Finite Group Databasebeta

← C7×#9039 #13186255

Subgroups — C7×#9039 #13186255

Order 1792.

834 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×#448C211yesmaximal, characteristic
2C7×#2325C211yesmaximal, characteristic
3C7×#1787C211yesmaximal, characteristic, complemented
4C2×C7×#327C211yesmaximal, characteristic
5C7×#1446C211yesmaximal, characteristic, complemented
6C7×#1294C211yesmaximal, characteristic, complemented
7C7×#1280C211yesmaximal, characteristic, complemented
8C23×C7×Q8C2211yescharacteristic
9C22×C7×SD16—21no—
10C2×C7×#121C2211yescharacteristic
11C2×C4×C7×D4—21no—
12C7×#448—22no—
13C7×#448C414yescomplemented
14C2×C7×#103—21no—
15C2×C7×#103—21no—
16C2×C7×#103—21no—
17C2×C7×C22⋊C8C2211yescharacteristic
18C2×C7×C22⋊C8C2211yescharacteristic
19C7×#388—21no—
20C7×#387—21no—
21C7×#386—21no—
22C7×#386—21no—
23C7×#385—21no—
24C7×#380C2211yescharacteristic
25C2×C7×#95—21no—
26C2×C7×#95—21no—
27C2×C7×#95C2211yescharacteristic
28C2×C7×#95C2211yescharacteristic
29C7×#327—22no—
30#9039C711yesmaximal, characteristic, direct product factor, Sylow
31C22×C7×Q8—21no—
32C22×C7×Q8—21no—
33C22×C7×Q8—21no—
34C22×C7×Q8—21no—
35C22×C7×Q8—21no—
36C22×C7×Q8—42no—
37C22×C7×Q8—42no—
38C22×C7×Q8C2×C412yescomplemented
39C22×C7×D4—21no—
40C22×C7×D4D411yescharacteristic
41C23×C4×C7—21no—
42C23×C4×C7—21no—
43C23×C4×C7—21no—
44C23×C4×C7D411yescharacteristic
45C23×C4×C7D411yescharacteristic
46C23×C4×C7C2311yescharacteristic
47C2×C7×SD16—24no—
48C2×C7×SD16—42no—
49C2×C7×SD16—42no—
50C22×C7×C8—21no—
51C22×C7×C8—21no—
52C7×#121—21no—
53C7×#121—21no—
54C7×#121C2×C412yes—
55C7×#121D412yes—
56C4×C7×D4—42no—
57C4×C7×D4—42no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4D411yescharacteristic
60C2×C7×C4⋊C4D411yescharacteristic
61C2×C7×C4⋊C4D411yescharacteristic
62C2×C7×C22⋊C4—21no—
63C2×C7×C22⋊C4—21no—
64C2×C7×C22⋊C4—21no—
65C2×C7×C22⋊C4—21no—
66C2×C7×C22⋊C4—21no—
67C2×C42×C7—21no—
68C7×#103—24no—
69C7×#103—42no—
70C7×#103—42no—
71C7×C22⋊C8—21no—
72C7×C22⋊C8—21no—
73C7×C22⋊C8—21no—
74C7×C22⋊C8C2×C412yescomplemented
75C7×#95—21no—
76C7×#95—21no—
77C7×#95—21no—
78C7×#95—21no—
79C7×#95—21no—
80C7×#95—21no—
81C7×#95—21no—
82C7×#95—42no—
83C7×#95—42no—
84C2×#448C2×C711yescharacteristic
85#2325C2×C711yescharacteristic
86#1787C2×C711yescharacteristic, complemented
87C2×#327C2×C711yescharacteristic
88#1446C2×C711yescharacteristic, complemented
89#1294C2×C711yescharacteristic, complemented
90#1280C2×C711yescharacteristic, complemented
91C24×C7—21no—
92C24×C7C2×D411yescharacteristic
93C2×C7×Q8—22no—
94C2×C7×Q8—22no—
95C2×C7×Q8—22no—
96C2×C7×Q8—22no—
97C2×C7×Q8—22no—
98C2×C7×Q8—22no—
99C2×C7×Q8—44no—
100C2×C7×Q8—41no—
101C2×C7×Q8—41no—
102C2×C7×Q8—41no—
103C2×C7×Q8—41no—
104C2×C7×Q8—44no—
105C2×C7×Q8—44no—
106C2×C7×Q8—44no—
107C2×C7×Q8—41no—
108C2×C7×Q8—41no—
109C2×C7×D4—22no—
110C2×C7×D4—22no—
111C2×C7×D4—21no—
112C2×C7×D4—41no—
113C2×C7×D4—41no—
114C2×C7×D4—41no—
115C2×C7×D4—41no—
116C2×C7×D4—41no—
117C2×C7×D4—41no—
118C2×C7×D4C22⋊C412yes—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—21no—
123C22×C4×C7—21no—
124C22×C4×C7—21no—
125C22×C4×C7—22no—
126C22×C4×C7—21no—
127C22×C4×C7—21no—
128C22×C4×C7—21no—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7—41no—
132C22×C4×C7—41no—
133C22×C4×C7—42no—
134C22×C4×C7—41no—
135C22×C4×C7—41no—
136C22×C4×C7—41no—
137C22×C4×C7—41no—
138C22×C4×C7—41no—
139C22×C4×C7—41no—
140C22×C4×C7—41no—
141C22×C4×C7—41no—
142C22×C4×C7—41no—
143C22×C4×C7—41no—
144C22×C4×C7—41no—
145C22×C4×C7—41no—
146C22×C4×C7—41no—
147C22×C4×C7—41no—
148C22×C4×C7—81no—
149C22×C4×C7—81no—
150C22×C4×C7SD1612yescomplemented
151C22×C4×C7C22×C411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7#4111yescharacteristic
156C22×C4×C7#4111yescharacteristic
157C7×SD16—48no—
158C2×C7×C8—22no—
159C2×C7×C8—41no—
160C2×C7×C8—41no—
161C2×C7×C8—41no—
162C2×C7×C8—41no—
163C2×C7×C8—41no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—41no—
166C7×C4⋊C4—41no—
167C7×C4⋊C4—42no—
168C7×C4⋊C4C22⋊C412yes—
169C7×C22⋊C4—22no—
170C7×C22⋊C4—22no—
171C7×C22⋊C4—42no—
172C7×C22⋊C4—41no—
173C7×C22⋊C4—41no—
174C7×C22⋊C4—81no—
175C7×C22⋊C4—81no—
176C42×C7—42no—
177C23×Q8C22×C711yescharacteristic
178C22×SD16—21no—
179C2×#121C22×C711yescharacteristic
180C2×C4×D4—21no—
181#448—22no—
182#448C4×C714yescomplemented
183C2×#103—21no—
184C2×#103—21no—
185C2×#103—21no—
186C2×C22⋊C8C22×C711yescharacteristic
187C2×C22⋊C8C22×C711yescharacteristic
188#388—21no—
189#387—21no—
190#386—21no—
191#386—21no—
192#385—21no—
193#380C22×C711yescharacteristic
194C2×#95—21no—
195C2×#95—21no—
196C2×#95C22×C711yescharacteristic
197C2×#95C22×C711yescharacteristic
198#327—22no—
199C23×C7—21no—
200C23×C7—21no—
201C23×C7—21no—
202C23×C7—21no—
203C23×C7—22no—
204C23×C7—41no—
205C23×C7—41no—
206C23×C7—41no—
207C23×C7—41no—
208C23×C7—41no—
209C23×C7—41no—
210C23×C7C4×D411yescharacteristic
211C23×C7#12011yescharacteristic
212C23×C7#12111yescharacteristic
213C23×C7C2×SD1611yescharacteristic
214C23×C7#13511yescharacteristic
215C23×C7AGL(1,ℤ/8)11yescharacteristic
216C23×C7#13711yescharacteristic
217C7×Q8—24no—
218C7×Q8—24no—
219C7×Q8—42no—
220C7×Q8—42no—
221C7×Q8—42no—
222C7×Q8—42no—
223C7×Q8—42no—
224C7×Q8—42no—
225C7×D4—42no—
226C7×D4—42no—
227C7×D4—42no—
228C7×D4—42no—
229C7×D4—42no—
230C7×D4—81no—
231C2×C4×C7—21no—
232C2×C4×C7—21no—
233C2×C4×C7—21no—
234C2×C4×C7—21no—
235C2×C4×C7—24no—
236C2×C4×C7—22no—
237C2×C4×C7—22no—
238C2×C4×C7—41no—
239C2×C4×C7—41no—
240C2×C4×C7—41no—
241C2×C4×C7—41no—
242C2×C4×C7—42no—
243C2×C4×C7—42no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7—41no—
247C2×C4×C7—41no—
248C2×C4×C7—41no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—41no—
252C2×C4×C7—42no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7—41no—
258C2×C4×C7—41no—
259C2×C4×C7—41no—
260C2×C4×C7—41no—
261C2×C4×C7—41no—
262C2×C4×C7—41no—
263C2×C4×C7—41no—
264C2×C4×C7—41no—
265C2×C4×C7—41no—
266C2×C4×C7—81no—
267C2×C4×C7—84no—
268C2×C4×C7—81no—
269C2×C4×C7—81no—
270C2×C4×C7—81no—
271C2×C4×C7—81no—
272C2×C4×C7—81no—
273C2×C4×C7C2×C22⋊C411yescharacteristic
274C2×C4×C7C4×D411yescharacteristic
275C2×C4×C7#12111yescharacteristic
276C2×C4×C7#12311yescharacteristic
277C7×C8—42no—
278C7×C8—81no—
279C22×Q8—21no—
280C22×Q8—21no—
281C22×Q8—21no—
282C22×Q8—21no—
283C22×Q8—21no—
284C22×Q8—42no—
285C22×Q8—42no—
286C22×Q8C2×C4×C712yescomplemented
287C22×D4—21no—
288C22×D4C7×D411yescharacteristic
289C23×C4—21no—
290C23×C4—21no—
291C23×C4—21no—
292C23×C4C7×D411yescharacteristic
293C23×C4C7×D411yescharacteristic
294C23×C4C23×C711yescharacteristic
295C2×SD16—24no—
296C2×SD16—42no—
297C2×SD16—42no—
298C22×C8—21no—
299C22×C8—21no—
300#121—21no—
301#121—21no—
302#121C2×C4×C712yes—
303#121C7×D412yes—
304C4×D4—42no—
305C4×D4—42no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4C7×D411yescharacteristic
308C2×C4⋊C4C7×D411yescharacteristic
309C2×C4⋊C4C7×D411yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C42—21no—
316#103—24no—
317#103—42no—
318#103—42no—
319C22⋊C8—21no—
320C22⋊C8—21no—
321C22⋊C8—21no—
322C22⋊C8C2×C4×C712yescomplemented
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—42no—
331#95—42no—
332C22×C7—21no—
333C22×C7—21no—
334C22×C7—21no—
335C22×C7—21no—
336C22×C7—21no—
337C22×C7—21no—
338C22×C7—21no—
339C22×C7—21no—
340C22×C7—21no—
341C22×C7—21no—
342C22×C7—21no—
343C22×C7—21no—
344C22×C7—42no—
345C22×C7—42no—
346C22×C7—42no—
347C22×C7—81no—
348C22×C7—81no—
349C22×C7—81no—
350C22×C7—81no—
351C22×C7#38611yescharacteristic
352C22×C7C4×SD1611yescharacteristic
353C22×C7#44011yescharacteristic
354C22×C7#45011yescharacteristic
355C22×C7#45211yescharacteristic
356C22×C7#45511yescharacteristic
357C22×C7#45611yescharacteristic
358C22×C7B(4,𝔽2)11yescharacteristic
359C22×C7B(4,𝔽2)11yescharacteristic
360C22×C7#46111yescharacteristic
361C22×C7#46311yescharacteristic
362C4×C7—21no—
363C4×C7—21no—
364C4×C7—21no—
365C4×C7—21no—
366C4×C7—42no—
367C4×C7—42no—
368C4×C7—44no—
369C4×C7—82no—
370C4×C7—81no—
371C4×C7—81no—
372C4×C7—81no—
373C24—21no—
374C24C2×C7×D411yescharacteristic
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—44no—
382C2×Q8—41no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×Q8—41no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—41no—
390C2×Q8—41no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—21no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4C7×C22⋊C412yes—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—42no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—81no—
431C22×C4—81no—
432C22×C4C7×SD1612yescomplemented
433C22×C4C22×C4×C711yescharacteristic
434C22×C4C2×C7×D411yescharacteristic
435C22×C4C2×C7×D411yescharacteristic
436C22×C4C2×C7×D411yescharacteristic
437C22×C4C7×#4111yescharacteristic
438C22×C4C7×#4111yescharacteristic
439SD16—48no—
440C2×C8—22no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C4⋊C4—21no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—42no—
450C4⋊C4C7×C22⋊C412yes—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—42no—
454C22⋊C4—41no—
455C22⋊C4—41no—
456C22⋊C4—81no—
457C22⋊C4—81no—
458C42—42no—
459C2×C7—21no—
460C2×C7—21no—
461C2×C7—21no—
462C2×C7—21no—
463C2×C7—82no—
464C2×C7#160111yescharacteristic
465C2×C7#160311yescharacteristic
466C2×C7#160411yescharacteristic
467C2×C7#160611yescharacteristic
468C2×C7#187711yescharacteristic
469C2×C7#188211yescharacteristic
470C2×C7#188911yescharacteristic
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—22no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23C4×C7×D411yescharacteristic
483C23C7×#12011yescharacteristic
484C23C7×#12111yescharacteristic
485C23C2×C7×SD1611yescharacteristic
486C23C7×#13511yescharacteristic
487C23C7×AGL(1,ℤ/8)11yescharacteristic
488C23C7×#13711yescharacteristic
489Q8—24no—
490Q8—24no—
491Q8—42no—
492Q8—42no—
493Q8—42no—
494Q8—42no—
495Q8—42no—
496Q8—42no—
497D4—42no—
498D4—42no—
499D4—42no—
500D4—42no—
501D4—42no—
502D4—81no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—81no—
539C2×C4—84no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4C2×C7×C22⋊C411yescharacteristic
546C2×C4C4×C7×D411yescharacteristic
547C2×C4C7×#12111yescharacteristic
548C2×C4C7×#12311yescharacteristic
549C8—42no—
550C8—81no—
551C7#903911yescharacteristic, direct product factor, Sylow
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22C7×#38611yescharacteristic
572C22C4×C7×SD1611yescharacteristic
573C22C7×#44011yescharacteristic
574C22C7×#45011yescharacteristic
575C22C7×#45211yescharacteristic
576C22C7×#45511yescharacteristic
577C22C7×#45611yescharacteristic
578C22C7×B(4,𝔽2)11yescharacteristic
579C22C7×B(4,𝔽2)11yescharacteristic
580C22C7×#46111yescharacteristic
581C22C7×#46311yescharacteristic
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—42no—
587C4—42no—
588C4—44no—
589C4—82no—
590C4—81no—
591C4—81no—
592C4—81no—
593C2—21no—
594C2—21no—
595C2—21no—
596C2—21no—
597C2—82no—
598C2C7×#160111yescharacteristic
599C2C7×#160311yescharacteristic
600C2C7×#160411yescharacteristic
601C2C7×#160611yescharacteristic
602C2C7×#187711yescharacteristic
603C2C7×#188211yescharacteristic
604C2C7×#188911yescharacteristic
subgroupG/H
center, socleC23×C7#120
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC2×C7×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#1787C2semidirect
5. C7×#1446C2semidirect
6. C7×#1294C2semidirect
7. C7×#1280C2semidirect
13. C7×#448C4semidirect
30. #9039C7direct (N × H)
38. C22×C7×Q8C2×C4semidirect
74. C7×C22⋊C8C2×C4semidirect
86. #1787C2×C7semidirect
88. #1446C2×C7semidirect
89. #1294C2×C7semidirect
90. #1280C2×C7semidirect
150. C22×C4×C7SD16semidirect
182. #448C4×C7semidirect
286. C22×Q8C2×C4×C7semidirect
322. C22⋊C8C2×C4×C7semidirect
432. C22×C4C7×SD16semidirect
551. C7#9039direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4482
C7×#23252
C7×#17872
C2×C7×#3272
C7×#14462
C7×#12942
C7×#12802
#90397