Finite Group Databasebeta

← C7×#8538 #13185754

Subgroups — C7×#8538 #13185754

Order 1792.

820 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3505C211yesmaximal, characteristic
2C7×#1901C211yesmaximal, characteristic, complemented
3C7×#1900C211yesmaximal, characteristic, complemented
4C7×#1870C211yesmaximal, characteristic, complemented
5C7×#1472C211yesmaximal, characteristic, complemented
6C7×#1471C211yesmaximal, characteristic, complemented
7C7×#1465C211yesmaximal, characteristic, complemented
8C22×C7×M16C2211yescharacteristic
9C7×#553—22no—
10C7×#553—21no—
11C7×#552—21no—
12C7×#551—22no—
13C7×#548—21no—
14C7×#546—21no—
15C7×#539—21no—
16C7×#538—21no—
17C7×#534C412yes—
18C7×#529—21no—
19C7×#525—21no—
20C2×C7×C22⋊Q8C2211yescharacteristic
21C2×C7×#121C2211yescharacteristic
22C7×#518C412yes—
23C2×C4×C7×D4C2211yescharacteristic
24C2×C7×#101—21no—
25C2×C7×#100—21no—
26C7×#407C2211yescharacteristic
27C7×#407C2211yescharacteristic
28C2×C7×C22⋊C8—21no—
29C2×C7×C22⋊C8C2211yescharacteristic
30C7×#343—21no—
31C7×#343—21no—
32C7×#336—21no—
33C7×#324—21no—
34C7×#324—22no—
35C7×#323—21no—
36C7×#323—22no—
37#8538C711yesmaximal, characteristic, direct product factor, Sylow
38C2×C7×#41—21no—
39C22×C7×Q8D411yescharacteristic
40C22×C7×D4C2×C411yescharacteristic
41C22×C7×D4C2×C412yes—
42C22×C7×D4D411yescharacteristic
43C23×C4×C7D411yescharacteristic
44C23×C4×C7C2311yescharacteristic
45C2×C7×M16—21no—
46C2×C7×M16—21no—
47C2×C7×M16—21no—
48C2×C7×M16—21no—
49C2×C7×M16—42no—
50C22×C7×C8—21no—
51C22×C7×C8—21no—
52C7×#126—22no—
53C7×#126—22no—
54C7×#126—22no—
55C7×#125—21no—
56C7×#125—22no—
57C7×#124—21no—
58C7×#124—22no—
59C7×#123—22no—
60C7×#123—22no—
61C7×#123—22no—
62C7×#123—22no—
63C7×#123—22no—
64C7×C22⋊Q8—21no—
65C7×C22⋊Q8—22no—
66C7×C22⋊Q8—21no—
67C7×C22⋊Q8—21no—
68C7×C22⋊Q8—21no—
69C7×C22⋊Q8—21no—
70C7×#121—21no—
71C7×#121—22no—
72C7×#121—22no—
73C7×#121—22no—
74C7×#121—21no—
75C7×#121—21no—
76C7×#121—21no—
77C7×#121—21no—
78C7×#120—22no—
79C7×#120—22no—
80C4×C7×Q8—21no—
81C4×C7×D4—22no—
82C4×C7×D4—21no—
83C4×C7×D4—22no—
84C4×C7×D4—22no—
85C4×C7×D4—22no—
86C4×C7×D4—22no—
87C4×C7×D4—21no—
88C4×C7×D4—21no—
89C4×C7×D4—21no—
90C4×C7×D4—21no—
91C7×#117—21no—
92C7×#117—22no—
93C7×#117—22no—
94C2×C7×C4⋊C4D411yescharacteristic
95C2×C7×C4⋊C4D412yes—
96C2×C7×C4⋊C4D411yescharacteristic
97C2×C7×C4⋊C4D411yescharacteristic
98C2×C7×C22⋊C4—22no—
99C2×C7×C22⋊C4C2×C411yescharacteristic
100C2×C7×C22⋊C4C2×C412yes—
101C2×C7×C22⋊C4D411yescharacteristic
102C2×C7×C22⋊C4D412yes—
103C2×C42×C7D411yescharacteristic
104C7×#101—42no—
105C7×#100—42no—
106C7×C22⋊C8—21no—
107C7×C22⋊C8—21no—
108C7×C22⋊C8—21no—
109C7×C22⋊C8—21no—
110C7×C22⋊C8—21no—
111C7×C22⋊C8—21no—
112C7×C22⋊C8—42no—
113#3505C2×C711yescharacteristic
114#1901C2×C711yescharacteristic, complemented
115#1900C2×C711yescharacteristic, complemented
116#1870C2×C711yescharacteristic, complemented
117#1472C2×C711yescharacteristic, complemented
118#1471C2×C711yescharacteristic, complemented
119#1465C2×C711yescharacteristic, complemented
120C24×C7C22⋊C411yescharacteristic
121C24×C7C22⋊C412yes—
122C24×C7C22×C411yescharacteristic
123C7×#41—81no—
124C7×#41—81no—
125C2×C7×Q8—21no—
126C2×C7×Q8—21no—
127C2×C7×Q8—41no—
128C2×C7×Q8—41no—
129C2×C7×D4—21no—
130C2×C7×D4—22no—
131C2×C7×D4—22no—
132C2×C7×D4—21no—
133C2×C7×D4—22no—
134C2×C7×D4—22no—
135C2×C7×D4—21no—
136C2×C7×D4—21no—
137C2×C7×D4—41no—
138C2×C7×D4—41no—
139C2×C7×D4—41no—
140C2×C7×D4—41no—
141C2×C7×D4—42no—
142C2×C7×D4—42no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—41no—
148C22×C4×C7—41no—
149C22×C4×C7—41no—
150C22×C4×C7—41no—
151C22×C4×C7C22⋊C411yescharacteristic
152C22×C4×C7C22⋊C411yescharacteristic
153C22×C4×C7C22⋊C412yes—
154C22×C4×C7C22⋊C411yescharacteristic
155C22×C4×C7C22⋊C412yes—
156C22×C4×C7C22⋊C412yes—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C7×M16—22no—
164C7×M16—22no—
165C7×M16—41no—
166C7×M16—41no—
167C7×M16—41no—
168C7×M16—41no—
169C7×M16—41no—
170C7×M16—41no—
171C2×C7×C8—22no—
172C2×C7×C8—41no—
173C2×C7×C8—41no—
174C2×C7×C8—41no—
175C2×C7×C8—41no—
176C2×C7×C8—41no—
177C2×C7×C8—41no—
178C2×C7×C8—41no—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—22no—
181C7×C4⋊C4—22no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—21no—
191C7×C22⋊C4—22no—
192C7×C22⋊C4—22no—
193C7×C22⋊C4—22no—
194C7×C22⋊C4—22no—
195C7×C22⋊C4—22no—
196C7×C22⋊C4—22no—
197C7×C22⋊C4—21no—
198C7×C22⋊C4—22no—
199C7×C22⋊C4—21no—
200C7×C22⋊C4—22no—
201C7×C22⋊C4—22no—
202C7×C22⋊C4—21no—
203C7×C22⋊C4—21no—
204C7×C22⋊C4—22no—
205C42×C7—22no—
206C42×C7—22no—
207C42×C7—21no—
208C42×C7—21no—
209C22×M16C22×C711yescharacteristic
210#553—22no—
211#553—21no—
212#552—21no—
213#551—22no—
214#548—21no—
215#546—21no—
216#539—21no—
217#538—21no—
218#534C4×C712yes—
219#529—21no—
220#525—21no—
221C2×C22⋊Q8C22×C711yescharacteristic
222C2×#121C22×C711yescharacteristic
223#518C4×C712yes—
224C2×C4×D4C22×C711yescharacteristic
225C2×#101—21no—
226C2×#100—21no—
227#407C22×C711yescharacteristic
228#407C22×C711yescharacteristic
229C2×C22⋊C8—21no—
230C2×C22⋊C8C22×C711yescharacteristic
231#343—21no—
232#343—21no—
233#336—21no—
234#324—21no—
235#324—22no—
236#323—21no—
237#323—22no—
238C23×C7—21no—
239C23×C7—21no—
240C23×C7—21no—
241C23×C7—22no—
242C23×C7—21no—
243C23×C7—21no—
244C23×C7—41no—
245C23×C7—41no—
246C23×C7—41no—
247C23×C7—41no—
248C23×C7—42no—
249C23×C7—42no—
250C23×C7—42no—
251C23×C7C2×C22⋊C411yescharacteristic
252C23×C7C2×C22⋊C411yescharacteristic
253C23×C7C2×C22⋊C411yescharacteristic
254C7×Q8—41no—
255C7×Q8—41no—
256C7×Q8—81no—
257C7×D4—42no—
258C7×D4—42no—
259C7×D4—44no—
260C7×D4—42no—
261C7×D4—42no—
262C7×D4—41no—
263C7×D4—41no—
264C7×D4—81no—
265C7×D4—82no—
266C7×D4—82no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—22no—
276C2×C4×C7—21no—
277C2×C4×C7—22no—
278C2×C4×C7—22no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—42no—
289C2×C4×C7—42no—
290C2×C4×C7—42no—
291C2×C4×C7—81no—
292C2×C4×C7—82no—
293C2×C4×C7—82no—
294C2×C4×C7#12011yescharacteristic
295C2×C4×C7#12011yescharacteristic
296C2×C4×C7#12011yescharacteristic
297C2×C4×C7#12011yescharacteristic
298C7×C8—42no—
299C7×C8—81no—
300C7×C8—81no—
301C7×C8—81no—
302C2×#41—21no—
303C22×Q8C7×D411yescharacteristic
304C22×D4C2×C4×C711yescharacteristic
305C22×D4C2×C4×C712yes—
306C22×D4C7×D411yescharacteristic
307C23×C4C7×D411yescharacteristic
308C23×C4C23×C711yescharacteristic
309C2×M16—21no—
310C2×M16—21no—
311C2×M16—21no—
312C2×M16—21no—
313C2×M16—42no—
314C22×C8—21no—
315C22×C8—21no—
316#126—22no—
317#126—22no—
318#126—22no—
319#125—21no—
320#125—22no—
321#124—21no—
322#124—22no—
323#123—22no—
324#123—22no—
325#123—22no—
326#123—22no—
327#123—22no—
328C22⋊Q8—21no—
329C22⋊Q8—22no—
330C22⋊Q8—21no—
331C22⋊Q8—21no—
332C22⋊Q8—21no—
333C22⋊Q8—21no—
334#121—21no—
335#121—22no—
336#121—22no—
337#121—22no—
338#121—21no—
339#121—21no—
340#121—21no—
341#121—21no—
342#120—22no—
343#120—22no—
344C4×Q8—21no—
345C4×D4—22no—
346C4×D4—21no—
347C4×D4—22no—
348C4×D4—22no—
349C4×D4—22no—
350C4×D4—22no—
351C4×D4—21no—
352C4×D4—21no—
353C4×D4—21no—
354C4×D4—21no—
355#117—21no—
356#117—22no—
357#117—22no—
358C2×C4⋊C4C7×D411yescharacteristic
359C2×C4⋊C4C7×D411yescharacteristic
360C2×C4⋊C4C7×D412yes—
361C2×C4⋊C4C7×D411yescharacteristic
362C2×C22⋊C4—22no—
363C2×C22⋊C4C2×C4×C711yescharacteristic
364C2×C22⋊C4C2×C4×C712yes—
365C2×C22⋊C4C7×D411yescharacteristic
366C2×C22⋊C4C7×D412yes—
367C2×C42C7×D411yescharacteristic
368#101—42no—
369#100—42no—
370C22⋊C8—21no—
371C22⋊C8—21no—
372C22⋊C8—21no—
373C22⋊C8—21no—
374C22⋊C8—21no—
375C22⋊C8—21no—
376C22⋊C8—42no—
377C22×C7—21no—
378C22×C7—21no—
379C22×C7—21no—
380C22×C7—21no—
381C22×C7—21no—
382C22×C7—21no—
383C22×C7—21no—
384C22×C7—21no—
385C22×C7—42no—
386C22×C7—41no—
387C22×C7—42no—
388C22×C7—41no—
389C22×C7—42no—
390C22×C7—42no—
391C22×C7—41no—
392C22×C7—81no—
393C22×C7—81no—
394C22×C7—81no—
395C22×C7—162no—
396C22×C7#37911yescharacteristic
397C22×C7#41311yescharacteristic
398C22×C7#41311yescharacteristic
399C4×C7—21no—
400C4×C7—21no—
401C4×C7—21no—
402C4×C7—41no—
403C4×C7—42no—
404C4×C7—41no—
405C4×C7—42no—
406C4×C7—41no—
407C4×C7—42no—
408C4×C7B(4,𝔽2)11yescharacteristic
409C4×C7#45811yescharacteristic
410C24C7×C22⋊C411yescharacteristic
411C24C7×C22⋊C412yes—
412C24C22×C4×C711yescharacteristic
413#41—81no—
414#41—81no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—41no—
418C2×Q8—41no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—41no—
428C2×D4—41no—
429C2×D4—41no—
430C2×D4—41no—
431C2×D4—42no—
432C2×D4—42no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4C7×C22⋊C411yescharacteristic
442C22×C4C7×C22⋊C411yescharacteristic
443C22×C4C7×C22⋊C412yes—
444C22×C4C7×C22⋊C411yescharacteristic
445C22×C4C7×C22⋊C412yes—
446C22×C4C7×C22⋊C412yes—
447C22×C4C2×C7×D411yescharacteristic
448C22×C4C2×C7×D411yescharacteristic
449C22×C4C2×C7×D411yescharacteristic
450C22×C4C2×C7×D411yescharacteristic
451C22×C4C2×C7×D411yescharacteristic
452C22×C4C2×C7×D411yescharacteristic
453M16—22no—
454M16—22no—
455M16—41no—
456M16—41no—
457M16—41no—
458M16—41no—
459M16—41no—
460M16—41no—
461C2×C8—22no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—21no—
488C22⋊C4—22no—
489C22⋊C4—21no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—22no—
495C42—22no—
496C42—22no—
497C42—21no—
498C42—21no—
499C2×C7—21no—
500C2×C7—21no—
501C2×C7—41no—
502C2×C7—41no—
503C2×C7—81no—
504C2×C7—82no—
505C2×C7#178511yescharacteristic
506C2×C7#179011yescharacteristic
507C2×C7#179111yescharacteristic
508C23—21no—
509C23—21no—
510C23—21no—
511C23—22no—
512C23—21no—
513C23—21no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23C2×C7×C22⋊C411yescharacteristic
522C23C2×C7×C22⋊C411yescharacteristic
523C23C2×C7×C22⋊C411yescharacteristic
524Q8—41no—
525Q8—41no—
526Q8—81no—
527D4—42no—
528D4—42no—
529D4—44no—
530D4—42no—
531D4—42no—
532D4—41no—
533D4—41no—
534D4—81no—
535D4—82no—
536D4—82no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—81no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4C7×#12011yescharacteristic
565C2×C4C7×#12011yescharacteristic
566C2×C4C7×#12011yescharacteristic
567C2×C4C7×#12011yescharacteristic
568C8—42no—
569C8—81no—
570C8—81no—
571C8—81no—
572C7#853811yescharacteristic, direct product factor, Sylow
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—42no—
582C22—41no—
583C22—42no—
584C22—41no—
585C22—42no—
586C22—42no—
587C22—41no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—162no—
592C22C7×#37911yescharacteristic
593C22C7×#41311yescharacteristic
594C22C7×#41311yescharacteristic
595C4—21no—
596C4—21no—
597C4—21no—
598C4—41no—
599C4—42no—
600C4—41no—
601C4—42no—
602C4—41no—
603C4—42no—
604C4C7×B(4,𝔽2)11yescharacteristic
605C4C7×#45811yescharacteristic
606C2—21no—
607C2—21no—
608C2—41no—
609C2—41no—
610C2—81no—
611C2—82no—
612C2C7×#178511yescharacteristic
613C2C7×#179011yescharacteristic
614C2C7×#179111yescharacteristic
subgroupG/H
center, socleC22×C7#379
commutatorC24C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC7×#3505C2
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#1901C2semidirect
3. C7×#1900C2semidirect
4. C7×#1870C2semidirect
5. C7×#1472C2semidirect
6. C7×#1471C2semidirect
7. C7×#1465C2semidirect
37. #8538C7direct (N × H)
114. #1901C2×C7semidirect
115. #1900C2×C7semidirect
116. #1870C2×C7semidirect
117. #1472C2×C7semidirect
118. #1471C2×C7semidirect
119. #1465C2×C7semidirect
572. C7#8538direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35052
C7×#19012
C7×#19002
C7×#18702
C7×#14722
C7×#14712
C7×#14652
#85387