Finite Group Databasebeta

← C7×#16563 #13193779

Subgroups — C7×#16563 #13193779

Order 1792.

1238 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#398C211yesmaximal, characteristic
2C2×C7×#387C212yesmaximal
3C2×C7×#384C211yesmaximal, characteristic
4C7×#1441C212yesmaximal
5C7×#1441C212yesmaximal
6C7×#1441C212yesmaximal
7C7×#1441C212yesmaximal
8C7×#1441C211yesmaximal, characteristic
9C7×#1441C212yesmaximal
10C22×C7×C4⋊C4C2212yes—
11C22×C7×C4⋊C4C2212yes—
12C22×C7×C4⋊C4C2212yes—
13C22×C7×C4⋊C4C2212yes—
14C22×C7×C4⋊C4C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16C22×C7×C4⋊C4C2211yescharacteristic
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×C42×C7C2211yescharacteristic
19C7×#398—24no—
20C7×#387—28no—
21C7×#384—24no—
22C2×C7×#95C2212yes—
23C2×C7×#95C2211yescharacteristic
24C2×C7×#95C2212yes—
25C2×C7×#95C2212yes—
26C2×C7×#95C2212yes—
27C2×C7×#95C2212yes—
28C2×C7×#95C2212yes—
29C2×C7×#95C2212yes—
30C2×C7×#95C2212yes—
31C2×C7×#95C2212yes—
32C2×C7×#95C2212yes—
33C2×C7×#95C2211yescharacteristic
34#16563C711yesmaximal, characteristic, direct product factor, Sylow
35C23×C4×C7C2312yes—
36C23×C4×C7C2312yes—
37C23×C4×C7C2312yes—
38C23×C4×C7C2311yescharacteristic
39C23×C4×C7C2311yescharacteristic
40C23×C4×C7C2312yes—
41C23×C4×C7C2312yes—
42C23×C4×C7C2311yescharacteristic
43C23×C4×C7C2312yes—
44C2×C7×C4⋊C4—24no—
45C2×C7×C4⋊C4—24no—
46C2×C7×C4⋊C4—24no—
47C2×C7×C4⋊C4—24no—
48C2×C7×C4⋊C4—22no—
49C2×C7×C4⋊C4—24no—
50C2×C7×C4⋊C4—22no—
51C2×C7×C4⋊C4—24no—
52C2×C7×C4⋊C4—24no—
53C2×C7×C4⋊C4—24no—
54C2×C7×C4⋊C4—24no—
55C2×C7×C4⋊C4—41no—
56C2×C7×C4⋊C4—41no—
57C2×C7×C4⋊C4—42no—
58C2×C7×C4⋊C4—42no—
59C2×C7×C4⋊C4—41no—
60C2×C7×C4⋊C4—41no—
61C2×C7×C4⋊C4—41no—
62C2×C7×C4⋊C4—41no—
63C2×C7×C4⋊C4—41no—
64C2×C7×C4⋊C4—42no—
65C2×C7×C4⋊C4D414yes—
66C2×C7×C4⋊C4Q818yes—
67C2×C42×C7—22no—
68C2×C42×C7—22no—
69C2×C42×C7D414yes—
70C7×#95—24no—
71C7×#95—24no—
72C7×#95—24no—
73C7×#95—24no—
74C7×#95—22no—
75C7×#95—24no—
76C7×#95—41no—
77C7×#95—42no—
78C7×#95—42no—
79C7×#95—42no—
80C7×#95—42no—
81C7×#95—42no—
82C2×#398C2×C711yescharacteristic
83C2×#387C2×C712yes—
84C2×#384C2×C711yescharacteristic
85#1441C2×C712yes—
86#1441C2×C712yes—
87#1441C2×C712yes—
88#1441C2×C712yes—
89#1441C2×C712yes—
90#1441C2×C711yescharacteristic
91C24×C7C2411yescharacteristic
92C22×C4×C7—22no—
93C22×C4×C7—22no—
94C22×C4×C7—21no—
95C22×C4×C7—21no—
96C22×C4×C7—22no—
97C22×C4×C7—21no—
98C22×C4×C7—21no—
99C22×C4×C7—22no—
100C22×C4×C7—22no—
101C22×C4×C7—22no—
102C22×C4×C7—22no—
103C22×C4×C7—22no—
104C22×C4×C7—22no—
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7—22no—
108C22×C4×C7—22no—
109C22×C4×C7—22no—
110C22×C4×C7—22no—
111C22×C4×C7—21no—
112C22×C4×C7—22no—
113C22×C4×C7—21no—
114C22×C4×C7—21no—
115C22×C4×C7—21no—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
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—22no—
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×C7C2×D412yes—
142C22×C4×C7C2×D412yes—
143C22×C4×C7C2×D412yes—
144C22×C4×C7C2×D412yes—
145C22×C4×C7C2×D414yes—
146C22×C4×C7C2×Q814yes—
147C22×C4×C7C2×Q814yes—
148C22×C4×C7C2×Q814yes—
149C22×C4×C7#4112yes—
150C22×C4×C7#4114yes—
151C7×C4⋊C4—216no—
152C7×C4⋊C4—44no—
153C7×C4⋊C4—48no—
154C7×C4⋊C4—48no—
155C7×C4⋊C4—44no—
156C7×C4⋊C4—44no—
157C7×C4⋊C4—44no—
158C7×C4⋊C4—48no—
159C42×C7—28no—
160C22×C4⋊C4C22×C712yes—
161C22×C4⋊C4C22×C712yes—
162C22×C4⋊C4C22×C712yes—
163C22×C4⋊C4C22×C712yes—
164C22×C4⋊C4C22×C711yescharacteristic
165C22×C4⋊C4C22×C711yescharacteristic
166C22×C4⋊C4C22×C711yescharacteristic
167C22×C4⋊C4C22×C711yescharacteristic
168C22×C42C22×C711yescharacteristic
169#398—24no—
170#387—28no—
171#384—24no—
172C2×#95C22×C712yes—
173C2×#95C22×C711yescharacteristic
174C2×#95C22×C712yes—
175C2×#95C22×C712yes—
176C2×#95C22×C712yes—
177C2×#95C22×C712yes—
178C2×#95C22×C712yes—
179C2×#95C22×C712yes—
180C2×#95C22×C712yes—
181C2×#95C22×C712yes—
182C2×#95C22×C712yes—
183C2×#95C22×C711yescharacteristic
184C23×C7C22×D411yescharacteristic
185C23×C7C22×D411yescharacteristic
186C23×C7C22×Q812yes—
187C23×C7C2×#4111yescharacteristic
188C23×C7C2×#4112yes—
189C23×C7H(5,𝔽2)11yescharacteristic
190C23×C7H(5,𝔽2)11yescharacteristic
191C23×C7H(5,𝔽2)12yes—
192C23×C72−1+411yescharacteristic
193C23×C72−1+411yescharacteristic
194C23×C72−1+412yes—
195C2×C4×C7—24no—
196C2×C4×C7—22no—
197C2×C4×C7—22no—
198C2×C4×C7—22no—
199C2×C4×C7—22no—
200C2×C4×C7—24no—
201C2×C4×C7—22no—
202C2×C4×C7—22no—
203C2×C4×C7—24no—
204C2×C4×C7—24no—
205C2×C4×C7—24no—
206C2×C4×C7—24no—
207C2×C4×C7—24no—
208C2×C4×C7—24no—
209C2×C4×C7—22no—
210C2×C4×C7—22no—
211C2×C4×C7—24no—
212C2×C4×C7—24no—
213C2×C4×C7—24no—
214C2×C4×C7—24no—
215C2×C4×C7—24no—
216C2×C4×C7—24no—
217C2×C4×C7—42no—
218C2×C4×C7—42no—
219C2×C4×C7—42no—
220C2×C4×C7—42no—
221C2×C4×C7—42no—
222C2×C4×C7—42no—
223C2×C4×C7—42no—
224C2×C4×C7—42no—
225C2×C4×C7—42no—
226C2×C4×C7—42no—
227C2×C4×C7—41no—
228C2×C4×C7—41no—
229C2×C4×C7—41no—
230C2×C4×C7—41no—
231C2×C4×C7—42no—
232C2×C4×C7—42no—
233C2×C4×C7—42no—
234C2×C4×C7—42no—
235C2×C4×C7—42no—
236C2×C4×C7—42no—
237C2×C4×C7—42no—
238C2×C4×C7—42no—
239C2×C4×C7—42no—
240C2×C4×C7—42no—
241C2×C4×C7—42no—
242C2×C4×C7—42no—
243C2×C4×C7—42no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7—42no—
247C2×C4×C7—42no—
248C2×C4×C7—42no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7#12114yes—
253C2×C4×C7C22⋊Q818yes—
254C23×C4C23×C712yes—
255C23×C4C23×C712yes—
256C23×C4C23×C712yes—
257C23×C4C23×C711yescharacteristic
258C23×C4C23×C711yescharacteristic
259C23×C4C23×C712yes—
260C23×C4C23×C712yes—
261C23×C4C23×C711yescharacteristic
262C23×C4C23×C712yes—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—41no—
275C2×C4⋊C4—41no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—41no—
279C2×C4⋊C4—41no—
280C2×C4⋊C4—41no—
281C2×C4⋊C4—41no—
282C2×C4⋊C4—41no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4C7×D414yes—
285C2×C4⋊C4C7×Q818yes—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42C7×D414yes—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—22no—
294#95—24no—
295#95—41no—
296#95—42no—
297#95—42no—
298#95—42no—
299#95—42no—
300#95—42no—
301C22×C7C2×#12111yescharacteristic
302C22×C7C2×C22⋊Q812yes—
303C22×C7#53411yescharacteristic
304C22×C7#53611yescharacteristic
305C22×C7#53711yescharacteristic
306C22×C7#53712yes—
307C22×C7#53711yescharacteristic
308C22×C7#53911yescharacteristic
309C22×C7#53911yescharacteristic
310C22×C7#54012yes—
311C22×C7#54112yes—
312C22×C7#54412yes—
313C22×C7#54412yes—
314C22×C7#54612yes—
315C22×C7D4×Q812yes—
316C22×C7#55212yes—
317C22×C7#55412yes—
318C22×C7#55712yes—
319C22×C7Q8211yescharacteristic
320C22×C7#55911yescharacteristic
321C22×C7#56011yescharacteristic
322C22×C7#56312yes—
323C22×C7#56311yescharacteristic
324C4×C7—24no—
325C4×C7—28no—
326C4×C7—44no—
327C4×C7—44no—
328C4×C7—42no—
329C4×C7—42no—
330C4×C7—82no—
331C4×C7—82no—
332C4×C7—82no—
333C24C24×C711yescharacteristic
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4C2×C7×D412yes—
384C22×C4C2×C7×D412yes—
385C22×C4C2×C7×D412yes—
386C22×C4C2×C7×D412yes—
387C22×C4C2×C7×D414yes—
388C22×C4C2×C7×Q814yes—
389C22×C4C2×C7×Q814yes—
390C22×C4C2×C7×Q814yes—
391C22×C4C7×#4112yes—
392C22×C4C7×#4114yes—
393C4⋊C4—216no—
394C4⋊C4—44no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—48no—
401C42—28no—
402C2×C7#241712yes—
403C2×C7#258211yescharacteristic
404C2×C7#261811yescharacteristic
405C2×C7#262912yes—
406C2×C7#270911yescharacteristic
407C2×C7#271312yes—
408C2×C7#273411yescharacteristic
409C2×C7#279112yes—
410C2×C7#279211yescharacteristic
411C2×C7#279411yescharacteristic
412C2×C7#282311yescharacteristic
413C23C22×C7×D411yescharacteristic
414C23C22×C7×D411yescharacteristic
415C23C22×C7×Q812yes—
416C23C2×C7×#4111yescharacteristic
417C23C2×C7×#4112yes—
418C23C7×H(5,𝔽2)11yescharacteristic
419C23C7×H(5,𝔽2)11yescharacteristic
420C23C7×H(5,𝔽2)12yes—
421C23C7×2−1+411yescharacteristic
422C23C7×2−1+411yescharacteristic
423C23C7×2−1+412yes—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4C7×#12114yes—
482C2×C4C7×C22⋊Q818yes—
483C7#1656311yescharacteristic, direct product factor, Sylow
484C22C2×C7×#12111yescharacteristic
485C22C2×C7×C22⋊Q812yes—
486C22C7×#53411yescharacteristic
487C22C7×#53611yescharacteristic
488C22C7×#53712yes—
489C22C7×#53711yescharacteristic
490C22C7×#53711yescharacteristic
491C22C7×#53911yescharacteristic
492C22C7×#53911yescharacteristic
493C22C7×#54012yes—
494C22C7×#54112yes—
495C22C7×#54412yes—
496C22C7×#54412yes—
497C22C7×#54612yes—
498C22C7×D4×Q812yes—
499C22C7×#55212yes—
500C22C7×#55412yes—
501C22C7×#55712yes—
502C22C7×Q8211yescharacteristic
503C22C7×#55911yescharacteristic
504C22C7×#56011yescharacteristic
505C22C7×#56312yes—
506C22C7×#56311yescharacteristic
507C4—28no—
508C4—24no—
509C4—44no—
510C4—42no—
511C4—42no—
512C4—44no—
513C4—82no—
514C4—82no—
515C4—82no—
516C2C7×#241712yes—
517C2C7×#258211yescharacteristic
518C2C7×#261811yescharacteristic
519C2C7×#262912yes—
520C2C7×#270911yescharacteristic
521C2C7×#271312yes—
522C2C7×#273411yescharacteristic
523C2C7×#279112yes—
524C2C7×#279211yescharacteristic
525C2C7×#279411yescharacteristic
526C2C7×#282311yescharacteristic
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
34. #16563C7direct (N × H)
483. C7#16563direct (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×#3872
C2×C7×#3842
C7×#14412
#165637