Finite Group Databasebeta

← C7×#12780 #13189996

Subgroups — C7×#12780 #13189996

Order 1792.

756 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3551C211yesmaximal, characteristic
2C7×#2129C211yesmaximal, characteristic, complemented
3C7×#2128C211yesmaximal, characteristic, complemented
4C7×#2126C211yesmaximal, characteristic, complemented
5C7×#1910C211yesmaximal, characteristic, complemented
6C7×#1909C211yesmaximal, characteristic, complemented
7C7×#1884C211yesmaximal, characteristic, complemented
8C7×#568C2211yescharacteristic
9C7×#563C412yescomplemented
10C7×#560—21no—
11C7×#555C412yescomplemented
12C7×#553—21no—
13C7×#552—21no—
14C7×#551—21no—
15C7×#548—21no—
16C7×#546—21no—
17C7×#540—22no—
18C7×#538—22no—
19C7×#537—21no—
20C7×#536C2211yescharacteristic
21C7×#535C2211yescharacteristic
22C7×#532C2211yescharacteristic
23C7×#529—21no—
24C7×#421—21no—
25C7×#419—21no—
26C7×#418—21no—
27C7×#410C2211yescharacteristic
28C7×#410C2211yescharacteristic
29C7×#410C2211yescharacteristic
30C7×#354—21no—
31C7×#354—22no—
32C7×#353—21no—
33C7×#353—22no—
34C7×#339—21no—
35C7×H(3,ℤ/4)—21no—
36C7×H(3,ℤ/4)—21no—
37#12780C711yesmaximal, characteristic, direct product factor, Sylow
38C2×C7×#41—21no—
39C2×C7×#41D411yescharacteristic
40C2×C7×#41C2311yescharacteristic
41C22×C7×Q8D411yescharacteristic
42C22×C7×D4D411yescharacteristic
43C7×#131—42no—
44C2×C7×M16—21no—
45C2×C7×M16—21no—
46C2×C7×M16—21no—
47C7×C4⋊Q8C2×C411yescharacteristic
48C7×C4⋊Q8C2×C412yes—
49C7×#127D411yescharacteristic
50C7×#127D412yes—
51C7×#126—22no—
52C7×#126—22no—
53C7×#126—22no—
54C7×#125—21no—
55C7×#125—22no—
56C7×#124—21no—
57C7×#124—22no—
58C7×#124C2×C411yescharacteristic
59C7×#124C2×C412yes—
60C7×#124D411yescharacteristic
61C7×#124D412yes—
62C7×#123—22no—
63C7×#123—22no—
64C7×#123—22no—
65C7×#123—22no—
66C7×#123—21no—
67C7×#123—21no—
68C7×C22⋊Q8—21no—
69C7×C22⋊Q8—22no—
70C7×C22⋊Q8—22no—
71C7×C22⋊Q8—21no—
72C7×C22⋊Q8—21no—
73C7×#121—22no—
74C7×#121—22no—
75C7×#121—21no—
76C7×#121—22no—
77C7×#121—22no—
78C7×#121—22no—
79C7×#121—22no—
80C7×#121—21no—
81C7×#121—21no—
82C7×#121—21no—
83C7×#121—21no—
84C7×#120—22no—
85C7×#120—21no—
86C7×#120—21no—
87C4×C7×Q8—21no—
88C4×C7×D4—22no—
89C4×C7×D4—21no—
90C4×C7×D4—22no—
91C4×C7×D4—21no—
92C4×C7×D4—21no—
93C4×C7×D4—22no—
94C7×#117—22no—
95C7×#117—22no—
96C7×#117—21no—
97C7×#117—21no—
98C7×#117—21no—
99C7×#117D411yescharacteristic
100C7×#117D411yescharacteristic
101C2×C7×C4⋊C4—21no—
102C2×C7×C22⋊C4—22no—
103C2×C42×C7D411yescharacteristic
104C7×#104—42no—
105C7×#103—42no—
106C7×#102—42no—
107C7×#99—21no—
108C7×#99—21no—
109C7×#99—21no—
110C7×#99—21no—
111C7×#99—21no—
112C7×#99—21no—
113#3551C2×C711yescharacteristic
114#2129C2×C711yescharacteristic, complemented
115#2128C2×C711yescharacteristic, complemented
116#2126C2×C711yescharacteristic, complemented
117#1910C2×C711yescharacteristic, complemented
118#1909C2×C711yescharacteristic, complemented
119#1884C2×C711yescharacteristic, complemented
120C24×C7—21no—
121C7×#41—41no—
122C7×#41—41no—
123C7×#41—41no—
124C7×#41—41no—
125C7×#41—81no—
126C7×#41—81no—
127C2×C7×Q8—21no—
128C2×C7×Q8—41no—
129C2×C7×Q8—41no—
130C2×C7×Q8C22⋊C411yescharacteristic
131C2×C7×Q8C22⋊C412yes—
132C2×C7×Q8C22×C411yescharacteristic
133C2×C7×D4—22no—
134C2×C7×D4—22no—
135C2×C7×D4—21no—
136C2×C7×D4—22no—
137C2×C7×D4—22no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4—21no—
141C2×C7×D4—41no—
142C2×C7×D4—41no—
143C2×C7×D4C22⋊C411yescharacteristic
144C2×C7×D4C22⋊C412yes—
145C2×C7×D4C2×D411yescharacteristic
146C2×C7×D4C2×D411yescharacteristic
147C2×C7×D4C2×D411yescharacteristic
148C22×C4×C7—22no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C7×M16—22no—
158C7×M16—41no—
159C7×M16—41no—
160C7×M16—41no—
161C2×C7×C8—22no—
162C2×C7×C8—22no—
163C2×C7×C8—41no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4—22no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—41no—
176C7×C4⋊C4—41no—
177C7×C4⋊C4—41no—
178C7×C22⋊C4—22no—
179C7×C22⋊C4—22no—
180C7×C22⋊C4—22no—
181C7×C22⋊C4—22no—
182C7×C22⋊C4—22no—
183C7×C22⋊C4—21no—
184C7×C22⋊C4—22no—
185C7×C22⋊C4—21no—
186C7×C22⋊C4—21no—
187C7×C22⋊C4—21no—
188C7×C22⋊C4—22no—
189C7×C22⋊C4—21no—
190C7×C22⋊C4—22no—
191C7×C22⋊C4—21no—
192C7×C22⋊C4—22no—
193C7×C22⋊C4—41no—
194C7×C22⋊C4—41no—
195C7×C22⋊C4—41no—
196C42×C7—22no—
197C42×C7—21no—
198C42×C7—41no—
199C42×C7—41no—
200C42×C7—41no—
201C42×C7C22⋊C411yescharacteristic
202C42×C7C22⋊C411yescharacteristic
203C42×C7C22⋊C412yes—
204C42×C7C22⋊C412yes—
205#568C22×C711yescharacteristic
206#563C4×C712yescomplemented
207#560—21no—
208#555C4×C712yescomplemented
209#553—21no—
210#552—21no—
211#551—21no—
212#548—21no—
213#546—21no—
214#540—22no—
215#538—22no—
216#537—21no—
217#536C22×C711yescharacteristic
218#535C22×C711yescharacteristic
219#532C22×C711yescharacteristic
220#529—21no—
221#421—21no—
222#419—21no—
223#418—21no—
224#410C22×C711yescharacteristic
225#410C22×C711yescharacteristic
226#410C22×C711yescharacteristic
227#354—21no—
228#354—22no—
229#353—21no—
230#353—22no—
231#339—21no—
232H(3,ℤ/4)—21no—
233H(3,ℤ/4)—21no—
234C23×C7—22no—
235C23×C7—21no—
236C23×C7—81no—
237C23×C7—82no—
238C23×C7#12011yescharacteristic
239C23×C7#12011yescharacteristic
240C23×C7#12011yescharacteristic
241C7×Q8—41no—
242C7×Q8—42no—
243C7×Q8—41no—
244C7×Q8—41no—
245C7×Q8—41no—
246C7×D4—41no—
247C7×D4—41no—
248C7×D4—41no—
249C7×D4—42no—
250C7×D4—41no—
251C7×D4—41no—
252C7×D4—41no—
253C7×D4—81no—
254C7×D4—82no—
255C7×D4—82no—
256C7×D4—82no—
257C7×D4—81no—
258C7×D4—81no—
259C7×D4—82no—
260C2×C4×C7—21no—
261C2×C4×C7—21no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—22no—
268C2×C4×C7—21no—
269C2×C4×C7—22no—
270C2×C4×C7—22no—
271C2×C4×C7—21no—
272C2×C4×C7—41no—
273C2×C4×C7—41no—
274C2×C4×C7—81no—
275C2×C4×C7—82no—
276C2×C4×C7—81no—
277C2×C4×C7—82no—
278C2×C4×C7—82no—
279C2×C4×C7—81no—
280C2×C4×C7—81no—
281C2×C4×C7—81no—
282C2×C4×C7C2×C22⋊C411yescharacteristic
283C2×C4×C7C2×C22⋊C411yescharacteristic
284C2×C4×C7C2×C22⋊C411yescharacteristic
285C2×C4×C7#12011yescharacteristic
286C7×C8—42no—
287C2×#41—21no—
288C2×#41C7×D411yescharacteristic
289C2×#41C23×C711yescharacteristic
290C22×Q8C7×D411yescharacteristic
291C22×D4C7×D411yescharacteristic
292#131—42no—
293C2×M16—21no—
294C2×M16—21no—
295C2×M16—21no—
296C4⋊Q8C2×C4×C711yescharacteristic
297C4⋊Q8C2×C4×C712yes—
298#127C7×D411yescharacteristic
299#127C7×D412yes—
300#126—22no—
301#126—22no—
302#126—22no—
303#125—21no—
304#125—22no—
305#124—21no—
306#124—22no—
307#124C2×C4×C711yescharacteristic
308#124C2×C4×C712yes—
309#124C7×D411yescharacteristic
310#124C7×D412yes—
311#123—22no—
312#123—22no—
313#123—22no—
314#123—22no—
315#123—21no—
316#123—21no—
317C22⋊Q8—21no—
318C22⋊Q8—22no—
319C22⋊Q8—22no—
320C22⋊Q8—21no—
321C22⋊Q8—21no—
322#121—22no—
323#121—22no—
324#121—21no—
325#121—22no—
326#121—22no—
327#121—22no—
328#121—22no—
329#121—21no—
330#121—21no—
331#121—21no—
332#121—21no—
333#120—22no—
334#120—21no—
335#120—21no—
336C4×Q8—21no—
337C4×D4—22no—
338C4×D4—21no—
339C4×D4—22no—
340C4×D4—21no—
341C4×D4—21no—
342C4×D4—22no—
343#117—22no—
344#117—22no—
345#117—21no—
346#117—21no—
347#117—21no—
348#117C7×D411yescharacteristic
349#117C7×D411yescharacteristic
350C2×C4⋊C4—21no—
351C2×C22⋊C4—22no—
352C2×C42C7×D411yescharacteristic
353#104—42no—
354#103—42no—
355#102—42no—
356#99—21no—
357#99—21no—
358#99—21no—
359#99—21no—
360#99—21no—
361#99—21no—
362C22×C7—21no—
363C22×C7—21no—
364C22×C7—21no—
365C22×C7—41no—
366C22×C7—41no—
367C22×C7—41no—
368C22×C7—41no—
369C22×C7—42no—
370C22×C7—41no—
371C22×C7—42no—
372C22×C7—41no—
373C22×C7—42no—
374C22×C7—162no—
375C22×C7#37911yescharacteristic
376C4×C7—21no—
377C4×C7—21no—
378C4×C7—21no—
379C4×C7—41no—
380C4×C7—42no—
381C4×C7—42no—
382C4×C7—41no—
383C4×C7—41no—
384C4×C7—42no—
385C4×C7—161no—
386C4×C7—161no—
387C4×C7—161no—
388C4×C7B(4,𝔽2)11yescharacteristic
389C4×C7#45811yescharacteristic
390C24—21no—
391#41—41no—
392#41—41no—
393#41—41no—
394#41—41no—
395#41—81no—
396#41—81no—
397C2×Q8—21no—
398C2×Q8—41no—
399C2×Q8—41no—
400C2×Q8C7×C22⋊C411yescharacteristic
401C2×Q8C7×C22⋊C412yes—
402C2×Q8C22×C4×C711yescharacteristic
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4C7×C22⋊C411yescharacteristic
414C2×D4C7×C22⋊C412yes—
415C2×D4C2×C7×D411yescharacteristic
416C2×D4C2×C7×D411yescharacteristic
417C2×D4C2×C7×D411yescharacteristic
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4C2×C7×D411yescharacteristic
425C22×C4C2×C7×D411yescharacteristic
426C22×C4C2×C7×D411yescharacteristic
427M16—22no—
428M16—41no—
429M16—41no—
430M16—41no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—41no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—21no—
454C22⋊C4—22no—
455C22⋊C4—21no—
456C22⋊C4—21no—
457C22⋊C4—21no—
458C22⋊C4—22no—
459C22⋊C4—21no—
460C22⋊C4—22no—
461C22⋊C4—21no—
462C22⋊C4—22no—
463C22⋊C4—41no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C42—22no—
467C42—21no—
468C42—41no—
469C42—41no—
470C42—41no—
471C42C7×C22⋊C411yescharacteristic
472C42C7×C22⋊C411yescharacteristic
473C42C7×C22⋊C412yes—
474C42C7×C22⋊C412yes—
475C2×C7—21no—
476C2×C7—41no—
477C2×C7—41no—
478C2×C7—41no—
479C2×C7—81no—
480C2×C7—82no—
481C2×C7#179011yescharacteristic
482C23—22no—
483C23—21no—
484C23—81no—
485C23—82no—
486C23C7×#12011yescharacteristic
487C23C7×#12011yescharacteristic
488C23C7×#12011yescharacteristic
489Q8—41no—
490Q8—42no—
491Q8—41no—
492Q8—41no—
493Q8—41no—
494D4—41no—
495D4—41no—
496D4—41no—
497D4—42no—
498D4—41no—
499D4—41no—
500D4—41no—
501D4—81no—
502D4—82no—
503D4—82no—
504D4—82no—
505D4—81no—
506D4—81no—
507D4—82no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—81no—
523C2×C4—82no—
524C2×C4—81no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4C2×C7×C22⋊C411yescharacteristic
531C2×C4C2×C7×C22⋊C411yescharacteristic
532C2×C4C2×C7×C22⋊C411yescharacteristic
533C2×C4C7×#12011yescharacteristic
534C8—42no—
535C7#1278011yescharacteristic, direct product factor, Sylow
536C22—21no—
537C22—21no—
538C22—21no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—42no—
544C22—41no—
545C22—42no—
546C22—41no—
547C22—42no—
548C22—162no—
549C22C7×#37911yescharacteristic
550C4—21no—
551C4—21no—
552C4—21no—
553C4—41no—
554C4—42no—
555C4—42no—
556C4—41no—
557C4—41no—
558C4—42no—
559C4—161no—
560C4—161no—
561C4—161no—
562C4C7×B(4,𝔽2)11yescharacteristic
563C4C7×#45811yescharacteristic
564C2—21no—
565C2—41no—
566C2—41no—
567C2—41no—
568C2—82no—
569C2—81no—
570C2C7×#179011yescharacteristic
subgroupG/H
center, socleC2×C7#1790
commutatorC2×Q8C22×C4×C7
Frattini, cosocleC2×#41C23×C7
prime radicalC7×#3551C2
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×#41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2129C2semidirect
3. C7×#2128C2semidirect
4. C7×#2126C2semidirect
5. C7×#1910C2semidirect
6. C7×#1909C2semidirect
7. C7×#1884C2semidirect
9. C7×#563C4semidirect
11. C7×#555C4semidirect
37. #12780C7direct (N × H)
114. #2129C2×C7semidirect
115. #2128C2×C7semidirect
116. #2126C2×C7semidirect
117. #1910C2×C7semidirect
118. #1909C2×C7semidirect
119. #1884C2×C7semidirect
206. #563C4×C7semidirect
208. #555C4×C7semidirect
535. C7#12780direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35512
C7×#21292
C7×#21282
C7×#21262
C7×#19102
C7×#19092
C7×#18842
#127807