Finite Group Databasebeta

← C7×#18330 #13195546

Subgroups — C7×#18330 #13195546

Order 1792.

1068 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3460C211yesmaximal, characteristic, complemented
2C7×#3194C212yesmaximal, complemented
3C7×#2543C211yesmaximal, characteristic, complemented
4C7×#1612C212yesmaximal, complemented
5C7×#1611C211yesmaximal, characteristic, complemented
6C7×#1610C211yesmaximal, characteristic, complemented
7C7×#1600C212yesmaximal, complemented
8C7×#1599C211yesmaximal, characteristic, complemented
9C7×#1598C211yesmaximal, characteristic, complemented
10C7×#1504C212yesmaximal, complemented
11C7×#1465C211yesmaximal, characteristic, complemented
12C7×#544C2211yescharacteristic
13C7×#542—22no—
14C7×#540—22no—
15C7×#538—21no—
16C7×#538C2212yes—
17C7×#537C2211yescharacteristic
18C7×#535—21no—
19C7×#532—22no—
20C7×#529—22no—
21C7×#525—21no—
22C7×#525C2212yes—
23C2×C7×#121—21no—
24C2×C7×#121C2211yescharacteristic
25C4×C7×#41—21no—
26C2×C4×C7×D4C2211yescharacteristic
27C7×#483C2212yes—
28C7×#482—22no—
29C7×#480C2212yes—
30C7×#469—22no—
31C7×#468—22no—
32C7×#450C2212yes—
33C7×#449—22no—
34C7×#447C2212yes—
35C7×#407C2212yes—
36C7×#398C2211yescharacteristic
37C7×#397C2211yescharacteristic
38C7×#394C2211yescharacteristic
39C7×#392C2211yescharacteristic
40C7×#388—21no—
41C7×#386—21no—
42C7×#386—21no—
43C2×C7×#95C2211yescharacteristic
44C7×#328C2212yes—
45C7×#328C2212yes—
46C7×#327C2212yes—
47C7×#327C2212yes—
48C7×#324C2211yescharacteristic
49C7×#324C2211yescharacteristic
50C7×#323C2212yes—
51#18330C711yesmaximal, characteristic, direct product factor, Sylow
52C2×C7×#41—21no—
53C2×C7×#41—21no—
54C2×C7×#41D412yescomplemented
55C22×C7×D4—21no—
56C22×C7×D4—21no—
57C22×C7×D4D411yescharacteristic
58C22×C7×D4C2311yescharacteristic
59C23×C4×C7—21no—
60C23×C4×C7D411yescharacteristic
61C23×C4×C7C2311yescharacteristic
62C2×C7×SD16—22no—
63C2×C7×D8—22no—
64C2×C7×M16—22no—
65C7×C4⋊Q8—22no—
66C7×#127—22no—
67C7×#126—22no—
68C7×#126—22no—
69C7×#125—22no—
70C7×#124—22no—
71C7×#124—22no—
72C7×#124—22no—
73C7×#123—22no—
74C7×#123—22no—
75C7×#123—22no—
76C7×#123—22no—
77C7×C22⋊Q8—21no—
78C7×C22⋊Q8—21no—
79C7×C22⋊Q8—22no—
80C7×C22⋊Q8D412yescomplemented
81C7×C22⋊Q8C2312yes—
82C7×#121—21no—
83C7×#121—22no—
84C7×#121—22no—
85C7×#121—22no—
86C7×#121—22no—
87C7×#121—22no—
88C7×#121—21no—
89C7×#121—22no—
90C7×#121—22no—
91C7×#121—44no—
92C7×#121D412yescomplemented
93C7×#121C2312yes—
94C7×#120—22no—
95C7×#120—22no—
96C4×C7×Q8—21no—
97C4×C7×Q8—21no—
98C4×C7×D4—21no—
99C4×C7×D4—21no—
100C4×C7×D4—22no—
101C4×C7×D4—22no—
102C4×C7×D4—22no—
103C4×C7×D4—22no—
104C4×C7×D4—22no—
105C4×C7×D4—22no—
106C7×#117—22no—
107C7×#117—21no—
108C7×#117—21no—
109C7×#117D412yescomplemented
110C2×C7×C4⋊C4—21no—
111C2×C7×C4⋊C4C2311yescharacteristic
112C2×C7×C4⋊C4C2311yescharacteristic
113C2×C7×C22⋊C4—21no—
114C2×C7×C22⋊C4—21no—
115C2×C7×C22⋊C4—21no—
116C2×C7×C22⋊C4—21no—
117C2×C7×C22⋊C4—21no—
118C2×C7×C22⋊C4D411yescharacteristic
119C2×C7×C22⋊C4C2311yescharacteristic
120C2×C42×C7—22no—
121C2×C42×C7D411yescharacteristic
122C7×#107—22no—
123C7×#106—22no—
124C7×#103—22no—
125C7×#102—22no—
126C7×#102—22no—
127C7×#102—22no—
128C7×C22⋊C8—22no—
129C7×C22⋊C8C2312yes—
130C7×C22⋊C8C2312yes—
131C7×#95—21no—
132C7×#95—21no—
133C7×#95—21no—
134C7×#95C2311yescharacteristic
135C7×#95C2311yescharacteristic
136#3460C2×C711yescharacteristic, complemented
137#3194C2×C712yescomplemented
138#2543C2×C711yescharacteristic, complemented
139#1612C2×C712yescomplemented
140#1611C2×C711yescharacteristic, complemented
141#1610C2×C711yescharacteristic, complemented
142#1600C2×C712yescomplemented
143#1599C2×C711yescharacteristic, complemented
144#1598C2×C711yescharacteristic, complemented
145#1504C2×C712yescomplemented
146#1465C2×C711yescharacteristic, complemented
147C24×C7—21no—
148C24×C7C2×D411yescharacteristic
149C24×C7C2×D411yescharacteristic
150C7×#41—42no—
151C7×#41—42no—
152C7×#41—82no—
153C7×#41—82no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8C2×D412yes—
157C2×C7×D4—21no—
158C2×C7×D4—22no—
159C2×C7×D4—22no—
160C2×C7×D4—22no—
161C2×C7×D4—22no—
162C2×C7×D4—21no—
163C2×C7×D4—22no—
164C2×C7×D4—22no—
165C2×C7×D4—22no—
166C2×C7×D4—22no—
167C2×C7×D4—41no—
168C2×C7×D4—42no—
169C2×C7×D4—42no—
170C2×C7×D4—42no—
171C2×C7×D4—42no—
172C2×C7×D4—42no—
173C2×C7×D4—81no—
174C2×C7×D4—81no—
175C2×C7×D4C2×D412yes—
176C22×C4×C7—22no—
177C22×C4×C7—21no—
178C22×C4×C7—22no—
179C22×C4×C7—22no—
180C22×C4×C7—21no—
181C22×C4×C7—21no—
182C22×C4×C7—21no—
183C22×C4×C7—21no—
184C22×C4×C7—42no—
185C22×C4×C7—42no—
186C22×C4×C7—81no—
187C22×C4×C7—81no—
188C22×C4×C7—81no—
189C22×C4×C7C2×D411yescharacteristic
190C22×C4×C7C2×D411yescharacteristic
191C22×C4×C7C2×D411yescharacteristic
192C22×C4×C7C2×D411yescharacteristic
193C22×C4×C7C2×D412yes—
194C22×C4×C7C2×D412yes—
195C22×C4×C7C2411yescharacteristic
196C7×SD16—82no—
197C7×D8—82no—
198C7×M16—82no—
199C2×C7×C8—22no—
200C2×C7×C8—22no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—22no—
204C7×C4⋊C4—21no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—22no—
207C7×C4⋊C4—22no—
208C7×C4⋊C4—22no—
209C7×C4⋊C4—22no—
210C7×C4⋊C4—22no—
211C7×C4⋊C4—42no—
212C7×C4⋊C4C2×D412yes—
213C7×C4⋊C4C2×D412yes—
214C7×C22⋊C4—22no—
215C7×C22⋊C4—22no—
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—22no—
223C7×C22⋊C4—22no—
224C7×C22⋊C4—22no—
225C7×C22⋊C4—42no—
226C7×C22⋊C4—42no—
227C7×C22⋊C4—41no—
228C7×C22⋊C4—41no—
229C7×C22⋊C4—42no—
230C7×C22⋊C4—81no—
231C42×C7—22no—
232C42×C7—22no—
233C42×C7—21no—
234C42×C7—22no—
235C42×C7—21no—
236#544C22×C711yescharacteristic
237#542—22no—
238#540—22no—
239#538—21no—
240#538C22×C712yes—
241#537C22×C711yescharacteristic
242#535—21no—
243#532—22no—
244#529—22no—
245#525—21no—
246#525C22×C712yes—
247C2×#121—21no—
248C2×#121C22×C711yescharacteristic
249C4×#41—21no—
250C2×C4×D4C22×C711yescharacteristic
251#483C22×C712yes—
252#482—22no—
253#480C22×C712yes—
254#469—22no—
255#468—22no—
256#450C22×C712yes—
257#449—22no—
258#447C22×C712yes—
259#407C22×C712yes—
260#398C22×C711yescharacteristic
261#397C22×C711yescharacteristic
262#394C22×C711yescharacteristic
263#392C22×C711yescharacteristic
264#388—21no—
265#386—21no—
266#386—21no—
267C2×#95C22×C711yescharacteristic
268#328C22×C712yes—
269#328C22×C712yes—
270#327C22×C712yes—
271#327C22×C712yes—
272#324C22×C711yescharacteristic
273#324C22×C711yescharacteristic
274#323C22×C712yes—
275C23×C7—21no—
276C23×C7—21no—
277C23×C7—21no—
278C23×C7—22no—
279C23×C7—21no—
280C23×C7—21no—
281C23×C7—41no—
282C23×C7—41no—
283C23×C7—41no—
284C23×C7—41no—
285C23×C7—41no—
286C23×C7—41no—
287C23×C7—41no—
288C23×C7—41no—
289C23×C7—41no—
290C23×C7—41no—
291C23×C7—41no—
292C23×C7#12012yes—
293C23×C7C22×D411yescharacteristic
294C7×Q8—42no—
295C7×Q8—42no—
296C7×Q8—81no—
297C7×D4—44no—
298C7×D4—42no—
299C7×D4—42no—
300C7×D4—42no—
301C7×D4—42no—
302C7×D4—42no—
303C7×D4—42no—
304C7×D4—82no—
305C7×D4—82no—
306C7×D4—82no—
307C7×D4—81no—
308C7×D4—82no—
309C7×D4—84no—
310C7×D4—84no—
311C2×C4×C7—21no—
312C2×C4×C7—21no—
313C2×C4×C7—22no—
314C2×C4×C7—21no—
315C2×C4×C7—22no—
316C2×C4×C7—22no—
317C2×C4×C7—22no—
318C2×C4×C7—21no—
319C2×C4×C7—42no—
320C2×C4×C7—41no—
321C2×C4×C7—42no—
322C2×C4×C7—41no—
323C2×C4×C7—42no—
324C2×C4×C7—44no—
325C2×C4×C7—44no—
326C2×C4×C7—41no—
327C2×C4×C7—41no—
328C2×C4×C7—41no—
329C2×C4×C7—41no—
330C2×C4×C7—41no—
331C2×C4×C7—41no—
332C2×C4×C7—41no—
333C2×C4×C7—41no—
334C2×C4×C7—41no—
335C2×C4×C7—41no—
336C2×C4×C7—81no—
337C2×C4×C7—82no—
338C2×C4×C7—84no—
339C2×C4×C7#12011yescharacteristic
340C2×C4×C7#12011yescharacteristic
341C2×C4×C7AGL(1,ℤ/8)12yes—
342C2×C4×C7C22×D412yes—
343C7×C8—42no—
344C7×C8—42no—
345C2×#41—21no—
346C2×#41—21no—
347C2×#41C7×D412yescomplemented
348C22×D4—21no—
349C22×D4—21no—
350C22×D4C7×D411yescharacteristic
351C22×D4C23×C711yescharacteristic
352C23×C4—21no—
353C23×C4C7×D411yescharacteristic
354C23×C4C23×C711yescharacteristic
355C2×SD16—22no—
356C2×D8—22no—
357C2×M16—22no—
358C4⋊Q8—22no—
359#127—22no—
360#126—22no—
361#126—22no—
362#125—22no—
363#124—22no—
364#124—22no—
365#124—22no—
366#123—22no—
367#123—22no—
368#123—22no—
369#123—22no—
370C22⋊Q8—21no—
371C22⋊Q8—21no—
372C22⋊Q8—22no—
373C22⋊Q8C7×D412yescomplemented
374C22⋊Q8C23×C712yes—
375#121—21no—
376#121—22no—
377#121—22no—
378#121—22no—
379#121—22no—
380#121—22no—
381#121—21no—
382#121—22no—
383#121—22no—
384#121—44no—
385#121C7×D412yescomplemented
386#121C23×C712yes—
387#120—22no—
388#120—22no—
389C4×Q8—21no—
390C4×Q8—21no—
391C4×D4—21no—
392C4×D4—21no—
393C4×D4—22no—
394C4×D4—22no—
395C4×D4—22no—
396C4×D4—22no—
397C4×D4—22no—
398C4×D4—22no—
399#117—22no—
400#117—21no—
401#117—21no—
402#117C7×D412yescomplemented
403C2×C4⋊C4—21no—
404C2×C4⋊C4C23×C711yescharacteristic
405C2×C4⋊C4C23×C711yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4C7×D411yescharacteristic
412C2×C22⋊C4C23×C711yescharacteristic
413C2×C42—22no—
414C2×C42C7×D411yescharacteristic
415#107—22no—
416#106—22no—
417#103—22no—
418#102—22no—
419#102—22no—
420#102—22no—
421C22⋊C8—22no—
422C22⋊C8C23×C712yes—
423C22⋊C8C23×C712yes—
424#95—21no—
425#95—21no—
426#95—21no—
427#95C23×C711yescharacteristic
428#95C23×C711yescharacteristic
429C22×C7—21no—
430C22×C7—21no—
431C22×C7—22no—
432C22×C7—22no—
433C22×C7—22no—
434C22×C7—42no—
435C22×C7—42no—
436C22×C7—42no—
437C22×C7—42no—
438C22×C7—41no—
439C22×C7—42no—
440C22×C7—41no—
441C22×C7—41no—
442C22×C7—42no—
443C22×C7—81no—
444C22×C7—81no—
445C22×C7—81no—
446C22×C7—81no—
447C22×C7—81no—
448C22×C7—81no—
449C22×C7—81no—
450C22×C7—81no—
451C22×C7C2×#12011yescharacteristic
452C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
453C22×C7#57511yescharacteristic
454C4×C7—22no—
455C4×C7—22no—
456C4×C7—21no—
457C4×C7—21no—
458C4×C7—42no—
459C4×C7—42no—
460C4×C7—41no—
461C4×C7—42no—
462C4×C7—41no—
463C4×C7—42no—
464C4×C7—81no—
465C4×C7—81no—
466C24—21no—
467C24C2×C7×D411yescharacteristic
468C24C2×C7×D411yescharacteristic
469#41—42no—
470#41—42no—
471#41—82no—
472#41—82no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8C2×C7×D412yes—
476C2×D4—21no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—21no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—41no—
487C2×D4—42no—
488C2×D4—42no—
489C2×D4—42no—
490C2×D4—42no—
491C2×D4—42no—
492C2×D4—81no—
493C2×D4—81no—
494C2×D4C2×C7×D412yes—
495C22×C4—22no—
496C22×C4—21no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—81no—
506C22×C4—81no—
507C22×C4—81no—
508C22×C4C2×C7×D411yescharacteristic
509C22×C4C2×C7×D411yescharacteristic
510C22×C4C2×C7×D411yescharacteristic
511C22×C4C2×C7×D411yescharacteristic
512C22×C4C2×C7×D412yes—
513C22×C4C2×C7×D412yes—
514C22×C4C24×C711yescharacteristic
515SD16—82no—
516D8—82no—
517M16—82no—
518C2×C8—22no—
519C2×C8—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—42no—
531C4⋊C4C2×C7×D412yes—
532C4⋊C4C2×C7×D412yes—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—41no—
547C22⋊C4—41no—
548C22⋊C4—42no—
549C22⋊C4—81no—
550C42—22no—
551C42—22no—
552C42—21no—
553C42—22no—
554C42—21no—
555C2×C7—21no—
556C2×C7—21no—
557C2×C7—42no—
558C2×C7—82no—
559C2×C7—81no—
560C2×C7—161no—
561C2×C7#300611yescharacteristic
562C2×C7#301811yescharacteristic
563C2×C7#302611yescharacteristic
564C23—21no—
565C23—21no—
566C23—21no—
567C23—22no—
568C23—21no—
569C23—21no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23C7×#12012yes—
582C23C22×C7×D411yescharacteristic
583Q8—42no—
584Q8—42no—
585Q8—81no—
586D4—44no—
587D4—42no—
588D4—42no—
589D4—42no—
590D4—42no—
591D4—42no—
592D4—42no—
593D4—82no—
594D4—82no—
595D4—82no—
596D4—81no—
597D4—82no—
598D4—84no—
599D4—84no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—81no—
626C2×C4—82no—
627C2×C4—84no—
628C2×C4C7×#12011yescharacteristic
629C2×C4C7×#12011yescharacteristic
630C2×C4C7×AGL(1,ℤ/8)12yes—
631C2×C4C22×C7×D412yes—
632C8—42no—
633C8—42no—
634C7#1833011yescharacteristic, direct product factor, Sylow
635C22—21no—
636C22—21no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—41no—
645C22—42no—
646C22—41no—
647C22—41no—
648C22—42no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22C2×C7×#12011yescharacteristic
658C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
659C22C7×#57511yescharacteristic
660C4—22no—
661C4—22no—
662C4—21no—
663C4—21no—
664C4—42no—
665C4—42no—
666C4—41no—
667C4—42no—
668C4—41no—
669C4—42no—
670C4—81no—
671C4—81no—
672C2—21no—
673C2—21no—
674C2—42no—
675C2—81no—
676C2—82no—
677C2—161no—
678C2C7×#300611yescharacteristic
679C2C7×#301811yescharacteristic
680C2C7×#302611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C7×#3460C2semidirect
2. C7×#3194C2semidirect
3. C7×#2543C2semidirect
4. C7×#1612C2semidirect
5. C7×#1611C2semidirect
6. C7×#1610C2semidirect
7. C7×#1600C2semidirect
8. C7×#1599C2semidirect
9. C7×#1598C2semidirect
10. C7×#1504C2semidirect
11. C7×#1465C2semidirect
51. #18330C7direct (N × H)
54. C2×C7×#41D4semidirect
80. C7×C22⋊Q8D4semidirect
92. C7×#121D4semidirect
109. C7×#117D4semidirect
136. #3460C2×C7semidirect
137. #3194C2×C7semidirect
138. #2543C2×C7semidirect
139. #1612C2×C7semidirect
140. #1611C2×C7semidirect
141. #1610C2×C7semidirect
142. #1600C2×C7semidirect
143. #1599C2×C7semidirect
144. #1598C2×C7semidirect
145. #1504C2×C7semidirect
146. #1465C2×C7semidirect
347. C2×#41C7×D4semidirect
373. C22⋊Q8C7×D4semidirect
385. #121C7×D4semidirect
402. #117C7×D4semidirect
634. C7#18330direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34602
C7×#31942
C7×#25432
C7×#16122
C7×#16112
C7×#16102
C7×#16002
C7×#15992
C7×#15982
C7×#15042
C7×#14652
#183307