Finite Group Databasebeta

← C7×#18961 #13196177

Subgroups — C7×#18961 #13196177

Order 1792.

818 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×#3555C211yesmaximal, characteristic
2C7×#3238C211yesmaximal, characteristic
3C7×#3230C211yesmaximal, characteristic, complemented
4C7×#3209C211yesmaximal, characteristic, complemented
5C7×#1638C212yesmaximal, complemented
6C7×#1636C212yesmaximal, complemented
7C7×#1633C212yesmaximal, complemented
8C7×#1632C212yesmaximal, complemented
9C7×#1554C211yesmaximal, characteristic, complemented
10C7×#1497C211yesmaximal, characteristic, complemented
11C7×#1482C211yesmaximal, characteristic
12C7×#563—22no—
13C7×Q82C2211yescharacteristic
14C7×#557—21no—
15C7×#557C2211yescharacteristic
16C7×#556—21no—
17C7×#556—21no—
18C7×#556C2211yescharacteristic
19C7×#554C2211yescharacteristic
20C7×#552—21no—
21C7×#544—21no—
22C7×#542—21no—
23C7×#541—21no—
24C7×#541—21no—
25C7×#541—22no—
26C7×#533C2211yescharacteristic
27C7×#533C2211yescharacteristic
28C2×C7×C4⋊Q8C2211yescharacteristic
29C7×#529C2211yescharacteristic
30C7×#501—21no—
31C7×#500—21no—
32C7×#494C2212yes—
33C7×#489—21no—
34C7×#487—21no—
35C7×#487C2212yes—
36C7×#484—21no—
37C7×#484—21no—
38C7×#483—21no—
39C7×#480—21no—
40C7×#477C2212yes—
41C7×#477C2212yes—
42C7×#477C2212yes—
43C7×#476C2212yes—
44C7×#462—22no—
45C7×#451—22no—
46C7×#432C2211yescharacteristic
47C7×#431C2211yescharacteristic
48C7×#431C2211yescharacteristic
49C7×#333C2212yes—
50C7×#332C2212yes—
51C7×Q8⋊C8C2212yes—
52C7×Q8⋊C8C2212yes—
53C7×Q8⋊C8C2212yes—
54C7×D4⋊C8C2212yes—
55#18961C711yesmaximal, characteristic, direct product factor, Sylow
56C22×C7×Q8—21no—
57C2×C7×Q16—24no—
58C7×C4⋊Q8—22no—
59C7×C4⋊Q8—21no—
60C7×C4⋊Q8—22no—
61C7×C4⋊Q8—21no—
62C7×C4⋊Q8—21no—
63C7×C4⋊Q8—21no—
64C7×C4⋊Q8—21no—
65C7×C4⋊Q8—21no—
66C7×C4⋊Q8D411yescharacteristic
67C7×C4⋊Q8D411yescharacteristic
68C7×C4⋊Q8C2312yes—
69C7×C4⋊Q8C2311yescharacteristic
70C7×C4⋊Q8C2311yescharacteristic
71C7×#126—22no—
72C7×#126—22no—
73C7×#126—22no—
74C7×#126—22no—
75C7×#126—22no—
76C7×#126—22no—
77C7×#126—21no—
78C7×#125—22no—
79C7×#125—22no—
80C7×#125—22no—
81C7×#125—22no—
82C7×#125—21no—
83C7×#125—22no—
84C7×#125—21no—
85C7×#125—21no—
86C7×#125—21no—
87C7×#125D411yescharacteristic
88C7×#124—22no—
89C7×#124—21no—
90C7×#124D411yescharacteristic
91C7×#123—22no—
92C7×#123—21no—
93C7×C22⋊Q8—22no—
94C7×C22⋊Q8—22no—
95C7×C22⋊Q8—21no—
96C7×C22⋊Q8—21no—
97C7×C22⋊Q8—22no—
98C7×C22⋊Q8—21no—
99C7×C22⋊Q8—21no—
100C7×C22⋊Q8D411yescharacteristic
101C7×C22⋊Q8D411yescharacteristic
102C7×C22⋊Q8D411yescharacteristic
103C7×C22⋊Q8D412yes—
104C7×#121D411yescharacteristic
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×Q8—21no—
112C4×C7×Q8—21no—
113C4×C7×Q8D412yes—
114C4×C7×Q8C2311yescharacteristic
115C4×C7×Q8C2311yescharacteristic
116C4×C7×Q8C2311yescharacteristic
117C4×C7×D4—21no—
118C4×C7×D4—21no—
119C4×C7×D4C2311yescharacteristic
120C7×#117—22no—
121C7×#117—21no—
122C7×#117—21no—
123C7×#117—21no—
124C7×#117—21no—
125C2×C7×C4⋊C4—21no—
126C2×C7×C4⋊C4—21no—
127C2×C7×C4⋊C4—21no—
128C2×C42×C7C2311yescharacteristic
129C7×#107—22no—
130C7×#107—22no—
131C7×#107—22no—
132C7×#106—22no—
133C7×C4⋊C8—21no—
134C7×C4⋊C8C2312yes—
135C7×C4⋊C8C2312yes—
136C7×#103—22no—
137C7×#103—22no—
138C7×#103—22no—
139C7×#103—24no—
140C7×#102—22no—
141C7×C22⋊C8—21no—
142C7×C22⋊C8—21no—
143C7×C22⋊C8—21no—
144C7×#97—21no—
145C4×C7×C8—21no—
146C4×C7×C8C2312yes—
147#3555C2×C711yescharacteristic
148#3238C2×C711yescharacteristic
149#3230C2×C711yescharacteristic, complemented
150#3209C2×C711yescharacteristic, complemented
151#1638C2×C712yescomplemented
152#1636C2×C712yescomplemented
153#1633C2×C712yescomplemented
154#1632C2×C712yescomplemented
155#1554C2×C711yescharacteristic, complemented
156#1497C2×C711yescharacteristic, complemented
157#1482C2×C711yescharacteristic
158C2×C7×Q8—22no—
159C2×C7×Q8—21no—
160C2×C7×Q8—21no—
161C2×C7×Q8—22no—
162C2×C7×Q8—41no—
163C2×C7×Q8—82no—
164C2×C7×Q8C2×D411yescharacteristic
165C2×C7×Q8C2×D411yescharacteristic
166C2×C7×Q8C2×D411yescharacteristic
167C2×C7×Q8C2×D412yes—
168C2×C7×D4—21no—
169C2×C7×D4C2×D411yescharacteristic
170C22×C4×C7—22no—
171C22×C4×C7—21no—
172C22×C4×C7—41no—
173C22×C4×C7C2×D411yescharacteristic
174C22×C4×C7C2×D411yescharacteristic
175C22×C4×C7C2×D411yescharacteristic
176C7×Q16—84no—
177C2×C7×C8—22no—
178C2×C7×C8—22no—
179C2×C7×C8—22no—
180C7×C4⋊C4—22no—
181C7×C4⋊C4—22no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4—21no—
194C7×C4⋊C4—21no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—21no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—21no—
204C7×C4⋊C4—22no—
205C7×C4⋊C4—22no—
206C7×C4⋊C4—41no—
207C7×C4⋊C4—41no—
208C7×C4⋊C4C2×D412yes—
209C7×C4⋊C4C2×D411yescharacteristic
210C7×C4⋊C4C2×D411yescharacteristic
211C7×C4⋊C4C2×D411yescharacteristic
212C7×C4⋊C4C2×D411yescharacteristic
213C7×C22⋊C4—22no—
214C7×C22⋊C4—22no—
215C7×C22⋊C4—22no—
216C7×C22⋊C4—21no—
217C7×C22⋊C4—22no—
218C7×C22⋊C4—21no—
219C7×C22⋊C4—21no—
220C7×C22⋊C4—21no—
221C7×C22⋊C4—21no—
222C7×C22⋊C4—21no—
223C7×C22⋊C4—21no—
224C42×C7—22no—
225C42×C7—22no—
226C42×C7—22no—
227C42×C7—22no—
228C42×C7—21no—
229C42×C7—21no—
230C42×C7—21no—
231C42×C7C2×D411yescharacteristic
232C42×C7C2×D411yescharacteristic
233C42×C7C2×D411yescharacteristic
234C42×C7C2411yescharacteristic
235#563—22no—
236Q82C22×C711yescharacteristic
237#557—21no—
238#557C22×C711yescharacteristic
239#556—21no—
240#556—21no—
241#556C22×C711yescharacteristic
242#554C22×C711yescharacteristic
243#552—21no—
244#544—21no—
245#542—21no—
246#541—21no—
247#541—21no—
248#541—22no—
249#533C22×C711yescharacteristic
250#533C22×C711yescharacteristic
251C2×C4⋊Q8C22×C711yescharacteristic
252#529C22×C711yescharacteristic
253#501—21no—
254#500—21no—
255#494C22×C712yes—
256#489—21no—
257#487—21no—
258#487C22×C712yes—
259#484—21no—
260#484—21no—
261#483—21no—
262#480—21no—
263#477C22×C712yes—
264#477C22×C712yes—
265#477C22×C712yes—
266#476C22×C712yes—
267#462—22no—
268#451—22no—
269#432C22×C711yescharacteristic
270#431C22×C711yescharacteristic
271#431C22×C711yescharacteristic
272#333C22×C712yes—
273#332C22×C712yes—
274Q8⋊C8C22×C712yes—
275Q8⋊C8C22×C712yes—
276Q8⋊C8C22×C712yes—
277D4⋊C8C22×C712yes—
278C23×C7—21no—
279C23×C7#12011yescharacteristic
280C7×Q8—22no—
281C7×Q8—22no—
282C7×Q8—22no—
283C7×Q8—42no—
284C7×Q8—44no—
285C7×Q8—42no—
286C7×Q8—81no—
287C7×Q8—82no—
288C7×Q8—82no—
289C7×D4—22no—
290C7×D4—81no—
291C2×C4×C7—22no—
292C2×C4×C7—22no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—22no—
297C2×C4×C7—21no—
298C2×C4×C7—21no—
299C2×C4×C7—21no—
300C2×C4×C7—22no—
301C2×C4×C7—21no—
302C2×C4×C7—41no—
303C2×C4×C7—42no—
304C2×C4×C7—42no—
305C2×C4×C7—82no—
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—42no—
314C7×C8—42no—
315C7×C8—42no—
316C22×Q8—21no—
317C2×Q16—24no—
318C4⋊Q8—22no—
319C4⋊Q8—21no—
320C4⋊Q8—22no—
321C4⋊Q8—21no—
322C4⋊Q8—21no—
323C4⋊Q8—21no—
324C4⋊Q8—21no—
325C4⋊Q8—21no—
326C4⋊Q8C7×D411yescharacteristic
327C4⋊Q8C7×D411yescharacteristic
328C4⋊Q8C23×C712yes—
329C4⋊Q8C23×C711yescharacteristic
330C4⋊Q8C23×C711yescharacteristic
331#126—22no—
332#126—22no—
333#126—22no—
334#126—22no—
335#126—22no—
336#126—22no—
337#126—21no—
338#125—22no—
339#125—22no—
340#125—22no—
341#125—22no—
342#125—21no—
343#125—22no—
344#125—21no—
345#125—21no—
346#125—21no—
347#125C7×D411yescharacteristic
348#124—22no—
349#124—21no—
350#124C7×D411yescharacteristic
351#123—22no—
352#123—21no—
353C22⋊Q8—22no—
354C22⋊Q8—22no—
355C22⋊Q8—21no—
356C22⋊Q8—21no—
357C22⋊Q8—22no—
358C22⋊Q8—21no—
359C22⋊Q8—21no—
360C22⋊Q8C7×D411yescharacteristic
361C22⋊Q8C7×D411yescharacteristic
362C22⋊Q8C7×D411yescharacteristic
363C22⋊Q8C7×D412yes—
364#121C7×D411yescharacteristic
365C4×Q8—22no—
366C4×Q8—21no—
367C4×Q8—21no—
368C4×Q8—21no—
369C4×Q8—21no—
370C4×Q8—21no—
371C4×Q8—21no—
372C4×Q8—21no—
373C4×Q8C7×D412yes—
374C4×Q8C23×C711yescharacteristic
375C4×Q8C23×C711yescharacteristic
376C4×Q8C23×C711yescharacteristic
377C4×D4—21no—
378C4×D4—21no—
379C4×D4C23×C711yescharacteristic
380#117—22no—
381#117—21no—
382#117—21no—
383#117—21no—
384#117—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C42C23×C711yescharacteristic
389#107—22no—
390#107—22no—
391#107—22no—
392#106—22no—
393C4⋊C8—21no—
394C4⋊C8C23×C712yes—
395C4⋊C8C23×C712yes—
396#103—22no—
397#103—22no—
398#103—22no—
399#103—24no—
400#102—22no—
401C22⋊C8—21no—
402C22⋊C8—21no—
403C22⋊C8—21no—
404#97—21no—
405C4×C8—21no—
406C4×C8C23×C712yes—
407C22×C7—21no—
408C22×C7—21no—
409C22×C7—21no—
410C22×C7—41no—
411C22×C7—41no—
412C22×C7—41no—
413C22×C7C2×#12011yescharacteristic
414C4×C7—21no—
415C4×C7—21no—
416C4×C7—21no—
417C4×C7—21no—
418C4×C7—41no—
419C4×C7—41no—
420C4×C7—41no—
421C4×C7—41no—
422C4×C7—42no—
423C4×C7—41no—
424C4×C7—42no—
425C4×C7—41no—
426C4×C7—41no—
427C4×C7—82no—
428C4×C7#57812yes—
429C4×C7#57812yes—
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—41no—
435C2×Q8—82no—
436C2×Q8C2×C7×D411yescharacteristic
437C2×Q8C2×C7×D411yescharacteristic
438C2×Q8C2×C7×D411yescharacteristic
439C2×Q8C2×C7×D412yes—
440C2×D4—21no—
441C2×D4C2×C7×D411yescharacteristic
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—41no—
445C22×C4C2×C7×D411yescharacteristic
446C22×C4C2×C7×D411yescharacteristic
447C22×C4C2×C7×D411yescharacteristic
448Q16—84no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4C2×C7×D412yes—
481C4⋊C4C2×C7×D411yescharacteristic
482C4⋊C4C2×C7×D411yescharacteristic
483C4⋊C4C2×C7×D411yescharacteristic
484C4⋊C4C2×C7×D411yescharacteristic
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—21no—
489C22⋊C4—22no—
490C22⋊C4—21no—
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—21no—
495C22⋊C4—21no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42C2×C7×D411yescharacteristic
504C42C2×C7×D411yescharacteristic
505C42C2×C7×D411yescharacteristic
506C42C24×C711yescharacteristic
507C2×C7—41no—
508C2×C7—81no—
509C2×C7#301111yescharacteristic
510C2×C7#301111yescharacteristic
511C2×C7#302111yescharacteristic
512C23—21no—
513C23C7×#12011yescharacteristic
514Q8—22no—
515Q8—22no—
516Q8—22no—
517Q8—42no—
518Q8—44no—
519Q8—42no—
520Q8—81no—
521Q8—82no—
522Q8—82no—
523D4—22no—
524D4—81no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—82no—
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—42no—
548C8—42no—
549C8—42no—
550C7#1896111yescharacteristic, direct product factor, Sylow
551C22—21no—
552C22—21no—
553C22—21no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22C2×C7×#12011yescharacteristic
558C4—21no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—42no—
567C4—41no—
568C4—42no—
569C4—41no—
570C4—41no—
571C4—82no—
572C4C7×#57812yes—
573C4C7×#57812yes—
574C2—41no—
575C2—81no—
576C2C7×#301111yescharacteristic
577C2C7×#301111yescharacteristic
578C2C7×#302111yescharacteristic
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
3. C7×#3230C2semidirect
4. C7×#3209C2semidirect
5. C7×#1638C2semidirect
6. C7×#1636C2semidirect
7. C7×#1633C2semidirect
8. C7×#1632C2semidirect
9. C7×#1554C2semidirect
10. C7×#1497C2semidirect
55. #18961C7direct (N × H)
149. #3230C2×C7semidirect
150. #3209C2×C7semidirect
151. #1638C2×C7semidirect
152. #1636C2×C7semidirect
153. #1633C2×C7semidirect
154. #1632C2×C7semidirect
155. #1554C2×C7semidirect
156. #1497C2×C7semidirect
550. C7#18961direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35552
C7×#32382
C7×#32302
C7×#32092
C7×#16382
C7×#16362
C7×#16332
C7×#16322
C7×#15542
C7×#14972
C7×#14822
#189617