Finite Group Databasebeta

← C7×#18286 #13195502

Subgroups — C7×#18286 #13195502

Order 1792.

1142 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#543C211yesmaximal, characteristic, complemented
2C7×#3187C212yesmaximal, complemented
3C7×#2617C211yesmaximal, characteristic
4C7×#1608C214yesmaximal, complemented
5C7×#1596C214yesmaximal, complemented
6C7×#1520C212yesmaximal, complemented
7C2×C7×#323C211yesmaximal, characteristic, complemented
8C7×#543—24no—
9C7×#543—22no—
10C7×#543C2214yes—
11C2×C7×C22⋊Q8—22no—
12C2×C7×C22⋊Q8—21no—
13C2×C7×C22⋊Q8—22no—
14C2×C7×C22⋊Q8C2212yes—
15C2×C7×#121C2211yescharacteristic
16C2×C7×#120—21no—
17C22×C7×C22⋊C4—21no—
18C22×C7×C22⋊C4C2211yescharacteristic
19C7×#483C2214yescomplemented
20C7×#481—22no—
21C7×#465—24no—
22C7×#450C2214yescomplemented
23C7×#448—22no—
24C2×C7×C22⋊C8C2212yes—
25C7×#397C2212yes—
26C7×#394—22no—
27C7×#392C2212yes—
28C7×#386—21no—
29C7×#380C2211yescharacteristic
30C7×#328C2218yescomplemented
31C7×#323C2214yes—
32#18286C711yesmaximal, characteristic, direct product factor, Sylow
33C25×C7D411yescharacteristic
34C22×C7×Q8—21no—
35C22×C7×Q8D412yescomplemented
36C22×C7×D4—21no—
37C22×C7×D4—22no—
38C22×C7×D4C2311yescharacteristic
39C23×C4×C7—21no—
40C23×C4×C7—21no—
41C23×C4×C7D411yescharacteristic
42C23×C4×C7C2311yescharacteristic
43C2×C7×SD16—24no—
44C22×C7×C8—22no—
45C7×C22⋊Q8—28no—
46C7×C22⋊Q8—28no—
47C7×C22⋊Q8—24no—
48C7×C22⋊Q8—28no—
49C7×C22⋊Q8—24no—
50C7×C22⋊Q8—22no—
51C7×C22⋊Q8—22no—
52C7×C22⋊Q8—28no—
53C7×C22⋊Q8D414yescomplemented
54C7×C22⋊Q8C2314yes—
55C7×#121—22no—
56C7×#121—22no—
57C7×#120—44no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4—22no—
60C2×C7×C4⋊C4—22no—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4D412yescomplemented
63C2×C7×C4⋊C4C2311yescharacteristic
64C2×C7×C22⋊C4—22no—
65C2×C7×C22⋊C4—24no—
66C2×C7×C22⋊C4—22no—
67C2×C7×C22⋊C4—24no—
68C2×C7×C22⋊C4—22no—
69C2×C7×C22⋊C4—22no—
70C2×C7×C22⋊C4—21no—
71C2×C7×C22⋊C4—22no—
72C2×C7×C22⋊C4—21no—
73C2×C7×C22⋊C4—22no—
74C2×C7×C22⋊C4D412yes—
75C2×C7×C22⋊C4C2312yes—
76C7×#106—24no—
77C7×#103—24no—
78C7×#102—24no—
79C7×C22⋊C8—22no—
80C7×C22⋊C8C2314yes—
81C7×#95—22no—
82C7×#95C2312yes—
83C2×#543C2×C711yescharacteristic, complemented
84#3187C2×C712yescomplemented
85#2617C2×C711yescharacteristic
86#1608C2×C714yescomplemented
87#1596C2×C714yescomplemented
88#1520C2×C712yescomplemented
89C2×#323C2×C711yescharacteristic, complemented
90C24×C7—21no—
91C24×C7—21no—
92C24×C7—22no—
93C24×C7—21no—
94C24×C7—81no—
95C24×C7—81no—
96C24×C7SD1614yescomplemented
97C24×C7C2×D412yes—
98C24×C7C2×D411yescharacteristic
99C2×C7×Q8—24no—
100C2×C7×Q8—22no—
101C2×C7×Q8—82no—
102C2×C7×Q8—82no—
103C2×C7×Q8C2×D414yes—
104C2×C7×D4—22no—
105C2×C7×D4—41no—
106C2×C7×D4—41no—
107C2×C7×D4—44no—
108C2×C7×D4—42no—
109C2×C7×D4—42no—
110C2×C7×D4—81no—
111C2×C7×D4—82no—
112C2×C7×D4—82no—
113C22×C4×C7—22no—
114C22×C4×C7—21no—
115C22×C4×C7—22no—
116C22×C4×C7—24no—
117C22×C4×C7—24no—
118C22×C4×C7—21no—
119C22×C4×C7—21no—
120C22×C4×C7—22no—
121C22×C4×C7—81no—
122C22×C4×C7—81no—
123C22×C4×C7—82no—
124C22×C4×C7—82no—
125C22×C4×C7C2×D412yes—
126C22×C4×C7C2×D411yescharacteristic
127C22×C4×C7C2×D412yes—
128C22×C4×C7C2×D412yes—
129C22×C4×C7C2411yescharacteristic
130C7×SD16—84no—
131C2×C7×C8—24no—
132C2×C7×C8—82no—
133C7×C4⋊C4—24no—
134C7×C4⋊C4—28no—
135C7×C4⋊C4—24no—
136C7×C4⋊C4—22no—
137C7×C4⋊C4—24no—
138C7×C4⋊C4—22no—
139C7×C4⋊C4—81no—
140C7×C4⋊C4C2×D414yes—
141C7×C22⋊C4—28no—
142C7×C22⋊C4—24no—
143C7×C22⋊C4—24no—
144C7×C22⋊C4—24no—
145C7×C22⋊C4—24no—
146C7×C22⋊C4—44no—
147C7×C22⋊C4—42no—
148C7×C22⋊C4—42no—
149C7×C22⋊C4—82no—
150#543—24no—
151#543—22no—
152#543C22×C714yes—
153C2×C22⋊Q8—22no—
154C2×C22⋊Q8—21no—
155C2×C22⋊Q8—22no—
156C2×C22⋊Q8C22×C712yes—
157C2×#121C22×C711yescharacteristic
158C2×#120—21no—
159C22×C22⋊C4—21no—
160C22×C22⋊C4C22×C711yescharacteristic
161#483C22×C714yescomplemented
162#481—22no—
163#465—24no—
164#450C22×C714yescomplemented
165#448—22no—
166C2×C22⋊C8C22×C712yes—
167#397C22×C712yes—
168#394—22no—
169#392C22×C712yes—
170#386—21no—
171#380C22×C711yescharacteristic
172#328C22×C718yescomplemented
173#323C22×C714yes—
174C23×C7—22no—
175C23×C7—24no—
176C23×C7—22no—
177C23×C7—24no—
178C23×C7—41no—
179C23×C7—41no—
180C23×C7—42no—
181C23×C7—42no—
182C23×C7—42no—
183C23×C7—42no—
184C23×C7—42no—
185C23×C7—42no—
186C23×C7—41no—
187C23×C7—41no—
188C23×C7—41no—
189C23×C7—81no—
190C23×C7—81no—
191C23×C7—84no—
192C23×C7—84no—
193C23×C7#12011yescharacteristic
194C23×C7#12011yescharacteristic
195C23×C7C2×SD1614yes—
196C23×C7C2×SD1612yes—
197C23×C7#13712yes—
198C23×C7C22×D411yescharacteristic
199C7×Q8—44no—
200C7×Q8—82no—
201C7×Q8—84no—
202C7×D4—44no—
203C7×D4—88no—
204C7×D4—82no—
205C7×D4—82no—
206C2×C4×C7—24no—
207C2×C4×C7—22no—
208C2×C4×C7—24no—
209C2×C4×C7—41no—
210C2×C4×C7—42no—
211C2×C4×C7—42no—
212C2×C4×C7—41no—
213C2×C4×C7—42no—
214C2×C4×C7—44no—
215C2×C4×C7—44no—
216C2×C4×C7—42no—
217C2×C4×C7—42no—
218C2×C4×C7—42no—
219C2×C4×C7—41no—
220C2×C4×C7—42no—
221C2×C4×C7—41no—
222C2×C4×C7—41no—
223C2×C4×C7—81no—
224C2×C4×C7—84no—
225C2×C4×C7—84no—
226C2×C4×C7#12012yes—
227C2×C4×C7C22×D412yes—
228C7×C8—44no—
229C25C7×D411yescharacteristic
230C22×Q8—21no—
231C22×Q8C7×D412yescomplemented
232C22×D4—21no—
233C22×D4—22no—
234C22×D4C23×C711yescharacteristic
235C23×C4—21no—
236C23×C4—21no—
237C23×C4C7×D411yescharacteristic
238C23×C4C23×C711yescharacteristic
239C2×SD16—24no—
240C22×C8—22no—
241C22⋊Q8—28no—
242C22⋊Q8—28no—
243C22⋊Q8—24no—
244C22⋊Q8—28no—
245C22⋊Q8—24no—
246C22⋊Q8—22no—
247C22⋊Q8—22no—
248C22⋊Q8—28no—
249C22⋊Q8C7×D414yescomplemented
250C22⋊Q8C23×C714yes—
251#121—22no—
252#121—22no—
253#120—44no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4C7×D412yescomplemented
259C2×C4⋊C4C23×C711yescharacteristic
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4C7×D412yes—
271C2×C22⋊C4C23×C712yes—
272#106—24no—
273#103—24no—
274#102—24no—
275C22⋊C8—22no—
276C22⋊C8C23×C714yes—
277#95—22no—
278#95C23×C712yes—
279C22×C7—21no—
280C22×C7—21no—
281C22×C7—21no—
282C22×C7—21no—
283C22×C7—21no—
284C22×C7—21no—
285C22×C7—24no—
286C22×C7—42no—
287C22×C7—42no—
288C22×C7—44no—
289C22×C7—48no—
290C22×C7—42no—
291C22×C7—42no—
292C22×C7—82no—
293C22×C7—84no—
294C22×C7—82no—
295C22×C7—82no—
296C22×C7—81no—
297C22×C7—81no—
298C22×C7—81no—
299C22×C7#44812yes—
300C22×C7#44812yes—
301C22×C7C2×#12011yescharacteristic
302C22×C7C22×SD1611yescharacteristic
303C22×C7C2×#13711yescharacteristic
304C4×C7—22no—
305C4×C7—22no—
306C4×C7—42no—
307C4×C7—44no—
308C4×C7—44no—
309C4×C7—82no—
310C4×C7—81no—
311C24—21no—
312C24—21no—
313C24—22no—
314C24—21no—
315C24—81no—
316C24—81no—
317C24C7×SD1614yescomplemented
318C24C2×C7×D412yes—
319C24C2×C7×D411yescharacteristic
320C2×Q8—24no—
321C2×Q8—22no—
322C2×Q8—82no—
323C2×Q8—82no—
324C2×Q8C2×C7×D414yes—
325C2×D4—22no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—44no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—81no—
332C2×D4—82no—
333C2×D4—82no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—81no—
343C22×C4—81no—
344C22×C4—82no—
345C22×C4—82no—
346C22×C4C2×C7×D412yes—
347C22×C4C2×C7×D411yescharacteristic
348C22×C4C2×C7×D412yes—
349C22×C4C2×C7×D412yes—
350C22×C4C24×C711yescharacteristic
351SD16—84no—
352C2×C8—24no—
353C2×C8—82no—
354C4⋊C4—24no—
355C4⋊C4—28no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—81no—
361C4⋊C4C2×C7×D414yes—
362C22⋊C4—28no—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—24no—
366C22⋊C4—24no—
367C22⋊C4—44no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—82no—
371C2×C7—21no—
372C2×C7—22no—
373C2×C7—21no—
374C2×C7—22no—
375C2×C7—44no—
376C2×C7—161no—
377C2×C7C2×#44811yescharacteristic
378C2×C7#301911yescharacteristic
379C2×C7#302611yescharacteristic
380C23—22no—
381C23—24no—
382C23—22no—
383C23—24no—
384C23—41no—
385C23—41no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—81no—
396C23—81no—
397C23—84no—
398C23—84no—
399C23C7×#12011yescharacteristic
400C23C7×#12011yescharacteristic
401C23C2×C7×SD1614yes—
402C23C2×C7×SD1612yes—
403C23C7×#13712yes—
404C23C22×C7×D411yescharacteristic
405Q8—44no—
406Q8—82no—
407Q8—84no—
408D4—44no—
409D4—88no—
410D4—82no—
411D4—82no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—81no—
430C2×C4—84no—
431C2×C4—84no—
432C2×C4C7×#12012yes—
433C2×C4C22×C7×D412yes—
434C8—44no—
435C7#1828611yescharacteristic, direct product factor, Sylow
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—24no—
443C22—42no—
444C22—42no—
445C22—44no—
446C22—48no—
447C22—42no—
448C22—42no—
449C22—82no—
450C22—84no—
451C22—82no—
452C22—82no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22C7×#44812yes—
457C22C7×#44812yes—
458C22C2×C7×#12011yescharacteristic
459C22C22×C7×SD1611yescharacteristic
460C22C2×C7×#13711yescharacteristic
461C4—22no—
462C4—22no—
463C4—42no—
464C4—44no—
465C4—44no—
466C4—82no—
467C4—81no—
468C2—21no—
469C2—22no—
470C2—21no—
471C2—22no—
472C2—44no—
473C2—161no—
474C2C2×C7×#44811yescharacteristic
475C2C7×#301911yescharacteristic
476C2C7×#302611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#2617C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#543C2semidirect
2. C7×#3187C2semidirect
4. C7×#1608C2semidirect
5. C7×#1596C2semidirect
6. C7×#1520C2semidirect
7. C2×C7×#323C2semidirect
19. C7×#483C22semidirect
22. C7×#450C22semidirect
30. C7×#328C22semidirect
32. #18286C7direct (N × H)
35. C22×C7×Q8D4semidirect
53. C7×C22⋊Q8D4semidirect
62. C2×C7×C4⋊C4D4semidirect
83. C2×#543C2×C7semidirect
84. #3187C2×C7semidirect
86. #1608C2×C7semidirect
87. #1596C2×C7semidirect
88. #1520C2×C7semidirect
89. C2×#323C2×C7semidirect
96. C24×C7SD16semidirect
161. #483C22×C7semidirect
164. #450C22×C7semidirect
172. #328C22×C7semidirect
231. C22×Q8C7×D4semidirect
249. C22⋊Q8C7×D4semidirect
258. C2×C4⋊C4C7×D4semidirect
317. C24C7×SD16semidirect
435. C7#18286direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5432
C7×#31872
C7×#26172
C7×#16082
C7×#15962
C7×#15202
C2×C7×#3232
#182867