Finite Group Databasebeta

← C7×#24509 #13201725

Subgroups — C7×#24509 #13201725

Order 1792.

1034 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#448C211yesmaximal, characteristic
2C2×C7×#432C211yesmaximal, characteristic, complemented
3C7×#2854C211yesmaximal, characteristic
4C2×C7×#391C211yesmaximal, characteristic, complemented
5C7×#2068C212yesmaximal, complemented
6C7×#2055C212yesmaximal, complemented
7C7×#1973C212yesmaximal, complemented
8C7×#1935C212yesmaximal, complemented
9C7×#1874C211yesmaximal, characteristic
10C7×#1852C211yesmaximal, characteristic, complemented
11C7×#1787C211yesmaximal, characteristic, complemented
12C23×C7×Q8C2211yescharacteristic
13C22×C7×SD16C2212yes—
14C2×C7×#123—21no—
15C2×C7×#121C2211yescharacteristic
16C22×C7×C4⋊C4C2211yescharacteristic
17C22×C42×C7C2211yescharacteristic
18C7×#448—24no—
19C7×#432—24no—
20C2×C7×#107C2212yes—
21C2×C7×C4⋊C8C2212yes—
22C2×C7×#103C2212yes—
23C2×C7×#103C2212yes—
24C2×C7×#103C2212yes—
25C2×C7×#102C2212yes—
26C2×C7×C22⋊C8C2211yescharacteristic
27C2×C7×C22⋊C8C2211yescharacteristic
28C2×C7×#97C2212yes—
29C7×#391—21no—
30C7×#391—21no—
31C7×#391C2212yes—
32C7×#391C2212yes—
33C7×#390C2211yescharacteristic
34C7×#390C2211yescharacteristic
35C7×#389C2211yescharacteristic
36C7×#389C2211yescharacteristic
37C7×#388—21no—
38C7×#388—21no—
39C7×#386—21no—
40C7×#380C2211yescharacteristic
41C2×C7×#95—21no—
42C2×C7×#95—21no—
43C7×#340C2212yes—
44C7×#340C2212yes—
45#24509C711yesmaximal, characteristic, direct product factor, Sylow
46C22×C7×Q8—24no—
47C22×C7×Q8—21no—
48C22×C7×Q8—22no—
49C22×C7×Q8—22no—
50C22×C7×Q8—42no—
51C22×C7×Q8C2312yes—
52C22×C7×D4—21no—
53C22×C7×D4C2311yescharacteristic
54C23×C4×C7—21no—
55C23×C4×C7—21no—
56C23×C4×C7D411yescharacteristic
57C23×C4×C7D411yescharacteristic
58C23×C4×C7C2311yescharacteristic
59C2×C7×SD16—24no—
60C2×C7×SD16—24no—
61C2×C7×SD16—42no—
62C22×C7×C8C2312yes—
63C22×C7×C8C2312yes—
64C7×#123—44no—
65C7×#121—22no—
66C7×#121—22no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—42no—
71C2×C7×C4⋊C4C2312yes—
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C22⋊C4—21no—
76C2×C7×C22⋊C4—21no—
77C2×C7×C22⋊C4—21no—
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4—21no—
80C2×C7×C22⋊C4—21no—
81C2×C42×C7—22no—
82C2×C42×C7—22no—
83C2×C42×C7D411yescharacteristic
84C2×C42×C7D411yescharacteristic
85C2×C42×C7C2311yescharacteristic
86C2×C42×C7C2311yescharacteristic
87C7×#107—42no—
88C7×C4⋊C8—24no—
89C7×#103—24no—
90C7×#103—24no—
91C7×#103—42no—
92C7×#102—42no—
93C7×C22⋊C8—22no—
94C7×C22⋊C8D414yes—
95C7×#97—24no—
96C7×#95—22no—
97C7×#95—22no—
98C7×#95—22no—
99C7×#95—22no—
100C7×#95—21no—
101C7×#95—21no—
102C7×#95—21no—
103C2×#448C2×C711yescharacteristic
104C2×#432C2×C711yescharacteristic, complemented
105#2854C2×C711yescharacteristic
106C2×#391C2×C711yescharacteristic, complemented
107#2068C2×C712yescomplemented
108#2055C2×C712yescomplemented
109#1973C2×C712yescomplemented
110#1935C2×C712yescomplemented
111#1874C2×C711yescharacteristic
112#1852C2×C711yescharacteristic, complemented
113#1787C2×C711yescharacteristic, complemented
114C24×C7—21no—
115C24×C7C2×D411yescharacteristic
116C2×C7×Q8—24no—
117C2×C7×Q8—22no—
118C2×C7×Q8—22no—
119C2×C7×Q8—22no—
120C2×C7×Q8—22no—
121C2×C7×Q8—22no—
122C2×C7×Q8—48no—
123C2×C7×Q8—48no—
124C2×C7×Q8—42no—
125C2×C7×Q8—42no—
126C2×C7×Q8#4114yes—
127C2×C7×D4—21no—
128C2×C7×D4—21no—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4—42no—
132C2×C7×D4—42no—
133C2×C7×D4—42no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—24no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—41no—
148C22×C4×C7—42no—
149C22×C4×C7—42no—
150C22×C4×C7—42no—
151C22×C4×C7—42no—
152C22×C4×C7—42no—
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D412yes—
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7#4112yes—
160C22×C4×C7C2411yescharacteristic
161C7×SD16—48no—
162C2×C7×C8—22no—
163C2×C7×C8—22no—
164C2×C7×C8—22no—
165C2×C7×C8—22no—
166C2×C7×C8—22no—
167C2×C7×C8C2×D414yes—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—42no—
175C7×C4⋊C4—42no—
176C7×C4⋊C4—42no—
177C7×C22⋊C4—42no—
178C7×C22⋊C4—42no—
179C7×C22⋊C4—42no—
180C7×C22⋊C4—42no—
181C7×C22⋊C4—42no—
182C7×C22⋊C4—42no—
183C42×C7—22no—
184C42×C7—22no—
185C42×C7—21no—
186C42×C7—21no—
187C42×C7—21no—
188C42×C7—21no—
189C23×Q8C22×C711yescharacteristic
190C22×SD16C22×C712yes—
191C2×#123—21no—
192C2×#121C22×C711yescharacteristic
193C22×C4⋊C4C22×C711yescharacteristic
194C22×C42C22×C711yescharacteristic
195#448—24no—
196#432—24no—
197C2×#107C22×C712yes—
198C2×C4⋊C8C22×C712yes—
199C2×#103C22×C712yes—
200C2×#103C22×C712yes—
201C2×#103C22×C712yes—
202C2×#102C22×C712yes—
203C2×C22⋊C8C22×C711yescharacteristic
204C2×C22⋊C8C22×C711yescharacteristic
205C2×#97C22×C712yes—
206#391—21no—
207#391—21no—
208#391C22×C712yes—
209#391C22×C712yes—
210#390C22×C711yescharacteristic
211#390C22×C711yescharacteristic
212#389C22×C711yescharacteristic
213#389C22×C711yescharacteristic
214#388—21no—
215#388—21no—
216#386—21no—
217#380C22×C711yescharacteristic
218C2×#95—21no—
219C2×#95—21no—
220#340C22×C712yes—
221#340C22×C712yes—
222C23×C7—21no—
223C23×C7—21no—
224C23×C7—21no—
225C23×C7—21no—
226C23×C7—41no—
227C23×C7—41no—
228C23×C7—41no—
229C23×C7—41no—
230C23×C7—41no—
231C23×C7—41no—
232C23×C7—41no—
233C23×C7#12112yes—
234C23×C7AGL(1,ℤ/8)12yes—
235C23×C7#13712yes—
236C23×C7C22×D411yescharacteristic
237C7×Q8—24no—
238C7×Q8—24no—
239C7×Q8—48no—
240C7×Q8—42no—
241C7×Q8—42no—
242C7×D4—42no—
243C7×D4—42no—
244C7×D4—84no—
245C2×C4×C7—21no—
246C2×C4×C7—21no—
247C2×C4×C7—21no—
248C2×C4×C7—21no—
249C2×C4×C7—24no—
250C2×C4×C7—24no—
251C2×C4×C7—21no—
252C2×C4×C7—21no—
253C2×C4×C7—21no—
254C2×C4×C7—24no—
255C2×C4×C7—21no—
256C2×C4×C7—21no—
257C2×C4×C7—21no—
258C2×C4×C7—21no—
259C2×C4×C7—21no—
260C2×C4×C7—21no—
261C2×C4×C7—44no—
262C2×C4×C7—48no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—44no—
266C2×C4×C7—41no—
267C2×C4×C7—41no—
268C2×C4×C7—42no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7—42no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—42no—
278C2×C4×C7—84no—
279C2×C4×C7#12112yes—
280C2×C4×C7#13712yes—
281C2×C4×C7C22×D411yescharacteristic
282C2×C4×C7C2×#4112yes—
283C2×C4×C7C2×#4111yescharacteristic
284C2×C4×C7H(5,𝔽2)12yes—
285C7×C8—24no—
286C7×C8—42no—
287C22×Q8—24no—
288C22×Q8—21no—
289C22×Q8—22no—
290C22×Q8—22no—
291C22×Q8—42no—
292C22×Q8C23×C712yes—
293C22×D4—21no—
294C22×D4C23×C711yescharacteristic
295C23×C4—21no—
296C23×C4—21no—
297C23×C4C7×D411yescharacteristic
298C23×C4C7×D411yescharacteristic
299C23×C4C23×C711yescharacteristic
300C2×SD16—24no—
301C2×SD16—24no—
302C2×SD16—42no—
303C22×C8C23×C712yes—
304C22×C8C23×C712yes—
305#123—44no—
306#121—22no—
307#121—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4C23×C712yes—
313C2×C4⋊C4C23×C711yescharacteristic
314C2×C4⋊C4C23×C711yescharacteristic
315C2×C4⋊C4C23×C711yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42C7×D411yescharacteristic
325C2×C42C7×D411yescharacteristic
326C2×C42C23×C711yescharacteristic
327C2×C42C23×C711yescharacteristic
328#107—42no—
329C4⋊C8—24no—
330#103—24no—
331#103—24no—
332#103—42no—
333#102—42no—
334C22⋊C8—22no—
335C22⋊C8C7×D414yes—
336#97—24no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—21no—
342#95—21no—
343#95—21no—
344C22×C7—22no—
345C22×C7—22no—
346C22×C7—22no—
347C22×C7—22no—
348C22×C7—22no—
349C22×C7—22no—
350C22×C7—81no—
351C22×C7—81no—
352C22×C7—81no—
353C22×C7—81no—
354C22×C7—81no—
355C22×C7—81no—
356C22×C7—81no—
357C22×C7#46814yes—
358C22×C7C2×#12111yescharacteristic
359C22×C7#52511yescharacteristic
360C22×C7#53511yescharacteristic
361C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
362C22×C7C2×#13711yescharacteristic
363C22×C7C2×#13711yescharacteristic
364C22×C7#57511yescharacteristic
365C4×C7—22no—
366C4×C7—22no—
367C4×C7—44no—
368C4×C7—41no—
369C4×C7—41no—
370C4×C7—41no—
371C4×C7—81no—
372C4×C7—81no—
373C4×C7—82no—
374C4×C7—81no—
375C4×C7#54612yes—
376C4×C7#55112yes—
377C24—21no—
378C24C2×C7×D411yescharacteristic
379C2×Q8—24no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—48no—
386C2×Q8—48no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8C7×#4114yes—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—42no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—41no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4C2×C7×D411yescharacteristic
417C22×C4C2×C7×D411yescharacteristic
418C22×C4C2×C7×D411yescharacteristic
419C22×C4C2×C7×D411yescharacteristic
420C22×C4C2×C7×D412yes—
421C22×C4C2×C7×D411yescharacteristic
422C22×C4C7×#4112yes—
423C22×C4C24×C711yescharacteristic
424SD16—48no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8C2×C7×D414yes—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C42—22no—
447C42—22no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C2×C7—24no—
453C2×C7—161no—
454C2×C7#241111yescharacteristic
455C2×C7C2×#46811yescharacteristic
456C2×C7#305611yescharacteristic
457C2×C7#311211yescharacteristic
458C2×C7#311711yescharacteristic
459C2×C7#321311yescharacteristic
460C2×C7#321811yescharacteristic
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23C7×#12112yes—
473C23C7×AGL(1,ℤ/8)12yes—
474C23C7×#13712yes—
475C23C22×C7×D411yescharacteristic
476Q8—24no—
477Q8—24no—
478Q8—48no—
479Q8—42no—
480Q8—42no—
481D4—42no—
482D4—42no—
483D4—84no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—44no—
501C2×C4—48no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—84no—
518C2×C4C7×#12112yes—
519C2×C4C7×#13712yes—
520C2×C4C22×C7×D411yescharacteristic
521C2×C4C2×C7×#4112yes—
522C2×C4C2×C7×#4111yescharacteristic
523C2×C4C7×H(5,𝔽2)12yes—
524C8—24no—
525C8—42no—
526C7#2450911yescharacteristic, direct product factor, Sylow
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22C7×#46814yes—
541C22C2×C7×#12111yescharacteristic
542C22C7×#52511yescharacteristic
543C22C7×#53511yescharacteristic
544C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
545C22C2×C7×#13711yescharacteristic
546C22C2×C7×#13711yescharacteristic
547C22C7×#57511yescharacteristic
548C4—22no—
549C4—22no—
550C4—44no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—82no—
555C4—81no—
556C4—81no—
557C4—81no—
558C4C7×#54612yes—
559C4C7×#55112yes—
560C2—24no—
561C2—161no—
562C2C7×#241111yescharacteristic
563C2C2×C7×#46811yescharacteristic
564C2C7×#305611yescharacteristic
565C2C7×#311211yescharacteristic
566C2C7×#311711yescharacteristic
567C2C7×#321311yescharacteristic
568C2C7×#321811yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C22×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#432C2semidirect
4. C2×C7×#391C2semidirect
5. C7×#2068C2semidirect
6. C7×#2055C2semidirect
7. C7×#1973C2semidirect
8. C7×#1935C2semidirect
10. C7×#1852C2semidirect
11. C7×#1787C2semidirect
45. #24509C7direct (N × H)
104. C2×#432C2×C7semidirect
106. C2×#391C2×C7semidirect
107. #2068C2×C7semidirect
108. #2055C2×C7semidirect
109. #1973C2×C7semidirect
110. #1935C2×C7semidirect
112. #1852C2×C7semidirect
113. #1787C2×C7semidirect
526. C7#24509direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4482
C2×C7×#4322
C7×#28542
C2×C7×#3912
C7×#20682
C7×#20552
C7×#19732
C7×#19352
C7×#18742
C7×#18522
C7×#17872
#245097