Finite Group Databasebeta

← C7×#24118 #13201334

Subgroups — C7×#24118 #13201334

Order 1792.

1218 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#554C211yesmaximal, characteristic, complemented
2C2×C7×#476C211yesmaximal, characteristic, complemented
3C2×C4×C7×SD16C211yesmaximal, characteristic, complemented
4C7×#2537C211yesmaximal, characteristic
5C7×#2035C212yesmaximal, complemented
6C7×#2000C212yesmaximal, complemented
7C7×#1990C211yesmaximal, characteristic, complemented
8C7×#1954C211yesmaximal, characteristic, complemented
9C7×#1874C212yesmaximal, complemented
10C7×#1853C212yesmaximal, complemented
11C7×#1803C211yesmaximal, characteristic, complemented
12C22×C7×SD16C2211yescharacteristic
13C7×#554—24no—
14C7×#554—24no—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#125—21no—
17C2×C7×C22⋊Q8—22no—
18C2×C7×C22⋊Q8C2212yes—
19C2×C7×#121—21no—
20C2×C7×#121C2212yes—
21C2×C4×C7×Q8C2211yescharacteristic
22C2×C4×C7×D4—21no—
23C2×C4×C7×D4C2211yescharacteristic
24C2×C4×C7×D4C2211yescharacteristic
25C22×C7×C4⋊C4C2212yes—
26C7×#476—24no—
27C4×C7×SD16—24no—
28C2×C7×#106C2211yescharacteristic
29C2×C7×#106C2212yes—
30C2×C7×C4⋊C8C2211yescharacteristic
31C2×C7×#103C2211yescharacteristic
32C2×C7×#103C2212yes—
33C2×C7×#102C2212yes—
34C2×C7×#102C2211yescharacteristic
35C2×C7×C22⋊C8C2212yes—
36C2×C7×C22⋊C8C2212yes—
37C2×C4×C7×C8C2211yescharacteristic
38C7×#394—22no—
39C7×#384C2211yescharacteristic
40C7×#380—21no—
41C7×#380C2212yes—
42C7×#340C2212yes—
43C7×#340C2212yes—
44C7×#340C2212yes—
45#24118C711yesmaximal, characteristic, direct product factor, Sylow
46C22×C7×Q8—21no—
47C22×C7×Q8C2311yescharacteristic
48C22×C7×D4—22no—
49C22×C7×D4D411yescharacteristic
50C22×C7×D4C2311yescharacteristic
51C23×C4×C7—22no—
52C23×C4×C7—21no—
53C23×C4×C7C2312yes—
54C2×C7×SD16—22no—
55C2×C7×SD16—22no—
56C2×C7×SD16—22no—
57C22×C7×C8C2311yescharacteristic
58C22×C7×C8C2312yes—
59C22×C7×C8C2311yescharacteristic
60C7×C4⋊Q8—22no—
61C7×C4⋊Q8D414yescomplemented
62C7×#125—28no—
63C7×C22⋊Q8—216no—
64C7×C22⋊Q8—24no—
65C7×C22⋊Q8—24no—
66C7×#121—24no—
67C7×#121—42no—
68C7×#121—44no—
69C4×C7×Q8—22no—
70C4×C7×Q8D414yescomplemented
71C4×C7×D4—28no—
72C4×C7×D4—22no—
73C4×C7×D4—22no—
74C4×C7×D4—22no—
75C4×C7×D4—22no—
76C2×C7×C4⋊C4—22no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4—22no—
80C2×C7×C4⋊C4—24no—
81C2×C7×C4⋊C4—24no—
82C2×C7×C4⋊C4—21no—
83C2×C7×C4⋊C4—21no—
84C2×C7×C4⋊C4D411yescharacteristic
85C2×C7×C4⋊C4C2312yes—
86C2×C7×C4⋊C4C2311yescharacteristic
87C2×C7×C4⋊C4C2311yescharacteristic
88C2×C7×C4⋊C4C2312yes—
89C2×C7×C22⋊C4—22no—
90C2×C7×C22⋊C4—22no—
91C2×C7×C22⋊C4—21no—
92C2×C7×C22⋊C4—22no—
93C2×C7×C22⋊C4D412yes—
94C2×C42×C7—21no—
95C2×C42×C7C2311yescharacteristic
96C7×#106—24no—
97C7×#106—22no—
98C7×C4⋊C8—22no—
99C7×#103—22no—
100C7×#103—42no—
101C7×#102—24no—
102C7×#102—22no—
103C7×C22⋊C8—24no—
104C7×C22⋊C8—42no—
105C4×C7×C8—22no—
106C7×#95—22no—
107C2×#554C2×C711yescharacteristic, complemented
108C2×#476C2×C711yescharacteristic, complemented
109C2×C4×SD16C2×C711yescharacteristic, complemented
110#2537C2×C711yescharacteristic
111#2035C2×C712yescomplemented
112#2000C2×C712yescomplemented
113#1990C2×C711yescharacteristic, complemented
114#1954C2×C711yescharacteristic, complemented
115#1874C2×C712yescomplemented
116#1853C2×C712yescomplemented
117#1803C2×C711yescharacteristic, complemented
118C24×C7—21no—
119C24×C7C2×D412yes—
120C2×C7×Q8—24no—
121C2×C7×Q8—21no—
122C2×C7×Q8—21no—
123C2×C7×Q8—21no—
124C2×C7×Q8—21no—
125C2×C7×Q8—21no—
126C2×C7×Q8—81no—
127C2×C7×Q8—81no—
128C2×C7×Q8C2×D412yes—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4—21no—
132C2×C7×D4—21no—
133C2×C7×D4—21no—
134C2×C7×D4—21no—
135C2×C7×D4—21no—
136C2×C7×D4—44no—
137C2×C7×D4—44no—
138C2×C7×D4—82no—
139C2×C7×D4SD1614yescomplemented
140C2×C7×D4#4112yes—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—22no—
144C22×C4×C7—28no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—21no—
152C22×C4×C7—22no—
153C22×C4×C7—42no—
154C22×C4×C7—42no—
155C22×C4×C7—42no—
156C22×C4×C7—82no—
157C22×C4×C7—82no—
158C22×C4×C7C2×D412yes—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2411yescharacteristic
162C7×SD16—28no—
163C2×C7×C8—21no—
164C2×C7×C8—21no—
165C2×C7×C8—21no—
166C2×C7×C8—22no—
167C2×C7×C8—22no—
168C2×C7×C8—21no—
169C2×C7×C8—22no—
170C2×C7×C8—21no—
171C2×C7×C8#4112yes—
172C7×C4⋊C4—28no—
173C7×C4⋊C4—28no—
174C7×C4⋊C4—24no—
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—42no—
184C7×C4⋊C4—81no—
185C7×C4⋊C4C2×D412yes—
186C7×C4⋊C4C2×D414yes—
187C7×C22⋊C4—28no—
188C7×C22⋊C4—24no—
189C7×C22⋊C4—44no—
190C7×C22⋊C4—44no—
191C7×C22⋊C4—42no—
192C42×C7—24no—
193C42×C7C2×D412yes—
194C42×C7C2×D412yes—
195C22×SD16C22×C711yescharacteristic
196#554—24no—
197#554—24no—
198C2×C4⋊Q8C22×C711yescharacteristic
199C2×#125—21no—
200C2×C22⋊Q8—22no—
201C2×C22⋊Q8C22×C712yes—
202C2×#121—21no—
203C2×#121C22×C712yes—
204C2×C4×Q8C22×C711yescharacteristic
205C2×C4×D4—21no—
206C2×C4×D4C22×C711yescharacteristic
207C2×C4×D4C22×C711yescharacteristic
208C22×C4⋊C4C22×C712yes—
209#476—24no—
210C4×SD16—24no—
211C2×#106C22×C711yescharacteristic
212C2×#106C22×C712yes—
213C2×C4⋊C8C22×C711yescharacteristic
214C2×#103C22×C711yescharacteristic
215C2×#103C22×C712yes—
216C2×#102C22×C711yescharacteristic
217C2×#102C22×C712yes—
218C2×C22⋊C8C22×C712yes—
219C2×C22⋊C8C22×C712yes—
220C2×C4×C8C22×C711yescharacteristic
221#394—22no—
222#384C22×C711yescharacteristic
223#380—21no—
224#380C22×C712yes—
225#340C22×C712yes—
226#340C22×C712yes—
227#340C22×C712yes—
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—41no—
240C23×C7—41no—
241C23×C7C2×SD1614yes—
242C23×C7C22×D411yescharacteristic
243C7×Q8—22no—
244C7×Q8—22no—
245C7×Q8—41no—
246C7×Q8—41no—
247C7×Q8—84no—
248C7×D4—22no—
249C7×D4—22no—
250C7×D4—28no—
251C7×D4—41no—
252C7×D4—41no—
253C7×D4—88no—
254C2×C4×C7—21no—
255C2×C4×C7—21no—
256C2×C4×C7—22no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—24no—
260C2×C4×C7—22no—
261C2×C4×C7—22no—
262C2×C4×C7—41no—
263C2×C4×C7—41no—
264C2×C4×C7—41no—
265C2×C4×C7—42no—
266C2×C4×C7—41no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7—42no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7—41no—
273C2×C4×C7—42no—
274C2×C4×C7—41no—
275C2×C4×C7—42no—
276C2×C4×C7—41no—
277C2×C4×C7—42no—
278C2×C4×C7—88no—
279C2×C4×C7—84no—
280C2×C4×C7#12012yes—
281C2×C4×C7#12012yes—
282C2×C4×C7C2×SD1612yes—
283C2×C4×C7#13712yes—
284C2×C4×C7C22×D411yescharacteristic
285C2×C4×C7C22×D411yescharacteristic
286C2×C4×C7C2×#4111yescharacteristic
287C2×C4×C7C2×#4111yescharacteristic
288C2×C4×C7H(5,𝔽2)11yescharacteristic
289C2×C4×C72−1+411yescharacteristic
290C7×C8—22no—
291C7×C8—41no—
292C7×C8—42no—
293C22×Q8—21no—
294C22×Q8C23×C711yescharacteristic
295C22×D4—22no—
296C22×D4C7×D411yescharacteristic
297C22×D4C23×C711yescharacteristic
298C23×C4—22no—
299C23×C4—21no—
300C23×C4C23×C712yes—
301C2×SD16—22no—
302C2×SD16—22no—
303C2×SD16—22no—
304C22×C8C23×C711yescharacteristic
305C22×C8C23×C712yes—
306C22×C8C23×C711yescharacteristic
307C4⋊Q8—22no—
308C4⋊Q8C7×D414yescomplemented
309#125—28no—
310C22⋊Q8—216no—
311C22⋊Q8—24no—
312C22⋊Q8—24no—
313#121—24no—
314#121—42no—
315#121—44no—
316C4×Q8—22no—
317C4×Q8C7×D414yescomplemented
318C4×D4—28no—
319C4×D4—22no—
320C4×D4—22no—
321C4×D4—22no—
322C4×D4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4C7×D411yescharacteristic
332C2×C4⋊C4C23×C711yescharacteristic
333C2×C4⋊C4C23×C712yes—
334C2×C4⋊C4C23×C711yescharacteristic
335C2×C4⋊C4C23×C712yes—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4C7×D412yes—
341C2×C42—21no—
342C2×C42C23×C711yescharacteristic
343#106—22no—
344#106—24no—
345C4⋊C8—22no—
346#103—22no—
347#103—42no—
348#102—22no—
349#102—24no—
350C22⋊C8—24no—
351C22⋊C8—42no—
352C4×C8—22no—
353#95—22no—
354C22×C7—24no—
355C22×C7—24no—
356C22×C7—24no—
357C22×C7—42no—
358C22×C7—42no—
359C22×C7—42no—
360C22×C7—42no—
361C22×C7—42no—
362C22×C7—42no—
363C22×C7—42no—
364C22×C7—81no—
365C22×C7—81no—
366C22×C7—81no—
367C22×C7—81no—
368C22×C7C2×#12011yescharacteristic
369C22×C7#54611yescharacteristic
370C22×C7#54711yescharacteristic
371C22×C7C22×SD1611yescharacteristic
372C22×C7C2×#13711yescharacteristic
373C22×C7#57611yescharacteristic
374C22×C7#57711yescharacteristic
375C4×C7—22no—
376C4×C7—42no—
377C4×C7—42no—
378C4×C7—42no—
379C4×C7—44no—
380C4×C7—81no—
381C4×C7—82no—
382C4×C7#44814yes—
383C4×C7#52511yescharacteristic
384C4×C7#53711yescharacteristic
385C4×C7#54811yescharacteristic
386C4×C7#55011yescharacteristic
387C24—21no—
388C24C2×C7×D412yes—
389C2×Q8—24no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—81no—
396C2×Q8—81no—
397C2×Q8C2×C7×D412yes—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—82no—
408C2×D4C7×SD1614yescomplemented
409C2×D4C7×#4112yes—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4C2×C7×D412yes—
428C22×C4C2×C7×D411yescharacteristic
429C22×C4C2×C7×D411yescharacteristic
430C22×C4C24×C711yescharacteristic
431SD16—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×#4112yes—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—42no—
453C4⋊C4—81no—
454C4⋊C4C2×C7×D412yes—
455C4⋊C4C2×C7×D414yes—
456C22⋊C4—28no—
457C22⋊C4—24no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—42no—
461C42—24no—
462C42C2×C7×D412yes—
463C42C2×C7×D412yes—
464C2×C7—44no—
465C2×C7—82no—
466C2×C7#240711yescharacteristic
467C2×C7C2×#44811yescharacteristic
468C2×C7#301211yescharacteristic
469C2×C7#329511yescharacteristic
470C2×C7#333111yescharacteristic
471C2×C7#333511yescharacteristic
472C2×C7#335711yescharacteristic
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23C2×C7×SD1614yes—
487C23C22×C7×D411yescharacteristic
488Q8—22no—
489Q8—22no—
490Q8—41no—
491Q8—41no—
492Q8—84no—
493D4—22no—
494D4—22no—
495D4—28no—
496D4—41no—
497D4—41no—
498D4—88no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—88no—
524C2×C4—84no—
525C2×C4C7×#12012yes—
526C2×C4C7×#12012yes—
527C2×C4C2×C7×SD1612yes—
528C2×C4C7×#13712yes—
529C2×C4C22×C7×D411yescharacteristic
530C2×C4C22×C7×D411yescharacteristic
531C2×C4C2×C7×#4111yescharacteristic
532C2×C4C2×C7×#4111yescharacteristic
533C2×C4C7×H(5,𝔽2)11yescharacteristic
534C2×C4C7×2−1+411yescharacteristic
535C8—22no—
536C8—41no—
537C8—42no—
538C7#2411811yescharacteristic, direct product factor, Sylow
539C22—24no—
540C22—24no—
541C22—24no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22C2×C7×#12011yescharacteristic
554C22C7×#54611yescharacteristic
555C22C7×#54711yescharacteristic
556C22C22×C7×SD1611yescharacteristic
557C22C2×C7×#13711yescharacteristic
558C22C7×#57611yescharacteristic
559C22C7×#57711yescharacteristic
560C4—22no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—44no—
565C4—81no—
566C4—82no—
567C4C7×#44814yes—
568C4C7×#52511yescharacteristic
569C4C7×#53711yescharacteristic
570C4C7×#54811yescharacteristic
571C4C7×#55011yescharacteristic
572C2—44no—
573C2—82no—
574C2C7×#240711yescharacteristic
575C2C2×C7×#44811yescharacteristic
576C2C7×#301211yescharacteristic
577C2C7×#329511yescharacteristic
578C2C7×#333111yescharacteristic
579C2C7×#333511yescharacteristic
580C2C7×#335711yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#2537C2
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×#554C2semidirect
2. C2×C7×#476C2semidirect
3. C2×C4×C7×SD16C2semidirect
5. C7×#2035C2semidirect
6. C7×#2000C2semidirect
7. C7×#1990C2semidirect
8. C7×#1954C2semidirect
9. C7×#1874C2semidirect
10. C7×#1853C2semidirect
11. C7×#1803C2semidirect
45. #24118C7direct (N × H)
61. C7×C4⋊Q8D4semidirect
70. C4×C7×Q8D4semidirect
107. C2×#554C2×C7semidirect
108. C2×#476C2×C7semidirect
109. C2×C4×SD16C2×C7semidirect
111. #2035C2×C7semidirect
112. #2000C2×C7semidirect
113. #1990C2×C7semidirect
114. #1954C2×C7semidirect
115. #1874C2×C7semidirect
116. #1853C2×C7semidirect
117. #1803C2×C7semidirect
139. C2×C7×D4SD16semidirect
308. C4⋊Q8C7×D4semidirect
317. C4×Q8C7×D4semidirect
408. C2×D4C7×SD16semidirect
538. C7#24118direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5542
C2×C7×#4762
C2×C4×C7×SD162
C7×#25372
C7×#20352
C7×#20002
C7×#19902
C7×#19542
C7×#18742
C7×#18532
C7×#18032
#241187