Finite Group Databasebeta

← C7×#15121 #13192337

Subgroups — C7×#15121 #13192337

Order 1792.

1182 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#389C211yesmaximal, characteristic
2C2×C7×#376C211yesmaximal, characteristic
3C7×#1445C211yesmaximal, characteristic
4C7×#1445C211yesmaximal, characteristic
5C7×#1444C211yesmaximal, characteristic
6C7×#1442C212yesmaximal
7C7×#1442C211yesmaximal, characteristic
8C7×#1441C212yesmaximal
9C7×#1436C212yesmaximal
10C7×#1434C211yesmaximal, characteristic
11C7×#1434C212yesmaximal
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C4⋊C4C2212yes—
15C22×C7×C4⋊C4C2212yes—
16C22×C42×C7C2211yescharacteristic
17C22×C42×C7C2211yescharacteristic
18C22×C42×C7C2211yescharacteristic
19C7×#389—24no—
20C7×#376—24no—
21C2×C7×#95C2211yescharacteristic
22C2×C7×#95C2211yescharacteristic
23C2×C7×#95C2211yescharacteristic
24C2×C7×#95C2211yescharacteristic
25C2×C7×#95C2212yes—
26C2×C7×#95C2212yes—
27C2×C7×#95C2212yes—
28C2×C7×#95C2212yes—
29C2×C7×#95C2212yes—
30C2×C7×#95C2211yescharacteristic
31C2×C7×#95C2211yescharacteristic
32C2×C7×#95C2212yes—
33C2×C7×#95C2212yes—
34C2×C7×#95C2212yes—
35C2×C7×#95C2212yes—
36C2×C7×#95C2212yes—
37#15121C711yesmaximal, characteristic, direct product factor, Sylow
38C23×C4×C7C2312yes—
39C23×C4×C7C2312yes—
40C23×C4×C7C2311yescharacteristic
41C23×C4×C7C2311yescharacteristic
42C23×C4×C7C2311yescharacteristic
43C23×C4×C7C2311yescharacteristic
44C23×C4×C7C2311yescharacteristic
45C23×C4×C7C2312yes—
46C23×C4×C7C2312yes—
47C23×C4×C7C2311yescharacteristic
48C23×C4×C7C2311yescharacteristic
49C2×C7×C4⋊C4—22no—
50C2×C7×C4⋊C4—24no—
51C2×C7×C4⋊C4—24no—
52C2×C7×C4⋊C4—22no—
53C2×C7×C4⋊C4—24no—
54C2×C7×C4⋊C4—24no—
55C2×C7×C4⋊C4—22no—
56C2×C7×C4⋊C4—22no—
57C2×C7×C4⋊C4—42no—
58C2×C7×C4⋊C4—42no—
59C2×C7×C4⋊C4—42no—
60C2×C42×C7—22no—
61C2×C42×C7—22no—
62C2×C42×C7—22no—
63C2×C42×C7—22no—
64C2×C42×C7—22no—
65C2×C42×C7—22no—
66C2×C42×C7—22no—
67C2×C42×C7D414yes—
68C2×C42×C7Q814yes—
69C7×#95—22no—
70C7×#95—22no—
71C7×#95—22no—
72C7×#95—22no—
73C7×#95—24no—
74C7×#95—22no—
75C7×#95—24no—
76C7×#95—24no—
77C7×#95—24no—
78C7×#95—24no—
79C7×#95—24no—
80C7×#95—24no—
81C7×#95—22no—
82C7×#95—42no—
83C7×#95—42no—
84C7×#95—42no—
85C2×#389C2×C711yescharacteristic
86C2×#376C2×C711yescharacteristic
87#1445C2×C711yescharacteristic
88#1445C2×C711yescharacteristic
89#1444C2×C711yescharacteristic
90#1442C2×C711yescharacteristic
91#1442C2×C712yes—
92#1441C2×C712yes—
93#1436C2×C712yes—
94#1434C2×C711yescharacteristic
95#1434C2×C712yes—
96C24×C7C2411yescharacteristic
97C22×C4×C7—21no—
98C22×C4×C7—21no—
99C22×C4×C7—21no—
100C22×C4×C7—21no—
101C22×C4×C7—22no—
102C22×C4×C7—22no—
103C22×C4×C7—21no—
104C22×C4×C7—21no—
105C22×C4×C7—21no—
106C22×C4×C7—21no—
107C22×C4×C7—22no—
108C22×C4×C7—22no—
109C22×C4×C7—22no—
110C22×C4×C7—22no—
111C22×C4×C7—22no—
112C22×C4×C7—22no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—21no—
116C22×C4×C7—21no—
117C22×C4×C7—21no—
118C22×C4×C7—21no—
119C22×C4×C7—21no—
120C22×C4×C7—21no—
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×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×C7C2×D412yes—
150C22×C4×C7C2×D412yes—
151C22×C4×C7C2×D412yes—
152C22×C4×C7C2×Q812yes—
153C22×C4×C7C2×Q812yes—
154C22×C4×C7C2×Q812yes—
155C22×C4×C7#4114yes—
156C22×C4×C7#4112yes—
157C22×C4×C7#4114yes—
158C22×C4×C7#4112yes—
159C22×C4×C7#4112yes—
160C22×C4×C7#4112yes—
161C22×C4×C7#4112yes—
162C7×C4⋊C4—28no—
163C7×C4⋊C4—48no—
164C7×C4⋊C4—48no—
165C7×C4⋊C4—44no—
166C42×C7—28no—
167C42×C7—28no—
168C42×C7—28no—
169C22×C4⋊C4C22×C711yescharacteristic
170C22×C4⋊C4C22×C711yescharacteristic
171C22×C4⋊C4C22×C712yes—
172C22×C4⋊C4C22×C712yes—
173C22×C42C22×C711yescharacteristic
174C22×C42C22×C711yescharacteristic
175C22×C42C22×C711yescharacteristic
176#389—24no—
177#376—24no—
178C2×#95C22×C712yes—
179C2×#95C22×C712yes—
180C2×#95C22×C712yes—
181C2×#95C22×C712yes—
182C2×#95C22×C712yes—
183C2×#95C22×C711yescharacteristic
184C2×#95C22×C711yescharacteristic
185C2×#95C22×C712yes—
186C2×#95C22×C712yes—
187C2×#95C22×C711yescharacteristic
188C2×#95C22×C711yescharacteristic
189C2×#95C22×C711yescharacteristic
190C2×#95C22×C711yescharacteristic
191C2×#95C22×C712yes—
192C2×#95C22×C712yes—
193C2×#95C22×C712yes—
194C23×C7C22×D411yescharacteristic
195C23×C7C22×Q811yescharacteristic
196C23×C7C2×#4111yescharacteristic
197C23×C7C2×#4111yescharacteristic
198C23×C7C2×#4111yescharacteristic
199C23×C7C2×#4111yescharacteristic
200C23×C7C2×#4112yes—
201C23×C7C2×#4111yescharacteristic
202C23×C7C2×#4112yes—
203C23×C7H(5,𝔽2)12yes—
204C23×C72−1+412yes—
205C2×C4×C7—22no—
206C2×C4×C7—22no—
207C2×C4×C7—22no—
208C2×C4×C7—22no—
209C2×C4×C7—22no—
210C2×C4×C7—22no—
211C2×C4×C7—22no—
212C2×C4×C7—22no—
213C2×C4×C7—24no—
214C2×C4×C7—24no—
215C2×C4×C7—22no—
216C2×C4×C7—22no—
217C2×C4×C7—22no—
218C2×C4×C7—22no—
219C2×C4×C7—22no—
220C2×C4×C7—24no—
221C2×C4×C7—24no—
222C2×C4×C7—22no—
223C2×C4×C7—24no—
224C2×C4×C7—22no—
225C2×C4×C7—24no—
226C2×C4×C7—24no—
227C2×C4×C7—24no—
228C2×C4×C7—24no—
229C2×C4×C7—24no—
230C2×C4×C7—24no—
231C2×C4×C7—24no—
232C2×C4×C7—42no—
233C2×C4×C7—41no—
234C2×C4×C7—41no—
235C2×C4×C7—42no—
236C2×C4×C7—42no—
237C2×C4×C7—41no—
238C2×C4×C7—41no—
239C2×C4×C7—42no—
240C2×C4×C7—41no—
241C2×C4×C7—41no—
242C2×C4×C7—41no—
243C2×C4×C7—41no—
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—42no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
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×C7C22⋊Q814yes—
269C2×C4×C7#12314yes—
270C23×C4C23×C712yes—
271C23×C4C23×C712yes—
272C23×C4C23×C711yescharacteristic
273C23×C4C23×C711yescharacteristic
274C23×C4C23×C711yescharacteristic
275C23×C4C23×C712yes—
276C23×C4C23×C711yescharacteristic
277C23×C4C23×C712yes—
278C23×C4C23×C711yescharacteristic
279C23×C4C23×C711yescharacteristic
280C23×C4C23×C711yescharacteristic
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42C7×D414yes—
300C2×C42C7×Q814yes—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—24no—
306#95—22no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—22no—
314#95—42no—
315#95—42no—
316#95—42no—
317C22×C7C2×C22⋊Q811yescharacteristic
318C22×C7C2×#12311yescharacteristic
319C22×C7#52511yescharacteristic
320C22×C7#52511yescharacteristic
321C22×C7#52911yescharacteristic
322C22×C7#52911yescharacteristic
323C22×C7#52912yes—
324C22×C7#53211yescharacteristic
325C22×C7#53311yescharacteristic
326C22×C7#53311yescharacteristic
327C22×C7#53312yes—
328C22×C7#53511yescharacteristic
329C22×C7#53611yescharacteristic
330C22×C7#54212yes—
331C22×C7#54412yes—
332C22×C7#55112yes—
333C22×C7#55212yes—
334C22×C7#55212yes—
335C22×C7#55212yes—
336C22×C7#55212yes—
337C22×C7#55312yes—
338C22×C7#55312yes—
339C22×C7#55512yes—
340C4×C7—24no—
341C4×C7—24no—
342C4×C7—42no—
343C4×C7—44no—
344C4×C7—42no—
345C4×C7—42no—
346C4×C7—42no—
347C4×C7—42no—
348C4×C7—44no—
349C4×C7—82no—
350C4×C7—82no—
351C24C24×C711yescharacteristic
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
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—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
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—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4C2×C7×D412yes—
405C22×C4C2×C7×D412yes—
406C22×C4C2×C7×D412yes—
407C22×C4C2×C7×Q812yes—
408C22×C4C2×C7×Q812yes—
409C22×C4C2×C7×Q812yes—
410C22×C4C7×#4114yes—
411C22×C4C7×#4112yes—
412C22×C4C7×#4114yes—
413C22×C4C7×#4112yes—
414C22×C4C7×#4112yes—
415C22×C4C7×#4112yes—
416C22×C4C7×#4112yes—
417C4⋊C4—28no—
418C4⋊C4—48no—
419C4⋊C4—48no—
420C4⋊C4—44no—
421C42—28no—
422C42—28no—
423C42—28no—
424C2×C7#239611yescharacteristic
425C2×C7#248411yescharacteristic
426C2×C7#250011yescharacteristic
427C2×C7#252912yes—
428C2×C7#253012yes—
429C2×C7#259811yescharacteristic
430C2×C7#260612yes—
431C2×C7#261111yescharacteristic
432C2×C7#261212yes—
433C2×C7#262811yescharacteristic
434C2×C7#285511yescharacteristic
435C23C22×C7×D411yescharacteristic
436C23C22×C7×Q811yescharacteristic
437C23C2×C7×#4111yescharacteristic
438C23C2×C7×#4111yescharacteristic
439C23C2×C7×#4111yescharacteristic
440C23C2×C7×#4111yescharacteristic
441C23C2×C7×#4112yes—
442C23C2×C7×#4111yescharacteristic
443C23C2×C7×#4112yes—
444C23C7×H(5,𝔽2)12yes—
445C23C7×2−1+412yes—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4C7×C22⋊Q814yes—
510C2×C4C7×#12314yes—
511C7#1512111yescharacteristic, direct product factor, Sylow
512C22C2×C7×C22⋊Q811yescharacteristic
513C22C2×C7×#12311yescharacteristic
514C22C7×#52511yescharacteristic
515C22C7×#52511yescharacteristic
516C22C7×#52912yes—
517C22C7×#52911yescharacteristic
518C22C7×#52911yescharacteristic
519C22C7×#53211yescharacteristic
520C22C7×#53312yes—
521C22C7×#53311yescharacteristic
522C22C7×#53311yescharacteristic
523C22C7×#53511yescharacteristic
524C22C7×#53611yescharacteristic
525C22C7×#54212yes—
526C22C7×#54412yes—
527C22C7×#55112yes—
528C22C7×#55212yes—
529C22C7×#55212yes—
530C22C7×#55212yes—
531C22C7×#55212yes—
532C22C7×#55312yes—
533C22C7×#55312yes—
534C22C7×#55512yes—
535C4—24no—
536C4—24no—
537C4—42no—
538C4—42no—
539C4—44no—
540C4—42no—
541C4—44no—
542C4—42no—
543C4—42no—
544C4—82no—
545C4—82no—
546C2C7×#239611yescharacteristic
547C2C7×#248411yescharacteristic
548C2C7×#250011yescharacteristic
549C2C7×#252912yes—
550C2C7×#253012yes—
551C2C7×#259811yescharacteristic
552C2C7×#260612yes—
553C2C7×#261111yescharacteristic
554C2C7×#261212yes—
555C2C7×#262811yescharacteristic
556C2C7×#285511yescharacteristic
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
37. #15121C7direct (N × H)
511. C7#15121direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#3892
C2×C7×#3762
C7×#14452
C7×#14442
C7×#14422
C7×#14412
C7×#14362
C7×#14342
#151217