Finite Group Databasebeta

← C7×#15166 #13192382

Subgroups — C7×#15166 #13192382

Order 1792.

1310 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#398C212yesmaximal
2C2×C7×#389C211yesmaximal, characteristic
3C2×C4×C7×C4⋊C4C211yesmaximal, characteristic
4C7×#1444C211yesmaximal, characteristic
5C7×#1442C211yesmaximal, characteristic
6C7×#1441C211yesmaximal, characteristic
7C7×#1441C211yesmaximal, characteristic
8C7×#1441C212yesmaximal
9C7×#1436C212yesmaximal
10C7×C4⋊#95C211yesmaximal, characteristic
11C7×#1434C212yesmaximal
12C22×C7×C4⋊C4C2212yes—
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C4⋊C4C2212yes—
15C22×C7×C4⋊C4C2212yes—
16C22×C7×C4⋊C4C2211yescharacteristic
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×C7×C4⋊C4C2211yescharacteristic
19C22×C42×C7C2211yescharacteristic
20C22×C42×C7C2211yescharacteristic
21C22×C42×C7C2211yescharacteristic
22C7×#398—28no—
23C7×#389—24no—
24C4×C7×C4⋊C4—24no—
25C2×C7×#95C2212yes—
26C2×C7×#95C2212yes—
27C2×C7×#95C2212yes—
28C2×C7×#95C2211yescharacteristic
29C2×C7×#95C2211yescharacteristic
30C2×C7×#95C2212yes—
31C2×C7×#95C2212yes—
32C2×C7×#95C2212yes—
33C2×C7×#95C2212yes—
34C2×C7×#95C2211yescharacteristic
35C2×C7×#95C2211yescharacteristic
36C2×C7×#95C2212yes—
37C2×C7×#95C2212yes—
38#15166C711yesmaximal, characteristic, direct product factor, Sylow
39C23×C4×C7C2312yes—
40C23×C4×C7C2312yes—
41C23×C4×C7C2311yescharacteristic
42C23×C4×C7C2311yescharacteristic
43C23×C4×C7C2311yescharacteristic
44C23×C4×C7C2312yes—
45C23×C4×C7C2312yes—
46C23×C4×C7C2311yescharacteristic
47C23×C4×C7C2311yescharacteristic
48C23×C4×C7C2311yescharacteristic
49C23×C4×C7C2311yescharacteristic
50C2×C7×C4⋊C4—22no—
51C2×C7×C4⋊C4—22no—
52C2×C7×C4⋊C4—22no—
53C2×C7×C4⋊C4—24no—
54C2×C7×C4⋊C4—22no—
55C2×C7×C4⋊C4—22no—
56C2×C7×C4⋊C4—22no—
57C2×C7×C4⋊C4—24no—
58C2×C7×C4⋊C4—24no—
59C2×C7×C4⋊C4—24no—
60C2×C7×C4⋊C4—24no—
61C2×C7×C4⋊C4—24no—
62C2×C7×C4⋊C4—24no—
63C2×C7×C4⋊C4—24no—
64C2×C7×C4⋊C4—42no—
65C2×C7×C4⋊C4—42no—
66C2×C7×C4⋊C4—42no—
67C2×C7×C4⋊C4D414yes—
68C2×C7×C4⋊C4Q814yes—
69C2×C42×C7—22no—
70C2×C42×C7—22no—
71C2×C42×C7—22no—
72C2×C42×C7—42no—
73C2×C42×C7—42no—
74C2×C42×C7D414yes—
75C2×C42×C7Q814yes—
76C7×#95—22no—
77C7×#95—22no—
78C7×#95—24no—
79C7×#95—24no—
80C7×#95—24no—
81C7×#95—24no—
82C7×#95—24no—
83C7×#95—22no—
84C7×#95—24no—
85C7×#95—24no—
86C7×#95—24no—
87C7×#95—22no—
88C7×#95—42no—
89C2×#398C2×C712yes—
90C2×#389C2×C711yescharacteristic
91C2×C4×C4⋊C4C2×C711yescharacteristic
92#1444C2×C711yescharacteristic
93#1442C2×C711yescharacteristic
94#1441C2×C712yes—
95#1441C2×C711yescharacteristic
96#1441C2×C711yescharacteristic
97#1436C2×C712yes—
98C4⋊#95C2×C711yescharacteristic
99#1434C2×C712yes—
100C24×C7C2411yescharacteristic
101C22×C4×C7—21no—
102C22×C4×C7—21no—
103C22×C4×C7—21no—
104C22×C4×C7—21no—
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—22no—
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—21no—
124C22×C4×C7—21no—
125C22×C4×C7—21no—
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×D412yes—
153C22×C4×C7C2×D412yes—
154C22×C4×C7C2×D412yes—
155C22×C4×C7C2×Q812yes—
156C22×C4×C7C2×Q812yes—
157C22×C4×C7C2×Q812yes—
158C22×C4×C7C2×Q812yes—
159C22×C4×C7C2×Q812yes—
160C22×C4×C7C2×Q812yes—
161C22×C4×C7#4114yes—
162C22×C4×C7#4114yes—
163C22×C4×C7#4112yes—
164C22×C4×C7#4112yes—
165C22×C4×C7#4112yes—
166C7×C4⋊C4—28no—
167C7×C4⋊C4—28no—
168C7×C4⋊C4—28no—
169C7×C4⋊C4—48no—
170C7×C4⋊C4—48no—
171C7×C4⋊C4—44no—
172C7×C4⋊C4—48no—
173C42×C7—28no—
174C42×C7—44no—
175C42×C7—44no—
176C22×C4⋊C4C22×C712yes—
177C22×C4⋊C4C22×C711yescharacteristic
178C22×C4⋊C4C22×C712yes—
179C22×C4⋊C4C22×C712yes—
180C22×C4⋊C4C22×C711yescharacteristic
181C22×C4⋊C4C22×C711yescharacteristic
182C22×C4⋊C4C22×C711yescharacteristic
183C22×C42C22×C711yescharacteristic
184C22×C42C22×C711yescharacteristic
185C22×C42C22×C711yescharacteristic
186#398—28no—
187#389—24no—
188C4×C4⋊C4—24no—
189C2×#95C22×C712yes—
190C2×#95C22×C712yes—
191C2×#95C22×C712yes—
192C2×#95C22×C712yes—
193C2×#95C22×C712yes—
194C2×#95C22×C712yes—
195C2×#95C22×C712yes—
196C2×#95C22×C711yescharacteristic
197C2×#95C22×C711yescharacteristic
198C2×#95C22×C712yes—
199C2×#95C22×C711yescharacteristic
200C2×#95C22×C711yescharacteristic
201C2×#95C22×C712yes—
202C23×C7C22×D411yescharacteristic
203C23×C7C22×D411yescharacteristic
204C23×C7C22×Q811yescharacteristic
205C23×C7C22×Q811yescharacteristic
206C23×C7C2×#4111yescharacteristic
207C23×C7C2×#4111yescharacteristic
208C23×C7C2×#4111yescharacteristic
209C23×C7C2×#4112yes—
210C23×C7C2×#4112yes—
211C23×C7H(5,𝔽2)12yes—
212C23×C72−1+412yes—
213C2×C4×C7—22no—
214C2×C4×C7—22no—
215C2×C4×C7—22no—
216C2×C4×C7—22no—
217C2×C4×C7—22no—
218C2×C4×C7—22no—
219C2×C4×C7—22no—
220C2×C4×C7—22no—
221C2×C4×C7—22no—
222C2×C4×C7—22no—
223C2×C4×C7—22no—
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—22no—
232C2×C4×C7—22no—
233C2×C4×C7—22no—
234C2×C4×C7—24no—
235C2×C4×C7—24no—
236C2×C4×C7—24no—
237C2×C4×C7—24no—
238C2×C4×C7—24no—
239C2×C4×C7—24no—
240C2×C4×C7—24no—
241C2×C4×C7—24no—
242C2×C4×C7—24no—
243C2×C4×C7—42no—
244C2×C4×C7—41no—
245C2×C4×C7—41no—
246C2×C4×C7—42no—
247C2×C4×C7—41no—
248C2×C4×C7—41no—
249C2×C4×C7—42no—
250C2×C4×C7—41no—
251C2×C4×C7—41no—
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×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#12114yes—
275C2×C4×C7C22⋊Q814yes—
276C2×C4×C7C22⋊Q814yes—
277C2×C4×C7C4⋊Q814yes—
278C23×C4C23×C712yes—
279C23×C4C23×C712yes—
280C23×C4C23×C711yescharacteristic
281C23×C4C23×C711yescharacteristic
282C23×C4C23×C711yescharacteristic
283C23×C4C23×C712yes—
284C23×C4C23×C712yes—
285C23×C4C23×C711yescharacteristic
286C23×C4C23×C711yescharacteristic
287C23×C4C23×C711yescharacteristic
288C23×C4C23×C711yescharacteristic
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4C7×D414yes—
307C2×C4⋊C4C7×Q814yes—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—42no—
312C2×C42—42no—
313C2×C42C7×D414yes—
314C2×C42C7×Q814yes—
315#95—22no—
316#95—22no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—22no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—22no—
327#95—42no—
328C22×C7C2×#12111yescharacteristic
329C22×C7C2×C22⋊Q811yescharacteristic
330C22×C7C2×C22⋊Q811yescharacteristic
331C22×C7#52511yescharacteristic
332C22×C7#52912yes—
333C22×C7C2×C4⋊Q811yescharacteristic
334C22×C7#53211yescharacteristic
335C22×C7#53211yescharacteristic
336C22×C7#53311yescharacteristic
337C22×C7#53311yescharacteristic
338C22×C7#53312yes—
339C22×C7#53411yescharacteristic
340C22×C7#53611yescharacteristic
341C22×C7#53912yes—
342C22×C7#54412yes—
343C22×C7#54612yes—
344C22×C7#54812yes—
345C22×C7#55212yes—
346C22×C7#55212yes—
347C22×C7#55312yes—
348C22×C7#55312yes—
349C22×C7#55412yes—
350C22×C7#55712yes—
351C4×C7—24no—
352C4×C7—24no—
353C4×C7—24no—
354C4×C7—24no—
355C4×C7—42no—
356C4×C7—42no—
357C4×C7—44no—
358C4×C7—42no—
359C4×C7—44no—
360C4×C7—82no—
361C4×C7—82no—
362C24C24×C711yescharacteristic
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
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—21no—
386C22×C4—21no—
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—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
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×C4C2×C7×D412yes—
412C22×C4C2×C7×D412yes—
413C22×C4C2×C7×D412yes—
414C22×C4C2×C7×D412yes—
415C22×C4C2×C7×D412yes—
416C22×C4C2×C7×D412yes—
417C22×C4C2×C7×Q812yes—
418C22×C4C2×C7×Q812yes—
419C22×C4C2×C7×Q812yes—
420C22×C4C2×C7×Q812yes—
421C22×C4C2×C7×Q812yes—
422C22×C4C2×C7×Q812yes—
423C22×C4C7×#4114yes—
424C22×C4C7×#4114yes—
425C22×C4C7×#4112yes—
426C22×C4C7×#4112yes—
427C22×C4C7×#4112yes—
428C4⋊C4—28no—
429C4⋊C4—28no—
430C4⋊C4—28no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—44no—
434C4⋊C4—48no—
435C42—28no—
436C42—44no—
437C42—44no—
438C2×C7#239611yescharacteristic
439C2×C7#253611yescharacteristic
440C2×C7#254311yescharacteristic
441C2×C7#255012yes—
442C2×C7#255812yes—
443C2×C7#256511yescharacteristic
444C2×C7#257911yescharacteristic
445C2×C7#259012yes—
446C2×C7#260612yes—
447C2×C7#262711yescharacteristic
448C2×C7#285711yescharacteristic
449C23C22×C7×D411yescharacteristic
450C23C22×C7×D411yescharacteristic
451C23C22×C7×Q811yescharacteristic
452C23C22×C7×Q811yescharacteristic
453C23C2×C7×#4111yescharacteristic
454C23C2×C7×#4111yescharacteristic
455C23C2×C7×#4111yescharacteristic
456C23C2×C7×#4112yes—
457C23C2×C7×#4112yes—
458C23C7×H(5,𝔽2)12yes—
459C23C7×2−1+412yes—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
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×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4C7×#12114yes—
522C2×C4C7×C22⋊Q814yes—
523C2×C4C7×C22⋊Q814yes—
524C2×C4C7×C4⋊Q814yes—
525C7#1516611yescharacteristic, direct product factor, Sylow
526C22C2×C7×#12111yescharacteristic
527C22C2×C7×C22⋊Q811yescharacteristic
528C22C2×C7×C22⋊Q811yescharacteristic
529C22C7×#52511yescharacteristic
530C22C7×#52912yes—
531C22C2×C7×C4⋊Q811yescharacteristic
532C22C7×#53211yescharacteristic
533C22C7×#53211yescharacteristic
534C22C7×#53312yes—
535C22C7×#53311yescharacteristic
536C22C7×#53311yescharacteristic
537C22C7×#53411yescharacteristic
538C22C7×#53611yescharacteristic
539C22C7×#53912yes—
540C22C7×#54412yes—
541C22C7×#54612yes—
542C22C7×#54812yes—
543C22C7×#55212yes—
544C22C7×#55212yes—
545C22C7×#55312yes—
546C22C7×#55312yes—
547C22C7×#55412yes—
548C22C7×#55712yes—
549C4—24no—
550C4—24no—
551C4—24no—
552C4—24no—
553C4—42no—
554C4—44no—
555C4—42no—
556C4—44no—
557C4—42no—
558C4—82no—
559C4—82no—
560C2C7×#239611yescharacteristic
561C2C7×#253611yescharacteristic
562C2C7×#254311yescharacteristic
563C2C7×#255012yes—
564C2C7×#255812yes—
565C2C7×#256511yescharacteristic
566C2C7×#257911yescharacteristic
567C2C7×#259012yes—
568C2C7×#260612yes—
569C2C7×#262711yescharacteristic
570C2C7×#285711yescharacteristic
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
38. #15166C7direct (N × H)
525. C7#15166direct (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×C4×C7×C4⋊C42
C7×#14442
C7×#14422
C7×#14412
C7×#14362
C7×C4⋊#952
C7×#14342
#151667