Finite Group Databasebeta

← C7×#24128 #13201344

Subgroups — C7×#24128 #13201344

Order 1792.

1400 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#547C211yesmaximal, characteristic, complemented
2C2×C7×#476C211yesmaximal, characteristic, complemented
3C2×C7×#442C211yesmaximal, characteristic, complemented
4C7×#2535C211yesmaximal, characteristic, complemented
5C7×#2034C212yesmaximal, complemented
6C7×#1999C212yesmaximal, complemented
7C7×#1993C211yesmaximal, characteristic, complemented
8C7×#1955C211yesmaximal, characteristic, complemented
9C7×#1874C212yesmaximal, complemented
10C7×#1853C212yesmaximal, complemented
11C7×#1804C211yesmaximal, characteristic, complemented
12C22×C7×#41—21no—
13C22×C7×D8C2211yescharacteristic
14C7×#547—24no—
15C7×#547—24no—
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×#123—22no—
18C2×C7×C22⋊Q8—21no—
19C2×C7×#121C2212yes—
20C2×C7×#121C2212yes—
21C2×C4×C7×D4C2211yescharacteristic
22C2×C4×C7×D4C2211yescharacteristic
23C2×C4×C7×D4C2211yescharacteristic
24C22×C7×C4⋊C4C2212yes—
25C7×#476—24no—
26C7×#442—24no—
27C2×C7×#106C2211yescharacteristic
28C2×C7×#106C2212yes—
29C2×C7×C4⋊C8C2211yescharacteristic
30C2×C7×#102C2212yes—
31C2×C7×#102C2211yescharacteristic
32C2×C7×#102C2211yescharacteristic
33C2×C7×#102C2212yes—
34C2×C7×C22⋊C8C2212yes—
35C2×C7×C22⋊C8C2212yes—
36C2×C7×#97C2211yescharacteristic
37C7×#397—21no—
38C7×#392—21no—
39C7×#386—22no—
40C7×#382C2211yescharacteristic
41C7×#380C2212yes—
42C7×#340C2212yes—
43C7×#340C2212yes—
44C7×#340C2212yes—
45#24128C711yesmaximal, characteristic, direct product factor, Sylow
46C2×C7×#41—28no—
47C2×C7×#41—48no—
48C22×C7×Q8—21no—
49C22×C7×D4—22no—
50C22×C7×D4—22no—
51C22×C7×D4D411yescharacteristic
52C22×C7×D4C2311yescharacteristic
53C22×C7×D4C2311yescharacteristic
54C23×C4×C7—22no—
55C23×C4×C7—21no—
56C23×C4×C7—21no—
57C23×C4×C7C2312yes—
58C2×C7×D8—22no—
59C2×C7×D8—22no—
60C2×C7×D8—22no—
61C22×C7×C8C2311yescharacteristic
62C22×C7×C8C2312yes—
63C22×C7×C8C2311yescharacteristic
64C7×C4⋊Q8—22no—
65C7×C4⋊Q8D414yescomplemented
66C7×#123—216no—
67C7×C22⋊Q8—28no—
68C7×#121—24no—
69C7×#121—24no—
70C7×#121—24no—
71C7×#121—42no—
72C4×C7×D4—22no—
73C4×C7×D4—22no—
74C4×C7×D4—22no—
75C4×C7×D4—22no—
76C4×C7×D4—22no—
77C4×C7×D4D414yescomplemented
78C2×C7×C4⋊C4—22no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4—24no—
81C2×C7×C4⋊C4—24no—
82C2×C7×C4⋊C4D411yescharacteristic
83C2×C7×C4⋊C4C2312yes—
84C2×C7×C4⋊C4C2311yescharacteristic
85C2×C7×C4⋊C4C2311yescharacteristic
86C2×C7×C4⋊C4C2312yes—
87C2×C7×C22⋊C4—21no—
88C2×C7×C22⋊C4—22no—
89C2×C7×C22⋊C4—21no—
90C2×C7×C22⋊C4—22no—
91C2×C7×C22⋊C4—22no—
92C2×C7×C22⋊C4D412yes—
93C2×C42×C7C2311yescharacteristic
94C7×#106—24no—
95C7×#106—22no—
96C7×C4⋊C8—22no—
97C7×#102—22no—
98C7×#102—24no—
99C7×#102—22no—
100C7×#102—42no—
101C7×C22⋊C8—24no—
102C7×C22⋊C8—42no—
103C7×#97—22no—
104C7×#95—22no—
105C7×#95—21no—
106C7×#95—21no—
107C2×#547C2×C711yescharacteristic, complemented
108C2×#476C2×C711yescharacteristic, complemented
109C2×#442C2×C711yescharacteristic, complemented
110#2535C2×C711yescharacteristic, complemented
111#2034C2×C712yescomplemented
112#1999C2×C712yescomplemented
113#1993C2×C711yescharacteristic, complemented
114#1955C2×C711yescharacteristic, complemented
115#1874C2×C712yescomplemented
116#1853C2×C712yescomplemented
117#1804C2×C711yescharacteristic, complemented
118C24×C7—21no—
119C24×C7—21no—
120C24×C7C2×D412yes—
121C7×#41—432no—
122C2×C7×Q8—24no—
123C2×C7×Q8—44no—
124C2×C7×D4—24no—
125C2×C7×D4—28no—
126C2×C7×D4—22no—
127C2×C7×D4—21no—
128C2×C7×D4—21no—
129C2×C7×D4—21no—
130C2×C7×D4—21no—
131C2×C7×D4—21no—
132C2×C7×D4—21no—
133C2×C7×D4—21no—
134C2×C7×D4—21no—
135C2×C7×D4—21no—
136C2×C7×D4—21no—
137C2×C7×D4—44no—
138C2×C7×D4—44no—
139C2×C7×D4—44no—
140C2×C7×D4—44no—
141C2×C7×D4—82no—
142C2×C7×D4C2×D412yes—
143C2×C7×D4#4112yescomplemented
144C22×C4×C7—28no—
145C22×C4×C7—21no—
146C22×C4×C7—22no—
147C22×C4×C7—24no—
148C22×C4×C7—22no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—22no—
154C22×C4×C7—22no—
155C22×C4×C7—21no—
156C22×C4×C7—44no—
157C22×C4×C7—44no—
158C22×C4×C7—44no—
159C22×C4×C7—42no—
160C22×C4×C7—82no—
161C22×C4×C7C2×D412yes—
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2411yescharacteristic
165C7×D8—28no—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C2×C7×C8—21no—
169C2×C7×C8—22no—
170C2×C7×C8—22no—
171C2×C7×C8—21no—
172C2×C7×C8—22no—
173C2×C7×C8—21no—
174C2×C7×C8#4112yescomplemented
175C7×C4⋊C4—28no—
176C7×C4⋊C4—24no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4C2×D412yes—
185C7×C4⋊C4C2×D414yes—
186C7×C22⋊C4—28no—
187C7×C22⋊C4—24no—
188C7×C22⋊C4—24no—
189C7×C22⋊C4—44no—
190C7×C22⋊C4—42no—
191C7×C22⋊C4—82no—
192C42×C7C2×D412yes—
193C42×C7C2×D412yes—
194C22×#41—21no—
195C22×D8C22×C711yescharacteristic
196#547—24no—
197#547—24no—
198C2×C4⋊Q8C22×C711yescharacteristic
199C2×#123—22no—
200C2×C22⋊Q8—21no—
201C2×#121C22×C712yes—
202C2×#121C22×C712yes—
203C2×C4×D4C22×C711yescharacteristic
204C2×C4×D4C22×C711yescharacteristic
205C2×C4×D4C22×C711yescharacteristic
206C22×C4⋊C4C22×C712yes—
207#476—24no—
208#442—24no—
209C2×#106C22×C711yescharacteristic
210C2×#106C22×C712yes—
211C2×C4⋊C8C22×C711yescharacteristic
212C2×#102C22×C711yescharacteristic
213C2×#102C22×C712yes—
214C2×#102C22×C711yescharacteristic
215C2×#102C22×C712yes—
216C2×C22⋊C8C22×C712yes—
217C2×C22⋊C8C22×C712yes—
218C2×#97C22×C711yescharacteristic
219#397—21no—
220#392—21no—
221#386—22no—
222#382C22×C711yescharacteristic
223#380C22×C712yes—
224#340C22×C712yes—
225#340C22×C712yes—
226#340C22×C712yes—
227C23×C7—22no—
228C23×C7—22no—
229C23×C7—22no—
230C23×C7—22no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—22no—
234C23×C7—22no—
235C23×C7—41no—
236C23×C7—41no—
237C23×C7—41no—
238C23×C7—41no—
239C23×C7—42no—
240C23×C7—42no—
241C23×C7—42no—
242C23×C7—42no—
243C23×C7AGL(1,ℤ/8)14yes—
244C23×C7C22×D411yescharacteristic
245C7×Q8—48no—
246C7×D4—28no—
247C7×D4—22no—
248C7×D4—22no—
249C7×D4—22no—
250C7×D4—22no—
251C7×D4—416no—
252C7×D4—41no—
253C7×D4—41no—
254C7×D4—41no—
255C7×D4—41no—
256C7×D4—88no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—22no—
260C2×C4×C7—22no—
261C2×C4×C7—24no—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
264C2×C4×C7—48no—
265C2×C4×C7—416no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7—42no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7—41no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—84no—
279C2×C4×C7#12012yes—
280C2×C4×C7#12012yes—
281C2×C4×C7C22×D411yescharacteristic
282C2×C4×C7C22×D411yescharacteristic
283C2×C4×C7C2×#4111yescharacteristic
284C2×C4×C7C2×#4111yescharacteristic
285C2×C4×C7H(5,𝔽2)11yescharacteristic
286C2×C4×C72−1+411yescharacteristic
287C7×C8—22no—
288C7×C8—41no—
289C7×C8—42no—
290C2×#41—28no—
291C2×#41—48no—
292C22×Q8—21no—
293C22×D4—22no—
294C22×D4—22no—
295C22×D4C7×D411yescharacteristic
296C22×D4C23×C711yescharacteristic
297C22×D4C23×C711yescharacteristic
298C23×C4—22no—
299C23×C4—21no—
300C23×C4—21no—
301C23×C4C23×C712yes—
302C2×D8—22no—
303C2×D8—22no—
304C2×D8—22no—
305C22×C8C23×C711yescharacteristic
306C22×C8C23×C712yes—
307C22×C8C23×C711yescharacteristic
308C4⋊Q8—22no—
309C4⋊Q8C7×D414yescomplemented
310#123—216no—
311C22⋊Q8—28no—
312#121—24no—
313#121—24no—
314#121—24no—
315#121—42no—
316C4×D4—22no—
317C4×D4—22no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—22no—
321C4×D4C7×D414yescomplemented
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4C7×D411yescharacteristic
327C2×C4⋊C4C23×C711yescharacteristic
328C2×C4⋊C4C23×C712yes—
329C2×C4⋊C4C23×C711yescharacteristic
330C2×C4⋊C4C23×C712yes—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4C7×D412yes—
337C2×C42C23×C711yescharacteristic
338#106—22no—
339#106—24no—
340C4⋊C8—22no—
341#102—22no—
342#102—22no—
343#102—24no—
344#102—42no—
345C22⋊C8—24no—
346C22⋊C8—42no—
347#97—22no—
348#95—22no—
349#95—21no—
350#95—21no—
351C22×C7—24no—
352C22×C7—24no—
353C22×C7—24no—
354C22×C7—42no—
355C22×C7—42no—
356C22×C7—42no—
357C22×C7—42no—
358C22×C7—42no—
359C22×C7—42no—
360C22×C7—42no—
361C22×C7—42no—
362C22×C7—44no—
363C22×C7—82no—
364C22×C7—82no—
365C22×C7—81no—
366C22×C7—81no—
367C22×C7—82no—
368C22×C7C2×#12011yescharacteristic
369C22×C7#54611yescharacteristic
370C22×C7#54711yescharacteristic
371C22×C7C2×AGL(1,ℤ/8)12yes—
372C22×C7#57712yes—
373C4×C7—22no—
374C4×C7—22no—
375C4×C7—42no—
376C4×C7—42no—
377C4×C7—44no—
378C4×C7—81no—
379C4×C7—82no—
380C4×C7#52511yescharacteristic
381C4×C7#53711yescharacteristic
382C4×C7#54811yescharacteristic
383C4×C7#55011yescharacteristic
384C24—21no—
385C24—21no—
386C24C2×C7×D412yes—
387#41—432no—
388C2×Q8—24no—
389C2×Q8—44no—
390C2×D4—24no—
391C2×D4—28no—
392C2×D4—22no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—82no—
408C2×D4C2×C7×D412yes—
409C2×D4C7×#4112yescomplemented
410C22×C4—28no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—82no—
427C22×C4C2×C7×D412yes—
428C22×C4C2×C7×D411yescharacteristic
429C22×C4C2×C7×D411yescharacteristic
430C22×C4C24×C711yescharacteristic
431D8—28no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—21no—
438C2×C8—22no—
439C2×C8—21no—
440C2×C8C7×#4112yescomplemented
441C4⋊C4—28no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4C2×C7×D412yes—
451C4⋊C4C2×C7×D414yes—
452C22⋊C4—28no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—44no—
456C22⋊C4—42no—
457C22⋊C4—82no—
458C42C2×C7×D412yes—
459C42C2×C7×D412yes—
460C2×C7—44no—
461C2×C7—82no—
462C2×C7—82no—
463C2×C7#240711yescharacteristic
464C2×C7#300311yescharacteristic
465C2×C7#301011yescharacteristic
466C2×C7#330612yes—
467C2×C7#334512yes—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23C7×AGL(1,ℤ/8)14yes—
485C23C22×C7×D411yescharacteristic
486Q8—48no—
487D4—28no—
488D4—22no—
489D4—22no—
490D4—22no—
491D4—22no—
492D4—416no—
493D4—41no—
494D4—41no—
495D4—41no—
496D4—41no—
497D4—88no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—48no—
506C2×C4—416no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—84no—
520C2×C4C7×#12012yes—
521C2×C4C7×#12012yes—
522C2×C4C22×C7×D411yescharacteristic
523C2×C4C22×C7×D411yescharacteristic
524C2×C4C2×C7×#4111yescharacteristic
525C2×C4C2×C7×#4111yescharacteristic
526C2×C4C7×H(5,𝔽2)11yescharacteristic
527C2×C4C7×2−1+411yescharacteristic
528C8—22no—
529C8—41no—
530C8—42no—
531C7#2412811yescharacteristic, direct product factor, Sylow
532C22—24no—
533C22—24no—
534C22—24no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—44no—
544C22—81no—
545C22—81no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22C2×C7×#12011yescharacteristic
550C22C7×#54611yescharacteristic
551C22C7×#54711yescharacteristic
552C22C2×C7×AGL(1,ℤ/8)12yes—
553C22C7×#57712yes—
554C4—22no—
555C4—22no—
556C4—42no—
557C4—42no—
558C4—44no—
559C4—81no—
560C4—82no—
561C4C7×#52511yescharacteristic
562C4C7×#53711yescharacteristic
563C4C7×#54811yescharacteristic
564C4C7×#55011yescharacteristic
565C2—44no—
566C2—82no—
567C2—82no—
568C2C7×#240711yescharacteristic
569C2C7×#300311yescharacteristic
570C2C7×#301011yescharacteristic
571C2C7×#330612yes—
572C2C7×#334512yes—
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#547C2semidirect
2. C2×C7×#476C2semidirect
3. C2×C7×#442C2semidirect
4. C7×#2535C2semidirect
5. C7×#2034C2semidirect
6. C7×#1999C2semidirect
7. C7×#1993C2semidirect
8. C7×#1955C2semidirect
9. C7×#1874C2semidirect
10. C7×#1853C2semidirect
11. C7×#1804C2semidirect
45. #24128C7direct (N × H)
65. C7×C4⋊Q8D4semidirect
77. C4×C7×D4D4semidirect
107. C2×#547C2×C7semidirect
108. C2×#476C2×C7semidirect
109. C2×#442C2×C7semidirect
110. #2535C2×C7semidirect
111. #2034C2×C7semidirect
112. #1999C2×C7semidirect
113. #1993C2×C7semidirect
114. #1955C2×C7semidirect
115. #1874C2×C7semidirect
116. #1853C2×C7semidirect
117. #1804C2×C7semidirect
143. C2×C7×D4#41semidirect
174. C2×C7×C8#41semidirect
309. C4⋊Q8C7×D4semidirect
321. C4×D4C7×D4semidirect
409. C2×D4C7×#41semidirect
440. C2×C8C7×#41semidirect
531. C7#24128direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5472
C2×C7×#4762
C2×C7×#4422
C7×#25352
C7×#20342
C7×#19992
C7×#19932
C7×#19552
C7×#18742
C7×#18532
C7×#18042
#241287