Finite Group Databasebeta

← C7×#23986 #13201202

Subgroups — C7×#23986 #13201202

Order 1792.

1426 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3491C211yesmaximal, characteristic, complemented
2C7×#2849C212yesmaximal, complemented
3C7×#2280C211yesmaximal, characteristic, complemented
4C7×#2030C211yesmaximal, characteristic, complemented
5C7×#2029C211yesmaximal, characteristic, complemented
6C7×#2022C211yesmaximal, characteristic, complemented
7C7×#2021C211yesmaximal, characteristic
8C7×#1899C212yesmaximal, complemented
9C7×#1890C212yesmaximal, complemented
10C7×#1889C212yesmaximal, complemented
11C7×#1820C211yesmaximal, characteristic, complemented
12C24×C4×C7C2211yescharacteristic
13C7×#552—24no—
14C7×#551—24no—
15C7×#543C2211yescharacteristic
16C7×#539—22no—
17C7×#534C2211yescharacteristic
18C7×#529—22no—
19C2×C7×#123C2212yes—
20C2×C7×#123C2212yes—
21C2×C7×#120C2212yes—
22C7×#518C2212yes—
23C2×C7×#117C2211yescharacteristic
24C2×C7×#117C2211yescharacteristic
25C2×C7×#99C2212yes—
26C2×C7×#99C2212yes—
27C7×#386—24no—
28C7×#385—24no—
29C7×#381C2211yescharacteristic
30C7×#381C2212yes—
31C7×#381C2211yescharacteristic
32C7×#381C2212yes—
33C7×#380—22no—
34C7×#379C2211yescharacteristic
35C7×#379C2212yes—
36C7×#379C2211yescharacteristic
37C4×C7×C22⋊C4—22no—
38C7×#342C2211yescharacteristic
39C7×#342C2211yescharacteristic
40C7×#342C2212yes—
41C7×#342C2212yes—
42C7×#342C2212yes—
43#23986C711yesmaximal, characteristic, direct product factor, Sylow
44C25×C7C2311yescharacteristic
45C22×C7×D4—22no—
46C22×C7×D4C2312yes—
47C23×C4×C7—22no—
48C23×C4×C7—21no—
49C23×C4×C7—21no—
50C23×C4×C7—22no—
51C23×C4×C7—24no—
52C23×C4×C7—42no—
53C23×C4×C7C2311yescharacteristic
54C23×C4×C7C2311yescharacteristic
55C7×#126—24no—
56C7×#126—24no—
57C7×#125—22no—
58C7×#125—24no—
59C7×#124—22no—
60C7×#123—24no—
61C7×#123—22no—
62C7×#123—22no—
63C7×#123—24no—
64C7×#123—22no—
65C7×#123—24no—
66C7×#123—42no—
67C7×C22⋊Q8—24no—
68C7×C22⋊Q8—22no—
69C7×C22⋊Q8—24no—
70C7×#121—22no—
71C7×#120—22no—
72C7×#120—24no—
73C7×#120—42no—
74C4×C7×Q8—22no—
75C4×C7×D4—22no—
76C4×C7×D4—24no—
77C4×C7×D4—24no—
78C7×#117—24no—
79C7×#117—24no—
80C7×#117—22no—
81C7×#117—42no—
82C2×C7×C4⋊C4—21no—
83C2×C7×C4⋊C4—22no—
84C2×C7×C4⋊C4D412yescomplemented
85C2×C7×C4⋊C4D414yescomplemented
86C2×C7×C22⋊C4—22no—
87C2×C7×C22⋊C4—22no—
88C2×C7×C22⋊C4—22no—
89C2×C7×C22⋊C4—22no—
90C2×C7×C22⋊C4—24no—
91C2×C7×C22⋊C4—22no—
92C2×C7×C22⋊C4D414yescomplemented
93C2×C7×C22⋊C4C2312yes—
94C2×C7×C22⋊C4C2312yes—
95C2×C7×C22⋊C4C2311yescharacteristic
96C2×C7×C22⋊C4C2311yescharacteristic
97C2×C7×C22⋊C4C2311yescharacteristic
98C2×C7×C22⋊C4C2311yescharacteristic
99C2×C7×C22⋊C4C2312yes—
100C2×C42×C7—21no—
101C2×C42×C7D412yescomplemented
102C7×#99—42no—
103C7×#99—42no—
104C7×#95—22no—
105C7×#95—22no—
106C7×#95—24no—
107C7×#95—24no—
108#3491C2×C711yescharacteristic, complemented
109#2849C2×C712yescomplemented
110#2280C2×C711yescharacteristic, complemented
111#2030C2×C711yescharacteristic, complemented
112#2029C2×C711yescharacteristic, complemented
113#2022C2×C711yescharacteristic, complemented
114#2021C2×C711yescharacteristic
115#1899C2×C712yescomplemented
116#1890C2×C712yescomplemented
117#1889C2×C712yescomplemented
118#1820C2×C711yescharacteristic, complemented
119C24×C7—21no—
120C24×C7—22no—
121C24×C7—21no—
122C24×C7—22no—
123C24×C7—44no—
124C24×C7C2×D412yes—
125C24×C7#4112yes—
126C24×C7C2411yescharacteristic
127C2×C7×Q8—22no—
128C2×C7×D4—22no—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4—22no—
132C2×C7×D4—22no—
133C2×C7×D4—22no—
134C2×C7×D4—22no—
135C2×C7×D4—44no—
136C2×C7×D4—48no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—22no—
140C22×C4×C7—24no—
141C22×C4×C7—24no—
142C22×C4×C7—21no—
143C22×C4×C7—22no—
144C22×C4×C7—21no—
145C22×C4×C7—22no—
146C22×C4×C7—24no—
147C22×C4×C7—22no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—22no—
151C22×C4×C7—48no—
152C22×C4×C7—48no—
153C22×C4×C7—42no—
154C22×C4×C7—42no—
155C22×C4×C7—42no—
156C22×C4×C7—42no—
157C22×C4×C7C2×D412yes—
158C22×C4×C7C2×D414yes—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D412yes—
162C22×C4×C7C2×D412yes—
163C22×C4×C7C2×D414yes—
164C22×C4×C7#4112yes—
165C7×C4⋊C4—24no—
166C7×C4⋊C4—24no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—24no—
170C7×C4⋊C4—24no—
171C7×C4⋊C4—24no—
172C7×C4⋊C4—44no—
173C7×C4⋊C4—42no—
174C7×C22⋊C4—24no—
175C7×C22⋊C4—24no—
176C7×C22⋊C4—22no—
177C7×C22⋊C4—22no—
178C7×C22⋊C4—22no—
179C7×C22⋊C4—22no—
180C7×C22⋊C4—24no—
181C7×C22⋊C4—24no—
182C7×C22⋊C4—22no—
183C7×C22⋊C4—22no—
184C7×C22⋊C4—22no—
185C7×C22⋊C4—22no—
186C7×C22⋊C4—22no—
187C7×C22⋊C4—22no—
188C7×C22⋊C4—44no—
189C7×C22⋊C4—44no—
190C7×C22⋊C4—44no—
191C7×C22⋊C4—44no—
192C7×C22⋊C4—48no—
193C42×C7—24no—
194C42×C7—22no—
195C42×C7—22no—
196C42×C7—42no—
197C24×C4C22×C711yescharacteristic
198#552—24no—
199#551—24no—
200#543C22×C711yescharacteristic
201#539—22no—
202#534C22×C711yescharacteristic
203#529—22no—
204C2×#123C22×C712yes—
205C2×#123C22×C712yes—
206C2×#120C22×C712yes—
207#518C22×C712yes—
208C2×#117C22×C711yescharacteristic
209C2×#117C22×C711yescharacteristic
210C2×#99C22×C712yes—
211C2×#99C22×C712yes—
212#386—24no—
213#385—24no—
214#381C22×C712yes—
215#381C22×C712yes—
216#381C22×C711yescharacteristic
217#381C22×C711yescharacteristic
218#380—22no—
219#379C22×C712yes—
220#379C22×C711yescharacteristic
221#379C22×C711yescharacteristic
222C4×C22⋊C4—22no—
223#342C22×C712yes—
224#342C22×C711yescharacteristic
225#342C22×C711yescharacteristic
226#342C22×C712yes—
227#342C22×C712yes—
228C23×C7—24no—
229C23×C7—28no—
230C23×C7—22no—
231C23×C7—24no—
232C23×C7—22no—
233C23×C7—22no—
234C23×C7—22no—
235C23×C7—44no—
236C23×C7—42no—
237C23×C7—42no—
238C23×C7—44no—
239C23×C7—48no—
240C23×C7—44no—
241C23×C7—44no—
242C23×C7—41no—
243C23×C7—41no—
244C23×C7—42no—
245C23×C7C22×D411yescharacteristic
246C23×C7C22×D412yes—
247C23×C7C2×#4111yescharacteristic
248C23×C7C2×#4111yescharacteristic
249C23×C7H(5,𝔽2)12yes—
250C7×Q8—82no—
251C7×D4—42no—
252C7×D4—42no—
253C7×D4—42no—
254C7×D4—42no—
255C7×D4—82no—
256C7×D4—88no—
257C2×C4×C7—21no—
258C2×C4×C7—21no—
259C2×C4×C7—21no—
260C2×C4×C7—21no—
261C2×C4×C7—24no—
262C2×C4×C7—24no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—24no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—24no—
270C2×C4×C7—22no—
271C2×C4×C7—48no—
272C2×C4×C7—48no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—44no—
278C2×C4×C7—42no—
279C2×C4×C7—42no—
280C2×C4×C7—42no—
281C2×C4×C7—44no—
282C2×C4×C7—44no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—42no—
286C2×C4×C7—42no—
287C2×C4×C7—84no—
288C2×C4×C7#12012yes—
289C2×C4×C7#12012yes—
290C25C23×C711yescharacteristic
291C22×D4—22no—
292C22×D4C23×C712yes—
293C23×C4—22no—
294C23×C4—21no—
295C23×C4—21no—
296C23×C4—22no—
297C23×C4—24no—
298C23×C4—42no—
299C23×C4C23×C711yescharacteristic
300C23×C4C23×C711yescharacteristic
301#126—24no—
302#126—24no—
303#125—22no—
304#125—24no—
305#124—22no—
306#123—24no—
307#123—22no—
308#123—22no—
309#123—24no—
310#123—22no—
311#123—24no—
312#123—42no—
313C22⋊Q8—24no—
314C22⋊Q8—22no—
315C22⋊Q8—24no—
316#121—22no—
317#120—22no—
318#120—24no—
319#120—42no—
320C4×Q8—22no—
321C4×D4—22no—
322C4×D4—24no—
323C4×D4—24no—
324#117—24no—
325#117—24no—
326#117—22no—
327#117—42no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4C7×D412yescomplemented
331C2×C4⋊C4C7×D414yescomplemented
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4C7×D414yescomplemented
339C2×C22⋊C4C23×C711yescharacteristic
340C2×C22⋊C4C23×C711yescharacteristic
341C2×C22⋊C4C23×C712yes—
342C2×C22⋊C4C23×C712yes—
343C2×C22⋊C4C23×C711yescharacteristic
344C2×C22⋊C4C23×C711yescharacteristic
345C2×C22⋊C4C23×C712yes—
346C2×C42—21no—
347C2×C42C7×D412yescomplemented
348#99—42no—
349#99—42no—
350#95—22no—
351#95—22no—
352#95—24no—
353#95—24no—
354C22×C7—22no—
355C22×C7—24no—
356C22×C7—28no—
357C22×C7—24no—
358C22×C7—41no—
359C22×C7—41no—
360C22×C7—42no—
361C22×C7—44no—
362C22×C7—44no—
363C22×C7—48no—
364C22×C7—42no—
365C22×C7—42no—
366C22×C7—44no—
367C22×C7—42no—
368C22×C7—44no—
369C22×C7—82no—
370C22×C7—82no—
371C22×C7C2×#12011yescharacteristic
372C22×C7#52511yescharacteristic
373C22×C7#53511yescharacteristic
374C22×C7#54612yes—
375C22×C7#54612yes—
376C4×C7—22no—
377C4×C7—22no—
378C4×C7—42no—
379C4×C7—42no—
380C4×C7—42no—
381C4×C7—44no—
382C4×C7—42no—
383C4×C7—81no—
384C4×C7—81no—
385C4×C7—82no—
386C4×C7—82no—
387C24—21no—
388C24—22no—
389C24—21no—
390C24—22no—
391C24—44no—
392C24C2×C7×D412yes—
393C24C7×#4112yes—
394C24C24×C711yescharacteristic
395C2×Q8—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—44no—
404C2×D4—48no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4C2×C7×D412yes—
426C22×C4C2×C7×D414yes—
427C22×C4C2×C7×D411yescharacteristic
428C22×C4C2×C7×D411yescharacteristic
429C22×C4C2×C7×D412yes—
430C22×C4C2×C7×D412yes—
431C22×C4C2×C7×D414yes—
432C22×C4C7×#4112yes—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—44no—
441C4⋊C4—42no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—48no—
461C42—24no—
462C42—22no—
463C42—22no—
464C42—42no—
465C2×C7—21no—
466C2×C7—22no—
467C2×C7—22no—
468C2×C7—21no—
469C2×C7—44no—
470C2×C7—82no—
471C2×C7#240411yescharacteristic
472C2×C7#302512yes—
473C23—24no—
474C23—28no—
475C23—22no—
476C23—24no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—44no—
481C23—42no—
482C23—42no—
483C23—44no—
484C23—48no—
485C23—44no—
486C23—44no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23C22×C7×D411yescharacteristic
491C23C22×C7×D412yes—
492C23C2×C7×#4111yescharacteristic
493C23C2×C7×#4111yescharacteristic
494C23C7×H(5,𝔽2)12yes—
495Q8—82no—
496D4—42no—
497D4—42no—
498D4—42no—
499D4—42no—
500D4—82no—
501D4—88no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—48no—
517C2×C4—48no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—84no—
533C2×C4C7×#12012yes—
534C2×C4C7×#12012yes—
535C7#2398611yescharacteristic, direct product factor, Sylow
536C22—22no—
537C22—24no—
538C22—28no—
539C22—24no—
540C22—41no—
541C22—41no—
542C22—42no—
543C22—44no—
544C22—44no—
545C22—48no—
546C22—42no—
547C22—42no—
548C22—44no—
549C22—42no—
550C22—44no—
551C22—82no—
552C22—82no—
553C22C2×C7×#12011yescharacteristic
554C22C7×#52511yescharacteristic
555C22C7×#53511yescharacteristic
556C22C7×#54612yes—
557C22C7×#54612yes—
558C4—22no—
559C4—22no—
560C4—42no—
561C4—42no—
562C4—42no—
563C4—44no—
564C4—42no—
565C4—82no—
566C4—81no—
567C4—81no—
568C4—82no—
569C2—21no—
570C2—22no—
571C2—22no—
572C2—21no—
573C2—44no—
574C2—82no—
575C2C7×#240411yescharacteristic
576C2C7×#302512yes—
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC24C24×C7
prime radicalC7×#2021C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3491C2semidirect
2. C7×#2849C2semidirect
3. C7×#2280C2semidirect
4. C7×#2030C2semidirect
5. C7×#2029C2semidirect
6. C7×#2022C2semidirect
8. C7×#1899C2semidirect
9. C7×#1890C2semidirect
10. C7×#1889C2semidirect
11. C7×#1820C2semidirect
43. #23986C7direct (N × H)
84. C2×C7×C4⋊C4D4semidirect
85. C2×C7×C4⋊C4D4semidirect
92. C2×C7×C22⋊C4D4semidirect
101. C2×C42×C7D4semidirect
108. #3491C2×C7semidirect
109. #2849C2×C7semidirect
110. #2280C2×C7semidirect
111. #2030C2×C7semidirect
112. #2029C2×C7semidirect
113. #2022C2×C7semidirect
115. #1899C2×C7semidirect
116. #1890C2×C7semidirect
117. #1889C2×C7semidirect
118. #1820C2×C7semidirect
330. C2×C4⋊C4C7×D4semidirect
331. C2×C4⋊C4C7×D4semidirect
338. C2×C22⋊C4C7×D4semidirect
347. C2×C42C7×D4semidirect
535. C7#23986direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34912
C7×#28492
C7×#22802
C7×#20302
C7×#20292
C7×#20222
C7×#20212
C7×#18992
C7×#18902
C7×#18892
C7×#18202
#239867