Finite Group Databasebeta

← C7×#23983 #13201199

Subgroups — C7×#23983 #13201199

Order 1792.

1228 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×#552C211yesmaximal, characteristic
2C2×C7×#451C211yesmaximal, characteristic, complemented
3C2×C7×#419C211yesmaximal, characteristic, complemented
4C7×#2409C211yesmaximal, characteristic
5C7×#2026C212yesmaximal, complemented
6C7×#2001C212yesmaximal, complemented
7C7×#1894C211yesmaximal, characteristic, complemented
8C7×#1878C212yesmaximal, complemented
9C7×#1872C212yesmaximal, complemented
10C7×#1870C211yesmaximal, characteristic
11C7×#1825C211yesmaximal, characteristic, complemented
12C23×C7×Q8C2211yescharacteristic
13C22×C7×Q16C2212yes—
14C22×C7×M16C2211yescharacteristic
15C7×#552—24no—
16C7×#552—24no—
17C2×C7×#126—21no—
18C2×C7×#125—21no—
19C2×C7×#125C2211yescharacteristic
20C2×C7×#123—21no—
21C2×C7×C22⋊Q8C2211yescharacteristic
22C2×C7×C22⋊Q8C2212yes—
23C2×C7×C22⋊Q8C2211yescharacteristic
24C2×C4×C7×Q8C2211yescharacteristic
25C2×C4×C7×D4C2211yescharacteristic
26C2×C7×#117—21no—
27C2×C7×#117C2211yescharacteristic
28C22×C7×C4⋊C4C2211yescharacteristic
29C7×#451—24no—
30C7×#419—24no—
31C2×C7×#103C2212yes—
32C2×C7×#103C2212yes—
33C2×C7×#103C2212yes—
34C2×C7×#103C2212yes—
35C2×C7×C22⋊C8C2211yescharacteristic
36C2×C7×C22⋊C8C2212yes—
37C7×#395—21no—
38C7×#393—21no—
39C7×#391C2212yes—
40C7×#386—21no—
41C7×#385—21no—
42C7×#380C2211yescharacteristic
43C7×#340C2212yes—
44C7×#340C2212yes—
45C7×#340C2212yes—
46C7×#336C2212yes—
47#23983C711yesmaximal, characteristic, direct product factor, Sylow
48C22×C7×Q8—21no—
49C22×C7×Q8—24no—
50C22×C7×Q8—22no—
51C22×C7×Q8—22no—
52C22×C7×Q8—42no—
53C22×C7×Q8C2311yescharacteristic
54C22×C7×Q8C2312yes—
55C22×C7×D4D411yescharacteristic
56C23×C4×C7—21no—
57C23×C4×C7—21no—
58C23×C4×C7C2311yescharacteristic
59C23×C4×C7C2311yescharacteristic
60C2×C7×Q16—24no—
61C2×C7×Q16—24no—
62C2×C7×Q16—42no—
63C2×C7×M16—22no—
64C2×C7×M16—42no—
65C22×C7×C8C2312yes—
66C22×C7×C8C2312yes—
67C7×#126—28no—
68C7×#125—28no—
69C7×#125—22no—
70C7×#125—22no—
71C7×#123—28no—
72C7×C22⋊Q8—22no—
73C7×C22⋊Q8—22no—
74C7×C22⋊Q8—22no—
75C7×C22⋊Q8—24no—
76C7×C22⋊Q8—42no—
77C7×C22⋊Q8D414yes—
78C4×C7×Q8—22no—
79C4×C7×Q8—22no—
80C4×C7×D4—22no—
81C4×C7×D4—22no—
82C7×#117—28no—
83C7×#117—22no—
84C7×#117D414yes—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4—21no—
87C2×C7×C4⋊C4—21no—
88C2×C7×C4⋊C4—21no—
89C2×C7×C4⋊C4—21no—
90C2×C7×C4⋊C4—21no—
91C2×C7×C4⋊C4—22no—
92C2×C7×C4⋊C4—22no—
93C2×C7×C4⋊C4—22no—
94C2×C7×C4⋊C4D411yescharacteristic
95C2×C7×C4⋊C4C2312yes—
96C2×C7×C4⋊C4C2311yescharacteristic
97C2×C7×C4⋊C4C2311yescharacteristic
98C2×C7×C4⋊C4C2311yescharacteristic
99C2×C7×C22⋊C4—21no—
100C2×C7×C22⋊C4—21no—
101C2×C7×C22⋊C4—21no—
102C2×C7×C22⋊C4—21no—
103C2×C7×C22⋊C4—22no—
104C2×C7×C22⋊C4D411yescharacteristic
105C2×C7×C22⋊C4D411yescharacteristic
106C2×C42×C7—21no—
107C2×C42×C7—21no—
108C2×C42×C7C2311yescharacteristic
109C7×#103—24no—
110C7×#103—24no—
111C7×#103—24no—
112C7×#103—42no—
113C7×C22⋊C8—22no—
114C7×C22⋊C8—42no—
115C7×#95—21no—
116C7×#95—22no—
117C7×#95—22no—
118C2×#552C2×C711yescharacteristic
119C2×#451C2×C711yescharacteristic, complemented
120C2×#419C2×C711yescharacteristic, complemented
121#2409C2×C711yescharacteristic
122#2026C2×C712yescomplemented
123#2001C2×C712yescomplemented
124#1894C2×C711yescharacteristic, complemented
125#1878C2×C712yescomplemented
126#1872C2×C712yescomplemented
127#1870C2×C711yescharacteristic
128#1825C2×C711yescharacteristic, complemented
129C24×C7C2×D411yescharacteristic
130C24×C7C2×D411yescharacteristic
131C2×C7×Q8—21no—
132C2×C7×Q8—22no—
133C2×C7×Q8—28no—
134C2×C7×Q8—22no—
135C2×C7×Q8—22no—
136C2×C7×Q8—22no—
137C2×C7×Q8—22no—
138C2×C7×Q8—22no—
139C2×C7×Q8—22no—
140C2×C7×Q8—44no—
141C2×C7×Q8—48no—
142C2×C7×Q8—44no—
143C2×C7×Q8—42no—
144C2×C7×Q8C2×D412yes—
145C2×C7×Q8#4114yes—
146C2×C7×D4—22no—
147C2×C7×D4—22no—
148C2×C7×D4—41no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—24no—
152C22×C4×C7—22no—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7—22no—
160C22×C4×C7—22no—
161C22×C4×C7—22no—
162C22×C4×C7—44no—
163C22×C4×C7—42no—
164C22×C4×C7—42no—
165C22×C4×C7—42no—
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2×D412yes—
171C22×C4×C7C2×D412yes—
172C22×C4×C7C2411yescharacteristic
173C7×Q16—48no—
174C7×M16—44no—
175C2×C7×C8—22no—
176C2×C7×C8—22no—
177C2×C7×C8—22no—
178C2×C7×C8—22no—
179C2×C7×C8—22no—
180C2×C7×C8—22no—
181C7×C4⋊C4—24no—
182C7×C4⋊C4—24no—
183C7×C4⋊C4—24no—
184C7×C4⋊C4—24no—
185C7×C4⋊C4—24no—
186C7×C4⋊C4—24no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—22no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—44no—
194C7×C4⋊C4C2×D412yes—
195C7×C4⋊C4C2×D412yes—
196C7×C4⋊C4C2×D412yes—
197C7×C22⋊C4—24no—
198C7×C22⋊C4—24no—
199C7×C22⋊C4—24no—
200C7×C22⋊C4—22no—
201C7×C22⋊C4—22no—
202C7×C22⋊C4—44no—
203C7×C22⋊C4—42no—
204C42×C7—24no—
205C42×C7—24no—
206C42×C7—21no—
207C42×C7—21no—
208C23×Q8C22×C711yescharacteristic
209C22×Q16C22×C712yes—
210C22×M16C22×C711yescharacteristic
211#552—24no—
212#552—24no—
213C2×#126—21no—
214C2×#125—21no—
215C2×#125C22×C711yescharacteristic
216C2×#123—21no—
217C2×C22⋊Q8C22×C711yescharacteristic
218C2×C22⋊Q8C22×C712yes—
219C2×C22⋊Q8C22×C711yescharacteristic
220C2×C4×Q8C22×C711yescharacteristic
221C2×C4×D4C22×C711yescharacteristic
222C2×#117—21no—
223C2×#117C22×C711yescharacteristic
224C22×C4⋊C4C22×C711yescharacteristic
225#451—24no—
226#419—24no—
227C2×#103C22×C712yes—
228C2×#103C22×C712yes—
229C2×#103C22×C712yes—
230C2×#103C22×C712yes—
231C2×C22⋊C8C22×C711yescharacteristic
232C2×C22⋊C8C22×C712yes—
233#395—21no—
234#393—21no—
235#391C22×C712yes—
236#386—21no—
237#385—21no—
238#380C22×C711yescharacteristic
239#340C22×C712yes—
240#340C22×C712yes—
241#340C22×C712yes—
242#336C22×C712yes—
243C23×C7—21no—
244C23×C7—22no—
245C23×C7—22no—
246C23×C7—21no—
247C23×C7—21no—
248C23×C7—21no—
249C23×C7—22no—
250C23×C7—22no—
251C23×C7#12012yes—
252C23×C7#13712yes—
253C23×C7C22×D411yescharacteristic
254C7×Q8—24no—
255C7×Q8—24no—
256C7×Q8—24no—
257C7×Q8—48no—
258C7×Q8—41no—
259C7×Q8—41no—
260C7×Q8—42no—
261C7×Q8—42no—
262C7×D4—44no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—24no—
267C2×C4×C7—21no—
268C2×C4×C7—22no—
269C2×C4×C7—22no—
270C2×C4×C7—22no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—22no—
274C2×C4×C7—24no—
275C2×C4×C7—22no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—22no—
279C2×C4×C7—48no—
280C2×C4×C7—48no—
281C2×C4×C7—42no—
282C2×C4×C7—42no—
283C2×C4×C7—42no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—42no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7—42no—
292C2×C4×C7—42no—
293C2×C4×C7—42no—
294C2×C4×C7—42no—
295C2×C4×C7#12012yes—
296C2×C4×C7C22×D411yescharacteristic
297C2×C4×C7C22×D411yescharacteristic
298C2×C4×C7C2×#4112yes—
299C2×C4×C7H(5,𝔽2)12yes—
300C7×C8—42no—
301C7×C8—42no—
302C22×Q8—21no—
303C22×Q8—24no—
304C22×Q8—22no—
305C22×Q8—22no—
306C22×Q8—42no—
307C22×Q8C23×C711yescharacteristic
308C22×Q8C23×C712yes—
309C22×D4C7×D411yescharacteristic
310C23×C4—21no—
311C23×C4—21no—
312C23×C4C23×C711yescharacteristic
313C23×C4C23×C711yescharacteristic
314C2×Q16—24no—
315C2×Q16—24no—
316C2×Q16—42no—
317C2×M16—22no—
318C2×M16—42no—
319C22×C8C23×C712yes—
320C22×C8C23×C712yes—
321#126—28no—
322#125—28no—
323#125—22no—
324#125—22no—
325#123—28no—
326C22⋊Q8—22no—
327C22⋊Q8—22no—
328C22⋊Q8—22no—
329C22⋊Q8—24no—
330C22⋊Q8—42no—
331C22⋊Q8C7×D414yes—
332C4×Q8—22no—
333C4×Q8—22no—
334C4×D4—22no—
335C4×D4—22no—
336#117—28no—
337#117—22no—
338#117C7×D414yes—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4C7×D411yescharacteristic
349C2×C4⋊C4C23×C712yes—
350C2×C4⋊C4C23×C711yescharacteristic
351C2×C4⋊C4C23×C711yescharacteristic
352C2×C4⋊C4C23×C711yescharacteristic
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4C7×D411yescharacteristic
359C2×C22⋊C4C7×D411yescharacteristic
360C2×C42—21no—
361C2×C42—21no—
362C2×C42C23×C711yescharacteristic
363#103—24no—
364#103—24no—
365#103—24no—
366#103—42no—
367C22⋊C8—22no—
368C22⋊C8—42no—
369#95—21no—
370#95—22no—
371#95—22no—
372C22×C7—22no—
373C22×C7—22no—
374C22×C7—22no—
375C22×C7—22no—
376C22×C7—24no—
377C22×C7—41no—
378C22×C7—41no—
379C22×C7—42no—
380C22×C7—42no—
381C22×C7—42no—
382C22×C7C2×#12011yescharacteristic
383C22×C7#54612yes—
384C22×C7C2×#13711yescharacteristic
385C22×C7#57511yescharacteristic
386C22×C7#57812yes—
387C4×C7—22no—
388C4×C7—41no—
389C4×C7—44no—
390C4×C7—42no—
391C4×C7—41no—
392C4×C7—42no—
393C4×C7—42no—
394C4×C7—41no—
395C4×C7—42no—
396C4×C7—82no—
397C4×C7#52511yescharacteristic
398C4×C7#53511yescharacteristic
399C4×C7#54612yes—
400C24C2×C7×D411yescharacteristic
401C24C2×C7×D411yescharacteristic
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—28no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—44no—
412C2×Q8—48no—
413C2×Q8—44no—
414C2×Q8—42no—
415C2×Q8C2×C7×D412yes—
416C2×Q8C7×#4114yes—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—41no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4C2×C7×D411yescharacteristic
438C22×C4C2×C7×D411yescharacteristic
439C22×C4C2×C7×D411yescharacteristic
440C22×C4C2×C7×D411yescharacteristic
441C22×C4C2×C7×D412yes—
442C22×C4C2×C7×D412yes—
443C22×C4C24×C711yescharacteristic
444Q16—48no—
445M16—44no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—44no—
465C4⋊C4C2×C7×D412yes—
466C4⋊C4C2×C7×D412yes—
467C4⋊C4C2×C7×D412yes—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—44no—
474C22⋊C4—42no—
475C42—24no—
476C42—24no—
477C42—21no—
478C42—21no—
479C2×C7—42no—
480C2×C7—42no—
481C2×C7#240411yescharacteristic
482C2×C7#300711yescharacteristic
483C2×C7#301111yescharacteristic
484C2×C7#331212yes—
485C2×C7#332712yes—
486C23—21no—
487C23—22no—
488C23—22no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—22no—
493C23—22no—
494C23C7×#12012yes—
495C23C7×#13712yes—
496C23C22×C7×D411yescharacteristic
497Q8—24no—
498Q8—24no—
499Q8—24no—
500Q8—48no—
501Q8—41no—
502Q8—41no—
503Q8—42no—
504Q8—42no—
505D4—44no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—48no—
523C2×C4—48no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4C7×#12012yes—
539C2×C4C22×C7×D411yescharacteristic
540C2×C4C22×C7×D411yescharacteristic
541C2×C4C2×C7×#4112yes—
542C2×C4C7×H(5,𝔽2)12yes—
543C8—42no—
544C8—42no—
545C7#2398311yescharacteristic, direct product factor, Sylow
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—24no—
551C22—41no—
552C22—41no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22C2×C7×#12011yescharacteristic
557C22C7×#54612yes—
558C22C2×C7×#13711yescharacteristic
559C22C7×#57511yescharacteristic
560C22C7×#57812yes—
561C4—22no—
562C4—41no—
563C4—44no—
564C4—42no—
565C4—41no—
566C4—42no—
567C4—42no—
568C4—41no—
569C4—42no—
570C4—82no—
571C4C7×#52511yescharacteristic
572C4C7×#53511yescharacteristic
573C4C7×#54612yes—
574C2—42no—
575C2—42no—
576C2C7×#240411yescharacteristic
577C2C7×#300711yescharacteristic
578C2C7×#301111yescharacteristic
579C2C7×#331212yes—
580C2C7×#332712yes—
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×#451C2semidirect
3. C2×C7×#419C2semidirect
5. C7×#2026C2semidirect
6. C7×#2001C2semidirect
7. C7×#1894C2semidirect
8. C7×#1878C2semidirect
9. C7×#1872C2semidirect
11. C7×#1825C2semidirect
47. #23983C7direct (N × H)
119. C2×#451C2×C7semidirect
120. C2×#419C2×C7semidirect
122. #2026C2×C7semidirect
123. #2001C2×C7semidirect
124. #1894C2×C7semidirect
125. #1878C2×C7semidirect
126. #1872C2×C7semidirect
128. #1825C2×C7semidirect
545. C7#23983direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5522
C2×C7×#4512
C2×C7×#4192
C7×#24092
C7×#20262
C7×#20012
C7×#18942
C7×#18782
C7×#18722
C7×#18702
C7×#18252
#239837