Finite Group Databasebeta

← C7×#15181 #13192397

Subgroups — C7×#15181 #13192397

Order 1792.

2314 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C7×D4C211yesmaximal, characteristic
2C2×C7×#389C211yesmaximal, characteristic, complemented
3C2×C7×#380C212yesmaximal, complemented
4C2×C7×#380C212yesmaximal, complemented
5C7×#1442C212yesmaximal, complemented
6C7×#1439C214yesmaximal, complemented
7C7×#1438C212yesmaximal
8C7×C4⋊#95C211yesmaximal, characteristic, complemented
9C23×C7×D4C2211yescharacteristic
10C24×C4×C7C2212yes—
11C2×C4×C7×D4—28no—
12C2×C4×C7×D4—24no—
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C22×C7×C4⋊C4C2212yes—
16C22×C7×C4⋊C4C2211yescharacteristic
17C22×C7×C4⋊C4C2212yes—
18C22×C7×C22⋊C4C2212yes—
19C22×C7×C22⋊C4C2214yes—
20C22×C7×C22⋊C4C2214yes—
21C22×C42×C7C2211yescharacteristic
22C7×#389—24no—
23C7×#380—28no—
24C7×#380—28no—
25C2×C7×#95C2214yes—
26C2×C7×#95C2214yes—
27C2×C7×#95C2212yes—
28C2×C7×#95C2212yes—
29C2×C7×#95C2212yes—
30#15181C711yesmaximal, characteristic, direct product factor, Sylow
31C25×C7C2312yes—
32C22×C7×D4—24no—
33C22×C7×D4—24no—
34C22×C7×D4—22no—
35C22×C7×D4—22no—
36C22×C7×D4Q814yes—
37C23×C4×C7—24no—
38C23×C4×C7—24no—
39C23×C4×C7—24no—
40C23×C4×C7—24no—
41C23×C4×C7—24no—
42C23×C4×C7—24no—
43C23×C4×C7D418yes—
44C23×C4×C7C2314yes—
45C23×C4×C7C2312yes—
46C23×C4×C7C2311yescharacteristic
47C23×C4×C7C2312yes—
48C23×C4×C7C2312yes—
49C23×C4×C7C2311yescharacteristic
50C23×C4×C7C2311yescharacteristic
51C4×C7×D4—232no—
52C2×C7×C4⋊C4—24no—
53C2×C7×C4⋊C4—22no—
54C2×C7×C4⋊C4—22no—
55C2×C7×C4⋊C4—22no—
56C2×C7×C4⋊C4—24no—
57C2×C7×C4⋊C4—24no—
58C2×C7×C4⋊C4—24no—
59C2×C7×C4⋊C4—24no—
60C2×C7×C4⋊C4—22no—
61C2×C7×C4⋊C4—24no—
62C2×C7×C4⋊C4—44no—
63C2×C7×C4⋊C4D414yescomplemented
64C2×C7×C22⋊C4—24no—
65C2×C7×C22⋊C4—24no—
66C2×C7×C22⋊C4—24no—
67C2×C7×C22⋊C4—28no—
68C2×C7×C22⋊C4—28no—
69C2×C7×C22⋊C4—28no—
70C2×C7×C22⋊C4—44no—
71C2×C7×C22⋊C4—44no—
72C2×C7×C22⋊C4—44no—
73C2×C42×C7—24no—
74C2×C42×C7D414yes—
75C7×#95—28no—
76C7×#95—28no—
77C7×#95—24no—
78C7×#95—24no—
79C7×#95—42no—
80C22×C4×D4C2×C711yescharacteristic
81C2×#389C2×C711yescharacteristic, complemented
82C2×#380C2×C712yescomplemented
83C2×#380C2×C712yescomplemented
84#1442C2×C712yescomplemented
85#1439C2×C714yescomplemented
86#1438C2×C712yes—
87C4⋊#95C2×C711yescharacteristic, complemented
88C24×C7—22no—
89C24×C7—22no—
90C24×C7—22no—
91C24×C7—22no—
92C24×C7—22no—
93C24×C7—22no—
94C24×C7—22no—
95C24×C7—22no—
96C24×C7—22no—
97C24×C7—22no—
98C24×C7—22no—
99C24×C7—22no—
100C24×C7C2×D414yes—
101C24×C7C2×Q814yes—
102C24×C7#4114yes—
103C24×C7C2411yescharacteristic
104C2×C7×D4—28no—
105C2×C7×D4—28no—
106C2×C7×D4—216no—
107C2×C7×D4—216no—
108C2×C7×D4—28no—
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—24no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—21no—
122C22×C4×C7—24no—
123C22×C4×C7—22no—
124C22×C4×C7—21no—
125C22×C4×C7—216no—
126C22×C4×C7—216no—
127C22×C4×C7—216no—
128C22×C4×C7—24no—
129C22×C4×C7—24no—
130C22×C4×C7—24no—
131C22×C4×C7—22no—
132C22×C4×C7—24no—
133C22×C4×C7—24no—
134C22×C4×C7—24no—
135C22×C4×C7—24no—
136C22×C4×C7—22no—
137C22×C4×C7—24no—
138C22×C4×C7—24no—
139C22×C4×C7—22no—
140C22×C4×C7—48no—
141C22×C4×C7—48no—
142C22×C4×C7—48no—
143C22×C4×C7—48no—
144C22×C4×C7C2×D414yes—
145C22×C4×C7C2×D414yes—
146C22×C4×C7C2×D412yes—
147C22×C4×C7C2×D412yes—
148C22×C4×C7C2×D414yes—
149C22×C4×C7C2×D412yes—
150C22×C4×C7C2×D412yes—
151C22×C4×C7C2×Q812yes—
152C22×C4×C7#4112yes—
153C22×C4×C7#4112yes—
154C22×C4×C7#4114yes—
155C7×C4⋊C4—28no—
156C7×C4⋊C4—28no—
157C7×C4⋊C4—48no—
158C7×C4⋊C4—44no—
159C7×C4⋊C4—48no—
160C7×C22⋊C4—216no—
161C7×C22⋊C4—416no—
162C7×C22⋊C4—416no—
163C42×C7—28no—
164C23×D4C22×C711yescharacteristic
165C24×C4C22×C712yes—
166C2×C4×D4—28no—
167C2×C4×D4—24no—
168C22×C4⋊C4C22×C711yescharacteristic
169C22×C4⋊C4C22×C712yes—
170C22×C4⋊C4C22×C711yescharacteristic
171C22×C4⋊C4C22×C711yescharacteristic
172C22×C4⋊C4C22×C712yes—
173C22×C22⋊C4C22×C712yes—
174C22×C22⋊C4C22×C714yes—
175C22×C22⋊C4C22×C714yes—
176C22×C42C22×C711yescharacteristic
177#389—24no—
178#380—28no—
179#380—28no—
180C2×#95C22×C712yes—
181C2×#95C22×C712yes—
182C2×#95C22×C714yes—
183C2×#95C22×C714yes—
184C2×#95C22×C712yes—
185C23×C7—24no—
186C23×C7—24no—
187C23×C7—24no—
188C23×C7—24no—
189C23×C7—24no—
190C23×C7—24no—
191C23×C7—24no—
192C23×C7—24no—
193C23×C7—24no—
194C23×C7—24no—
195C23×C7—24no—
196C23×C7—24no—
197C23×C7—24no—
198C23×C7—24no—
199C23×C7—24no—
200C23×C7—24no—
201C23×C7—24no—
202C23×C7—24no—
203C23×C7—42no—
204C23×C7—42no—
205C23×C7—42no—
206C23×C7—42no—
207C23×C7—42no—
208C23×C7—42no—
209C23×C7—42no—
210C23×C7—42no—
211C23×C7—42no—
212C23×C7—42no—
213C23×C7—42no—
214C23×C7—42no—
215C23×C7—42no—
216C23×C7—42no—
217C23×C7—42no—
218C23×C7—42no—
219C23×C7C22⋊Q818yes—
220C23×C7C22×D412yes—
221C23×C7C22×D411yescharacteristic
222C23×C7C22×D411yescharacteristic
223C23×C7C22×Q811yescharacteristic
224C23×C7C2×#4111yescharacteristic
225C23×C7C2×#4112yes—
226C23×C7C2×#4112yes—
227C23×C7C2×#4111yescharacteristic
228C23×C7H(5,𝔽2)12yes—
229C23×C7H(5,𝔽2)11yescharacteristic
230C23×C72−1+411yescharacteristic
231C7×D4—232no—
232C2×C4×C7—22no—
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—24no—
241C2×C4×C7—22no—
242C2×C4×C7—22no—
243C2×C4×C7—28no—
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—44no—
253C2×C4×C7—432no—
254C2×C4×C7—42no—
255C2×C4×C7—44no—
256C2×C4×C7—44no—
257C2×C4×C7—44no—
258C2×C4×C7—44no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—44no—
262C2×C4×C7—44no—
263C2×C4×C7—44no—
264C2×C4×C7—44no—
265C2×C4×C7—44no—
266C2×C4×C7#12014yes—
267C2×C4×C7#12114yes—
268C2×C4×C7#12118yes—
269C2×C4×C7C22⋊Q814yes—
270C2×C4×C7#12318yes—
271C2×C4×C7#12414yes—
272C25C23×C712yes—
273C22×D4—24no—
274C22×D4—24no—
275C22×D4—22no—
276C22×D4—22no—
277C22×D4C7×Q814yes—
278C23×C4—24no—
279C23×C4—24no—
280C23×C4—24no—
281C23×C4—24no—
282C23×C4—24no—
283C23×C4—24no—
284C23×C4C7×D418yes—
285C23×C4C23×C712yes—
286C23×C4C23×C714yes—
287C23×C4C23×C712yes—
288C23×C4C23×C711yescharacteristic
289C23×C4C23×C712yes—
290C23×C4C23×C711yescharacteristic
291C23×C4C23×C711yescharacteristic
292C4×D4—232no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4C7×D414yescomplemented
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C42—24no—
315C2×C42C7×D414yes—
316#95—28no—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—42no—
321C22×C7—28no—
322C22×C7—28no—
323C22×C7—28no—
324C22×C7—44no—
325C22×C7—44no—
326C22×C7—44no—
327C22×C7—44no—
328C22×C7—44no—
329C22×C7—44no—
330C22×C7—44no—
331C22×C7—44no—
332C22×C7—44no—
333C22×C7—44no—
334C22×C7—44no—
335C22×C7—44no—
336C22×C7C2×#12011yescharacteristic
337C22×C7C2×#12112yes—
338C22×C7C2×#12111yescharacteristic
339C22×C7C2×C22⋊Q811yescharacteristic
340C22×C7C2×C22⋊Q812yes—
341C22×C7C2×#12312yes—
342C22×C7#52511yescharacteristic
343C22×C7#52512yes—
344C22×C7C2×#12411yescharacteristic
345C22×C7#53511yescharacteristic
346C22×C7#53512yes—
347C22×C7#53711yescharacteristic
348C22×C7#53712yes—
349C22×C7#54612yes—
350C22×C7D4×Q811yescharacteristic
351C22×C7#55011yescharacteristic
352C22×C7#55212yes—
353C22×C7#55312yes—
354C22×C7#55312yes—
355C22×C7#55412yes—
356C22×C7#55411yescharacteristic
357C22×C7#55512yes—
358C22×C7#55611yescharacteristic
359C4×C7—24no—
360C4×C7—42no—
361C4×C7—44no—
362C4×C7—44no—
363C4×C7—82no—
364C4×C7—84no—
365C4×C7#39418yes—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24C2×C7×D414yes—
379C24C2×C7×Q814yes—
380C24C7×#4114yes—
381C24C24×C711yescharacteristic
382C2×D4—28no—
383C2×D4—28no—
384C2×D4—216no—
385C2×D4—216no—
386C2×D4—28no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—216no—
404C22×C4—216no—
405C22×C4—216no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4C2×C7×D414yes—
423C22×C4C2×C7×D414yes—
424C22×C4C2×C7×D412yes—
425C22×C4C2×C7×D412yes—
426C22×C4C2×C7×D414yes—
427C22×C4C2×C7×D412yes—
428C22×C4C2×C7×D412yes—
429C22×C4C2×C7×Q812yes—
430C22×C4C7×#4112yes—
431C22×C4C7×#4112yes—
432C22×C4C7×#4114yes—
433C4⋊C4—28no—
434C4⋊C4—28no—
435C4⋊C4—48no—
436C4⋊C4—44no—
437C4⋊C4—48no—
438C22⋊C4—216no—
439C22⋊C4—416no—
440C22⋊C4—416no—
441C42—28no—
442C2×C7—48no—
443C2×C7C2×#39411yescharacteristic
444C2×C7#241412yes—
445C2×C7#241712yes—
446C2×C7#248512yes—
447C2×C7#248712yes—
448C2×C7#253511yescharacteristic
449C2×C7#253811yescharacteristic
450C2×C7#256211yescharacteristic
451C2×C7#256311yescharacteristic
452C2×C7#261511yescharacteristic
453C2×C7#284811yescharacteristic
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23C7×C22⋊Q818yes—
489C23C22×C7×D412yes—
490C23C22×C7×D411yescharacteristic
491C23C22×C7×D411yescharacteristic
492C23C22×C7×Q811yescharacteristic
493C23C2×C7×#4111yescharacteristic
494C23C2×C7×#4112yes—
495C23C2×C7×#4112yes—
496C23C2×C7×#4111yescharacteristic
497C23C7×H(5,𝔽2)12yes—
498C23C7×H(5,𝔽2)11yescharacteristic
499C23C7×2−1+411yescharacteristic
500D4—232no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—28no—
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×C4—44no—
522C2×C4—432no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4C7×#12014yes—
536C2×C4C7×#12114yes—
537C2×C4C7×#12118yes—
538C2×C4C7×C22⋊Q814yes—
539C2×C4C7×#12318yes—
540C2×C4C7×#12414yes—
541C7#1518111yescharacteristic, direct product factor, Sylow
542C22—28no—
543C22—28no—
544C22—28no—
545C22—44no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—44no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22—44no—
557C22C2×C7×#12011yescharacteristic
558C22C2×C7×#12112yes—
559C22C2×C7×#12111yescharacteristic
560C22C2×C7×C22⋊Q812yes—
561C22C2×C7×C22⋊Q811yescharacteristic
562C22C2×C7×#12312yes—
563C22C7×#52512yes—
564C22C7×#52511yescharacteristic
565C22C2×C7×#12411yescharacteristic
566C22C7×#53512yes—
567C22C7×#53511yescharacteristic
568C22C7×#53712yes—
569C22C7×#53711yescharacteristic
570C22C7×#54612yes—
571C22C7×D4×Q811yescharacteristic
572C22C7×#55011yescharacteristic
573C22C7×#55212yes—
574C22C7×#55312yes—
575C22C7×#55312yes—
576C22C7×#55412yes—
577C22C7×#55411yescharacteristic
578C22C7×#55512yes—
579C22C7×#55611yescharacteristic
580C4—24no—
581C4—44no—
582C4—42no—
583C4—44no—
584C4—84no—
585C4—82no—
586C4C7×#39418yes—
587C2—48no—
588C2C2×C7×#39411yescharacteristic
589C2C7×#241412yes—
590C2C7×#241712yes—
591C2C7×#248512yes—
592C2C7×#248712yes—
593C2C7×#253511yescharacteristic
594C2C7×#253811yescharacteristic
595C2C7×#256211yescharacteristic
596C2C7×#256311yescharacteristic
597C2C7×#261511yescharacteristic
598C2C7×#284811yescharacteristic
subgroupG/H
center, socleC24×C7C24
commutator, Frattini, cosocleC24C24×C7
prime radicalC23×C7×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#389C2semidirect
3. C2×C7×#380C2semidirect
4. C2×C7×#380C2semidirect
5. C7×#1442C2semidirect
6. C7×#1439C2semidirect
8. C7×C4⋊#95C2semidirect
30. #15181C7direct (N × H)
63. C2×C7×C4⋊C4D4semidirect
81. C2×#389C2×C7semidirect
82. C2×#380C2×C7semidirect
83. C2×#380C2×C7semidirect
84. #1442C2×C7semidirect
85. #1439C2×C7semidirect
87. C4⋊#95C2×C7semidirect
304. C2×C4⋊C4C7×D4semidirect
541. C7#15181direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C7×D42
C2×C7×#3892
C2×C7×#3802
C7×#14422
C7×#14392
C7×#14382
C7×C4⋊#952
#151817