharacteristic, complemented
15C7×#557C2211yescharacteristic, complemented
16C7×#554—21no—
17C7×#550C2211yescharacteristic, complemented
18C7×D4×Q8C2211yescharacteristic, complemented
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×#127C2211yescharacteristic
29C7×#529C2211yescharacteristic
30C7×#496—21no—
31C7×#493C2212yes—
32C7×#492—21no—
33C7×#490—21no—
34C7×#486—21no—
35C7×#486C2212yes—
36C7×#460C2212yes—
37C7×#460C2212yes—
38C7×#460C2212yes—
39C7×#459—22no—
40C7×#459C2212yes—
41C7×#450—21no—
42C7×#450—21no—
43C7×#450—21no—
44C7×#447—22no—
45C7×#447—21no—
46C7×#432C2211yescharacteristic
47C7×#431C2211yescharacteristic
48C7×#431C2211yescharacteristic
49C7×#332C2212yescomplemented
50C7×#331C2212yescomplemented
51C7×Q8⋊C8C2212yescomplemented
52C7×Q8⋊C8C2212yescomplemented
53C7×Q8⋊C8C2212yescomplemented
54C7×D4⋊C8C2212yescomplemented
55#18958C711yesmaximal, characteristic, direct product factor, Sylow
56C2×C7×#41—21no—
57C2×C7×#41—21no—
58C22×C7×Q8—21no—
59C22×C7×D4—21no—
60C22×C7×D4—21no—
61C22×C7×D4—21no—
62C2×C7×SD16—22no—
63C2×C7×SD16—22no—
64C2×C7×SD16—22no—
65C2×C7×D8—24no—
66C2×C7×D8—22no—
67C7×C4⋊Q8—21no—
68C7×C4⋊Q8—21no—
69C7×C4⋊Q8D411yescharacteristic
70C7×C4⋊Q8D411yescharacteristic
71C7×C4⋊Q8C2311yescharacteristic
72C7×#127—21no—
73C7×#127—21no—
74C7×#127—21no—
75C7×#127—21no—
76C7×#127C2312yes—
77C7×#127C2311yescharacteristic
78C7×#126—22no—
79C7×#126—21no—
80C7×#125—22no—
81C7×#125—21no—
82C7×#125—21no—
83C7×#125D411yescharacteristic
84C7×#124—22no—
85C7×#124—22no—
86C7×#124—21no—
87C7×#124—21no—
88C7×#124—21no—
89C7×#124D411yescharacteristic
90C7×#123—22no—
91C7×#123—22no—
92C7×#123—22no—
93C7×#123—22no—
94C7×#123—22no—
95C7×#123—22no—
96C7×#123—21no—
97C7×C22⋊Q8—22no—
98C7×C22⋊Q8—22no—
99C7×C22⋊Q8—22no—
100C7×C22⋊Q8—22no—
101C7×C22⋊Q8—21no—
102C7×C22⋊Q8—22no—
103C7×C22⋊Q8—21no—
104C7×C22⋊Q8D411yescharacteristic
105C7×C22⋊Q8D411yescharacteristic
106C7×C22⋊Q8D411yescharacteristic
107C7×#121—22no—
108C7×#121—22no—
109C7×#121—22no—
110C7×#121—21no—
111C7×#121—21no—
112C7×#121D411yescharacteristic
113C7×#121D412yes—
114C4×C7×Q8—21no—
115C4×C7×Q8—21no—
116C4×C7×Q8—21no—
117C4×C7×Q8C2311yescharacteristic
118C4×C7×Q8C2311yescharacteristic
119C4×C7×Q8C2311yescharacteristic
120C4×C7×D4—22no—
121C4×C7×D4—21no—
122C4×C7×D4—21no—
123C4×C7×D4—21no—
124C4×C7×D4—21no—
125C4×C7×D4—21no—
126C4×C7×D4—21no—
127C4×C7×D4D412yes—
128C4×C7×D4C2311yescharacteristic
129C7×#117—21no—
130C7×#117—21no—
131C2×C7×C4⋊C4—22no—
132C2×C42×C7C2311yescharacteristic
133C7×C4⋊C8—21no—
134C7×C4⋊C8C2312yes—
135C7×C4⋊C8C2312yes—
136C7×#102—22no—
137C7×#102—22no—
138C7×#102—24no—
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#3554C2×C711yescharacteristic, complemented
148#3217C2×C711yescharacteristic, complemented
149#3208C2×C711yescharacteristic, complemented
150#3205C2×C711yescharacteristic, complemented
151#1627C2×C712yescomplemented
152#1626C2×C712yescomplemented
153#1621C2×C712yescomplemented
154#1620C2×C712yescomplemented
155#1553C2×C711yescharacteristic, complemented
156#1497C2×C711yescharacteristic, complemented
157#1482C2×C711yescharacteristic, complemented
158C24×C7—41no—
159C7×#41—82no—
160C7×#41—82no—
161C2×C7×Q8—22no—
162C2×C7×Q8—21no—
163C2×C7×Q8—21no—
164C2×C7×Q8—82no—
165C2×C7×Q8C2×D411yescharacteristic
166C2×C7×Q8C2×D411yescharacteristic
167C2×C7×Q8C2×D411yescharacteristic
168C2×C7×D4—22no—
169C2×C7×D4—22no—
170C2×C7×D4—21no—
171C2×C7×D4—22no—
172C2×C7×D4—21no—
173C2×C7×D4—21no—
174C2×C7×D4—22no—
175C2×C7×D4—22no—
176C2×C7×D4—22no—
177C2×C7×D4—41no—
178C2×C7×D4—41no—
179C2×C7×D4—41no—
180C2×C7×D4—82no—
181C2×C7×D4—82no—
182C2×C7×D4—82no—
183C2×C7×D4C2×D411yescharacteristic
184C2×C7×D4C2×D412yes—
185C22×C4×C7—22no—
186C22×C4×C7—22no—
187C22×C4×C7—22no—
188C22×C4×C7—21no—
189C22×C4×C7—22no—
190C22×C4×C7C2×D411yescharacteristic
191C22×C4×C7C2×D411yescharacteristic
192C22×C4×C7C2×D411yescharacteristic
193C7×SD16—82no—
194C7×SD16—82no—
195C7×SD16—82no—
196C7×D8—82no—
197C7×D8—84no—
198C2×C7×C8—22no—
199C2×C7×C8—22no—
200C2×C7×C8—22no—
201C7×C4⋊C4—22no—
202C7×C4⋊C4—22no—
203C7×C4⋊C4—22no—
204C7×C4⋊C4—22no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—21no—
207C7×C4⋊C4—21no—
208C7×C4⋊C4—21no—
209C7×C4⋊C4—21no—
210C7×C4⋊C4—21no—
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⋊C4C2×D411yescharacteristic
218C7×C4⋊C4C2×D411yescharacteristic
219C7×C4⋊C4C2×D411yescharacteristic
220C7×C4⋊C4C2×D412yes—
221C7×C4⋊C4C2×D411yescharacteristic
222C7×C22⋊C4—22no—
223C7×C22⋊C4—22no—
224C7×C22⋊C4—22no—
225C7×C22⋊C4—22no—
226C7×C22⋊C4—22no—
227C7×C22⋊C4—21no—
228C7×C22⋊C4—21no—
229C7×C22⋊C4—21no—
230C7×C22⋊C4—21no—
231C7×C22⋊C4—21no—
232C7×C22⋊C4—21no—
233C7×C22⋊C4—22no—
234C7×C22⋊C4—22no—
235C7×C22⋊C4—21no—
236C42×C7—22no—
237C42×C7—21no—
238C42×C7—21no—
239C42×C7—21no—
240C42×C7C2×D411yescharacteristic
241C42×C7C2×D411yescharacteristic
242C42×C7C2×D411yescharacteristic
243C42×C7C2411yescharacteristic
244#562—22no—
245#559—21no—
246#559C22×C711yescharacteristic, complemented
247#557C22×C711yescharacteristic, complemented
248#554—21no—
249#550C22×C711yescharacteristic, complemented
250D4×Q8C22×C711yescharacteristic, complemented
251#548—21no—
252#547—21no—
253#542—21no—
254#542—21no—
255#540—21no—
256#539—22no—
257#536—21no—
258#533C22×C711yescharacteristic
259#532C22×C711yescharacteristic
260C2×#127C22×C711yescharacteristic
261#529C22×C711yescharacteristic
262#496—21no—
263#493C22×C712yes—
264#492—21no—
265#490—21no—
266#486—21no—
267#486C22×C712yes—
268#460C22×C712yes—
269#460C22×C712yes—
270#460C22×C712yes—
271#459—22no—
272#459C22×C712yes—
273#450—21no—
274#450—21no—
275#450—21no—
276#447—22no—
277#447—21no—
278#432C22×C711yescharacteristic
279#431C22×C711yescharacteristic
280#431C22×C711yescharacteristic
281#332C22×C712yescomplemented
282#331C22×C712yescomplemented
283Q8⋊C8C22×C712yescomplemented
284Q8⋊C8C22×C712yescomplemented
285Q8⋊C8C22×C712yescomplemented
286D4⋊C8C22×C712yescomplemented
287C23×C7—21no—
288C23×C7—22no—
289C23×C7—42no—
290C23×C7—82no—
291C23×C7—82no—
292C23×C7—82no—
293C23×C7#12011yescharacteristic
294C7×Q8—22no—
295C7×Q8—22no—
296C7×Q8—22no—
297C7×Q8—42no—
298C7×Q8—81no—
299C7×Q8—82no—
300C7×D4—22no—
301C7×D4—42no—
302C7×D4—44no—
303C7×D4—42no—
304C7×D4—42no—
305C7×D4—42no—
306C7×D4—42no—
307C7×D4—42no—
308C7×D4—42no—
309C7×D4—82no—
310C7×D4—82no—
311C7×D4—82no—
312C7×D4—81no—
313C7×D4—82no—
314C7×D4—82no—
315C7×D4—82no—
316C2×C4×C7—22no—
317C2×C4×C7—22no—
318C2×C4×C7—21no—
319C2×C4×C7—21no—
320C2×C4×C7—21no—
321C2×C4×C7—21no—
322C2×C4×C7—21no—
323C2×C4×C7—22no—
324C2×C4×C7—21no—
325C2×C4×C7—21no—
326C2×C4×C7—41no—
327C2×C4×C7—42no—
328C2×C4×C7—82no—
329C2×C4×C7—82no—
330C2×C4×C7—82no—
331C2×C4×C7—82no—
332C2×C4×C7#12011yescharacteristic
333C2×C4×C7#12011yescharacteristic
334C2×C4×C7#12011yescharacteristic
335C2×C4×C7C22×D411yescharacteristic
336C2×C4×C7C22×D411yescharacteristic
337C2×C4×C7C22×D411yescharacteristic
338C7×C8—42no—
339C7×C8—42no—
340C7×C8—42no—
341C2×#41—21no—
342C2×#41—21no—
343C22×Q8—21no—
344C22×D4—21no—
345C22×D4—21no—
346C22×D4—21no—
347C2×SD16—22no—
348C2×SD16—22no—
349C2×SD16—22no—
350C2×D8—24no—
351C2×D8—22no—
352C4⋊Q8—21no—
353C4⋊Q8—21no—
354C4⋊Q8C7×D411yescharacteristic
355C4⋊Q8C7×D411yescharacteristic
356C4⋊Q8C23×C711yescharacteristic
357#127—21no—
358#127—21no—
359#127—21no—
360#127—21no—
361#127C23×C712yes—
362#127C23×C711yescharacteristic
363#126—22no—
364#126—21no—
365#125—22no—
366#125—21no—
367#125—21no—
368#125C7×D411yescharacteristic
369#124—22no—
370#124—22no—
371#124—21no—
372#124—21no—
373#124—21no—
374#124C7×D411yescharacteristic
375#123—22no—
376#123—22no—
377#123—22no—
378#123—22no—
379#123—22no—
380#123—22no—
381#123—21no—
382C22⋊Q8—22no—
383C22⋊Q8—22no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8—21no—
387C22⋊Q8—22no—
388C22⋊Q8—21no—
389C22⋊Q8C7×D411yescharacteristic
390C22⋊Q8C7×D411yescharacteristic
391C22⋊Q8C7×D411yescharacteristic
392#121—22no—
393#121—22no—
394#121—22no—
395#121—21no—
396#121—21no—
397#121C7×D411yescharacteristic
398#121C7×D412yes—
399C4×Q8—21no—
400C4×Q8—21no—
401C4×Q8—21no—
402C4×Q8C23×C711yescharacteristic
403C4×Q8C23×C711yescharacteristic
404C4×Q8C23×C711yescharacteristic
405C4×D4—22no—
406C4×D4—21no—
407C4×D4—21no—
408C4×D4—21no—
409C4×D4—21no—
410C4×D4—21no—
411C4×D4—21no—
412C4×D4C7×D412yes—
413C4×D4C23×C711yescharacteristic
414#117—21no—
415#117—21no—
416C2×C4⋊C4—22no—
417C2×C42C23×C711yescharacteristic
418C4⋊C8—21no—
419C4⋊C8C23×C712yes—
420C4⋊C8C23×C712yes—
421#102—24no—
422#102—22no—
423#102—22no—
424#102—22no—
425#102—22no—
426C22⋊C8—21no—
427C22⋊C8—21no—
428C22⋊C8—21no—
429#97—21no—
430C4×C8—21no—
431C4×C8C23×C712yes—
432C22×C7—21no—
433C22×C7—21no—
434C22×C7—21no—
435C22×C7—41no—
436C22×C7—42no—
437C22×C7—41no—
438C22×C7—42no—
439C22×C7—42no—
440C22×C7—41no—
441C22×C7—82no—
442C22×C7—82no—
443C22×C7—82no—
444C22×C7—164no—
445C22×C7C2×#12011yescharacteristic
446C4×C7—21no—
447C4×C7—21no—
448C4×C7—21no—
449C4×C7—21no—
450C4×C7—41no—
451C4×C7—41no—
452C4×C7—42no—
453C4×C7—41no—
454C4×C7—41no—
455C4×C7—41no—
456C4×C7—41no—
457C4×C7—41no—
458C4×C7#57612yes—
459C4×C7#57712yes—
460C24—41no—
461#41—82no—
462#41—82no—
463C2×Q8—22no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—82no—
467C2×Q8C2×C7×D411yescharacteristic
468C2×Q8C2×C7×D411yescharacteristic
469C2×Q8C2×C7×D411yescharacteristic
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—21no—
473C2×D4—22no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—41no—
480C2×D4—41no—
481C2×D4—41no—
482C2×D4—82no—
483C2×D4—82no—
484C2×D4—82no—
485C2×D4C2×C7×D411yescharacteristic
486C2×D4C2×C7×D412yes—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4C2×C7×D411yescharacteristic
493C22×C4C2×C7×D411yescharacteristic
494C22×C4C2×C7×D411yescharacteristic
495SD16—82no—
496SD16—82no—
497SD16—82no—
498D8—82no—
499D8—84no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4C2×C7×D411yescharacteristic
520C4⋊C4C2×C7×D411yescharacteristic
521C4⋊C4C2×C7×D411yescharacteristic
522C4⋊C4C2×C7×D412yes—
523C4⋊C4C2×C7×D411yescharacteristic
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—21no—
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—21no—
533C22⋊C4—21no—
534C22⋊C4—21no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—21no—
538C42—22no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42C2×C7×D411yescharacteristic
543C42C2×C7×D411yescharacteristic
544C42C2×C7×D411yescharacteristic
545C42C24×C711yescharacteristic
546C2×C7—41no—
547C2×C7—81no—
548C2×C7—82no—
549C2×C7—162no—
550C2×C7#300811yescharacteristic
551C2×C7#301011yescharacteristic
552C2×C7#302111yescharacteristic
553C23—21no—
554C23—22no—
555C23—42no—
556C23—82no—
557C23—82no—
558C23—82no—
559C23C7×#12011yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—22no—
563Q8—42no—
564Q8—81no—
565Q8—82no—
566D4—22no—
567D4—42no—
568D4—44no—
569D4—42no—
570D4—42no—
571D4—42no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—82no—
576D4—82no—
577D4—82no—
578D4—81no—
579D4—82no—
580D4—82no—
581D4—82no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4C7×#12011yescharacteristic
599C2×C4C7×#12011yescharacteristic
600C2×C4C7×#12011yescharacteristic
601C2×C4C22×C7×D411yescharacteristic
602C2×C4C22×C7×D411yescharacteristic
603C2×C4C22×C7×D411yescharacteristic
604C8—42no—
605C8—42no—
606C8—42no—
607C7#1895811yescharacteristic, direct product factor, Sylow
608C22—21no—
609C22—21no—
610C22—21no—
611C22—41no—
612C22—42no—
613C22—41no—
614C22—42no—
615C22—42no—
616C22—41no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—164no—
621C22C2×C7×#12011yescharacteristic
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—41no—
627C4—41no—
628C4—42no—
629C4—41no—
630C4—41no—
631C4—41no—
632C4—41no—
633C4—41no—
634C4C7×#57612yes—
635C4C7×#57712yes—
636C2—41no—
637C2—81no—
638C2—82no—
639C2—162no—
640C2C7×#300811yescharacteristic
641C2C7×#301011yescharacteristic
642C2C7×#302111yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
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
1. C7×#3554C2semidirect
2. C7×#3217C2semidirect
3. C7×#3208C2semidirect
4. C7×#3205C2semidirect
5. C7×#1627C2semidirect
6. C7×#1626C2semidirect
7. C7×#1621C2semidirect
8. C7×#1620C2semidirect
9. C7×#1553C2semidirect
10. C7×#1497C2semidirect
11. C7×#1482C2semidirect
14. C7×#559C22semidirect
15. C7×#557C22semidirect
17. C7×#550C22semidirect
18. C7×D4×Q8C22semidirect
49. C7×#332C22semidirect
50. C7×#331C22semidirect
51. C7×Q8⋊C8C22semidirect
52. C7×Q8⋊C8C22semidirect
53. C7×Q8⋊C8C22semidirect
54. C7×D4⋊C8C22semidirect
55. #18958C7direct (N × H)
147. #3554C2×C7semidirect
148. #3217C2×C7semidirect
149. #3208C2×C7semidirect
150. #3205C2×C7semidirect
151. #1627C2×C7semidirect
152. #1626C2×C7semidirect
153. #1621C2×C7semidirect
154. #1620C2×C7semidirect
155. #1553C2×C7semidirect
156. #1497C2×C7semidirect
157. #1482C2×C7semidirect
246. #559C22×C7semidirect
247. #557C22×C7semidirect
249. #550C22×C7semidirect
250. D4×Q8C22×C7semidirect
281. #332C22×C7semidirect
282. #331C22×C7semidirect
283. Q8⋊C8C22×C7semidirect
284. Q8⋊C8C22×C7semidirect
285. Q8⋊C8C22×C7semidirect
286. D4⋊C8C22×C7semidirect
607. C7#18958direct (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×#32172
C7×#32082
C7×#32052
C7×#16272
C7×#16262
C7×#16212
C7×#16202
C7×#15532
C7×#14972
C7×#14822
#189587