Finite Group Databasebeta

← C7×#12770 #13189986

Subgroups — C7×#12770 #13189986

Order 1792.

752 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3551C211yesmaximal, characteristic
2C7×#2132C211yesmaximal, characteristic, complemented
3C7×#2123C211yesmaximal, characteristic, complemented
4C7×#2122C211yesmaximal, characteristic, complemented
5C7×#1915C211yesmaximal, characteristic, complemented
6C7×#1914C211yesmaximal, characteristic, complemented
7C7×#1884C211yesmaximal, characteristic, complemented
8C7×#568C2211yescharacteristic
9C7×#563—21no—
10C7×#560C412yescomplemented
11C7×#555—21no—
12C7×#553—21no—
13C7×#552—21no—
14C7×#551C412yescomplemented
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×#413C2211yescharacteristic
26C7×#413C2211yescharacteristic
27C2×C7×#101—21no—
28C2×C7×#100—21no—
29C7×#410C2211yescharacteristic
30C7×#356—21no—
31C7×#355—21no—
32C7×#352—22no—
33C7×AΣL(1,𝔽16)—22no—
34C7×#344—21no—
35C7×#344—21no—
36C7×#339—21no—
37#12770C711yesmaximal, 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—
47C2×C7×M16—21no—
48C2×C7×M16—21no—
49C7×C4⋊Q8—21no—
50C7×C4⋊Q8D411yescharacteristic
51C7×#127—21no—
52C7×#127C2×C411yescharacteristic
53C7×#126—22no—
54C7×#126—22no—
55C7×#126—22no—
56C7×#125—21no—
57C7×#125—22no—
58C7×#124—21no—
59C7×#124—22no—
60C7×#124—21no—
61C7×#124—21no—
62C7×#124C2×C411yescharacteristic
63C7×#124D411yescharacteristic
64C7×#123—22no—
65C7×#123—22no—
66C7×#123—22no—
67C7×#123—22no—
68C7×#123C2×C412yes—
69C7×#123D412yes—
70C7×C22⋊Q8—21no—
71C7×C22⋊Q8—22no—
72C7×C22⋊Q8—22no—
73C7×C22⋊Q8—21no—
74C7×C22⋊Q8—21no—
75C7×#121—22no—
76C7×#121—22no—
77C7×#121—21no—
78C7×#121—22no—
79C7×#121—22no—
80C7×#121—21no—
81C7×#121—21no—
82C7×#121—22no—
83C7×#121—21no—
84C7×#121—21no—
85C7×#121—22no—
86C7×#120—22no—
87C7×#120C2×C412yes—
88C7×#120D412yes—
89C4×C7×Q8—21no—
90C4×C7×D4—22no—
91C4×C7×D4—21no—
92C4×C7×D4—22no—
93C4×C7×D4—22no—
94C4×C7×D4—21no—
95C4×C7×D4—21no—
96C7×#117—22no—
97C7×#117—22no—
98C7×#117—21no—
99C7×#117D411yescharacteristic
100C7×#117D411yescharacteristic
101C2×C7×C4⋊C4—21no—
102C2×C7×C22⋊C4—22no—
103C2×C42×C7D411yescharacteristic
104C7×#104—42no—
105C7×#101—21no—
106C7×#101—21no—
107C7×#101—42no—
108C7×#100—21no—
109C7×#100—21no—
110C7×#100—42no—
111C7×#99—21no—
112C7×#99—21no—
113#3551C2×C711yescharacteristic
114#2132C2×C711yescharacteristic, complemented
115#2123C2×C711yescharacteristic, complemented
116#2122C2×C711yescharacteristic, complemented
117#1915C2×C711yescharacteristic, complemented
118#1914C2×C711yescharacteristic, complemented
119#1884C2×C711yescharacteristic, complemented
120C24×C7C22⋊C412yes—
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—21no—
129C2×C7×Q8—41no—
130C2×C7×Q8—41no—
131C2×C7×Q8C22⋊C411yescharacteristic
132C2×C7×Q8C2×D411yescharacteristic
133C2×C7×D4—22no—
134C2×C7×D4—22no—
135C2×C7×D4—21no—
136C2×C7×D4—22no—
137C2×C7×D4—21no—
138C2×C7×D4—22no—
139C2×C7×D4—21no—
140C2×C7×D4—21no—
141C2×C7×D4—21no—
142C2×C7×D4—41no—
143C2×C7×D4—41no—
144C2×C7×D4C22⋊C411yescharacteristic
145C2×C7×D4C22×C411yescharacteristic
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×C7C22⋊C412yes—
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C7×M16—22no—
158C7×M16—22no—
159C7×M16—22no—
160C7×M16—41no—
161C7×M16—41no—
162C7×M16—41no—
163C7×M16—41no—
164C7×M16—41no—
165C2×C7×C8—41no—
166C2×C7×C8—41no—
167C2×C7×C8—41no—
168C2×C7×C8—41no—
169C2×C7×C8—41no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—41no—
182C7×C22⋊C4—22no—
183C7×C22⋊C4—22no—
184C7×C22⋊C4—22no—
185C7×C22⋊C4—22no—
186C7×C22⋊C4—22no—
187C7×C22⋊C4—21no—
188C7×C22⋊C4—22no—
189C7×C22⋊C4—21no—
190C7×C22⋊C4—22no—
191C7×C22⋊C4—22no—
192C7×C22⋊C4—21no—
193C7×C22⋊C4—21no—
194C7×C22⋊C4—22no—
195C7×C22⋊C4—41no—
196C7×C22⋊C4C22⋊C412yes—
197C7×C22⋊C4C22⋊C412yes—
198C42×C7—22no—
199C42×C7—21no—
200C42×C7—21no—
201C42×C7—21no—
202C42×C7—41no—
203C42×C7C22⋊C411yescharacteristic
204C42×C7C22⋊C411yescharacteristic
205#568C22×C711yescharacteristic
206#563—21no—
207#560C4×C712yescomplemented
208#555—21no—
209#553—21no—
210#552—21no—
211#551C4×C712yescomplemented
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#413C22×C711yescharacteristic
223#413C22×C711yescharacteristic
224C2×#101—21no—
225C2×#100—21no—
226#410C22×C711yescharacteristic
227#356—21no—
228#355—21no—
229#352—22no—
230AΣL(1,𝔽16)—22no—
231#344—21no—
232#344—21no—
233#339—21no—
234C23×C7—22no—
235C23×C7—21no—
236C23×C7—42no—
237C23×C7—82no—
238C23×C7C2×C22⋊C411yescharacteristic
239C23×C7C2×C22⋊C411yescharacteristic
240C23×C7#12011yescharacteristic
241C7×Q8—41no—
242C7×Q8—41no—
243C7×Q8—41no—
244C7×Q8—41no—
245C7×Q8—81no—
246C7×D4—41no—
247C7×D4—41no—
248C7×D4—41no—
249C7×D4—41no—
250C7×D4—41no—
251C7×D4—41no—
252C7×D4—81no—
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—42no—
275C2×C4×C7—82no—
276C2×C4×C7—81no—
277C2×C4×C7—82no—
278C2×C4×C7—82no—
279C2×C4×C7—81no—
280C2×C4×C7C2×C22⋊C411yescharacteristic
281C2×C4×C7#12011yescharacteristic
282C2×C4×C7#12011yescharacteristic
283C2×C4×C7#12011yescharacteristic
284C7×C8—42no—
285C7×C8—81no—
286C7×C8—81no—
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—
296C2×M16—21no—
297C2×M16—21no—
298C4⋊Q8—21no—
299C4⋊Q8C7×D411yescharacteristic
300#127—21no—
301#127C2×C4×C711yescharacteristic
302#126—22no—
303#126—22no—
304#126—22no—
305#125—21no—
306#125—22no—
307#124—21no—
308#124—22no—
309#124—21no—
310#124—21no—
311#124C2×C4×C711yescharacteristic
312#124C7×D411yescharacteristic
313#123—22no—
314#123—22no—
315#123—22no—
316#123—22no—
317#123C2×C4×C712yes—
318#123C7×D412yes—
319C22⋊Q8—21no—
320C22⋊Q8—22no—
321C22⋊Q8—22no—
322C22⋊Q8—21no—
323C22⋊Q8—21no—
324#121—22no—
325#121—22no—
326#121—21no—
327#121—22no—
328#121—22no—
329#121—21no—
330#121—21no—
331#121—22no—
332#121—21no—
333#121—21no—
334#121—22no—
335#120—22no—
336#120C2×C4×C712yes—
337#120C7×D412yes—
338C4×Q8—21no—
339C4×D4—22no—
340C4×D4—21no—
341C4×D4—22no—
342C4×D4—22no—
343C4×D4—21no—
344C4×D4—21no—
345#117—22no—
346#117—22no—
347#117—21no—
348#117C7×D411yescharacteristic
349#117C7×D411yescharacteristic
350C2×C4⋊C4—21no—
351C2×C22⋊C4—22no—
352C2×C42C7×D411yescharacteristic
353#104—42no—
354#101—21no—
355#101—21no—
356#101—42no—
357#100—21no—
358#100—21no—
359#100—42no—
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×C7B(4,𝔽2)11yescharacteristic
387C4×C7#45811yescharacteristic
388C24C7×C22⋊C412yes—
389#41—41no—
390#41—41no—
391#41—41no—
392#41—41no—
393#41—81no—
394#41—81no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—41no—
398C2×Q8—41no—
399C2×Q8C7×C22⋊C411yescharacteristic
400C2×Q8C2×C7×D411yescharacteristic
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—21no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4C7×C22⋊C411yescharacteristic
413C2×D4C22×C4×C711yescharacteristic
414C2×D4C2×C7×D411yescharacteristic
415C2×D4C2×C7×D411yescharacteristic
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4C7×C22⋊C412yes—
422C22×C4C2×C7×D411yescharacteristic
423C22×C4C2×C7×D411yescharacteristic
424C22×C4C2×C7×D411yescharacteristic
425M16—22no—
426M16—22no—
427M16—22no—
428M16—41no—
429M16—41no—
430M16—41no—
431M16—41no—
432M16—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—41no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—21no—
456C22⋊C4—22no—
457C22⋊C4—21no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—21no—
461C22⋊C4—21no—
462C22⋊C4—22no—
463C22⋊C4—41no—
464C22⋊C4C7×C22⋊C412yes—
465C22⋊C4C7×C22⋊C412yes—
466C42—22no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—41no—
471C42C7×C22⋊C411yescharacteristic
472C42C7×C22⋊C411yescharacteristic
473C2×C7—21no—
474C2×C7—41no—
475C2×C7—41no—
476C2×C7—41no—
477C2×C7—81no—
478C2×C7—82no—
479C2×C7#179011yescharacteristic
480C23—22no—
481C23—21no—
482C23—42no—
483C23—82no—
484C23C2×C7×C22⋊C411yescharacteristic
485C23C2×C7×C22⋊C411yescharacteristic
486C23C7×#12011yescharacteristic
487Q8—41no—
488Q8—41no—
489Q8—41no—
490Q8—41no—
491Q8—81no—
492D4—41no—
493D4—41no—
494D4—41no—
495D4—41no—
496D4—41no—
497D4—41no—
498D4—81no—
499D4—81no—
500D4—82no—
501D4—82no—
502D4—82no—
503D4—81no—
504D4—81no—
505D4—82no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—82no—
522C2×C4—81no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—81no—
526C2×C4C2×C7×C22⋊C411yescharacteristic
527C2×C4C7×#12011yescharacteristic
528C2×C4C7×#12011yescharacteristic
529C2×C4C7×#12011yescharacteristic
530C8—42no—
531C8—81no—
532C8—81no—
533C7#1277011yescharacteristic, direct product factor, Sylow
534C22—21no—
535C22—21no—
536C22—21no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—42no—
542C22—41no—
543C22—42no—
544C22—41no—
545C22—42no—
546C22—162no—
547C22C7×#37911yescharacteristic
548C4—21no—
549C4—21no—
550C4—21no—
551C4—41no—
552C4—42no—
553C4—42no—
554C4—41no—
555C4—41no—
556C4—42no—
557C4—161no—
558C4C7×B(4,𝔽2)11yescharacteristic
559C4C7×#45811yescharacteristic
560C2—21no—
561C2—41no—
562C2—41no—
563C2—41no—
564C2—82no—
565C2—81no—
566C2C7×#179011yescharacteristic
subgroupG/H
center, socleC2×C7#1790
commutatorC2×D4C22×C4×C7
Frattini, cosocleC2×#41C23×C7
prime radicalC7×#3551C2
LCSG ▷ C2×D4 ▷ 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×#2132C2semidirect
3. C7×#2123C2semidirect
4. C7×#2122C2semidirect
5. C7×#1915C2semidirect
6. C7×#1914C2semidirect
7. C7×#1884C2semidirect
10. C7×#560C4semidirect
14. C7×#551C4semidirect
37. #12770C7direct (N × H)
114. #2132C2×C7semidirect
115. #2123C2×C7semidirect
116. #2122C2×C7semidirect
117. #1915C2×C7semidirect
118. #1914C2×C7semidirect
119. #1884C2×C7semidirect
207. #560C4×C7semidirect
211. #551C4×C7semidirect
533. C7#12770direct (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×#21322
C7×#21232
C7×#21222
C7×#19152
C7×#19142
C7×#18842
#127707