Finite Group Databasebeta

← C7×#16517 #13193733

Subgroups — C7×#16517 #13193733

Order 1792.

1234 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#398C212yesmaximal
2C2×C7×#398C211yesmaximal, characteristic
3C2×C7×#389C211yesmaximal, characteristic
4C2×C7×#387C211yesmaximal, characteristic
5C2×C7×#382C211yesmaximal, characteristic
6C7×#1445C211yesmaximal, characteristic
7C7×#1443C212yesmaximal
8C7×#1442C212yesmaximal
9C7×#1442C211yesmaximal, characteristic
10C7×#1441C211yesmaximal, characteristic
11C7×#1441C212yesmaximal
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C7×C4⋊C4C2212yes—
14C22×C7×C4⋊C4C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16C22×C7×C4⋊C4C2212yes—
17C22×C7×C4⋊C4C2212yes—
18C22×C7×C4⋊C4C2211yescharacteristic
19C22×C42×C7C2211yescharacteristic
20C7×#398—28no—
21C7×#398—24no—
22C7×#389—24no—
23C7×#387—24no—
24C7×#382—24no—
25C2×C7×#95C2212yes—
26C2×C7×#95C2212yes—
27C2×C7×#95C2212yes—
28C2×C7×#95C2212yes—
29C2×C7×#95C2211yescharacteristic
30C2×C7×#95C2212yes—
31C2×C7×#95C2211yescharacteristic
32C2×C7×#95C2211yescharacteristic
33C2×C7×#95C2211yescharacteristic
34C2×C7×#95C2211yescharacteristic
35C2×C7×#95C2212yes—
36C2×C7×#95C2212yes—
37C2×C7×#95C2212yes—
38C2×C7×#95C2212yes—
39C2×C7×#95C2211yescharacteristic
40#16517C711yesmaximal, characteristic, direct product factor, Sylow
41C23×C4×C7C2311yescharacteristic
42C23×C4×C7C2311yescharacteristic
43C23×C4×C7C2312yes—
44C23×C4×C7C2312yes—
45C23×C4×C7C2311yescharacteristic
46C23×C4×C7C2311yescharacteristic
47C23×C4×C7C2312yes—
48C23×C4×C7C2311yescharacteristic
49C23×C4×C7C2312yes—
50C23×C4×C7C2311yescharacteristic
51C23×C4×C7C2311yescharacteristic
52C2×C7×C4⋊C4—22no—
53C2×C7×C4⋊C4—22no—
54C2×C7×C4⋊C4—24no—
55C2×C7×C4⋊C4—22no—
56C2×C7×C4⋊C4—24no—
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4—24no—
59C2×C7×C4⋊C4—22no—
60C2×C7×C4⋊C4—24no—
61C2×C7×C4⋊C4—24no—
62C2×C7×C4⋊C4—24no—
63C2×C7×C4⋊C4—24no—
64C2×C7×C4⋊C4—22no—
65C2×C7×C4⋊C4—22no—
66C2×C7×C4⋊C4—22no—
67C2×C7×C4⋊C4—24no—
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4—42no—
70C2×C7×C4⋊C4—42no—
71C2×C7×C4⋊C4D414yes—
72C2×C42×C7—22no—
73C2×C42×C7—22no—
74C2×C42×C7D414yes—
75C7×#95—24no—
76C7×#95—22no—
77C7×#95—22no—
78C7×#95—22no—
79C7×#95—22no—
80C7×#95—22no—
81C7×#95—24no—
82C7×#95—24no—
83C7×#95—24no—
84C7×#95—24no—
85C7×#95—24no—
86C7×#95—24no—
87C7×#95—24no—
88C7×#95—42no—
89C7×#95D414yes—
90C2×#398C2×C711yescharacteristic
91C2×#398C2×C712yes—
92C2×#389C2×C711yescharacteristic
93C2×#387C2×C711yescharacteristic
94C2×#382C2×C711yescharacteristic
95#1445C2×C711yescharacteristic
96#1443C2×C712yes—
97#1442C2×C711yescharacteristic
98#1442C2×C712yes—
99#1441C2×C711yescharacteristic
100#1441C2×C712yes—
101C24×C7C2411yescharacteristic
102C22×C4×C7—21no—
103C22×C4×C7—21no—
104C22×C4×C7—22no—
105C22×C4×C7—21no—
106C22×C4×C7—22no—
107C22×C4×C7—21no—
108C22×C4×C7—21no—
109C22×C4×C7—21no—
110C22×C4×C7—22no—
111C22×C4×C7—21no—
112C22×C4×C7—22no—
113C22×C4×C7—22no—
114C22×C4×C7—21no—
115C22×C4×C7—22no—
116C22×C4×C7—21no—
117C22×C4×C7—21no—
118C22×C4×C7—21no—
119C22×C4×C7—22no—
120C22×C4×C7—21no—
121C22×C4×C7—22no—
122C22×C4×C7—21no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—21no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—21no—
135C22×C4×C7—21no—
136C22×C4×C7—21no—
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—21no—
148C22×C4×C7—21no—
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—21no—
157C22×C4×C7—22no—
158C22×C4×C7C2×D412yes—
159C22×C4×C7C2×D412yes—
160C22×C4×C7C2×D414yes—
161C22×C4×C7C2×D412yes—
162C22×C4×C7C2×D412yes—
163C22×C4×C7C2×D412yes—
164C22×C4×C7C2×D412yes—
165C22×C4×C7C2×D412yes—
166C22×C4×C7#4114yes—
167C22×C4×C7#4112yes—
168C22×C4×C7#4112yes—
169C7×C4⋊C4—28no—
170C7×C4⋊C4—28no—
171C7×C4⋊C4—48no—
172C7×C4⋊C4—48no—
173C7×C4⋊C4—44no—
174C7×C4⋊C4—48no—
175C7×C4⋊C4—44no—
176C42×C7—28no—
177C22×C4⋊C4C22×C711yescharacteristic
178C22×C4⋊C4C22×C712yes—
179C22×C4⋊C4C22×C711yescharacteristic
180C22×C4⋊C4C22×C711yescharacteristic
181C22×C4⋊C4C22×C712yes—
182C22×C4⋊C4C22×C712yes—
183C22×C4⋊C4C22×C711yescharacteristic
184C22×C42C22×C711yescharacteristic
185#398—28no—
186#398—24no—
187#389—24no—
188#387—24no—
189#382—24no—
190C2×#95C22×C712yes—
191C2×#95C22×C712yes—
192C2×#95C22×C712yes—
193C2×#95C22×C712yes—
194C2×#95C22×C712yes—
195C2×#95C22×C711yescharacteristic
196C2×#95C22×C712yes—
197C2×#95C22×C711yescharacteristic
198C2×#95C22×C711yescharacteristic
199C2×#95C22×C712yes—
200C2×#95C22×C712yes—
201C2×#95C22×C711yescharacteristic
202C2×#95C22×C711yescharacteristic
203C2×#95C22×C712yes—
204C2×#95C22×C711yescharacteristic
205C23×C7C22×D411yescharacteristic
206C23×C7C22×D411yescharacteristic
207C23×C7C22×D411yescharacteristic
208C23×C7C2×#4111yescharacteristic
209C23×C7C2×#4111yescharacteristic
210C23×C7C2×#4112yes—
211C23×C7H(5,𝔽2)11yescharacteristic
212C23×C72−1+412yes—
213C23×C72−1+412yes—
214C23×C72−1+412yes—
215C23×C72−1+411yescharacteristic
216C2×C4×C7—22no—
217C2×C4×C7—22no—
218C2×C4×C7—22no—
219C2×C4×C7—22no—
220C2×C4×C7—22no—
221C2×C4×C7—24no—
222C2×C4×C7—22no—
223C2×C4×C7—22no—
224C2×C4×C7—24no—
225C2×C4×C7—22no—
226C2×C4×C7—22no—
227C2×C4×C7—22no—
228C2×C4×C7—22no—
229C2×C4×C7—24no—
230C2×C4×C7—22no—
231C2×C4×C7—22no—
232C2×C4×C7—24no—
233C2×C4×C7—24no—
234C2×C4×C7—22no—
235C2×C4×C7—24no—
236C2×C4×C7—22no—
237C2×C4×C7—24no—
238C2×C4×C7—24no—
239C2×C4×C7—24no—
240C2×C4×C7—41no—
241C2×C4×C7—42no—
242C2×C4×C7—41no—
243C2×C4×C7—41no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7—42no—
247C2×C4×C7—41no—
248C2×C4×C7—42no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—41no—
252C2×C4×C7—42no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—41no—
256C2×C4×C7—42no—
257C2×C4×C7—41no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—41no—
261C2×C4×C7—41no—
262C2×C4×C7—41no—
263C2×C4×C7—41no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—41no—
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#12114yes—
282C2×C4×C7#12114yes—
283C23×C4C23×C712yes—
284C23×C4C23×C711yescharacteristic
285C23×C4C23×C712yes—
286C23×C4C23×C711yescharacteristic
287C23×C4C23×C711yescharacteristic
288C23×C4C23×C712yes—
289C23×C4C23×C711yescharacteristic
290C23×C4C23×C711yescharacteristic
291C23×C4C23×C712yes—
292C23×C4C23×C711yescharacteristic
293C23×C4C23×C711yescharacteristic
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4C7×D414yes—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42C7×D414yes—
317#95—24no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—42no—
331#95C7×D414yes—
332C22×C7C2×#12111yescharacteristic
333C22×C7C2×#12111yescharacteristic
334C22×C7#52511yescharacteristic
335C22×C7#53611yescharacteristic
336C22×C7#53611yescharacteristic
337C22×C7#53611yescharacteristic
338C22×C7#53612yes—
339C22×C7#53612yes—
340C22×C7#53711yescharacteristic
341C22×C7#54112yes—
342C22×C7#54112yes—
343C22×C7#54112yes—
344C22×C7#54212yes—
345C22×C7D4211yescharacteristic
346C22×C7#54712yes—
347C22×C7#54711yescharacteristic
348C22×C7#54811yescharacteristic
349C22×C7#54812yes—
350C22×C7#55212yes—
351C22×C7#55212yes—
352C22×C7#55311yescharacteristic
353C22×C7#56312yes—
354C22×C7#56412yes—
355C4×C7—24no—
356C4×C7—24no—
357C4×C7—42no—
358C4×C7—42no—
359C4×C7—42no—
360C4×C7—44no—
361C4×C7—44no—
362C4×C7—81no—
363C4×C7—81no—
364C4×C7—82no—
365C4×C7—82no—
366C24C24×C711yescharacteristic
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
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—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4C2×C7×D412yes—
424C22×C4C2×C7×D412yes—
425C22×C4C2×C7×D414yes—
426C22×C4C2×C7×D412yes—
427C22×C4C2×C7×D412yes—
428C22×C4C2×C7×D412yes—
429C22×C4C2×C7×D412yes—
430C22×C4C2×C7×D412yes—
431C22×C4C7×#4114yes—
432C22×C4C7×#4112yes—
433C22×C4C7×#4112yes—
434C4⋊C4—28no—
435C4⋊C4—28no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—44no—
439C4⋊C4—48no—
440C4⋊C4—44no—
441C42—28no—
442C2×C7#241311yescharacteristic
443C2×C7#241611yescharacteristic
444C2×C7#253611yescharacteristic
445C2×C7#262111yescharacteristic
446C2×C7#262111yescharacteristic
447C2×C7#263711yescharacteristic
448C2×C7#267412yes—
449C2×C7#271912yes—
450C2×C7#271912yes—
451C2×C7#276612yes—
452C2×C7#282211yescharacteristic
453C23C22×C7×D411yescharacteristic
454C23C22×C7×D411yescharacteristic
455C23C22×C7×D411yescharacteristic
456C23C2×C7×#4111yescharacteristic
457C23C2×C7×#4111yescharacteristic
458C23C2×C7×#4112yes—
459C23C7×H(5,𝔽2)11yescharacteristic
460C23C7×2−1+412yes—
461C23C7×2−1+412yes—
462C23C7×2−1+412yes—
463C23C7×2−1+411yescharacteristic
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—41no—
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×C4C7×#12114yes—
530C2×C4C7×#12114yes—
531C7#1651711yescharacteristic, direct product factor, Sylow
532C22C2×C7×#12111yescharacteristic
533C22C2×C7×#12111yescharacteristic
534C22C7×#52511yescharacteristic
535C22C7×#53611yescharacteristic
536C22C7×#53612yes—
537C22C7×#53612yes—
538C22C7×#53611yescharacteristic
539C22C7×#53611yescharacteristic
540C22C7×#53711yescharacteristic
541C22C7×#54112yes—
542C22C7×#54112yes—
543C22C7×#54112yes—
544C22C7×#54212yes—
545C22C7×D4211yescharacteristic
546C22C7×#54712yes—
547C22C7×#54711yescharacteristic
548C22C7×#54811yescharacteristic
549C22C7×#54812yes—
550C22C7×#55212yes—
551C22C7×#55212yes—
552C22C7×#55311yescharacteristic
553C22C7×#56312yes—
554C22C7×#56412yes—
555C4—24no—
556C4—24no—
557C4—42no—
558C4—44no—
559C4—44no—
560C4—42no—
561C4—42no—
562C4—81no—
563C4—82no—
564C4—82no—
565C4—81no—
566C2C7×#241311yescharacteristic
567C2C7×#241611yescharacteristic
568C2C7×#253611yescharacteristic
569C2C7×#262111yescharacteristic
570C2C7×#262111yescharacteristic
571C2C7×#263711yescharacteristic
572C2C7×#267412yes—
573C2C7×#271912yes—
574C2C7×#271912yes—
575C2C7×#276612yes—
576C2C7×#282211yescharacteristic
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
40. #16517C7direct (N × H)
531. C7#16517direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#3982
C2×C7×#3892
C2×C7×#3872
C2×C7×#3822
C7×#14452
C7×#14432
C7×#14422
C7×#14412
#165177