Finite Group Databasebeta

← C2×C7×#3092 #13237723

Subgroups — C2×C7×#3092 #13237723

Order 1792.

2650 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×#525C211yesmaximal, characteristic
2C23×C7×C4⋊C4C211yesmaximal, characteristic, complemented
3C7×#3092C2116yesmaximal, direct product factor
4C2×C7×#484C212yesmaximal, complemented
5C2×C7×#481C212yesmaximal, complemented
6C2×C7×#483C212yesmaximal, complemented
7C2×C7×#480C212yesmaximal, complemented
8C2×C7×#428C212yesmaximal, complemented
9C2×C7×#417C212yesmaximal, complemented
10C2×C7×#407C211yesmaximal, characteristic, complemented
11C22×C7×#41C2211yescharacteristic
12C24×C4×C7—21no—
13C24×C4×C7C2211yescharacteristic
14C22×C7×M16C2212yes—
15C7×#525—24no—
16C7×#525C2218yes—
17C2×C7×#123—21no—
18C2×C7×C22⋊Q8C2211yescharacteristic
19C2×C7×C22⋊Q8C2211yescharacteristic
20C2×C7×#121C2211yescharacteristic
21C2×C7×#121C2211yescharacteristic
22C2×C7×#120—21no—
23C2×C4×C7×D4—21no—
24C2×C4×C7×D4—21no—
25C2×C7×#117C2211yescharacteristic
26C22×C7×C4⋊C4—24no—
27C22×C7×C4⋊C4—21no—
28C22×C7×C4⋊C4—21no—
29C22×C7×C4⋊C4—21no—
30C22×C7×C4⋊C4C2212yes—
31C22×C7×C4⋊C4C2212yes—
32C22×C7×C4⋊C4C2212yes—
33C22×C7×C4⋊C4C2218yescomplemented
34C7×#484C22116yescomplemented
35C7×#483C22116yescomplemented
36C7×#481C22116yescomplemented
37C7×#480C22116yescomplemented
38C7×#428C22116yescomplemented
39C2×C7×#107C2214yescomplemented
40C2×C7×#106C2214yescomplemented
41C7×#417C22116yescomplemented
42C2×C7×#103C2214yescomplemented
43C2×C7×#102C2214yescomplemented
44C7×#407C2218yescomplemented
45C2×C7×C22⋊C8C2212yes—
46C2×C7×C22⋊C8C2212yes—
47C2×#3092C711yesmaximal, characteristic, direct product factor, Sylow
48C25×C7D411yescharacteristic
49C2×C7×#41—22no—
50C2×C7×#41—28no—
51C2×C7×#41C2314yes—
52C22×C7×Q8C2311yescharacteristic
53C22×C7×D4—21no—
54C22×C7×D4—21no—
55C22×C7×D4—21no—
56C22×C7×D4C2311yescharacteristic
57C23×C4×C7—28no—
58C23×C4×C7—22no—
59C23×C4×C7—22no—
60C23×C4×C7—22no—
61C23×C4×C7—28no—
62C23×C4×C7—21no—
63C23×C4×C7—48no—
64C23×C4×C7D411yescharacteristic
65C23×C4×C7D411yescharacteristic
66C23×C4×C7D411yescharacteristic
67C23×C4×C7D414yes—
68C23×C4×C7C2311yescharacteristic
69C23×C4×C7C2311yescharacteristic
70C23×C4×C7C2314yes—
71C23×C4×C7C2311yescharacteristic
72C2×C7×M16—28no—
73C2×C7×M16C2318yes—
74C22×C7×C8C2314yes—
75C7×#123—28no—
76C7×C22⋊Q8—22no—
77C7×C22⋊Q8—22no—
78C7×C22⋊Q8C2314yes—
79C7×C22⋊Q8C2314yes—
80C7×#121—22no—
81C7×#121—22no—
82C7×#121C2314yes—
83C7×#121C2314yes—
84C7×#120—28no—
85C4×C7×D4—28no—
86C4×C7×D4—28no—
87C7×#117—22no—
88C7×#117C2314yes—
89C2×C7×C4⋊C4—24no—
90C2×C7×C4⋊C4—24no—
91C2×C7×C4⋊C4—24no—
92C2×C7×C4⋊C4—22no—
93C2×C7×C4⋊C4—22no—
94C2×C7×C4⋊C4—22no—
95C2×C7×C4⋊C4—28no—
96C2×C7×C4⋊C4—28no—
97C2×C7×C4⋊C4—28no—
98C2×C7×C4⋊C4—21no—
99C2×C7×C4⋊C4—21no—
100C2×C7×C4⋊C4C2318yes—
101C2×C7×C4⋊C4C2318yes—
102C2×C7×C4⋊C4C2314yes—
103C2×C7×C4⋊C4C2318yes—
104C2×C7×C4⋊C4C2311yescharacteristic
105C2×C7×C4⋊C4C2311yescharacteristic
106C2×C7×C22⋊C4—21no—
107C2×C7×C22⋊C4—21no—
108C2×C7×C22⋊C4—21no—
109C2×C7×C22⋊C4—21no—
110C2×C7×C22⋊C4—21no—
111C2×C42×C7—21no—
112C7×#107C23116yescomplemented
113C7×#106C23116yescomplemented
114C7×#103C23116yescomplemented
115C7×#102C23116yescomplemented
116C7×C22⋊C8C2318yes—
117C7×C22⋊C8C2318yes—
118C2×#525C2×C711yescharacteristic
119C23×C4⋊C4C2×C711yescharacteristic, complemented
120#3092C2×C7116yesdirect product factor
121C2×#484C2×C712yescomplemented
122C2×#481C2×C712yescomplemented
123C2×#483C2×C712yescomplemented
124C2×#480C2×C712yescomplemented
125C2×#428C2×C712yescomplemented
126C2×#417C2×C712yescomplemented
127C2×#407C2×C711yescharacteristic, complemented
128C24×C7—24no—
129C24×C7—24no—
130C24×C7—24no—
131C24×C7—21no—
132C24×C7C2×D414yes—
133C24×C7C2×D411yescharacteristic
134C24×C7C2×D411yescharacteristic
135C24×C7C2×D411yescharacteristic
136C7×#41—216no—
137C7×#41—48no—
138C2×C7×Q8—21no—
139C2×C7×Q8—24no—
140C2×C7×Q8C2412yes—
141C2×C7×D4—24no—
142C2×C7×D4—24no—
143C2×C7×D4—24no—
144C2×C7×D4—21no—
145C2×C7×D4—24no—
146C2×C7×D4—44no—
147C2×C7×D4—44no—
148C2×C7×D4—44no—
149C2×C7×D4C2412yes—
150C22×C4×C7—24no—
151C22×C4×C7—24no—
152C22×C4×C7—24no—
153C22×C4×C7—216no—
154C22×C4×C7—24no—
155C22×C4×C7—24no—
156C22×C4×C7—216no—
157C22×C4×C7—28no—
158C22×C4×C7—28no—
159C22×C4×C7—28no—
160C22×C4×C7—21no—
161C22×C4×C7—21no—
162C22×C4×C7—21no—
163C22×C4×C7—24no—
164C22×C4×C7—24no—
165C22×C4×C7—48no—
166C22×C4×C7—48no—
167C22×C4×C7—432no—
168C22×C4×C7—48no—
169C22×C4×C7—44no—
170C22×C4×C7C2×D414yes—
171C22×C4×C7C2×D414yes—
172C22×C4×C7C2×D412yes—
173C22×C4×C7C2×D412yes—
174C22×C4×C7C2×D414yes—
175C22×C4×C7C2×D411yescharacteristic
176C22×C4×C7C2×D411yescharacteristic
177C22×C4×C7C2×D412yes—
178C22×C4×C7C2×D411yescharacteristic
179C22×C4×C7#4118yes—
180C22×C4×C7C2412yes—
181C22×C4×C7C2412yes—
182C22×C4×C7C2412yes—
183C22×C4×C7C2411yescharacteristic
184C7×M16—216no—
185C2×C7×C8—28no—
186C2×C7×C8C2418yes—
187C7×C4⋊C4—28no—
188C7×C4⋊C4—28no—
189C7×C4⋊C4—28no—
190C7×C4⋊C4—24no—
191C7×C4⋊C4—24no—
192C7×C4⋊C4—24no—
193C7×C4⋊C4—21no—
194C7×C4⋊C4—21no—
195C7×C4⋊C4C2418yes—
196C7×C4⋊C4C2412yes—
197C7×C4⋊C4C2412yes—
198C7×C22⋊C4—24no—
199C7×C22⋊C4—24no—
200C7×C22⋊C4—24no—
201C7×C22⋊C4—24no—
202C7×C22⋊C4—24no—
203C42×C7—24no—
204C22×#41C22×C711yescharacteristic
205C24×C4—21no—
206C24×C4C22×C711yescharacteristic
207C22×M16C22×C712yes—
208#525—24no—
209#525C22×C718yes—
210C2×#123—21no—
211C2×C22⋊Q8C22×C711yescharacteristic
212C2×C22⋊Q8C22×C711yescharacteristic
213C2×#121C22×C711yescharacteristic
214C2×#121C22×C711yescharacteristic
215C2×#120—21no—
216C2×C4×D4—21no—
217C2×C4×D4—21no—
218C2×#117C22×C711yescharacteristic
219C22×C4⋊C4—24no—
220C22×C4⋊C4—21no—
221C22×C4⋊C4—21no—
222C22×C4⋊C4—21no—
223C22×C4⋊C4C22×C712yes—
224C22×C4⋊C4C22×C712yes—
225C22×C4⋊C4C22×C712yes—
226C22×C4⋊C4C22×C718yescomplemented
227#484C22×C7116yescomplemented
228#483C22×C7116yescomplemented
229#481C22×C7116yescomplemented
230#480C22×C7116yescomplemented
231#428C22×C7116yescomplemented
232C2×#107C22×C714yescomplemented
233C2×#106C22×C714yescomplemented
234#417C22×C7116yescomplemented
235C2×#103C22×C714yescomplemented
236C2×#102C22×C714yescomplemented
237#407C22×C718yescomplemented
238C2×C22⋊C8C22×C712yes—
239C2×C22⋊C8C22×C712yes—
240C23×C7—232no—
241C23×C7—28no—
242C23×C7—22no—
243C23×C7—22no—
244C23×C7—28no—
245C23×C7—22no—
246C23×C7—22no—
247C23×C7—28no—
248C23×C7—22no—
249C23×C7—22no—
250C23×C7—22no—
251C23×C7—42no—
252C23×C7—42no—
253C23×C7—42no—
254C23×C7#12314yes—
255C23×C7AGL(1,ℤ/8)14yes—
256C23×C7#13714yes—
257C23×C7C22×D412yes—
258C23×C7C22×D412yes—
259C23×C7C22×D412yes—
260C23×C7C22×D411yescharacteristic
261C7×Q8—24no—
262C7×Q8—42no—
263C7×D4—24no—
264C7×D4—48no—
265C7×D4—48no—
266C7×D4—48no—
267C7×D4—42no—
268C2×C4×C7—28no—
269C2×C4×C7—22no—
270C2×C4×C7—22no—
271C2×C4×C7—28no—
272C2×C4×C7—28no—
273C2×C4×C7—24no—
274C2×C4×C7—24no—
275C2×C4×C7—24no—
276C2×C4×C7—24no—
277C2×C4×C7—22no—
278C2×C4×C7—22no—
279C2×C4×C7—22no—
280C2×C4×C7—28no—
281C2×C4×C7—416no—
282C2×C4×C7—416no—
283C2×C4×C7—42no—
284C2×C4×C7—416no—
285C2×C4×C7—48no—
286C2×C4×C7—42no—
287C2×C4×C7—42no—
288C2×C4×C7—48no—
289C2×C4×C7#12314yes—
290C2×C4×C7#12318yes—
291C2×C4×C7C22×D412yes—
292C2×C4×C7C22×D412yes—
293C2×C4×C7C22×D412yes—
294C2×C4×C7C22×D411yescharacteristic
295C2×C4×C7C2×#4114yes—
296C2×C4×C7C2×#4118yes—
297C2×C4×C7C2511yescharacteristic
298C7×C8—28no—
299C25C7×D411yescharacteristic
300C2×#41—22no—
301C2×#41—28no—
302C2×#41C23×C714yes—
303C22×Q8C23×C711yescharacteristic
304C22×D4—21no—
305C22×D4—21no—
306C22×D4—21no—
307C22×D4C23×C711yescharacteristic
308C23×C4—28no—
309C23×C4—22no—
310C23×C4—22no—
311C23×C4—22no—
312C23×C4—28no—
313C23×C4—21no—
314C23×C4—48no—
315C23×C4C7×D411yescharacteristic
316C23×C4C7×D411yescharacteristic
317C23×C4C7×D411yescharacteristic
318C23×C4C7×D414yes—
319C23×C4C23×C711yescharacteristic
320C23×C4C23×C711yescharacteristic
321C23×C4C23×C714yes—
322C23×C4C23×C711yescharacteristic
323C2×M16—28no—
324C2×M16C23×C718yes—
325C22×C8C23×C714yes—
326#123—28no—
327C22⋊Q8—22no—
328C22⋊Q8—22no—
329C22⋊Q8C23×C714yes—
330C22⋊Q8C23×C714yes—
331#121—22no—
332#121—22no—
333#121C23×C714yes—
334#121C23×C714yes—
335#120—28no—
336C4×D4—28no—
337C4×D4—28no—
338#117—22no—
339#117C23×C714yes—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—28no—
347C2×C4⋊C4—28no—
348C2×C4⋊C4—28no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4C23×C714yes—
352C2×C4⋊C4C23×C718yes—
353C2×C4⋊C4C23×C718yes—
354C2×C4⋊C4C23×C718yes—
355C2×C4⋊C4C23×C711yescharacteristic
356C2×C4⋊C4C23×C711yescharacteristic
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C42—21no—
363#107C23×C7116yescomplemented
364#106C23×C7116yescomplemented
365#103C23×C7116yescomplemented
366#102C23×C7116yescomplemented
367C22⋊C8C23×C718yes—
368C22⋊C8C23×C718yes—
369C22×C7—232no—
370C22×C7—28no—
371C22×C7—28no—
372C22×C7—28no—
373C22×C7—22no—
374C22×C7—22no—
375C22×C7—22no—
376C22×C7—22no—
377C22×C7—22no—
378C22×C7—22no—
379C22×C7—42no—
380C22×C7—42no—
381C22×C7—42no—
382C22×C7—84no—
383C22×C7C2×#12314yes—
384C22×C7C2×#12312yes—
385C22×C7C2×AGL(1,ℤ/8)14yes—
386C22×C7C2×AGL(1,ℤ/8)12yes—
387C22×C7C2×#13714yes—
388C22×C7C2×#13712yes—
389C22×C7C23×D411yescharacteristic
390C4×C7—22no—
391C4×C7—22no—
392C4×C7—42no—
393C4×C7—48no—
394C4×C7—42no—
395C4×C7—42no—
396C4×C7C2×#12314yes—
397C4×C7C2×#12312yes—
398C4×C7C22×#4112yes—
399C24—24no—
400C24—24no—
401C24—24no—
402C24—21no—
403C24C2×C7×D414yes—
404C24C2×C7×D411yescharacteristic
405C24C2×C7×D411yescharacteristic
406C24C2×C7×D411yescharacteristic
407#41—216no—
408#41—48no—
409C2×Q8—21no—
410C2×Q8—24no—
411C2×Q8C24×C712yes—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—21no—
416C2×D4—24no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4C24×C712yes—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—216no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—216no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—432no—
439C22×C4—48no—
440C22×C4—44no—
441C22×C4C2×C7×D411yescharacteristic
442C22×C4C2×C7×D411yescharacteristic
443C22×C4C2×C7×D414yes—
444C22×C4C2×C7×D414yes—
445C22×C4C2×C7×D412yes—
446C22×C4C2×C7×D412yes—
447C22×C4C2×C7×D414yes—
448C22×C4C2×C7×D411yescharacteristic
449C22×C4C2×C7×D412yes—
450C22×C4C7×#4118yes—
451C22×C4C24×C712yes—
452C22×C4C24×C712yes—
453C22×C4C24×C711yescharacteristic
454C22×C4C24×C712yes—
455M16—216no—
456C2×C8—28no—
457C2×C8C24×C718yes—
458C4⋊C4—28no—
459C4⋊C4—28no—
460C4⋊C4—28no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4C24×C718yes—
467C4⋊C4C24×C712yes—
468C4⋊C4C24×C712yes—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C42—24no—
475C2×C7—24no—
476C2×C7—24no—
477C2×C7—24no—
478C2×C7—82no—
479C2×C7#309214yesdirect product factor
480C2×C7C22×#12311yescharacteristic
481C2×C7C22×AGL(1,ℤ/8)11yescharacteristic
482C2×C7C22×#13711yescharacteristic
483C23—232no—
484C23—28no—
485C23—22no—
486C23—22no—
487C23—28no—
488C23—22no—
489C23—22no—
490C23—28no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23C7×#12314yes—
498C23C7×AGL(1,ℤ/8)14yes—
499C23C7×#13714yes—
500C23C22×C7×D412yes—
501C23C22×C7×D412yes—
502C23C22×C7×D412yes—
503C23C22×C7×D411yescharacteristic
504Q8—24no—
505Q8—42no—
506D4—24no—
507D4—48no—
508D4—48no—
509D4—48no—
510D4—42no—
511C2×C4—28no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—28no—
524C2×C4—416no—
525C2×C4—416no—
526C2×C4—42no—
527C2×C4—416no—
528C2×C4—48no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—48no—
532C2×C4C7×#12314yes—
533C2×C4C7×#12318yes—
534C2×C4C22×C7×D412yes—
535C2×C4C22×C7×D412yes—
536C2×C4C22×C7×D412yes—
537C2×C4C22×C7×D411yescharacteristic
538C2×C4C2×C7×#4114yes—
539C2×C4C2×C7×#4118yes—
540C2×C4C25×C711yescharacteristic
541C8—28no—
542C7C2×#309211yescharacteristic, direct product factor, Sylow
543C22—232no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—28no—
551C22—28no—
552C22—28no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—84no—
557C22C2×C7×#12314yes—
558C22C2×C7×#12312yes—
559C22C2×C7×AGL(1,ℤ/8)14yes—
560C22C2×C7×AGL(1,ℤ/8)12yes—
561C22C2×C7×#13714yes—
562C22C2×C7×#13712yes—
563C22C23×C7×D411yescharacteristic
564C4—22no—
565C4—22no—
566C4—48no—
567C4—42no—
568C4—42no—
569C4—42no—
570C4C2×C7×#12314yes—
571C4C2×C7×#12312yes—
572C4C22×C7×#4112yes—
573C2—24no—
574C2—24no—
575C2—24no—
576C2—82no—
577C2C7×#309214yesdirect product factor
578C2C22×C7×#12311yescharacteristic
579C2C22×C7×AGL(1,ℤ/8)11yescharacteristic
580C2C22×C7×#13711yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC2×C4C25×C7
prime radicalC2×C7×#525C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C24×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×C7×C4⋊C4C2semidirect
3. C7×#3092C2direct (N × H)
4. C2×C7×#484C2semidirect
5. C2×C7×#481C2semidirect
6. C2×C7×#483C2semidirect
7. C2×C7×#480C2semidirect
8. C2×C7×#428C2semidirect
9. C2×C7×#417C2semidirect
10. C2×C7×#407C2semidirect
33. C22×C7×C4⋊C4C22semidirect
34. C7×#484C22semidirect
35. C7×#483C22semidirect
36. C7×#481C22semidirect
37. C7×#480C22semidirect
38. C7×#428C22semidirect
39. C2×C7×#107C22semidirect
40. C2×C7×#106C22semidirect
41. C7×#417C22semidirect
42. C2×C7×#103C22semidirect
43. C2×C7×#102C22semidirect
44. C7×#407C22semidirect
47. C2×#3092C7direct (N × H)
112. C7×#107C23semidirect
113. C7×#106C23semidirect
114. C7×#103C23semidirect
115. C7×#102C23semidirect
119. C23×C4⋊C4C2×C7semidirect
120. #3092C2×C7direct (N × H)
121. C2×#484C2×C7semidirect
122. C2×#481C2×C7semidirect
123. C2×#483C2×C7semidirect
124. C2×#480C2×C7semidirect
125. C2×#428C2×C7semidirect
126. C2×#417C2×C7semidirect
127. C2×#407C2×C7semidirect
226. C22×C4⋊C4C22×C7semidirect
227. #484C22×C7semidirect
228. #483C22×C7semidirect
229. #481C22×C7semidirect
230. #480C22×C7semidirect
231. #428C22×C7semidirect
232. C2×#107C22×C7semidirect
233. C2×#106C22×C7semidirect
234. #417C22×C7semidirect
235. C2×#103C22×C7semidirect
236. C2×#102C22×C7semidirect
237. #407C22×C7semidirect
363. #107C23×C7semidirect
364. #106C23×C7semidirect
365. #103C23×C7semidirect
366. #102C23×C7semidirect
479. C2×C7#3092direct (N × H)
542. C7C2×#3092direct (N × H)
577. C2C7×#3092direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5252
C23×C7×C4⋊C42
C7×#30922
C2×C7×#4842
C2×C7×#4812
C2×C7×#4832
C2×C7×#4802
C2×C7×#4282
C2×C7×#4172
C2×C7×#4072
C2×#30927