Finite Group Databasebeta

← C7×#23951 #13201167

Subgroups — C7×#23951 #13201167

Order 1792.

1234 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3492C211yesmaximal, characteristic, complemented
2C7×#3005C212yesmaximal, complemented
3C7×#2311C211yesmaximal, characteristic, complemented
4C7×#2028C211yesmaximal, characteristic, complemented
5C7×#2027C211yesmaximal, characteristic, complemented
6C7×#2007C211yesmaximal, characteristic, complemented
7C7×#2006C211yesmaximal, characteristic, complemented
8C7×#1897C212yesmaximal, complemented
9C7×#1883C212yesmaximal, complemented
10C7×#1882C212yesmaximal, complemented
11C7×#1821C211yesmaximal, characteristic, complemented
12C22×C7×#41C2211yescharacteristic
13C2×C7×#137C2212yes—
14C2×C7×AGL(1,ℤ/8)C2212yes—
15C7×#554—22no—
16C7×#553—22no—
17C7×#552—22no—
18C7×#547—22no—
19C7×#544C2211yescharacteristic
20C7×#539—22no—
21C7×#537C2211yescharacteristic
22C7×#525—22no—
23C7×#525C2212yes—
24C7×#520C2212yes—
25C2×C7×#117C2211yescharacteristic
26C22×C7×C4⋊C4C2211yescharacteristic
27C7×#452—22no—
28C7×#452—22no—
29C7×#449—22no—
30C7×#449—22no—
31C2×C7×#104C2212yes—
32C2×C7×#104C2212yes—
33C7×#419C2212yes—
34C7×#418C2212yes—
35C7×#407C2212yes—
36C7×#391C2211yescharacteristic
37C7×#391C2211yescharacteristic
38C7×#390C2211yescharacteristic
39C7×#390C2211yescharacteristic
40C7×#386—22no—
41C7×#380—22no—
42C7×#343C2211yescharacteristic
43C7×#343C2211yescharacteristic
44C7×#343C2212yes—
45C7×#339C2212yes—
46C7×#339C2212yes—
47#23951C711yesmaximal, characteristic, direct product factor, Sylow
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41—22no—
51C2×C7×#41—22no—
52C2×C7×#41—22no—
53C2×C7×#41—42no—
54C2×C7×#41C2312yes—
55C22×C7×Q8C2311yescharacteristic
56C22×C7×D4—21no—
57C22×C7×D4C2311yescharacteristic
58C23×C4×C7—21no—
59C23×C4×C7D412yes—
60C23×C4×C7C2311yescharacteristic
61C7×#137—24no—
62C7×#137—42no—
63C7×AGL(1,ℤ/8)—24no—
64C7×AGL(1,ℤ/8)—42no—
65C2×C7×Q16—22no—
66C2×C7×SD16—22no—
67C2×C7×SD16—22no—
68C2×C7×D8—22no—
69C2×C7×M16C2312yes—
70C2×C7×M16C2312yes—
71C7×C4⋊Q8—22no—
72C7×#126—24no—
73C7×#125—24no—
74C7×#125—22no—
75C7×#123—24no—
76C7×#123—24no—
77C7×#123—22no—
78C7×C22⋊Q8—22no—
79C7×C22⋊Q8—24no—
80C7×C22⋊Q8—22no—
81C7×C22⋊Q8—22no—
82C7×C22⋊Q8D412yes—
83C7×C22⋊Q8D412yes—
84C7×#121—22no—
85C7×#121—22no—
86C7×#121—22no—
87C7×#121D412yes—
88C7×#121D412yes—
89C7×#120—22no—
90C4×C7×Q8—22no—
91C4×C7×D4—24no—
92C4×C7×D4—22no—
93C4×C7×D4—24no—
94C4×C7×D4—22no—
95C4×C7×D4—22no—
96C7×#117—22no—
97C7×#117—22no—
98C7×#117—42no—
99C7×#117C2312yes—
100C2×C7×C4⋊C4—22no—
101C2×C7×C4⋊C4—22no—
102C2×C7×C4⋊C4—22no—
103C2×C7×C4⋊C4—21no—
104C2×C7×C4⋊C4D412yes—
105C2×C7×C4⋊C4C2311yescharacteristic
106C2×C7×C4⋊C4C2311yescharacteristic
107C2×C7×C22⋊C4—21no—
108C2×C7×C22⋊C4—22no—
109C2×C7×C22⋊C4—22no—
110C2×C42×C7C2311yescharacteristic
111C2×C42×C7C2311yescharacteristic
112C7×#104—42no—
113C7×#104—42no—
114C7×#103—22no—
115C7×#103—22no—
116C7×#102—22no—
117C7×#102—22no—
118C7×C22⋊C8—22no—
119C7×C22⋊C8—22no—
120C7×#95—22no—
121C7×#95—22no—
122#3492C2×C711yescharacteristic, complemented
123#3005C2×C712yescomplemented
124#2311C2×C711yescharacteristic, complemented
125#2028C2×C711yescharacteristic, complemented
126#2027C2×C711yescharacteristic, complemented
127#2007C2×C711yescharacteristic, complemented
128#2006C2×C711yescharacteristic, complemented
129#1897C2×C712yescomplemented
130#1883C2×C712yescomplemented
131#1882C2×C712yescomplemented
132#1821C2×C711yescharacteristic, complemented
133C24×C7—21no—
134C24×C7C2×D411yescharacteristic
135C7×#41—22no—
136C7×#41—22no—
137C7×#41—22no—
138C7×#41—22no—
139C7×#41—44no—
140C7×#41—44no—
141C7×#41—44no—
142C7×#41—44no—
143C7×#41—84no—
144C2×C7×Q8—22no—
145C2×C7×Q8—21no—
146C2×C7×Q8—22no—
147C2×C7×Q8—22no—
148C2×C7×Q8C2×D412yes—
149C2×C7×Q8#4112yescomplemented
150C2×C7×D4—24no—
151C2×C7×D4—22no—
152C2×C7×D4—22no—
153C2×C7×D4—22no—
154C2×C7×D4—22no—
155C2×C7×D4—22no—
156C2×C7×D4—21no—
157C2×C7×D4—22no—
158C2×C7×D4—42no—
159C2×C7×D4—41no—
160C2×C7×D4—41no—
161C2×C7×D4C2×D412yes—
162C2×C7×D4#4112yescomplemented
163C22×C4×C7—21no—
164C22×C4×C7—21no—
165C22×C4×C7—22no—
166C22×C4×C7—24no—
167C22×C4×C7—22no—
168C22×C4×C7—21no—
169C22×C4×C7—24no—
170C22×C4×C7—22no—
171C22×C4×C7—22no—
172C22×C4×C7—21no—
173C22×C4×C7—21no—
174C22×C4×C7—42no—
175C22×C4×C7—42no—
176C22×C4×C7—82no—
177C22×C4×C7C2×D411yescharacteristic
178C22×C4×C7C2×D412yes—
179C22×C4×C7C2×D412yes—
180C22×C4×C7C2×D412yes—
181C22×C4×C7C2×D412yes—
182C22×C4×C7C2411yescharacteristic
183C7×Q16—44no—
184C7×SD16—44no—
185C7×SD16—44no—
186C7×D8—44no—
187C7×M16—22no—
188C7×M16—22no—
189C7×M16—22no—
190C7×M16—22no—
191C2×C7×C8—22no—
192C2×C7×C8—22no—
193C7×C4⋊C4—24no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—24no—
197C7×C4⋊C4—24no—
198C7×C4⋊C4—22no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—22no—
201C7×C4⋊C4—22no—
202C7×C4⋊C4—42no—
203C7×C4⋊C4C2×D412yes—
204C7×C4⋊C4C2×D412yes—
205C7×C22⋊C4—24no—
206C7×C22⋊C4—24no—
207C7×C22⋊C4—22no—
208C7×C22⋊C4—22no—
209C7×C22⋊C4—22no—
210C7×C22⋊C4—22no—
211C7×C22⋊C4—22no—
212C7×C22⋊C4—42no—
213C7×C22⋊C4—82no—
214C7×C22⋊C4—82no—
215C42×C7—24no—
216C42×C7—22no—
217C42×C7—22no—
218C42×C7—22no—
219C22×#41C22×C711yescharacteristic
220C2×#137C22×C712yes—
221C2×AGL(1,ℤ/8)C22×C712yes—
222#554—22no—
223#553—22no—
224#552—22no—
225#547—22no—
226#544C22×C711yescharacteristic
227#539—22no—
228#537C22×C711yescharacteristic
229#525—22no—
230#525C22×C712yes—
231#520C22×C712yes—
232C2×#117C22×C711yescharacteristic
233C22×C4⋊C4C22×C711yescharacteristic
234#452—22no—
235#452—22no—
236#449—22no—
237#449—22no—
238C2×#104C22×C712yes—
239C2×#104C22×C712yes—
240#419C22×C712yes—
241#418C22×C712yes—
242#407C22×C712yes—
243#391C22×C711yescharacteristic
244#391C22×C711yescharacteristic
245#390C22×C711yescharacteristic
246#390C22×C711yescharacteristic
247#386—22no—
248#380—22no—
249#343C22×C712yes—
250#343C22×C711yescharacteristic
251#343C22×C711yescharacteristic
252#339C22×C712yes—
253#339C22×C712yes—
254C23×C7—21no—
255C23×C7—21no—
256C23×C7—22no—
257C23×C7—22no—
258C23×C7—41no—
259C23×C7—41no—
260C23×C7—42no—
261C23×C7—42no—
262C23×C7—41no—
263C23×C7—41no—
264C23×C7#12011yescharacteristic
265C23×C7#12011yescharacteristic
266C23×C7C22×D411yescharacteristic
267C7×Q8—22no—
268C7×Q8—24no—
269C7×Q8—22no—
270C7×Q8—42no—
271C7×Q8—82no—
272C7×D4—22no—
273C7×D4—24no—
274C7×D4—22no—
275C7×D4—44no—
276C7×D4—44no—
277C7×D4—44no—
278C7×D4—44no—
279C7×D4—42no—
280C7×D4—42no—
281C7×D4—42no—
282C7×D4—82no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
287C2×C4×C7—22no—
288C2×C4×C7—22no—
289C2×C4×C7—22no—
290C2×C4×C7—22no—
291C2×C4×C7—22no—
292C2×C4×C7—22no—
293C2×C4×C7—22no—
294C2×C4×C7—22no—
295C2×C4×C7—48no—
296C2×C4×C7—44no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—42no—
300C2×C4×C7—42no—
301C2×C4×C7—42no—
302C2×C4×C7—42no—
303C2×C4×C7—41no—
304C2×C4×C7—41no—
305C2×C4×C7—41no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
308C2×C4×C7—41no—
309C2×C4×C7—82no—
310C2×C4×C7—84no—
311C2×C4×C7#12012yes—
312C2×C4×C7C22×D412yes—
313C2×C4×C7C2×#4111yescharacteristic
314C2×C4×C7C2×#4111yescharacteristic
315C2×C4×C7H(5,𝔽2)12yes—
316C7×C8—42no—
317C7×C8—42no—
318C2×#41—22no—
319C2×#41—22no—
320C2×#41—22no—
321C2×#41—22no—
322C2×#41—22no—
323C2×#41—42no—
324C2×#41C23×C712yes—
325C22×Q8C23×C711yescharacteristic
326C22×D4—21no—
327C22×D4C23×C711yescharacteristic
328C23×C4—21no—
329C23×C4C7×D412yes—
330C23×C4C23×C711yescharacteristic
331#137—24no—
332#137—42no—
333AGL(1,ℤ/8)—24no—
334AGL(1,ℤ/8)—42no—
335C2×Q16—22no—
336C2×SD16—22no—
337C2×SD16—22no—
338C2×D8—22no—
339C2×M16C23×C712yes—
340C2×M16C23×C712yes—
341C4⋊Q8—22no—
342#126—24no—
343#125—24no—
344#125—22no—
345#123—24no—
346#123—24no—
347#123—22no—
348C22⋊Q8—22no—
349C22⋊Q8—24no—
350C22⋊Q8—22no—
351C22⋊Q8—22no—
352C22⋊Q8C7×D412yes—
353C22⋊Q8C7×D412yes—
354#121—22no—
355#121—22no—
356#121—22no—
357#121C7×D412yes—
358#121C7×D412yes—
359#120—22no—
360C4×Q8—22no—
361C4×D4—24no—
362C4×D4—22no—
363C4×D4—24no—
364C4×D4—22no—
365C4×D4—22no—
366#117—22no—
367#117—22no—
368#117—42no—
369#117C23×C712yes—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4C7×D412yes—
375C2×C4⋊C4C23×C711yescharacteristic
376C2×C4⋊C4C23×C711yescharacteristic
377C2×C22⋊C4—21no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C42C23×C711yescharacteristic
381C2×C42C23×C711yescharacteristic
382#104—42no—
383#104—42no—
384#103—22no—
385#103—22no—
386#102—22no—
387#102—22no—
388C22⋊C8—22no—
389C22⋊C8—22no—
390#95—22no—
391#95—22no—
392C22×C7—21no—
393C22×C7—21no—
394C22×C7—21no—
395C22×C7—21no—
396C22×C7—21no—
397C22×C7—21no—
398C22×C7—21no—
399C22×C7—21no—
400C22×C7—42no—
401C22×C7—42no—
402C22×C7—44no—
403C22×C7—42no—
404C22×C7—42no—
405C22×C7—42no—
406C22×C7—42no—
407C22×C7—82no—
408C22×C7—81no—
409C22×C7—81no—
410C22×C7C2×#12011yescharacteristic
411C22×C7#52511yescharacteristic
412C22×C7#53511yescharacteristic
413C4×C7—22no—
414C4×C7—42no—
415C4×C7—42no—
416C4×C7—42no—
417C4×C7—42no—
418C4×C7—42no—
419C4×C7—42no—
420C4×C7—81no—
421C4×C7—81no—
422C4×C7#54612yes—
423C4×C7#54612yes—
424C24—21no—
425C24C2×C7×D411yescharacteristic
426#41—22no—
427#41—22no—
428#41—22no—
429#41—22no—
430#41—44no—
431#41—44no—
432#41—44no—
433#41—44no—
434#41—84no—
435C2×Q8—22no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8C2×C7×D412yes—
440C2×Q8C7×#4112yescomplemented
441C2×D4—24no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—21no—
448C2×D4—22no—
449C2×D4—42no—
450C2×D4—41no—
451C2×D4—41no—
452C2×D4C2×C7×D412yes—
453C2×D4C7×#4112yescomplemented
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—82no—
468C22×C4C2×C7×D411yescharacteristic
469C22×C4C2×C7×D412yes—
470C22×C4C2×C7×D412yes—
471C22×C4C2×C7×D412yes—
472C22×C4C2×C7×D412yes—
473C22×C4C24×C711yescharacteristic
474Q16—44no—
475SD16—44no—
476SD16—44no—
477D8—44no—
478M16—22no—
479M16—22no—
480M16—22no—
481M16—22no—
482C2×C8—22no—
483C2×C8—22no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—42no—
494C4⋊C4C2×C7×D412yes—
495C4⋊C4C2×C7×D412yes—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—42no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C42—24no—
507C42—22no—
508C42—22no—
509C42—22no—
510C2×C7—21no—
511C2×C7—21no—
512C2×C7—41no—
513C2×C7—41no—
514C2×C7—82no—
515C2×C7—82no—
516C2×C7#240411yescharacteristic
517C2×C7#301911yescharacteristic
518C2×C7#302011yescharacteristic
519C23—21no—
520C23—21no—
521C23—22no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—41no—
528C23—41no—
529C23C7×#12011yescharacteristic
530C23C7×#12011yescharacteristic
531C23C22×C7×D411yescharacteristic
532Q8—22no—
533Q8—24no—
534Q8—22no—
535Q8—42no—
536Q8—82no—
537D4—22no—
538D4—24no—
539D4—22no—
540D4—44no—
541D4—44no—
542D4—44no—
543D4—44no—
544D4—42no—
545D4—42no—
546D4—42no—
547D4—82no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—48no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—82no—
575C2×C4—84no—
576C2×C4C7×#12012yes—
577C2×C4C22×C7×D412yes—
578C2×C4C2×C7×#4111yescharacteristic
579C2×C4C2×C7×#4111yescharacteristic
580C2×C4C7×H(5,𝔽2)12yes—
581C8—42no—
582C8—42no—
583C7#2395111yescharacteristic, direct product factor, Sylow
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—42no—
593C22—42no—
594C22—44no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—82no—
600C22—81no—
601C22—81no—
602C22C2×C7×#12011yescharacteristic
603C22C7×#52511yescharacteristic
604C22C7×#53511yescharacteristic
605C4—22no—
606C4—42no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—81no—
613C4—81no—
614C4C7×#54612yes—
615C4C7×#54612yes—
616C2—21no—
617C2—21no—
618C2—41no—
619C2—41no—
620C2—82no—
621C2—82no—
622C2C7×#240411yescharacteristic
623C2C7×#301911yescharacteristic
624C2C7×#302011yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3492C2semidirect
2. C7×#3005C2semidirect
3. C7×#2311C2semidirect
4. C7×#2028C2semidirect
5. C7×#2027C2semidirect
6. C7×#2007C2semidirect
7. C7×#2006C2semidirect
8. C7×#1897C2semidirect
9. C7×#1883C2semidirect
10. C7×#1882C2semidirect
11. C7×#1821C2semidirect
47. #23951C7direct (N × H)
122. #3492C2×C7semidirect
123. #3005C2×C7semidirect
124. #2311C2×C7semidirect
125. #2028C2×C7semidirect
126. #2027C2×C7semidirect
127. #2007C2×C7semidirect
128. #2006C2×C7semidirect
129. #1897C2×C7semidirect
130. #1883C2×C7semidirect
131. #1882C2×C7semidirect
132. #1821C2×C7semidirect
149. C2×C7×Q8#41semidirect
162. C2×C7×D4#41semidirect
440. C2×Q8C7×#41semidirect
453. C2×D4C7×#41semidirect
583. C7#23951direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34922
C7×#30052
C7×#23112
C7×#20282
C7×#20272
C7×#20072
C7×#20062
C7×#18972
C7×#18832
C7×#18822
C7×#18212
#239517