Finite Group Databasebeta

← C7×#14537 #13191753

Subgroups — C7×#14537 #13191753

Order 1792.

1554 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×#393C211yesmaximal, characteristic
2C2×C7×#391C212yesmaximal, complemented
3C2×C7×#388C212yesmaximal
4C7×#1442C212yesmaximal, complemented
5C7×#1440C211yesmaximal, characteristic
6C7×#1439C212yesmaximal
7C7×#1437C211yesmaximal, characteristic
8C7×#1436C212yesmaximal, complemented
9C7×#1436C212yesmaximal, complemented
10C23×C7×Q8C2211yescharacteristic
11C22×C7×C4⋊C4C2212yes—
12C22×C7×C4⋊C4C2212yes—
13C22×C7×C22⋊C4C2211yescharacteristic
14C22×C7×C22⋊C4C2212yes—
15C22×C7×C22⋊C4C2212yes—
16C22×C7×C22⋊C4C2211yescharacteristic
17C22×C7×C22⋊C4C2211yescharacteristic
18C22×C42×C7C2212yes—
19C22×C42×C7C2211yescharacteristic
20C7×#393C418yescomplemented
21C7×#391—28no—
22C7×#388—28no—
23C2×C7×#95C2212yes—
24C2×C7×#95C2212yes—
25C2×C7×#95C2212yes—
26C2×C7×#95C2212yes—
27C2×C7×#95C2212yes—
28C2×C7×#95C2211yescharacteristic
29C2×C7×#95C2212yes—
30C2×C7×#95C2212yes—
31C2×C7×#95C2212yes—
32C2×C7×#95C2211yescharacteristic
33C2×C7×#95C2212yes—
34#14537C711yesmaximal, characteristic, direct product factor, Sylow
35C25×C7C2311yescharacteristic
36C22×C7×Q8—24no—
37C22×C7×Q8—24no—
38C22×C7×Q8—41no—
39C22×C7×Q8—41no—
40C22×C7×Q8C2×C414yes—
41C23×C4×C7C2312yes—
42C23×C4×C7C2311yescharacteristic
43C23×C4×C7C2311yescharacteristic
44C23×C4×C7C2312yes—
45C23×C4×C7C2312yes—
46C23×C4×C7C2312yes—
47C23×C4×C7C2312yes—
48C23×C4×C7C2312yes—
49C2×C7×C4⋊C4—24no—
50C2×C7×C4⋊C4—24no—
51C2×C7×C4⋊C4—24no—
52C2×C7×C4⋊C4—24no—
53C2×C7×C4⋊C4—42no—
54C2×C7×C4⋊C4—42no—
55C2×C7×C22⋊C4—24no—
56C2×C7×C22⋊C4—24no—
57C2×C7×C22⋊C4—24no—
58C2×C7×C22⋊C4—24no—
59C2×C7×C22⋊C4—24no—
60C2×C7×C22⋊C4—41no—
61C2×C7×C22⋊C4—41no—
62C2×C7×C22⋊C4—41no—
63C2×C7×C22⋊C4—42no—
64C2×C7×C22⋊C4—42no—
65C2×C7×C22⋊C4—41no—
66C2×C7×C22⋊C4C2×C418yes—
67C2×C7×C22⋊C4C2×C414yes—
68C2×C42×C7—24no—
69C2×C42×C7—24no—
70C2×C42×C7—42no—
71C2×C42×C7—42no—
72C2×C42×C7—41no—
73C7×#95—24no—
74C7×#95—24no—
75C7×#95—24no—
76C7×#95—24no—
77C7×#95—24no—
78C7×#95—41no—
79C7×#95—42no—
80C7×#95—42no—
81C7×#95C2×C414yes—
82C7×#95C2×C418yes—
83C7×#95D418yes—
84C2×#393C2×C711yescharacteristic
85C2×#391C2×C712yescomplemented
86C2×#388C2×C712yes—
87#1442C2×C712yescomplemented
88#1440C2×C711yescharacteristic
89#1439C2×C712yes—
90#1437C2×C711yescharacteristic
91#1436C2×C712yescomplemented
92#1436C2×C712yescomplemented
93C24×C7—22no—
94C24×C7—22no—
95C24×C7—22no—
96C24×C7—21no—
97C24×C7—21no—
98C24×C7—22no—
99C24×C7—22no—
100C24×C7—22no—
101C24×C7C22×C412yes—
102C24×C7C2411yescharacteristic
103C2×C7×Q8—216no—
104C2×C7×Q8—44no—
105C2×C7×Q8—44no—
106C2×C7×Q8—48no—
107C2×C7×Q8—44no—
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—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—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—21no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—21no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
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—22no—
148C22×C4×C7—22no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7C22×C414yes—
152C22×C4×C7C22×C412yes—
153C22×C4×C7C22×C412yes—
154C22×C4×C7C22×C414yes—
155C22×C4×C7C2×D414yes—
156C22×C4×C7C2×D414yes—
157C22×C4×C7C2×D414yes—
158C22×C4×C7#4114yes—
159C22×C4×C7#4114yes—
160C7×C4⋊C4—48no—
161C7×C4⋊C4—48no—
162C7×C22⋊C4—216no—
163C7×C22⋊C4—44no—
164C7×C22⋊C4—44no—
165C7×C22⋊C4—48no—
166C7×C22⋊C4—44no—
167C42×C7—48no—
168C42×C7—44no—
169C23×Q8C22×C711yescharacteristic
170C22×C4⋊C4C22×C712yes—
171C22×C4⋊C4C22×C712yes—
172C22×C22⋊C4C22×C712yes—
173C22×C22⋊C4C22×C711yescharacteristic
174C22×C22⋊C4C22×C712yes—
175C22×C22⋊C4C22×C711yescharacteristic
176C22×C22⋊C4C22×C711yescharacteristic
177C22×C42C22×C712yes—
178C22×C42C22×C711yescharacteristic
179#393C4×C718yescomplemented
180#391—28no—
181#388—28no—
182C2×#95C22×C712yes—
183C2×#95C22×C712yes—
184C2×#95C22×C712yes—
185C2×#95C22×C712yes—
186C2×#95C22×C712yes—
187C2×#95C22×C711yescharacteristic
188C2×#95C22×C712yes—
189C2×#95C22×C712yes—
190C2×#95C22×C712yes—
191C2×#95C22×C711yescharacteristic
192C2×#95C22×C712yes—
193C23×C7—24no—
194C23×C7—22no—
195C23×C7—22no—
196C23×C7—22no—
197C23×C7—24no—
198C23×C7—42no—
199C23×C7—42no—
200C23×C7—42no—
201C23×C7—42no—
202C23×C7—42no—
203C23×C7—42no—
204C23×C7—41no—
205C23×C7—42no—
206C23×C7—42no—
207C23×C7—42no—
208C23×C7—41no—
209C23×C7—42no—
210C23×C7—42no—
211C23×C7—41no—
212C23×C7—42no—
213C23×C7—41no—
214C23×C7C23×C411yescharacteristic
215C23×C7C22×D412yes—
216C23×C7C2×#4112yes—
217C23×C7C2×#4112yes—
218C23×C7H(5,𝔽2)11yescharacteristic
219C23×C7H(5,𝔽2)11yescharacteristic
220C23×C7H(5,𝔽2)12yes—
221C23×C72−1+412yes—
222C23×C72−1+412yes—
223C7×Q8—416no—
224C2×C4×C7—24no—
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—24no—
233C2×C4×C7—22no—
234C2×C4×C7—22no—
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—24no—
244C2×C4×C7—24no—
245C2×C4×C7—24no—
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—41no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—41no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—41no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—41no—
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×C7C4×D418yes—
276C25C23×C711yescharacteristic
277C22×Q8—24no—
278C22×Q8—24no—
279C22×Q8—41no—
280C22×Q8—41no—
281C22×Q8C2×C4×C714yes—
282C23×C4C23×C712yes—
283C23×C4C23×C712yes—
284C23×C4C23×C711yescharacteristic
285C23×C4C23×C711yescharacteristic
286C23×C4C23×C712yes—
287C23×C4C23×C712yes—
288C23×C4C23×C712yes—
289C23×C4C23×C712yes—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—42no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—41no—
302C2×C22⋊C4—41no—
303C2×C22⋊C4—41no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—41no—
307C2×C22⋊C4C2×C4×C718yes—
308C2×C22⋊C4C2×C4×C714yes—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—42no—
312C2×C42—42no—
313C2×C42—41no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—41no—
320#95—42no—
321#95—42no—
322#95C2×C4×C714yes—
323#95C2×C4×C718yes—
324#95C7×D418yes—
325C22×C7—44no—
326C22×C7—42no—
327C22×C7—42no—
328C22×C7—44no—
329C22×C7—42no—
330C22×C7—82no—
331C22×C7—82no—
332C22×C7—82no—
333C22×C7—82no—
334C22×C7C2×C4×D412yes—
335C22×C7C4×#4111yescharacteristic
336C22×C7#51811yescharacteristic
337C22×C7#51911yescharacteristic
338C22×C7#52012yes—
339C22×C7#53612yes—
340C22×C7#53712yes—
341C22×C7#53912yes—
342C22×C7#54112yes—
343C22×C7#54212yes—
344C22×C7#54212yes—
345C22×C7D4211yescharacteristic
346C22×C7#54612yes—
347C22×C7#54612yes—
348C22×C7#54812yes—
349C22×C7#55111yescharacteristic
350C22×C7#55612yes—
351C22×C7#55912yes—
352C22×C7H(3,𝔽4)11yescharacteristic
353C22×C7#56212yes—
354C22×C7#56311yescharacteristic
355C4×C7—28no—
356C4×C7—42no—
357C4×C7—44no—
358C4×C7—42no—
359C4×C7—44no—
360C4×C7—44no—
361C4×C7—44no—
362C4×C7—82no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—21no—
367C24—21no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24C22×C4×C712yes—
372C24C24×C711yescharacteristic
373C2×Q8—216no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8—48no—
377C2×Q8—44no—
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—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4C22×C4×C714yes—
422C22×C4C22×C4×C712yes—
423C22×C4C22×C4×C712yes—
424C22×C4C22×C4×C714yes—
425C22×C4C2×C7×D414yes—
426C22×C4C2×C7×D414yes—
427C22×C4C2×C7×D414yes—
428C22×C4C7×#4114yes—
429C22×C4C7×#4114yes—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C22⋊C4—216no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—48no—
436C22⋊C4—44no—
437C42—48no—
438C42—44no—
439C2×C7—82no—
440C2×C7#232312yes—
441C2×C7#233211yescharacteristic
442C2×C7#235811yescharacteristic
443C2×C7#236712yes—
444C2×C7#237911yescharacteristic
445C2×C7#268112yes—
446C2×C7#271212yes—
447C2×C7#271612yes—
448C2×C7#279812yes—
449C23—24no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—24no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—41no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—41no—
465C23—42no—
466C23—42no—
467C23—41no—
468C23—42no—
469C23—41no—
470C23C23×C4×C711yescharacteristic
471C23C22×C7×D412yes—
472C23C2×C7×#4112yes—
473C23C2×C7×#4112yes—
474C23C7×H(5,𝔽2)11yescharacteristic
475C23C7×H(5,𝔽2)11yescharacteristic
476C23C7×H(5,𝔽2)12yes—
477C23C7×2−1+412yes—
478C23C7×2−1+412yes—
479Q8—416no—
480C2×C4—24no—
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—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4C4×C7×D418yes—
532C7#1453711yescharacteristic, direct product factor, Sylow
533C22—44no—
534C22—42no—
535C22—42no—
536C22—44no—
537C22—42no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22C2×C4×C7×D412yes—
543C22C4×C7×#4111yescharacteristic
544C22C7×#51811yescharacteristic
545C22C7×#51911yescharacteristic
546C22C7×#52012yes—
547C22C7×#53612yes—
548C22C7×#53712yes—
549C22C7×#53912yes—
550C22C7×#54112yes—
551C22C7×#54212yes—
552C22C7×#54212yes—
553C22C7×D4211yescharacteristic
554C22C7×#54612yes—
555C22C7×#54612yes—
556C22C7×#54812yes—
557C22C7×#55111yescharacteristic
558C22C7×#55612yes—
559C22C7×#55912yes—
560C22C7×H(3,𝔽4)11yescharacteristic
561C22C7×#56212yes—
562C22C7×#56311yescharacteristic
563C4—28no—
564C4—44no—
565C4—44no—
566C4—44no—
567C4—42no—
568C4—44no—
569C4—42no—
570C4—82no—
571C2—82no—
572C2C7×#232312yes—
573C2C7×#233211yescharacteristic
574C2C7×#235811yescharacteristic
575C2C7×#236712yes—
576C2C7×#237911yescharacteristic
577C2C7×#268112yes—
578C2C7×#271212yes—
579C2C7×#271612yes—
580C2C7×#279812yes—
subgroupG/H
center, socleC24×C7C24
commutatorC23C23×C4×C7
Frattini, cosocleC24C24×C7
prime radicalC25×C7C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C7 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C7×#391C2semidirect
4. C7×#1442C2semidirect
8. C7×#1436C2semidirect
9. C7×#1436C2semidirect
20. C7×#393C4semidirect
34. #14537C7direct (N × H)
85. C2×#391C2×C7semidirect
87. #1442C2×C7semidirect
91. #1436C2×C7semidirect
92. #1436C2×C7semidirect
179. #393C4×C7semidirect
532. C7#14537direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#3932
C2×C7×#3912
C2×C7×#3882
C7×#14422
C7×#14402
C7×#14392
C7×#14372
C7×#14362
#145377