Finite Group Databasebeta

← C7×#18270 #13195486

Subgroups — C7×#18270 #13195486

Order 1792.

1000 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3536C211yesmaximal, characteristic
2C7×#3092C211yesmaximal, characteristic, complemented
3C2×C7×#480C211yesmaximal, characteristic, complemented
4C7×#2502C211yesmaximal, characteristic, complemented
5C7×#1610C214yesmaximal, complemented
6C7×#1609C214yesmaximal, complemented
7C7×#1507C211yesmaximal, characteristic, complemented
8C7×#1501C211yesmaximal, characteristic, complemented
9C7×#1472C211yesmaximal, characteristic, complemented
10C7×#562—22no—
11C7×#554—21no—
12C7×#548—21no—
13C7×#547—21no—
14C7×#546—21no—
15C7×#546—22no—
16C7×#543—21no—
17C7×#542—21no—
18C7×#539C2212yes—
19C7×#538—21no—
20C7×#537—21no—
21C7×#537C2212yes—
22C7×#525C2211yescharacteristic
23C2×C7×C22⋊Q8C2211yescharacteristic
24C2×C7×#121C2211yescharacteristic
25C22×C7×C4⋊C4C2211yescharacteristic
26C7×#484—21no—
27C7×#483C2212yescomplemented
28C7×#481—21no—
29C7×#480—22no—
30C7×#480C2214yes—
31C7×#480C2212yescomplemented
32C7×#428—21no—
33C2×C7×#107—21no—
34C7×#417—21no—
35C2×C7×#102—21no—
36C7×#407C2211yescharacteristic
37C2×C7×C22⋊C8C2211yescharacteristic
38C7×#398C2214yescomplemented
39C7×#391—21no—
40C7×#385—22no—
41C7×#384—21no—
42C4×C7×C22⋊C4C2211yescharacteristic
43C7×#328C2212yescomplemented
44C7×#327C2214yes—
45C7×#327C2212yescomplemented
46C7×#324C2214yescomplemented
47#18270C711yesmaximal, characteristic, direct product factor, Sylow
48C2×C7×#41—21no—
49C2×C7×#41—22no—
50C2×C7×#41D411yescharacteristic
51C22×C7×Q8D411yescharacteristic
52C22×C7×D4—21no—
53C22×C7×D4D411yescharacteristic
54C23×C4×C7—21no—
55C23×C4×C7C2311yescharacteristic
56C2×C7×M16—21no—
57C22×C7×C8—21no—
58C7×C4⋊Q8—21no—
59C7×#126—22no—
60C7×#125—22no—
61C7×#124—22no—
62C7×#124—21no—
63C7×#124—22no—
64C7×#123—22no—
65C7×#123—22no—
66C7×#123—22no—
67C7×#123—21no—
68C7×#123—22no—
69C7×C22⋊Q8—21no—
70C7×C22⋊Q8—22no—
71C7×C22⋊Q8—22no—
72C7×C22⋊Q8—22no—
73C7×C22⋊Q8—22no—
74C7×C22⋊Q8—21no—
75C7×C22⋊Q8—21no—
76C7×C22⋊Q8—21no—
77C7×C22⋊Q8—21no—
78C7×C22⋊Q8—22no—
79C7×C22⋊Q8—22no—
80C7×C22⋊Q8D411yescharacteristic
81C7×C22⋊Q8C2311yescharacteristic
82C7×#121—21no—
83C7×#121—22no—
84C7×#121—22no—
85C7×#121—22no—
86C7×#121—22no—
87C7×#121—21no—
88C7×#121—21no—
89C7×#121—22no—
90C7×#121D412yes—
91C7×#121D411yescharacteristic
92C7×#121C2312yes—
93C7×#121C2311yescharacteristic
94C7×#120—22no—
95C7×#120—21no—
96C4×C7×Q8—21no—
97C4×C7×D4—22no—
98C4×C7×D4—21no—
99C4×C7×D4—21no—
100C4×C7×D4—21no—
101C4×C7×D4—22no—
102C7×#117D411yescharacteristic
103C2×C7×C4⋊C4—21no—
104C2×C7×C4⋊C4—21no—
105C2×C7×C4⋊C4—21no—
106C2×C7×C4⋊C4—21no—
107C2×C7×C4⋊C4—21no—
108C2×C7×C4⋊C4—21no—
109C2×C7×C4⋊C4—24no—
110C2×C7×C4⋊C4D411yescharacteristic
111C2×C7×C4⋊C4D411yescharacteristic
112C2×C7×C4⋊C4C2312yes—
113C2×C7×C4⋊C4C2312yes—
114C2×C7×C22⋊C4—22no—
115C2×C7×C22⋊C4—21no—
116C2×C7×C22⋊C4—21no—
117C2×C7×C22⋊C4D412yes—
118C2×C42×C7—21no—
119C7×#107—24no—
120C7×#107—22no—
121C7×#106—22no—
122C7×#103—22no—
123C7×#102—24no—
124C7×#102—22no—
125C7×C22⋊C8—21no—
126C7×C22⋊C8—21no—
127C7×C22⋊C8C2312yes—
128C7×C22⋊C8C2312yes—
129C7×#95—24no—
130C7×#95C2312yes—
131#3536C2×C711yescharacteristic
132#3092C2×C711yescharacteristic, complemented
133C2×#480C2×C711yescharacteristic, complemented
134#2502C2×C711yescharacteristic, complemented
135#1610C2×C714yescomplemented
136#1609C2×C714yescomplemented
137#1507C2×C711yescharacteristic, complemented
138#1501C2×C711yescharacteristic, complemented
139#1472C2×C711yescharacteristic, complemented
140C24×C7—21no—
141C24×C7C2×D411yescharacteristic
142C7×#41—42no—
143C7×#41—84no—
144C7×#41—82no—
145C2×C7×Q8—22no—
146C2×C7×Q8—21no—
147C2×C7×Q8—21no—
148C2×C7×Q8—42no—
149C2×C7×Q8C2×D411yescharacteristic
150C2×C7×D4—21no—
151C2×C7×D4—21no—
152C2×C7×D4—22no—
153C2×C7×D4—22no—
154C2×C7×D4—22no—
155C2×C7×D4—21no—
156C2×C7×D4—22no—
157C2×C7×D4—21no—
158C2×C7×D4—42no—
159C2×C7×D4—82no—
160C2×C7×D4C2×D411yescharacteristic
161C2×C7×D4C2×D412yes—
162C22×C4×C7—22no—
163C22×C4×C7—22no—
164C22×C4×C7—22no—
165C22×C4×C7—22no—
166C22×C4×C7—21no—
167C22×C4×C7—21no—
168C22×C4×C7—22no—
169C22×C4×C7—42no—
170C22×C4×C7—41no—
171C22×C4×C7—41no—
172C22×C4×C7—82no—
173C22×C4×C7C2×D412yes—
174C22×C4×C7C2×D411yescharacteristic
175C22×C4×C7C2×D411yescharacteristic
176C22×C4×C7C2×D411yescharacteristic
177C22×C4×C7C2×D412yes—
178C22×C4×C7C2×D411yescharacteristic
179C22×C4×C7C2×D411yescharacteristic
180C22×C4×C7C2411yescharacteristic
181C7×M16—42no—
182C2×C7×C8—22no—
183C2×C7×C8—22no—
184C2×C7×C8—42no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4—41no—
199C7×C4⋊C4—41no—
200C7×C4⋊C4—84no—
201C7×C4⋊C4C2×D411yescharacteristic
202C7×C4⋊C4C2×D412yes—
203C7×C4⋊C4C2×D411yescharacteristic
204C7×C22⋊C4—24no—
205C7×C22⋊C4—22no—
206C7×C22⋊C4—22no—
207C7×C22⋊C4—22no—
208C7×C22⋊C4—22no—
209C7×C22⋊C4—22no—
210C7×C22⋊C4—21no—
211C7×C22⋊C4—22no—
212C7×C22⋊C4—22no—
213C7×C22⋊C4—21no—
214C7×C22⋊C4—21no—
215C7×C22⋊C4—22no—
216C7×C22⋊C4—21no—
217C7×C22⋊C4—22no—
218C7×C22⋊C4—21no—
219C42×C7—22no—
220C42×C7—21no—
221C42×C7—42no—
222#562—22no—
223#554—21no—
224#548—21no—
225#547—21no—
226#546—21no—
227#546—22no—
228#543—21no—
229#542—21no—
230#539C22×C712yes—
231#538—21no—
232#537—21no—
233#537C22×C712yes—
234#525C22×C711yescharacteristic
235C2×C22⋊Q8C22×C711yescharacteristic
236C2×#121C22×C711yescharacteristic
237C22×C4⋊C4C22×C711yescharacteristic
238#484—21no—
239#483C22×C712yescomplemented
240#481—21no—
241#480—22no—
242#480C22×C714yes—
243#480C22×C712yescomplemented
244#428—21no—
245C2×#107—21no—
246#417—21no—
247C2×#102—21no—
248#407C22×C711yescharacteristic
249C2×C22⋊C8C22×C711yescharacteristic
250#398C22×C714yescomplemented
251#391—21no—
252#385—22no—
253#384—21no—
254C4×C22⋊C4C22×C711yescharacteristic
255#328C22×C712yescomplemented
256#327C22×C714yes—
257#327C22×C712yescomplemented
258#324C22×C714yescomplemented
259C23×C7—22no—
260C23×C7—22no—
261C23×C7—22no—
262C23×C7—21no—
263C23×C7—41no—
264C23×C7—42no—
265C23×C7—81no—
266C23×C7—81no—
267C23×C7#12011yescharacteristic
268C23×C7#12011yescharacteristic
269C23×C7C22×D411yescharacteristic
270C7×Q8—41no—
271C7×Q8—42no—
272C7×Q8—81no—
273C7×Q8—82no—
274C7×D4—42no—
275C7×D4—41no—
276C7×D4—44no—
277C7×D4—42no—
278C7×D4—42no—
279C7×D4—42no—
280C7×D4—82no—
281C7×D4—81no—
282C7×D4—82no—
283C7×D4—82no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
287C2×C4×C7—24no—
288C2×C4×C7—21no—
289C2×C4×C7—22no—
290C2×C4×C7—21no—
291C2×C4×C7—22no—
292C2×C4×C7—21no—
293C2×C4×C7—22no—
294C2×C4×C7—24no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—42no—
299C2×C4×C7—42no—
300C2×C4×C7—42no—
301C2×C4×C7—42no—
302C2×C4×C7—42no—
303C2×C4×C7—44no—
304C2×C4×C7—81no—
305C2×C4×C7—82no—
306C2×C4×C7—82no—
307C2×C4×C7—82no—
308C2×C4×C7—82no—
309C2×C4×C7—82no—
310C2×C4×C7#12011yescharacteristic
311C2×C4×C7#12011yescharacteristic
312C2×C4×C7C22×D411yescharacteristic
313C2×C4×C7C22×D411yescharacteristic
314C7×C8—42no—
315C7×C8—42no—
316C2×#41—21no—
317C2×#41—22no—
318C2×#41C7×D411yescharacteristic
319C22×Q8C7×D411yescharacteristic
320C22×D4—21no—
321C22×D4C7×D411yescharacteristic
322C23×C4—21no—
323C23×C4C23×C711yescharacteristic
324C2×M16—21no—
325C22×C8—21no—
326C4⋊Q8—21no—
327#126—22no—
328#125—22no—
329#124—22no—
330#124—21no—
331#124—22no—
332#123—22no—
333#123—22no—
334#123—22no—
335#123—21no—
336#123—22no—
337C22⋊Q8—21no—
338C22⋊Q8—22no—
339C22⋊Q8—22no—
340C22⋊Q8—22no—
341C22⋊Q8—22no—
342C22⋊Q8—21no—
343C22⋊Q8—21no—
344C22⋊Q8—21no—
345C22⋊Q8—21no—
346C22⋊Q8—22no—
347C22⋊Q8—22no—
348C22⋊Q8C7×D411yescharacteristic
349C22⋊Q8C23×C711yescharacteristic
350#121—21no—
351#121—22no—
352#121—22no—
353#121—22no—
354#121—22no—
355#121—21no—
356#121—21no—
357#121—22no—
358#121C7×D412yes—
359#121C7×D411yescharacteristic
360#121C23×C712yes—
361#121C23×C711yescharacteristic
362#120—22no—
363#120—21no—
364C4×Q8—21no—
365C4×D4—22no—
366C4×D4—21no—
367C4×D4—21no—
368C4×D4—21no—
369C4×D4—22no—
370#117C7×D411yescharacteristic
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—24no—
378C2×C4⋊C4C7×D411yescharacteristic
379C2×C4⋊C4C7×D411yescharacteristic
380C2×C4⋊C4C23×C712yes—
381C2×C4⋊C4C23×C712yes—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4C7×D412yes—
386C2×C42—21no—
387#107—24no—
388#107—22no—
389#106—22no—
390#103—22no—
391#102—24no—
392#102—22no—
393C22⋊C8—21no—
394C22⋊C8—21no—
395C22⋊C8C23×C712yes—
396C22⋊C8C23×C712yes—
397#95—24no—
398#95C23×C712yes—
399C22×C7—21no—
400C22×C7—21no—
401C22×C7—21no—
402C22×C7—21no—
403C22×C7—21no—
404C22×C7—21no—
405C22×C7—21no—
406C22×C7—44no—
407C22×C7—41no—
408C22×C7—42no—
409C22×C7—41no—
410C22×C7—42no—
411C22×C7—41no—
412C22×C7—42no—
413C22×C7—162no—
414C22×C7#45612yes—
415C22×C7C2×#12011yescharacteristic
416C22×C7#57612yes—
417C4×C7—21no—
418C4×C7—21no—
419C4×C7—21no—
420C4×C7—44no—
421C4×C7—42no—
422C4×C7—42no—
423C4×C7—41no—
424C4×C7—41no—
425C4×C7—41no—
426C4×C7—42no—
427C4×C7—82no—
428C4×C7#45812yes—
429C24—21no—
430C24C2×C7×D411yescharacteristic
431#41—42no—
432#41—84no—
433#41—82no—
434C2×Q8—22no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—42no—
438C2×Q8C2×C7×D411yescharacteristic
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—21no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—42no—
448C2×D4—82no—
449C2×D4C2×C7×D411yescharacteristic
450C2×D4C2×C7×D412yes—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—42no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—82no—
462C22×C4C2×C7×D412yes—
463C22×C4C2×C7×D411yescharacteristic
464C22×C4C2×C7×D411yescharacteristic
465C22×C4C2×C7×D411yescharacteristic
466C22×C4C2×C7×D412yes—
467C22×C4C2×C7×D411yescharacteristic
468C22×C4C2×C7×D411yescharacteristic
469C22×C4C24×C711yescharacteristic
470M16—42no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—42no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—84no—
490C4⋊C4C2×C7×D411yescharacteristic
491C4⋊C4C2×C7×D412yes—
492C4⋊C4C2×C7×D411yescharacteristic
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—24no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—21no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—22no—
505C22⋊C4—21no—
506C22⋊C4—22no—
507C22⋊C4—21no—
508C42—22no—
509C42—21no—
510C42—42no—
511C2×C7—22no—
512C2×C7—22no—
513C2×C7—41no—
514C2×C7—81no—
515C2×C7—82no—
516C2×C7#300811yescharacteristic
517C2×C7C2×#45611yescharacteristic
518C2×C7C2×#45811yescharacteristic
519C23—22no—
520C23—22no—
521C23—22no—
522C23—21no—
523C23—41no—
524C23—42no—
525C23—81no—
526C23—81no—
527C23C7×#12011yescharacteristic
528C23C7×#12011yescharacteristic
529C23C22×C7×D411yescharacteristic
530Q8—41no—
531Q8—42no—
532Q8—81no—
533Q8—82no—
534D4—42no—
535D4—41no—
536D4—44no—
537D4—42no—
538D4—42no—
539D4—42no—
540D4—82no—
541D4—81no—
542D4—82no—
543D4—82no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—81no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4C7×#12011yescharacteristic
571C2×C4C7×#12011yescharacteristic
572C2×C4C22×C7×D411yescharacteristic
573C2×C4C22×C7×D411yescharacteristic
574C8—42no—
575C8—42no—
576C7#1827011yescharacteristic, direct product factor, Sylow
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—44no—
585C22—41no—
586C22—42no—
587C22—41no—
588C22—42no—
589C22—41no—
590C22—42no—
591C22—162no—
592C22C7×#45612yes—
593C22C2×C7×#12011yescharacteristic
594C22C7×#57612yes—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—44no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—42no—
605C4—82no—
606C4C7×#45812yes—
607C2—22no—
608C2—22no—
609C2—41no—
610C2—82no—
611C2—81no—
612C2C7×#300811yescharacteristic
613C2C2×C7×#45611yescharacteristic
614C2C2×C7×#45811yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3536C2
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
2. C7×#3092C2semidirect
3. C2×C7×#480C2semidirect
4. C7×#2502C2semidirect
5. C7×#1610C2semidirect
6. C7×#1609C2semidirect
7. C7×#1507C2semidirect
8. C7×#1501C2semidirect
9. C7×#1472C2semidirect
27. C7×#483C22semidirect
31. C7×#480C22semidirect
38. C7×#398C22semidirect
43. C7×#328C22semidirect
45. C7×#327C22semidirect
46. C7×#324C22semidirect
47. #18270C7direct (N × H)
132. #3092C2×C7semidirect
133. C2×#480C2×C7semidirect
134. #2502C2×C7semidirect
135. #1610C2×C7semidirect
136. #1609C2×C7semidirect
137. #1507C2×C7semidirect
138. #1501C2×C7semidirect
139. #1472C2×C7semidirect
239. #483C22×C7semidirect
243. #480C22×C7semidirect
250. #398C22×C7semidirect
255. #328C22×C7semidirect
257. #327C22×C7semidirect
258. #324C22×C7semidirect
576. C7#18270direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35362
C7×#30922
C2×C7×#4802
C7×#25022
C7×#16102
C7×#16092
C7×#15072
C7×#15012
C7×#14722
#182707