Finite Group Databasebeta

← C7×#19025 #13196241

Subgroups — C7×#19025 #13196241

Order 1792.

872 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×#3561C211yesmaximal, characteristic
2C7×#3237C211yesmaximal, characteristic, complemented
3C7×#3172C211yesmaximal, characteristic, complemented
4C7×#3140C211yesmaximal, characteristic, complemented
5C7×#1664C212yesmaximal, complemented
6C7×#1664C212yesmaximal, complemented
7C7×#1662C212yesmaximal, complemented
8C7×#1661C212yesmaximal, complemented
9C7×#1529C211yesmaximal, characteristic, complemented
10C7×#1527C211yesmaximal, characteristic, complemented
11C7×#1526C211yesmaximal, 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×#501C2211yescharacteristic, complemented
29C7×#501C2212yescomplemented
30C7×#497C2211yescharacteristic, complemented
31C7×#491C2211yescharacteristic, complemented
32C7×#491C2212yescomplemented
33C7×#489C2212yes—
34C7×#489C2212yes—
35C7×#489C2211yescharacteristic, complemented
36C7×#488—22no—
37C7×#485—22no—
38C7×#477—21no—
39C7×#475—21no—
40C7×#470—21no—
41C7×#468—21no—
42C4×C7×Q16—21no—
43C4×C7×SD16—21no—
44C7×#432C2211yescharacteristic
45C7×#419—21no—
46C7×#418—21no—
47C4×C7×M16C2211yescharacteristic
48C4×C7×M16C2211yescharacteristic
49C7×#330C2212yescomplemented
50C7×#330C2212yescomplemented
51C7×#330C2212yescomplemented
52C7×#329C2212yes—
53C7×#329C2212yes—
54C7×#329C2212yescomplemented
55#19025C711yesmaximal, 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×SD16—21no—
62C2×C7×M16—21no—
63C2×C7×M16—21no—
64C7×C4⋊Q8—21no—
65C7×C4⋊Q8—22no—
66C7×C4⋊Q8D411yescharacteristic
67C7×C4⋊Q8D411yescharacteristic
68C7×C4⋊Q8C2311yescharacteristic
69C7×C4⋊Q8C2311yescharacteristic
70C7×#126—22no—
71C7×#126—22no—
72C7×#126—22no—
73C7×#126—21no—
74C7×#125—21no—
75C7×#125C2311yescharacteristic
76C7×#125C2312yes—
77C7×#124—21no—
78C7×#124—22no—
79C7×#124—22no—
80C7×#124—21no—
81C7×#124—21no—
82C7×#124D412yes—
83C7×#124C2312yes—
84C7×#124C2311yescharacteristic
85C7×#123—22no—
86C7×#123—22no—
87C7×#123—21no—
88C7×#123—22no—
89C7×#123—22no—
90C7×C22⋊Q8—22no—
91C7×C22⋊Q8—22no—
92C7×C22⋊Q8—21no—
93C7×C22⋊Q8—22no—
94C7×C22⋊Q8—22no—
95C7×C22⋊Q8—22no—
96C7×C22⋊Q8—21no—
97C7×C22⋊Q8—21no—
98C7×C22⋊Q8—22no—
99C7×C22⋊Q8—22no—
100C7×C22⋊Q8—21no—
101C7×C22⋊Q8—21no—
102C7×C22⋊Q8D411yescharacteristic
103C7×#121—21no—
104C7×#121—22no—
105C7×#121—22no—
106C7×#121D411yescharacteristic
107C7×#120—22no—
108C4×C7×Q8—21no—
109C4×C7×Q8—21no—
110C4×C7×Q8—21no—
111C4×C7×Q8—21no—
112C4×C7×Q8—21no—
113C4×C7×Q8D411yescharacteristic
114C4×C7×D4—22no—
115C4×C7×D4—21no—
116C4×C7×D4—22no—
117C4×C7×D4—21no—
118C4×C7×D4D411yescharacteristic
119C7×#117—22no—
120C7×#117—21no—
121C7×#117—21no—
122C7×#117D412yes—
123C2×C7×C22⋊C4—22no—
124C2×C7×C22⋊C4—21no—
125C2×C7×C22⋊C4—21no—
126C2×C42×C7C2311yescharacteristic
127C7×#107—21no—
128C7×#107—22no—
129C7×#106—21no—
130C7×#106—22no—
131C7×C4⋊C8—21no—
132C7×#103—21no—
133C7×#103—22no—
134C7×#103—22no—
135C7×#103—21no—
136C7×#103—21no—
137C7×#103—22no—
138C7×#103—22no—
139C7×#102—21no—
140C7×#102—22no—
141C7×#102—22no—
142C7×C22⋊C8—21no—
143C7×#97C2312yes—
144C7×#97C2311yescharacteristic
145C7×#97C2312yes—
146C7×#97C2311yescharacteristic
147C4×C7×C8—21no—
148C4×C7×C8—21no—
149#3561C2×C711yescharacteristic
150#3237C2×C711yescharacteristic, complemented
151#3172C2×C711yescharacteristic, complemented
152#3140C2×C711yescharacteristic, complemented
153#1664C2×C712yescomplemented
154#1664C2×C712yescomplemented
155#1662C2×C712yescomplemented
156#1661C2×C712yescomplemented
157#1529C2×C711yescharacteristic, complemented
158#1527C2×C711yescharacteristic, complemented
159#1526C2×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—42no—
169C2×C7×Q8—82no—
170C2×C7×Q8C2×D412yes—
171C2×C7×Q8C2×D411yescharacteristic
172C2×C7×D4—21no—
173C2×C7×D4—22no—
174C2×C7×D4—22no—
175C2×C7×D4—21no—
176C2×C7×D4—42no—
177C2×C7×D4C2×D412yes—
178C2×C7×D4C2×D411yescharacteristic
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×SD16—42no—
189C7×M16—42no—
190C7×M16—42no—
191C2×C7×C8—21no—
192C2×C7×C8—22no—
193C2×C7×C8—21no—
194C2×C7×C8—22no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4—22no—
199C7×C4⋊C4—21no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—21no—
204C7×C4⋊C4—21no—
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—41no—
212C7×C4⋊C4—41no—
213C7×C4⋊C4C2×D412yes—
214C7×C4⋊C4C2×D411yescharacteristic
215C7×C4⋊C4C2×D412yes—
216C7×C4⋊C4C2×D411yescharacteristic
217C7×C22⋊C4—22no—
218C7×C22⋊C4—22no—
219C7×C22⋊C4—22no—
220C7×C22⋊C4—22no—
221C7×C22⋊C4—22no—
222C7×C22⋊C4—21no—
223C7×C22⋊C4—22no—
224C7×C22⋊C4—22no—
225C7×C22⋊C4—22no—
226C7×C22⋊C4—21no—
227C7×C22⋊C4—21no—
228C7×C22⋊C4—22no—
229C7×C22⋊C4—22no—
230C7×C22⋊C4—21no—
231C7×C22⋊C4—21no—
232C7×C22⋊C4—41no—
233C42×C7—22no—
234C42×C7—22no—
235C42×C7—21no—
236C42×C7—41no—
237C42×C7C2×D411yescharacteristic
238C42×C7C2×D411yescharacteristic
239C42×C7C2×D411yescharacteristic
240C42×C7C2411yescharacteristic
241#563C22×C712yes—
242#562C22×C712yes—
243#556—21no—
244#551—22no—
245D4×Q8—21no—
246#548—21no—
247#546—21no—
248#543—21no—
249#542—21no—
250#542—22no—
251#538—21no—
252#536—21no—
253#533C22×C711yescharacteristic
254#533C22×C711yescharacteristic
255#529C22×C711yescharacteristic
256C2×#124C22×C711yescharacteristic
257#501C22×C711yescharacteristic, complemented
258#501C22×C712yescomplemented
259#497C22×C711yescharacteristic, complemented
260#491C22×C711yescharacteristic, complemented
261#491C22×C712yescomplemented
262#489C22×C712yes—
263#489C22×C712yes—
264#489C22×C711yescharacteristic, complemented
265#488—22no—
266#485—22no—
267#477—21no—
268#475—21no—
269#470—21no—
270#468—21no—
271C4×Q16—21no—
272C4×SD16—21no—
273#432C22×C711yescharacteristic
274#419—21no—
275#418—21no—
276C4×M16C22×C711yescharacteristic
277C4×M16C22×C711yescharacteristic
278#330C22×C712yescomplemented
279#330C22×C712yescomplemented
280#330C22×C712yescomplemented
281#329C22×C712yes—
282#329C22×C712yes—
283#329C22×C712yescomplemented
284C23×C7—22no—
285C23×C7—21no—
286C23×C7—42no—
287C23×C7—81no—
288C23×C7—81no—
289C23×C7#12011yescharacteristic
290C7×Q8—41no—
291C7×Q8—42no—
292C7×Q8—41no—
293C7×Q8—41no—
294C7×Q8—42no—
295C7×Q8—42no—
296C7×Q8—42no—
297C7×Q8—81no—
298C7×Q8—82no—
299C7×D4—41no—
300C7×D4—42no—
301C7×D4—42no—
302C7×D4—42no—
303C7×D4—82no—
304C7×D4—82no—
305C2×C4×C7—22no—
306C2×C4×C7—21no—
307C2×C4×C7—22no—
308C2×C4×C7—21no—
309C2×C4×C7—22no—
310C2×C4×C7—22no—
311C2×C4×C7—21no—
312C2×C4×C7—41no—
313C2×C4×C7—41no—
314C2×C4×C7—42no—
315C2×C4×C7—41no—
316C2×C4×C7—41no—
317C2×C4×C7—81no—
318C2×C4×C7—82no—
319C2×C4×C7—82no—
320C2×C4×C7—82no—
321C2×C4×C7#12011yescharacteristic
322C2×C4×C7#12011yescharacteristic
323C2×C4×C7#12011yescharacteristic
324C2×C4×C7C22×D411yescharacteristic
325C2×C4×C7C22×D411yescharacteristic
326C2×C4×C7C22×D411yescharacteristic
327C7×C8—22no—
328C7×C8—42no—
329C7×C8—42no—
330C7×C8—41no—
331C2×#41—21no—
332C22×Q8—21no—
333C22×Q8C7×D411yescharacteristic
334C22×D4C7×D411yescharacteristic
335C2×Q16—21no—
336C2×SD16—21no—
337C2×M16—21no—
338C2×M16—21no—
339C4⋊Q8—21no—
340C4⋊Q8—22no—
341C4⋊Q8C7×D411yescharacteristic
342C4⋊Q8C7×D411yescharacteristic
343C4⋊Q8C23×C711yescharacteristic
344C4⋊Q8C23×C711yescharacteristic
345#126—22no—
346#126—22no—
347#126—22no—
348#126—21no—
349#125—21no—
350#125C23×C711yescharacteristic
351#125C23×C712yes—
352#124—21no—
353#124—22no—
354#124—22no—
355#124—21no—
356#124—21no—
357#124C7×D412yes—
358#124C23×C712yes—
359#124C23×C711yescharacteristic
360#123—22no—
361#123—22no—
362#123—21no—
363#123—22no—
364#123—22no—
365C22⋊Q8—22no—
366C22⋊Q8—22no—
367C22⋊Q8—21no—
368C22⋊Q8—22no—
369C22⋊Q8—22no—
370C22⋊Q8—22no—
371C22⋊Q8—21no—
372C22⋊Q8—21no—
373C22⋊Q8—22no—
374C22⋊Q8—22no—
375C22⋊Q8—21no—
376C22⋊Q8—21no—
377C22⋊Q8C7×D411yescharacteristic
378#121—21no—
379#121—22no—
380#121—22no—
381#121C7×D411yescharacteristic
382#120—22no—
383C4×Q8—21no—
384C4×Q8—21no—
385C4×Q8—21no—
386C4×Q8—21no—
387C4×Q8—21no—
388C4×Q8C7×D411yescharacteristic
389C4×D4—22no—
390C4×D4—21no—
391C4×D4—22no—
392C4×D4—21no—
393C4×D4C7×D411yescharacteristic
394#117—22no—
395#117—21no—
396#117—21no—
397#117C7×D412yes—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C42C23×C711yescharacteristic
402#107—21no—
403#107—22no—
404#106—21no—
405#106—22no—
406C4⋊C8—21no—
407#103—22no—
408#103—22no—
409#103—21no—
410#103—21no—
411#103—21no—
412#103—22no—
413#103—22no—
414#102—22no—
415#102—22no—
416#102—21no—
417C22⋊C8—21no—
418#97C23×C712yes—
419#97C23×C711yescharacteristic
420#97C23×C711yescharacteristic
421#97C23×C712yes—
422C4×C8—21no—
423C4×C8—21no—
424C22×C7—21no—
425C22×C7—21no—
426C22×C7—21no—
427C22×C7—41no—
428C22×C7—42no—
429C22×C7—42no—
430C22×C7—41no—
431C22×C7—41no—
432C22×C7—42no—
433C22×C7—162no—
434C22×C7C2×#12011yescharacteristic
435C4×C7—21no—
436C4×C7—21no—
437C4×C7—21no—
438C4×C7—21no—
439C4×C7—21no—
440C4×C7—41no—
441C4×C7—42no—
442C4×C7—42no—
443C4×C7—41no—
444C4×C7—42no—
445C4×C7—41no—
446C4×C7—81no—
447C4×C7—81no—
448C4×C7#45512yes—
449C24—21no—
450#41—82no—
451C2×Q8—22no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—42no—
458C2×Q8—82no—
459C2×Q8C2×C7×D412yes—
460C2×Q8C2×C7×D411yescharacteristic
461C2×D4—21no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—21no—
465C2×D4—42no—
466C2×D4C2×C7×D412yes—
467C2×D4C2×C7×D411yescharacteristic
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4C2×C7×D411yescharacteristic
474C22×C4C2×C7×D411yescharacteristic
475C22×C4C2×C7×D411yescharacteristic
476Q16—42no—
477SD16—42no—
478M16—42no—
479M16—42no—
480C2×C8—21no—
481C2×C8—22no—
482C2×C8—21no—
483C2×C8—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4C2×C7×D412yes—
503C4⋊C4C2×C7×D411yescharacteristic
504C4⋊C4C2×C7×D412yes—
505C4⋊C4C2×C7×D411yescharacteristic
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—21no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—41no—
522C42—22no—
523C42—22no—
524C42—21no—
525C42—41no—
526C42C2×C7×D411yescharacteristic
527C42C2×C7×D411yescharacteristic
528C42C2×C7×D411yescharacteristic
529C42C24×C711yescharacteristic
530C2×C7—41no—
531C2×C7—81no—
532C2×C7—82no—
533C2×C7C2×#45511yescharacteristic
534C2×C7#301811yescharacteristic
535C2×C7#302011yescharacteristic
536C23—22no—
537C23—21no—
538C23—42no—
539C23—81no—
540C23—81no—
541C23C7×#12011yescharacteristic
542Q8—41no—
543Q8—42no—
544Q8—41no—
545Q8—41no—
546Q8—42no—
547Q8—42no—
548Q8—42no—
549Q8—81no—
550Q8—82no—
551D4—41no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—82no—
556D4—82no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—81no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4C7×#12011yescharacteristic
574C2×C4C7×#12011yescharacteristic
575C2×C4C7×#12011yescharacteristic
576C2×C4C22×C7×D411yescharacteristic
577C2×C4C22×C7×D411yescharacteristic
578C2×C4C22×C7×D411yescharacteristic
579C8—22no—
580C8—42no—
581C8—42no—
582C8—41no—
583C7#1902511yescharacteristic, direct product factor, Sylow
584C22—21no—
585C22—21no—
586C22—21no—
587C22—41no—
588C22—42no—
589C22—42no—
590C22—41no—
591C22—41no—
592C22—42no—
593C22—162no—
594C22C2×C7×#12011yescharacteristic
595C4—21no—
596C4—21no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—41no—
601C4—42no—
602C4—42no—
603C4—41no—
604C4—42no—
605C4—41no—
606C4—81no—
607C4—81no—
608C4C7×#45512yes—
609C2—41no—
610C2—82no—
611C2—81no—
612C2C2×C7×#45511yescharacteristic
613C2C7×#301811yescharacteristic
614C2C7×#302011yescharacteristic
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×#3237C2semidirect
3. C7×#3172C2semidirect
4. C7×#3140C2semidirect
5. C7×#1664C2semidirect
6. C7×#1664C2semidirect
7. C7×#1662C2semidirect
8. C7×#1661C2semidirect
9. C7×#1529C2semidirect
10. C7×#1527C2semidirect
11. C7×#1526C2semidirect
28. C7×#501C22semidirect
29. C7×#501C22semidirect
30. C7×#497C22semidirect
31. C7×#491C22semidirect
32. C7×#491C22semidirect
35. C7×#489C22semidirect
49. C7×#330C22semidirect
50. C7×#330C22semidirect
51. C7×#330C22semidirect
54. C7×#329C22semidirect
55. #19025C7direct (N × H)
150. #3237C2×C7semidirect
151. #3172C2×C7semidirect
152. #3140C2×C7semidirect
153. #1664C2×C7semidirect
154. #1664C2×C7semidirect
155. #1662C2×C7semidirect
156. #1661C2×C7semidirect
157. #1529C2×C7semidirect
158. #1527C2×C7semidirect
159. #1526C2×C7semidirect
257. #501C22×C7semidirect
258. #501C22×C7semidirect
259. #497C22×C7semidirect
260. #491C22×C7semidirect
261. #491C22×C7semidirect
264. #489C22×C7semidirect
278. #330C22×C7semidirect
279. #330C22×C7semidirect
280. #330C22×C7semidirect
283. #329C22×C7semidirect
583. C7#19025direct (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×#32372
C7×#31722
C7×#31402
C7×#16642
C7×#16622
C7×#16612
C7×#15292
C7×#15272
C7×#15262
#190257