Finite Group Databasebeta

← C7×#18967 #13196183

Subgroups — C7×#18967 #13196183

Order 1792.

938 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3546C211yesmaximal, characteristic
2C7×#3227C211yesmaximal, characteristic, complemented
3C7×#3226C211yesmaximal, characteristic, complemented
4C7×#3218C211yesmaximal, characteristic, complemented
5C7×#1654C212yesmaximal, complemented
6C7×#1652C212yesmaximal, complemented
7C7×#1641C212yesmaximal, complemented
8C7×#1639C212yesmaximal, complemented
9C7×#1553C211yesmaximal, characteristic, complemented
10C7×#1496C211yesmaximal, characteristic, complemented
11C7×#1482C211yesmaximal, characteristic, complemented
12C7×#560—22no—
13C7×#556C2211yescharacteristic
14C7×#553—21no—
15C7×#550—21no—
16C7×#550C2211yescharacteristic
17C7×#547C2211yescharacteristic
18C7×#546—21no—
19C7×#546—21no—
20C7×D42C2211yescharacteristic
21C7×#542—21no—
22C7×#538—22no—
23C7×#537—21no—
24C7×#535—21no—
25C7×#535—21no—
26C7×#532C2211yescharacteristic
27C7×#532C2211yescharacteristic
28C2×C7×C4⋊Q8C2211yescharacteristic, complemented
29C7×#529C2211yescharacteristic
30C7×#500—21no—
31C7×#498C2212yescomplemented
32C7×#497—21no—
33C7×#491—21no—
34C7×#486—21no—
35C7×#486C2212yes—
36C7×#481—22no—
37C7×#480—21no—
38C7×#480—21no—
39C7×#476—22no—
40C7×#474C2212yes—
41C7×#474C2212yescomplemented
42C7×#462C2212yescomplemented
43C7×#461C2212yes—
44C7×#451—21no—
45C7×#448—21no—
46C7×#432C2211yescharacteristic, complemented
47C7×#431C2211yescharacteristic, complemented
48C7×#431C2211yescharacteristic, complemented
49C7×#332C2212yescomplemented
50C7×#331C2212yes—
51C7×Q8⋊C8C2212yescomplemented
52C7×D4⋊C8C2212yes—
53C7×D4⋊C8C2212yes—
54C7×D4⋊C8C2212yescomplemented
55#18967C711yesmaximal, characteristic, direct product factor, Sylow
56C2×C7×#41—21no—
57C2×C7×#41—21no—
58C2×C7×#41—21no—
59C2×C7×#41—21no—
60C22×C7×Q8—21no—
61C22×C7×D4—21no—
62C22×C7×D4—21no—
63C2×C7×Q16—22no—
64C2×C7×SD16—22no—
65C7×C4⋊Q8—21no—
66C7×C4⋊Q8—21no—
67C7×C4⋊Q8—21no—
68C7×C4⋊Q8D411yescharacteristic
69C7×C4⋊Q8C2312yes—
70C7×C4⋊Q8C2311yescharacteristic
71C7×#127—22no—
72C7×#127—21no—
73C7×#127D411yescharacteristic
74C7×#127C2311yescharacteristic
75C7×#126—22no—
76C7×#126—22no—
77C7×#126—21no—
78C7×#125D411yescharacteristic
79C7×#124—22no—
80C7×#124—21no—
81C7×#124—21no—
82C7×#124—21no—
83C7×#124D411yescharacteristic
84C7×#123—22no—
85C7×#123—22no—
86C7×#123—21no—
87C7×#123—22no—
88C7×C22⋊Q8—22no—
89C7×C22⋊Q8—21no—
90C7×C22⋊Q8—21no—
91C7×C22⋊Q8D411yescharacteristic
92C7×#121—22no—
93C7×#121—22no—
94C7×#121—22no—
95C7×#121—22no—
96C7×#121—22no—
97C7×#121—21no—
98C7×#121—22no—
99C7×#121—21no—
100C7×#121—21no—
101C7×#121—21no—
102C7×#121—21no—
103C7×#121—21no—
104C7×#121D411yescharacteristic
105C7×#121D411yescharacteristic
106C7×#121D412yes—
107C7×#121D411yescharacteristic
108C7×#120—22no—
109C7×#120—22no—
110C7×#120—22no—
111C7×#120—22no—
112C4×C7×Q8—21no—
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×D4D412yes—
120C4×C7×D4C2311yescharacteristic
121C4×C7×D4C2311yescharacteristic
122C4×C7×D4C2311yescharacteristic
123C7×#117—21no—
124C7×#117—21no—
125C2×C7×C4⋊C4—21no—
126C2×C7×C4⋊C4—21no—
127C2×C7×C4⋊C4—21no—
128C2×C7×C22⋊C4—22no—
129C2×C42×C7C2311yescharacteristic
130C7×#107—22no—
131C7×#107—22no—
132C7×#106—24no—
133C7×C4⋊C8—21no—
134C7×C4⋊C8C2312yes—
135C7×C4⋊C8C2312yes—
136C7×#103—22no—
137C7×#103—22no—
138C7×#102—22no—
139C7×#102—22no—
140C7×#102—24no—
141C7×C22⋊C8—21no—
142C7×C22⋊C8—21no—
143C7×C22⋊C8—21no—
144C7×#97—21no—
145C4×C7×C8—21no—
146C4×C7×C8C2312yes—
147#3546C2×C711yescharacteristic
148#3227C2×C711yescharacteristic, complemented
149#3226C2×C711yescharacteristic, complemented
150#3218C2×C711yescharacteristic, complemented
151#1654C2×C712yescomplemented
152#1652C2×C712yescomplemented
153#1641C2×C712yescomplemented
154#1639C2×C712yescomplemented
155#1553C2×C711yescharacteristic, complemented
156#1496C2×C711yescharacteristic, complemented
157#1482C2×C711yescharacteristic, complemented
158C24×C7—22no—
159C7×#41—82no—
160C7×#41—82no—
161C7×#41—82no—
162C7×#41—82no—
163C2×C7×Q8—21no—
164C2×C7×Q8—21no—
165C2×C7×Q8—22no—
166C2×C7×Q8—41no—
167C2×C7×Q8—82no—
168C2×C7×Q8C2×D411yescharacteristic
169C2×C7×D4—22no—
170C2×C7×D4—22no—
171C2×C7×D4—22no—
172C2×C7×D4—22no—
173C2×C7×D4—22no—
174C2×C7×D4—21no—
175C2×C7×D4—21no—
176C2×C7×D4—22no—
177C2×C7×D4—21no—
178C2×C7×D4—21no—
179C2×C7×D4—21no—
180C2×C7×D4—21no—
181C2×C7×D4—21no—
182C2×C7×D4—82no—
183C2×C7×D4—82no—
184C2×C7×D4C2×D411yescharacteristic
185C2×C7×D4C2×D411yescharacteristic
186C2×C7×D4C2×D411yescharacteristic
187C2×C7×D4C2×D412yes—
188C22×C4×C7—22no—
189C22×C4×C7—22no—
190C22×C4×C7—21no—
191C22×C4×C7—21no—
192C22×C4×C7—21no—
193C22×C4×C7—22no—
194C22×C4×C7—41no—
195C22×C4×C7C2×D411yescharacteristic
196C22×C4×C7C2×D411yescharacteristic
197C22×C4×C7C2×D411yescharacteristic
198C7×Q16—82no—
199C7×SD16—82no—
200C2×C7×C8—22no—
201C2×C7×C8—22no—
202C2×C7×C8—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—22no—
212C7×C4⋊C4—22no—
213C7×C4⋊C4—41no—
214C7×C4⋊C4—41no—
215C7×C4⋊C4C2×D411yescharacteristic
216C7×C4⋊C4C2×D411yescharacteristic
217C7×C4⋊C4C2×D412yes—
218C7×C4⋊C4C2×D411yescharacteristic
219C7×C4⋊C4C2×D411yescharacteristic
220C7×C22⋊C4—22no—
221C7×C22⋊C4—22no—
222C7×C22⋊C4—22no—
223C7×C22⋊C4—22no—
224C7×C22⋊C4—22no—
225C7×C22⋊C4—21no—
226C7×C22⋊C4—21no—
227C7×C22⋊C4—21no—
228C7×C22⋊C4—21no—
229C7×C22⋊C4—21no—
230C7×C22⋊C4—21no—
231C7×C22⋊C4—22no—
232C7×C22⋊C4—22no—
233C7×C22⋊C4—21no—
234C7×C22⋊C4—21no—
235C7×C22⋊C4—21no—
236C42×C7—22no—
237C42×C7—21no—
238C42×C7C2×D411yescharacteristic
239C42×C7C2×D411yescharacteristic
240C42×C7C2×D411yescharacteristic
241C42×C7C2411yescharacteristic
242#560—22no—
243#556C22×C711yescharacteristic
244#553—21no—
245#550—21no—
246#550C22×C711yescharacteristic
247#547C22×C711yescharacteristic
248#546—21no—
249#546—21no—
250D42C22×C711yescharacteristic
251#542—21no—
252#538—22no—
253#537—21no—
254#535—21no—
255#535—21no—
256#532C22×C711yescharacteristic
257#532C22×C711yescharacteristic
258C2×C4⋊Q8C22×C711yescharacteristic, complemented
259#529C22×C711yescharacteristic
260#500—21no—
261#498C22×C712yescomplemented
262#497—21no—
263#491—21no—
264#486—21no—
265#486C22×C712yes—
266#481—22no—
267#480—21no—
268#480—21no—
269#476—22no—
270#474C22×C712yes—
271#474C22×C712yescomplemented
272#462C22×C712yescomplemented
273#461C22×C712yes—
274#451—21no—
275#448—21no—
276#432C22×C711yescharacteristic, complemented
277#431C22×C711yescharacteristic, complemented
278#431C22×C711yescharacteristic, complemented
279#332C22×C712yescomplemented
280#331C22×C712yes—
281Q8⋊C8C22×C712yescomplemented
282D4⋊C8C22×C712yes—
283D4⋊C8C22×C712yes—
284D4⋊C8C22×C712yescomplemented
285C23×C7—21no—
286C23×C7—21no—
287C23×C7—22no—
288C23×C7—21no—
289C23×C7—82no—
290C23×C7—82no—
291C23×C7—82no—
292C23×C7#12011yescharacteristic
293C7×Q8—22no—
294C7×Q8—42no—
295C7×Q8—42no—
296C7×Q8—42no—
297C7×Q8—81no—
298C7×Q8—82no—
299C7×D4—22no—
300C7×D4—22no—
301C7×D4—22no—
302C7×D4—42no—
303C7×D4—42no—
304C7×D4—42no—
305C7×D4—81no—
306C7×D4—81no—
307C7×D4—81no—
308C7×D4—81no—
309C7×D4—81no—
310C7×D4—82no—
311C7×D4—82no—
312C7×D4—82no—
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—22no—
322C2×C4×C7—21no—
323C2×C4×C7—21no—
324C2×C4×C7—41no—
325C2×C4×C7—42no—
326C2×C4×C7—42no—
327C2×C4×C7—42no—
328C2×C4×C7—42no—
329C2×C4×C7—82no—
330C2×C4×C7—82no—
331C2×C4×C7—82no—
332C2×C4×C7—82no—
333C2×C4×C7#12011yescharacteristic
334C2×C4×C7#12011yescharacteristic
335C2×C4×C7#12011yescharacteristic
336C2×C4×C7C22×D411yescharacteristic
337C2×C4×C7C22×D411yescharacteristic
338C2×C4×C7C22×D411yescharacteristic
339C7×C8—42no—
340C7×C8—42no—
341C7×C8—42no—
342C2×#41—21no—
343C2×#41—21no—
344C2×#41—21no—
345C2×#41—21no—
346C22×Q8—21no—
347C22×D4—21no—
348C22×D4—21no—
349C2×Q16—22no—
350C2×SD16—22no—
351C4⋊Q8—21no—
352C4⋊Q8—21no—
353C4⋊Q8—21no—
354C4⋊Q8C7×D411yescharacteristic
355C4⋊Q8C23×C712yes—
356C4⋊Q8C23×C711yescharacteristic
357#127—22no—
358#127—21no—
359#127C7×D411yescharacteristic
360#127C23×C711yescharacteristic
361#126—22no—
362#126—22no—
363#126—21no—
364#125C7×D411yescharacteristic
365#124—22no—
366#124—21no—
367#124—21no—
368#124—21no—
369#124C7×D411yescharacteristic
370#123—22no—
371#123—22no—
372#123—21no—
373#123—22no—
374C22⋊Q8—22no—
375C22⋊Q8—21no—
376C22⋊Q8—21no—
377C22⋊Q8C7×D411yescharacteristic
378#121—22no—
379#121—22no—
380#121—22no—
381#121—22no—
382#121—22no—
383#121—21no—
384#121—22no—
385#121—21no—
386#121—21no—
387#121—21no—
388#121—21no—
389#121—21no—
390#121C7×D411yescharacteristic
391#121C7×D411yescharacteristic
392#121C7×D411yescharacteristic
393#121C7×D412yes—
394#120—22no—
395#120—22no—
396#120—22no—
397#120—22no—
398C4×Q8—21no—
399C4×Q8C23×C711yescharacteristic
400C4×D4—22no—
401C4×D4—21no—
402C4×D4—21no—
403C4×D4—21no—
404C4×D4—21no—
405C4×D4C7×D412yes—
406C4×D4C23×C711yescharacteristic
407C4×D4C23×C711yescharacteristic
408C4×D4C23×C711yescharacteristic
409#117—21no—
410#117—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C22⋊C4—22no—
415C2×C42C23×C711yescharacteristic
416#107—22no—
417#107—22no—
418#106—24no—
419C4⋊C8—21no—
420C4⋊C8C23×C712yes—
421C4⋊C8C23×C712yes—
422#103—22no—
423#103—22no—
424#102—22no—
425#102—22no—
426#102—24no—
427C22⋊C8—21no—
428C22⋊C8—21no—
429C22⋊C8—21no—
430#97—21no—
431C4×C8—21no—
432C4×C8C23×C712yes—
433C22×C7—21no—
434C22×C7—21no—
435C22×C7—21no—
436C22×C7—41no—
437C22×C7—41no—
438C22×C7—42no—
439C22×C7—41no—
440C22×C7—41no—
441C22×C7—41no—
442C22×C7—42no—
443C22×C7—41no—
444C22×C7—42no—
445C22×C7—41no—
446C22×C7—41no—
447C22×C7—41no—
448C22×C7—164no—
449C22×C7C2×#12011yescharacteristic
450C4×C7—21no—
451C4×C7—21no—
452C4×C7—21no—
453C4×C7—21no—
454C4×C7—41no—
455C4×C7—41no—
456C4×C7—41no—
457C4×C7—41no—
458C4×C7—42no—
459C4×C7—41no—
460C4×C7—82no—
461C4×C7#57612yes—
462C4×C7#57712yes—
463C24—22no—
464#41—82no—
465#41—82no—
466#41—82no—
467#41—82no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—22no—
471C2×Q8—41no—
472C2×Q8—82no—
473C2×Q8C2×C7×D411yescharacteristic
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—22no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—82no—
488C2×D4—82no—
489C2×D4C2×C7×D411yescharacteristic
490C2×D4C2×C7×D411yescharacteristic
491C2×D4C2×C7×D411yescharacteristic
492C2×D4C2×C7×D412yes—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—22no—
499C22×C4—41no—
500C22×C4C2×C7×D411yescharacteristic
501C22×C4C2×C7×D411yescharacteristic
502C22×C4C2×C7×D411yescharacteristic
503Q16—82no—
504SD16—82no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4C2×C7×D411yescharacteristic
521C4⋊C4C2×C7×D411yescharacteristic
522C4⋊C4C2×C7×D412yes—
523C4⋊C4C2×C7×D411yescharacteristic
524C4⋊C4C2×C7×D411yescharacteristic
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—21no—
533C22⋊C4—21no—
534C22⋊C4—21no—
535C22⋊C4—21no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C42—22no—
542C42—21no—
543C42C2×C7×D411yescharacteristic
544C42C2×C7×D411yescharacteristic
545C42C2×C7×D411yescharacteristic
546C42C24×C711yescharacteristic
547C2×C7—41no—
548C2×C7—81no—
549C2×C7—82no—
550C2×C7—81no—
551C2×C7—81no—
552C2×C7#300811yescharacteristic
553C2×C7#300911yescharacteristic
554C2×C7#302211yescharacteristic
555C23—21no—
556C23—21no—
557C23—22no—
558C23—21no—
559C23—82no—
560C23—82no—
561C23—82no—
562C23C7×#12011yescharacteristic
563Q8—22no—
564Q8—42no—
565Q8—42no—
566Q8—42no—
567Q8—81no—
568Q8—82no—
569D4—22no—
570D4—22no—
571D4—22no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—81no—
576D4—81no—
577D4—81no—
578D4—81no—
579D4—81no—
580D4—82no—
581D4—82no—
582D4—82no—
583D4—82no—
584D4—82no—
585D4—82no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4C7×#12011yescharacteristic
604C2×C4C7×#12011yescharacteristic
605C2×C4C7×#12011yescharacteristic
606C2×C4C22×C7×D411yescharacteristic
607C2×C4C22×C7×D411yescharacteristic
608C2×C4C22×C7×D411yescharacteristic
609C8—42no—
610C8—42no—
611C8—42no—
612C7#1896711yescharacteristic, direct product factor, Sylow
613C22—21no—
614C22—21no—
615C22—21no—
616C22—41no—
617C22—41no—
618C22—42no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—42no—
623C22—41no—
624C22—42no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—164no—
629C22C2×C7×#12011yescharacteristic
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—41no—
635C4—41no—
636C4—41no—
637C4—41no—
638C4—42no—
639C4—41no—
640C4—82no—
641C4C7×#57612yes—
642C4C7×#57712yes—
643C2—41no—
644C2—81no—
645C2—81no—
646C2—82no—
647C2—81no—
648C2C7×#300811yescharacteristic
649C2C7×#300911yescharacteristic
650C2C7×#302211yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3546C2
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×#3227C2semidirect
3. C7×#3226C2semidirect
4. C7×#3218C2semidirect
5. C7×#1654C2semidirect
6. C7×#1652C2semidirect
7. C7×#1641C2semidirect
8. C7×#1639C2semidirect
9. C7×#1553C2semidirect
10. C7×#1496C2semidirect
11. C7×#1482C2semidirect
28. C2×C7×C4⋊Q8C22semidirect
31. C7×#498C22semidirect
41. C7×#474C22semidirect
42. C7×#462C22semidirect
46. C7×#432C22semidirect
47. C7×#431C22semidirect
48. C7×#431C22semidirect
49. C7×#332C22semidirect
51. C7×Q8⋊C8C22semidirect
54. C7×D4⋊C8C22semidirect
55. #18967C7direct (N × H)
148. #3227C2×C7semidirect
149. #3226C2×C7semidirect
150. #3218C2×C7semidirect
151. #1654C2×C7semidirect
152. #1652C2×C7semidirect
153. #1641C2×C7semidirect
154. #1639C2×C7semidirect
155. #1553C2×C7semidirect
156. #1496C2×C7semidirect
157. #1482C2×C7semidirect
258. C2×C4⋊Q8C22×C7semidirect
261. #498C22×C7semidirect
271. #474C22×C7semidirect
272. #462C22×C7semidirect
276. #432C22×C7semidirect
277. #431C22×C7semidirect
278. #431C22×C7semidirect
279. #332C22×C7semidirect
281. Q8⋊C8C22×C7semidirect
284. D4⋊C8C22×C7semidirect
612. C7#18967direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35462
C7×#32272
C7×#32262
C7×#32182
C7×#16542
C7×#16522
C7×#16412
C7×#16392
C7×#15532
C7×#14962
C7×#14822
#189677