Finite Group Databasebeta

← C7×#18336 #13195552

Subgroups — C7×#18336 #13195552

Order 1792.

1002 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3459C211yesmaximal, characteristic
2C7×#3218C212yesmaximal, complemented
3C7×#2563C211yesmaximal, characteristic, complemented
4C7×#1606C214yesmaximal, complemented
5C7×#1602C212yesmaximal, complemented
6C7×#1601C212yesmaximal, complemented
7C7×#1509C212yesmaximal, complemented
8C7×#1466C211yesmaximal, characteristic, complemented
9C7×#543C2211yescharacteristic
10C7×#539—22no—
11C7×#539—21no—
12C7×#538—22no—
13C7×#538C2212yes—
14C7×#534—21no—
15C7×#534C2211yescharacteristic
16C2×C7×C4⋊Q8C2211yescharacteristic, complemented
17C7×#529—22no—
18C7×#529C2212yes—
19C7×#525—21no—
20C7×#525—22no—
21C2×C4×C7×D4—21no—
22C2×C4×C7×D4C2211yescharacteristic
23C7×#497—22no—
24C7×#491—22no—
25C7×#462—22no—
26C7×#461—22no—
27C7×#451C2214yescomplemented
28C7×#448C2214yes—
29C7×#432C2212yescomplemented
30C7×#393C2212yes—
31C7×#393C2212yescomplemented
32C7×#390—21no—
33C7×#382—21no—
34C7×#380—21no—
35C7×#380—21no—
36C4×C7×C4⋊C4C2211yescharacteristic, complemented
37C7×#328C2214yescomplemented
38C7×#327C2214yes—
39C7×#323C2212yes—
40C7×#323C2212yescomplemented
41#18336C711yesmaximal, characteristic, direct product factor, Sylow
42C2×C7×#41—21no—
43C22×C7×Q8C2312yes—
44C22×C7×D4—21no—
45C22×C7×D4D411yescharacteristic
46C23×C4×C7—21no—
47C23×C4×C7D411yescharacteristic
48C23×C4×C7D411yescharacteristic
49C2×C7×Q16—24no—
50C2×C7×SD16—24no—
51C7×C4⋊Q8—22no—
52C7×C4⋊Q8—22no—
53C7×#126—22no—
54C7×#126—22no—
55C7×#126—22no—
56C7×#125—22no—
57C7×#125D412yes—
58C7×#124—22no—
59C7×#124D412yes—
60C7×#123—22no—
61C7×#123—22no—
62C7×#123—22no—
63C7×#123—22no—
64C7×#123—22no—
65C7×#123—22no—
66C7×C22⋊Q8—21no—
67C7×C22⋊Q8—22no—
68C7×C22⋊Q8—22no—
69C7×C22⋊Q8—21no—
70C7×C22⋊Q8—22no—
71C7×C22⋊Q8C2312yes—
72C7×#121—22no—
73C7×#121—21no—
74C7×#121—22no—
75C7×#121—22no—
76C7×#121—21no—
77C7×#121C2312yes—
78C7×#120—22no—
79C7×#120—22no—
80C7×#120—22no—
81C4×C7×Q8D412yes—
82C4×C7×D4—22no—
83C4×C7×D4—22no—
84C4×C7×D4—22no—
85C4×C7×D4—22no—
86C4×C7×D4—22no—
87C4×C7×D4—22no—
88C4×C7×D4—22no—
89C4×C7×D4D412yes—
90C7×#117—21no—
91C7×#117—22no—
92C2×C7×C4⋊C4—21no—
93C2×C7×C4⋊C4—21no—
94C2×C7×C4⋊C4—21no—
95C2×C7×C4⋊C4—21no—
96C2×C7×C4⋊C4D411yescharacteristic
97C2×C7×C22⋊C4—21no—
98C2×C7×C22⋊C4—21no—
99C2×C7×C22⋊C4—22no—
100C2×C7×C22⋊C4—22no—
101C2×C7×C22⋊C4C2311yescharacteristic
102C2×C7×C22⋊C4C2311yescharacteristic
103C2×C42×C7—21no—
104C2×C42×C7—21no—
105C2×C42×C7C2311yescharacteristic
106C7×C4⋊C8—22no—
107C7×#103—24no—
108C7×#103—24no—
109C7×C22⋊C8C2314yes—
110C7×#97—22no—
111C7×#95—22no—
112C7×#95—22no—
113C7×#95C2312yes—
114#3459C2×C711yescharacteristic
115#3218C2×C712yescomplemented
116#2563C2×C711yescharacteristic, complemented
117#1606C2×C714yescomplemented
118#1602C2×C712yescomplemented
119#1601C2×C712yescomplemented
120#1509C2×C712yescomplemented
121#1466C2×C711yescharacteristic, complemented
122C24×C7—21no—
123C24×C7C2×D411yescharacteristic
124C24×C7C2×D411yescharacteristic
125C7×#41—82no—
126C2×C7×Q8—21no—
127C2×C7×Q8—24no—
128C2×C7×Q8—82no—
129C2×C7×Q8C2×D412yes—
130C2×C7×D4—22no—
131C2×C7×D4—22no—
132C2×C7×D4—22no—
133C2×C7×D4—22no—
134C2×C7×D4—21no—
135C2×C7×D4—22no—
136C2×C7×D4—22no—
137C2×C7×D4—41no—
138C2×C7×D4—42no—
139C2×C7×D4—42no—
140C2×C7×D4C2×D412yes—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—22no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—22no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—42no—
153C22×C4×C7—42no—
154C22×C4×C7—82no—
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2411yescharacteristic
160C7×Q16—84no—
161C7×SD16—84no—
162C2×C7×C8—24no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—42no—
173C7×C4⋊C4—42no—
174C7×C4⋊C4—41no—
175C7×C4⋊C4—41no—
176C7×C4⋊C4C2×D412yes—
177C7×C4⋊C4C2×D412yes—
178C7×C22⋊C4—22no—
179C7×C22⋊C4—22no—
180C7×C22⋊C4—22no—
181C7×C22⋊C4—22no—
182C7×C22⋊C4—22no—
183C7×C22⋊C4—22no—
184C7×C22⋊C4—22no—
185C7×C22⋊C4—22no—
186C7×C22⋊C4—22no—
187C7×C22⋊C4—22no—
188C7×C22⋊C4—22no—
189C7×C22⋊C4—22no—
190C7×C22⋊C4—82no—
191C7×C22⋊C4—82no—
192C42×C7—22no—
193C42×C7—22no—
194C42×C7—42no—
195C42×C7C2×D412yes—
196C42×C7C2×D412yes—
197#543C22×C711yescharacteristic
198#539—22no—
199#539—21no—
200#538—22no—
201#538C22×C712yes—
202#534—21no—
203#534C22×C711yescharacteristic
204C2×C4⋊Q8C22×C711yescharacteristic, complemented
205#529—22no—
206#529C22×C712yes—
207#525—21no—
208#525—22no—
209C2×C4×D4—21no—
210C2×C4×D4C22×C711yescharacteristic
211#497—22no—
212#491—22no—
213#462—22no—
214#461—22no—
215#451C22×C714yescomplemented
216#448C22×C714yes—
217#432C22×C712yescomplemented
218#393C22×C712yes—
219#393C22×C712yescomplemented
220#390—21no—
221#382—21no—
222#380—21no—
223#380—21no—
224C4×C4⋊C4C22×C711yescharacteristic, complemented
225#328C22×C714yescomplemented
226#327C22×C714yes—
227#323C22×C712yes—
228#323C22×C712yescomplemented
229C23×C7—21no—
230C23×C7—21no—
231C23×C7—21no—
232C23×C7—22no—
233C23×C7—21no—
234C23×C7—21no—
235C23×C7—41no—
236C23×C7—41no—
237C23×C7—41no—
238C23×C7—41no—
239C23×C7—42no—
240C23×C7—81no—
241C23×C7—81no—
242C23×C7#13712yes—
243C23×C7C22×D411yescharacteristic
244C7×Q8—42no—
245C7×Q8—44no—
246C7×Q8—44no—
247C7×Q8—81no—
248C7×D4—44no—
249C7×D4—42no—
250C7×D4—44no—
251C7×D4—42no—
252C7×D4—42no—
253C7×D4—82no—
254C7×D4—81no—
255C7×D4—82no—
256C2×C4×C7—21no—
257C2×C4×C7—21no—
258C2×C4×C7—22no—
259C2×C4×C7—22no—
260C2×C4×C7—22no—
261C2×C4×C7—21no—
262C2×C4×C7—21no—
263C2×C4×C7—22no—
264C2×C4×C7—41no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—44no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—81no—
278C2×C4×C7—81no—
279C2×C4×C7—81no—
280C2×C4×C7—82no—
281C2×C4×C7—81no—
282C2×C4×C7#12012yes—
283C2×C4×C7#12011yescharacteristic
284C2×C4×C7#12011yescharacteristic
285C2×C4×C7C22×D412yes—
286C7×C8—44no—
287C2×#41—21no—
288C22×Q8C23×C712yes—
289C22×D4—21no—
290C22×D4C7×D411yescharacteristic
291C23×C4—21no—
292C23×C4C7×D411yescharacteristic
293C23×C4C7×D411yescharacteristic
294C2×Q16—24no—
295C2×SD16—24no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298#126—22no—
299#126—22no—
300#126—22no—
301#125—22no—
302#125C7×D412yes—
303#124—22no—
304#124C7×D412yes—
305#123—22no—
306#123—22no—
307#123—22no—
308#123—22no—
309#123—22no—
310#123—22no—
311C22⋊Q8—21no—
312C22⋊Q8—22no—
313C22⋊Q8—22no—
314C22⋊Q8—21no—
315C22⋊Q8—22no—
316C22⋊Q8C23×C712yes—
317#121—22no—
318#121—21no—
319#121—22no—
320#121—22no—
321#121—21no—
322#121C23×C712yes—
323#120—22no—
324#120—22no—
325#120—22no—
326C4×Q8C7×D412yes—
327C4×D4—22no—
328C4×D4—22no—
329C4×D4—22no—
330C4×D4—22no—
331C4×D4—22no—
332C4×D4—22no—
333C4×D4—22no—
334C4×D4C7×D412yes—
335#117—21no—
336#117—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4C7×D411yescharacteristic
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4C23×C711yescharacteristic
347C2×C22⋊C4C23×C711yescharacteristic
348C2×C42—21no—
349C2×C42—21no—
350C2×C42C23×C711yescharacteristic
351C4⋊C8—22no—
352#103—24no—
353#103—24no—
354C22⋊C8C23×C714yes—
355#97—22no—
356#95—22no—
357#95—22no—
358#95C23×C712yes—
359C22×C7—21no—
360C22×C7—21no—
361C22×C7—22no—
362C22×C7—42no—
363C22×C7—42no—
364C22×C7—41no—
365C22×C7—41no—
366C22×C7—41no—
367C22×C7—42no—
368C22×C7—41no—
369C22×C7—42no—
370C22×C7—42no—
371C22×C7—41no—
372C22×C7—81no—
373C22×C7—81no—
374C22×C7—84no—
375C22×C7C2×#12011yescharacteristic
376C22×C7C2×#13711yescharacteristic
377C22×C7#57511yescharacteristic
378C4×C7—22no—
379C4×C7—21no—
380C4×C7—22no—
381C4×C7—42no—
382C4×C7—41no—
383C4×C7—41no—
384C4×C7—42no—
385C4×C7—42no—
386C4×C7—82no—
387C4×C7—82no—
388C4×C7#45512yes—
389C24—21no—
390C24C2×C7×D411yescharacteristic
391C24C2×C7×D411yescharacteristic
392#41—82no—
393C2×Q8—21no—
394C2×Q8—24no—
395C2×Q8—82no—
396C2×Q8C2×C7×D412yes—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—41no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4C2×C7×D412yes—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—82no—
422C22×C4C2×C7×D411yescharacteristic
423C22×C4C2×C7×D411yescharacteristic
424C22×C4C2×C7×D411yescharacteristic
425C22×C4C2×C7×D411yescharacteristic
426C22×C4C24×C711yescharacteristic
427Q16—84no—
428SD16—84no—
429C2×C8—24no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4C2×C7×D412yes—
444C4⋊C4C2×C7×D412yes—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—82no—
458C22⋊C4—82no—
459C42—22no—
460C42—22no—
461C42—42no—
462C42C2×C7×D412yes—
463C42C2×C7×D412yes—
464C2×C7—21no—
465C2×C7—21no—
466C2×C7—42no—
467C2×C7—81no—
468C2×C7—82no—
469C2×C7#300711yescharacteristic
470C2×C7C2×#45511yescharacteristic
471C2×C7#302711yescharacteristic
472C23—21no—
473C23—21no—
474C23—21no—
475C23—22no—
476C23—21no—
477C23—21no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—81no—
484C23—81no—
485C23C7×#13712yes—
486C23C22×C7×D411yescharacteristic
487Q8—42no—
488Q8—44no—
489Q8—44no—
490Q8—81no—
491D4—44no—
492D4—42no—
493D4—44no—
494D4—42no—
495D4—42no—
496D4—82no—
497D4—81no—
498D4—82no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—82no—
524C2×C4—81no—
525C2×C4C7×#12012yes—
526C2×C4C7×#12011yescharacteristic
527C2×C4C7×#12011yescharacteristic
528C2×C4C22×C7×D412yes—
529C8—44no—
530C7#1833611yescharacteristic, direct product factor, Sylow
531C22—21no—
532C22—21no—
533C22—22no—
534C22—42no—
535C22—42no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—42no—
540C22—41no—
541C22—42no—
542C22—42no—
543C22—41no—
544C22—81no—
545C22—81no—
546C22—84no—
547C22C2×C7×#12011yescharacteristic
548C22C2×C7×#13711yescharacteristic
549C22C7×#57511yescharacteristic
550C4—22no—
551C4—21no—
552C4—22no—
553C4—42no—
554C4—41no—
555C4—41no—
556C4—42no—
557C4—42no—
558C4—82no—
559C4—82no—
560C4C7×#45512yes—
561C2—21no—
562C2—21no—
563C2—42no—
564C2—81no—
565C2—82no—
566C2C7×#300711yescharacteristic
567C2C2×C7×#45511yescharacteristic
568C2C7×#302711yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3459C2
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×#2563C2semidirect
4. C7×#1606C2semidirect
5. C7×#1602C2semidirect
6. C7×#1601C2semidirect
7. C7×#1509C2semidirect
8. C7×#1466C2semidirect
16. C2×C7×C4⋊Q8C22semidirect
27. C7×#451C22semidirect
29. C7×#432C22semidirect
31. C7×#393C22semidirect
36. C4×C7×C4⋊C4C22semidirect
37. C7×#328C22semidirect
40. C7×#323C22semidirect
41. #18336C7direct (N × H)
115. #3218C2×C7semidirect
116. #2563C2×C7semidirect
117. #1606C2×C7semidirect
118. #1602C2×C7semidirect
119. #1601C2×C7semidirect
120. #1509C2×C7semidirect
121. #1466C2×C7semidirect
204. C2×C4⋊Q8C22×C7semidirect
215. #451C22×C7semidirect
217. #432C22×C7semidirect
219. #393C22×C7semidirect
224. C4×C4⋊C4C22×C7semidirect
225. #328C22×C7semidirect
228. #323C22×C7semidirect
530. C7#18336direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34592
C7×#32182
C7×#25632
C7×#16062
C7×#16022
C7×#16012
C7×#15092
C7×#14662
#183367