Finite Group Databasebeta

← C7×#23946 #13201162

Subgroups — C7×#23946 #13201162

Order 1792.

1222 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#542C211yesmaximal, characteristic
2C2×C7×#451C211yesmaximal, characteristic, complemented
3C2×C7×#448C211yesmaximal, characteristic
4C7×#2310C211yesmaximal, characteristic, complemented
5C7×#2024C212yesmaximal, complemented
6C7×#2015C212yesmaximal, complemented
7C7×#1878C212yesmaximal, complemented
8C7×#1877C212yesmaximal, complemented
9C7×#1875C211yesmaximal, characteristic
10C7×#1873C211yesmaximal, characteristic, complemented
11C7×#1818C211yesmaximal, characteristic, complemented
12C23×C7×Q8C2211yescharacteristic
13C22×C7×Q16C2212yes—
14C22×C7×SD16C2212yes—
15C7×#542—24no—
16C7×#542—24no—
17C2×C7×C4⋊Q8C2211yescharacteristic
18C2×C7×#126—21no—
19C2×C7×#124—21no—
20C2×C7×#124C2211yescharacteristic
21C2×C7×#123—21no—
22C2×C7×C22⋊Q8—21no—
23C2×C7×C22⋊Q8C2211yescharacteristic
24C2×C7×#121C2211yescharacteristic
25C2×C4×C7×Q8C2211yescharacteristic
26C2×C4×C7×D4C2211yescharacteristic
27C2×C7×#117C2211yescharacteristic
28C2×C7×#117C2211yescharacteristic
29C7×#451—24no—
30C7×#448—24no—
31C2×C7×#103C2212yes—
32C2×C7×#103C2212yes—
33C2×C7×#103C2212yes—
34C2×C7×#102C2212yes—
35C2×C7×C22⋊C8C2211yescharacteristic
36C2×C7×C22⋊C8C2211yescharacteristic
37C7×#391—21no—
38C7×#391—21no—
39C7×#391C2212yes—
40C7×#391C2212yes—
41C7×#381—21no—
42C7×#381—21no—
43C7×#340C2212yes—
44C7×#340C2212yes—
45C7×#336C2212yes—
46C7×#336C2212yes—
47#23946C711yesmaximal, characteristic, direct product factor, Sylow
48C22×C7×Q8—24no—
49C22×C7×Q8—21no—
50C22×C7×Q8—21no—
51C22×C7×Q8—22no—
52C22×C7×Q8—22no—
53C22×C7×Q8—42no—
54C22×C7×Q8C2311yescharacteristic
55C22×C7×Q8C2312yes—
56C22×C7×D4—21no—
57C22×C7×D4C2311yescharacteristic
58C23×C4×C7—21no—
59C23×C4×C7—21no—
60C23×C4×C7C2311yescharacteristic
61C2×C7×Q16—24no—
62C2×C7×Q16—24no—
63C2×C7×Q16—42no—
64C2×C7×SD16—24no—
65C2×C7×SD16—24no—
66C2×C7×SD16—42no—
67C22×C7×C8C2312yes—
68C22×C7×C8C2312yes—
69C7×C4⋊Q8—22no—
70C7×C4⋊Q8—22no—
71C7×#126—28no—
72C7×#124—28no—
73C7×#124—22no—
74C7×#124—22no—
75C7×#123—28no—
76C7×C22⋊Q8—28no—
77C7×C22⋊Q8—22no—
78C7×C22⋊Q8D414yes—
79C7×#121—22no—
80C7×#121D414yes—
81C4×C7×Q8—22no—
82C4×C7×Q8—22no—
83C4×C7×D4—22no—
84C4×C7×D4—22no—
85C7×#117—22no—
86C7×#117—22no—
87C7×#117—42no—
88C2×C7×C4⋊C4—21no—
89C2×C7×C4⋊C4—21no—
90C2×C7×C4⋊C4—21no—
91C2×C7×C4⋊C4D411yescharacteristic
92C2×C7×C4⋊C4D411yescharacteristic
93C2×C7×C4⋊C4C2312yes—
94C2×C7×C4⋊C4C2311yescharacteristic
95C2×C7×C4⋊C4C2311yescharacteristic
96C2×C7×C22⋊C4—21no—
97C2×C7×C22⋊C4—21no—
98C2×C7×C22⋊C4—21no—
99C2×C7×C22⋊C4—21no—
100C2×C7×C22⋊C4—21no—
101C2×C7×C22⋊C4—21no—
102C2×C7×C22⋊C4D411yescharacteristic
103C2×C7×C22⋊C4D411yescharacteristic
104C2×C42×C7—21no—
105C2×C42×C7—21no—
106C2×C42×C7C2311yescharacteristic
107C2×C42×C7C2311yescharacteristic
108C7×#103—24no—
109C7×#103—24no—
110C7×#103—42no—
111C7×#102—42no—
112C7×C22⋊C8—22no—
113C7×C22⋊C8—22no—
114C7×#95—22no—
115C7×#95—22no—
116C7×#95—22no—
117C7×#95—22no—
118C2×#542C2×C711yescharacteristic
119C2×#451C2×C711yescharacteristic, complemented
120C2×#448C2×C711yescharacteristic
121#2310C2×C711yescharacteristic, complemented
122#2024C2×C712yescomplemented
123#2015C2×C712yescomplemented
124#1878C2×C712yescomplemented
125#1877C2×C712yescomplemented
126#1875C2×C711yescharacteristic
127#1873C2×C711yescharacteristic, complemented
128#1818C2×C711yescharacteristic, complemented
129C24×C7—21no—
130C24×C7C2×D411yescharacteristic
131C2×C7×Q8—24no—
132C2×C7×Q8—24no—
133C2×C7×Q8—21no—
134C2×C7×Q8—22no—
135C2×C7×Q8—22no—
136C2×C7×Q8—22no—
137C2×C7×Q8—22no—
138C2×C7×Q8—22no—
139C2×C7×Q8—22no—
140C2×C7×Q8—22no—
141C2×C7×Q8—48no—
142C2×C7×Q8—48no—
143C2×C7×Q8—42no—
144C2×C7×Q8—42no—
145C2×C7×Q8—82no—
146C2×C7×Q8C2×D412yes—
147C2×C7×Q8#4114yes—
148C2×C7×D4—24no—
149C2×C7×D4—21no—
150C2×C7×D4—22no—
151C2×C7×D4—22no—
152C2×C7×D4—82no—
153C2×C7×D4C2×D412yes—
154C22×C4×C7—22no—
155C22×C4×C7—22no—
156C22×C4×C7—24no—
157C22×C4×C7—22no—
158C22×C4×C7—21no—
159C22×C4×C7—21no—
160C22×C4×C7—21no—
161C22×C4×C7—24no—
162C22×C4×C7—22no—
163C22×C4×C7—42no—
164C22×C4×C7—42no—
165C22×C4×C7—82no—
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2×D412yes—
172C22×C4×C7C2×D412yes—
173C22×C4×C7C2411yescharacteristic
174C7×Q16—48no—
175C7×SD16—48no—
176C2×C7×C8—22no—
177C2×C7×C8—22no—
178C2×C7×C8—22no—
179C2×C7×C8—22no—
180C2×C7×C8—22no—
181C2×C7×C8—22no—
182C7×C4⋊C4—24no—
183C7×C4⋊C4—24no—
184C7×C4⋊C4—24no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—22no—
191C7×C4⋊C4C2×D412yes—
192C7×C4⋊C4C2×D412yes—
193C7×C22⋊C4—24no—
194C7×C22⋊C4—24no—
195C7×C22⋊C4—24no—
196C7×C22⋊C4—24no—
197C7×C22⋊C4—22no—
198C7×C22⋊C4—22no—
199C7×C22⋊C4—24no—
200C7×C22⋊C4—42no—
201C42×C7—24no—
202C42×C7—21no—
203C42×C7—21no—
204C42×C7—21no—
205C42×C7—21no—
206C42×C7—42no—
207C23×Q8C22×C711yescharacteristic
208C22×Q16C22×C712yes—
209C22×SD16C22×C712yes—
210#542—24no—
211#542—24no—
212C2×C4⋊Q8C22×C711yescharacteristic
213C2×#126—21no—
214C2×#124—21no—
215C2×#124C22×C711yescharacteristic
216C2×#123—21no—
217C2×C22⋊Q8—21no—
218C2×C22⋊Q8C22×C711yescharacteristic
219C2×#121C22×C711yescharacteristic
220C2×C4×Q8C22×C711yescharacteristic
221C2×C4×D4C22×C711yescharacteristic
222C2×#117C22×C711yescharacteristic
223C2×#117C22×C711yescharacteristic
224#451—24no—
225#448—24no—
226C2×#103C22×C712yes—
227C2×#103C22×C712yes—
228C2×#103C22×C712yes—
229C2×#102C22×C712yes—
230C2×C22⋊C8C22×C711yescharacteristic
231C2×C22⋊C8C22×C711yescharacteristic
232#391—21no—
233#391—21no—
234#391C22×C712yes—
235#391C22×C712yes—
236#381—21no—
237#381—21no—
238#340C22×C712yes—
239#340C22×C712yes—
240#336C22×C712yes—
241#336C22×C712yes—
242C23×C7—21no—
243C23×C7—21no—
244C23×C7—22no—
245C23×C7—22no—
246C23×C7—22no—
247C23×C7—41no—
248C23×C7—41no—
249C23×C7—42no—
250C23×C7—42no—
251C23×C7#12012yes—
252C23×C7C22×D411yescharacteristic
253C7×Q8—24no—
254C7×Q8—24no—
255C7×Q8—24no—
256C7×Q8—48no—
257C7×Q8—42no—
258C7×Q8—42no—
259C7×Q8—42no—
260C7×Q8—84no—
261C7×D4—24no—
262C7×D4—42no—
263C7×D4—84no—
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—24no—
267C2×C4×C7—24no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—22no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—22no—
274C2×C4×C7—22no—
275C2×C4×C7—24no—
276C2×C4×C7—22no—
277C2×C4×C7—22no—
278C2×C4×C7—22no—
279C2×C4×C7—22no—
280C2×C4×C7—48no—
281C2×C4×C7—42no—
282C2×C4×C7—41no—
283C2×C4×C7—42no—
284C2×C4×C7—41no—
285C2×C4×C7—42no—
286C2×C4×C7—42no—
287C2×C4×C7—42no—
288C2×C4×C7—42no—
289C2×C4×C7—42no—
290C2×C4×C7—84no—
291C2×C4×C7#12012yes—
292C2×C4×C7C22×D411yescharacteristic
293C2×C4×C7C22×D411yescharacteristic
294C2×C4×C7C2×#4112yes—
295C2×C4×C7H(5,𝔽2)12yes—
296C7×C8—42no—
297C7×C8—42no—
298C22×Q8—24no—
299C22×Q8—21no—
300C22×Q8—21no—
301C22×Q8—22no—
302C22×Q8—22no—
303C22×Q8—42no—
304C22×Q8C23×C711yescharacteristic
305C22×Q8C23×C712yes—
306C22×D4—21no—
307C22×D4C23×C711yescharacteristic
308C23×C4—21no—
309C23×C4—21no—
310C23×C4C23×C711yescharacteristic
311C2×Q16—24no—
312C2×Q16—24no—
313C2×Q16—42no—
314C2×SD16—24no—
315C2×SD16—24no—
316C2×SD16—42no—
317C22×C8C23×C712yes—
318C22×C8C23×C712yes—
319C4⋊Q8—22no—
320C4⋊Q8—22no—
321#126—28no—
322#124—28no—
323#124—22no—
324#124—22no—
325#123—28no—
326C22⋊Q8—28no—
327C22⋊Q8—22no—
328C22⋊Q8C7×D414yes—
329#121—22no—
330#121C7×D414yes—
331C4×Q8—22no—
332C4×Q8—22no—
333C4×D4—22no—
334C4×D4—22no—
335#117—22no—
336#117—22no—
337#117—42no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4C7×D411yescharacteristic
342C2×C4⋊C4C7×D411yescharacteristic
343C2×C4⋊C4C23×C712yes—
344C2×C4⋊C4C23×C711yescharacteristic
345C2×C4⋊C4C23×C711yescharacteristic
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4C7×D411yescharacteristic
353C2×C22⋊C4C7×D411yescharacteristic
354C2×C42—21no—
355C2×C42—21no—
356C2×C42C23×C711yescharacteristic
357C2×C42C23×C711yescharacteristic
358#103—24no—
359#103—24no—
360#103—42no—
361#102—42no—
362C22⋊C8—22no—
363C22⋊C8—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368C22×C7—22no—
369C22×C7—22no—
370C22×C7—22no—
371C22×C7—42no—
372C22×C7—42no—
373C22×C7—42no—
374C22×C7—42no—
375C22×C7—42no—
376C22×C7—82no—
377C22×C7—82no—
378C22×C7C2×#12011yescharacteristic
379C22×C7#52511yescharacteristic
380C22×C7#53511yescharacteristic
381C22×C7#57712yes—
382C22×C7#57812yes—
383C4×C7—22no—
384C4×C7—44no—
385C4×C7—42no—
386C4×C7—41no—
387C4×C7—41no—
388C4×C7—42no—
389C4×C7—42no—
390C4×C7—41no—
391C4×C7—41no—
392C4×C7—82no—
393C4×C7#54612yes—
394C4×C7#54612yes—
395C24—21no—
396C24C2×C7×D411yescharacteristic
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—48no—
408C2×Q8—48no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—82no—
412C2×Q8C2×C7×D412yes—
413C2×Q8C7×#4114yes—
414C2×D4—24no—
415C2×D4—21no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—82no—
419C2×D4C2×C7×D412yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—82no—
432C22×C4C2×C7×D411yescharacteristic
433C22×C4C2×C7×D411yescharacteristic
434C22×C4C2×C7×D411yescharacteristic
435C22×C4C2×C7×D411yescharacteristic
436C22×C4C2×C7×D411yescharacteristic
437C22×C4C2×C7×D412yes—
438C22×C4C2×C7×D412yes—
439C22×C4C24×C711yescharacteristic
440Q16—48no—
441SD16—48no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4C2×C7×D412yes—
458C4⋊C4C2×C7×D412yes—
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—24no—
466C22⋊C4—42no—
467C42—24no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—42no—
473C2×C7—42no—
474C2×C7—82no—
475C2×C7#240411yescharacteristic
476C2×C7#300911yescharacteristic
477C2×C7#301111yescharacteristic
478C2×C7#312112yes—
479C2×C7#322512yes—
480C23—21no—
481C23—21no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23C7×#12012yes—
490C23C22×C7×D411yescharacteristic
491Q8—24no—
492Q8—24no—
493Q8—24no—
494Q8—48no—
495Q8—42no—
496Q8—42no—
497Q8—42no—
498Q8—84no—
499D4—24no—
500D4—42no—
501D4—84no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—48no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—84no—
529C2×C4C7×#12012yes—
530C2×C4C22×C7×D411yescharacteristic
531C2×C4C22×C7×D411yescharacteristic
532C2×C4C2×C7×#4112yes—
533C2×C4C7×H(5,𝔽2)12yes—
534C8—42no—
535C8—42no—
536C7#2394611yescharacteristic, direct product factor, Sylow
537C22—22no—
538C22—22no—
539C22—22no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—82no—
546C22—82no—
547C22C2×C7×#12011yescharacteristic
548C22C7×#52511yescharacteristic
549C22C7×#53511yescharacteristic
550C22C7×#57712yes—
551C22C7×#57812yes—
552C4—22no—
553C4—44no—
554C4—42no—
555C4—41no—
556C4—41no—
557C4—42no—
558C4—42no—
559C4—41no—
560C4—41no—
561C4—82no—
562C4C7×#54612yes—
563C4C7×#54612yes—
564C2—42no—
565C2—82no—
566C2C7×#240411yescharacteristic
567C2C7×#300911yescharacteristic
568C2C7×#301111yescharacteristic
569C2C7×#312112yes—
570C2C7×#322512yes—
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C7×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#451C2semidirect
4. C7×#2310C2semidirect
5. C7×#2024C2semidirect
6. C7×#2015C2semidirect
7. C7×#1878C2semidirect
8. C7×#1877C2semidirect
10. C7×#1873C2semidirect
11. C7×#1818C2semidirect
47. #23946C7direct (N × H)
119. C2×#451C2×C7semidirect
121. #2310C2×C7semidirect
122. #2024C2×C7semidirect
123. #2015C2×C7semidirect
124. #1878C2×C7semidirect
125. #1877C2×C7semidirect
127. #1873C2×C7semidirect
128. #1818C2×C7semidirect
536. C7#23946direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5422
C2×C7×#4512
C2×C7×#4482
C7×#23102
C7×#20242
C7×#20152
C7×#18782
C7×#18772
C7×#18752
C7×#18732
C7×#18182
#239467