Finite Group Databasebeta

← C7×#18285 #13195501

Subgroups — C7×#18285 #13195501

Order 1792.

1408 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#534C211yesmaximal, characteristic
2C7×#3187C212yesmaximal, complemented
3C7×#2626C211yesmaximal, characteristic, complemented
4C7×#1613C214yesmaximal, complemented
5C7×#1601C214yesmaximal, complemented
6C7×#1519C212yesmaximal, complemented
7C2×C7×#323C211yesmaximal, characteristic, complemented
8C23×C7×D4—21no—
9C7×#534—24no—
10C7×#534—22no—
11C7×#534C2214yes—
12C2×C7×#123—22no—
13C2×C7×C22⋊Q8C2211yescharacteristic, complemented
14C2×C7×#121—21no—
15C2×C7×#121C2212yes—
16C2×C7×#120—22no—
17C22×C7×C22⋊C4C2211yescharacteristic
18C7×#483—22no—
19C7×#481C2214yescomplemented
20C7×#465—24no—
21C7×#450—22no—
22C7×#448C2214yescomplemented
23C2×C7×C22⋊C8C2212yescomplemented
24C7×#399C2212yescomplemented
25C7×#396—22no—
26C7×#393C2212yescomplemented
27C7×#385—21no—
28C7×#380C2211yescharacteristic, complemented
29C7×#379—21no—
30C7×#327C2218yescomplemented
31C7×#323C2214yescomplemented
32#18285C711yesmaximal, characteristic, direct product factor, Sylow
33C25×C7—21no—
34C25×C7D411yescharacteristic
35C22×C7×Q8C2311yescharacteristic, complemented
36C22×C7×D4—22no—
37C22×C7×D4—24no—
38C22×C7×D4—22no—
39C22×C7×D4—24no—
40C22×C7×D4—21no—
41C22×C7×D4—44no—
42C22×C7×D4—44no—
43C22×C7×D4D412yes—
44C23×C4×C7—21no—
45C23×C4×C7D411yescharacteristic
46C23×C4×C7C2311yescharacteristic
47C2×C7×SD16—24no—
48C22×C7×C8—22no—
49C7×#123—28no—
50C7×#123—28no—
51C7×C22⋊Q8—22no—
52C7×C22⋊Q8—22no—
53C7×#121—24no—
54C7×#121—24no—
55C7×#121—22no—
56C7×#121—22no—
57C7×#121D414yes—
58C7×#121C2314yes—
59C7×#120—28no—
60C7×#120—28no—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4—21no—
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4—22no—
65C2×C7×C4⋊C4D412yes—
66C2×C7×C4⋊C4C2311yescharacteristic, complemented
67C2×C7×C22⋊C4—22no—
68C2×C7×C22⋊C4—22no—
69C2×C7×C22⋊C4—22no—
70C2×C7×C22⋊C4—22no—
71C2×C7×C22⋊C4—21no—
72C2×C7×C22⋊C4—21no—
73C2×C7×C22⋊C4—22no—
74C2×C7×C22⋊C4—22no—
75C2×C7×C22⋊C4D412yes—
76C2×C7×C22⋊C4C2312yes—
77C7×#106—24no—
78C7×#103—24no—
79C7×#102—24no—
80C7×C22⋊C8—22no—
81C7×C22⋊C8C2314yescomplemented
82C7×#95—22no—
83C7×#95C2312yescomplemented
84C2×#534C2×C711yescharacteristic
85#3187C2×C712yescomplemented
86#2626C2×C711yescharacteristic, complemented
87#1613C2×C714yescomplemented
88#1601C2×C714yescomplemented
89#1519C2×C712yescomplemented
90C2×#323C2×C711yescharacteristic, complemented
91C24×C7—21no—
92C24×C7—21no—
93C24×C7—22no—
94C24×C7—24no—
95C24×C7—22no—
96C24×C7—44no—
97C24×C7—81no—
98C24×C7—81no—
99C24×C7—82no—
100C24×C7—82no—
101C24×C7SD1614yescomplemented
102C24×C7C2×D412yes—
103C24×C7C2×D411yescharacteristic
104C2×C7×Q8—22no—
105C2×C7×Q8—81no—
106C2×C7×D4—28no—
107C2×C7×D4—24no—
108C2×C7×D4—24no—
109C2×C7×D4—24no—
110C2×C7×D4—22no—
111C2×C7×D4—48no—
112C2×C7×D4—416no—
113C2×C7×D4—48no—
114C2×C7×D4—48no—
115C2×C7×D4—44no—
116C2×C7×D4—44no—
117C2×C7×D4—82no—
118C2×C7×D4—82no—
119C2×C7×D4C2×D414yes—
120C22×C4×C7—24no—
121C22×C4×C7—21no—
122C22×C4×C7—22no—
123C22×C4×C7—21no—
124C22×C4×C7—21no—
125C22×C4×C7—22no—
126C22×C4×C7—21no—
127C22×C4×C7—44no—
128C22×C4×C7—81no—
129C22×C4×C7—81no—
130C22×C4×C7C2×D412yes—
131C22×C4×C7C2×D411yescharacteristic
132C22×C4×C7C2×D412yes—
133C22×C4×C7C2×D412yes—
134C22×C4×C7C2411yescharacteristic
135C7×SD16—84no—
136C2×C7×C8—24no—
137C2×C7×C8—82no—
138C7×C4⋊C4—24no—
139C7×C4⋊C4—22no—
140C7×C4⋊C4—22no—
141C7×C4⋊C4—42no—
142C7×C4⋊C4—81no—
143C7×C4⋊C4—82no—
144C7×C4⋊C4C2×D414yes—
145C7×C22⋊C4—28no—
146C7×C22⋊C4—24no—
147C7×C22⋊C4—24no—
148C7×C22⋊C4—24no—
149C7×C22⋊C4—24no—
150C7×C22⋊C4—41no—
151C7×C22⋊C4—41no—
152C7×C22⋊C4—42no—
153C7×C22⋊C4—82no—
154C7×C22⋊C4—82no—
155C23×D4—21no—
156#534—24no—
157#534—22no—
158#534C22×C714yes—
159C2×#123—22no—
160C2×C22⋊Q8C22×C711yescharacteristic, complemented
161C2×#121—21no—
162C2×#121C22×C712yes—
163C2×#120—22no—
164C22×C22⋊C4C22×C711yescharacteristic
165#483—22no—
166#481C22×C714yescomplemented
167#465—24no—
168#450—22no—
169#448C22×C714yescomplemented
170C2×C22⋊C8C22×C712yescomplemented
171#399C22×C712yescomplemented
172#396—22no—
173#393C22×C712yescomplemented
174#385—21no—
175#380C22×C711yescharacteristic, complemented
176#379—21no—
177#327C22×C718yescomplemented
178#323C22×C714yescomplemented
179C23×C7—22no—
180C23×C7—24no—
181C23×C7—22no—
182C23×C7—24no—
183C23×C7—24no—
184C23×C7—41no—
185C23×C7—41no—
186C23×C7—42no—
187C23×C7—42no—
188C23×C7—42no—
189C23×C7—42no—
190C23×C7—48no—
191C23×C7—44no—
192C23×C7—44no—
193C23×C7—44no—
194C23×C7—81no—
195C23×C7—81no—
196C23×C7—84no—
197C23×C7—84no—
198C23×C7—84no—
199C23×C7—84no—
200C23×C7—816no—
201C23×C7#12011yescharacteristic
202C23×C7#12011yescharacteristic
203C23×C7C2×SD1614yes—
204C23×C7C2×SD1612yes—
205C23×C7AGL(1,ℤ/8)12yes—
206C23×C7C22×D411yescharacteristic
207C7×Q8—44no—
208C7×D4—44no—
209C7×D4—416no—
210C7×D4—44no—
211C7×D4—44no—
212C7×D4—48no—
213C7×D4—82no—
214C7×D4—84no—
215C2×C4×C7—22no—
216C2×C4×C7—24no—
217C2×C4×C7—41no—
218C2×C4×C7—42no—
219C2×C4×C7—42no—
220C2×C4×C7—41no—
221C2×C4×C7—48no—
222C2×C4×C7—42no—
223C2×C4×C7—44no—
224C2×C4×C7—42no—
225C2×C4×C7—42no—
226C2×C4×C7—42no—
227C2×C4×C7—42no—
228C2×C4×C7—42no—
229C2×C4×C7—41no—
230C2×C4×C7—41no—
231C2×C4×C7—42no—
232C2×C4×C7—81no—
233C2×C4×C7#12012yes—
234C2×C4×C7C22×D412yes—
235C7×C8—44no—
236C25—21no—
237C25C7×D411yescharacteristic
238C22×Q8C23×C711yescharacteristic, complemented
239C22×D4—22no—
240C22×D4—24no—
241C22×D4—22no—
242C22×D4—24no—
243C22×D4—21no—
244C22×D4—44no—
245C22×D4—44no—
246C22×D4C7×D412yes—
247C23×C4—21no—
248C23×C4C7×D411yescharacteristic
249C23×C4C23×C711yescharacteristic
250C2×SD16—24no—
251C22×C8—22no—
252#123—28no—
253#123—28no—
254C22⋊Q8—22no—
255C22⋊Q8—22no—
256#121—24no—
257#121—24no—
258#121—22no—
259#121—22no—
260#121C7×D414yes—
261#121C23×C714yes—
262#120—28no—
263#120—28no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4C7×D412yes—
269C2×C4⋊C4C23×C711yescharacteristic, complemented
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4C7×D412yes—
279C2×C22⋊C4C23×C712yes—
280#106—24no—
281#103—24no—
282#102—24no—
283C22⋊C8—22no—
284C22⋊C8C23×C714yescomplemented
285#95—22no—
286#95C23×C712yescomplemented
287C22×C7—21no—
288C22×C7—21no—
289C22×C7—21no—
290C22×C7—21no—
291C22×C7—21no—
292C22×C7—21no—
293C22×C7—24no—
294C22×C7—42no—
295C22×C7—42no—
296C22×C7—44no—
297C22×C7—48no—
298C22×C7—42no—
299C22×C7—42no—
300C22×C7—44no—
301C22×C7—44no—
302C22×C7—44no—
303C22×C7—82no—
304C22×C7—84no—
305C22×C7—88no—
306C22×C7—816no—
307C22×C7#45012yes—
308C22×C7#45012yes—
309C22×C7C2×#12011yescharacteristic
310C22×C7C22×SD1611yescharacteristic
311C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
312C4×C7—22no—
313C4×C7—22no—
314C4×C7—44no—
315C4×C7—42no—
316C4×C7—82no—
317C4×C7—81no—
318C4×C7—81no—
319C24—21no—
320C24—21no—
321C24—22no—
322C24—24no—
323C24—22no—
324C24—44no—
325C24—81no—
326C24—81no—
327C24—82no—
328C24—82no—
329C24C7×SD1614yescomplemented
330C24C2×C7×D412yes—
331C24C2×C7×D411yescharacteristic
332C2×Q8—22no—
333C2×Q8—81no—
334C2×D4—28no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—22no—
339C2×D4—48no—
340C2×D4—416no—
341C2×D4—48no—
342C2×D4—48no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—82no—
346C2×D4—82no—
347C2×D4C2×C7×D414yes—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—44no—
356C22×C4—81no—
357C22×C4—81no—
358C22×C4C2×C7×D412yes—
359C22×C4C2×C7×D411yescharacteristic
360C22×C4C2×C7×D412yes—
361C22×C4C2×C7×D412yes—
362C22×C4C24×C711yescharacteristic
363SD16—84no—
364C2×C8—24no—
365C2×C8—82no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—42no—
370C4⋊C4—81no—
371C4⋊C4—82no—
372C4⋊C4C2×C7×D414yes—
373C22⋊C4—28no—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—24no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C22⋊C4—42no—
381C22⋊C4—82no—
382C22⋊C4—82no—
383C2×C7—21no—
384C2×C7—22no—
385C2×C7—21no—
386C2×C7—22no—
387C2×C7—44no—
388C2×C7—84no—
389C2×C7C2×#45011yescharacteristic
390C2×C7#301911yescharacteristic
391C2×C7#302611yescharacteristic
392C23—22no—
393C23—24no—
394C23—22no—
395C23—24no—
396C23—24no—
397C23—41no—
398C23—41no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—48no—
404C23—44no—
405C23—44no—
406C23—44no—
407C23—81no—
408C23—81no—
409C23—84no—
410C23—84no—
411C23—84no—
412C23—84no—
413C23—816no—
414C23C7×#12011yescharacteristic
415C23C7×#12011yescharacteristic
416C23C2×C7×SD1614yes—
417C23C2×C7×SD1612yes—
418C23C7×AGL(1,ℤ/8)12yes—
419C23C22×C7×D411yescharacteristic
420Q8—44no—
421D4—44no—
422D4—416no—
423D4—44no—
424D4—44no—
425D4—48no—
426D4—82no—
427D4—84no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—48no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—81no—
446C2×C4C7×#12012yes—
447C2×C4C22×C7×D412yes—
448C8—44no—
449C7#1828511yescharacteristic, direct product factor, Sylow
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—24no—
457C22—42no—
458C22—42no—
459C22—44no—
460C22—48no—
461C22—42no—
462C22—42no—
463C22—44no—
464C22—44no—
465C22—44no—
466C22—82no—
467C22—84no—
468C22—88no—
469C22—816no—
470C22C7×#45012yes—
471C22C7×#45012yes—
472C22C2×C7×#12011yescharacteristic
473C22C22×C7×SD1611yescharacteristic
474C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
475C4—22no—
476C4—22no—
477C4—44no—
478C4—42no—
479C4—82no—
480C4—81no—
481C4—81no—
482C2—21no—
483C2—22no—
484C2—21no—
485C2—22no—
486C2—44no—
487C2—84no—
488C2C2×C7×#45011yescharacteristic
489C2C7×#301911yescharacteristic
490C2C7×#302611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#534C2
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
2. C7×#3187C2semidirect
3. C7×#2626C2semidirect
4. C7×#1613C2semidirect
5. C7×#1601C2semidirect
6. C7×#1519C2semidirect
7. C2×C7×#323C2semidirect
13. C2×C7×C22⋊Q8C22semidirect
19. C7×#481C22semidirect
22. C7×#448C22semidirect
23. C2×C7×C22⋊C8C22semidirect
24. C7×#399C22semidirect
26. C7×#393C22semidirect
28. C7×#380C22semidirect
30. C7×#327C22semidirect
31. C7×#323C22semidirect
32. #18285C7direct (N × H)
35. C22×C7×Q8C23semidirect
66. C2×C7×C4⋊C4C23semidirect
81. C7×C22⋊C8C23semidirect
83. C7×#95C23semidirect
85. #3187C2×C7semidirect
86. #2626C2×C7semidirect
87. #1613C2×C7semidirect
88. #1601C2×C7semidirect
89. #1519C2×C7semidirect
90. C2×#323C2×C7semidirect
101. C24×C7SD16semidirect
160. C2×C22⋊Q8C22×C7semidirect
166. #481C22×C7semidirect
169. #448C22×C7semidirect
170. C2×C22⋊C8C22×C7semidirect
171. #399C22×C7semidirect
173. #393C22×C7semidirect
175. #380C22×C7semidirect
177. #327C22×C7semidirect
178. #323C22×C7semidirect
238. C22×Q8C23×C7semidirect
269. C2×C4⋊C4C23×C7semidirect
284. C22⋊C8C23×C7semidirect
286. #95C23×C7semidirect
329. C24C7×SD16semidirect
449. C7#18285direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5342
C7×#31872
C7×#26262
C7×#16132
C7×#16012
C7×#15192
C2×C7×#3232
#182857