Finite Group Databasebeta

← C7×#15129 #13192345

Subgroups — C7×#15129 #13192345

Order 1792.

2354 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C7×D4C211yesmaximal, characteristic
2C2×C7×#394C212yesmaximal
3C2×C7×#389C211yesmaximal, characteristic, complemented
4C2×C7×#380C212yesmaximal, complemented
5C2×C7×#380C212yesmaximal, complemented
6C7×#1441C212yesmaximal, complemented
7C7×#1440C214yesmaximal, complemented
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⋊C4C2212yes—
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×C7×C22⋊C4C2212yes—
19C22×C7×C22⋊C4C2214yes—
20C22×C7×C22⋊C4C2214yes—
21C22×C42×C7C2211yescharacteristic
22C7×#394—28no—
23C7×#389—24no—
24C7×#380—28no—
25C7×#380—28no—
26C2×C7×#95C2214yes—
27C2×C7×#95C2214yes—
28C2×C7×#95C2212yes—
29C2×C7×#95C2212yes—
30C2×C7×#95C2212yes—
31#15129C711yesmaximal, characteristic, direct product factor, Sylow
32C25×C7C2312yes—
33C22×C7×D4—24no—
34C22×C7×D4—24no—
35C22×C7×D4—22no—
36C22×C7×D4—22no—
37C22×C7×D4D414yes—
38C23×C4×C7—24no—
39C23×C4×C7—24no—
40C23×C4×C7—24no—
41C23×C4×C7—24no—
42C23×C4×C7—24no—
43C23×C4×C7—24no—
44C23×C4×C7D418yes—
45C23×C4×C7C2314yes—
46C23×C4×C7C2312yes—
47C23×C4×C7C2311yescharacteristic
48C23×C4×C7C2312yes—
49C23×C4×C7C2312yes—
50C23×C4×C7C2311yescharacteristic
51C23×C4×C7C2311yescharacteristic
52C4×C7×D4—232no—
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—22no—
61C2×C7×C4⋊C4—24no—
62C2×C7×C4⋊C4—24no—
63C2×C7×C4⋊C4—24no—
64C2×C7×C4⋊C4—24no—
65C2×C7×C4⋊C4D414yescomplemented
66C2×C7×C22⋊C4—24no—
67C2×C7×C22⋊C4—24no—
68C2×C7×C22⋊C4—24no—
69C2×C7×C22⋊C4—28no—
70C2×C7×C22⋊C4—28no—
71C2×C7×C22⋊C4—28no—
72C2×C7×C22⋊C4—28no—
73C2×C7×C22⋊C4—28no—
74C2×C7×C22⋊C4—44no—
75C2×C42×C7—24no—
76C2×C42×C7D414yes—
77C7×#95—28no—
78C7×#95—28no—
79C7×#95—24no—
80C7×#95—24no—
81C7×#95—42no—
82C22×C4×D4C2×C711yescharacteristic
83C2×#394C2×C712yes—
84C2×#389C2×C711yescharacteristic, complemented
85C2×#380C2×C712yescomplemented
86C2×#380C2×C712yescomplemented
87#1441C2×C712yescomplemented
88#1440C2×C714yescomplemented
89C4⋊#95C2×C711yescharacteristic, complemented
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×C7—22no—
101C24×C7—22no—
102C24×C7C2×D414yes—
103C24×C7C2×D414yes—
104C24×C7#4114yes—
105C24×C7C2411yescharacteristic
106C2×C7×D4—28no—
107C2×C7×D4—28no—
108C2×C7×D4—216no—
109C2×C7×D4—216no—
110C2×C7×D4—28no—
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—21no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—21no—
128C22×C4×C7—22no—
129C22×C4×C7—216no—
130C22×C4×C7—216no—
131C22×C4×C7—216no—
132C22×C4×C7—24no—
133C22×C4×C7—24no—
134C22×C4×C7—24no—
135C22×C4×C7—24no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—24no—
139C22×C4×C7—24no—
140C22×C4×C7—22no—
141C22×C4×C7—24no—
142C22×C4×C7—24no—
143C22×C4×C7—24no—
144C22×C4×C7—48no—
145C22×C4×C7—48no—
146C22×C4×C7—48no—
147C22×C4×C7—48no—
148C22×C4×C7C2×D412yes—
149C22×C4×C7C2×D414yes—
150C22×C4×C7C2×D414yes—
151C22×C4×C7C2×D412yes—
152C22×C4×C7C2×D412yes—
153C22×C4×C7C2×D414yes—
154C22×C4×C7C2×D412yes—
155C22×C4×C7C2×D412yes—
156C22×C4×C7#4112yes—
157C22×C4×C7#4112yes—
158C22×C4×C7#4114yes—
159C7×C4⋊C4—28no—
160C7×C4⋊C4—28no—
161C7×C4⋊C4—44no—
162C7×C4⋊C4—48no—
163C7×C4⋊C4—48no—
164C7×C22⋊C4—216no—
165C7×C22⋊C4—416no—
166C7×C22⋊C4—416no—
167C42×C7—28no—
168C23×D4C22×C711yescharacteristic
169C24×C4C22×C712yes—
170C2×C4×D4—28no—
171C2×C4×D4—24no—
172C22×C4⋊C4C22×C712yes—
173C22×C4⋊C4C22×C711yescharacteristic
174C22×C4⋊C4C22×C712yes—
175C22×C4⋊C4C22×C711yescharacteristic
176C22×C4⋊C4C22×C711yescharacteristic
177C22×C22⋊C4C22×C714yes—
178C22×C22⋊C4C22×C712yes—
179C22×C22⋊C4C22×C714yes—
180C22×C42C22×C711yescharacteristic
181#394—28no—
182#389—24no—
183#380—28no—
184#380—28no—
185C2×#95C22×C712yes—
186C2×#95C22×C712yes—
187C2×#95C22×C714yes—
188C2×#95C22×C714yes—
189C2×#95C22×C712yes—
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—24no—
204C23×C7—24no—
205C23×C7—24no—
206C23×C7—24no—
207C23×C7—24no—
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×C7—42no—
220C23×C7—42no—
221C23×C7—42no—
222C23×C7—42no—
223C23×C7—42no—
224C23×C7#12118yes—
225C23×C7C22×D412yes—
226C23×C7C22×D411yescharacteristic
227C23×C7C22×D411yescharacteristic
228C23×C7C22×D411yescharacteristic
229C23×C7C2×#4111yescharacteristic
230C23×C7C2×#4112yes—
231C23×C7C2×#4111yescharacteristic
232C23×C7C2×#4112yes—
233C23×C7H(5,𝔽2)11yescharacteristic
234C23×C72−1+411yescharacteristic
235C23×C72−1+412yes—
236C7×D4—232no—
237C2×C4×C7—22no—
238C2×C4×C7—24no—
239C2×C4×C7—22no—
240C2×C4×C7—24no—
241C2×C4×C7—22no—
242C2×C4×C7—24no—
243C2×C4×C7—24no—
244C2×C4×C7—24no—
245C2×C4×C7—24no—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—24no—
249C2×C4×C7—24no—
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—432no—
261C2×C4×C7—42no—
262C2×C4×C7—44no—
263C2×C4×C7—44no—
264C2×C4×C7—42no—
265C2×C4×C7—44no—
266C2×C4×C7—42no—
267C2×C4×C7—44no—
268C2×C4×C7—44no—
269C2×C4×C7—44no—
270C2×C4×C7—44no—
271C2×C4×C7—44no—
272C2×C4×C7—44no—
273C2×C4×C7#12014yes—
274C2×C4×C7#12114yes—
275C2×C4×C7#12114yes—
276C2×C4×C7#12118yes—
277C2×C4×C7#12318yes—
278C2×C4×C7#12414yes—
279C25C23×C712yes—
280C22×D4—24no—
281C22×D4—24no—
282C22×D4—22no—
283C22×D4—22no—
284C22×D4C7×D414yes—
285C23×C4—24no—
286C23×C4—24no—
287C23×C4—24no—
288C23×C4—24no—
289C23×C4—24no—
290C23×C4—24no—
291C23×C4C7×D418yes—
292C23×C4C23×C712yes—
293C23×C4C23×C714yes—
294C23×C4C23×C712yes—
295C23×C4C23×C711yescharacteristic
296C23×C4C23×C712yes—
297C23×C4C23×C711yescharacteristic
298C23×C4C23×C711yescharacteristic
299C4×D4—232no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4C7×D414yescomplemented
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—28no—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—44no—
322C2×C42—24no—
323C2×C42C7×D414yes—
324#95—28no—
325#95—28no—
326#95—24no—
327#95—24no—
328#95—42no—
329C22×C7—28no—
330C22×C7—28no—
331C22×C7—28no—
332C22×C7—44no—
333C22×C7—44no—
334C22×C7—44no—
335C22×C7—44no—
336C22×C7—44no—
337C22×C7—44no—
338C22×C7—44no—
339C22×C7—44no—
340C22×C7—44no—
341C22×C7—44no—
342C22×C7—44no—
343C22×C7—44no—
344C22×C7C2×#12011yescharacteristic
345C22×C7C2×#12112yes—
346C22×C7C2×#12111yescharacteristic
347C22×C7C2×#12112yes—
348C22×C7C2×#12111yescharacteristic
349C22×C7C2×#12312yes—
350C22×C7#52511yescharacteristic
351C22×C7#52512yes—
352C22×C7C2×#12411yescharacteristic
353C22×C7#53612yes—
354C22×C7#53611yescharacteristic
355C22×C7#53711yescharacteristic
356C22×C7#53712yes—
357C22×C7D4211yescharacteristic
358C22×C7#54711yescharacteristic
359C22×C7#54712yes—
360C22×C7#54711yescharacteristic
361C22×C7#54812yes—
362C22×C7#55212yes—
363C22×C7#55212yes—
364C22×C7#55312yes—
365C22×C7#55511yescharacteristic
366C22×C7#55612yes—
367C4×C7—24no—
368C4×C7—42no—
369C4×C7—44no—
370C4×C7—44no—
371C4×C7—82no—
372C4×C7—84no—
373C4×C7#39418yescomplemented
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24C2×C7×D414yes—
387C24C2×C7×D414yes—
388C24C7×#4114yes—
389C24C24×C711yescharacteristic
390C2×D4—28no—
391C2×D4—28no—
392C2×D4—216no—
393C2×D4—216no—
394C2×D4—28no—
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—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—216no—
414C22×C4—216no—
415C22×C4—216no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4C2×C7×D412yes—
433C22×C4C2×C7×D414yes—
434C22×C4C2×C7×D414yes—
435C22×C4C2×C7×D412yes—
436C22×C4C2×C7×D412yes—
437C22×C4C2×C7×D414yes—
438C22×C4C2×C7×D412yes—
439C22×C4C2×C7×D412yes—
440C22×C4C7×#4112yes—
441C22×C4C7×#4112yes—
442C22×C4C7×#4114yes—
443C4⋊C4—28no—
444C4⋊C4—28no—
445C4⋊C4—44no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C22⋊C4—216no—
449C22⋊C4—416no—
450C22⋊C4—416no—
451C42—28no—
452C2×C7—48no—
453C2×C7C2×#39411yescharacteristic
454C2×C7#241512yes—
455C2×C7#241612yes—
456C2×C7#248512yes—
457C2×C7#248612yes—
458C2×C7#253611yescharacteristic
459C2×C7#253711yescharacteristic
460C2×C7#254711yescharacteristic
461C2×C7#256211yescharacteristic
462C2×C7#261611yescharacteristic
463C2×C7#284711yescharacteristic
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23C7×#12118yes—
499C23C22×C7×D412yes—
500C23C22×C7×D411yescharacteristic
501C23C22×C7×D411yescharacteristic
502C23C22×C7×D411yescharacteristic
503C23C2×C7×#4111yescharacteristic
504C23C2×C7×#4112yes—
505C23C2×C7×#4111yescharacteristic
506C23C2×C7×#4112yes—
507C23C7×H(5,𝔽2)11yescharacteristic
508C23C7×2−1+411yescharacteristic
509C23C7×2−1+412yes—
510D4—232no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—432no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4C7×#12014yes—
548C2×C4C7×#12114yes—
549C2×C4C7×#12114yes—
550C2×C4C7×#12118yes—
551C2×C4C7×#12318yes—
552C2×C4C7×#12414yes—
553C7#1512911yescharacteristic, direct product factor, Sylow
554C22—28no—
555C22—28no—
556C22—28no—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22C2×C7×#12011yescharacteristic
570C22C2×C7×#12112yes—
571C22C2×C7×#12112yes—
572C22C2×C7×#12111yescharacteristic
573C22C2×C7×#12111yescharacteristic
574C22C2×C7×#12312yes—
575C22C7×#52512yes—
576C22C7×#52511yescharacteristic
577C22C2×C7×#12411yescharacteristic
578C22C7×#53612yes—
579C22C7×#53611yescharacteristic
580C22C7×#53712yes—
581C22C7×#53711yescharacteristic
582C22C7×D4211yescharacteristic
583C22C7×#54711yescharacteristic
584C22C7×#54712yes—
585C22C7×#54711yescharacteristic
586C22C7×#54812yes—
587C22C7×#55212yes—
588C22C7×#55212yes—
589C22C7×#55312yes—
590C22C7×#55511yescharacteristic
591C22C7×#55612yes—
592C4—24no—
593C4—44no—
594C4—42no—
595C4—44no—
596C4—84no—
597C4—82no—
598C4C7×#39418yescomplemented
599C2—48no—
600C2C2×C7×#39411yescharacteristic
601C2C7×#241512yes—
602C2C7×#241612yes—
603C2C7×#248512yes—
604C2C7×#248612yes—
605C2C7×#253611yescharacteristic
606C2C7×#253711yescharacteristic
607C2C7×#254711yescharacteristic
608C2C7×#256211yescharacteristic
609C2C7×#261611yescharacteristic
610C2C7×#284711yescharacteristic
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
3. C2×C7×#389C2semidirect
4. C2×C7×#380C2semidirect
5. C2×C7×#380C2semidirect
6. C7×#1441C2semidirect
7. C7×#1440C2semidirect
8. C7×C4⋊#95C2semidirect
31. #15129C7direct (N × H)
65. C2×C7×C4⋊C4D4semidirect
84. C2×#389C2×C7semidirect
85. C2×#380C2×C7semidirect
86. C2×#380C2×C7semidirect
87. #1441C2×C7semidirect
88. #1440C2×C7semidirect
89. C4⋊#95C2×C7semidirect
312. C2×C4⋊C4C7×D4semidirect
373. C4×C7#394semidirect
553. C7#15129direct (N × H)
598. C4C7×#394semidirect

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×#3942
C2×C7×#3892
C2×C7×#3802
C7×#14412
C7×#14402
C7×C4⋊#952
#151297