Finite Group Databasebeta

← C7×#19049 #13196265

Subgroups — C7×#19049 #13196265

Order 1792.

886 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3561C211yesmaximal, characteristic
2C7×#3215C211yesmaximal, characteristic, complemented
3C7×#3181C211yesmaximal, characteristic, complemented
4C7×#3162C211yesmaximal, characteristic, complemented
5C7×#1664C212yesmaximal, complemented
6C7×#1661C212yesmaximal, complemented
7C7×#1661C212yesmaximal, complemented
8C7×#1660C212yesmaximal, complemented
9C7×#1532C211yesmaximal, characteristic, complemented
10C7×#1531C211yesmaximal, characteristic, complemented
11C2×C7×#329C211yesmaximal, characteristic, complemented
12C7×#563C2212yes—
13C7×#562C2212yes—
14C7×#556—21no—
15C7×#551—22no—
16C7×D4×Q8—21no—
17C7×#548—21no—
18C7×#546—21no—
19C7×#543—21no—
20C7×#542—21no—
21C7×#542—22no—
22C7×#538—21no—
23C7×#536—21no—
24C7×#533C2211yescharacteristic
25C7×#533C2211yescharacteristic
26C7×#529C2211yescharacteristic
27C2×C7×#124C2211yescharacteristic
28C7×#501—21no—
29C7×#501—21no—
30C7×#501C2212yescomplemented
31C7×#501C2212yescomplemented
32C7×#501C2211yescharacteristic, complemented
33C7×#497C2212yes—
34C7×#497C2211yescharacteristic, complemented
35C7×#491C2211yescharacteristic, complemented
36C7×#489C2212yes—
37C7×#489C2211yescharacteristic, complemented
38C7×#467—21no—
39C7×#465—21no—
40C7×#464—22no—
41C7×#451—21no—
42C7×#450—21no—
43C7×#441—21no—
44C7×#440—21no—
45C7×#432C2211yescharacteristic
46C7×#427—22no—
47C2×C7×#97C2211yescharacteristic
48C2×C7×#97C2211yescharacteristic
49C7×#330C2212yescomplemented
50C7×#330C2212yescomplemented
51C7×#330C2212yescomplemented
52C7×#329C2212yes—
53C7×#329C2212yes—
54C7×#329C2212yescomplemented
55#19049C711yesmaximal, characteristic, direct product factor, Sylow
56C2×C7×#41—21no—
57C22×C7×Q8—21no—
58C22×C7×Q8D411yescharacteristic
59C22×C7×D4D411yescharacteristic
60C2×C7×Q16—21no—
61C2×C7×Q16—22no—
62C2×C7×SD16—21no—
63C2×C7×SD16—22no—
64C22×C7×C8—21no—
65C22×C7×C8—21no—
66C7×C4⋊Q8—21no—
67C7×C4⋊Q8—22no—
68C7×C4⋊Q8D411yescharacteristic
69C7×C4⋊Q8D411yescharacteristic
70C7×C4⋊Q8C2311yescharacteristic
71C7×C4⋊Q8C2311yescharacteristic
72C7×#126—22no—
73C7×#126—22no—
74C7×#126—22no—
75C7×#126—21no—
76C7×#125—21no—
77C7×#125C2311yescharacteristic
78C7×#125C2312yes—
79C7×#124—21no—
80C7×#124—22no—
81C7×#124—22no—
82C7×#124—21no—
83C7×#124—21no—
84C7×#124D412yes—
85C7×#124C2312yes—
86C7×#124C2311yescharacteristic
87C7×#123—22no—
88C7×#123—22no—
89C7×#123—21no—
90C7×#123—22no—
91C7×#123—22no—
92C7×C22⋊Q8—22no—
93C7×C22⋊Q8—22no—
94C7×C22⋊Q8—21no—
95C7×C22⋊Q8—22no—
96C7×C22⋊Q8—22no—
97C7×C22⋊Q8—22no—
98C7×C22⋊Q8—21no—
99C7×C22⋊Q8—21no—
100C7×C22⋊Q8—22no—
101C7×C22⋊Q8—22no—
102C7×C22⋊Q8—21no—
103C7×C22⋊Q8—21no—
104C7×C22⋊Q8D411yescharacteristic
105C7×#121—21no—
106C7×#121—22no—
107C7×#121—22no—
108C7×#121D411yescharacteristic
109C7×#120—22no—
110C4×C7×Q8—21no—
111C4×C7×Q8—21no—
112C4×C7×Q8—21no—
113C4×C7×Q8—21no—
114C4×C7×Q8—21no—
115C4×C7×Q8D411yescharacteristic
116C4×C7×D4—22no—
117C4×C7×D4—21no—
118C4×C7×D4—22no—
119C4×C7×D4—21no—
120C4×C7×D4D411yescharacteristic
121C7×#117—22no—
122C7×#117—21no—
123C7×#117—21no—
124C7×#117D412yes—
125C2×C7×C22⋊C4—22no—
126C2×C7×C22⋊C4—21no—
127C2×C7×C22⋊C4—21no—
128C2×C42×C7C2311yescharacteristic
129C7×#107—21no—
130C7×#107—22no—
131C7×#107—22no—
132C7×#106—22no—
133C7×#106—22no—
134C7×#106—21no—
135C7×C4⋊C8—21no—
136C7×#103—21no—
137C7×#103—22no—
138C7×#103—21no—
139C7×#103—21no—
140C7×#102—22no—
141C7×#102—21no—
142C7×C22⋊C8—21no—
143C7×#97—21no—
144C7×#97—21no—
145C7×#97C2312yes—
146C7×#97C2311yescharacteristic
147C7×#97C2312yes—
148C7×#97C2311yescharacteristic
149#3561C2×C711yescharacteristic
150#3215C2×C711yescharacteristic, complemented
151#3181C2×C711yescharacteristic, complemented
152#3162C2×C711yescharacteristic, complemented
153#1664C2×C712yescomplemented
154#1661C2×C712yescomplemented
155#1661C2×C712yescomplemented
156#1660C2×C712yescomplemented
157#1532C2×C711yescharacteristic, complemented
158#1531C2×C711yescharacteristic, complemented
159C2×#329C2×C711yescharacteristic, complemented
160C24×C7—21no—
161C7×#41—82no—
162C2×C7×Q8—22no—
163C2×C7×Q8—21no—
164C2×C7×Q8—21no—
165C2×C7×Q8—21no—
166C2×C7×Q8—21no—
167C2×C7×Q8—21no—
168C2×C7×Q8—82no—
169C2×C7×Q8—81no—
170C2×C7×Q8C2×D411yescharacteristic
171C2×C7×Q8C2×D412yes—
172C2×C7×D4—21no—
173C2×C7×D4—22no—
174C2×C7×D4—22no—
175C2×C7×D4—21no—
176C2×C7×D4—81no—
177C2×C7×D4C2×D411yescharacteristic
178C2×C7×D4C2×D412yes—
179C22×C4×C7—22no—
180C22×C4×C7—22no—
181C22×C4×C7—22no—
182C22×C4×C7—21no—
183C22×C4×C7—21no—
184C22×C4×C7C2×D411yescharacteristic
185C22×C4×C7C2×D411yescharacteristic
186C22×C4×C7C2×D411yescharacteristic
187C7×Q16—42no—
188C7×Q16—82no—
189C7×SD16—42no—
190C7×SD16—82no—
191C2×C7×C8—24no—
192C2×C7×C8—22no—
193C2×C7×C8—21no—
194C2×C7×C8—21no—
195C2×C7×C8—22no—
196C2×C7×C8—42no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4—22no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—22no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—21no—
204C7×C4⋊C4—21no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—21no—
207C7×C4⋊C4—22no—
208C7×C4⋊C4—22no—
209C7×C4⋊C4—21no—
210C7×C4⋊C4—21no—
211C7×C4⋊C4—21no—
212C7×C4⋊C4—21no—
213C7×C4⋊C4—41no—
214C7×C4⋊C4—41no—
215C7×C4⋊C4C2×D411yescharacteristic
216C7×C4⋊C4C2×D411yescharacteristic
217C7×C4⋊C4C2×D412yes—
218C7×C4⋊C4C2×D412yes—
219C7×C22⋊C4—22no—
220C7×C22⋊C4—22no—
221C7×C22⋊C4—22no—
222C7×C22⋊C4—22no—
223C7×C22⋊C4—22no—
224C7×C22⋊C4—21no—
225C7×C22⋊C4—22no—
226C7×C22⋊C4—22no—
227C7×C22⋊C4—22no—
228C7×C22⋊C4—21no—
229C7×C22⋊C4—21no—
230C7×C22⋊C4—22no—
231C7×C22⋊C4—22no—
232C7×C22⋊C4—21no—
233C7×C22⋊C4—21no—
234C7×C22⋊C4—41no—
235C42×C7—22no—
236C42×C7—22no—
237C42×C7—21no—
238C42×C7—41no—
239C42×C7C2×D411yescharacteristic
240C42×C7C2×D411yescharacteristic
241C42×C7C2×D411yescharacteristic
242C42×C7C2411yescharacteristic
243#563C22×C712yes—
244#562C22×C712yes—
245#556—21no—
246#551—22no—
247D4×Q8—21no—
248#548—21no—
249#546—21no—
250#543—21no—
251#542—21no—
252#542—22no—
253#538—21no—
254#536—21no—
255#533C22×C711yescharacteristic
256#533C22×C711yescharacteristic
257#529C22×C711yescharacteristic
258C2×#124C22×C711yescharacteristic
259#501—21no—
260#501—21no—
261#501C22×C712yescomplemented
262#501C22×C712yescomplemented
263#501C22×C711yescharacteristic, complemented
264#497C22×C712yes—
265#497C22×C711yescharacteristic, complemented
266#491C22×C711yescharacteristic, complemented
267#489C22×C712yes—
268#489C22×C711yescharacteristic, complemented
269#467—21no—
270#465—21no—
271#464—22no—
272#451—21no—
273#450—21no—
274#441—21no—
275#440—21no—
276#432C22×C711yescharacteristic
277#427—22no—
278C2×#97C22×C711yescharacteristic
279C2×#97C22×C711yescharacteristic
280#330C22×C712yescomplemented
281#330C22×C712yescomplemented
282#330C22×C712yescomplemented
283#329C22×C712yes—
284#329C22×C712yes—
285#329C22×C712yescomplemented
286C23×C7—22no—
287C23×C7—21no—
288C23×C7—42no—
289C23×C7—81no—
290C23×C7—81no—
291C23×C7#12011yescharacteristic
292C7×Q8—41no—
293C7×Q8—42no—
294C7×Q8—42no—
295C7×Q8—42no—
296C7×Q8—41no—
297C7×Q8—42no—
298C7×Q8—41no—
299C7×Q8—81no—
300C7×Q8—82no—
301C7×D4—41no—
302C7×D4—42no—
303C7×D4—42no—
304C7×D4—42no—
305C7×D4—82no—
306C7×D4—82no—
307C2×C4×C7—22no—
308C2×C4×C7—21no—
309C2×C4×C7—22no—
310C2×C4×C7—21no—
311C2×C4×C7—22no—
312C2×C4×C7—22no—
313C2×C4×C7—21no—
314C2×C4×C7—41no—
315C2×C4×C7—41no—
316C2×C4×C7—42no—
317C2×C4×C7—41no—
318C2×C4×C7—41no—
319C2×C4×C7—81no—
320C2×C4×C7—82no—
321C2×C4×C7—82no—
322C2×C4×C7—82no—
323C2×C4×C7#12011yescharacteristic
324C2×C4×C7#12011yescharacteristic
325C2×C4×C7#12011yescharacteristic
326C2×C4×C7C22×D411yescharacteristic
327C2×C4×C7C22×D411yescharacteristic
328C2×C4×C7C22×D411yescharacteristic
329C7×C8—24no—
330C7×C8—22no—
331C7×C8—41no—
332C7×C8—42no—
333C2×#41—21no—
334C22×Q8—21no—
335C22×Q8C7×D411yescharacteristic
336C22×D4C7×D411yescharacteristic
337C2×Q16—21no—
338C2×Q16—22no—
339C2×SD16—21no—
340C2×SD16—22no—
341C22×C8—21no—
342C22×C8—21no—
343C4⋊Q8—21no—
344C4⋊Q8—22no—
345C4⋊Q8C7×D411yescharacteristic
346C4⋊Q8C7×D411yescharacteristic
347C4⋊Q8C23×C711yescharacteristic
348C4⋊Q8C23×C711yescharacteristic
349#126—22no—
350#126—22no—
351#126—22no—
352#126—21no—
353#125—21no—
354#125C23×C711yescharacteristic
355#125C23×C712yes—
356#124—21no—
357#124—22no—
358#124—22no—
359#124—21no—
360#124—21no—
361#124C7×D412yes—
362#124C23×C712yes—
363#124C23×C711yescharacteristic
364#123—22no—
365#123—22no—
366#123—21no—
367#123—22no—
368#123—22no—
369C22⋊Q8—22no—
370C22⋊Q8—22no—
371C22⋊Q8—21no—
372C22⋊Q8—22no—
373C22⋊Q8—22no—
374C22⋊Q8—22no—
375C22⋊Q8—21no—
376C22⋊Q8—21no—
377C22⋊Q8—22no—
378C22⋊Q8—22no—
379C22⋊Q8—21no—
380C22⋊Q8—21no—
381C22⋊Q8C7×D411yescharacteristic
382#121—21no—
383#121—22no—
384#121—22no—
385#121C7×D411yescharacteristic
386#120—22no—
387C4×Q8—21no—
388C4×Q8—21no—
389C4×Q8—21no—
390C4×Q8—21no—
391C4×Q8—21no—
392C4×Q8C7×D411yescharacteristic
393C4×D4—22no—
394C4×D4—21no—
395C4×D4—22no—
396C4×D4—21no—
397C4×D4C7×D411yescharacteristic
398#117—22no—
399#117—21no—
400#117—21no—
401#117C7×D412yes—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C42C23×C711yescharacteristic
406#107—22no—
407#107—22no—
408#107—21no—
409#106—22no—
410#106—22no—
411#106—21no—
412C4⋊C8—21no—
413#103—21no—
414#103—22no—
415#103—21no—
416#103—21no—
417#102—21no—
418#102—22no—
419C22⋊C8—21no—
420#97—21no—
421#97—21no—
422#97C23×C712yes—
423#97C23×C711yescharacteristic
424#97C23×C711yescharacteristic
425#97C23×C712yes—
426C22×C7—21no—
427C22×C7—21no—
428C22×C7—41no—
429C22×C7—42no—
430C22×C7—42no—
431C22×C7—41no—
432C22×C7—41no—
433C22×C7—42no—
434C22×C7—162no—
435C22×C7#45612yes—
436C22×C7C2×#12011yescharacteristic
437C4×C7—21no—
438C4×C7—21no—
439C4×C7—21no—
440C4×C7—21no—
441C4×C7—21no—
442C4×C7—21no—
443C4×C7—41no—
444C4×C7—41no—
445C4×C7—42no—
446C4×C7—41no—
447C4×C7—42no—
448C4×C7—42no—
449C4×C7—81no—
450C4×C7—81no—
451C24—21no—
452#41—82no—
453C2×Q8—22no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—82no—
460C2×Q8—81no—
461C2×Q8C2×C7×D411yescharacteristic
462C2×Q8C2×C7×D412yes—
463C2×D4—21no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—21no—
467C2×D4—81no—
468C2×D4C2×C7×D411yescharacteristic
469C2×D4C2×C7×D412yes—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4C2×C7×D411yescharacteristic
476C22×C4C2×C7×D411yescharacteristic
477C22×C4C2×C7×D411yescharacteristic
478Q16—42no—
479Q16—82no—
480SD16—42no—
481SD16—82no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—22no—
487C2×C8—42no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4C2×C7×D411yescharacteristic
507C4⋊C4C2×C7×D411yescharacteristic
508C4⋊C4C2×C7×D412yes—
509C4⋊C4C2×C7×D412yes—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—21no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—41no—
526C42—22no—
527C42—22no—
528C42—21no—
529C42—41no—
530C42C2×C7×D411yescharacteristic
531C42C2×C7×D411yescharacteristic
532C42C2×C7×D411yescharacteristic
533C42C24×C711yescharacteristic
534C2×C7—22no—
535C2×C7—81no—
536C2×C7—82no—
537C2×C7C2×#45611yescharacteristic
538C2×C7#301811yescharacteristic
539C2×C7#301911yescharacteristic
540C23—22no—
541C23—21no—
542C23—42no—
543C23—81no—
544C23—81no—
545C23C7×#12011yescharacteristic
546Q8—41no—
547Q8—42no—
548Q8—42no—
549Q8—42no—
550Q8—41no—
551Q8—42no—
552Q8—41no—
553Q8—81no—
554Q8—82no—
555D4—41no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—82no—
560D4—82no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—81no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4C7×#12011yescharacteristic
578C2×C4C7×#12011yescharacteristic
579C2×C4C7×#12011yescharacteristic
580C2×C4C22×C7×D411yescharacteristic
581C2×C4C22×C7×D411yescharacteristic
582C2×C4C22×C7×D411yescharacteristic
583C8—24no—
584C8—22no—
585C8—41no—
586C8—42no—
587C7#1904911yescharacteristic, direct product factor, Sylow
588C22—21no—
589C22—21no—
590C22—41no—
591C22—42no—
592C22—42no—
593C22—41no—
594C22—41no—
595C22—42no—
596C22—162no—
597C22C7×#45612yes—
598C22C2×C7×#12011yescharacteristic
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—21no—
605C4—41no—
606C4—41no—
607C4—42no—
608C4—41no—
609C4—42no—
610C4—42no—
611C4—81no—
612C4—81no—
613C2—22no—
614C2—82no—
615C2—81no—
616C2C2×C7×#45611yescharacteristic
617C2C7×#301811yescharacteristic
618C2C7×#301911yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3561C2
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×#3215C2semidirect
3. C7×#3181C2semidirect
4. C7×#3162C2semidirect
5. C7×#1664C2semidirect
6. C7×#1661C2semidirect
7. C7×#1661C2semidirect
8. C7×#1660C2semidirect
9. C7×#1532C2semidirect
10. C7×#1531C2semidirect
11. C2×C7×#329C2semidirect
30. C7×#501C22semidirect
31. C7×#501C22semidirect
32. C7×#501C22semidirect
34. C7×#497C22semidirect
35. C7×#491C22semidirect
37. C7×#489C22semidirect
49. C7×#330C22semidirect
50. C7×#330C22semidirect
51. C7×#330C22semidirect
54. C7×#329C22semidirect
55. #19049C7direct (N × H)
150. #3215C2×C7semidirect
151. #3181C2×C7semidirect
152. #3162C2×C7semidirect
153. #1664C2×C7semidirect
154. #1661C2×C7semidirect
155. #1661C2×C7semidirect
156. #1660C2×C7semidirect
157. #1532C2×C7semidirect
158. #1531C2×C7semidirect
159. C2×#329C2×C7semidirect
261. #501C22×C7semidirect
262. #501C22×C7semidirect
263. #501C22×C7semidirect
265. #497C22×C7semidirect
266. #491C22×C7semidirect
268. #489C22×C7semidirect
280. #330C22×C7semidirect
281. #330C22×C7semidirect
282. #330C22×C7semidirect
285. #329C22×C7semidirect
587. C7#19049direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35612
C7×#32152
C7×#31812
C7×#31622
C7×#16642
C7×#16612
C7×#16602
C7×#15322
C7×#15312
C2×C7×#3292
#190497