Finite Group Databasebeta

← C7×#18288 #13195504

Subgroups — C7×#18288 #13195504

Order 1792.

1034 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×#539C211yesmaximal, characteristic, complemented
2C7×#3187C211yesmaximal, characteristic, complemented
3C7×#3186C211yesmaximal, characteristic
4C7×#2618C211yesmaximal, characteristic, complemented
5C7×#1609C214yesmaximal, complemented
6C7×#1597C214yesmaximal, complemented
7C7×#1520C211yesmaximal, characteristic, complemented
8C7×#1519C211yesmaximal, characteristic, complemented
9C2×C7×#324C211yesmaximal, characteristic, complemented
10C7×#539—22no—
11C7×#539—22no—
12C7×#539—22no—
13C7×#539C2214yes—
14C2×C7×#126—21no—
15C2×C7×#125—21no—
16C2×C7×#123—21no—
17C2×C7×#123—21no—
18C2×C7×#123—21no—
19C2×C7×C22⋊Q8—21no—
20C2×C7×C22⋊Q8C2211yescharacteristic
21C2×C7×#121C2211yescharacteristic
22C2×C7×#121C2211yescharacteristic
23C2×C4×C7×D4—21no—
24C22×C7×C4⋊C4C2211yescharacteristic
25C7×#483C2212yescomplemented
26C7×#481—21no—
27C7×#480—21no—
28C7×#480C2212yes—
29C7×#466—22no—
30C7×#465—22no—
31C7×#450C2212yescomplemented
32C7×#448—21no—
33C7×#447—21no—
34C7×#447C2212yes—
35C2×C7×C22⋊C8C2211yescharacteristic
36C2×C7×C22⋊C8C2211yescharacteristic
37C7×#398C2212yescomplemented
38C7×#396—22no—
39C7×#394C2212yescomplemented
40C7×#386—21no—
41C7×#380C2211yescharacteristic
42C7×#328C2214yescomplemented
43C7×#327C2214yes—
44C7×#324C2214yescomplemented
45#18288C711yesmaximal, characteristic, direct product factor, Sylow
46C22×C7×Q8D411yescharacteristic
47C22×C7×D4—21no—
48C22×C7×D4—21no—
49C22×C7×D4D411yescharacteristic
50C22×C7×D4C2311yescharacteristic
51C23×C4×C7—21no—
52C23×C4×C7D411yescharacteristic
53C23×C4×C7D411yescharacteristic
54C23×C4×C7C2311yescharacteristic
55C2×C7×SD16—22no—
56C2×C7×D8—22no—
57C22×C7×C8—21no—
58C22×C7×C8—21no—
59C7×#126—24no—
60C7×#126—24no—
61C7×#125—24no—
62C7×#125—24no—
63C7×#123—24no—
64C7×#123—24no—
65C7×#123—24no—
66C7×#123—24no—
67C7×#123—44no—
68C7×C22⋊Q8—24no—
69C7×C22⋊Q8—24no—
70C7×C22⋊Q8—21no—
71C7×C22⋊Q8—21no—
72C7×C22⋊Q8D412yes—
73C7×C22⋊Q8C2312yes—
74C7×#121—21no—
75C7×#121—21no—
76C7×#121—21no—
77C7×#121—21no—
78C7×#121—21no—
79C7×#121—21no—
80C7×#121D412yes—
81C7×#121C2312yes—
82C4×C7×D4—24no—
83C4×C7×D4—24no—
84C2×C7×C4⋊C4—21no—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4—21no—
87C2×C7×C4⋊C4—21no—
88C2×C7×C4⋊C4—21no—
89C2×C7×C4⋊C4—21no—
90C2×C7×C4⋊C4—21no—
91C2×C7×C4⋊C4—21no—
92C2×C7×C4⋊C4—22no—
93C2×C7×C4⋊C4—21no—
94C2×C7×C4⋊C4D412yes—
95C2×C7×C4⋊C4D411yescharacteristic
96C2×C7×C4⋊C4D411yescharacteristic
97C2×C7×C4⋊C4C2311yescharacteristic
98C2×C7×C4⋊C4C2312yes—
99C2×C7×C22⋊C4—21no—
100C2×C7×C22⋊C4—21no—
101C2×C7×C22⋊C4—21no—
102C2×C7×C22⋊C4—21no—
103C2×C7×C22⋊C4—21no—
104C2×C7×C22⋊C4—21no—
105C2×C7×C22⋊C4—21no—
106C2×C7×C22⋊C4—22no—
107C2×C7×C22⋊C4—22no—
108C2×C42×C7—21no—
109C7×#107—22no—
110C7×#106—22no—
111C7×#103—22no—
112C7×#102—22no—
113C7×#102—22no—
114C7×#102—22no—
115C7×C22⋊C8—21no—
116C7×C22⋊C8—21no—
117C7×C22⋊C8C2312yes—
118C7×C22⋊C8C2312yes—
119C7×#95—22no—
120C7×#95C2312yes—
121C2×#539C2×C711yescharacteristic, complemented
122#3187C2×C711yescharacteristic, complemented
123#3186C2×C711yescharacteristic
124#2618C2×C711yescharacteristic, complemented
125#1609C2×C714yescomplemented
126#1597C2×C714yescomplemented
127#1520C2×C711yescharacteristic, complemented
128#1519C2×C711yescharacteristic, complemented
129C2×#324C2×C711yescharacteristic, complemented
130C24×C7—21no—
131C24×C7—21no—
132C24×C7C2×D411yescharacteristic
133C2×C7×Q8—21no—
134C2×C7×Q8—81no—
135C2×C7×Q8C2×D412yes—
136C2×C7×D4—22no—
137C2×C7×D4—22no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4—21no—
141C2×C7×D4—42no—
142C2×C7×D4—42no—
143C2×C7×D4—41no—
144C2×C7×D4—41no—
145C2×C7×D4—42no—
146C2×C7×D4—42no—
147C2×C7×D4—81no—
148C2×C7×D4—81no—
149C2×C7×D4C2×D412yes—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—22no—
157C22×C4×C7—22no—
158C22×C4×C7—24no—
159C22×C4×C7—21no—
160C22×C4×C7—21no—
161C22×C4×C7—22no—
162C22×C4×C7—81no—
163C22×C4×C7—81no—
164C22×C4×C7—82no—
165C22×C4×C7C2×D412yes—
166C22×C4×C7C2×D412yes—
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2411yescharacteristic
173C7×SD16—82no—
174C7×D8—82no—
175C2×C7×C8—22no—
176C2×C7×C8—22no—
177C2×C7×C8—81no—
178C2×C7×C8—81no—
179C7×C4⋊C4—24no—
180C7×C4⋊C4—24no—
181C7×C4⋊C4—24no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—44no—
193C7×C4⋊C4—81no—
194C7×C4⋊C4C2×D412yes—
195C7×C4⋊C4C2×D412yes—
196C7×C22⋊C4—24no—
197C7×C22⋊C4—24no—
198C7×C22⋊C4—24no—
199C7×C22⋊C4—22no—
200C7×C22⋊C4—22no—
201C7×C22⋊C4—22no—
202C7×C22⋊C4—22no—
203C7×C22⋊C4—44no—
204C7×C22⋊C4—42no—
205C7×C22⋊C4—41no—
206C7×C22⋊C4—41no—
207C7×C22⋊C4—82no—
208C42×C7—24no—
209#539—22no—
210#539—22no—
211#539—22no—
212#539C22×C714yes—
213C2×#126—21no—
214C2×#125—21no—
215C2×#123—21no—
216C2×#123—21no—
217C2×#123—21no—
218C2×C22⋊Q8—21no—
219C2×C22⋊Q8C22×C711yescharacteristic
220C2×#121C22×C711yescharacteristic
221C2×#121C22×C711yescharacteristic
222C2×C4×D4—21no—
223C22×C4⋊C4C22×C711yescharacteristic
224#483C22×C712yescomplemented
225#481—21no—
226#480—21no—
227#480C22×C712yes—
228#466—22no—
229#465—22no—
230#450C22×C712yescomplemented
231#448—21no—
232#447—21no—
233#447C22×C712yes—
234C2×C22⋊C8C22×C711yescharacteristic
235C2×C22⋊C8C22×C711yescharacteristic
236#398C22×C712yescomplemented
237#396—22no—
238#394C22×C712yescomplemented
239#386—21no—
240#380C22×C711yescharacteristic
241#328C22×C714yescomplemented
242#327C22×C714yes—
243#324C22×C714yescomplemented
244C23×C7—22no—
245C23×C7—41no—
246C23×C7—41no—
247C23×C7—42no—
248C23×C7—42no—
249C23×C7—42no—
250C23×C7—42no—
251C23×C7—42no—
252C23×C7—41no—
253C23×C7—41no—
254C23×C7—41no—
255C23×C7#12011yescharacteristic
256C23×C7#12011yescharacteristic
257C23×C7#13512yes—
258C23×C7AGL(1,ℤ/8)12yes—
259C23×C7C22×D411yescharacteristic
260C7×Q8—42no—
261C7×Q8—81no—
262C7×D4—42no—
263C7×D4—42no—
264C7×D4—42no—
265C7×D4—42no—
266C7×D4—42no—
267C7×D4—44no—
268C7×D4—81no—
269C7×D4—82no—
270C7×D4—82no—
271C2×C4×C7—22no—
272C2×C4×C7—22no—
273C2×C4×C7—22no—
274C2×C4×C7—22no—
275C2×C4×C7—22no—
276C2×C4×C7—22no—
277C2×C4×C7—22no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—42no—
285C2×C4×C7—42no—
286C2×C4×C7—42no—
287C2×C4×C7—42no—
288C2×C4×C7—42no—
289C2×C4×C7—42no—
290C2×C4×C7—42no—
291C2×C4×C7—42no—
292C2×C4×C7—42no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—42no—
296C2×C4×C7—41no—
297C2×C4×C7—81no—
298C2×C4×C7—84no—
299C2×C4×C7#12011yescharacteristic
300C2×C4×C7#12011yescharacteristic
301C2×C4×C7C22×D411yescharacteristic
302C2×C4×C7C22×D411yescharacteristic
303C7×C8—42no—
304C7×C8—42no—
305C22×Q8C7×D411yescharacteristic
306C22×D4—21no—
307C22×D4—21no—
308C22×D4C7×D411yescharacteristic
309C22×D4C23×C711yescharacteristic
310C23×C4—21no—
311C23×C4C7×D411yescharacteristic
312C23×C4C7×D411yescharacteristic
313C23×C4C23×C711yescharacteristic
314C2×SD16—22no—
315C2×D8—22no—
316C22×C8—21no—
317C22×C8—21no—
318#126—24no—
319#126—24no—
320#125—24no—
321#125—24no—
322#123—24no—
323#123—24no—
324#123—24no—
325#123—24no—
326#123—44no—
327C22⋊Q8—24no—
328C22⋊Q8—24no—
329C22⋊Q8—21no—
330C22⋊Q8—21no—
331C22⋊Q8C7×D412yes—
332C22⋊Q8C23×C712yes—
333#121—21no—
334#121—21no—
335#121—21no—
336#121—21no—
337#121—21no—
338#121—21no—
339#121C7×D412yes—
340#121C23×C712yes—
341C4×D4—24no—
342C4×D4—24no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4C7×D412yes—
354C2×C4⋊C4C7×D411yescharacteristic
355C2×C4⋊C4C7×D411yescharacteristic
356C2×C4⋊C4C23×C711yescharacteristic
357C2×C4⋊C4C23×C712yes—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C42—21no—
368#107—22no—
369#106—22no—
370#103—22no—
371#102—22no—
372#102—22no—
373#102—22no—
374C22⋊C8—21no—
375C22⋊C8—21no—
376C22⋊C8C23×C712yes—
377C22⋊C8C23×C712yes—
378#95—22no—
379#95C23×C712yes—
380C22×C7—21no—
381C22×C7—21no—
382C22×C7—21no—
383C22×C7—21no—
384C22×C7—21no—
385C22×C7—21no—
386C22×C7—42no—
387C22×C7—42no—
388C22×C7—42no—
389C22×C7—42no—
390C22×C7—42no—
391C22×C7—84no—
392C22×C7—82no—
393C22×C7—82no—
394C22×C7—81no—
395C22×C7—81no—
396C22×C7—81no—
397C22×C7#44912yes—
398C22×C7#44912yes—
399C22×C7C2×#12011yescharacteristic
400C22×C7C2×#13511yescharacteristic
401C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
402C4×C7—21no—
403C4×C7—21no—
404C4×C7—21no—
405C4×C7—21no—
406C4×C7—42no—
407C4×C7—42no—
408C4×C7—42no—
409C4×C7—42no—
410C4×C7—42no—
411C4×C7—82no—
412C4×C7—81no—
413C24—21no—
414C24—21no—
415C24C2×C7×D411yescharacteristic
416C2×Q8—21no—
417C2×Q8—81no—
418C2×Q8C2×C7×D412yes—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—81no—
431C2×D4—81no—
432C2×D4C2×C7×D412yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—81no—
446C22×C4—81no—
447C22×C4—82no—
448C22×C4C2×C7×D412yes—
449C22×C4C2×C7×D412yes—
450C22×C4C2×C7×D411yescharacteristic
451C22×C4C2×C7×D411yescharacteristic
452C22×C4C2×C7×D411yescharacteristic
453C22×C4C2×C7×D411yescharacteristic
454C22×C4C2×C7×D411yescharacteristic
455C22×C4C24×C711yescharacteristic
456SD16—82no—
457D8—82no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—81no—
461C2×C8—81no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—44no—
476C4⋊C4—81no—
477C4⋊C4C2×C7×D412yes—
478C4⋊C4C2×C7×D412yes—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—44no—
487C22⋊C4—42no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—82no—
491C42—24no—
492C2×C7—21no—
493C2×C7—21no—
494C2×C7—22no—
495C2×C7—22no—
496C2×C7—82no—
497C2×C7—161no—
498C2×C7C2×#44911yescharacteristic
499C2×C7#302011yescharacteristic
500C2×C7#302611yescharacteristic
501C23—22no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23C7×#12011yescharacteristic
513C23C7×#12011yescharacteristic
514C23C7×#13512yes—
515C23C7×AGL(1,ℤ/8)12yes—
516C23C22×C7×D411yescharacteristic
517Q8—42no—
518Q8—81no—
519D4—42no—
520D4—42no—
521D4—42no—
522D4—42no—
523D4—42no—
524D4—44no—
525D4—81no—
526D4—82no—
527D4—82no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—81no—
555C2×C4—84no—
556C2×C4C7×#12011yescharacteristic
557C2×C4C7×#12011yescharacteristic
558C2×C4C22×C7×D411yescharacteristic
559C2×C4C22×C7×D411yescharacteristic
560C8—42no—
561C8—42no—
562C7#1828811yescharacteristic, direct product factor, Sylow
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—84no—
575C22—82no—
576C22—82no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22C7×#44912yes—
581C22C7×#44912yes—
582C22C2×C7×#12011yescharacteristic
583C22C2×C7×#13511yescharacteristic
584C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—42no—
590C4—42no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—82no—
595C4—81no—
596C2—21no—
597C2—21no—
598C2—22no—
599C2—22no—
600C2—82no—
601C2—161no—
602C2C2×C7×#44911yescharacteristic
603C2C7×#302011yescharacteristic
604C2C7×#302611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3186C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#539C2semidirect
2. C7×#3187C2semidirect
4. C7×#2618C2semidirect
5. C7×#1609C2semidirect
6. C7×#1597C2semidirect
7. C7×#1520C2semidirect
8. C7×#1519C2semidirect
9. C2×C7×#324C2semidirect
25. C7×#483C22semidirect
31. C7×#450C22semidirect
37. C7×#398C22semidirect
39. C7×#394C22semidirect
42. C7×#328C22semidirect
44. C7×#324C22semidirect
45. #18288C7direct (N × H)
121. C2×#539C2×C7semidirect
122. #3187C2×C7semidirect
124. #2618C2×C7semidirect
125. #1609C2×C7semidirect
126. #1597C2×C7semidirect
127. #1520C2×C7semidirect
128. #1519C2×C7semidirect
129. C2×#324C2×C7semidirect
224. #483C22×C7semidirect
230. #450C22×C7semidirect
236. #398C22×C7semidirect
238. #394C22×C7semidirect
241. #328C22×C7semidirect
243. #324C22×C7semidirect
562. C7#18288direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5392
C7×#31872
C7×#31862
C7×#26182
C7×#16092
C7×#15972
C7×#15202
C7×#15192
C2×C7×#3242
#182887