Finite Group Databasebeta

← C7×#18429 #13195645

Subgroups — C7×#18429 #13195645

Order 1792.

932 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3561C211yesmaximal, characteristic
2C7×#3218C211yesmaximal, characteristic, complemented
3C7×#3209C211yesmaximal, characteristic, complemented
4C7×#2640C211yesmaximal, characteristic, complemented
5C7×#1606C214yesmaximal, complemented
6C7×#1602C214yesmaximal, complemented
7C7×#1506C211yesmaximal, characteristic, complemented
8C7×#1500C211yesmaximal, characteristic, complemented
9C7×#1469C211yesmaximal, characteristic, complemented
10C7×#563—21no—
11C7×#562—21no—
12C7×#556—21no—
13C7×#551—22no—
14C7×D4×Q8—21no—
15C7×#548—21no—
16C7×#546—21no—
17C7×#543C2212yes—
18C7×#542—21no—
19C7×#542—22no—
20C7×#538C2212yes—
21C7×#536—21no—
22C7×#533C2211yescharacteristic
23C2×C7×C4⋊Q8C2211yescharacteristic
24C7×#529C2211yescharacteristic
25C2×C7×#124C2211yescharacteristic
26C7×#497—21no—
27C7×#494—21no—
28C7×#491—21no—
29C7×#487—21no—
30C7×#462—22no—
31C7×#462—21no—
32C7×#461—21no—
33C7×#451C2214yes—
34C7×#451C2212yes—
35C7×#448C2212yes—
36C7×#432C2211yescharacteristic
37C7×#431C2211yescharacteristic
38C7×#397—22no—
39C7×#393C2214yes—
40C7×#391—21no—
41C7×#390—21no—
42C7×#376C2211yescharacteristic
43C7×#328C2212yes—
44C7×#328C2214yes—
45C7×#327C2212yes—
46C7×#323C2214yes—
47#18429C711yesmaximal, characteristic, direct product factor, Sylow
48C2×C7×#41—21no—
49C22×C7×Q8—21no—
50C22×C7×Q8D411yescharacteristic
51C22×C7×Q8C2312yes—
52C22×C7×D4D411yescharacteristic
53C2×C7×Q16—24no—
54C2×C7×Q16—22no—
55C2×C7×SD16—22no—
56C7×C4⋊Q8—21no—
57C7×C4⋊Q8—22no—
58C7×C4⋊Q8—21no—
59C7×C4⋊Q8—21no—
60C7×C4⋊Q8—21no—
61C7×C4⋊Q8—21no—
62C7×C4⋊Q8D411yescharacteristic
63C7×C4⋊Q8D411yescharacteristic
64C7×#126—22no—
65C7×#126—22no—
66C7×#126—22no—
67C7×#126—21no—
68C7×#125—21no—
69C7×#125D411yescharacteristic
70C7×#124—21no—
71C7×#124—22no—
72C7×#124—22no—
73C7×#124—21no—
74C7×#124—21no—
75C7×#124—21no—
76C7×#124—21no—
77C7×#124D411yescharacteristic
78C7×#123—22no—
79C7×#123—22no—
80C7×#123—22no—
81C7×#123—21no—
82C7×#123—22no—
83C7×C22⋊Q8—22no—
84C7×C22⋊Q8—21no—
85C7×C22⋊Q8—22no—
86C7×C22⋊Q8—22no—
87C7×C22⋊Q8—22no—
88C7×C22⋊Q8—21no—
89C7×C22⋊Q8—22no—
90C7×C22⋊Q8—22no—
91C7×C22⋊Q8—22no—
92C7×C22⋊Q8C2311yescharacteristic
93C7×C22⋊Q8C2312yes—
94C7×#121—21no—
95C7×#121—22no—
96C7×#121—22no—
97C7×#121C2311yescharacteristic
98C7×#120—22no—
99C4×C7×Q8—21no—
100C4×C7×Q8—21no—
101C4×C7×Q8D411yescharacteristic
102C4×C7×Q8D412yes—
103C4×C7×D4—21no—
104C4×C7×D4—22no—
105C4×C7×D4—21no—
106C4×C7×D4—22no—
107C4×C7×D4D411yescharacteristic
108C7×#117—22no—
109C7×#117—21no—
110C7×#117—21no—
111C2×C7×C4⋊C4—21no—
112C2×C7×C4⋊C4—21no—
113C2×C7×C22⋊C4—22no—
114C2×C7×C22⋊C4—24no—
115C2×C7×C22⋊C4D412yes—
116C2×C7×C22⋊C4C2312yes—
117C2×C42×C7—21no—
118C2×C42×C7C2311yescharacteristic
119C7×C4⋊C8—21no—
120C7×C4⋊C8—21no—
121C7×#103—24no—
122C7×#103—22no—
123C7×#103—22no—
124C7×C22⋊C8C2312yes—
125C7×C22⋊C8C2312yes—
126C7×#97—21no—
127C4×C7×C8—21no—
128C7×#95—21no—
129C7×#95—24no—
130C7×#95C2312yes—
131#3561C2×C711yescharacteristic
132#3218C2×C711yescharacteristic, complemented
133#3209C2×C711yescharacteristic, complemented
134#2640C2×C711yescharacteristic, complemented
135#1606C2×C714yescomplemented
136#1602C2×C714yescomplemented
137#1506C2×C711yescharacteristic, complemented
138#1500C2×C711yescharacteristic, complemented
139#1469C2×C711yescharacteristic, complemented
140C24×C7C2×D412yes—
141C7×#41—82no—
142C2×C7×Q8—22no—
143C2×C7×Q8—21no—
144C2×C7×Q8—21no—
145C2×C7×Q8—21no—
146C2×C7×Q8—22no—
147C2×C7×Q8—22no—
148C2×C7×Q8—42no—
149C2×C7×Q8—82no—
150C2×C7×Q8—82no—
151C2×C7×Q8C2×D411yescharacteristic
152C2×C7×Q8C2×D412yes—
153C2×C7×D4—21no—
154C2×C7×D4—22no—
155C2×C7×D4—22no—
156C2×C7×D4—21no—
157C2×C7×D4—21no—
158C2×C7×D4—42no—
159C2×C7×D4C2×D411yescharacteristic
160C22×C4×C7—22no—
161C22×C4×C7—22no—
162C22×C4×C7—22no—
163C22×C4×C7—21no—
164C22×C4×C7—22no—
165C22×C4×C7—22no—
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D412yes—
169C22×C4×C7C2411yescharacteristic
170C7×Q16—82no—
171C7×Q16—84no—
172C7×SD16—82no—
173C2×C7×C8—22no—
174C2×C7×C8—22no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—41no—
189C7×C4⋊C4—41no—
190C7×C4⋊C4—41no—
191C7×C4⋊C4—41no—
192C7×C4⋊C4C2×D411yescharacteristic
193C7×C4⋊C4C2×D411yescharacteristic
194C7×C4⋊C4C2×D412yes—
195C7×C22⋊C4—22no—
196C7×C22⋊C4—22no—
197C7×C22⋊C4—22no—
198C7×C22⋊C4—22no—
199C7×C22⋊C4—22no—
200C7×C22⋊C4—21no—
201C7×C22⋊C4—22no—
202C7×C22⋊C4—22no—
203C7×C22⋊C4—22no—
204C7×C22⋊C4—21no—
205C7×C22⋊C4—21no—
206C7×C22⋊C4—21no—
207C7×C22⋊C4—22no—
208C7×C22⋊C4—22no—
209C7×C22⋊C4—21no—
210C7×C22⋊C4—84no—
211C42×C7—22no—
212C42×C7—22no—
213C42×C7—21no—
214C42×C7—42no—
215C42×C7C2×D411yescharacteristic
216C42×C7C2×D411yescharacteristic
217C42×C7C2×D411yescharacteristic
218C42×C7C2×D411yescharacteristic
219#563—21no—
220#562—21no—
221#556—21no—
222#551—22no—
223D4×Q8—21no—
224#548—21no—
225#546—21no—
226#543C22×C712yes—
227#542—21no—
228#542—22no—
229#538C22×C712yes—
230#536—21no—
231#533C22×C711yescharacteristic
232C2×C4⋊Q8C22×C711yescharacteristic
233#529C22×C711yescharacteristic
234C2×#124C22×C711yescharacteristic
235#497—21no—
236#494—21no—
237#491—21no—
238#487—21no—
239#462—22no—
240#462—21no—
241#461—21no—
242#451C22×C712yes—
243#451C22×C714yes—
244#448C22×C712yes—
245#432C22×C711yescharacteristic
246#431C22×C711yescharacteristic
247#397—22no—
248#393C22×C714yes—
249#391—21no—
250#390—21no—
251#376C22×C711yescharacteristic
252#328C22×C712yes—
253#328C22×C714yes—
254#327C22×C712yes—
255#323C22×C714yes—
256C23×C7—22no—
257C23×C7—21no—
258C23×C7—24no—
259C23×C7—42no—
260C23×C7—42no—
261C23×C7C22×D411yescharacteristic
262C7×Q8—41no—
263C7×Q8—42no—
264C7×Q8—42no—
265C7×Q8—42no—
266C7×Q8—44no—
267C7×Q8—42no—
268C7×Q8—42no—
269C7×Q8—81no—
270C7×Q8—82no—
271C7×D4—41no—
272C7×D4—42no—
273C7×D4—42no—
274C7×D4—81no—
275C7×D4—82no—
276C7×D4—82no—
277C2×C4×C7—22no—
278C2×C4×C7—22no—
279C2×C4×C7—21no—
280C2×C4×C7—22no—
281C2×C4×C7—21no—
282C2×C4×C7—22no—
283C2×C4×C7—22no—
284C2×C4×C7—21no—
285C2×C4×C7—41no—
286C2×C4×C7—42no—
287C2×C4×C7—42no—
288C2×C4×C7—42no—
289C2×C4×C7—42no—
290C2×C4×C7—44no—
291C2×C4×C7—42no—
292C2×C4×C7—42no—
293C2×C4×C7—82no—
294C2×C4×C7—82no—
295C2×C4×C7—82no—
296C2×C4×C7#12011yescharacteristic
297C2×C4×C7#12011yescharacteristic
298C2×C4×C7#12011yescharacteristic
299C2×C4×C7#12011yescharacteristic
300C2×C4×C7C22×D411yescharacteristic
301C2×C4×C7C22×D411yescharacteristic
302C7×C8—42no—
303C7×C8—42no—
304C2×#41—21no—
305C22×Q8—21no—
306C22×Q8C7×D411yescharacteristic
307C22×Q8C23×C712yes—
308C22×D4C7×D411yescharacteristic
309C2×Q16—24no—
310C2×Q16—22no—
311C2×SD16—22no—
312C4⋊Q8—21no—
313C4⋊Q8—22no—
314C4⋊Q8—21no—
315C4⋊Q8—21no—
316C4⋊Q8—21no—
317C4⋊Q8—21no—
318C4⋊Q8C7×D411yescharacteristic
319C4⋊Q8C7×D411yescharacteristic
320#126—22no—
321#126—22no—
322#126—22no—
323#126—21no—
324#125—21no—
325#125C7×D411yescharacteristic
326#124—21no—
327#124—22no—
328#124—22no—
329#124—21no—
330#124—21no—
331#124—21no—
332#124—21no—
333#124C7×D411yescharacteristic
334#123—22no—
335#123—22no—
336#123—22no—
337#123—21no—
338#123—22no—
339C22⋊Q8—22no—
340C22⋊Q8—21no—
341C22⋊Q8—22no—
342C22⋊Q8—22no—
343C22⋊Q8—22no—
344C22⋊Q8—21no—
345C22⋊Q8—22no—
346C22⋊Q8—22no—
347C22⋊Q8—22no—
348C22⋊Q8C23×C711yescharacteristic
349C22⋊Q8C23×C712yes—
350#121—21no—
351#121—22no—
352#121—22no—
353#121C23×C711yescharacteristic
354#120—22no—
355C4×Q8—21no—
356C4×Q8—21no—
357C4×Q8C7×D411yescharacteristic
358C4×Q8C7×D412yes—
359C4×D4—21no—
360C4×D4—22no—
361C4×D4—21no—
362C4×D4—22no—
363C4×D4C7×D411yescharacteristic
364#117—22no—
365#117—21no—
366#117—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4C7×D412yes—
372C2×C22⋊C4C23×C712yes—
373C2×C42—21no—
374C2×C42C23×C711yescharacteristic
375C4⋊C8—21no—
376C4⋊C8—21no—
377#103—22no—
378#103—22no—
379#103—24no—
380C22⋊C8C23×C712yes—
381C22⋊C8C23×C712yes—
382#97—21no—
383C4×C8—21no—
384#95—21no—
385#95—24no—
386#95C23×C712yes—
387C22×C7—21no—
388C22×C7—21no—
389C22×C7—41no—
390C22×C7—42no—
391C22×C7—41no—
392C22×C7—42no—
393C22×C7—42no—
394C22×C7—41no—
395C22×C7—84no—
396C22×C7C2×#12011yescharacteristic
397C22×C7#57812yes—
398C4×C7—21no—
399C4×C7—21no—
400C4×C7—21no—
401C4×C7—21no—
402C4×C7—21no—
403C4×C7—41no—
404C4×C7—41no—
405C4×C7—42no—
406C4×C7—42no—
407C4×C7—41no—
408C4×C7—42no—
409C4×C7—82no—
410C4×C7—82no—
411C4×C7#45512yes—
412C24C2×C7×D412yes—
413#41—82no—
414C2×Q8—22no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—42no—
421C2×Q8—82no—
422C2×Q8—82no—
423C2×Q8C2×C7×D411yescharacteristic
424C2×Q8C2×C7×D412yes—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—42no—
431C2×D4C2×C7×D411yescharacteristic
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4C2×C7×D411yescharacteristic
439C22×C4C2×C7×D411yescharacteristic
440C22×C4C2×C7×D412yes—
441C22×C4C24×C711yescharacteristic
442Q16—82no—
443Q16—84no—
444SD16—82no—
445C2×C8—22no—
446C2×C8—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4C2×C7×D411yescharacteristic
465C4⋊C4C2×C7×D411yescharacteristic
466C4⋊C4C2×C7×D412yes—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—21no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—21no—
482C22⋊C4—84no—
483C42—22no—
484C42—22no—
485C42—21no—
486C42—42no—
487C42C2×C7×D411yescharacteristic
488C42C2×C7×D411yescharacteristic
489C42C2×C7×D411yescharacteristic
490C42C2×C7×D411yescharacteristic
491C2×C7—22no—
492C2×C7—81no—
493C2×C7—82no—
494C2×C7#301111yescharacteristic
495C2×C7C2×#45511yescharacteristic
496C2×C7#302711yescharacteristic
497C23—22no—
498C23—21no—
499C23—24no—
500C23—42no—
501C23—42no—
502C23C22×C7×D411yescharacteristic
503Q8—41no—
504Q8—42no—
505Q8—42no—
506Q8—42no—
507Q8—44no—
508Q8—42no—
509Q8—42no—
510Q8—81no—
511Q8—82no—
512D4—41no—
513D4—42no—
514D4—42no—
515D4—81no—
516D4—82no—
517D4—82no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4C7×#12011yescharacteristic
538C2×C4C7×#12011yescharacteristic
539C2×C4C7×#12011yescharacteristic
540C2×C4C7×#12011yescharacteristic
541C2×C4C22×C7×D411yescharacteristic
542C2×C4C22×C7×D411yescharacteristic
543C8—42no—
544C8—42no—
545C7#1842911yescharacteristic, direct product factor, Sylow
546C22—21no—
547C22—21no—
548C22—41no—
549C22—42no—
550C22—41no—
551C22—42no—
552C22—42no—
553C22—41no—
554C22—84no—
555C22C2×C7×#12011yescharacteristic
556C22C7×#57812yes—
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—41no—
563C4—41no—
564C4—42no—
565C4—42no—
566C4—41no—
567C4—42no—
568C4—82no—
569C4—82no—
570C4C7×#45512yes—
571C2—22no—
572C2—82no—
573C2—81no—
574C2C7×#301111yescharacteristic
575C2C2×C7×#45511yescharacteristic
576C2C7×#302711yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3561C2
LCSG ▷ C22×C4 ▷ 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×#3218C2semidirect
3. C7×#3209C2semidirect
4. C7×#2640C2semidirect
5. C7×#1606C2semidirect
6. C7×#1602C2semidirect
7. C7×#1506C2semidirect
8. C7×#1500C2semidirect
9. C7×#1469C2semidirect
47. #18429C7direct (N × H)
132. #3218C2×C7semidirect
133. #3209C2×C7semidirect
134. #2640C2×C7semidirect
135. #1606C2×C7semidirect
136. #1602C2×C7semidirect
137. #1506C2×C7semidirect
138. #1500C2×C7semidirect
139. #1469C2×C7semidirect
545. C7#18429direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35612
C7×#32182
C7×#32092
C7×#26402
C7×#16062
C7×#16022
C7×#15062
C7×#15002
C7×#14692
#184297