Finite Group Databasebeta

← C7×#13977 #13191193

Subgroups — C7×#13977 #13191193

Order 1792.

2558 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C7×D4C211yesmaximal, characteristic
2C2×C7×#390C211yesmaximal, characteristic
3C2×C7×#390C211yesmaximal, characteristic
4C2×C7×#389C211yesmaximal, characteristic, complemented
5C2×C7×#385C214yesmaximal, complemented
6C2×C7×#384C211yesmaximal, characteristic, complemented
7C7×#1439C214yesmaximal, complemented
8C7×C4⋊#95C211yesmaximal, characteristic, complemented
9C4×C7×#95C211yesmaximal, characteristic, complemented
10C23×C7×D4C2211yescharacteristic
11C24×C4×C7C2212yes—
12C2×C4×C7×D4—24no—
13C2×C4×C7×D4—24no—
14C2×C4×C7×D4—24no—
15C22×C7×C4⋊C4C2211yescharacteristic
16C22×C7×C4⋊C4C2212yes—
17C22×C7×C4⋊C4C2214yes—
18C22×C7×C4⋊C4C2211yescharacteristic
19C22×C7×C4⋊C4C2211yescharacteristic
20C22×C7×C22⋊C4C2212yes—
21C22×C7×C22⋊C4C2214yes—
22C22×C7×C22⋊C4C2214yes—
23C22×C42×C7C2211yescharacteristic
24C22×C42×C7C2211yescharacteristic
25C22×C42×C7C2211yescharacteristic
26C7×#390—24no—
27C7×#390C418yescomplemented
28C7×#389—24no—
29C7×#385—216no—
30C7×#384—24no—
31C2×C7×#95C2214yes—
32C2×C7×#95C2212yes—
33C2×C7×#95C2212yes—
34C2×C7×#95C2212yes—
35#13977C711yesmaximal, characteristic, direct product factor, Sylow
36C25×C7C2312yes—
37C22×C7×D4—22no—
38C22×C7×D4—22no—
39C22×C7×D4—22no—
40C22×C7×D4—22no—
41C22×C7×D4C2×C414yes—
42C22×C7×D4D414yes—
43C22×C7×D4Q814yes—
44C23×C4×C7—24no—
45C23×C4×C7—24no—
46C23×C4×C7—24no—
47C23×C4×C7—28no—
48C23×C4×C7—44no—
49C23×C4×C7C2312yes—
50C23×C4×C7C2312yes—
51C23×C4×C7C2311yescharacteristic
52C23×C4×C7C2312yes—
53C23×C4×C7C2312yes—
54C23×C4×C7C2312yes—
55C23×C4×C7C2311yescharacteristic
56C23×C4×C7C2311yescharacteristic
57C4×C7×D4—232no—
58C2×C7×C4⋊C4—22no—
59C2×C7×C4⋊C4—22no—
60C2×C7×C4⋊C4—22no—
61C2×C7×C4⋊C4—22no—
62C2×C7×C4⋊C4—28no—
63C2×C7×C4⋊C4—22no—
64C2×C7×C4⋊C4—28no—
65C2×C7×C4⋊C4—28no—
66C2×C7×C4⋊C4—22no—
67C2×C7×C4⋊C4—24no—
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4—44no—
70C2×C7×C4⋊C4C2×C414yescomplemented
71C2×C7×C4⋊C4D414yes—
72C2×C7×C22⋊C4—24no—
73C2×C7×C22⋊C4—24no—
74C2×C7×C22⋊C4—24no—
75C2×C7×C22⋊C4—28no—
76C2×C7×C22⋊C4—28no—
77C2×C7×C22⋊C4—28no—
78C2×C7×C22⋊C4—28no—
79C2×C7×C22⋊C4—44no—
80C2×C7×C22⋊C4C2×C4116yescomplemented
81C2×C42×C7—22no—
82C2×C42×C7—22no—
83C2×C42×C7—22no—
84C2×C42×C7—22no—
85C2×C42×C7—22no—
86C2×C42×C7—22no—
87C2×C42×C7—22no—
88C2×C42×C7C2×C414yescomplemented
89C2×C42×C7D414yes—
90C7×#95—28no—
91C7×#95—24no—
92C7×#95—24no—
93C7×#95D418yescomplemented
94C22×C4×D4C2×C711yescharacteristic
95C2×#390C2×C711yescharacteristic
96C2×#390C2×C711yescharacteristic
97C2×#389C2×C711yescharacteristic, complemented
98C2×#385C2×C714yescomplemented
99C2×#384C2×C711yescharacteristic, complemented
100#1439C2×C714yescomplemented
101C4⋊#95C2×C711yescharacteristic, complemented
102C4×#95C2×C711yescharacteristic, complemented
103C24×C7—24no—
104C24×C7—22no—
105C24×C7—22no—
106C24×C7—24no—
107C24×C7—22no—
108C24×C7—22no—
109C24×C7—24no—
110C24×C7—24no—
111C24×C7C22×C414yes—
112C24×C7C2×D414yes—
113C24×C7C2×Q814yes—
114C24×C7C2411yescharacteristic
115C2×C7×D4—28no—
116C2×C7×D4—28no—
117C2×C7×D4—28no—
118C2×C7×D4—28no—
119C2×C7×D4—28no—
120C2×C7×D4—28no—
121C2×C7×D4C4⋊C4116yescomplemented
122C22×C4×C7—22no—
123C22×C4×C7—21no—
124C22×C4×C7—21no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—216no—
130C22×C4×C7—24no—
131C22×C4×C7—22no—
132C22×C4×C7—24no—
133C22×C4×C7—21no—
134C22×C4×C7—21no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—24no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—24no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7—22no—
150C22×C4×C7—24no—
151C22×C4×C7—22no—
152C22×C4×C7—24no—
153C22×C4×C7—416no—
154C22×C4×C7—416no—
155C22×C4×C7—48no—
156C22×C4×C7—48no—
157C22×C4×C7C22×C414yes—
158C22×C4×C7C22×C414yes—
159C22×C4×C7C22×C412yes—
160C22×C4×C7C2×D414yes—
161C22×C4×C7C2×D412yes—
162C22×C4×C7C2×D414yes—
163C22×C4×C7C2×D414yes—
164C22×C4×C7C2×D414yes—
165C22×C4×C7C2×D412yes—
166C22×C4×C7C2×D412yes—
167C22×C4×C7C2×D412yes—
168C22×C4×C7C2×D412yes—
169C22×C4×C7C2×Q812yes—
170C22×C4×C7#4114yes—
171C22×C4×C7#4112yes—
172C22×C4×C7#4112yes—
173C7×C4⋊C4—28no—
174C7×C4⋊C4—28no—
175C7×C4⋊C4—416no—
176C7×C4⋊C4—44no—
177C7×C4⋊C4—48no—
178C7×C22⋊C4—216no—
179C7×C22⋊C4—232no—
180C7×C22⋊C4—416no—
181C42×C7—28no—
182C42×C7—28no—
183C42×C7—44no—
184C23×D4C22×C711yescharacteristic
185C24×C4C22×C712yes—
186C2×C4×D4—24no—
187C2×C4×D4—24no—
188C2×C4×D4—24no—
189C22×C4⋊C4C22×C711yescharacteristic
190C22×C4⋊C4C22×C712yes—
191C22×C4⋊C4C22×C714yes—
192C22×C4⋊C4C22×C711yescharacteristic
193C22×C4⋊C4C22×C711yescharacteristic
194C22×C22⋊C4C22×C712yes—
195C22×C22⋊C4C22×C714yes—
196C22×C22⋊C4C22×C714yes—
197C22×C42C22×C711yescharacteristic
198C22×C42C22×C711yescharacteristic
199C22×C42C22×C711yescharacteristic
200#390—24no—
201#390C4×C718yescomplemented
202#389—24no—
203#385—216no—
204#384—24no—
205C2×#95C22×C712yes—
206C2×#95C22×C714yes—
207C2×#95C22×C712yes—
208C2×#95C22×C712yes—
209C23×C7—28no—
210C23×C7—24no—
211C23×C7—24no—
212C23×C7—28no—
213C23×C7—24no—
214C23×C7—24no—
215C23×C7—24no—
216C23×C7—24no—
217C23×C7—28no—
218C23×C7—28no—
219C23×C7—28no—
220C23×C7—28no—
221C23×C7—42no—
222C23×C7—42no—
223C23×C7—44no—
224C23×C7—42no—
225C23×C7—42no—
226C23×C7—44no—
227C23×C7—44no—
228C23×C7—44no—
229C23×C7—44no—
230C23×C7—44no—
231C23×C7C2×C4⋊C418yes—
232C23×C7C23×C411yescharacteristic
233C23×C7C22×D411yescharacteristic
234C23×C7C22×D411yescharacteristic
235C23×C7C22×D411yescharacteristic
236C23×C7C22×D412yes—
237C23×C7C22×Q811yescharacteristic
238C23×C7C2×#4111yescharacteristic
239C23×C7C2×#4111yescharacteristic
240C23×C7C2×#4112yes—
241C23×C7H(5,𝔽2)12yes—
242C23×C72−1+412yes—
243C7×D4—232no—
244C2×C4×C7—24no—
245C2×C4×C7—24no—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—24no—
249C2×C4×C7—24no—
250C2×C4×C7—24no—
251C2×C4×C7—24no—
252C2×C4×C7—24no—
253C2×C4×C7—24no—
254C2×C4×C7—28no—
255C2×C4×C7—24no—
256C2×C4×C7—22no—
257C2×C4×C7—24no—
258C2×C4×C7—24no—
259C2×C4×C7—22no—
260C2×C4×C7—22no—
261C2×C4×C7—24no—
262C2×C4×C7—28no—
263C2×C4×C7—24no—
264C2×C4×C7—432no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7—44no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7—44no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—42no—
280C2×C4×C7—44no—
281C2×C4×C7—44no—
282C2×C4×C7C2×C4⋊C414yes—
283C2×C4×C7C4×D414yes—
284C2×C4×C7C4×D414yes—
285C2×C4×C7C4×D418yescomplemented
286C2×C4×C7#12014yes—
287C2×C4×C7C22⋊Q814yes—
288C2×C4×C7C22⋊Q814yes—
289C2×C4×C7#12314yes—
290C2×C4×C7#12714yes—
291C25C23×C712yes—
292C22×D4—22no—
293C22×D4—22no—
294C22×D4—22no—
295C22×D4—22no—
296C22×D4C2×C4×C714yes—
297C22×D4C7×D414yes—
298C22×D4C7×Q814yes—
299C23×C4—24no—
300C23×C4—24no—
301C23×C4—24no—
302C23×C4—28no—
303C23×C4—44no—
304C23×C4C23×C712yes—
305C23×C4C23×C712yes—
306C23×C4C23×C711yescharacteristic
307C23×C4C23×C712yes—
308C23×C4C23×C712yes—
309C23×C4C23×C712yes—
310C23×C4C23×C711yescharacteristic
311C23×C4C23×C711yescharacteristic
312C4×D4—232no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—28no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—28no—
320C2×C4⋊C4—28no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4C2×C4×C714yescomplemented
326C2×C4⋊C4C7×D414yes—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—28no—
331C2×C22⋊C4—28no—
332C2×C22⋊C4—28no—
333C2×C22⋊C4—28no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4C2×C4×C7116yescomplemented
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42C2×C4×C714yescomplemented
344C2×C42C7×D414yes—
345#95—28no—
346#95—24no—
347#95—24no—
348#95C7×D418yescomplemented
349C22×C7—216no—
350C22×C7—28no—
351C22×C7—44no—
352C22×C7—44no—
353C22×C7—44no—
354C22×C7—48no—
355C22×C7—48no—
356C22×C7—44no—
357C22×C7—44no—
358C22×C7—48no—
359C22×C7—44no—
360C22×C7C22×C4⋊C411yescharacteristic
361C22×C7C2×C4×D411yescharacteristic
362C22×C7C2×C4×D412yes—
363C22×C7C2×C4×D411yescharacteristic
364C22×C7#52012yes—
365C22×C7C2×#12011yescharacteristic
366C22×C7C2×C22⋊Q811yescharacteristic
367C22×C7C2×C22⋊Q811yescharacteristic
368C22×C7C2×#12311yescharacteristic
369C22×C7C2×#12711yescharacteristic
370C22×C7#53211yescharacteristic
371C22×C7#53212yes—
372C22×C7#53511yescharacteristic
373C22×C7#53611yescharacteristic
374C22×C7#54212yes—
375C22×C7D4212yes—
376C22×C7D4212yes—
377C22×C7#54712yes—
378C22×C7#54712yes—
379C22×C7#54812yes—
380C22×C7D4×Q812yes—
381C22×C7#55312yes—
382C22×C7#55412yes—
383C4×C7—28no—
384C4×C7—24no—
385C4×C7—44no—
386C4×C7—44no—
387C4×C7—42no—
388C4×C7—44no—
389C4×C7—82no—
390C4×C7#38518yescomplemented
391C24—24no—
392C24—22no—
393C24—22no—
394C24—24no—
395C24—22no—
396C24—22no—
397C24—24no—
398C24—24no—
399C24C22×C4×C714yes—
400C24C2×C7×D414yes—
401C24C2×C7×Q814yes—
402C24C24×C711yescharacteristic
403C2×D4—28no—
404C2×D4—28no—
405C2×D4—28no—
406C2×D4—28no—
407C2×D4—28no—
408C2×D4—28no—
409C2×D4C7×C4⋊C4116yescomplemented
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—216no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4C22×C4×C714yes—
446C22×C4C22×C4×C714yes—
447C22×C4C22×C4×C712yes—
448C22×C4C2×C7×D414yes—
449C22×C4C2×C7×D412yes—
450C22×C4C2×C7×D414yes—
451C22×C4C2×C7×D414yes—
452C22×C4C2×C7×D414yes—
453C22×C4C2×C7×D412yes—
454C22×C4C2×C7×D412yes—
455C22×C4C2×C7×D412yes—
456C22×C4C2×C7×D412yes—
457C22×C4C2×C7×Q812yes—
458C22×C4C7×#4114yes—
459C22×C4C7×#4112yes—
460C22×C4C7×#4112yes—
461C4⋊C4—28no—
462C4⋊C4—28no—
463C4⋊C4—416no—
464C4⋊C4—44no—
465C4⋊C4—48no—
466C22⋊C4—216no—
467C22⋊C4—232no—
468C22⋊C4—416no—
469C42—28no—
470C42—28no—
471C42—44no—
472C2×C7—48no—
473C2×C7C2×#38511yescharacteristic
474C2×C7C4×#12711yescharacteristic
475C2×C7#232611yescharacteristic
476C2×C7D4×C4⋊C412yes—
477C2×C7#236112yes—
478C2×C7#242212yes—
479C2×C7#243211yescharacteristic
480C2×C7#243311yescharacteristic
481C2×C7#245212yes—
482C2×C7#254211yescharacteristic
483C2×C7#256611yescharacteristic
484C23—28no—
485C23—24no—
486C23—24no—
487C23—28no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—42no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23C2×C7×C4⋊C418yes—
507C23C23×C4×C711yescharacteristic
508C23C22×C7×D411yescharacteristic
509C23C22×C7×D411yescharacteristic
510C23C22×C7×D411yescharacteristic
511C23C22×C7×D412yes—
512C23C22×C7×Q811yescharacteristic
513C23C2×C7×#4111yescharacteristic
514C23C2×C7×#4111yescharacteristic
515C23C2×C7×#4112yes—
516C23C7×H(5,𝔽2)12yes—
517C23C7×2−1+412yes—
518D4—232no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—432no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4C2×C7×C4⋊C414yes—
558C2×C4C4×C7×D414yes—
559C2×C4C4×C7×D414yes—
560C2×C4C4×C7×D418yescomplemented
561C2×C4C7×#12014yes—
562C2×C4C7×C22⋊Q814yes—
563C2×C4C7×C22⋊Q814yes—
564C2×C4C7×#12314yes—
565C2×C4C7×#12714yes—
566C7#1397711yescharacteristic, direct product factor, Sylow
567C22—28no—
568C22—216no—
569C22—48no—
570C22—44no—
571C22—44no—
572C22—48no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—48no—
578C22C22×C7×C4⋊C411yescharacteristic
579C22C2×C4×C7×D411yescharacteristic
580C22C2×C4×C7×D411yescharacteristic
581C22C2×C4×C7×D412yes—
582C22C7×#52012yes—
583C22C2×C7×#12011yescharacteristic
584C22C2×C7×C22⋊Q811yescharacteristic
585C22C2×C7×C22⋊Q811yescharacteristic
586C22C2×C7×#12311yescharacteristic
587C22C2×C7×#12711yescharacteristic
588C22C7×#53212yes—
589C22C7×#53211yescharacteristic
590C22C7×#53511yescharacteristic
591C22C7×#53611yescharacteristic
592C22C7×#54212yes—
593C22C7×D4212yes—
594C22C7×D4212yes—
595C22C7×#54712yes—
596C22C7×#54712yes—
597C22C7×#54812yes—
598C22C7×D4×Q812yes—
599C22C7×#55312yes—
600C22C7×#55412yes—
601C4—24no—
602C4—28no—
603C4—44no—
604C4—42no—
605C4—44no—
606C4—44no—
607C4—82no—
608C4C7×#38518yescomplemented
609C2—48no—
610C2C2×C7×#38511yescharacteristic
611C2C4×C7×#12711yescharacteristic
612C2C7×#232611yescharacteristic
613C2C7×D4×C4⋊C412yes—
614C2C7×#236112yes—
615C2C7×#242212yes—
616C2C7×#243211yescharacteristic
617C2C7×#243311yescharacteristic
618C2C7×#245212yes—
619C2C7×#254211yescharacteristic
620C2C7×#256611yescharacteristic
subgroupG/H
center, socleC24×C7C24
commutatorC23C23×C4×C7
Frattini, cosocleC24C24×C7
prime radicalC23×C7×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C7 ◁ G

Complements

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

NH ≅ G/Ntype
4. C2×C7×#389C2semidirect
5. C2×C7×#385C2semidirect
6. C2×C7×#384C2semidirect
7. C7×#1439C2semidirect
8. C7×C4⋊#95C2semidirect
9. C4×C7×#95C2semidirect
27. C7×#390C4semidirect
35. #13977C7direct (N × H)
70. C2×C7×C4⋊C4C2×C4semidirect
80. C2×C7×C22⋊C4C2×C4semidirect
88. C2×C42×C7C2×C4semidirect
93. C7×#95D4semidirect
97. C2×#389C2×C7semidirect
98. C2×#385C2×C7semidirect
99. C2×#384C2×C7semidirect
100. #1439C2×C7semidirect
101. C4⋊#95C2×C7semidirect
102. C4×#95C2×C7semidirect
121. C2×C7×D4C4⋊C4semidirect
201. #390C4×C7semidirect
285. C2×C4×C7C4×D4semidirect
325. C2×C4⋊C4C2×C4×C7semidirect
335. C2×C22⋊C4C2×C4×C7semidirect
343. C2×C42C2×C4×C7semidirect
348. #95C7×D4semidirect
390. C4×C7#385semidirect
409. C2×D4C7×C4⋊C4semidirect
560. C2×C4C4×C7×D4semidirect
566. C7#13977direct (N × H)
608. C4C7×#385semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C7×D42
C2×C7×#3902
C2×C7×#3892
C2×C7×#3852
C2×C7×#3842
C7×#14392
C7×C4⋊#952
C4×C7×#952
#139777