Finite Group Databasebeta

← C7×#18956 #13196172

Subgroups — C7×#18956 #13196172

Order 1792.

950 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3547C211yesmaximal, characteristic, complemented
2C7×#3217C211yesmaximal, characteristic, complemented
3C7×#3208C211yesmaximal, characteristic, complemented
4C7×#3205C211yesmaximal, characteristic, complemented
5C7×#1627C212yesmaximal, complemented
6C7×#1625C212yesmaximal, complemented
7C7×#1624C212yesmaximal, complemented
8C7×#1622C212yesmaximal, complemented
9C7×#1554C211yesmaximal, characteristic, complemented
10C7×#1496C211yesmaximal, characteristic, complemented
11C7×#1482C211yesmaximal, characteristic, complemented
12C7×#562—22no—
13C7×#555—21no—
14C7×#555C2211yescharacteristic, complemented
15C7×#554C2211yescharacteristic, complemented
16C7×#553—21no—
17C7×#552—21no—
18C7×D4×Q8C2211yescharacteristic, complemented
19C7×#548—21no—
20C7×#547C2211yescharacteristic, complemented
21C7×#544—21no—
22C7×#542—22no—
23C7×#540—21no—
24C7×#537—21no—
25C7×#536—21no—
26C7×#533C2211yescharacteristic
27C7×#532C2211yescharacteristic
28C2×C7×#127C2211yescharacteristic
29C7×#529C2211yescharacteristic
30C7×#496—21no—
31C7×#493—21no—
32C7×#492C2212yes—
33C7×#490—21no—
34C7×#486—21no—
35C7×#486C2212yes—
36C7×#460—22no—
37C7×#460C2212yes—
38C7×#459C2212yes—
39C7×#459C2212yes—
40C7×#459C2212yes—
41C7×#450—22no—
42C7×#450—21no—
43C7×#447—21no—
44C7×#447—21no—
45C7×#447—21no—
46C7×#432C2211yescharacteristic
47C7×#431C2211yescharacteristic
48C7×#431C2211yescharacteristic
49C7×#333C2212yescomplemented
50C7×#332C2212yescomplemented
51C7×Q8⋊C8C2212yescomplemented
52C7×D4⋊C8C2212yescomplemented
53C7×D4⋊C8C2212yescomplemented
54C7×D4⋊C8C2212yescomplemented
55#18956C711yesmaximal, 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—24no—
63C2×C7×SD16—22no—
64C2×C7×D8—22no—
65C2×C7×D8—22no—
66C2×C7×D8—22no—
67C7×C4⋊Q8—22no—
68C7×C4⋊Q8—21no—
69C7×C4⋊Q8D411yescharacteristic
70C7×C4⋊Q8C2311yescharacteristic
71C7×C4⋊Q8C2311yescharacteristic
72C7×#127—21no—
73C7×#127—21no—
74C7×#127—21no—
75C7×#127D411yescharacteristic
76C7×#127C2312yes—
77C7×#126—22no—
78C7×#126—21no—
79C7×#125—22no—
80C7×#125—21no—
81C7×#125—21no—
82C7×#125D411yescharacteristic
83C7×#124—22no—
84C7×#124—22no—
85C7×#124—22no—
86C7×#124—21no—
87C7×#124D411yescharacteristic
88C7×#123—22no—
89C7×#123—22no—
90C7×#123—22no—
91C7×#123—21no—
92C7×#123—22no—
93C7×C22⋊Q8—22no—
94C7×C22⋊Q8—22no—
95C7×C22⋊Q8—21no—
96C7×C22⋊Q8—22no—
97C7×C22⋊Q8—21no—
98C7×C22⋊Q8—21no—
99C7×C22⋊Q8—21no—
100C7×C22⋊Q8—22no—
101C7×C22⋊Q8D411yescharacteristic
102C7×C22⋊Q8D412yes—
103C7×#121—22no—
104C7×#121—22no—
105C7×#121—22no—
106C7×#121—22no—
107C7×#121—21no—
108C7×#121—21no—
109C7×#121—21no—
110C7×#121—21no—
111C7×#121D411yescharacteristic
112C7×#121D411yescharacteristic
113C7×#121D411yescharacteristic
114C4×C7×Q8—21no—
115C4×C7×Q8D412yes—
116C4×C7×Q8C2311yescharacteristic
117C4×C7×D4—22no—
118C4×C7×D4—21no—
119C4×C7×D4—21no—
120C4×C7×D4—21no—
121C4×C7×D4—21no—
122C4×C7×D4C2311yescharacteristic
123C4×C7×D4C2311yescharacteristic
124C4×C7×D4C2311yescharacteristic
125C7×#117—22no—
126C7×#117—21no—
127C7×#117—21no—
128C7×#117—21no—
129C7×#117—21no—
130C2×C7×C4⋊C4—21no—
131C2×C7×C4⋊C4—21no—
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#3547C2×C711yescharacteristic, complemented
148#3217C2×C711yescharacteristic, complemented
149#3208C2×C711yescharacteristic, complemented
150#3205C2×C711yescharacteristic, complemented
151#1627C2×C712yescomplemented
152#1625C2×C712yescomplemented
153#1624C2×C712yescomplemented
154#1622C2×C712yescomplemented
155#1554C2×C711yescharacteristic, complemented
156#1496C2×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×D412yes—
167C2×C7×D4—22no—
168C2×C7×D4—22no—
169C2×C7×D4—21no—
170C2×C7×D4—21no—
171C2×C7×D4—21no—
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×D411yescharacteristic
185C2×C7×D4C2×D411yescharacteristic
186C22×C4×C7—22no—
187C22×C4×C7—22no—
188C22×C4×C7—22no—
189C22×C4×C7—21no—
190C22×C4×C7—21no—
191C22×C4×C7—21no—
192C22×C4×C7C2×D411yescharacteristic
193C22×C4×C7C2×D411yescharacteristic
194C22×C4×C7C2×D411yescharacteristic
195C7×SD16—82no—
196C7×SD16—84no—
197C7×D8—82no—
198C7×D8—82no—
199C7×D8—82no—
200C2×C7×C8—22no—
201C2×C7×C8—22no—
202C2×C7×C8—22no—
203C7×C4⋊C4—22no—
204C7×C4⋊C4—22no—
205C7×C4⋊C4—22no—
206C7×C4⋊C4—22no—
207C7×C4⋊C4—21no—
208C7×C4⋊C4—21no—
209C7×C4⋊C4—21no—
210C7×C4⋊C4—21no—
211C7×C4⋊C4—22no—
212C7×C4⋊C4—22no—
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×D411yescharacteristic
221C7×C4⋊C4C2×D412yes—
222C7×C22⋊C4—22no—
223C7×C22⋊C4—22no—
224C7×C22⋊C4—22no—
225C7×C22⋊C4—22no—
226C7×C22⋊C4—22no—
227C7×C22⋊C4—22no—
228C7×C22⋊C4—21no—
229C7×C22⋊C4—21no—
230C7×C22⋊C4—21no—
231C7×C22⋊C4—21no—
232C7×C22⋊C4—21no—
233C7×C22⋊C4—21no—
234C7×C22⋊C4—21no—
235C7×C22⋊C4—21no—
236C7×C22⋊C4—21no—
237C42×C7—22no—
238C42×C7—22no—
239C42×C7—21no—
240C42×C7C2×D411yescharacteristic
241C42×C7C2×D411yescharacteristic
242C42×C7C2×D411yescharacteristic
243C42×C7C2411yescharacteristic
244#562—22no—
245#555—21no—
246#555C22×C711yescharacteristic, complemented
247#554C22×C711yescharacteristic, complemented
248#553—21no—
249#552—21no—
250D4×Q8C22×C711yescharacteristic, complemented
251#548—21no—
252#547C22×C711yescharacteristic, complemented
253#544—21no—
254#542—22no—
255#540—21no—
256#537—21no—
257#536—21no—
258#533C22×C711yescharacteristic
259#532C22×C711yescharacteristic
260C2×#127C22×C711yescharacteristic
261#529C22×C711yescharacteristic
262#496—21no—
263#493—21no—
264#492C22×C712yes—
265#490—21no—
266#486—21no—
267#486C22×C712yes—
268#460—22no—
269#460C22×C712yes—
270#459C22×C712yes—
271#459C22×C712yes—
272#459C22×C712yes—
273#450—22no—
274#450—21no—
275#447—21no—
276#447—21no—
277#447—21no—
278#432C22×C711yescharacteristic
279#431C22×C711yescharacteristic
280#431C22×C711yescharacteristic
281#333C22×C712yescomplemented
282#332C22×C712yescomplemented
283Q8⋊C8C22×C712yescomplemented
284D4⋊C8C22×C712yescomplemented
285D4⋊C8C22×C712yescomplemented
286D4⋊C8C22×C712yescomplemented
287C23×C7—21no—
288C23×C7—21no—
289C23×C7—21no—
290C23×C7—42no—
291C23×C7—82no—
292C23×C7—82no—
293C23×C7—82no—
294C23×C7#12011yescharacteristic
295C7×Q8—22no—
296C7×Q8—42no—
297C7×Q8—42no—
298C7×Q8—81no—
299C7×Q8—82no—
300C7×D4—22no—
301C7×D4—22no—
302C7×D4—22no—
303C7×D4—44no—
304C7×D4—42no—
305C7×D4—42no—
306C7×D4—42no—
307C7×D4—42no—
308C7×D4—81no—
309C7×D4—81no—
310C7×D4—82no—
311C7×D4—82no—
312C7×D4—82no—
313C7×D4—81no—
314C7×D4—81no—
315C7×D4—81no—
316C7×D4—82no—
317C7×D4—82no—
318C2×C4×C7—22no—
319C2×C4×C7—22no—
320C2×C4×C7—21no—
321C2×C4×C7—22no—
322C2×C4×C7—21no—
323C2×C4×C7—21no—
324C2×C4×C7—21no—
325C2×C4×C7—22no—
326C2×C4×C7—21no—
327C2×C4×C7—41no—
328C2×C4×C7—42no—
329C2×C4×C7—42no—
330C2×C4×C7—42no—
331C2×C4×C7—82no—
332C2×C4×C7—82no—
333C2×C4×C7—82no—
334C2×C4×C7#12011yescharacteristic
335C2×C4×C7#12011yescharacteristic
336C2×C4×C7#12011yescharacteristic
337C2×C4×C7C22×D411yescharacteristic
338C2×C4×C7C22×D411yescharacteristic
339C2×C4×C7C22×D411yescharacteristic
340C7×C8—42no—
341C7×C8—42no—
342C7×C8—42no—
343C2×#41—21no—
344C2×#41—21no—
345C22×Q8—21no—
346C22×D4—21no—
347C22×D4—21no—
348C22×D4—21no—
349C2×SD16—24no—
350C2×SD16—22no—
351C2×D8—22no—
352C2×D8—22no—
353C2×D8—22no—
354C4⋊Q8—22no—
355C4⋊Q8—21no—
356C4⋊Q8C7×D411yescharacteristic
357C4⋊Q8C23×C711yescharacteristic
358C4⋊Q8C23×C711yescharacteristic
359#127—21no—
360#127—21no—
361#127—21no—
362#127C7×D411yescharacteristic
363#127C23×C712yes—
364#126—22no—
365#126—21no—
366#125—22no—
367#125—21no—
368#125—21no—
369#125C7×D411yescharacteristic
370#124—22no—
371#124—22no—
372#124—22no—
373#124—21no—
374#124C7×D411yescharacteristic
375#123—22no—
376#123—22no—
377#123—22no—
378#123—21no—
379#123—22no—
380C22⋊Q8—22no—
381C22⋊Q8—22no—
382C22⋊Q8—21no—
383C22⋊Q8—22no—
384C22⋊Q8—21no—
385C22⋊Q8—21no—
386C22⋊Q8—21no—
387C22⋊Q8—22no—
388C22⋊Q8C7×D411yescharacteristic
389C22⋊Q8C7×D412yes—
390#121—22no—
391#121—22no—
392#121—22no—
393#121—22no—
394#121—21no—
395#121—21no—
396#121—21no—
397#121—21no—
398#121C7×D411yescharacteristic
399#121C7×D411yescharacteristic
400#121C7×D411yescharacteristic
401C4×Q8—21no—
402C4×Q8C7×D412yes—
403C4×Q8C23×C711yescharacteristic
404C4×D4—22no—
405C4×D4—21no—
406C4×D4—21no—
407C4×D4—21no—
408C4×D4—21no—
409C4×D4C23×C711yescharacteristic
410C4×D4C23×C711yescharacteristic
411C4×D4C23×C711yescharacteristic
412#117—22no—
413#117—21no—
414#117—21no—
415#117—21no—
416#117—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C42C23×C711yescharacteristic
420C4⋊C8—21no—
421C4⋊C8C23×C712yes—
422C4⋊C8C23×C712yes—
423#102—22no—
424#102—22no—
425#102—22no—
426#102—22no—
427#102—24no—
428C22⋊C8—21no—
429C22⋊C8—21no—
430C22⋊C8—21no—
431#97—21no—
432C4×C8—21no—
433C4×C8C23×C712yes—
434C22×C7—21no—
435C22×C7—21no—
436C22×C7—21no—
437C22×C7—41no—
438C22×C7—41no—
439C22×C7—41no—
440C22×C7—41no—
441C22×C7—41no—
442C22×C7—41no—
443C22×C7—41no—
444C22×C7—41no—
445C22×C7—41no—
446C22×C7—82no—
447C22×C7—82no—
448C22×C7—82no—
449C22×C7—164no—
450C22×C7C2×#12011yescharacteristic
451C4×C7—21no—
452C4×C7—21no—
453C4×C7—21no—
454C4×C7—21no—
455C4×C7—41no—
456C4×C7—41no—
457C4×C7—42no—
458C4×C7—41no—
459C4×C7—41no—
460C4×C7—41no—
461C4×C7—42no—
462C4×C7#57612yes—
463C4×C7#57712yes—
464C24—41no—
465#41—82no—
466#41—82no—
467C2×Q8—22no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—82no—
471C2×Q8C2×C7×D411yescharacteristic
472C2×Q8C2×C7×D412yes—
473C2×D4—22no—
474C2×D4—22no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—41no—
484C2×D4—41no—
485C2×D4—41no—
486C2×D4—82no—
487C2×D4—82no—
488C2×D4—82no—
489C2×D4C2×C7×D411yescharacteristic
490C2×D4C2×C7×D411yescharacteristic
491C2×D4C2×C7×D411yescharacteristic
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4C2×C7×D411yescharacteristic
499C22×C4C2×C7×D411yescharacteristic
500C22×C4C2×C7×D411yescharacteristic
501SD16—82no—
502SD16—84no—
503D8—82no—
504D8—82no—
505D8—82no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4C2×C7×D411yescharacteristic
524C4⋊C4C2×C7×D411yescharacteristic
525C4⋊C4C2×C7×D411yescharacteristic
526C4⋊C4C2×C7×D411yescharacteristic
527C4⋊C4C2×C7×D412yes—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—21no—
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C22⋊C4—21no—
543C42—22no—
544C42—22no—
545C42—21no—
546C42C2×C7×D411yescharacteristic
547C42C2×C7×D411yescharacteristic
548C42C2×C7×D411yescharacteristic
549C42C24×C711yescharacteristic
550C2×C7—41no—
551C2×C7—81no—
552C2×C7—81no—
553C2×C7—81no—
554C2×C7—162no—
555C2×C7#300811yescharacteristic
556C2×C7#301011yescharacteristic
557C2×C7#302111yescharacteristic
558C23—21no—
559C23—21no—
560C23—21no—
561C23—42no—
562C23—82no—
563C23—82no—
564C23—82no—
565C23C7×#12011yescharacteristic
566Q8—22no—
567Q8—42no—
568Q8—42no—
569Q8—81no—
570Q8—82no—
571D4—22no—
572D4—22no—
573D4—22no—
574D4—44no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—81no—
580D4—81no—
581D4—82no—
582D4—82no—
583D4—82no—
584D4—81no—
585D4—81no—
586D4—81no—
587D4—82no—
588D4—82no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4C7×#12011yescharacteristic
606C2×C4C7×#12011yescharacteristic
607C2×C4C7×#12011yescharacteristic
608C2×C4C22×C7×D411yescharacteristic
609C2×C4C22×C7×D411yescharacteristic
610C2×C4C22×C7×D411yescharacteristic
611C8—42no—
612C8—42no—
613C8—42no—
614C7#1895611yescharacteristic, direct product factor, Sylow
615C22—21no—
616C22—21no—
617C22—21no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—164no—
631C22C2×C7×#12011yescharacteristic
632C4—21no—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—41no—
637C4—41no—
638C4—42no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—42no—
643C4C7×#57612yes—
644C4C7×#57712yes—
645C2—41no—
646C2—81no—
647C2—81no—
648C2—81no—
649C2—162no—
650C2C7×#300811yescharacteristic
651C2C7×#301011yescharacteristic
652C2C7×#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×#3547C2semidirect
2. C7×#3217C2semidirect
3. C7×#3208C2semidirect
4. C7×#3205C2semidirect
5. C7×#1627C2semidirect
6. C7×#1625C2semidirect
7. C7×#1624C2semidirect
8. C7×#1622C2semidirect
9. C7×#1554C2semidirect
10. C7×#1496C2semidirect
11. C7×#1482C2semidirect
14. C7×#555C22semidirect
15. C7×#554C22semidirect
18. C7×D4×Q8C22semidirect
20. C7×#547C22semidirect
49. C7×#333C22semidirect
50. C7×#332C22semidirect
51. C7×Q8⋊C8C22semidirect
52. C7×D4⋊C8C22semidirect
53. C7×D4⋊C8C22semidirect
54. C7×D4⋊C8C22semidirect
55. #18956C7direct (N × H)
147. #3547C2×C7semidirect
148. #3217C2×C7semidirect
149. #3208C2×C7semidirect
150. #3205C2×C7semidirect
151. #1627C2×C7semidirect
152. #1625C2×C7semidirect
153. #1624C2×C7semidirect
154. #1622C2×C7semidirect
155. #1554C2×C7semidirect
156. #1496C2×C7semidirect
157. #1482C2×C7semidirect
246. #555C22×C7semidirect
247. #554C22×C7semidirect
250. D4×Q8C22×C7semidirect
252. #547C22×C7semidirect
281. #333C22×C7semidirect
282. #332C22×C7semidirect
283. Q8⋊C8C22×C7semidirect
284. D4⋊C8C22×C7semidirect
285. D4⋊C8C22×C7semidirect
286. D4⋊C8C22×C7semidirect
614. C7#18956direct (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×#32172
C7×#32082
C7×#32052
C7×#16272
C7×#16252
C7×#16242
C7×#16222
C7×#15542
C7×#14962
C7×#14822
#189567