Finite Group Databasebeta

← C7×#19047 #13196263

Subgroups — C7×#19047 #13196263

Order 1792.

814 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3563C211yesmaximal, characteristic
2C7×#3236C211yesmaximal, characteristic, complemented
3C7×#3181C211yesmaximal, characteristic
4C7×#3162C211yesmaximal, characteristic, complemented
5C7×#1664C212yesmaximal, complemented
6C7×#1663C212yesmaximal, complemented
7C7×#1663C212yesmaximal, complemented
8C7×#1660C212yesmaximal, complemented
9C7×#1532C211yesmaximal, characteristic
10C7×#1531C211yesmaximal, characteristic, complemented
11C2×C7×#330C211yesmaximal, characteristic, complemented
12C7×#564C2212yes—
13C7×#563C2212yes—
14C7×#557—21no—
15C7×#554—21no—
16C7×#552—21no—
17C7×#552—22no—
18C7×#552—21no—
19C7×#544—21no—
20C7×#544—21no—
21C7×#541—21no—
22C7×#541—22no—
23C7×#539—21no—
24C7×#533C2211yescharacteristic
25C7×#533C2211yescharacteristic
26C7×#529C2211yescharacteristic
27C2×C7×#125C2211yescharacteristic
28C7×#501—21no—
29C7×#501—21no—
30C7×#501C2212yes—
31C7×#501C2212yes—
32C7×#501C2211yescharacteristic
33C7×#501C2212yes—
34C7×#497C2211yescharacteristic
35C7×#491C2211yescharacteristic
36C7×#491C2212yes—
37C7×#489C2211yescharacteristic
38C7×#484—21no—
39C7×#483—21no—
40C7×#479—22no—
41C7×#467—21no—
42C7×#465—21no—
43C7×#441—21no—
44C7×#440—21no—
45C7×#432C2211yescharacteristic
46C7×#427—22no—
47C2×C7×#97C2211yescharacteristic
48C2×C7×#97C2211yescharacteristic
49C7×#330C2212yes—
50C7×#330C2212yes—
51C7×#330C2212yes—
52C7×#330C2212yes—
53C7×#330C2212yes—
54C7×#329C2212yes—
55#19047C711yesmaximal, characteristic, direct product factor, Sylow
56C2×C7×Q16—21no—
57C2×C7×SD16—21no—
58C22×C7×C8—21no—
59C22×C7×C8—21no—
60C7×C4⋊Q8—22no—
61C7×C4⋊Q8—21no—
62C7×C4⋊Q8—22no—
63C7×C4⋊Q8D411yescharacteristic
64C7×C4⋊Q8D411yescharacteristic
65C7×C4⋊Q8C2311yescharacteristic
66C7×C4⋊Q8C2311yescharacteristic
67C7×#126—22no—
68C7×#126—22no—
69C7×#126—22no—
70C7×#126—22no—
71C7×#126—21no—
72C7×#125—22no—
73C7×#125—22no—
74C7×#125—21no—
75C7×#125—22no—
76C7×#125—22no—
77C7×#125—22no—
78C7×#125—22no—
79C7×#125—22no—
80C7×#125—22no—
81C7×#125—21no—
82C7×#125—21no—
83C7×#125—21no—
84C7×#125D412yes—
85C7×#125C2312yes—
86C7×#125C2311yescharacteristic
87C7×#125C2312yes—
88C7×#124C2311yescharacteristic
89C7×#123—22no—
90C7×#123—22no—
91C7×#123—21no—
92C7×C22⋊Q8—22no—
93C7×C22⋊Q8—21no—
94C7×C22⋊Q8—21no—
95C7×C22⋊Q8—22no—
96C7×C22⋊Q8—21no—
97C7×C22⋊Q8—21no—
98C7×C22⋊Q8—22no—
99C7×C22⋊Q8—22no—
100C7×C22⋊Q8—21no—
101C7×C22⋊Q8—21no—
102C7×C22⋊Q8D411yescharacteristic
103C7×#121D411yescharacteristic
104C4×C7×Q8—21no—
105C4×C7×Q8—22no—
106C4×C7×Q8—21no—
107C4×C7×Q8—21no—
108C4×C7×Q8—21no—
109C4×C7×Q8—21no—
110C4×C7×Q8—21no—
111C4×C7×Q8D411yescharacteristic
112C4×C7×D4—22no—
113C4×C7×D4—21no—
114C4×C7×D4D411yescharacteristic
115C7×#117—22no—
116C7×#117—21no—
117C7×#117—22no—
118C7×#117—21no—
119C7×#117—21no—
120C7×#117D412yes—
121C2×C7×C4⋊C4—21no—
122C2×C7×C4⋊C4—21no—
123C2×C7×C4⋊C4—21no—
124C2×C7×C4⋊C4D411yescharacteristic
125C2×C7×C4⋊C4D411yescharacteristic
126C2×C42×C7C2311yescharacteristic
127C7×#107—21no—
128C7×#107—22no—
129C7×#107—22no—
130C7×#107—22no—
131C7×#106—22no—
132C7×#106—22no—
133C7×#106—22no—
134C7×#106—21no—
135C7×C4⋊C8—21no—
136C7×#103—21no—
137C7×#103—21no—
138C7×#103—22no—
139C7×#103—22no—
140C7×#103—21no—
141C7×#102—21no—
142C7×C22⋊C8—21no—
143C7×#97—21no—
144C7×#97—21no—
145C7×#97C2311yescharacteristic
146C7×#97C2312yes—
147C7×#97C2312yes—
148C7×#97C2311yescharacteristic
149#3563C2×C711yescharacteristic
150#3236C2×C711yescharacteristic, complemented
151#3181C2×C711yescharacteristic
152#3162C2×C711yescharacteristic, complemented
153#1664C2×C712yescomplemented
154#1663C2×C712yescomplemented
155#1663C2×C712yescomplemented
156#1660C2×C712yescomplemented
157#1532C2×C711yescharacteristic
158#1531C2×C711yescharacteristic, complemented
159C2×#330C2×C711yescharacteristic, complemented
160C2×C7×Q8—22no—
161C2×C7×Q8—21no—
162C2×C7×Q8—21no—
163C2×C7×Q8—21no—
164C2×C7×Q8—21no—
165C2×C7×Q8C2×D411yescharacteristic
166C2×C7×D4—21no—
167C2×C7×D4C2×D411yescharacteristic
168C22×C4×C7—21no—
169C22×C4×C7—22no—
170C22×C4×C7—21no—
171C22×C4×C7—21no—
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2×D411yescharacteristic
174C22×C4×C7C2×D411yescharacteristic
175C7×Q16—42no—
176C7×SD16—42no—
177C2×C7×C8—24no—
178C2×C7×C8—22no—
179C2×C7×C8—21no—
180C2×C7×C8—21no—
181C2×C7×C8—22no—
182C2×C7×C8—42no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—22no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—22no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—22no—
204C7×C4⋊C4—22no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—21no—
207C7×C4⋊C4—21no—
208C7×C4⋊C4—21no—
209C7×C4⋊C4—41no—
210C7×C4⋊C4—41no—
211C7×C4⋊C4C2×D412yes—
212C7×C4⋊C4C2×D411yescharacteristic
213C7×C4⋊C4C2×D412yes—
214C7×C4⋊C4C2×D411yescharacteristic
215C7×C4⋊C4C2×D412yes—
216C7×C4⋊C4C2×D412yes—
217C7×C22⋊C4—22no—
218C7×C22⋊C4—22no—
219C7×C22⋊C4—22no—
220C7×C22⋊C4—21no—
221C7×C22⋊C4—22no—
222C7×C22⋊C4—21no—
223C7×C22⋊C4—21no—
224C7×C22⋊C4—21no—
225C7×C22⋊C4—21no—
226C7×C22⋊C4—41no—
227C42×C7—22no—
228C42×C7—22no—
229C42×C7—22no—
230C42×C7—22no—
231C42×C7—21no—
232C42×C7—41no—
233C42×C7C2×D411yescharacteristic
234C42×C7C2×D411yescharacteristic
235C42×C7C2×D411yescharacteristic
236C42×C7C2411yescharacteristic
237#564C22×C712yes—
238#563C22×C712yes—
239#557—21no—
240#554—21no—
241#552—21no—
242#552—22no—
243#552—21no—
244#544—21no—
245#544—21no—
246#541—21no—
247#541—22no—
248#539—21no—
249#533C22×C711yescharacteristic
250#533C22×C711yescharacteristic
251#529C22×C711yescharacteristic
252C2×#125C22×C711yescharacteristic
253#501—21no—
254#501—21no—
255#501C22×C712yes—
256#501C22×C712yes—
257#501C22×C711yescharacteristic
258#501C22×C712yes—
259#497C22×C711yescharacteristic
260#491C22×C711yescharacteristic
261#491C22×C712yes—
262#489C22×C711yescharacteristic
263#484—21no—
264#483—21no—
265#479—22no—
266#467—21no—
267#465—21no—
268#441—21no—
269#440—21no—
270#432C22×C711yescharacteristic
271#427—22no—
272C2×#97C22×C711yescharacteristic
273C2×#97C22×C711yescharacteristic
274#330C22×C712yes—
275#330C22×C712yes—
276#330C22×C712yes—
277#330C22×C712yes—
278#330C22×C712yes—
279#329C22×C712yes—
280C23×C7—21no—
281C23×C7#12011yescharacteristic
282C7×Q8—41no—
283C7×Q8—42no—
284C7×Q8—41no—
285C7×Q8—41no—
286C7×Q8—42no—
287C7×Q8—81no—
288C7×Q8—82no—
289C7×D4—41no—
290C7×D4—42no—
291C2×C4×C7—22no—
292C2×C4×C7—22no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—22no—
296C2×C4×C7—21no—
297C2×C4×C7—22no—
298C2×C4×C7—22no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7—42no—
302C2×C4×C7—42no—
303C2×C4×C7—41no—
304C2×C4×C7—41no—
305C2×C4×C7—81no—
306C2×C4×C7—82no—
307C2×C4×C7#12011yescharacteristic
308C2×C4×C7#12011yescharacteristic
309C2×C4×C7#12011yescharacteristic
310C2×C4×C7C22×D411yescharacteristic
311C2×C4×C7C22×D411yescharacteristic
312C2×C4×C7C22×D411yescharacteristic
313C7×C8—24no—
314C7×C8—22no—
315C7×C8—42no—
316C7×C8—41no—
317C2×Q16—21no—
318C2×SD16—21no—
319C22×C8—21no—
320C22×C8—21no—
321C4⋊Q8—22no—
322C4⋊Q8—21no—
323C4⋊Q8—22no—
324C4⋊Q8C7×D411yescharacteristic
325C4⋊Q8C7×D411yescharacteristic
326C4⋊Q8C23×C711yescharacteristic
327C4⋊Q8C23×C711yescharacteristic
328#126—22no—
329#126—22no—
330#126—22no—
331#126—22no—
332#126—21no—
333#125—22no—
334#125—22no—
335#125—21no—
336#125—22no—
337#125—22no—
338#125—22no—
339#125—22no—
340#125—22no—
341#125—22no—
342#125—21no—
343#125—21no—
344#125—21no—
345#125C7×D412yes—
346#125C23×C712yes—
347#125C23×C711yescharacteristic
348#125C23×C712yes—
349#124C23×C711yescharacteristic
350#123—22no—
351#123—22no—
352#123—21no—
353C22⋊Q8—22no—
354C22⋊Q8—21no—
355C22⋊Q8—21no—
356C22⋊Q8—22no—
357C22⋊Q8—21no—
358C22⋊Q8—21no—
359C22⋊Q8—22no—
360C22⋊Q8—22no—
361C22⋊Q8—21no—
362C22⋊Q8—21no—
363C22⋊Q8C7×D411yescharacteristic
364#121C7×D411yescharacteristic
365C4×Q8—21no—
366C4×Q8—22no—
367C4×Q8—21no—
368C4×Q8—21no—
369C4×Q8—21no—
370C4×Q8—21no—
371C4×Q8—21no—
372C4×Q8C7×D411yescharacteristic
373C4×D4—22no—
374C4×D4—21no—
375C4×D4C7×D411yescharacteristic
376#117—22no—
377#117—21no—
378#117—22no—
379#117—21no—
380#117—21no—
381#117C7×D412yes—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4C7×D411yescharacteristic
386C2×C4⋊C4C7×D411yescharacteristic
387C2×C42C23×C711yescharacteristic
388#107—22no—
389#107—22no—
390#107—21no—
391#107—22no—
392#106—22no—
393#106—22no—
394#106—21no—
395#106—22no—
396C4⋊C8—21no—
397#103—21no—
398#103—21no—
399#103—21no—
400#103—22no—
401#103—22no—
402#102—21no—
403C22⋊C8—21no—
404#97—21no—
405#97—21no—
406#97C23×C712yes—
407#97C23×C712yes—
408#97C23×C711yescharacteristic
409#97C23×C711yescharacteristic
410C22×C7—21no—
411C22×C7—21no—
412C22×C7—41no—
413C22×C7—41no—
414C22×C7—41no—
415C22×C7#45612yes—
416C22×C7C2×#12011yescharacteristic
417C4×C7—21no—
418C4×C7—21no—
419C4×C7—21no—
420C4×C7—21no—
421C4×C7—21no—
422C4×C7—21no—
423C4×C7—42no—
424C4×C7—41no—
425C4×C7—42no—
426C4×C7—42no—
427C4×C7—41no—
428C4×C7—41no—
429C4×C7—42no—
430C4×C7—81no—
431C4×C7—81no—
432C2×Q8—22no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8C2×C7×D411yescharacteristic
438C2×D4—21no—
439C2×D4C2×C7×D411yescharacteristic
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4C2×C7×D411yescharacteristic
445C22×C4C2×C7×D411yescharacteristic
446C22×C4C2×C7×D411yescharacteristic
447Q16—42no—
448SD16—42no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—22no—
454C2×C8—42no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4C2×C7×D412yes—
484C4⋊C4C2×C7×D411yescharacteristic
485C4⋊C4C2×C7×D412yes—
486C4⋊C4C2×C7×D411yescharacteristic
487C4⋊C4C2×C7×D412yes—
488C4⋊C4C2×C7×D412yes—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—21no—
493C22⋊C4—22no—
494C22⋊C4—21no—
495C22⋊C4—21no—
496C22⋊C4—21no—
497C22⋊C4—21no—
498C22⋊C4—41no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—21no—
504C42—41no—
505C42C2×C7×D411yescharacteristic
506C42C2×C7×D411yescharacteristic
507C42C2×C7×D411yescharacteristic
508C42C24×C711yescharacteristic
509C2×C7—22no—
510C2×C7—81no—
511C2×C7C2×#45611yescharacteristic
512C2×C7#301811yescharacteristic
513C2×C7#302011yescharacteristic
514C23—21no—
515C23C7×#12011yescharacteristic
516Q8—41no—
517Q8—42no—
518Q8—41no—
519Q8—41no—
520Q8—42no—
521Q8—81no—
522Q8—82no—
523D4—41no—
524D4—42no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—81no—
540C2×C4—82no—
541C2×C4C7×#12011yescharacteristic
542C2×C4C7×#12011yescharacteristic
543C2×C4C7×#12011yescharacteristic
544C2×C4C22×C7×D411yescharacteristic
545C2×C4C22×C7×D411yescharacteristic
546C2×C4C22×C7×D411yescharacteristic
547C8—24no—
548C8—22no—
549C8—42no—
550C8—41no—
551C7#1904711yescharacteristic, direct product factor, Sylow
552C22—21no—
553C22—21no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22C7×#45612yes—
558C22C2×C7×#12011yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—42no—
566C4—41no—
567C4—42no—
568C4—42no—
569C4—41no—
570C4—41no—
571C4—42no—
572C4—81no—
573C4—81no—
574C2—22no—
575C2—81no—
576C2C2×C7×#45611yescharacteristic
577C2C7×#301811yescharacteristic
578C2C7×#302011yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#121D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#3236C2semidirect
4. C7×#3162C2semidirect
5. C7×#1664C2semidirect
6. C7×#1663C2semidirect
7. C7×#1663C2semidirect
8. C7×#1660C2semidirect
10. C7×#1531C2semidirect
11. C2×C7×#330C2semidirect
55. #19047C7direct (N × H)
150. #3236C2×C7semidirect
152. #3162C2×C7semidirect
153. #1664C2×C7semidirect
154. #1663C2×C7semidirect
155. #1663C2×C7semidirect
156. #1660C2×C7semidirect
158. #1531C2×C7semidirect
159. C2×#330C2×C7semidirect
551. C7#19047direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35632
C7×#32362
C7×#31812
C7×#31622
C7×#16642
C7×#16632
C7×#16602
C7×#15322
C7×#15312
C2×C7×#3302
#190477