Finite Group Databasebeta

← C7×#18970 #13196186

Subgroups — C7×#18970 #13196186

Order 1792.

874 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3554C211yesmaximal, characteristic
2C7×#3239C211yesmaximal, characteristic, complemented
3C7×#3226C211yesmaximal, characteristic, complemented
4C7×#3209C211yesmaximal, characteristic, complemented
5C7×#1649C212yesmaximal, complemented
6C7×#1648C212yesmaximal, complemented
7C7×#1641C212yesmaximal, complemented
8C7×#1640C212yesmaximal, complemented
9C7×#1553C211yesmaximal, characteristic, complemented
10C7×#1497C211yesmaximal, characteristic, complemented
11C7×#1482C211yesmaximal, characteristic, complemented
12C7×#562—22no—
13C7×#559—21no—
14C7×#559C2211yescharacteristic
15C7×#557C2211yescharacteristic
16C7×#554—21no—
17C7×#550C2211yescharacteristic
18C7×D4×Q8C2211yescharacteristic
19C7×#548—21no—
20C7×#547—21no—
21C7×#542—21no—
22C7×#542—21no—
23C7×#540—21no—
24C7×#539—22no—
25C7×#536—21no—
26C7×#533C2211yescharacteristic
27C7×#532C2211yescharacteristic
28C2×C7×C4⋊Q8C2211yescharacteristic
29C7×#529C2211yescharacteristic
30C7×#501—21no—
31C7×#500C2212yes—
32C7×#494—21no—
33C7×#489—21no—
34C7×#487—21no—
35C7×#486C2212yes—
36C7×#484—21no—
37C7×#481—21no—
38C7×#480—22no—
39C7×#475C2212yes—
40C7×#474—22no—
41C7×#474C2212yes—
42C7×#462C2212yes—
43C7×#462C2212yes—
44C7×#451—21no—
45C7×#451—21no—
46C7×#432C2211yescharacteristic
47C7×#431C2211yescharacteristic
48C7×#431C2211yescharacteristic
49C7×#332C2212yes—
50C7×#331C2212yes—
51C7×Q8⋊C8C2212yes—
52C7×Q8⋊C8C2212yes—
53C7×Q8⋊C8C2212yes—
54C7×D4⋊C8C2212yes—
55#18970C711yesmaximal, characteristic, direct product factor, Sylow
56C2×C7×#41—21no—
57C2×C7×#41—21no—
58C22×C7×Q8—21no—
59C22×C7×Q8—21no—
60C2×C7×Q16—22no—
61C2×C7×Q16—22no—
62C7×C4⋊Q8—21no—
63C7×C4⋊Q8—21no—
64C7×C4⋊Q8—21no—
65C7×C4⋊Q8—21no—
66C7×C4⋊Q8—21no—
67C7×C4⋊Q8D411yescharacteristic
68C7×C4⋊Q8D411yescharacteristic
69C7×C4⋊Q8C2312yes—
70C7×C4⋊Q8C2311yescharacteristic
71C7×#127—21no—
72C7×#127C2311yescharacteristic
73C7×#126—22no—
74C7×#126—21no—
75C7×#125—22no—
76C7×#125—21no—
77C7×#125—21no—
78C7×#125D411yescharacteristic
79C7×#124—22no—
80C7×#124—22no—
81C7×#124—21no—
82C7×#124—21no—
83C7×#124—21no—
84C7×#124D411yescharacteristic
85C7×#123—22no—
86C7×#123—22no—
87C7×#123—22no—
88C7×#123—22no—
89C7×#123—22no—
90C7×#123—22no—
91C7×#123—21no—
92C7×C22⋊Q8—22no—
93C7×C22⋊Q8—22no—
94C7×C22⋊Q8—22no—
95C7×C22⋊Q8—22no—
96C7×C22⋊Q8—21no—
97C7×C22⋊Q8—22no—
98C7×C22⋊Q8—21no—
99C7×C22⋊Q8D411yescharacteristic
100C7×C22⋊Q8D411yescharacteristic
101C7×C22⋊Q8D411yescharacteristic
102C7×#121—22no—
103C7×#121—22no—
104C7×#121—22no—
105C7×#121—21no—
106C7×#121—21no—
107C7×#121D412yes—
108C7×#121D411yescharacteristic
109C4×C7×Q8—21no—
110C4×C7×Q8—21no—
111C4×C7×Q8—21no—
112C4×C7×Q8C2311yescharacteristic
113C4×C7×Q8C2311yescharacteristic
114C4×C7×Q8C2311yescharacteristic
115C4×C7×D4—22no—
116C4×C7×D4—21no—
117C4×C7×D4—21no—
118C4×C7×D4—21no—
119C4×C7×D4—21no—
120C4×C7×D4—21no—
121C4×C7×D4—21no—
122C4×C7×D4D412yes—
123C4×C7×D4C2311yescharacteristic
124C7×#117—21no—
125C7×#117—21no—
126C2×C7×C4⋊C4—22no—
127C2×C7×C4⋊C4—21no—
128C2×C7×C4⋊C4—21no—
129C2×C42×C7C2311yescharacteristic
130C7×#107—24no—
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×#102—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#3554C2×C711yescharacteristic
148#3239C2×C711yescharacteristic, complemented
149#3226C2×C711yescharacteristic, complemented
150#3209C2×C711yescharacteristic, complemented
151#1649C2×C712yescomplemented
152#1648C2×C712yescomplemented
153#1641C2×C712yescomplemented
154#1640C2×C712yescomplemented
155#1553C2×C711yescharacteristic, complemented
156#1497C2×C711yescharacteristic, complemented
157#1482C2×C711yescharacteristic, complemented
158C7×#41—82no—
159C7×#41—82no—
160C2×C7×Q8—22no—
161C2×C7×Q8—21no—
162C2×C7×Q8—21no—
163C2×C7×Q8—22no—
164C2×C7×Q8—41no—
165C2×C7×Q8—82no—
166C2×C7×Q8—82no—
167C2×C7×Q8C2×D411yescharacteristic
168C2×C7×Q8C2×D411yescharacteristic
169C2×C7×Q8C2×D411yescharacteristic
170C2×C7×D4—22no—
171C2×C7×D4—22no—
172C2×C7×D4—21no—
173C2×C7×D4—22no—
174C2×C7×D4—21no—
175C2×C7×D4—21no—
176C2×C7×D4C2×D411yescharacteristic
177C2×C7×D4C2×D412yes—
178C22×C4×C7—22no—
179C22×C4×C7—22no—
180C22×C4×C7—22no—
181C22×C4×C7—21no—
182C22×C4×C7—22no—
183C22×C4×C7—41no—
184C22×C4×C7C2×D411yescharacteristic
185C22×C4×C7C2×D411yescharacteristic
186C22×C4×C7C2×D411yescharacteristic
187C7×Q16—82no—
188C7×Q16—82no—
189C2×C7×C8—22no—
190C2×C7×C8—22no—
191C2×C7×C8—22no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—21no—
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—21no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—21no—
207C7×C4⋊C4—21no—
208C7×C4⋊C4—22no—
209C7×C4⋊C4—22no—
210C7×C4⋊C4—41no—
211C7×C4⋊C4—41no—
212C7×C4⋊C4C2×D411yescharacteristic
213C7×C4⋊C4C2×D411yescharacteristic
214C7×C4⋊C4C2×D411yescharacteristic
215C7×C4⋊C4C2×D411yescharacteristic
216C7×C4⋊C4C2×D412yes—
217C7×C22⋊C4—22no—
218C7×C22⋊C4—22no—
219C7×C22⋊C4—22no—
220C7×C22⋊C4—22no—
221C7×C22⋊C4—22no—
222C7×C22⋊C4—21no—
223C7×C22⋊C4—21no—
224C7×C22⋊C4—21no—
225C7×C22⋊C4—21no—
226C7×C22⋊C4—21no—
227C7×C22⋊C4—21no—
228C7×C22⋊C4—22no—
229C7×C22⋊C4—22no—
230C7×C22⋊C4—21no—
231C42×C7—22no—
232C42×C7—21no—
233C42×C7—21no—
234C42×C7—21no—
235C42×C7C2×D411yescharacteristic
236C42×C7C2×D411yescharacteristic
237C42×C7C2×D411yescharacteristic
238C42×C7C2411yescharacteristic
239#562—22no—
240#559—21no—
241#559C22×C711yescharacteristic
242#557C22×C711yescharacteristic
243#554—21no—
244#550C22×C711yescharacteristic
245D4×Q8C22×C711yescharacteristic
246#548—21no—
247#547—21no—
248#542—21no—
249#542—21no—
250#540—21no—
251#539—22no—
252#536—21no—
253#533C22×C711yescharacteristic
254#532C22×C711yescharacteristic
255C2×C4⋊Q8C22×C711yescharacteristic
256#529C22×C711yescharacteristic
257#501—21no—
258#500C22×C712yes—
259#494—21no—
260#489—21no—
261#487—21no—
262#486C22×C712yes—
263#484—21no—
264#481—21no—
265#480—22no—
266#475C22×C712yes—
267#474—22no—
268#474C22×C712yes—
269#462C22×C712yes—
270#462C22×C712yes—
271#451—21no—
272#451—21no—
273#432C22×C711yescharacteristic
274#431C22×C711yescharacteristic
275#431C22×C711yescharacteristic
276#332C22×C712yes—
277#331C22×C712yes—
278Q8⋊C8C22×C712yes—
279Q8⋊C8C22×C712yes—
280Q8⋊C8C22×C712yes—
281D4⋊C8C22×C712yes—
282C23×C7—21no—
283C23×C7—22no—
284C23×C7#12011yescharacteristic
285C7×Q8—22no—
286C7×Q8—22no—
287C7×Q8—22no—
288C7×Q8—42no—
289C7×Q8—42no—
290C7×Q8—42no—
291C7×Q8—81no—
292C7×Q8—82no—
293C7×Q8—82no—
294C7×D4—22no—
295C7×D4—42no—
296C7×D4—42no—
297C7×D4—42no—
298C7×D4—81no—
299C7×D4—82no—
300C7×D4—82no—
301C7×D4—82no—
302C2×C4×C7—22no—
303C2×C4×C7—22no—
304C2×C4×C7—21no—
305C2×C4×C7—21no—
306C2×C4×C7—21no—
307C2×C4×C7—21no—
308C2×C4×C7—21no—
309C2×C4×C7—22no—
310C2×C4×C7—21no—
311C2×C4×C7—21no—
312C2×C4×C7—41no—
313C2×C4×C7—42no—
314C2×C4×C7—42no—
315C2×C4×C7—82no—
316C2×C4×C7—82no—
317C2×C4×C7—82no—
318C2×C4×C7—82no—
319C2×C4×C7—82no—
320C2×C4×C7#12011yescharacteristic
321C2×C4×C7#12011yescharacteristic
322C2×C4×C7#12011yescharacteristic
323C2×C4×C7C22×D411yescharacteristic
324C2×C4×C7C22×D411yescharacteristic
325C2×C4×C7C22×D411yescharacteristic
326C7×C8—42no—
327C7×C8—42no—
328C7×C8—42no—
329C2×#41—21no—
330C2×#41—21no—
331C22×Q8—21no—
332C22×Q8—21no—
333C2×Q16—22no—
334C2×Q16—22no—
335C4⋊Q8—21no—
336C4⋊Q8—21no—
337C4⋊Q8—21no—
338C4⋊Q8—21no—
339C4⋊Q8—21no—
340C4⋊Q8C7×D411yescharacteristic
341C4⋊Q8C7×D411yescharacteristic
342C4⋊Q8C23×C712yes—
343C4⋊Q8C23×C711yescharacteristic
344#127—21no—
345#127C23×C711yescharacteristic
346#126—22no—
347#126—21no—
348#125—22no—
349#125—21no—
350#125—21no—
351#125C7×D411yescharacteristic
352#124—22no—
353#124—22no—
354#124—21no—
355#124—21no—
356#124—21no—
357#124C7×D411yescharacteristic
358#123—22no—
359#123—22no—
360#123—22no—
361#123—22no—
362#123—22no—
363#123—22no—
364#123—21no—
365C22⋊Q8—22no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C22⋊Q8—22no—
369C22⋊Q8—21no—
370C22⋊Q8—22no—
371C22⋊Q8—21no—
372C22⋊Q8C7×D411yescharacteristic
373C22⋊Q8C7×D411yescharacteristic
374C22⋊Q8C7×D411yescharacteristic
375#121—22no—
376#121—22no—
377#121—22no—
378#121—21no—
379#121—21no—
380#121C7×D411yescharacteristic
381#121C7×D412yes—
382C4×Q8—21no—
383C4×Q8—21no—
384C4×Q8—21no—
385C4×Q8C23×C711yescharacteristic
386C4×Q8C23×C711yescharacteristic
387C4×Q8C23×C711yescharacteristic
388C4×D4—22no—
389C4×D4—21no—
390C4×D4—21no—
391C4×D4—21no—
392C4×D4—21no—
393C4×D4—21no—
394C4×D4—21no—
395C4×D4C7×D412yes—
396C4×D4C23×C711yescharacteristic
397#117—21no—
398#117—21no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C42C23×C711yescharacteristic
403#107—24no—
404#107—22no—
405#106—22no—
406C4⋊C8—21no—
407C4⋊C8C23×C712yes—
408C4⋊C8C23×C712yes—
409#103—22no—
410#103—22no—
411#103—22no—
412#102—24no—
413#102—22no—
414C22⋊C8—21no—
415C22⋊C8—21no—
416C22⋊C8—21no—
417#97—21no—
418C4×C8—21no—
419C4×C8C23×C712yes—
420C22×C7—21no—
421C22×C7—21no—
422C22×C7—21no—
423C22×C7—41no—
424C22×C7—42no—
425C22×C7—41no—
426C22×C7—42no—
427C22×C7—42no—
428C22×C7—41no—
429C22×C7C2×#12011yescharacteristic
430C4×C7—21no—
431C4×C7—21no—
432C4×C7—21no—
433C4×C7—21no—
434C4×C7—41no—
435C4×C7—41no—
436C4×C7—42no—
437C4×C7—41no—
438C4×C7—41no—
439C4×C7—41no—
440C4×C7—41no—
441C4×C7—41no—
442C4×C7—82no—
443C4×C7#57612yes—
444C4×C7#57812yes—
445#41—82no—
446#41—82no—
447C2×Q8—22no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—22no—
451C2×Q8—41no—
452C2×Q8—82no—
453C2×Q8—82no—
454C2×Q8C2×C7×D411yescharacteristic
455C2×Q8C2×C7×D411yescharacteristic
456C2×Q8C2×C7×D411yescharacteristic
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—21no—
460C2×D4—22no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4C2×C7×D411yescharacteristic
464C2×D4C2×C7×D412yes—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—41no—
471C22×C4C2×C7×D411yescharacteristic
472C22×C4C2×C7×D411yescharacteristic
473C22×C4C2×C7×D411yescharacteristic
474Q16—82no—
475Q16—82no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4C2×C7×D411yescharacteristic
500C4⋊C4C2×C7×D411yescharacteristic
501C4⋊C4C2×C7×D411yescharacteristic
502C4⋊C4C2×C7×D411yescharacteristic
503C4⋊C4C2×C7×D412yes—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—21no—
518C42—22no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42C2×C7×D411yescharacteristic
523C42C2×C7×D411yescharacteristic
524C42C2×C7×D411yescharacteristic
525C42C24×C711yescharacteristic
526C2×C7—41no—
527C2×C7—81no—
528C2×C7—82no—
529C2×C7#300811yescharacteristic
530C2×C7#301111yescharacteristic
531C2×C7#302211yescharacteristic
532C23—21no—
533C23—22no—
534C23C7×#12011yescharacteristic
535Q8—22no—
536Q8—22no—
537Q8—22no—
538Q8—42no—
539Q8—42no—
540Q8—42no—
541Q8—81no—
542Q8—82no—
543Q8—82no—
544D4—22no—
545D4—42no—
546D4—42no—
547D4—42no—
548D4—81no—
549D4—82no—
550D4—82no—
551D4—82no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4C7×#12011yescharacteristic
571C2×C4C7×#12011yescharacteristic
572C2×C4C7×#12011yescharacteristic
573C2×C4C22×C7×D411yescharacteristic
574C2×C4C22×C7×D411yescharacteristic
575C2×C4C22×C7×D411yescharacteristic
576C8—42no—
577C8—42no—
578C8—42no—
579C7#1897011yescharacteristic, direct product factor, Sylow
580C22—21no—
581C22—21no—
582C22—21no—
583C22—41no—
584C22—42no—
585C22—41no—
586C22—42no—
587C22—42no—
588C22—41no—
589C22C2×C7×#12011yescharacteristic
590C4—21no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—41no—
595C4—41no—
596C4—42no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—82no—
603C4C7×#57612yes—
604C4C7×#57812yes—
605C2—41no—
606C2—81no—
607C2—82no—
608C2C7×#300811yescharacteristic
609C2C7×#301111yescharacteristic
610C2C7×#302211yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3554C2
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×#3239C2semidirect
3. C7×#3226C2semidirect
4. C7×#3209C2semidirect
5. C7×#1649C2semidirect
6. C7×#1648C2semidirect
7. C7×#1641C2semidirect
8. C7×#1640C2semidirect
9. C7×#1553C2semidirect
10. C7×#1497C2semidirect
11. C7×#1482C2semidirect
55. #18970C7direct (N × H)
148. #3239C2×C7semidirect
149. #3226C2×C7semidirect
150. #3209C2×C7semidirect
151. #1649C2×C7semidirect
152. #1648C2×C7semidirect
153. #1641C2×C7semidirect
154. #1640C2×C7semidirect
155. #1553C2×C7semidirect
156. #1497C2×C7semidirect
157. #1482C2×C7semidirect
579. C7#18970direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35542
C7×#32392
C7×#32262
C7×#32092
C7×#16492
C7×#16482
C7×#16412
C7×#16402
C7×#15532
C7×#14972
C7×#14822
#189707