Finite Group Databasebeta

← C7×#19051 #13196267

Subgroups — C7×#19051 #13196267

Order 1792.

950 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3545C211yesmaximal, characteristic
2C7×#3238C212yesmaximal, complemented
3C2×C7×#497C211yesmaximal, characteristic, complemented
4C7×#1661C214yesmaximal, complemented
5C7×#1660C212yesmaximal, complemented
6C7×#1659C212yesmaximal, complemented
7C7×#1533C211yesmaximal, characteristic, complemented
8C7×#1532C212yesmaximal, complemented
9C22×C7×Q16—21no—
10C22×C7×SD16—21no—
11C7×#563C2211yescharacteristic
12C7×#562C2212yes—
13C7×H(3,𝔽4)C2211yescharacteristic
14C7×#555—21no—
15C7×#551—21no—
16C7×#551—22no—
17C7×#546—22no—
18C7×#542—22no—
19C7×#538—22no—
20C7×#536—21no—
21C7×#534—21no—
22C2×C7×C4⋊Q8C2211yescharacteristic, complemented
23C7×#529C2212yes—
24C2×C7×#124C2211yescharacteristic
25C7×#501C2214yescomplemented
26C7×#497—21no—
27C7×#497—21no—
28C7×#497C2212yes—
29C7×#497C2212yescomplemented
30C7×#489C2214yescomplemented
31C7×#483—22no—
32C7×#480—22no—
33C7×#477—22no—
34C7×#476—22no—
35C7×#432C2212yescomplemented
36C2×C7×#97C2211yescharacteristic, complemented
37C7×#330C2214yescomplemented
38C7×#329C2212yes—
39C7×#329C2214yescomplemented
40C7×#329C2212yescomplemented
41#19051C711yesmaximal, characteristic, direct product factor, Sylow
42C2×C7×#41—21no—
43C22×C7×Q8—21no—
44C22×C7×Q8D411yescharacteristic
45C22×C7×D4—21no—
46C22×C7×D4D411yescharacteristic
47C2×C7×Q16—22no—
48C2×C7×Q16—22no—
49C2×C7×Q16—44no—
50C2×C7×SD16—22no—
51C2×C7×SD16—22no—
52C2×C7×SD16—44no—
53C22×C7×C8—21no—
54C7×C4⋊Q8—21no—
55C7×C4⋊Q8—21no—
56C7×C4⋊Q8—22no—
57C7×C4⋊Q8—21no—
58C7×C4⋊Q8C2312yes—
59C7×#126—22no—
60C7×#126—22no—
61C7×#126—22no—
62C7×#125C2312yes—
63C7×#124—21no—
64C7×#124—22no—
65C7×#124—21no—
66C7×#124—22no—
67C7×#124—22no—
68C7×#124D411yescharacteristic
69C7×#124D411yescharacteristic
70C7×#124C2312yes—
71C7×#124C2311yescharacteristic
72C7×#124C2311yescharacteristic
73C7×#123—21no—
74C7×#123—22no—
75C7×#123—22no—
76C7×#123—21no—
77C7×#123—22no—
78C7×#123—22no—
79C7×#123—22no—
80C7×C22⋊Q8—22no—
81C7×C22⋊Q8—22no—
82C7×C22⋊Q8—22no—
83C7×C22⋊Q8—22no—
84C7×C22⋊Q8D412yes—
85C7×#121—22no—
86C7×#121—22no—
87C7×#121—22no—
88C7×#121—22no—
89C7×#121D412yes—
90C7×#120—21no—
91C7×#120—22no—
92C7×#120—21no—
93C7×#120—22no—
94C4×C7×Q8D412yes—
95C4×C7×D4—22no—
96C4×C7×D4—22no—
97C4×C7×D4—22no—
98C4×C7×D4D412yes—
99C7×#117—21no—
100C7×#117—22no—
101C7×#117—22no—
102C2×C7×C4⋊C4—22no—
103C2×C7×C22⋊C4—21no—
104C2×C7×C22⋊C4—22no—
105C2×C7×C22⋊C4—22no—
106C2×C42×C7C2311yescharacteristic
107C7×#107—24no—
108C7×#106—24no—
109C7×C4⋊C8—22no—
110C7×#103—24no—
111C7×#102—24no—
112C7×C22⋊C8—22no—
113C7×#97—21no—
114C7×#97C2314yes—
115C7×#97C2312yes—
116#3545C2×C711yescharacteristic
117#3238C2×C712yescomplemented
118C2×#497C2×C711yescharacteristic, complemented
119#1661C2×C714yescomplemented
120#1660C2×C712yescomplemented
121#1659C2×C712yescomplemented
122#1533C2×C711yescharacteristic, complemented
123#1532C2×C712yescomplemented
124C24×C7—21no—
125C24×C7—21no—
126C7×#41—82no—
127C2×C7×Q8—21no—
128C2×C7×Q8—21no—
129C2×C7×Q8—22no—
130C2×C7×Q8—42no—
131C2×C7×Q8—42no—
132C2×C7×Q8—42no—
133C2×C7×Q8—41no—
134C2×C7×Q8C2×D412yes—
135C2×C7×Q8C2×D411yescharacteristic
136C2×C7×Q8C2×D411yescharacteristic
137C2×C7×D4—22no—
138C2×C7×D4—22no—
139C2×C7×D4—22no—
140C2×C7×D4—21no—
141C2×C7×D4—22no—
142C2×C7×D4—21no—
143C2×C7×D4—42no—
144C2×C7×D4—82no—
145C2×C7×D4C2×D412yes—
146C2×C7×D4C2×D411yescharacteristic
147C2×C7×D4C2×D411yescharacteristic
148C22×C4×C7—21no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—21no—
152C22×C4×C7—22no—
153C22×C4×C7—41no—
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D412yes—
156C7×Q16—44no—
157C7×Q16—44no—
158C7×SD16—44no—
159C7×SD16—44no—
160C2×C7×C8—24no—
161C2×C7×C8—22no—
162C2×C7×C8—24no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4—22no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—24no—
171C7×C4⋊C4—42no—
172C7×C4⋊C4C2×D412yes—
173C7×C4⋊C4C2×D412yes—
174C7×C22⋊C4—22no—
175C7×C22⋊C4—21no—
176C7×C22⋊C4—21no—
177C7×C22⋊C4—22no—
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—
189C42×C7—21no—
190C42×C7—21no—
191C42×C7—22no—
192C42×C7C2×D411yescharacteristic
193C42×C7C2×D412yes—
194C42×C7C2411yescharacteristic
195C22×Q16—21no—
196C22×SD16—21no—
197#563C22×C711yescharacteristic
198#562C22×C712yes—
199H(3,𝔽4)C22×C711yescharacteristic
200#555—21no—
201#551—21no—
202#551—22no—
203#546—22no—
204#542—22no—
205#538—22no—
206#536—21no—
207#534—21no—
208C2×C4⋊Q8C22×C711yescharacteristic, complemented
209#529C22×C712yes—
210C2×#124C22×C711yescharacteristic
211#501C22×C714yescomplemented
212#497—21no—
213#497—21no—
214#497C22×C712yes—
215#497C22×C712yescomplemented
216#489C22×C714yescomplemented
217#483—22no—
218#480—22no—
219#477—22no—
220#476—22no—
221#432C22×C712yescomplemented
222C2×#97C22×C711yescharacteristic, complemented
223#330C22×C714yescomplemented
224#329C22×C712yes—
225#329C22×C714yescomplemented
226#329C22×C712yescomplemented
227C23×C7—21no—
228C23×C7—22no—
229C23×C7—21no—
230C23×C7—42no—
231C23×C7—82no—
232C23×C7—81no—
233C23×C7—82no—
234C23×C7#12011yescharacteristic
235C7×Q8—41no—
236C7×Q8—41no—
237C7×Q8—42no—
238C7×Q8—42no—
239C7×Q8—42no—
240C7×Q8—81no—
241C7×Q8—82no—
242C7×Q8—81no—
243C7×D4—41no—
244C7×D4—41no—
245C7×D4—42no—
246C7×D4—44no—
247C7×D4—81no—
248C7×D4—82no—
249C7×D4—82no—
250C7×D4—82no—
251C7×D4—81no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—21no—
255C2×C4×C7—22no—
256C2×C4×C7—22no—
257C2×C4×C7—21no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—82no—
262C2×C4×C7—82no—
263C2×C4×C7—81no—
264C2×C4×C7—82no—
265C2×C4×C7—82no—
266C2×C4×C7#12012yes—
267C2×C4×C7#12011yescharacteristic
268C2×C4×C7C22×D412yes—
269C2×C4×C7C22×D411yescharacteristic
270C7×C8—24no—
271C7×C8—44no—
272C2×#41—21no—
273C22×Q8—21no—
274C22×Q8C7×D411yescharacteristic
275C22×D4—21no—
276C22×D4C7×D411yescharacteristic
277C2×Q16—22no—
278C2×Q16—22no—
279C2×Q16—44no—
280C2×SD16—22no—
281C2×SD16—22no—
282C2×SD16—44no—
283C22×C8—21no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8—22no—
287C4⋊Q8—21no—
288C4⋊Q8C23×C712yes—
289#126—22no—
290#126—22no—
291#126—22no—
292#125C23×C712yes—
293#124—21no—
294#124—22no—
295#124—21no—
296#124—22no—
297#124—22no—
298#124C7×D411yescharacteristic
299#124C7×D411yescharacteristic
300#124C23×C711yescharacteristic
301#124C23×C711yescharacteristic
302#124C23×C712yes—
303#123—21no—
304#123—22no—
305#123—22no—
306#123—21no—
307#123—22no—
308#123—22no—
309#123—22no—
310C22⋊Q8—22no—
311C22⋊Q8—22no—
312C22⋊Q8—22no—
313C22⋊Q8—22no—
314C22⋊Q8C7×D412yes—
315#121—22no—
316#121—22no—
317#121—22no—
318#121—22no—
319#121C7×D412yes—
320#120—21no—
321#120—22no—
322#120—21no—
323#120—22no—
324C4×Q8C7×D412yes—
325C4×D4—22no—
326C4×D4—22no—
327C4×D4—22no—
328C4×D4C7×D412yes—
329#117—21no—
330#117—22no—
331#117—22no—
332C2×C4⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C42C23×C711yescharacteristic
337#107—24no—
338#106—24no—
339C4⋊C8—22no—
340#103—24no—
341#102—24no—
342C22⋊C8—22no—
343#97—21no—
344#97C23×C714yes—
345#97C23×C712yes—
346C22×C7—22no—
347C22×C7—42no—
348C22×C7—41no—
349C22×C7—41no—
350C22×C7—42no—
351C22×C7—42no—
352C22×C7—42no—
353C22×C7—42no—
354C22×C7—162no—
355C22×C7—162no—
356C22×C7#45312yes—
357C22×C7C2×#12011yescharacteristic
358C4×C7—21no—
359C4×C7—21no—
360C4×C7—22no—
361C4×C7—22no—
362C4×C7—41no—
363C4×C7—41no—
364C4×C7—42no—
365C4×C7—42no—
366C4×C7—42no—
367C4×C7—82no—
368C24—21no—
369C24—21no—
370#41—82no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—22no—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×Q8—42no—
377C2×Q8—41no—
378C2×Q8C2×C7×D412yes—
379C2×Q8C2×C7×D411yescharacteristic
380C2×Q8C2×C7×D411yescharacteristic
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—22no—
386C2×D4—21no—
387C2×D4—42no—
388C2×D4—82no—
389C2×D4C2×C7×D412yes—
390C2×D4C2×C7×D411yescharacteristic
391C2×D4C2×C7×D411yescharacteristic
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—41no—
398C22×C4C2×C7×D411yescharacteristic
399C22×C4C2×C7×D412yes—
400Q16—44no—
401Q16—44no—
402SD16—44no—
403SD16—44no—
404C2×C8—24no—
405C2×C8—22no—
406C2×C8—24no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—42no—
416C4⋊C4C2×C7×D412yes—
417C4⋊C4C2×C7×D412yes—
418C22⋊C4—22no—
419C22⋊C4—21no—
420C22⋊C4—21no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C42—21no—
434C42—21no—
435C42—22no—
436C42C2×C7×D411yescharacteristic
437C42C2×C7×D412yes—
438C42C24×C711yescharacteristic
439C2×C7—22no—
440C2×C7—82no—
441C2×C7—81no—
442C2×C7—81no—
443C2×C7C2×#45311yescharacteristic
444C2×C7#302212yes—
445C23—21no—
446C23—22no—
447C23—21no—
448C23—42no—
449C23—82no—
450C23—81no—
451C23—82no—
452C23C7×#12011yescharacteristic
453Q8—41no—
454Q8—41no—
455Q8—42no—
456Q8—42no—
457Q8—42no—
458Q8—81no—
459Q8—82no—
460Q8—81no—
461D4—41no—
462D4—41no—
463D4—42no—
464D4—44no—
465D4—81no—
466D4—82no—
467D4—82no—
468D4—82no—
469D4—81no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4C7×#12012yes—
485C2×C4C7×#12011yescharacteristic
486C2×C4C22×C7×D412yes—
487C2×C4C22×C7×D411yescharacteristic
488C8—24no—
489C8—44no—
490C7#1905111yescharacteristic, direct product factor, Sylow
491C22—22no—
492C22—42no—
493C22—41no—
494C22—41no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—162no—
500C22—162no—
501C22C7×#45312yes—
502C22C2×C7×#12011yescharacteristic
503C4—21no—
504C4—21no—
505C4—22no—
506C4—22no—
507C4—41no—
508C4—41no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—82no—
513C2—22no—
514C2—81no—
515C2—82no—
516C2—81no—
517C2C2×C7×#45311yescharacteristic
518C2C7×#302212yes—
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3545C2
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×#3238C2semidirect
3. C2×C7×#497C2semidirect
4. C7×#1661C2semidirect
5. C7×#1660C2semidirect
6. C7×#1659C2semidirect
7. C7×#1533C2semidirect
8. C7×#1532C2semidirect
22. C2×C7×C4⋊Q8C22semidirect
25. C7×#501C22semidirect
29. C7×#497C22semidirect
30. C7×#489C22semidirect
35. C7×#432C22semidirect
36. C2×C7×#97C22semidirect
37. C7×#330C22semidirect
39. C7×#329C22semidirect
40. C7×#329C22semidirect
41. #19051C7direct (N × H)
117. #3238C2×C7semidirect
118. C2×#497C2×C7semidirect
119. #1661C2×C7semidirect
120. #1660C2×C7semidirect
121. #1659C2×C7semidirect
122. #1533C2×C7semidirect
123. #1532C2×C7semidirect
208. C2×C4⋊Q8C22×C7semidirect
211. #501C22×C7semidirect
215. #497C22×C7semidirect
216. #489C22×C7semidirect
221. #432C22×C7semidirect
222. C2×#97C22×C7semidirect
223. #330C22×C7semidirect
225. #329C22×C7semidirect
226. #329C22×C7semidirect
490. C7#19051direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35452
C7×#32382
C2×C7×#4972
C7×#16612
C7×#16602
C7×#16592
C7×#15332
C7×#15322
#190517