Finite Group Databasebeta

← C7×#19046 #13196262

Subgroups — C7×#19046 #13196262

Order 1792.

836 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3563C211yesmaximal, characteristic, complemented
2C7×#3234C211yesmaximal, characteristic, complemented
3C7×#3180C211yesmaximal, characteristic
4C7×#3135C211yesmaximal, characteristic, complemented
5C7×#1664C212yesmaximal, complemented
6C7×#1663C212yesmaximal, complemented
7C7×#1658C212yesmaximal, complemented
8C7×#1657C212yesmaximal, complemented
9C7×#1532C211yesmaximal, characteristic, complemented
10C7×#1531C211yesmaximal, characteristic, complemented
11C2×C7×#330C211yesmaximal, characteristic, complemented
12C7×#564C2212yes—
13C7×#563C2212yes—
14C7×#557—21no—
15C7×#554—21no—
16C7×#552—21no—
17C7×#552—22no—
18C7×#552—21no—
19C7×#544—21no—
20C7×#544—21no—
21C7×#541—21no—
22C7×#541—22no—
23C7×#539—21no—
24C7×#533C2211yescharacteristic
25C7×#532C2211yescharacteristic
26C7×#529C2211yescharacteristic
27C2×C7×#125C2211yescharacteristic
28C7×#501C2211yescharacteristic
29C7×#501C2212yes—
30C7×#496C2211yescharacteristic
31C7×#491C2212yes—
32C7×#490C2211yescharacteristic
33C7×#490C2212yes—
34C7×#490C2212yes—
35C7×#489—21no—
36C7×#489—21no—
37C7×#489C2211yescharacteristic
38C7×#481—21no—
39C7×#480—21no—
40C7×#478—22no—
41C7×#466—21no—
42C7×#465—21no—
43C7×#442—21no—
44C7×#440—21no—
45C7×#432C2211yescharacteristic
46C7×#416—22no—
47C2×C7×#97C2211yescharacteristic
48C2×C7×#97C2211yescharacteristic
49C7×#330C2212yes—
50C7×#330C2212yes—
51C7×#330C2212yes—
52C7×#330C2212yes—
53C7×#330C2212yes—
54C7×#329C2212yes—
55#19046C711yesmaximal, characteristic, direct product factor, Sylow
56C2×C7×#41—21no—
57C2×C7×SD16—21no—
58C2×C7×D8—21no—
59C22×C7×C8—21no—
60C22×C7×C8—21no—
61C7×C4⋊Q8—22no—
62C7×C4⋊Q8—21no—
63C7×C4⋊Q8—22no—
64C7×C4⋊Q8D411yescharacteristic
65C7×C4⋊Q8D411yescharacteristic
66C7×C4⋊Q8C2311yescharacteristic
67C7×#127C2311yescharacteristic
68C7×#126—22no—
69C7×#126—22no—
70C7×#126—22no—
71C7×#126—22no—
72C7×#126—21no—
73C7×#125—22no—
74C7×#125—22no—
75C7×#125—21no—
76C7×#125—22no—
77C7×#125—22no—
78C7×#125—22no—
79C7×#125—22no—
80C7×#125—22no—
81C7×#125—22no—
82C7×#125—21no—
83C7×#125—21no—
84C7×#125D412yes—
85C7×#125C2312yes—
86C7×#125C2312yes—
87C7×#125C2311yescharacteristic
88C7×#124—21no—
89C7×#124C2311yescharacteristic
90C7×#123—22no—
91C7×#123—22no—
92C7×#123—21no—
93C7×C22⋊Q8—22no—
94C7×C22⋊Q8—21no—
95C7×C22⋊Q8—21no—
96C7×C22⋊Q8—22no—
97C7×C22⋊Q8—21no—
98C7×C22⋊Q8—21no—
99C7×C22⋊Q8—22no—
100C7×C22⋊Q8—22no—
101C7×C22⋊Q8D411yescharacteristic
102C7×#121—21no—
103C7×#121—21no—
104C7×#121D411yescharacteristic
105C4×C7×Q8—21no—
106C4×C7×Q8—22no—
107C4×C7×Q8—21no—
108C4×C7×Q8—21no—
109C4×C7×Q8—21no—
110C4×C7×Q8D411yescharacteristic
111C4×C7×D4—22no—
112C4×C7×D4—21no—
113C4×C7×D4—21no—
114C4×C7×D4—21no—
115C4×C7×D4D411yescharacteristic
116C7×#117—22no—
117C7×#117—21no—
118C7×#117—22no—
119C7×#117—21no—
120C7×#117D412yes—
121C2×C7×C4⋊C4—21no—
122C2×C7×C4⋊C4—21no—
123C2×C7×C4⋊C4—21no—
124C2×C7×C4⋊C4D411yescharacteristic
125C2×C7×C4⋊C4D411yescharacteristic
126C2×C42×C7C2311yescharacteristic
127C7×#107—21no—
128C7×#107—22no—
129C7×#106—22no—
130C7×#106—21no—
131C7×C4⋊C8—21no—
132C7×#103—21no—
133C7×#103—22no—
134C7×#103—22no—
135C7×#102—21no—
136C7×#102—21no—
137C7×#102—22no—
138C7×#102—22no—
139C7×#102—22no—
140C7×#102—22no—
141C7×#102—21no—
142C7×C22⋊C8—21no—
143C7×#97—21no—
144C7×#97—21no—
145C7×#97C2311yescharacteristic
146C7×#97C2312yes—
147C7×#97C2312yes—
148C7×#97C2311yescharacteristic
149#3563C2×C711yescharacteristic, complemented
150#3234C2×C711yescharacteristic, complemented
151#3180C2×C711yescharacteristic
152#3135C2×C711yescharacteristic, complemented
153#1664C2×C712yescomplemented
154#1663C2×C712yescomplemented
155#1658C2×C712yescomplemented
156#1657C2×C712yescomplemented
157#1532C2×C711yescharacteristic, complemented
158#1531C2×C711yescharacteristic, complemented
159C2×#330C2×C711yescharacteristic, complemented
160C7×#41—44no—
161C2×C7×Q8—22no—
162C2×C7×Q8—21no—
163C2×C7×Q8—21no—
164C2×C7×Q8—21no—
165C2×C7×Q8C2×D411yescharacteristic
166C2×C7×D4—21no—
167C2×C7×D4—21no—
168C2×C7×D4—21no—
169C2×C7×D4—21no—
170C2×C7×D4—41no—
171C2×C7×D4C2×D411yescharacteristic
172C22×C4×C7—21no—
173C22×C4×C7—22no—
174C22×C4×C7—21no—
175C22×C4×C7—21no—
176C22×C4×C7—41no—
177C22×C4×C7C2×D411yescharacteristic
178C22×C4×C7C2×D411yescharacteristic
179C22×C4×C7C2×D411yescharacteristic
180C7×SD16—42no—
181C7×D8—42no—
182C2×C7×C8—21no—
183C2×C7×C8—21no—
184C2×C7×C8—22no—
185C2×C7×C8—22no—
186C2×C7×C8—42no—
187C2×C7×C8—42no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—22no—
191C7×C4⋊C4—22no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—22no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—21no—
204C7×C4⋊C4—22no—
205C7×C4⋊C4—22no—
206C7×C4⋊C4—21no—
207C7×C4⋊C4—21no—
208C7×C4⋊C4—22no—
209C7×C4⋊C4—22no—
210C7×C4⋊C4—21no—
211C7×C4⋊C4—21no—
212C7×C4⋊C4C2×D412yes—
213C7×C4⋊C4C2×D412yes—
214C7×C4⋊C4C2×D412yes—
215C7×C4⋊C4C2×D411yescharacteristic
216C7×C4⋊C4C2×D412yes—
217C7×C4⋊C4C2×D411yescharacteristic
218C7×C22⋊C4—22no—
219C7×C22⋊C4—22no—
220C7×C22⋊C4—22no—
221C7×C22⋊C4—21no—
222C7×C22⋊C4—22no—
223C7×C22⋊C4—21no—
224C7×C22⋊C4—21no—
225C7×C22⋊C4—21no—
226C7×C22⋊C4—21no—
227C7×C22⋊C4—41no—
228C7×C22⋊C4—41no—
229C42×C7—22no—
230C42×C7—22no—
231C42×C7—22no—
232C42×C7—22no—
233C42×C7—21no—
234C42×C7C2×D411yescharacteristic
235C42×C7C2×D411yescharacteristic
236C42×C7C2×D411yescharacteristic
237C42×C7C2411yescharacteristic
238#564C22×C712yes—
239#563C22×C712yes—
240#557—21no—
241#554—21no—
242#552—21no—
243#552—22no—
244#552—21no—
245#544—21no—
246#544—21no—
247#541—21no—
248#541—22no—
249#539—21no—
250#533C22×C711yescharacteristic
251#532C22×C711yescharacteristic
252#529C22×C711yescharacteristic
253C2×#125C22×C711yescharacteristic
254#501C22×C711yescharacteristic
255#501C22×C712yes—
256#496C22×C711yescharacteristic
257#491C22×C712yes—
258#490C22×C712yes—
259#490C22×C711yescharacteristic
260#490C22×C712yes—
261#489—21no—
262#489—21no—
263#489C22×C711yescharacteristic
264#481—21no—
265#480—21no—
266#478—22no—
267#466—21no—
268#465—21no—
269#442—21no—
270#440—21no—
271#432C22×C711yescharacteristic
272#416—22no—
273C2×#97C22×C711yescharacteristic
274C2×#97C22×C711yescharacteristic
275#330C22×C712yes—
276#330C22×C712yes—
277#330C22×C712yes—
278#330C22×C712yes—
279#330C22×C712yes—
280#329C22×C712yes—
281C23×C7—21no—
282C23×C7—41no—
283C23×C7#12011yescharacteristic
284C7×Q8—41no—
285C7×Q8—42no—
286C7×Q8—42no—
287C7×Q8—81no—
288C7×Q8—82no—
289C7×D4—41no—
290C7×D4—41no—
291C7×D4—41no—
292C7×D4—42no—
293C7×D4—42no—
294C7×D4—42no—
295C7×D4—82no—
296C2×C4×C7—22no—
297C2×C4×C7—22no—
298C2×C4×C7—21no—
299C2×C4×C7—21no—
300C2×C4×C7—22no—
301C2×C4×C7—21no—
302C2×C4×C7—22no—
303C2×C4×C7—22no—
304C2×C4×C7—41no—
305C2×C4×C7—41no—
306C2×C4×C7—42no—
307C2×C4×C7—42no—
308C2×C4×C7—41no—
309C2×C4×C7—82no—
310C2×C4×C7—81no—
311C2×C4×C7—82no—
312C2×C4×C7#12011yescharacteristic
313C2×C4×C7#12011yescharacteristic
314C2×C4×C7#12011yescharacteristic
315C2×C4×C7C22×D411yescharacteristic
316C2×C4×C7C22×D411yescharacteristic
317C2×C4×C7C22×D411yescharacteristic
318C7×C8—22no—
319C7×C8—42no—
320C7×C8—42no—
321C7×C8—41no—
322C2×#41—21no—
323C2×SD16—21no—
324C2×D8—21no—
325C22×C8—21no—
326C22×C8—21no—
327C4⋊Q8—22no—
328C4⋊Q8—21no—
329C4⋊Q8—22no—
330C4⋊Q8C7×D411yescharacteristic
331C4⋊Q8C7×D411yescharacteristic
332C4⋊Q8C23×C711yescharacteristic
333#127C23×C711yescharacteristic
334#126—22no—
335#126—22no—
336#126—22no—
337#126—22no—
338#126—21no—
339#125—22no—
340#125—22no—
341#125—21no—
342#125—22no—
343#125—22no—
344#125—22no—
345#125—22no—
346#125—22no—
347#125—22no—
348#125—21no—
349#125—21no—
350#125C7×D412yes—
351#125C23×C712yes—
352#125C23×C711yescharacteristic
353#125C23×C712yes—
354#124—21no—
355#124C23×C711yescharacteristic
356#123—22no—
357#123—22no—
358#123—21no—
359C22⋊Q8—22no—
360C22⋊Q8—21no—
361C22⋊Q8—21no—
362C22⋊Q8—22no—
363C22⋊Q8—21no—
364C22⋊Q8—21no—
365C22⋊Q8—22no—
366C22⋊Q8—22no—
367C22⋊Q8C7×D411yescharacteristic
368#121—21no—
369#121—21no—
370#121C7×D411yescharacteristic
371C4×Q8—21no—
372C4×Q8—22no—
373C4×Q8—21no—
374C4×Q8—21no—
375C4×Q8—21no—
376C4×Q8C7×D411yescharacteristic
377C4×D4—22no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—21no—
381C4×D4C7×D411yescharacteristic
382#117—22no—
383#117—21no—
384#117—22no—
385#117—21no—
386#117C7×D412yes—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4C7×D411yescharacteristic
391C2×C4⋊C4C7×D411yescharacteristic
392C2×C42C23×C711yescharacteristic
393#107—21no—
394#107—22no—
395#106—21no—
396#106—22no—
397C4⋊C8—21no—
398#103—21no—
399#103—22no—
400#103—22no—
401#102—21no—
402#102—22no—
403#102—22no—
404#102—21no—
405#102—21no—
406#102—22no—
407#102—22no—
408C22⋊C8—21no—
409#97—21no—
410#97—21no—
411#97C23×C712yes—
412#97C23×C712yes—
413#97C23×C711yescharacteristic
414#97C23×C711yescharacteristic
415C22×C7—21no—
416C22×C7—21no—
417C22×C7—41no—
418C22×C7—41no—
419C22×C7—41no—
420C22×C7—81no—
421C22×C7—81no—
422C22×C7—81no—
423C22×C7#45412yes—
424C22×C7C2×#12011yescharacteristic
425C4×C7—21no—
426C4×C7—21no—
427C4×C7—21no—
428C4×C7—21no—
429C4×C7—21no—
430C4×C7—21no—
431C4×C7—42no—
432C4×C7—41no—
433C4×C7—42no—
434C4×C7—42no—
435C4×C7—41no—
436C4×C7—41no—
437C4×C7—42no—
438C4×C7—81no—
439#41—44no—
440C2×Q8—22no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8C2×C7×D411yescharacteristic
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—41no—
450C2×D4C2×C7×D411yescharacteristic
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—41no—
456C22×C4C2×C7×D411yescharacteristic
457C22×C4C2×C7×D411yescharacteristic
458C22×C4C2×C7×D411yescharacteristic
459SD16—42no—
460D8—42no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—42no—
466C2×C8—42no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4C2×C7×D412yes—
492C4⋊C4C2×C7×D412yes—
493C4⋊C4C2×C7×D412yes—
494C4⋊C4C2×C7×D411yescharacteristic
495C4⋊C4C2×C7×D412yes—
496C4⋊C4C2×C7×D411yescharacteristic
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—21no—
501C22⋊C4—22no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—21no—
513C42C2×C7×D411yescharacteristic
514C42C2×C7×D411yescharacteristic
515C42C2×C7×D411yescharacteristic
516C42C24×C711yescharacteristic
517C2×C7—22no—
518C2×C7—81no—
519C2×C7—161no—
520C2×C7C2×#45411yescharacteristic
521C2×C7#301811yescharacteristic
522C2×C7#302011yescharacteristic
523C23—21no—
524C23—41no—
525C23C7×#12011yescharacteristic
526Q8—41no—
527Q8—42no—
528Q8—42no—
529Q8—81no—
530Q8—82no—
531D4—41no—
532D4—41no—
533D4—41no—
534D4—42no—
535D4—42no—
536D4—42no—
537D4—82no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—82no—
552C2×C4—81no—
553C2×C4—82no—
554C2×C4C7×#12011yescharacteristic
555C2×C4C7×#12011yescharacteristic
556C2×C4C7×#12011yescharacteristic
557C2×C4C22×C7×D411yescharacteristic
558C2×C4C22×C7×D411yescharacteristic
559C2×C4C22×C7×D411yescharacteristic
560C8—22no—
561C8—42no—
562C8—42no—
563C8—41no—
564C7#1904611yescharacteristic, direct product factor, Sylow
565C22—21no—
566C22—21no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22C7×#45412yes—
574C22C2×C7×#12011yescharacteristic
575C4—21no—
576C4—21no—
577C4—21no—
578C4—21no—
579C4—21no—
580C4—21no—
581C4—42no—
582C4—41no—
583C4—42no—
584C4—42no—
585C4—41no—
586C4—41no—
587C4—42no—
588C4—81no—
589C2—22no—
590C2—81no—
591C2—161no—
592C2C2×C7×#45411yescharacteristic
593C2C7×#301811yescharacteristic
594C2C7×#302011yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3180C2
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
1. C7×#3563C2semidirect
2. C7×#3234C2semidirect
4. C7×#3135C2semidirect
5. C7×#1664C2semidirect
6. C7×#1663C2semidirect
7. C7×#1658C2semidirect
8. C7×#1657C2semidirect
9. C7×#1532C2semidirect
10. C7×#1531C2semidirect
11. C2×C7×#330C2semidirect
55. #19046C7direct (N × H)
149. #3563C2×C7semidirect
150. #3234C2×C7semidirect
152. #3135C2×C7semidirect
153. #1664C2×C7semidirect
154. #1663C2×C7semidirect
155. #1658C2×C7semidirect
156. #1657C2×C7semidirect
157. #1532C2×C7semidirect
158. #1531C2×C7semidirect
159. C2×#330C2×C7semidirect
564. C7#19046direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35632
C7×#32342
C7×#31802
C7×#31352
C7×#16642
C7×#16632
C7×#16582
C7×#16572
C7×#15322
C7×#15312
C2×C7×#3302
#190467