Finite Group Databasebeta

← C7×#18342 #13195558

Subgroups — C7×#18342 #13195558

Order 1792.

998 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3459C211yesmaximal, characteristic
2C7×#3238C212yesmaximal, complemented
3C7×#2547C211yesmaximal, characteristic, complemented
4C7×#1612C214yesmaximal, complemented
5C7×#1608C212yesmaximal, complemented
6C7×#1607C212yesmaximal, 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—
19C2×C7×#124—21no—
20C7×#525—21no—
21C7×#525—22no—
22C2×C4×C7×D4—21no—
23C2×C4×C7×D4C2211yescharacteristic
24C7×#501—22no—
25C7×#489—22no—
26C7×#483C2214yescomplemented
27C7×#480C2214yes—
28C7×#477—22no—
29C7×#476—22no—
30C7×#432C2212yescomplemented
31C7×#397C2212yes—
32C7×#397C2212yescomplemented
33C7×#391—21no—
34C7×#380—21no—
35C7×#380—21no—
36C4×C7×C4⋊C4C2211yescharacteristic, complemented
37C7×#328C2214yescomplemented
38C7×#327C2214yes—
39C7×#323C2212yes—
40C7×#323C2212yescomplemented
41#18342C711yesmaximal, characteristic, direct product factor, Sylow
42C2×C7×#41—21no—
43C22×C7×Q8—21no—
44C22×C7×D4—21no—
45C22×C7×D4D411yescharacteristic
46C23×C4×C7—21no—
47C23×C4×C7D411yescharacteristic
48C23×C4×C7D411yescharacteristic
49C7×C4⋊Q8—22no—
50C7×C4⋊Q8—22no—
51C7×#126—22no—
52C7×#126—22no—
53C7×#126—22no—
54C7×#125—22no—
55C7×#125D412yes—
56C7×#124—22no—
57C7×#124—44no—
58C7×#124D412yes—
59C7×#123—22no—
60C7×#123—22no—
61C7×#123—22no—
62C7×#123—22no—
63C7×#123—22no—
64C7×#123—22no—
65C7×C22⋊Q8—21no—
66C7×C22⋊Q8—22no—
67C7×C22⋊Q8—22no—
68C7×C22⋊Q8—21no—
69C7×C22⋊Q8—22no—
70C7×C22⋊Q8C2312yes—
71C7×#121—22no—
72C7×#121—21no—
73C7×#121—22no—
74C7×#121—22no—
75C7×#121—21no—
76C7×#121C2312yes—
77C7×#120—22no—
78C7×#120—22no—
79C7×#120—22no—
80C4×C7×Q8D412yes—
81C4×C7×D4—22no—
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×D4D412yes—
89C7×#117—21no—
90C7×#117—22no—
91C2×C7×C4⋊C4—21no—
92C2×C7×C4⋊C4—21no—
93C2×C7×C4⋊C4—21no—
94C2×C7×C4⋊C4D411yescharacteristic
95C2×C7×C4⋊C4C2312yes—
96C2×C7×C22⋊C4—21no—
97C2×C7×C22⋊C4—21no—
98C2×C7×C22⋊C4—22no—
99C2×C7×C22⋊C4—22no—
100C2×C7×C22⋊C4C2311yescharacteristic
101C2×C7×C22⋊C4C2311yescharacteristic
102C2×C42×C7—21no—
103C2×C42×C7—21no—
104C2×C42×C7C2311yescharacteristic
105C7×#107—24no—
106C7×#106—24no—
107C7×C4⋊C8—22no—
108C7×#103—24no—
109C7×#102—24no—
110C7×C22⋊C8C2314yes—
111C7×#97—22no—
112C7×#95—22no—
113C7×#95—22no—
114C7×#95C2312yes—
115#3459C2×C711yescharacteristic
116#3238C2×C712yescomplemented
117#2547C2×C711yescharacteristic, complemented
118#1612C2×C714yescomplemented
119#1608C2×C712yescomplemented
120#1607C2×C712yescomplemented
121#1509C2×C712yescomplemented
122#1466C2×C711yescharacteristic, complemented
123C24×C7—21no—
124C24×C7C2×D411yescharacteristic
125C24×C7C2×D411yescharacteristic
126C7×#41—82no—
127C2×C7×Q8—21no—
128C2×C7×Q8—42no—
129C2×C7×Q8—42no—
130C2×C7×Q8—42no—
131C2×C7×Q8C2×D412yes—
132C2×C7×D4—22no—
133C2×C7×D4—22no—
134C2×C7×D4—22no—
135C2×C7×D4—22no—
136C2×C7×D4—21no—
137C2×C7×D4—22no—
138C2×C7×D4—22no—
139C2×C7×D4—41no—
140C2×C7×D4—42no—
141C2×C7×D4—42no—
142C2×C7×D4C2×D412yes—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—22no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—22no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—22no—
153C22×C4×C7—22no—
154C22×C4×C7—42no—
155C22×C4×C7—42no—
156C22×C4×C7—82no—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2411yescharacteristic
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—24no—
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—44no—
191C7×C22⋊C4—44no—
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—
207C2×#124—21no—
208#525—21no—
209#525—22no—
210C2×C4×D4—21no—
211C2×C4×D4C22×C711yescharacteristic
212#501—22no—
213#489—22no—
214#483C22×C714yescomplemented
215#480C22×C714yes—
216#477—22no—
217#476—22no—
218#432C22×C712yescomplemented
219#397C22×C712yes—
220#397C22×C712yescomplemented
221#391—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×C7AGL(1,ℤ/8)12yes—
243C23×C7C22×D411yescharacteristic
244C7×Q8—42no—
245C7×Q8—84no—
246C7×Q8—81no—
247C7×D4—44no—
248C7×D4—42no—
249C7×D4—44no—
250C7×D4—42no—
251C7×D4—42no—
252C7×D4—82no—
253C7×D4—81no—
254C7×D4—82no—
255C2×C4×C7—21no—
256C2×C4×C7—21no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—22no—
260C2×C4×C7—21no—
261C2×C4×C7—21no—
262C2×C4×C7—22no—
263C2×C4×C7—41no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—44no—
268C2×C4×C7—41no—
269C2×C4×C7—41no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—81no—
277C2×C4×C7—81no—
278C2×C4×C7—81no—
279C2×C4×C7—82no—
280C2×C4×C7—81no—
281C2×C4×C7#12012yes—
282C2×C4×C7#12011yescharacteristic
283C2×C4×C7#12011yescharacteristic
284C2×C4×C7C22×D412yes—
285C7×C8—44no—
286C2×#41—21no—
287C22×Q8—21no—
288C22×D4—21no—
289C22×D4C7×D411yescharacteristic
290C23×C4—21no—
291C23×C4C7×D411yescharacteristic
292C23×C4C7×D411yescharacteristic
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295#126—22no—
296#126—22no—
297#126—22no—
298#125—22no—
299#125C7×D412yes—
300#124—22no—
301#124—44no—
302#124C7×D412yes—
303#123—22no—
304#123—22no—
305#123—22no—
306#123—22no—
307#123—22no—
308#123—22no—
309C22⋊Q8—21no—
310C22⋊Q8—22no—
311C22⋊Q8—22no—
312C22⋊Q8—21no—
313C22⋊Q8—22no—
314C22⋊Q8C23×C712yes—
315#121—22no—
316#121—21no—
317#121—22no—
318#121—22no—
319#121—21no—
320#121C23×C712yes—
321#120—22no—
322#120—22no—
323#120—22no—
324C4×Q8C7×D412yes—
325C4×D4—22no—
326C4×D4—22no—
327C4×D4—22no—
328C4×D4—22no—
329C4×D4—22no—
330C4×D4—22no—
331C4×D4—22no—
332C4×D4C7×D412yes—
333#117—21no—
334#117—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4C7×D411yescharacteristic
339C2×C4⋊C4C23×C712yes—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4C23×C711yescharacteristic
345C2×C22⋊C4C23×C711yescharacteristic
346C2×C42—21no—
347C2×C42—21no—
348C2×C42C23×C711yescharacteristic
349#107—24no—
350#106—24no—
351C4⋊C8—22no—
352#103—24no—
353#102—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×AGL(1,ℤ/8)11yescharacteristic
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—42no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8C2×C7×D412yes—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—41no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4C2×C7×D412yes—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—82no—
423C22×C4C2×C7×D411yescharacteristic
424C22×C4C2×C7×D411yescharacteristic
425C22×C4C2×C7×D411yescharacteristic
426C22×C4C2×C7×D411yescharacteristic
427C22×C4C24×C711yescharacteristic
428C2×C8—24no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—42no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4C2×C7×D412yes—
443C4⋊C4C2×C7×D412yes—
444C22⋊C4—22no—
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—44no—
457C22⋊C4—44no—
458C42—22no—
459C42—22no—
460C42—42no—
461C42C2×C7×D412yes—
462C42C2×C7×D412yes—
463C2×C7—21no—
464C2×C7—21no—
465C2×C7—42no—
466C2×C7—81no—
467C2×C7—82no—
468C2×C7#300611yescharacteristic
469C2×C7C2×#45511yescharacteristic
470C2×C7#302811yescharacteristic
471C23—21no—
472C23—21no—
473C23—21no—
474C23—22no—
475C23—21no—
476C23—21no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—42no—
482C23—81no—
483C23—81no—
484C23C7×AGL(1,ℤ/8)12yes—
485C23C22×C7×D411yescharacteristic
486Q8—42no—
487Q8—84no—
488Q8—81no—
489D4—44no—
490D4—42no—
491D4—44no—
492D4—42no—
493D4—42no—
494D4—82no—
495D4—81no—
496D4—82no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—82no—
522C2×C4—81no—
523C2×C4C7×#12012yes—
524C2×C4C7×#12011yescharacteristic
525C2×C4C7×#12011yescharacteristic
526C2×C4C22×C7×D412yes—
527C8—44no—
528C7#1834211yescharacteristic, direct product factor, Sylow
529C22—21no—
530C22—21no—
531C22—22no—
532C22—42no—
533C22—42no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—42no—
538C22—41no—
539C22—42no—
540C22—42no—
541C22—41no—
542C22—81no—
543C22—81no—
544C22—84no—
545C22C2×C7×#12011yescharacteristic
546C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
547C22C7×#57511yescharacteristic
548C4—22no—
549C4—21no—
550C4—22no—
551C4—42no—
552C4—41no—
553C4—41no—
554C4—42no—
555C4—42no—
556C4—82no—
557C4—82no—
558C4C7×#45512yes—
559C2—21no—
560C2—21no—
561C2—42no—
562C2—81no—
563C2—82no—
564C2C7×#300611yescharacteristic
565C2C2×C7×#45511yescharacteristic
566C2C7×#302811yescharacteristic
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×#3238C2semidirect
3. C7×#2547C2semidirect
4. C7×#1612C2semidirect
5. C7×#1608C2semidirect
6. C7×#1607C2semidirect
7. C7×#1509C2semidirect
8. C7×#1466C2semidirect
16. C2×C7×C4⋊Q8C22semidirect
26. C7×#483C22semidirect
30. C7×#432C22semidirect
32. C7×#397C22semidirect
36. C4×C7×C4⋊C4C22semidirect
37. C7×#328C22semidirect
40. C7×#323C22semidirect
41. #18342C7direct (N × H)
116. #3238C2×C7semidirect
117. #2547C2×C7semidirect
118. #1612C2×C7semidirect
119. #1608C2×C7semidirect
120. #1607C2×C7semidirect
121. #1509C2×C7semidirect
122. #1466C2×C7semidirect
204. C2×C4⋊Q8C22×C7semidirect
214. #483C22×C7semidirect
218. #432C22×C7semidirect
220. #397C22×C7semidirect
224. C4×C4⋊C4C22×C7semidirect
225. #328C22×C7semidirect
228. #323C22×C7semidirect
528. C7#18342direct (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×#32382
C7×#25472
C7×#16122
C7×#16082
C7×#16072
C7×#15092
C7×#14662
#183427