Finite Group Databasebeta

← C7×#13879 #13191095

Subgroups — C7×#13879 #13191095

Order 1792.

2006 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#117C211yesmaximal, characteristic
2C2×C7×#384C211yesmaximal, characteristic, complemented
3C2×C7×#382C211yesmaximal, characteristic, complemented
4C2×C7×#380C211yesmaximal, characteristic
5C2×C4×C7×C22⋊C4C211yesmaximal, characteristic
6C7×#1439C214yesmaximal
7C7×#1436C214yesmaximal, complemented
8C7×C4⋊#95C211yesmaximal, characteristic, complemented
9C7×C4⋊#95C211yesmaximal, characteristic, complemented
10C24×C4×C7C2211yescharacteristic
11C2×C7×#117—24no—
12C2×C7×#117—24no—
13C2×C7×#117—24no—
14C22×C7×C4⋊C4C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16C22×C7×C4⋊C4C2214yes—
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×C7×C4⋊C4C2211yescharacteristic
19C22×C7×C22⋊C4C2212yes—
20C22×C7×C22⋊C4C2212yes—
21C22×C7×C22⋊C4C2212yes—
22C22×C42×C7C2212yes—
23C22×C42×C7C2211yescharacteristic
24C22×C42×C7C2211yescharacteristic
25C7×#384—24no—
26C7×#382—24no—
27C7×#380—24no—
28C4×C7×C22⋊C4C418yescomplemented
29C2×C7×#95C2212yes—
30C2×C7×#95C2214yes—
31C2×C7×#95C2214yes—
32C2×C7×#95C2214yes—
33C2×C7×#95C2212yes—
34#13879C711yesmaximal, characteristic, direct product factor, Sylow
35C25×C7C2311yescharacteristic
36C23×C4×C7—22no—
37C23×C4×C7—22no—
38C23×C4×C7—22no—
39C23×C4×C7—22no—
40C23×C4×C7C2×C414yes—
41C23×C4×C7D414yes—
42C23×C4×C7Q814yes—
43C23×C4×C7C2312yes—
44C23×C4×C7C2312yes—
45C23×C4×C7C2312yes—
46C23×C4×C7C2312yes—
47C23×C4×C7C2312yes—
48C23×C4×C7C2311yescharacteristic
49C23×C4×C7C2312yes—
50C23×C4×C7C2311yescharacteristic
51C7×#117—232no—
52C2×C7×C4⋊C4—22no—
53C2×C7×C4⋊C4—22no—
54C2×C7×C4⋊C4—22no—
55C2×C7×C4⋊C4—22no—
56C2×C7×C4⋊C4—22no—
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4—22no—
59C2×C7×C4⋊C4—28no—
60C2×C7×C4⋊C4—22no—
61C2×C7×C4⋊C4—28no—
62C2×C7×C4⋊C4—22no—
63C2×C7×C4⋊C4—22no—
64C2×C7×C4⋊C4—44no—
65C2×C7×C4⋊C4D414yes—
66C2×C7×C4⋊C4D414yes—
67C2×C7×C22⋊C4—24no—
68C2×C7×C22⋊C4—24no—
69C2×C7×C22⋊C4—24no—
70C2×C7×C22⋊C4—24no—
71C2×C7×C22⋊C4—24no—
72C2×C7×C22⋊C4—24no—
73C2×C7×C22⋊C4—44no—
74C2×C7×C22⋊C4C2×C418yes—
75C2×C42×C7—24no—
76C2×C42×C7—24no—
77C2×C42×C7—24no—
78C2×C42×C7—22no—
79C2×C42×C7—22no—
80C2×C42×C7—22no—
81C2×C42×C7—22no—
82C2×C42×C7C2×C414yes—
83C2×C42×C7C2×C414yes—
84C7×#95—24no—
85C7×#95—28no—
86C7×#95—28no—
87C7×#95—44no—
88C7×#95C2×C418yes—
89C22×#117C2×C711yescharacteristic
90C2×#384C2×C711yescharacteristic, complemented
91C2×#382C2×C711yescharacteristic, complemented
92C2×#380C2×C711yescharacteristic
93C2×C4×C22⋊C4C2×C711yescharacteristic
94#1439C2×C714yes—
95#1436C2×C714yescomplemented
96C4⋊#95C2×C711yescharacteristic, complemented
97C4⋊#95C2×C711yescharacteristic, complemented
98C24×C7—21no—
99C24×C7—21no—
100C24×C7—22no—
101C24×C7—22no—
102C24×C7—21no—
103C24×C7—21no—
104C24×C7—22no—
105C24×C7—22no—
106C24×C7C22×C412yes—
107C24×C7C2×D412yes—
108C24×C7C2×Q812yes—
109C24×C7C2411yescharacteristic
110C22×C4×C7—21no—
111C22×C4×C7—21no—
112C22×C4×C7—22no—
113C22×C4×C7—22no—
114C22×C4×C7—21no—
115C22×C4×C7—21no—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—28no—
119C22×C4×C7—28no—
120C22×C4×C7—28no—
121C22×C4×C7—28no—
122C22×C4×C7—28no—
123C22×C4×C7—28no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—24no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—24no—
134C22×C4×C7—22no—
135C22×C4×C7—24no—
136C22×C4×C7—22no—
137C22×C4×C7—24no—
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—24no—
147C22×C4×C7—24no—
148C22×C4×C7C4⋊C4116yescomplemented
149C22×C4×C7C22×C414yes—
150C22×C4×C7C22×C412yes—
151C22×C4×C7C22×C414yes—
152C22×C4×C7C22×C412yes—
153C22×C4×C7C2×D414yes—
154C22×C4×C7C2×D414yes—
155C22×C4×C7C2×D412yes—
156C22×C4×C7C2×D412yes—
157C22×C4×C7C2×D412yes—
158C22×C4×C7C2×D412yes—
159C22×C4×C7C2×Q812yes—
160C22×C4×C7C2×Q812yes—
161C22×C4×C7#4114yes—
162C22×C4×C7#4114yes—
163C22×C4×C7#4112yes—
164C22×C4×C7#4112yes—
165C7×C4⋊C4—28no—
166C7×C4⋊C4—28no—
167C7×C4⋊C4—416no—
168C7×C4⋊C4—44no—
169C7×C4⋊C4—44no—
170C7×C22⋊C4—216no—
171C7×C22⋊C4—216no—
172C7×C22⋊C4—48no—
173C42×C7—216no—
174C42×C7—28no—
175C42×C7—28no—
176C24×C4C22×C711yescharacteristic
177C2×#117—24no—
178C2×#117—24no—
179C2×#117—24no—
180C22×C4⋊C4C22×C711yescharacteristic
181C22×C4⋊C4C22×C711yescharacteristic
182C22×C4⋊C4C22×C714yes—
183C22×C4⋊C4C22×C711yescharacteristic
184C22×C4⋊C4C22×C711yescharacteristic
185C22×C22⋊C4C22×C712yes—
186C22×C22⋊C4C22×C712yes—
187C22×C22⋊C4C22×C712yes—
188C22×C42C22×C712yes—
189C22×C42C22×C711yescharacteristic
190C22×C42C22×C711yescharacteristic
191#384—24no—
192#382—24no—
193#380—24no—
194C4×C22⋊C4C4×C718yescomplemented
195C2×#95C22×C714yes—
196C2×#95C22×C712yes—
197C2×#95C22×C714yes—
198C2×#95C22×C714yes—
199C2×#95C22×C712yes—
200C23×C7—24no—
201C23×C7—22no—
202C23×C7—22no—
203C23×C7—24no—
204C23×C7—22no—
205C23×C7—22no—
206C23×C7—22no—
207C23×C7—22no—
208C23×C7—22no—
209C23×C7—22no—
210C23×C7—24no—
211C23×C7—22no—
212C23×C7—22no—
213C23×C7—24no—
214C23×C7—42no—
215C23×C7—42no—
216C23×C7—42no—
217C23×C7—42no—
218C23×C7—42no—
219C23×C7—42no—
220C23×C7—42no—
221C23×C7—42no—
222C23×C7C2×C4⋊C414yes—
223C23×C7C23×C411yescharacteristic
224C23×C7C22×D411yescharacteristic
225C23×C7C22×D411yescharacteristic
226C23×C7C22×D411yescharacteristic
227C23×C7C22×Q811yescharacteristic
228C23×C7C2×#4112yes—
229C23×C7C2×#4112yes—
230C23×C7C2×#4111yescharacteristic
231C23×C7C2×#4111yescharacteristic
232C23×C7H(5,𝔽2)12yes—
233C23×C72−1+412yes—
234C2×C4×C7—24no—
235C2×C4×C7—22no—
236C2×C4×C7—22no—
237C2×C4×C7—24no—
238C2×C4×C7—22no—
239C2×C4×C7—22no—
240C2×C4×C7—22no—
241C2×C4×C7—22no—
242C2×C4×C7—24no—
243C2×C4×C7—24no—
244C2×C4×C7—24no—
245C2×C4×C7—24no—
246C2×C4×C7—232no—
247C2×C4×C7—24no—
248C2×C4×C7—24no—
249C2×C4×C7—24no—
250C2×C4×C7—24no—
251C2×C4×C7—24no—
252C2×C4×C7—28no—
253C2×C4×C7—28no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—44no—
260C2×C4×C7—42no—
261C2×C4×C7—44no—
262C2×C4×C7—42no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7—42no—
270C2×C4×C7—44no—
271C2×C4×C7—44no—
272C2×C4×C7—44no—
273C2×C4×C7—44no—
274C2×C4×C7C2×C4⋊C414yes—
275C2×C4×C7C2×C4⋊C414yes—
276C2×C4×C7#11718yescomplemented
277C2×C4×C7C4×D414yes—
278C2×C4×C7C4×D414yes—
279C2×C4×C7#12014yes—
280C2×C4×C7C22⋊Q814yes—
281C2×C4×C7C22⋊Q814yes—
282C2×C4×C7#12314yes—
283C25C23×C711yescharacteristic
284C23×C4—22no—
285C23×C4—22no—
286C23×C4—22no—
287C23×C4—22no—
288C23×C4C2×C4×C714yes—
289C23×C4C7×D414yes—
290C23×C4C7×Q814yes—
291C23×C4C23×C712yes—
292C23×C4C23×C712yes—
293C23×C4C23×C712yes—
294C23×C4C23×C712yes—
295C23×C4C23×C712yes—
296C23×C4C23×C711yescharacteristic
297C23×C4C23×C712yes—
298C23×C4C23×C711yescharacteristic
299#117—232no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4C7×D414yes—
314C2×C4⋊C4C7×D414yes—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4C2×C4×C718yes—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42C2×C4×C714yes—
331C2×C42C2×C4×C714yes—
332#95—24no—
333#95—28no—
334#95—28no—
335#95—44no—
336#95C2×C4×C718yes—
337C22×C7—24no—
338C22×C7—24no—
339C22×C7—24no—
340C22×C7—44no—
341C22×C7—44no—
342C22×C7—42no—
343C22×C7—42no—
344C22×C7—44no—
345C22×C7—44no—
346C22×C7—42no—
347C22×C7—42no—
348C22×C7C22×C4⋊C411yescharacteristic
349C22×C7C2×#11712yes—
350C22×C7C2×C4×D411yescharacteristic
351C22×C7C2×C4×D411yescharacteristic
352C22×C7#52012yes—
353C22×C7C2×#12011yescharacteristic
354C22×C7C2×C22⋊Q811yescharacteristic
355C22×C7C2×C22⋊Q811yescharacteristic
356C22×C7C2×#12311yescharacteristic
357C22×C7#53211yescharacteristic
358C22×C7#53211yescharacteristic
359C22×C7#53312yes—
360C22×C7#53511yescharacteristic
361C22×C7#53611yescharacteristic
362C22×C7#54412yes—
363C22×C7#54612yes—
364C22×C7#54712yes—
365C22×C7#54812yes—
366C22×C7#55012yes—
367C22×C7#55212yes—
368C22×C7#55312yes—
369C22×C7#55512yes—
370C22×C7#55612yes—
371C4×C7—28no—
372C4×C7—24no—
373C4×C7—44no—
374C4×C7—44no—
375C4×C7—44no—
376C4×C7—82no—
377C4×C7—82no—
378C4×C7#38518yes—
379C24—21no—
380C24—21no—
381C24—22no—
382C24—22no—
383C24—21no—
384C24—21no—
385C24—22no—
386C24—22no—
387C24C22×C4×C712yes—
388C24C2×C7×D412yes—
389C24C2×C7×Q812yes—
390C24C24×C711yescharacteristic
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4C7×C4⋊C4116yescomplemented
430C22×C4C22×C4×C714yes—
431C22×C4C22×C4×C712yes—
432C22×C4C22×C4×C714yes—
433C22×C4C22×C4×C712yes—
434C22×C4C2×C7×D414yes—
435C22×C4C2×C7×D414yes—
436C22×C4C2×C7×D412yes—
437C22×C4C2×C7×D412yes—
438C22×C4C2×C7×D412yes—
439C22×C4C2×C7×D412yes—
440C22×C4C2×C7×Q812yes—
441C22×C4C2×C7×Q812yes—
442C22×C4C7×#4114yes—
443C22×C4C7×#4114yes—
444C22×C4C7×#4112yes—
445C22×C4C7×#4112yes—
446C4⋊C4—28no—
447C4⋊C4—28no—
448C4⋊C4—416no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C22⋊C4—216no—
452C22⋊C4—216no—
453C22⋊C4—48no—
454C42—216no—
455C42—28no—
456C42—28no—
457C2×C7—44no—
458C2×C7#228511yescharacteristic
459C2×C7C2×#38511yescharacteristic
460C2×C7#231711yescharacteristic
461C2×C7#235212yes—
462C2×C7#235412yes—
463C2×C7#243111yescharacteristic
464C2×C7#243211yescharacteristic
465C2×C7#243912yes—
466C2×C7#244112yes—
467C2×C7#249911yescharacteristic
468C2×C7#250011yescharacteristic
469C23—24no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23C2×C7×C4⋊C414yes—
492C23C23×C4×C711yescharacteristic
493C23C22×C7×D411yescharacteristic
494C23C22×C7×D411yescharacteristic
495C23C22×C7×D411yescharacteristic
496C23C22×C7×Q811yescharacteristic
497C23C2×C7×#4112yes—
498C23C2×C7×#4112yes—
499C23C2×C7×#4111yescharacteristic
500C23C2×C7×#4111yescharacteristic
501C23C7×H(5,𝔽2)12yes—
502C23C7×2−1+412yes—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—232no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4C2×C7×C4⋊C414yes—
544C2×C4C2×C7×C4⋊C414yes—
545C2×C4C7×#11718yescomplemented
546C2×C4C4×C7×D414yes—
547C2×C4C4×C7×D414yes—
548C2×C4C7×#12014yes—
549C2×C4C7×C22⋊Q814yes—
550C2×C4C7×C22⋊Q814yes—
551C2×C4C7×#12314yes—
552C7#1387911yescharacteristic, direct product factor, Sylow
553C22—24no—
554C22—24no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—44no—
559C22—44no—
560C22—42no—
561C22—42no—
562C22—44no—
563C22—44no—
564C22C22×C7×C4⋊C411yescharacteristic
565C22C2×C7×#11712yes—
566C22C2×C4×C7×D411yescharacteristic
567C22C2×C4×C7×D411yescharacteristic
568C22C7×#52012yes—
569C22C2×C7×#12011yescharacteristic
570C22C2×C7×C22⋊Q811yescharacteristic
571C22C2×C7×C22⋊Q811yescharacteristic
572C22C2×C7×#12311yescharacteristic
573C22C7×#53211yescharacteristic
574C22C7×#53211yescharacteristic
575C22C7×#53312yes—
576C22C7×#53511yescharacteristic
577C22C7×#53611yescharacteristic
578C22C7×#54412yes—
579C22C7×#54612yes—
580C22C7×#54712yes—
581C22C7×#54812yes—
582C22C7×#55012yes—
583C22C7×#55212yes—
584C22C7×#55312yes—
585C22C7×#55512yes—
586C22C7×#55612yes—
587C4—24no—
588C4—28no—
589C4—44no—
590C4—44no—
591C4—44no—
592C4—82no—
593C4—82no—
594C4C7×#38518yes—
595C2—44no—
596C2C7×#228511yescharacteristic
597C2C2×C7×#38511yescharacteristic
598C2C7×#231711yescharacteristic
599C2C7×#235212yes—
600C2C7×#235412yes—
601C2C7×#243111yescharacteristic
602C2C7×#243211yescharacteristic
603C2C7×#243912yes—
604C2C7×#244112yes—
605C2C7×#249911yescharacteristic
606C2C7×#250011yescharacteristic
subgroupG/H
center, socleC24×C7C24
commutatorC23C23×C4×C7
Frattini, cosocleC24C24×C7
prime radicalC25×C7C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#384C2semidirect
3. C2×C7×#382C2semidirect
7. C7×#1436C2semidirect
8. C7×C4⋊#95C2semidirect
9. C7×C4⋊#95C2semidirect
28. C4×C7×C22⋊C4C4semidirect
34. #13879C7direct (N × H)
90. C2×#384C2×C7semidirect
91. C2×#382C2×C7semidirect
95. #1436C2×C7semidirect
96. C4⋊#95C2×C7semidirect
97. C4⋊#95C2×C7semidirect
148. C22×C4×C7C4⋊C4semidirect
194. C4×C22⋊C4C4×C7semidirect
276. C2×C4×C7#117semidirect
429. C22×C4C7×C4⋊C4semidirect
545. C2×C4C7×#117semidirect
552. C7#13879direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×#1172
C2×C7×#3842
C2×C7×#3822
C2×C7×#3802
C2×C4×C7×C22⋊C42
C7×#14392
C7×#14362
C7×C4⋊#952
#138797