Finite Group Databasebeta

← C7×#24120 #13201336

Subgroups — C7×#24120 #13201336

Order 1792.

1350 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#547C211yesmaximal, characteristic
2C2×C7×#475C211yesmaximal, characteristic, complemented
3C2×C4×C7×SD16C211yesmaximal, characteristic, complemented
4C7×#2538C211yesmaximal, characteristic, complemented
5C7×#2034C212yesmaximal, complemented
6C7×#2000C212yesmaximal, complemented
7C7×#1990C211yesmaximal, characteristic, complemented
8C7×#1954C211yesmaximal, characteristic, complemented
9C7×#1872C212yesmaximal, complemented
10C7×#1853C212yesmaximal, complemented
11C7×#1802C211yesmaximal, characteristic, complemented
12C22×C7×#41—21no—
13C22×C7×SD16C2211yescharacteristic
14C7×#547—24no—
15C7×#547—24no—
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×#123—22no—
18C2×C7×C22⋊Q8—21no—
19C2×C7×C22⋊Q8C2212yes—
20C2×C7×#121C2212yes—
21C2×C4×C7×Q8C2211yescharacteristic
22C2×C4×C7×D4C2211yescharacteristic
23C2×C4×C7×D4C2211yescharacteristic
24C22×C7×C4⋊C4C2212yes—
25C7×#475—24no—
26C4×C7×SD16—24no—
27C2×C7×#106C2211yescharacteristic
28C2×C7×#106C2212yes—
29C2×C7×C4⋊C8C2211yescharacteristic
30C2×C7×#103C2212yes—
31C2×C7×#103C2211yescharacteristic
32C2×C7×#102C2211yescharacteristic
33C2×C7×#102C2212yes—
34C2×C7×C22⋊C8C2212yes—
35C2×C7×C22⋊C8C2212yes—
36C2×C4×C7×C8C2211yescharacteristic
37C7×#396—22no—
38C7×#390—21no—
39C7×#389—21no—
40C7×#384C2211yescharacteristic
41C7×#380C2212yes—
42C7×#340C2212yes—
43C7×#340C2212yes—
44C7×#340C2212yes—
45#24120C711yesmaximal, characteristic, direct product factor, Sylow
46C2×C7×#41—28no—
47C2×C7×#41—44no—
48C2×C7×#41—44no—
49C22×C7×Q8—21no—
50C22×C7×Q8C2311yescharacteristic
51C22×C7×D4—22no—
52C22×C7×D4D411yescharacteristic
53C22×C7×D4C2311yescharacteristic
54C23×C4×C7—22no—
55C23×C4×C7—21no—
56C23×C4×C7C2312yes—
57C2×C7×SD16—22no—
58C2×C7×SD16—22no—
59C2×C7×SD16—22no—
60C22×C7×C8C2311yescharacteristic
61C22×C7×C8C2312yes—
62C22×C7×C8C2311yescharacteristic
63C7×C4⋊Q8—22no—
64C7×C4⋊Q8D414yes—
65C7×#123—216no—
66C7×C22⋊Q8—28no—
67C7×C22⋊Q8—24no—
68C7×C22⋊Q8—42no—
69C7×#121—24no—
70C7×#121—24no—
71C4×C7×Q8—22no—
72C4×C7×Q8—22no—
73C4×C7×D4—22no—
74C4×C7×D4—22no—
75C4×C7×D4—22no—
76C4×C7×D4D414yes—
77C2×C7×C4⋊C4—22no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4—24no—
80C2×C7×C4⋊C4—24no—
81C2×C7×C4⋊C4—21no—
82C2×C7×C4⋊C4—21no—
83C2×C7×C4⋊C4—22no—
84C2×C7×C4⋊C4—21no—
85C2×C7×C4⋊C4D411yescharacteristic
86C2×C7×C4⋊C4C2312yes—
87C2×C7×C4⋊C4C2311yescharacteristic
88C2×C7×C4⋊C4C2311yescharacteristic
89C2×C7×C4⋊C4C2312yes—
90C2×C7×C22⋊C4—21no—
91C2×C7×C22⋊C4—22no—
92C2×C7×C22⋊C4—22no—
93C2×C7×C22⋊C4—22no—
94C2×C7×C22⋊C4D412yes—
95C2×C42×C7—21no—
96C2×C42×C7C2311yescharacteristic
97C7×#106—24no—
98C7×#106—22no—
99C7×C4⋊C8—22no—
100C7×#103—24no—
101C7×#103—22no—
102C7×#102—22no—
103C7×#102—42no—
104C7×C22⋊C8—24no—
105C7×C22⋊C8—42no—
106C4×C7×C8—22no—
107C7×#95—22no—
108C2×#547C2×C711yescharacteristic
109C2×#475C2×C711yescharacteristic, complemented
110C2×C4×SD16C2×C711yescharacteristic, complemented
111#2538C2×C711yescharacteristic, complemented
112#2034C2×C712yescomplemented
113#2000C2×C712yescomplemented
114#1990C2×C711yescharacteristic, complemented
115#1954C2×C711yescharacteristic, complemented
116#1872C2×C712yescomplemented
117#1853C2×C712yescomplemented
118#1802C2×C711yescharacteristic, complemented
119C24×C7—21no—
120C24×C7C2×D412yes—
121C7×#41—432no—
122C2×C7×Q8—24no—
123C2×C7×Q8—21no—
124C2×C7×Q8—21no—
125C2×C7×Q8—21no—
126C2×C7×Q8—21no—
127C2×C7×Q8—21no—
128C2×C7×Q8—42no—
129C2×C7×Q8—42no—
130C2×C7×Q8#4112yescomplemented
131C2×C7×D4—28no—
132C2×C7×D4—22no—
133C2×C7×D4—22no—
134C2×C7×D4—21no—
135C2×C7×D4—21no—
136C2×C7×D4—21no—
137C2×C7×D4—21no—
138C2×C7×D4—21no—
139C2×C7×D4—44no—
140C2×C7×D4—44no—
141C2×C7×D4SD1614yescomplemented
142C2×C7×D4C2×D412yes—
143C22×C4×C7—28no—
144C22×C4×C7—21no—
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—21no—
155C22×C4×C7—22no—
156C22×C4×C7—44no—
157C22×C4×C7—44no—
158C22×C4×C7—42no—
159C22×C4×C7—42no—
160C22×C4×C7C2×D412yes—
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2411yescharacteristic
164C7×SD16—28no—
165C2×C7×C8—21no—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C2×C7×C8—22no—
169C2×C7×C8—22no—
170C2×C7×C8—21no—
171C2×C7×C8—22no—
172C2×C7×C8—21no—
173C2×C7×C8#4112yescomplemented
174C7×C4⋊C4—28no—
175C7×C4⋊C4—24no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—22no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—44no—
184C7×C4⋊C4—42no—
185C7×C4⋊C4—42no—
186C7×C4⋊C4—81no—
187C7×C4⋊C4C2×D412yes—
188C7×C4⋊C4C2×D414yes—
189C7×C22⋊C4—28no—
190C7×C22⋊C4—24no—
191C7×C22⋊C4—24no—
192C7×C22⋊C4—44no—
193C7×C22⋊C4—82no—
194C42×C7—42no—
195C42×C7C2×D412yes—
196C42×C7C2×D412yes—
197C22×#41—21no—
198C22×SD16C22×C711yescharacteristic
199#547—24no—
200#547—24no—
201C2×C4⋊Q8C22×C711yescharacteristic
202C2×#123—22no—
203C2×C22⋊Q8—21no—
204C2×C22⋊Q8C22×C712yes—
205C2×#121C22×C712yes—
206C2×C4×Q8C22×C711yescharacteristic
207C2×C4×D4C22×C711yescharacteristic
208C2×C4×D4C22×C711yescharacteristic
209C22×C4⋊C4C22×C712yes—
210#475—24no—
211C4×SD16—24no—
212C2×#106C22×C712yes—
213C2×#106C22×C711yescharacteristic
214C2×C4⋊C8C22×C711yescharacteristic
215C2×#103C22×C712yes—
216C2×#103C22×C711yescharacteristic
217C2×#102C22×C711yescharacteristic
218C2×#102C22×C712yes—
219C2×C22⋊C8C22×C712yes—
220C2×C22⋊C8C22×C712yes—
221C2×C4×C8C22×C711yescharacteristic
222#396—22no—
223#390—21no—
224#389—21no—
225#384C22×C711yescharacteristic
226#380C22×C712yes—
227#340C22×C712yes—
228#340C22×C712yes—
229#340C22×C712yes—
230C23×C7—22no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—22no—
234C23×C7—22no—
235C23×C7—22no—
236C23×C7—22no—
237C23×C7—41no—
238C23×C7—41no—
239C23×C7—41no—
240C23×C7—41no—
241C23×C7—41no—
242C23×C7—41no—
243C23×C7C2×SD1614yes—
244C23×C7C22×D411yescharacteristic
245C7×Q8—22no—
246C7×Q8—22no—
247C7×Q8—41no—
248C7×Q8—41no—
249C7×Q8—48no—
250C7×D4—22no—
251C7×D4—22no—
252C7×D4—28no—
253C7×D4—416no—
254C7×D4—41no—
255C7×D4—41no—
256C2×C4×C7—21no—
257C2×C4×C7—21no—
258C2×C4×C7—22no—
259C2×C4×C7—22no—
260C2×C4×C7—24no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
264C2×C4×C7—416no—
265C2×C4×C7—41no—
266C2×C4×C7—41no—
267C2×C4×C7—41no—
268C2×C4×C7—42no—
269C2×C4×C7—41no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—48no—
276C2×C4×C7—41no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—41no—
280C2×C4×C7—42no—
281C2×C4×C7—41no—
282C2×C4×C7#12012yes—
283C2×C4×C7#12012yes—
284C2×C4×C7C2×SD1612yes—
285C2×C4×C7AGL(1,ℤ/8)12yes—
286C2×C4×C7C22×D411yescharacteristic
287C2×C4×C7C22×D411yescharacteristic
288C2×C4×C7C2×#4111yescharacteristic
289C2×C4×C7C2×#4111yescharacteristic
290C2×C4×C7H(5,𝔽2)11yescharacteristic
291C2×C4×C72−1+411yescharacteristic
292C7×C8—22no—
293C7×C8—41no—
294C7×C8—42no—
295C2×#41—28no—
296C2×#41—44no—
297C2×#41—44no—
298C22×Q8—21no—
299C22×Q8C23×C711yescharacteristic
300C22×D4—22no—
301C22×D4C7×D411yescharacteristic
302C22×D4C23×C711yescharacteristic
303C23×C4—22no—
304C23×C4—21no—
305C23×C4C23×C712yes—
306C2×SD16—22no—
307C2×SD16—22no—
308C2×SD16—22no—
309C22×C8C23×C711yescharacteristic
310C22×C8C23×C712yes—
311C22×C8C23×C711yescharacteristic
312C4⋊Q8—22no—
313C4⋊Q8C7×D414yes—
314#123—216no—
315C22⋊Q8—28no—
316C22⋊Q8—24no—
317C22⋊Q8—42no—
318#121—24no—
319#121—24no—
320C4×Q8—22no—
321C4×Q8—22no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—22no—
325C4×D4C7×D414yes—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4C7×D411yescharacteristic
335C2×C4⋊C4C23×C711yescharacteristic
336C2×C4⋊C4C23×C712yes—
337C2×C4⋊C4C23×C711yescharacteristic
338C2×C4⋊C4C23×C712yes—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4C7×D412yes—
344C2×C42—21no—
345C2×C42C23×C711yescharacteristic
346#106—22no—
347#106—24no—
348C4⋊C8—22no—
349#103—22no—
350#103—24no—
351#102—22no—
352#102—42no—
353C22⋊C8—24no—
354C22⋊C8—42no—
355C4×C8—22no—
356#95—22no—
357C22×C7—24no—
358C22×C7—24no—
359C22×C7—24no—
360C22×C7—42no—
361C22×C7—42no—
362C22×C7—42no—
363C22×C7—42no—
364C22×C7—42no—
365C22×C7—42no—
366C22×C7—42no—
367C22×C7—81no—
368C22×C7—81no—
369C22×C7—81no—
370C22×C7—81no—
371C22×C7C2×#12011yescharacteristic
372C22×C7#54611yescharacteristic
373C22×C7#54711yescharacteristic
374C22×C7C22×SD1611yescharacteristic
375C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
376C22×C7#57711yescharacteristic
377C22×C7#57811yescharacteristic
378C4×C7—22no—
379C4×C7—42no—
380C4×C7—42no—
381C4×C7—42no—
382C4×C7—44no—
383C4×C7—82no—
384C4×C7—81no—
385C4×C7#45014yes—
386C4×C7#52511yescharacteristic
387C4×C7#53711yescharacteristic
388C4×C7#54811yescharacteristic
389C4×C7#55011yescharacteristic
390C24—21no—
391C24C2×C7×D412yes—
392#41—432no—
393C2×Q8—24no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8C7×#4112yescomplemented
402C2×D4—28no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4C7×SD1614yescomplemented
413C2×D4C2×C7×D412yes—
414C22×C4—28no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4C2×C7×D412yes—
432C22×C4C2×C7×D411yescharacteristic
433C22×C4C2×C7×D411yescharacteristic
434C22×C4C24×C711yescharacteristic
435SD16—28no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—21no—
442C2×C8—22no—
443C2×C8—21no—
444C2×C8C7×#4112yescomplemented
445C4⋊C4—28no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—44no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—81no—
458C4⋊C4C2×C7×D412yes—
459C4⋊C4C2×C7×D414yes—
460C22⋊C4—28no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—44no—
464C22⋊C4—82no—
465C42—42no—
466C42C2×C7×D412yes—
467C42C2×C7×D412yes—
468C2×C7—44no—
469C2×C7—82no—
470C2×C7#240711yescharacteristic
471C2×C7C2×#45011yescharacteristic
472C2×C7#301311yescharacteristic
473C2×C7#329811yescharacteristic
474C2×C7#333211yescharacteristic
475C2×C7#333511yescharacteristic
476C2×C7#335311yescharacteristic
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23C2×C7×SD1614yes—
491C23C22×C7×D411yescharacteristic
492Q8—22no—
493Q8—22no—
494Q8—41no—
495Q8—41no—
496Q8—48no—
497D4—22no—
498D4—22no—
499D4—28no—
500D4—416no—
501D4—41no—
502D4—41no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—416no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—48no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4C7×#12012yes—
530C2×C4C7×#12012yes—
531C2×C4C2×C7×SD1612yes—
532C2×C4C7×AGL(1,ℤ/8)12yes—
533C2×C4C22×C7×D411yescharacteristic
534C2×C4C22×C7×D411yescharacteristic
535C2×C4C2×C7×#4111yescharacteristic
536C2×C4C2×C7×#4111yescharacteristic
537C2×C4C7×H(5,𝔽2)11yescharacteristic
538C2×C4C7×2−1+411yescharacteristic
539C8—22no—
540C8—41no—
541C8—42no—
542C7#2412011yescharacteristic, direct product factor, Sylow
543C22—24no—
544C22—24no—
545C22—24no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C2×C7×#12011yescharacteristic
558C22C7×#54611yescharacteristic
559C22C7×#54711yescharacteristic
560C22C22×C7×SD1611yescharacteristic
561C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
562C22C7×#57711yescharacteristic
563C22C7×#57811yescharacteristic
564C4—22no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—44no—
569C4—81no—
570C4—82no—
571C4C7×#45014yes—
572C4C7×#52511yescharacteristic
573C4C7×#53711yescharacteristic
574C4C7×#54811yescharacteristic
575C4C7×#55011yescharacteristic
576C2—44no—
577C2—82no—
578C2C7×#240711yescharacteristic
579C2C2×C7×#45011yescharacteristic
580C2C7×#301311yescharacteristic
581C2C7×#329811yescharacteristic
582C2C7×#333211yescharacteristic
583C2C7×#333511yescharacteristic
584C2C7×#335311yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#547C2
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
2. C2×C7×#475C2semidirect
3. C2×C4×C7×SD16C2semidirect
4. C7×#2538C2semidirect
5. C7×#2034C2semidirect
6. C7×#2000C2semidirect
7. C7×#1990C2semidirect
8. C7×#1954C2semidirect
9. C7×#1872C2semidirect
10. C7×#1853C2semidirect
11. C7×#1802C2semidirect
45. #24120C7direct (N × H)
109. C2×#475C2×C7semidirect
110. C2×C4×SD16C2×C7semidirect
111. #2538C2×C7semidirect
112. #2034C2×C7semidirect
113. #2000C2×C7semidirect
114. #1990C2×C7semidirect
115. #1954C2×C7semidirect
116. #1872C2×C7semidirect
117. #1853C2×C7semidirect
118. #1802C2×C7semidirect
130. C2×C7×Q8#41semidirect
141. C2×C7×D4SD16semidirect
173. C2×C7×C8#41semidirect
401. C2×Q8C7×#41semidirect
412. C2×D4C7×SD16semidirect
444. C2×C8C7×#41semidirect
542. C7#24120direct (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×#4752
C2×C4×C7×SD162
C7×#25382
C7×#20342
C7×#20002
C7×#19902
C7×#19542
C7×#18722
C7×#18532
C7×#18022
#241207