Finite Group Databasebeta

← C7×#24303 #13201519

Subgroups — C7×#24303 #13201519

Order 1792.

1066 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#533C211yesmaximal, characteristic
2C2×C7×#477C211yesmaximal, characteristic, complemented
3C2×C7×#475C211yesmaximal, characteristic, complemented
4C4×C7×C22⋊Q8C211yesmaximal, characteristic
5C7×#2059C212yesmaximal, complemented
6C7×#2026C212yesmaximal, complemented
7C7×#1988C211yesmaximal, characteristic, complemented
8C7×#1987C211yesmaximal, characteristic, complemented
9C7×#1873C212yesmaximal
10C7×#1854C212yesmaximal
11C7×#1797C211yesmaximal, characteristic
12C7×#533—24no—
13C7×#533—24no—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#125C2212yes—
17C2×C7×C22⋊Q8—21no—
18C2×C7×C22⋊Q8—21no—
19C2×C7×C22⋊Q8C2211yescharacteristic
20C2×C4×C7×Q8—21no—
21C2×C4×C7×Q8—21no—
22C2×C4×C7×Q8C2211yescharacteristic
23C2×C7×#117C2212yes—
24C22×C42×C7C2211yescharacteristic
25C7×#477—24no—
26C7×#475—24no—
27C2×C7×#107C2212yes—
28C2×C7×#106C2212yes—
29C2×C7×C4⋊C8C2211yescharacteristic
30C2×C7×C4⋊C8C2211yescharacteristic
31C2×C7×#103C2212yes—
32C2×C7×#103C2212yes—
33C2×C7×C22⋊C8C2212yes—
34C7×#391C2211yescharacteristic
35C7×#389C2211yescharacteristic
36C7×#387—21no—
37C7×#385—21no—
38C7×#380C2211yescharacteristic
39C4×C7×C4⋊C4—21no—
40C4×C7×C4⋊C4C2211yescharacteristic
41C4×C7×C22⋊C4—21no—
42C7×#340C2212yes—
43C7×#340C2212yes—
44C7×#340C2212yes—
45C7×#340C2212yes—
46C7×#336C2212yes—
47#24303C711yesmaximal, characteristic, direct product factor, Sylow
48C22×C7×Q8—21no—
49C22×C7×Q8—21no—
50C22×C7×Q8C2311yescharacteristic
51C23×C4×C7D411yescharacteristic
52C23×C4×C7D411yescharacteristic
53C23×C4×C7C2311yescharacteristic
54C22×C7×C8C2312yes—
55C22×C7×C8C2312yes—
56C7×C4⋊Q8—22no—
57C7×C4⋊Q8—22no—
58C7×C4⋊Q8D414yes—
59C7×C4⋊Q8D414yes—
60C7×#125—24no—
61C7×#125—24no—
62C7×C22⋊Q8—28no—
63C7×C22⋊Q8—28no—
64C7×C22⋊Q8—22no—
65C7×C22⋊Q8Q814yescomplemented
66C4×C7×Q8—28no—
67C4×C7×Q8—28no—
68C4×C7×Q8—22no—
69C4×C7×Q8—22no—
70C7×#117—24no—
71C7×#117—24no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—22no—
74C2×C7×C4⋊C4—22no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C4⋊C4C2312yes—
82C2×C7×C4⋊C4C2312yes—
83C2×C7×C4⋊C4C2311yescharacteristic
84C2×C7×C22⋊C4—21no—
85C2×C7×C22⋊C4—22no—
86C2×C7×C22⋊C4—21no—
87C2×C42×C7—22no—
88C2×C42×C7—22no—
89C2×C42×C7—22no—
90C2×C42×C7—21no—
91C2×C42×C7—21no—
92C2×C42×C7D411yescharacteristic
93C2×C42×C7D411yescharacteristic
94C2×C42×C7C2311yescharacteristic
95C2×C42×C7C2311yescharacteristic
96C7×#107—24no—
97C7×#106—24no—
98C7×C4⋊C8—22no—
99C7×C4⋊C8—22no—
100C7×#103—24no—
101C7×#103—24no—
102C7×C22⋊C8—24no—
103C7×#95—21no—
104C7×#95—21no—
105C7×#95—21no—
106C7×#95—21no—
107C7×#95—21no—
108C2×#533C2×C711yescharacteristic
109C2×#477C2×C711yescharacteristic, complemented
110C2×#475C2×C711yescharacteristic, complemented
111C4×C22⋊Q8C2×C711yescharacteristic
112#2059C2×C712yescomplemented
113#2026C2×C712yescomplemented
114#1988C2×C711yescharacteristic, complemented
115#1987C2×C711yescharacteristic, complemented
116#1873C2×C712yes—
117#1854C2×C712yes—
118#1797C2×C711yescharacteristic
119C24×C7C2×D411yescharacteristic
120C2×C7×Q8—24no—
121C2×C7×Q8—24no—
122C2×C7×Q8—21no—
123C2×C7×Q8—21no—
124C2×C7×Q8—21no—
125C2×C7×Q8—21no—
126C2×C7×Q8—21no—
127C2×C7×Q8—42no—
128C2×C7×Q8—42no—
129C2×C7×Q8—42no—
130C2×C7×Q8—42no—
131C2×C7×Q8C2×Q812yes—
132C22×C4×C7—21no—
133C22×C4×C7—21no—
134C22×C4×C7—21no—
135C22×C4×C7—21no—
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—41no—
147C22×C4×C7—41no—
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×Q812yes—
154C22×C4×C7#4112yes—
155C22×C4×C7C2411yescharacteristic
156C2×C7×C8—22no—
157C2×C7×C8—22no—
158C2×C7×C8—22no—
159C2×C7×C8—22no—
160C2×C7×C8—22no—
161C2×C7×C8—22no—
162C7×C4⋊C4—28no—
163C7×C4⋊C4—28no—
164C7×C4⋊C4—24no—
165C7×C4⋊C4—24no—
166C7×C4⋊C4—24no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—42no—
175C7×C4⋊C4—42no—
176C7×C4⋊C4C2×D414yes—
177C7×C4⋊C4C2×D414yes—
178C7×C4⋊C4C2×Q812yes—
179C7×C22⋊C4—28no—
180C7×C22⋊C4—24no—
181C7×C22⋊C4—42no—
182C42×C7—28no—
183C42×C7—22no—
184C42×C7—22no—
185C42×C7—21no—
186C42×C7—21no—
187C42×C7—42no—
188C42×C7—42no—
189C42×C7C2×D412yes—
190C42×C7C2×D412yes—
191#533—24no—
192#533—24no—
193C2×C4⋊Q8C22×C711yescharacteristic
194C2×C4⋊Q8C22×C711yescharacteristic
195C2×#125C22×C712yes—
196C2×C22⋊Q8—21no—
197C2×C22⋊Q8—21no—
198C2×C22⋊Q8C22×C711yescharacteristic
199C2×C4×Q8—21no—
200C2×C4×Q8—21no—
201C2×C4×Q8C22×C711yescharacteristic
202C2×#117C22×C712yes—
203C22×C42C22×C711yescharacteristic
204#477—24no—
205#475—24no—
206C2×#107C22×C712yes—
207C2×#106C22×C712yes—
208C2×C4⋊C8C22×C711yescharacteristic
209C2×C4⋊C8C22×C711yescharacteristic
210C2×#103C22×C712yes—
211C2×#103C22×C712yes—
212C2×C22⋊C8C22×C712yes—
213#391C22×C711yescharacteristic
214#389C22×C711yescharacteristic
215#387—21no—
216#385—21no—
217#380C22×C711yescharacteristic
218C4×C4⋊C4—21no—
219C4×C4⋊C4C22×C711yescharacteristic
220C4×C22⋊C4—21no—
221#340C22×C712yes—
222#340C22×C712yes—
223#340C22×C712yes—
224#340C22×C712yes—
225#336C22×C712yes—
226C23×C7—21no—
227C23×C7—21no—
228C23×C7—21no—
229C23×C7—21no—
230C23×C7—21no—
231C23×C7—21no—
232C23×C7C22⋊Q812yes—
233C23×C7C22×D411yescharacteristic
234C7×Q8—22no—
235C7×Q8—22no—
236C7×Q8—48no—
237C7×Q8—41no—
238C7×Q8—41no—
239C7×Q8—48no—
240C2×C4×C7—21no—
241C2×C4×C7—21no—
242C2×C4×C7—21no—
243C2×C4×C7—21no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—21no—
249C2×C4×C7—21no—
250C2×C4×C7—24no—
251C2×C4×C7—24no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—21no—
255C2×C4×C7—21no—
256C2×C4×C7—21no—
257C2×C4×C7—21no—
258C2×C4×C7—44no—
259C2×C4×C7—41no—
260C2×C4×C7—41no—
261C2×C4×C7—41no—
262C2×C4×C7—42no—
263C2×C4×C7—42no—
264C2×C4×C7—41no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—44no—
268C2×C4×C7—41no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7—41no—
274C2×C4×C7#12012yes—
275C2×C4×C7#12012yes—
276C2×C4×C7C22⋊Q812yes—
277C2×C4×C7#13512yes—
278C2×C4×C7#13712yes—
279C2×C4×C7C22×D411yescharacteristic
280C2×C4×C7C22×D411yescharacteristic
281C2×C4×C7C22×Q811yescharacteristic
282C2×C4×C7C2×#4111yescharacteristic
283C2×C4×C72−1+411yescharacteristic
284C2×C4×C72−1+411yescharacteristic
285C7×C8—42no—
286C7×C8—42no—
287C22×Q8—21no—
288C22×Q8—21no—
289C22×Q8C23×C711yescharacteristic
290C23×C4C7×D411yescharacteristic
291C23×C4C7×D411yescharacteristic
292C23×C4C23×C711yescharacteristic
293C22×C8C23×C712yes—
294C22×C8C23×C712yes—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8C7×D414yes—
298C4⋊Q8C7×D414yes—
299#125—24no—
300#125—24no—
301C22⋊Q8—28no—
302C22⋊Q8—28no—
303C22⋊Q8—22no—
304C22⋊Q8C7×Q814yescomplemented
305C4×Q8—28no—
306C4×Q8—28no—
307C4×Q8—22no—
308C4×Q8—22no—
309#117—24no—
310#117—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4C23×C711yescharacteristic
319C2×C4⋊C4C23×C711yescharacteristic
320C2×C4⋊C4C23×C712yes—
321C2×C4⋊C4C23×C712yes—
322C2×C4⋊C4C23×C711yescharacteristic
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42C7×D411yescharacteristic
332C2×C42C7×D411yescharacteristic
333C2×C42C23×C711yescharacteristic
334C2×C42C23×C711yescharacteristic
335#107—24no—
336#106—24no—
337C4⋊C8—22no—
338C4⋊C8—22no—
339#103—24no—
340#103—24no—
341C22⋊C8—24no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347C22×C7—22no—
348C22×C7—22no—
349C22×C7—22no—
350C22×C7—41no—
351C22×C7—41no—
352C22×C7—41no—
353C22×C7—41no—
354C22×C7C2×#12011yescharacteristic
355C22×C7C2×C22⋊Q811yescharacteristic
356C22×C7#53611yescharacteristic
357C22×C7C2×#13511yescharacteristic
358C22×C7C2×#13711yescharacteristic
359C22×C7#57511yescharacteristic
360C22×C7#57511yescharacteristic
361C4×C7—22no—
362C4×C7—22no—
363C4×C7—42no—
364C4×C7—42no—
365C4×C7—41no—
366C4×C7—44no—
367C4×C7—41no—
368C4×C7—44no—
369C4×C7—81no—
370C4×C7—81no—
371C4×C7#45214yes—
372C4×C7#54811yescharacteristic
373C4×C7#54811yescharacteristic
374C4×C7D4×Q811yescharacteristic
375C4×C7D4×Q811yescharacteristic
376C24C2×C7×D411yescharacteristic
377C2×Q8—24no—
378C2×Q8—24no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8C2×C7×Q812yes—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4C2×C7×D411yescharacteristic
406C22×C4C2×C7×D411yescharacteristic
407C22×C4C2×C7×D411yescharacteristic
408C22×C4C2×C7×D411yescharacteristic
409C22×C4C2×C7×D411yescharacteristic
410C22×C4C2×C7×Q812yes—
411C22×C4C7×#4112yes—
412C22×C4C24×C711yescharacteristic
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C4⋊C4—28no—
420C4⋊C4—28no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4C2×C7×D414yes—
434C4⋊C4C2×C7×D414yes—
435C4⋊C4C2×C7×Q812yes—
436C22⋊C4—28no—
437C22⋊C4—24no—
438C22⋊C4—42no—
439C42—28no—
440C42—22no—
441C42—22no—
442C42—21no—
443C42—21no—
444C42—42no—
445C42—42no—
446C42C2×C7×D412yes—
447C42C2×C7×D412yes—
448C2×C7—42no—
449C2×C7#240911yescharacteristic
450C2×C7C2×#45211yescharacteristic
451C2×C7#300511yescharacteristic
452C2×C7#307611yescharacteristic
453C2×C7#307811yescharacteristic
454C2×C7#323311yescharacteristic
455C2×C7#323711yescharacteristic
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23C7×C22⋊Q812yes—
463C23C22×C7×D411yescharacteristic
464Q8—22no—
465Q8—22no—
466Q8—48no—
467Q8—41no—
468Q8—41no—
469Q8—48no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—44no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4C7×#12012yes—
505C2×C4C7×#12012yes—
506C2×C4C7×C22⋊Q812yes—
507C2×C4C7×#13512yes—
508C2×C4C7×#13712yes—
509C2×C4C22×C7×D411yescharacteristic
510C2×C4C22×C7×D411yescharacteristic
511C2×C4C22×C7×Q811yescharacteristic
512C2×C4C2×C7×#4111yescharacteristic
513C2×C4C7×2−1+411yescharacteristic
514C2×C4C7×2−1+411yescharacteristic
515C8—42no—
516C8—42no—
517C7#2430311yescharacteristic, direct product factor, Sylow
518C22—22no—
519C22—22no—
520C22—22no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22C2×C7×#12011yescharacteristic
526C22C2×C7×C22⋊Q811yescharacteristic
527C22C7×#53611yescharacteristic
528C22C2×C7×#13511yescharacteristic
529C22C2×C7×#13711yescharacteristic
530C22C7×#57511yescharacteristic
531C22C7×#57511yescharacteristic
532C4—22no—
533C4—22no—
534C4—42no—
535C4—41no—
536C4—44no—
537C4—41no—
538C4—44no—
539C4—42no—
540C4—81no—
541C4—81no—
542C4C7×#45214yes—
543C4C7×#54811yescharacteristic
544C4C7×#54811yescharacteristic
545C4C7×D4×Q811yescharacteristic
546C4C7×D4×Q811yescharacteristic
547C2—42no—
548C2C7×#240911yescharacteristic
549C2C2×C7×#45211yescharacteristic
550C2C7×#300511yescharacteristic
551C2C7×#307611yescharacteristic
552C2C7×#307811yescharacteristic
553C2C7×#323311yescharacteristic
554C2C7×#323711yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC24×C7C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C22×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#477C2semidirect
3. C2×C7×#475C2semidirect
5. C7×#2059C2semidirect
6. C7×#2026C2semidirect
7. C7×#1988C2semidirect
8. C7×#1987C2semidirect
47. #24303C7direct (N × H)
65. C7×C22⋊Q8Q8semidirect
109. C2×#477C2×C7semidirect
110. C2×#475C2×C7semidirect
112. #2059C2×C7semidirect
113. #2026C2×C7semidirect
114. #1988C2×C7semidirect
115. #1987C2×C7semidirect
304. C22⋊Q8C7×Q8semidirect
517. C7#24303direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5332
C2×C7×#4772
C2×C7×#4752
C4×C7×C22⋊Q82
C7×#20592
C7×#20262
C7×#19882
C7×#19872
C7×#18732
C7×#18542
C7×#17972
#243037