Finite Group Databasebeta

← C7×#15164 #13192380

Subgroups — C7×#15164 #13192380

Order 1792.

1374 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×#400C211yesmaximal, characteristic
2C2×C7×#395C211yesmaximal, characteristic
3C2×C7×#391C211yesmaximal, characteristic
4C2×C4×C7×C4⋊C4C211yesmaximal, characteristic
5C7×#1446C211yesmaximal, characteristic
6C7×#1445C211yesmaximal, characteristic
7C7×#1441C212yesmaximal
8C7×#1441C212yesmaximal
9C7×#1436C212yesmaximal
10C7×C4⋊#95C211yesmaximal, characteristic
11C7×#1434C212yesmaximal
12C23×C7×Q8C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C4⋊C4C2212yes—
15C22×C7×C4⋊C4C2212yes—
16C22×C7×C4⋊C4C2211yescharacteristic
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×C42×C7C2211yescharacteristic
19C22×C42×C7C2211yescharacteristic
20C22×C42×C7C2211yescharacteristic
21C7×#400—24no—
22C7×#395—24no—
23C7×#391—24no—
24C4×C7×C4⋊C4—24no—
25C2×C7×#95C2212yes—
26C2×C7×#95C2211yescharacteristic
27C2×C7×#95C2212yes—
28C2×C7×#95C2211yescharacteristic
29C2×C7×#95C2212yes—
30C2×C7×#95C2211yescharacteristic
31C2×C7×#95C2211yescharacteristic
32C2×C7×#95C2212yes—
33C2×C7×#95C2212yes—
34C2×C7×#95C2212yes—
35C2×C7×#95C2212yes—
36C2×C7×#95C2212yes—
37C2×C7×#95C2212yes—
38C2×C7×#95C2212yes—
39#15164C711yesmaximal, characteristic, direct product factor, Sylow
40C22×C7×Q8—22no—
41C22×C7×Q8—22no—
42C22×C7×Q8—24no—
43C22×C7×Q8—22no—
44C22×C7×Q8—42no—
45C23×C4×C7C2312yes—
46C23×C4×C7C2312yes—
47C23×C4×C7C2311yescharacteristic
48C23×C4×C7C2311yescharacteristic
49C23×C4×C7C2311yescharacteristic
50C23×C4×C7C2312yes—
51C23×C4×C7C2312yes—
52C23×C4×C7C2311yescharacteristic
53C23×C4×C7C2311yescharacteristic
54C23×C4×C7C2311yescharacteristic
55C23×C4×C7C2311yescharacteristic
56C2×C7×C4⋊C4—22no—
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4—22no—
59C2×C7×C4⋊C4—22no—
60C2×C7×C4⋊C4—22no—
61C2×C7×C4⋊C4—24no—
62C2×C7×C4⋊C4—22no—
63C2×C7×C4⋊C4—24no—
64C2×C7×C4⋊C4—24no—
65C2×C7×C4⋊C4—24no—
66C2×C7×C4⋊C4—22no—
67C2×C7×C4⋊C4—42no—
68C2×C7×C4⋊C4—42no—
69C2×C7×C4⋊C4D414yes—
70C2×C7×C4⋊C4Q814yescomplemented
71C2×C42×C7—22no—
72C2×C42×C7—22no—
73C2×C42×C7—22no—
74C2×C42×C7—42no—
75C2×C42×C7—42no—
76C2×C42×C7D414yes—
77C2×C42×C7Q814yescomplemented
78C7×#95—22no—
79C7×#95—22no—
80C7×#95—24no—
81C7×#95—24no—
82C7×#95—22no—
83C7×#95—24no—
84C7×#95—22no—
85C7×#95—24no—
86C7×#95—24no—
87C7×#95—24no—
88C7×#95—24no—
89C7×#95—24no—
90C7×#95—24no—
91C7×#95—42no—
92C2×#400C2×C711yescharacteristic
93C2×#395C2×C711yescharacteristic
94C2×#391C2×C711yescharacteristic
95C2×C4×C4⋊C4C2×C711yescharacteristic
96#1446C2×C711yescharacteristic
97#1445C2×C711yescharacteristic
98#1441C2×C712yes—
99#1441C2×C712yes—
100#1436C2×C712yes—
101C4⋊#95C2×C711yescharacteristic
102#1434C2×C712yes—
103C24×C7C2411yescharacteristic
104C2×C7×Q8—28no—
105C2×C7×Q8—48no—
106C2×C7×Q8—48no—
107C2×C7×Q8—44no—
108C2×C7×Q8—44no—
109C22×C4×C7—21no—
110C22×C4×C7—21no—
111C22×C4×C7—21no—
112C22×C4×C7—21no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—21no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—21no—
123C22×C4×C7—21no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
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—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
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—22no—
152C22×C4×C7—22no—
153C22×C4×C7—22no—
154C22×C4×C7—22no—
155C22×C4×C7—22no—
156C22×C4×C7—22no—
157C22×C4×C7C2×D412yes—
158C22×C4×C7C2×D412yes—
159C22×C4×C7C2×D412yes—
160C22×C4×C7C2×D412yes—
161C22×C4×C7C2×D412yes—
162C22×C4×C7C2×D412yes—
163C22×C4×C7C2×Q812yes—
164C22×C4×C7C2×Q812yes—
165C22×C4×C7C2×Q812yes—
166C22×C4×C7C2×Q812yes—
167C22×C4×C7C2×Q812yes—
168C22×C4×C7C2×Q812yes—
169C22×C4×C7#4114yes—
170C22×C4×C7#4114yes—
171C22×C4×C7#4112yes—
172C22×C4×C7#4112yes—
173C22×C4×C7#4112yes—
174C7×C4⋊C4—28no—
175C7×C4⋊C4—28no—
176C7×C4⋊C4—48no—
177C7×C4⋊C4—48no—
178C7×C4⋊C4—44no—
179C42×C7—28no—
180C42×C7—44no—
181C42×C7—44no—
182C23×Q8C22×C711yescharacteristic
183C22×C4⋊C4C22×C711yescharacteristic
184C22×C4⋊C4C22×C712yes—
185C22×C4⋊C4C22×C712yes—
186C22×C4⋊C4C22×C711yescharacteristic
187C22×C4⋊C4C22×C711yescharacteristic
188C22×C42C22×C711yescharacteristic
189C22×C42C22×C711yescharacteristic
190C22×C42C22×C711yescharacteristic
191#400—24no—
192#395—24no—
193#391—24no—
194C4×C4⋊C4—24no—
195C2×#95C22×C712yes—
196C2×#95C22×C711yescharacteristic
197C2×#95C22×C712yes—
198C2×#95C22×C711yescharacteristic
199C2×#95C22×C712yes—
200C2×#95C22×C712yes—
201C2×#95C22×C712yes—
202C2×#95C22×C712yes—
203C2×#95C22×C712yes—
204C2×#95C22×C712yes—
205C2×#95C22×C711yescharacteristic
206C2×#95C22×C712yes—
207C2×#95C22×C711yescharacteristic
208C2×#95C22×C712yes—
209C23×C7C22×D411yescharacteristic
210C23×C7C22×D411yescharacteristic
211C23×C7C22×Q811yescharacteristic
212C23×C7C22×Q811yescharacteristic
213C23×C7C2×#4111yescharacteristic
214C23×C7C2×#4111yescharacteristic
215C23×C7C2×#4111yescharacteristic
216C23×C7C2×#4112yes—
217C23×C7C2×#4112yes—
218C23×C7H(5,𝔽2)12yes—
219C23×C72−1+412yes—
220C7×Q8—416no—
221C2×C4×C7—22no—
222C2×C4×C7—22no—
223C2×C4×C7—22no—
224C2×C4×C7—22no—
225C2×C4×C7—22no—
226C2×C4×C7—22no—
227C2×C4×C7—22no—
228C2×C4×C7—22no—
229C2×C4×C7—22no—
230C2×C4×C7—22no—
231C2×C4×C7—22no—
232C2×C4×C7—22no—
233C2×C4×C7—24no—
234C2×C4×C7—24no—
235C2×C4×C7—24no—
236C2×C4×C7—24no—
237C2×C4×C7—24no—
238C2×C4×C7—24no—
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—24no—
247C2×C4×C7—24no—
248C2×C4×C7—24no—
249C2×C4×C7—24no—
250C2×C4×C7—24no—
251C2×C4×C7—42no—
252C2×C4×C7—41no—
253C2×C4×C7—41no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—41no—
257C2×C4×C7—41no—
258C2×C4×C7—41no—
259C2×C4×C7—41no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
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—42no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
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—42no—
281C2×C4×C7—42no—
282C2×C4×C7#12114yes—
283C2×C4×C7C22⋊Q814yes—
284C2×C4×C7C22⋊Q814yes—
285C2×C4×C7C4⋊Q814yes—
286C22×Q8—22no—
287C22×Q8—22no—
288C22×Q8—24no—
289C22×Q8—22no—
290C22×Q8—42no—
291C23×C4C23×C712yes—
292C23×C4C23×C712yes—
293C23×C4C23×C711yescharacteristic
294C23×C4C23×C711yescharacteristic
295C23×C4C23×C711yescharacteristic
296C23×C4C23×C712yes—
297C23×C4C23×C712yes—
298C23×C4C23×C711yescharacteristic
299C23×C4C23×C711yescharacteristic
300C23×C4C23×C711yescharacteristic
301C23×C4C23×C711yescharacteristic
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4C7×D414yes—
316C2×C4⋊C4C7×Q814yescomplemented
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—42no—
321C2×C42—42no—
322C2×C42C7×D414yes—
323C2×C42C7×Q814yescomplemented
324#95—22no—
325#95—22no—
326#95—24no—
327#95—24no—
328#95—22no—
329#95—24no—
330#95—22no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—42no—
338C22×C7C2×#12111yescharacteristic
339C22×C7C2×C22⋊Q811yescharacteristic
340C22×C7C2×C22⋊Q811yescharacteristic
341C22×C7#52511yescharacteristic
342C22×C7#52912yes—
343C22×C7C2×C4⋊Q811yescharacteristic
344C22×C7#53211yescharacteristic
345C22×C7#53211yescharacteristic
346C22×C7#53311yescharacteristic
347C22×C7#53311yescharacteristic
348C22×C7#53312yes—
349C22×C7#53411yescharacteristic
350C22×C7#53611yescharacteristic
351C22×C7#53912yes—
352C22×C7#54412yes—
353C22×C7#54612yes—
354C22×C7#54812yes—
355C22×C7#55112yes—
356C22×C7#55112yes—
357C22×C7#55412yes—
358C22×C7#55612yes—
359C22×C7#55612yes—
360C22×C7#55712yes—
361C4×C7—24no—
362C4×C7—24no—
363C4×C7—24no—
364C4×C7—24no—
365C4×C7—44no—
366C4×C7—42no—
367C4×C7—42no—
368C4×C7—42no—
369C4×C7—44no—
370C4×C7—82no—
371C4×C7—82no—
372C24C24×C711yescharacteristic
373C2×Q8—28no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—44no—
377C2×Q8—44no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4C2×C7×D412yes—
427C22×C4C2×C7×D412yes—
428C22×C4C2×C7×D412yes—
429C22×C4C2×C7×D412yes—
430C22×C4C2×C7×D412yes—
431C22×C4C2×C7×D412yes—
432C22×C4C2×C7×Q812yes—
433C22×C4C2×C7×Q812yes—
434C22×C4C2×C7×Q812yes—
435C22×C4C2×C7×Q812yes—
436C22×C4C2×C7×Q812yes—
437C22×C4C2×C7×Q812yes—
438C22×C4C7×#4114yes—
439C22×C4C7×#4114yes—
440C22×C4C7×#4112yes—
441C22×C4C7×#4112yes—
442C22×C4C7×#4112yes—
443C4⋊C4—28no—
444C4⋊C4—28no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—44no—
448C42—28no—
449C42—44no—
450C42—44no—
451C2×C7#239611yescharacteristic
452C2×C7#253411yescharacteristic
453C2×C7#254511yescharacteristic
454C2×C7#255012yes—
455C2×C7#256012yes—
456C2×C7#256411yescharacteristic
457C2×C7#258111yescharacteristic
458C2×C7#258812yes—
459C2×C7#260512yes—
460C2×C7#262711yescharacteristic
461C2×C7#285711yescharacteristic
462C23C22×C7×D411yescharacteristic
463C23C22×C7×D411yescharacteristic
464C23C22×C7×Q811yescharacteristic
465C23C22×C7×Q811yescharacteristic
466C23C2×C7×#4111yescharacteristic
467C23C2×C7×#4111yescharacteristic
468C23C2×C7×#4111yescharacteristic
469C23C2×C7×#4112yes—
470C23C2×C7×#4112yes—
471C23C7×H(5,𝔽2)12yes—
472C23C7×2−1+412yes—
473Q8—416no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4C7×#12114yes—
536C2×C4C7×C22⋊Q814yes—
537C2×C4C7×C22⋊Q814yes—
538C2×C4C7×C4⋊Q814yes—
539C7#1516411yescharacteristic, direct product factor, Sylow
540C22C2×C7×#12111yescharacteristic
541C22C2×C7×C22⋊Q811yescharacteristic
542C22C2×C7×C22⋊Q811yescharacteristic
543C22C7×#52511yescharacteristic
544C22C7×#52912yes—
545C22C2×C7×C4⋊Q811yescharacteristic
546C22C7×#53211yescharacteristic
547C22C7×#53211yescharacteristic
548C22C7×#53312yes—
549C22C7×#53311yescharacteristic
550C22C7×#53311yescharacteristic
551C22C7×#53411yescharacteristic
552C22C7×#53611yescharacteristic
553C22C7×#53912yes—
554C22C7×#54412yes—
555C22C7×#54612yes—
556C22C7×#54812yes—
557C22C7×#55112yes—
558C22C7×#55112yes—
559C22C7×#55412yes—
560C22C7×#55612yes—
561C22C7×#55612yes—
562C22C7×#55712yes—
563C4—24no—
564C4—24no—
565C4—24no—
566C4—24no—
567C4—42no—
568C4—44no—
569C4—42no—
570C4—44no—
571C4—42no—
572C4—82no—
573C4—82no—
574C2C7×#239611yescharacteristic
575C2C7×#253411yescharacteristic
576C2C7×#254511yescharacteristic
577C2C7×#255012yes—
578C2C7×#256012yes—
579C2C7×#256411yescharacteristic
580C2C7×#258111yescharacteristic
581C2C7×#258812yes—
582C2C7×#260512yes—
583C2C7×#262711yescharacteristic
584C2C7×#285711yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C7C24
commutator, Frattini, cosocleC24C24×C7
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C7 ◁ G

Complements

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

NH ≅ G/Ntype
39. #15164C7direct (N × H)
70. C2×C7×C4⋊C4Q8semidirect
77. C2×C42×C7Q8semidirect
316. C2×C4⋊C4C7×Q8semidirect
323. C2×C42C7×Q8semidirect
539. C7#15164direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4002
C2×C7×#3952
C2×C7×#3912
C2×C4×C7×C4⋊C42
C7×#14462
C7×#14452
C7×#14412
C7×#14362
C7×C4⋊#952
C7×#14342
#151647