Finite Group Databasebeta

← C7×#24243 #13201459

Subgroups — C7×#24243 #13201459

Order 1792.

1406 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×D4×Q8C211yesmaximal, characteristic
2C2×C7×#462C211yesmaximal, characteristic, complemented
3C2×C7×#451C212yesmaximal, complemented
4C2×C4×C7×Q16C211yesmaximal, characteristic, complemented
5C7×#2538C211yesmaximal, characteristic
6C7×#2036C212yesmaximal, complemented
7C7×#2001C212yesmaximal, complemented
8C7×#1956C211yesmaximal, characteristic, complemented
9C7×#1954C211yesmaximal, characteristic
10C7×#1803C211yesmaximal, characteristic, complemented
11C7×#1787C212yesmaximal, complemented
12C23×C7×Q8C2212yes—
13C22×C7×Q16C2212yes—
14C22×C7×Q16C2211yescharacteristic
15C7×D4×Q8—24no—
16C7×D4×Q8—24no—
17C2×C7×C4⋊Q8—21no—
18C2×C7×C4⋊Q8C2211yescharacteristic
19C2×C7×C22⋊Q8—22no—
20C2×C7×C22⋊Q8C2212yes—
21C2×C7×C22⋊Q8C2212yes—
22C2×C4×C7×Q8C2211yescharacteristic
23C2×C4×C7×Q8C2211yescharacteristic
24C2×C4×C7×D4—21no—
25C2×C4×C7×D4C2211yescharacteristic
26C7×#462—24no—
27C7×#451—28no—
28C4×C7×Q16—24no—
29C2×C7×#107C2211yescharacteristic
30C2×C7×C4⋊C8C2211yescharacteristic
31C2×C7×#103C2211yescharacteristic
32C2×C7×#103C2211yescharacteristic
33C2×C7×#103C2212yes—
34C2×C7×#103C2212yes—
35C2×C7×#103C2212yes—
36C2×C7×#103C2212yes—
37C2×C7×C22⋊C8C2212yes—
38C2×C7×C22⋊C8C2212yes—
39C2×C4×C7×C8C2211yescharacteristic
40C7×#396—22no—
41C7×#390—21no—
42C7×#389—21no—
43C7×#384C2211yescharacteristic
44C7×#380C2212yes—
45C7×#340C2212yes—
46#24243C711yesmaximal, characteristic, direct product factor, Sylow
47C22×C7×Q8—24no—
48C22×C7×Q8—24no—
49C22×C7×Q8—22no—
50C22×C7×Q8—24no—
51C22×C7×Q8—24no—
52C22×C7×Q8—42no—
53C22×C7×Q8—42no—
54C22×C7×Q8C2312yes—
55C22×C7×Q8C2311yescharacteristic
56C22×C7×Q8C2311yescharacteristic
57C22×C7×D4D411yescharacteristic
58C23×C4×C7—22no—
59C23×C4×C7C2312yes—
60C2×C7×Q16—22no—
61C2×C7×Q16—22no—
62C2×C7×Q16—24no—
63C2×C7×Q16—24no—
64C2×C7×Q16—42no—
65C2×C7×Q16D414yescomplemented
66C22×C7×C8C2312yes—
67C22×C7×C8C2311yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C7×C4⋊Q8—28no—
70C7×C4⋊Q8—22no—
71C7×C4⋊Q8D414yes—
72C7×C22⋊Q8—216no—
73C7×C22⋊Q8—24no—
74C7×C22⋊Q8—24no—
75C7×C22⋊Q8—24no—
76C7×C22⋊Q8—42no—
77C4×C7×Q8—22no—
78C4×C7×Q8—22no—
79C4×C7×Q8—22no—
80C4×C7×Q8D414yes—
81C4×C7×D4—28no—
82C4×C7×D4—22no—
83C4×C7×D4—22no—
84C2×C7×C4⋊C4—21no—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4—21no—
87C2×C7×C4⋊C4—22no—
88C2×C7×C4⋊C4—21no—
89C2×C7×C4⋊C4—21no—
90C2×C7×C4⋊C4—22no—
91C2×C7×C4⋊C4—21no—
92C2×C7×C4⋊C4D411yescharacteristic
93C2×C7×C4⋊C4C2312yes—
94C2×C7×C4⋊C4C2311yescharacteristic
95C2×C7×C4⋊C4C2311yescharacteristic
96C2×C7×C22⋊C4—22no—
97C2×C7×C22⋊C4—22no—
98C2×C7×C22⋊C4—22no—
99C2×C7×C22⋊C4D412yes—
100C2×C42×C7—21no—
101C2×C42×C7—21no—
102C2×C42×C7C2311yescharacteristic
103C7×#107—22no—
104C7×C4⋊C8—22no—
105C7×#103—22no—
106C7×#103—24no—
107C7×#103—24no—
108C7×#103—24no—
109C7×#103—22no—
110C7×#103—42no—
111C7×C22⋊C8—24no—
112C7×C22⋊C8—42no—
113C4×C7×C8—22no—
114C7×#95—22no—
115C2×D4×Q8C2×C711yescharacteristic
116C2×#462C2×C711yescharacteristic, complemented
117C2×#451C2×C712yescomplemented
118C2×C4×Q16C2×C711yescharacteristic, complemented
119#2538C2×C711yescharacteristic
120#2036C2×C712yescomplemented
121#2001C2×C712yescomplemented
122#1956C2×C711yescharacteristic, complemented
123#1954C2×C711yescharacteristic
124#1803C2×C711yescharacteristic, complemented
125#1787C2×C712yescomplemented
126C24×C7C2×D412yes—
127C2×C7×Q8—28no—
128C2×C7×Q8—21no—
129C2×C7×Q8—22no—
130C2×C7×Q8—21no—
131C2×C7×Q8—21no—
132C2×C7×Q8—21no—
133C2×C7×Q8—21no—
134C2×C7×Q8—22no—
135C2×C7×Q8—21no—
136C2×C7×Q8—21no—
137C2×C7×Q8—22no—
138C2×C7×Q8—22no—
139C2×C7×Q8—21no—
140C2×C7×Q8—22no—
141C2×C7×Q8—44no—
142C2×C7×Q8—44no—
143C2×C7×Q8—48no—
144C2×C7×Q8—48no—
145C2×C7×Q8—48no—
146C2×C7×Q8—48no—
147C2×C7×Q8C2×D414yes—
148C2×C7×Q8C2×D412yes—
149C2×C7×Q8C2×D412yes—
150C2×C7×Q8C2×D412yes—
151C2×C7×Q8#4112yes—
152C2×C7×D4—22no—
153C2×C7×D4—22no—
154C2×C7×D4Q1614yescomplemented
155C22×C4×C7—21no—
156C22×C4×C7—28no—
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7—22no—
160C22×C4×C7—22no—
161C22×C4×C7—22no—
162C22×C4×C7—22no—
163C22×C4×C7—22no—
164C22×C4×C7—22no—
165C22×C4×C7—22no—
166C22×C4×C7—21no—
167C22×C4×C7—22no—
168C22×C4×C7—44no—
169C22×C4×C7—44no—
170C22×C4×C7C2×D412yes—
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2411yescharacteristic
174C7×Q16—28no—
175C7×Q16—48no—
176C2×C7×C8—21no—
177C2×C7×C8—21no—
178C2×C7×C8—21no—
179C2×C7×C8—22no—
180C2×C7×C8—22no—
181C2×C7×C8—21no—
182C2×C7×C8—22no—
183C2×C7×C8—21no—
184C2×C7×C8C2×D412yes—
185C7×C4⋊C4—28no—
186C7×C4⋊C4—24no—
187C7×C4⋊C4—24no—
188C7×C4⋊C4—24no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—21no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—44no—
196C7×C4⋊C4—42no—
197C7×C4⋊C4—42no—
198C7×C4⋊C4—81no—
199C7×C4⋊C4C2×D412yes—
200C7×C22⋊C4—28no—
201C7×C22⋊C4—24no—
202C7×C22⋊C4—44no—
203C7×C22⋊C4—82no—
204C42×C7—24no—
205C42×C7—42no—
206C42×C7C2×D412yes—
207C42×C7C2×D412yes—
208C23×Q8C22×C712yes—
209C22×Q16C22×C712yes—
210C22×Q16C22×C711yescharacteristic
211D4×Q8—24no—
212D4×Q8—24no—
213C2×C4⋊Q8—21no—
214C2×C4⋊Q8C22×C711yescharacteristic
215C2×C22⋊Q8—22no—
216C2×C22⋊Q8C22×C712yes—
217C2×C22⋊Q8C22×C712yes—
218C2×C4×Q8C22×C711yescharacteristic
219C2×C4×Q8C22×C711yescharacteristic
220C2×C4×D4—21no—
221C2×C4×D4C22×C711yescharacteristic
222#462—24no—
223#451—28no—
224C4×Q16—24no—
225C2×#107C22×C711yescharacteristic
226C2×C4⋊C8C22×C711yescharacteristic
227C2×#103C22×C712yes—
228C2×#103C22×C711yescharacteristic
229C2×#103C22×C711yescharacteristic
230C2×#103C22×C712yes—
231C2×#103C22×C712yes—
232C2×#103C22×C712yes—
233C2×C22⋊C8C22×C712yes—
234C2×C22⋊C8C22×C712yes—
235C2×C4×C8C22×C711yescharacteristic
236#396—22no—
237#390—21no—
238#389—21no—
239#384C22×C711yescharacteristic
240#380C22×C712yes—
241#340C22×C712yes—
242C23×C7—22no—
243C23×C7—22no—
244C23×C7—22no—
245C23×C7—22no—
246C23×C7—22no—
247C23×C7—22no—
248C23×C7C2×Q1614yes—
249C23×C7C22×D411yescharacteristic
250C7×Q8—22no—
251C7×Q8—22no—
252C7×Q8—22no—
253C7×Q8—22no—
254C7×Q8—22no—
255C7×Q8—24no—
256C7×Q8—24no—
257C7×Q8—416no—
258C7×Q8—42no—
259C7×Q8—41no—
260C7×Q8—42no—
261C7×Q8—41no—
262C7×Q8#12114yes—
263C7×D4—28no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—22no—
270C2×C4×C7—24no—
271C2×C4×C7—22no—
272C2×C4×C7—22no—
273C2×C4×C7—22no—
274C2×C4×C7—22no—
275C2×C4×C7—416no—
276C2×C4×C7—41no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—42no—
283C2×C4×C7—42no—
284C2×C4×C7—42no—
285C2×C4×C7—42no—
286C2×C4×C7—41no—
287C2×C4×C7—42no—
288C2×C4×C7—42no—
289C2×C4×C7—41no—
290C2×C4×C7—42no—
291C2×C4×C7—41no—
292C2×C4×C7#12012yes—
293C2×C4×C7#12012yes—
294C2×C4×C7C2×Q1612yes—
295C2×C4×C7#13712yes—
296C2×C4×C7C22×D411yescharacteristic
297C2×C4×C7C22×D411yescharacteristic
298C2×C4×C7C22×D411yescharacteristic
299C2×C4×C7C2×#4111yescharacteristic
300C2×C4×C7H(5,𝔽2)11yescharacteristic
301C2×C4×C7H(5,𝔽2)11yescharacteristic
302C7×C8—22no—
303C7×C8—41no—
304C7×C8—42no—
305C22×Q8—24no—
306C22×Q8—24no—
307C22×Q8—22no—
308C22×Q8—24no—
309C22×Q8—24no—
310C22×Q8—42no—
311C22×Q8—42no—
312C22×Q8C23×C711yescharacteristic
313C22×Q8C23×C712yes—
314C22×Q8C23×C711yescharacteristic
315C22×D4C7×D411yescharacteristic
316C23×C4—22no—
317C23×C4C23×C712yes—
318C2×Q16—22no—
319C2×Q16—22no—
320C2×Q16—24no—
321C2×Q16—24no—
322C2×Q16—42no—
323C2×Q16C7×D414yescomplemented
324C22×C8C23×C712yes—
325C22×C8C23×C711yescharacteristic
326C22×C8C23×C711yescharacteristic
327C4⋊Q8—28no—
328C4⋊Q8—22no—
329C4⋊Q8C7×D414yes—
330C22⋊Q8—216no—
331C22⋊Q8—24no—
332C22⋊Q8—24no—
333C22⋊Q8—24no—
334C22⋊Q8—42no—
335C4×Q8—22no—
336C4×Q8—22no—
337C4×Q8—22no—
338C4×Q8C7×D414yes—
339C4×D4—28no—
340C4×D4—22no—
341C4×D4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4C7×D411yescharacteristic
351C2×C4⋊C4C23×C711yescharacteristic
352C2×C4⋊C4C23×C712yes—
353C2×C4⋊C4C23×C711yescharacteristic
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4C7×D412yes—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42C23×C711yescharacteristic
361#107—22no—
362C4⋊C8—22no—
363#103—22no—
364#103—24no—
365#103—24no—
366#103—24no—
367#103—22no—
368#103—42no—
369C22⋊C8—24no—
370C22⋊C8—42no—
371C4×C8—22no—
372#95—22no—
373C22×C7—24no—
374C22×C7—24no—
375C22×C7—24no—
376C22×C7—42no—
377C22×C7—42no—
378C22×C7—42no—
379C22×C7—42no—
380C22×C7C2×#12011yescharacteristic
381C22×C7D4211yescharacteristic
382C22×C7#54611yescharacteristic
383C22×C7C22×Q1611yescharacteristic
384C22×C7C2×#13711yescharacteristic
385C22×C7#57711yescharacteristic
386C22×C7#57811yescharacteristic
387C4×C7—24no—
388C4×C7—22no—
389C4×C7—42no—
390C4×C7—44no—
391C4×C7—42no—
392C4×C7—42no—
393C4×C7—81no—
394C4×C7—82no—
395C4×C7#45114yescomplemented
396C4×C7C2×#12111yescharacteristic
397C4×C7#53511yescharacteristic
398C4×C7D4211yescharacteristic
399C4×C7#54611yescharacteristic
400C24C2×C7×D412yes—
401C2×Q8—28no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—22no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—48no—
418C2×Q8—48no—
419C2×Q8—48no—
420C2×Q8—48no—
421C2×Q8C2×C7×D414yes—
422C2×Q8C2×C7×D412yes—
423C2×Q8C2×C7×D412yes—
424C2×Q8C2×C7×D412yes—
425C2×Q8C7×#4112yes—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4C7×Q1614yescomplemented
429C22×C4—21no—
430C22×C4—28no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4C2×C7×D412yes—
445C22×C4C2×C7×D411yescharacteristic
446C22×C4C2×C7×D411yescharacteristic
447C22×C4C24×C711yescharacteristic
448Q16—28no—
449Q16—48no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—21no—
456C2×C8—22no—
457C2×C8—21no—
458C2×C8C2×C7×D412yes—
459C4⋊C4—28no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—81no—
473C4⋊C4C2×C7×D412yes—
474C22⋊C4—28no—
475C22⋊C4—24no—
476C22⋊C4—44no—
477C22⋊C4—82no—
478C42—24no—
479C42—42no—
480C42C2×C7×D412yes—
481C42C2×C7×D412yes—
482C2×C7—44no—
483C2×C7#240811yescharacteristic
484C2×C7C2×#45111yescharacteristic
485C2×C7#301311yescharacteristic
486C2×C7#327811yescharacteristic
487C2×C7D4×Q1611yescharacteristic
488C2×C7#329911yescharacteristic
489C2×C7#332811yescharacteristic
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23C2×C7×Q1614yes—
497C23C22×C7×D411yescharacteristic
498Q8—22no—
499Q8—22no—
500Q8—22no—
501Q8—22no—
502Q8—22no—
503Q8—24no—
504Q8—24no—
505Q8—416no—
506Q8—42no—
507Q8—41no—
508Q8—42no—
509Q8—41no—
510Q8C7×#12114yes—
511D4—28no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—416no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4C7×#12012yes—
541C2×C4C7×#12012yes—
542C2×C4C2×C7×Q1612yes—
543C2×C4C7×#13712yes—
544C2×C4C22×C7×D411yescharacteristic
545C2×C4C22×C7×D411yescharacteristic
546C2×C4C22×C7×D411yescharacteristic
547C2×C4C2×C7×#4111yescharacteristic
548C2×C4C7×H(5,𝔽2)11yescharacteristic
549C2×C4C7×H(5,𝔽2)11yescharacteristic
550C8—22no—
551C8—41no—
552C8—42no—
553C7#2424311yescharacteristic, direct product factor, Sylow
554C22—24no—
555C22—24no—
556C22—24no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22C2×C7×#12011yescharacteristic
562C22C7×D4211yescharacteristic
563C22C7×#54611yescharacteristic
564C22C22×C7×Q1611yescharacteristic
565C22C2×C7×#13711yescharacteristic
566C22C7×#57711yescharacteristic
567C22C7×#57811yescharacteristic
568C4—24no—
569C4—22no—
570C4—42no—
571C4—44no—
572C4—42no—
573C4—42no—
574C4—81no—
575C4—82no—
576C4C7×#45114yescomplemented
577C4C2×C7×#12111yescharacteristic
578C4C7×#53511yescharacteristic
579C4C7×D4211yescharacteristic
580C4C7×#54611yescharacteristic
581C2—44no—
582C2C7×#240811yescharacteristic
583C2C2×C7×#45111yescharacteristic
584C2C7×#301311yescharacteristic
585C2C7×#327811yescharacteristic
586C2C7×D4×Q1611yescharacteristic
587C2C7×#329911yescharacteristic
588C2C7×#332811yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC22×C7×D4D4
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
2. C2×C7×#462C2semidirect
3. C2×C7×#451C2semidirect
4. C2×C4×C7×Q16C2semidirect
6. C7×#2036C2semidirect
7. C7×#2001C2semidirect
8. C7×#1956C2semidirect
10. C7×#1803C2semidirect
11. C7×#1787C2semidirect
46. #24243C7direct (N × H)
65. C2×C7×Q16D4semidirect
116. C2×#462C2×C7semidirect
117. C2×#451C2×C7semidirect
118. C2×C4×Q16C2×C7semidirect
120. #2036C2×C7semidirect
121. #2001C2×C7semidirect
122. #1956C2×C7semidirect
124. #1803C2×C7semidirect
125. #1787C2×C7semidirect
154. C2×C7×D4Q16semidirect
323. C2×Q16C7×D4semidirect
395. C4×C7#451semidirect
428. C2×D4C7×Q16semidirect
553. C7#24243direct (N × H)
576. C4C7×#451semidirect

Maximal subgroups

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

subgroupindex
C2×C7×D4×Q82
C2×C7×#4622
C2×C7×#4512
C2×C4×C7×Q162
C7×#25382
C7×#20362
C7×#20012
C7×#19562
C7×#19542
C7×#18032
C7×#17872
#242437