Finite Group Databasebeta

← C7×#23394 #13200610

Subgroups — C7×#23394 #13200610

Order 1792.

2014 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#137C211yesmaximal, characteristic, complemented
2C2×C7×#537C211yesmaximal, characteristic
3C2×C7×#536C211yesmaximal, characteristic, complemented
4C2×C7×#416C211yesmaximal, characteristic, complemented
5C22×C7×#103C211yesmaximal, characteristic, complemented
6C2×C7×#408C211yesmaximal, characteristic, complemented
7C7×#2001C214yesmaximal, complemented
8C7×#2000C214yesmaximal, complemented
9C7×#1736C211yesmaximal, characteristic, complemented
10C22×C7×#41C2211yescharacteristic
11C22×C7×#41C2211yescharacteristic
12C23×C7×Q8C2211yescharacteristic
13C2×C7×#137—24no—
14C2×C7×#137—24no—
15C2×C7×#137—24no—
16C22×C7×Q16C2212yes—
17C22×C7×SD16C2212yes—
18C22×C7×M16C2211yescharacteristic
19C23×C7×C8C2211yescharacteristic
20C7×#537—24no—
21C7×#537—24no—
22C7×#536—24no—
23C7×#536—24no—
24C2×C7×C4⋊Q8—21no—
25C2×C7×#124—21no—
26C2×C7×#123—22no—
27C2×C7×C22⋊Q8—21no—
28C2×C7×C22⋊Q8C2214yes—
29C2×C7×C22⋊Q8C2212yes—
30C2×C7×#121—21no—
31C2×C7×#121—22no—
32C2×C7×#121C2212yes—
33C2×C7×#117C2211yescharacteristic
34C22×C7×C4⋊C4C2211yescharacteristic
35C7×#416—24no—
36C2×C7×#103—24no—
37C2×C7×#103C2214yes—
38C2×C7×#103C2212yes—
39C2×C7×#102C2212yes—
40C7×#408—24no—
41C2×C7×C22⋊C8C2214yes—
42C7×#340C2214yes—
43#23394C711yesmaximal, characteristic, direct product factor, Sylow
44C2×C7×#41—22no—
45C2×C7×#41—22no—
46C2×C7×#41—22no—
47C2×C7×#41—22no—
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41—22no—
51C2×C7×#41—22no—
52C2×C7×#41—22no—
53C2×C7×#41—22no—
54C2×C7×#41—22no—
55C2×C7×#41D414yes—
56C22×C7×Q8—21no—
57C22×C7×Q8—22no—
58C22×C7×Q8—22no—
59C22×C7×Q8—22no—
60C22×C7×Q8—22no—
61C22×C7×Q8—22no—
62C22×C7×Q8D411yescharacteristic
63C22×C7×Q8D414yescomplemented
64C22×C7×Q8C2312yes—
65C22×C7×Q8C2311yescharacteristic
66C22×C7×D4—21no—
67C22×C7×D4—22no—
68C22×C7×D4D412yes—
69C22×C7×D4D411yescharacteristic
70C22×C7×D4C2311yescharacteristic
71C23×C4×C7—21no—
72C23×C4×C7—21no—
73C23×C4×C7—21no—
74C23×C4×C7C2312yes—
75C23×C4×C7C2311yescharacteristic
76C7×#137—232no—
77C2×C7×Q16—24no—
78C2×C7×Q16—24no—
79C2×C7×Q16—24no—
80C2×C7×SD16—24no—
81C2×C7×SD16—24no—
82C2×C7×SD16—24no—
83C2×C7×M16—22no—
84C2×C7×M16—22no—
85C2×C7×M16D414yescomplemented
86C22×C7×C8—22no—
87C22×C7×C8—22no—
88C22×C7×C8—22no—
89C22×C7×C8C2312yes—
90C22×C7×C8C2312yes—
91C7×C4⋊Q8—28no—
92C7×#124—28no—
93C7×#123—216no—
94C7×C22⋊Q8—28no—
95C7×C22⋊Q8—28no—
96C7×C22⋊Q8—24no—
97C7×C22⋊Q8—28no—
98C7×C22⋊Q8—24no—
99C7×#121—216no—
100C7×#121—28no—
101C7×#121—24no—
102C7×#121—24no—
103C7×#117—22no—
104C7×#117D414yescomplemented
105C2×C7×C4⋊C4—21no—
106C2×C7×C4⋊C4—22no—
107C2×C7×C4⋊C4—22no—
108C2×C7×C4⋊C4—24no—
109C2×C7×C4⋊C4D414yes—
110C2×C7×C4⋊C4C2312yes—
111C2×C7×C4⋊C4C2312yes—
112C2×C7×C22⋊C4—21no—
113C2×C7×C22⋊C4—22no—
114C2×C7×C22⋊C4—24no—
115C2×C7×C22⋊C4—22no—
116C2×C42×C7—21no—
117C7×#103—28no—
118C7×#103—24no—
119C7×#102—24no—
120C7×C22⋊C8—28no—
121C22×#137C2×C711yescharacteristic, complemented
122C2×#537C2×C711yescharacteristic
123C2×#536C2×C711yescharacteristic, complemented
124C2×#416C2×C711yescharacteristic, complemented
125C22×#103C2×C711yescharacteristic, complemented
126C2×#408C2×C711yescharacteristic, complemented
127#2001C2×C714yescomplemented
128#2000C2×C714yescomplemented
129#1736C2×C711yescharacteristic, complemented
130C24×C7—21no—
131C24×C7C2×D412yes—
132C24×C7C2×D411yescharacteristic
133C7×#41—28no—
134C7×#41—28no—
135C7×#41—44no—
136C7×#41—44no—
137C7×#41—44no—
138C7×#41—44no—
139C7×#41—44no—
140C7×#41—44no—
141C2×C7×Q8—22no—
142C2×C7×Q8—24no—
143C2×C7×Q8—22no—
144C2×C7×Q8—22no—
145C2×C7×Q8—22no—
146C2×C7×Q8—21no—
147C2×C7×Q8—28no—
148C2×C7×Q8—28no—
149C2×C7×Q8—22no—
150C2×C7×Q8—21no—
151C2×C7×Q8—22no—
152C2×C7×Q8—22no—
153C2×C7×Q8—21no—
154C2×C7×Q8—21no—
155C2×C7×Q8—22no—
156C2×C7×Q8—21no—
157C2×C7×Q8—42no—
158C2×C7×Q8—42no—
159C2×C7×Q8—44no—
160C2×C7×Q8—44no—
161C2×C7×Q8C2×D414yes—
162C2×C7×Q8C2×D412yes—
163C2×C7×D4—24no—
164C2×C7×D4—24no—
165C2×C7×D4—24no—
166C2×C7×D4—24no—
167C2×C7×D4—28no—
168C2×C7×D4—22no—
169C2×C7×D4—22no—
170C2×C7×D4—22no—
171C2×C7×D4—21no—
172C2×C7×D4—21no—
173C2×C7×D4—21no—
174C2×C7×D4—21no—
175C2×C7×D4—21no—
176C2×C7×D4—42no—
177C2×C7×D4—42no—
178C2×C7×D4—84no—
179C2×C7×D4C2×D412yes—
180C22×C4×C7—24no—
181C22×C4×C7—22no—
182C22×C4×C7—22no—
183C22×C4×C7—21no—
184C22×C4×C7—22no—
185C22×C4×C7—24no—
186C22×C4×C7—24no—
187C22×C4×C7—22no—
188C22×C4×C7—22no—
189C22×C4×C7—22no—
190C22×C4×C7—22no—
191C22×C4×C7—22no—
192C22×C4×C7—22no—
193C22×C4×C7—42no—
194C22×C4×C7—42no—
195C22×C4×C7—42no—
196C22×C4×C7—42no—
197C22×C4×C7—82no—
198C22×C4×C7C2×D412yes—
199C22×C4×C7C2×D411yescharacteristic
200C22×C4×C7C2×D412yes—
201C22×C4×C7C2×D412yes—
202C22×C4×C7C2×D412yes—
203C22×C4×C7C2×D412yes—
204C22×C4×C7C2×D412yes—
205C22×C4×C7#4112yes—
206C22×C4×C7C2411yescharacteristic
207C7×Q16—216no—
208C7×SD16—216no—
209C7×M16—28no—
210C2×C7×C8—22no—
211C2×C7×C8—22no—
212C2×C7×C8—22no—
213C2×C7×C8—22no—
214C2×C7×C8—22no—
215C2×C7×C8—44no—
216C2×C7×C8C2×D414yes—
217C7×C4⋊C4—28no—
218C7×C4⋊C4—216no—
219C7×C4⋊C4—24no—
220C7×C4⋊C4—22no—
221C7×C4⋊C4—22no—
222C7×C4⋊C4C2×D414yes—
223C7×C4⋊C4C2×D414yes—
224C7×C22⋊C4—28no—
225C7×C22⋊C4—28no—
226C7×C22⋊C4—216no—
227C7×C22⋊C4—24no—
228C42×C7—24no—
229C22×#41C22×C711yescharacteristic
230C22×#41C22×C711yescharacteristic
231C23×Q8C22×C711yescharacteristic
232C2×#137—24no—
233C2×#137—24no—
234C2×#137—24no—
235C22×Q16C22×C712yes—
236C22×SD16C22×C712yes—
237C22×M16C22×C711yescharacteristic
238C23×C8C22×C711yescharacteristic
239#537—24no—
240#537—24no—
241#536—24no—
242#536—24no—
243C2×C4⋊Q8—21no—
244C2×#124—21no—
245C2×#123—22no—
246C2×C22⋊Q8—21no—
247C2×C22⋊Q8C22×C714yes—
248C2×C22⋊Q8C22×C712yes—
249C2×#121—21no—
250C2×#121—22no—
251C2×#121C22×C712yes—
252C2×#117C22×C711yescharacteristic
253C22×C4⋊C4C22×C711yescharacteristic
254#416—24no—
255C2×#103—24no—
256C2×#103C22×C714yes—
257C2×#103C22×C712yes—
258C2×#102C22×C712yes—
259#408—24no—
260C2×C22⋊C8C22×C714yes—
261#340C22×C714yes—
262C23×C7—22no—
263C23×C7—22no—
264C23×C7—22no—
265C23×C7—22no—
266C23×C7—22no—
267C23×C7—22no—
268C23×C7—24no—
269C23×C7—24no—
270C23×C7—41no—
271C23×C7—41no—
272C23×C7—42no—
273C23×C7—42no—
274C23×C7#12012yes—
275C23×C7#12012yes—
276C23×C7#12112yes—
277C23×C7C22×D411yescharacteristic
278C7×Q8—24no—
279C7×Q8—22no—
280C7×Q8—24no—
281C7×Q8—22no—
282C7×Q8—42no—
283C7×Q8—41no—
284C7×Q8—42no—
285C7×Q8—41no—
286C7×Q8—48no—
287C7×Q8—44no—
288C7×D4—22no—
289C7×D4—22no—
290C7×D4—48no—
291C7×D4—48no—
292C7×D4—41no—
293C7×D4—41no—
294C7×D4—44no—
295C7×D4—88no—
296C2×C4×C7—24no—
297C2×C4×C7—24no—
298C2×C4×C7—22no—
299C2×C4×C7—24no—
300C2×C4×C7—22no—
301C2×C4×C7—22no—
302C2×C4×C7—22no—
303C2×C4×C7—41no—
304C2×C4×C7—42no—
305C2×C4×C7—41no—
306C2×C4×C7—42no—
307C2×C4×C7—42no—
308C2×C4×C7—44no—
309C2×C4×C7—42no—
310C2×C4×C7—48no—
311C2×C4×C7—42no—
312C2×C4×C7—42no—
313C2×C4×C7—42no—
314C2×C4×C7—42no—
315C2×C4×C7—48no—
316C2×C4×C7—84no—
317C2×C4×C7#12014yes—
318C2×C4×C7#12012yes—
319C2×C4×C7#12012yes—
320C2×C4×C7#12114yes—
321C2×C4×C7#12114yes—
322C2×C4×C7#12112yes—
323C2×C4×C7#12714yes—
324C2×C4×C7C22×D411yescharacteristic
325C2×C4×C7C22×D411yescharacteristic
326C2×C4×C7C22×D411yescharacteristic
327C2×C4×C7C22×D411yescharacteristic
328C2×C4×C7C22×D411yescharacteristic
329C2×C4×C7C2×#4111yescharacteristic
330C7×C8—24no—
331C7×C8—42no—
332C2×#41—22no—
333C2×#41—22no—
334C2×#41—22no—
335C2×#41—22no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41—22no—
339C2×#41—22no—
340C2×#41—22no—
341C2×#41—22no—
342C2×#41—22no—
343C2×#41C7×D414yes—
344C22×Q8—21no—
345C22×Q8—22no—
346C22×Q8—22no—
347C22×Q8—22no—
348C22×Q8—22no—
349C22×Q8—22no—
350C22×Q8C7×D411yescharacteristic
351C22×Q8C7×D414yescomplemented
352C22×Q8C23×C712yes—
353C22×Q8C23×C711yescharacteristic
354C22×D4—21no—
355C22×D4—22no—
356C22×D4C7×D412yes—
357C22×D4C7×D411yescharacteristic
358C22×D4C23×C711yescharacteristic
359C23×C4—21no—
360C23×C4—21no—
361C23×C4—21no—
362C23×C4C23×C712yes—
363C23×C4C23×C711yescharacteristic
364#137—232no—
365C2×Q16—24no—
366C2×Q16—24no—
367C2×Q16—24no—
368C2×SD16—24no—
369C2×SD16—24no—
370C2×SD16—24no—
371C2×M16—22no—
372C2×M16—22no—
373C2×M16C7×D414yescomplemented
374C22×C8—22no—
375C22×C8—22no—
376C22×C8—22no—
377C22×C8C23×C712yes—
378C22×C8C23×C712yes—
379C4⋊Q8—28no—
380#124—28no—
381#123—216no—
382C22⋊Q8—28no—
383C22⋊Q8—28no—
384C22⋊Q8—24no—
385C22⋊Q8—28no—
386C22⋊Q8—24no—
387#121—216no—
388#121—28no—
389#121—24no—
390#121—24no—
391#117—22no—
392#117C7×D414yescomplemented
393C2×C4⋊C4—21no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—24no—
397C2×C4⋊C4C7×D414yes—
398C2×C4⋊C4C23×C712yes—
399C2×C4⋊C4C23×C712yes—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—24no—
403C2×C22⋊C4—22no—
404C2×C42—21no—
405#103—28no—
406#103—24no—
407#102—24no—
408C22⋊C8—28no—
409C22×C7—22no—
410C22×C7—22no—
411C22×C7—24no—
412C22×C7—24no—
413C22×C7—42no—
414C22×C7—42no—
415C22×C7—42no—
416C22×C7—42no—
417C22×C7—42no—
418C22×C7—42no—
419C22×C7—44no—
420C22×C7—44no—
421C22×C7—82no—
422C22×C7—82no—
423C22×C7#39214yes—
424C22×C7C2×#12011yescharacteristic
425C22×C7C2×#12011yescharacteristic
426C22×C7C2×#12111yescharacteristic
427C22×C7#57712yes—
428C22×C7#57812yes—
429C4×C7—42no—
430C4×C7—42no—
431C4×C7—44no—
432C4×C7—44no—
433C4×C7—44no—
434C4×C7#39214yes—
435C4×C7C2×#12011yescharacteristic
436C4×C7C2×#12111yescharacteristic
437C4×C7C2×#12111yescharacteristic
438C4×C7C2×#12711yescharacteristic
439C24—21no—
440C24C2×C7×D412yes—
441C24C2×C7×D411yescharacteristic
442#41—28no—
443#41—28no—
444#41—44no—
445#41—44no—
446#41—44no—
447#41—44no—
448#41—44no—
449#41—44no—
450C2×Q8—22no—
451C2×Q8—24no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—21no—
456C2×Q8—28no—
457C2×Q8—28no—
458C2×Q8—22no—
459C2×Q8—21no—
460C2×Q8—22no—
461C2×Q8—22no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—22no—
465C2×Q8—21no—
466C2×Q8—42no—
467C2×Q8—42no—
468C2×Q8—44no—
469C2×Q8—44no—
470C2×Q8C2×C7×D414yes—
471C2×Q8C2×C7×D412yes—
472C2×D4—24no—
473C2×D4—24no—
474C2×D4—24no—
475C2×D4—24no—
476C2×D4—28no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—42no—
486C2×D4—42no—
487C2×D4—84no—
488C2×D4C2×C7×D412yes—
489C22×C4—24no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4—22no—
494C22×C4—24no—
495C22×C4—24no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—82no—
507C22×C4C2×C7×D412yes—
508C22×C4C2×C7×D411yescharacteristic
509C22×C4C2×C7×D412yes—
510C22×C4C2×C7×D412yes—
511C22×C4C2×C7×D412yes—
512C22×C4C2×C7×D412yes—
513C22×C4C2×C7×D412yes—
514C22×C4C7×#4112yes—
515C22×C4C24×C711yescharacteristic
516Q16—216no—
517SD16—216no—
518M16—28no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—44no—
525C2×C8C2×C7×D414yes—
526C4⋊C4—28no—
527C4⋊C4—216no—
528C4⋊C4—24no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4C2×C7×D414yes—
532C4⋊C4C2×C7×D414yes—
533C22⋊C4—28no—
534C22⋊C4—28no—
535C22⋊C4—216no—
536C22⋊C4—24no—
537C42—24no—
538C2×C7—24no—
539C2×C7—82no—
540C2×C7—82no—
541C2×C7C2×#39211yescharacteristic
542C2×C7#300911yescharacteristic
543C2×C7#301111yescharacteristic
544C2×C7#301312yes—
545C2×C7#306312yes—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—24no—
553C23—24no—
554C23—41no—
555C23—41no—
556C23—42no—
557C23—42no—
558C23C7×#12012yes—
559C23C7×#12012yes—
560C23C7×#12112yes—
561C23C22×C7×D411yescharacteristic
562Q8—24no—
563Q8—22no—
564Q8—24no—
565Q8—22no—
566Q8—42no—
567Q8—41no—
568Q8—48no—
569Q8—44no—
570Q8—42no—
571Q8—41no—
572D4—22no—
573D4—22no—
574D4—48no—
575D4—48no—
576D4—41no—
577D4—44no—
578D4—41no—
579D4—88no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—48no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—48no—
599C2×C4—42no—
600C2×C4—84no—
601C2×C4C7×#12014yes—
602C2×C4C7×#12012yes—
603C2×C4C7×#12012yes—
604C2×C4C7×#12114yes—
605C2×C4C7×#12114yes—
606C2×C4C7×#12112yes—
607C2×C4C7×#12714yes—
608C2×C4C22×C7×D411yescharacteristic
609C2×C4C22×C7×D411yescharacteristic
610C2×C4C22×C7×D411yescharacteristic
611C2×C4C22×C7×D411yescharacteristic
612C2×C4C22×C7×D411yescharacteristic
613C2×C4C2×C7×#4111yescharacteristic
614C8—24no—
615C8—42no—
616C7#2339411yescharacteristic, direct product factor, Sylow
617C22—22no—
618C22—22no—
619C22—24no—
620C22—24no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—44no—
626C22—44no—
627C22—42no—
628C22—42no—
629C22—82no—
630C22—82no—
631C22C7×#39214yes—
632C22C2×C7×#12011yescharacteristic
633C22C2×C7×#12011yescharacteristic
634C22C2×C7×#12111yescharacteristic
635C22C7×#57712yes—
636C22C7×#57812yes—
637C4—44no—
638C4—44no—
639C4—42no—
640C4—42no—
641C4—44no—
642C4C7×#39214yes—
643C4C2×C7×#12011yescharacteristic
644C4C2×C7×#12111yescharacteristic
645C4C2×C7×#12111yescharacteristic
646C4C2×C7×#12711yescharacteristic
647C2—24no—
648C2—82no—
649C2—82no—
650C2C2×C7×#39211yescharacteristic
651C2C7×#300911yescharacteristic
652C2C7×#301111yescharacteristic
653C2C7×#301312yes—
654C2C7×#306312yes—
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#537C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C22×C7×#41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C7×#137C2semidirect
3. C2×C7×#536C2semidirect
4. C2×C7×#416C2semidirect
5. C22×C7×#103C2semidirect
6. C2×C7×#408C2semidirect
7. C7×#2001C2semidirect
8. C7×#2000C2semidirect
9. C7×#1736C2semidirect
43. #23394C7direct (N × H)
63. C22×C7×Q8D4semidirect
85. C2×C7×M16D4semidirect
104. C7×#117D4semidirect
121. C22×#137C2×C7semidirect
123. C2×#536C2×C7semidirect
124. C2×#416C2×C7semidirect
125. C22×#103C2×C7semidirect
126. C2×#408C2×C7semidirect
127. #2001C2×C7semidirect
128. #2000C2×C7semidirect
129. #1736C2×C7semidirect
351. C22×Q8C7×D4semidirect
373. C2×M16C7×D4semidirect
392. #117C7×D4semidirect
616. C7#23394direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×#1372
C2×C7×#5372
C2×C7×#5362
C2×C7×#4162
C22×C7×#1032
C2×C7×#4082
C7×#20012
C7×#20002
C7×#17362
#233947