Finite Group Databasebeta

← C7×#15134 #13192350

Subgroups — C7×#15134 #13192350

Order 1792.

2006 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#396C214yesmaximal, complemented
2C2×C7×#390C211yesmaximal, characteristic
3C2×C7×#390C211yesmaximal, characteristic
4C2×C7×#390C211yesmaximal, characteristic
5C2×C7×#389C211yesmaximal, characteristic, complemented
6C2×C7×#384C211yesmaximal, characteristic, complemented
7C2×C4×C7×C4⋊C4C211yesmaximal, characteristic, complemented
8C7×#1439C214yesmaximal, complemented
9C7×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×C7×D4C2211yescharacteristic
11C22×C7×C4⋊C4C2212yes—
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C4⋊C4C2214yes—
15C22×C7×C4⋊C4C2211yescharacteristic
16C22×C7×C4⋊C4C2212yes—
17C22×C7×C22⋊C4C2212yes—
18C22×C7×C22⋊C4C2214yes—
19C22×C7×C22⋊C4C2214yes—
20C22×C7×C22⋊C4C2212yes—
21C22×C42×C7C2211yescharacteristic
22C22×C42×C7C2211yescharacteristic
23C22×C42×C7C2211yescharacteristic
24C7×#396—216no—
25C7×#390—24no—
26C7×#390—24no—
27C7×#390—24no—
28C7×#389—24no—
29C7×#384—24no—
30C4×C7×C4⋊C4—24no—
31C2×C7×#95C2214yes—
32C2×C7×#95C2212yes—
33C2×C7×#95C2212yes—
34#15134C711yesmaximal, characteristic, direct product factor, Sylow
35C25×C7C2312yes—
36C22×C7×D4—22no—
37C22×C7×D4—22no—
38C22×C7×D4—22no—
39C22×C7×D4—22no—
40C22×C7×D4—22no—
41C22×C7×D4—22no—
42C22×C7×D4Q814yes—
43C23×C4×C7C2312yes—
44C23×C4×C7C2312yes—
45C23×C4×C7C2311yescharacteristic
46C23×C4×C7C2312yes—
47C23×C4×C7C2312yes—
48C23×C4×C7C2311yescharacteristic
49C23×C4×C7C2312yes—
50C23×C4×C7C2311yescharacteristic
51C2×C7×C4⋊C4—24no—
52C2×C7×C4⋊C4—22no—
53C2×C7×C4⋊C4—22no—
54C2×C7×C4⋊C4—24no—
55C2×C7×C4⋊C4—28no—
56C2×C7×C4⋊C4—22no—
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4—24no—
59C2×C7×C4⋊C4—22no—
60C2×C7×C4⋊C4—22no—
61C2×C7×C4⋊C4—28no—
62C2×C7×C4⋊C4—28no—
63C2×C7×C4⋊C4—22no—
64C2×C7×C4⋊C4—22no—
65C2×C7×C4⋊C4—44no—
66C2×C7×C4⋊C4D414yes—
67C2×C7×C4⋊C4D418yescomplemented
68C2×C7×C22⋊C4—28no—
69C2×C7×C22⋊C4—24no—
70C2×C7×C22⋊C4—28no—
71C2×C7×C22⋊C4—24no—
72C2×C7×C22⋊C4—28no—
73C2×C7×C22⋊C4—28no—
74C2×C7×C22⋊C4—28no—
75C2×C7×C22⋊C4—28no—
76C2×C7×C22⋊C4—44no—
77C2×C7×C22⋊C4—44no—
78C2×C42×C7—22no—
79C2×C42×C7—22no—
80C2×C42×C7—22no—
81C2×C42×C7—22no—
82C2×C42×C7—22no—
83C2×C42×C7—22no—
84C2×C42×C7—22no—
85C2×C42×C7D414yes—
86C2×C42×C7D414yes—
87C7×#95—24no—
88C7×#95—24no—
89C7×#95—28no—
90C2×#396C2×C714yescomplemented
91C2×#390C2×C711yescharacteristic
92C2×#390C2×C711yescharacteristic
93C2×#390C2×C711yescharacteristic
94C2×#389C2×C711yescharacteristic, complemented
95C2×#384C2×C711yescharacteristic, complemented
96C2×C4×C4⋊C4C2×C711yescharacteristic, complemented
97#1439C2×C714yescomplemented
98C4⋊#95C2×C711yescharacteristic, complemented
99C24×C7—24no—
100C24×C7—22no—
101C24×C7—22no—
102C24×C7—22no—
103C24×C7—24no—
104C24×C7—22no—
105C24×C7—22no—
106C24×C7—22no—
107C24×C7—24no—
108C24×C7—24no—
109C24×C7C2×Q814yes—
110C24×C7C2411yescharacteristic
111C2×C7×D4—28no—
112C2×C7×D4—28no—
113C2×C7×D4—28no—
114C2×C7×D4—44no—
115C2×C7×D4—44no—
116C2×C7×D4—44no—
117C2×C7×D4—44no—
118C22×C4×C7—21no—
119C22×C4×C7—22no—
120C22×C4×C7—21no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—21no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—24no—
130C22×C4×C7—21no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—24no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—24no—
140C22×C4×C7—24no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—24no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—24no—
149C22×C4×C7C2×D412yes—
150C22×C4×C7C2×D414yes—
151C22×C4×C7C2×D414yes—
152C22×C4×C7C2×D414yes—
153C22×C4×C7C2×D414yes—
154C22×C4×C7C2×D412yes—
155C22×C4×C7C2×D414yes—
156C22×C4×C7C2×D412yes—
157C22×C4×C7C2×D412yes—
158C22×C4×C7C2×D412yes—
159C22×C4×C7C2×Q812yes—
160C22×C4×C7#4114yes—
161C22×C4×C7#4112yes—
162C22×C4×C7#4112yes—
163C22×C4×C7#4112yes—
164C7×C4⋊C4—28no—
165C7×C4⋊C4—216no—
166C7×C4⋊C4—416no—
167C7×C4⋊C4—44no—
168C7×C4⋊C4—44no—
169C7×C4⋊C4—48no—
170C7×C22⋊C4—416no—
171C7×C22⋊C4—48no—
172C7×C22⋊C4—416no—
173C7×C22⋊C4—48no—
174C42×C7—28no—
175C42×C7—28no—
176C42×C7—44no—
177C23×D4C22×C711yescharacteristic
178C22×C4⋊C4C22×C712yes—
179C22×C4⋊C4C22×C711yescharacteristic
180C22×C4⋊C4C22×C711yescharacteristic
181C22×C4⋊C4C22×C714yes—
182C22×C4⋊C4C22×C711yescharacteristic
183C22×C4⋊C4C22×C712yes—
184C22×C22⋊C4C22×C712yes—
185C22×C22⋊C4C22×C714yes—
186C22×C22⋊C4C22×C714yes—
187C22×C22⋊C4C22×C712yes—
188C22×C42C22×C711yescharacteristic
189C22×C42C22×C711yescharacteristic
190C22×C42C22×C711yescharacteristic
191#396—216no—
192#390—24no—
193#390—24no—
194#390—24no—
195#389—24no—
196#384—24no—
197C4×C4⋊C4—24no—
198C2×#95C22×C714yes—
199C2×#95C22×C712yes—
200C2×#95C22×C712yes—
201C23×C7—24no—
202C23×C7—24no—
203C23×C7—24no—
204C23×C7—28no—
205C23×C7—28no—
206C23×C7—44no—
207C23×C7—42no—
208C23×C7—42no—
209C23×C7—42no—
210C23×C7—42no—
211C23×C7—42no—
212C23×C7—42no—
213C23×C7—44no—
214C23×C7—42no—
215C23×C7—42no—
216C23×C7—44no—
217C23×C7—44no—
218C23×C7—44no—
219C23×C7—44no—
220C23×C7—44no—
221C23×C7—44no—
222C23×C7—44no—
223C23×C7—44no—
224C23×C7C22×D411yescharacteristic
225C23×C7C22×D411yescharacteristic
226C23×C7C22×D411yescharacteristic
227C23×C7C22×D412yes—
228C23×C7C22×Q811yescharacteristic
229C23×C7C2×#4111yescharacteristic
230C23×C7C2×#4111yescharacteristic
231C23×C7C2×#4111yescharacteristic
232C23×C7C2×#4112yes—
233C23×C7H(5,𝔽2)12yes—
234C23×C72−1+412yes—
235C7×D4—416no—
236C2×C4×C7—22no—
237C2×C4×C7—24no—
238C2×C4×C7—24no—
239C2×C4×C7—22no—
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—22no—
247C2×C4×C7—24no—
248C2×C4×C7—24no—
249C2×C4×C7—22no—
250C2×C4×C7—22no—
251C2×C4×C7—24no—
252C2×C4×C7—28no—
253C2×C4×C7—24no—
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—44no—
262C2×C4×C7—42no—
263C2×C4×C7—44no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7—44no—
270C2×C4×C7—42no—
271C2×C4×C7—44no—
272C2×C4×C7—44no—
273C2×C4×C7#12118yes—
274C2×C4×C7#12114yes—
275C2×C4×C7#12114yes—
276C2×C4×C7#12114yes—
277C2×C4×C7C22⋊Q814yes—
278C2×C4×C7C22⋊Q814yes—
279C2×C4×C7C22⋊Q814yes—
280C2×C4×C7#12614yes—
281C2×C4×C7#12714yes—
282C25C23×C712yes—
283C22×D4—22no—
284C22×D4—22no—
285C22×D4—22no—
286C22×D4—22no—
287C22×D4—22no—
288C22×D4—22no—
289C22×D4C7×Q814yes—
290C23×C4C23×C712yes—
291C23×C4C23×C712yes—
292C23×C4C23×C712yes—
293C23×C4C23×C712yes—
294C23×C4C23×C711yescharacteristic
295C23×C4C23×C711yescharacteristic
296C23×C4C23×C712yes—
297C23×C4C23×C711yescharacteristic
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—28no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—28no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4C7×D414yes—
314C2×C4⋊C4C7×D418yescomplemented
315C2×C22⋊C4—28no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—28no—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—28no—
322C2×C22⋊C4—28no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42C7×D414yes—
333C2×C42C7×D414yes—
334#95—24no—
335#95—24no—
336#95—28no—
337C22×C7—44no—
338C22×C7—44no—
339C22×C7—48no—
340C22×C7—48no—
341C22×C7—44no—
342C22×C7—82no—
343C22×C7—82no—
344C22×C7—84no—
345C22×C7—84no—
346C22×C7—82no—
347C22×C7—82no—
348C22×C7C2×#12111yescharacteristic
349C22×C7C2×#12112yes—
350C22×C7C2×#12111yescharacteristic
351C22×C7C2×#12111yescharacteristic
352C22×C7C2×C22⋊Q811yescharacteristic
353C22×C7C2×C22⋊Q811yescharacteristic
354C22×C7C2×C22⋊Q811yescharacteristic
355C22×C7C2×#12611yescharacteristic
356C22×C7C2×#12711yescharacteristic
357C22×C7#53211yescharacteristic
358C22×C7#53212yes—
359C22×C7#53511yescharacteristic
360C22×C7#53611yescharacteristic
361C22×C7#53712yes—
362C22×C7#54212yes—
363C22×C7D4212yes—
364C22×C7#54712yes—
365C22×C7#54812yes—
366C22×C7D4×Q812yes—
367C22×C7#55012yes—
368C22×C7#55312yes—
369C22×C7#55412yes—
370C22×C7#55612yes—
371C4×C7—24no—
372C4×C7—28no—
373C4×C7—42no—
374C4×C7—44no—
375C4×C7—44no—
376C4×C7—82no—
377C4×C7—82no—
378C4×C7#39618yescomplemented
379C24—24no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—24no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—24no—
388C24—24no—
389C24C2×C7×Q814yes—
390C24C24×C711yescharacteristic
391C2×D4—28no—
392C2×D4—28no—
393C2×D4—28no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4C2×C7×D412yes—
430C22×C4C2×C7×D414yes—
431C22×C4C2×C7×D414yes—
432C22×C4C2×C7×D414yes—
433C22×C4C2×C7×D414yes—
434C22×C4C2×C7×D412yes—
435C22×C4C2×C7×D414yes—
436C22×C4C2×C7×D412yes—
437C22×C4C2×C7×D412yes—
438C22×C4C2×C7×D412yes—
439C22×C4C2×C7×Q812yes—
440C22×C4C7×#4114yes—
441C22×C4C7×#4112yes—
442C22×C4C7×#4112yes—
443C22×C4C7×#4112yes—
444C4⋊C4—28no—
445C4⋊C4—216no—
446C4⋊C4—416no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—48no—
450C22⋊C4—416no—
451C22⋊C4—48no—
452C22⋊C4—416no—
453C22⋊C4—48no—
454C42—28no—
455C42—28no—
456C42—44no—
457C2×C7—84no—
458C2×C7C2×#39611yescharacteristic
459C2×C7#241612yes—
460C2×C7#245212yes—
461C2×C7#251212yes—
462C2×C7#253812yes—
463C2×C7#254111yescharacteristic
464C2×C7#254211yescharacteristic
465C2×C7#254511yescharacteristic
466C2×C7#256611yescharacteristic
467C2×C7#262011yescharacteristic
468C2×C7#285611yescharacteristic
469C23—24no—
470C23—24no—
471C23—24no—
472C23—28no—
473C23—28no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23C22×C7×D411yescharacteristic
493C23C22×C7×D411yescharacteristic
494C23C22×C7×D411yescharacteristic
495C23C22×C7×D412yes—
496C23C22×C7×Q811yescharacteristic
497C23C2×C7×#4111yescharacteristic
498C23C2×C7×#4111yescharacteristic
499C23C2×C7×#4111yescharacteristic
500C23C2×C7×#4112yes—
501C23C7×H(5,𝔽2)12yes—
502C23C7×2−1+412yes—
503D4—416no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4C7×#12118yes—
542C2×C4C7×#12114yes—
543C2×C4C7×#12114yes—
544C2×C4C7×#12114yes—
545C2×C4C7×C22⋊Q814yes—
546C2×C4C7×C22⋊Q814yes—
547C2×C4C7×C22⋊Q814yes—
548C2×C4C7×#12614yes—
549C2×C4C7×#12714yes—
550C7#1513411yescharacteristic, direct product factor, Sylow
551C22—44no—
552C22—48no—
553C22—48no—
554C22—44no—
555C22—44no—
556C22—84no—
557C22—84no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22C2×C7×#12111yescharacteristic
563C22C2×C7×#12111yescharacteristic
564C22C2×C7×#12111yescharacteristic
565C22C2×C7×#12112yes—
566C22C2×C7×C22⋊Q811yescharacteristic
567C22C2×C7×C22⋊Q811yescharacteristic
568C22C2×C7×C22⋊Q811yescharacteristic
569C22C2×C7×#12611yescharacteristic
570C22C2×C7×#12711yescharacteristic
571C22C7×#53212yes—
572C22C7×#53211yescharacteristic
573C22C7×#53511yescharacteristic
574C22C7×#53611yescharacteristic
575C22C7×#53712yes—
576C22C7×#54212yes—
577C22C7×D4212yes—
578C22C7×#54712yes—
579C22C7×#54812yes—
580C22C7×D4×Q812yes—
581C22C7×#55012yes—
582C22C7×#55312yes—
583C22C7×#55412yes—
584C22C7×#55612yes—
585C4—28no—
586C4—24no—
587C4—44no—
588C4—44no—
589C4—42no—
590C4—82no—
591C4—82no—
592C4C7×#39618yescomplemented
593C2—84no—
594C2C2×C7×#39611yescharacteristic
595C2C7×#241612yes—
596C2C7×#245212yes—
597C2C7×#251212yes—
598C2C7×#253812yes—
599C2C7×#254111yescharacteristic
600C2C7×#254211yescharacteristic
601C2C7×#254511yescharacteristic
602C2C7×#256611yescharacteristic
603C2C7×#262011yescharacteristic
604C2C7×#285611yescharacteristic
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
1. C2×C7×#396C2semidirect
5. C2×C7×#389C2semidirect
6. C2×C7×#384C2semidirect
7. C2×C4×C7×C4⋊C4C2semidirect
8. C7×#1439C2semidirect
9. C7×C4⋊#95C2semidirect
34. #15134C7direct (N × H)
67. C2×C7×C4⋊C4D4semidirect
90. C2×#396C2×C7semidirect
94. C2×#389C2×C7semidirect
95. C2×#384C2×C7semidirect
96. C2×C4×C4⋊C4C2×C7semidirect
97. #1439C2×C7semidirect
98. C4⋊#95C2×C7semidirect
314. C2×C4⋊C4C7×D4semidirect
378. C4×C7#396semidirect
550. C7#15134direct (N × H)
592. C4C7×#396semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#3962
C2×C7×#3902
C2×C7×#3892
C2×C7×#3842
C2×C4×C7×C4⋊C42
C7×#14392
C7×C4⋊#952
#151347