Finite Group Databasebeta

← C7×#18762 #13195978

Subgroups — C7×#18762 #13195978

Order 1792.

964 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#556C211yesmaximal, characteristic, complemented
2C7×#3227C211yesmaximal, characteristic, complemented
3C7×#3100C211yesmaximal, characteristic
4C7×#3037C211yesmaximal, characteristic, complemented
5C7×#1652C214yesmaximal, complemented
6C7×#1628C214yesmaximal, complemented
7C7×#1549C211yesmaximal, characteristic, complemented
8C7×#1486C211yesmaximal, characteristic, complemented
9C7×#1479C211yesmaximal, characteristic, complemented
10C2×C7×#137—21no—
11C7×#556—22no—
12C7×#556—22no—
13C7×#556—22no—
14C7×#556C2214yes—
15C7×#532C2211yescharacteristic
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×#126—22no—
18C2×C7×#124—21no—
19C2×C7×C22⋊Q8—21no—
20C2×C4×C7×Q8—21no—
21C2×C4×C7×Q8C2211yescharacteristic
22C2×C4×C7×D4C2211yescharacteristic
23C2×C7×#117—21no—
24C7×#498C2212yescomplemented
25C7×#486C2212yescomplemented
26C7×#481—22no—
27C7×#480—21no—
28C7×#476—22no—
29C7×#474C2212yes—
30C7×#464—22no—
31C7×#462C2212yescomplemented
32C7×#460C2212yescomplemented
33C7×#459C2212yes—
34C7×#447—21no—
35C4×C7×D8—22no—
36C7×#431C2211yescharacteristic
37C7×#431C2211yescharacteristic
38C7×#423C2211yescharacteristic
39C7×#417—21no—
40C7×#332C2214yescomplemented
41C7×Q8⋊C8C2214yescomplemented
42C7×D4⋊C8C2214yes—
43#18762C711yesmaximal, characteristic, direct product factor, Sylow
44C2×C7×#41—21no—
45C22×C7×Q8—21no—
46C22×C7×Q8D411yescharacteristic
47C22×C7×D4D411yescharacteristic
48C23×C4×C7—21no—
49C7×#137—44no—
50C2×C7×Q16—22no—
51C2×C7×SD16—22no—
52C2×C7×D8—22no—
53C2×C7×M16—21no—
54C7×C4⋊Q8—21no—
55C7×C4⋊Q8—21no—
56C7×C4⋊Q8D412yes—
57C7×C4⋊Q8C2311yescharacteristic
58C7×C4⋊Q8C2312yes—
59C7×#127C2311yescharacteristic
60C7×#126—28no—
61C7×#126—28no—
62C7×#124—24no—
63C7×#124—24no—
64C7×#124—21no—
65C7×C22⋊Q8—24no—
66C7×C22⋊Q8—24no—
67C7×#121—21no—
68C7×#121—21no—
69C4×C7×Q8—24no—
70C4×C7×Q8—24no—
71C4×C7×Q8—21no—
72C4×C7×Q8—21no—
73C4×C7×Q8D412yes—
74C4×C7×Q8C2312yes—
75C4×C7×D4—21no—
76C4×C7×D4—21no—
77C4×C7×D4—21no—
78C4×C7×D4—21no—
79C4×C7×D4D412yes—
80C4×C7×D4C2312yes—
81C7×#117—24no—
82C7×#117—24no—
83C2×C7×C4⋊C4—22no—
84C2×C7×C4⋊C4—21no—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4D411yescharacteristic
87C2×C7×C4⋊C4D412yes—
88C2×C7×C4⋊C4D411yescharacteristic
89C2×C7×C22⋊C4—22no—
90C2×C7×C22⋊C4—22no—
91C2×C7×C22⋊C4—21no—
92C2×C42×C7—22no—
93C2×C42×C7—21no—
94C2×C42×C7C2311yescharacteristic
95C7×#107—22no—
96C7×#106—24no—
97C7×C4⋊C8—21no—
98C7×C4⋊C8—21no—
99C7×C4⋊C8C2312yes—
100C7×C4⋊C8C2312yes—
101C7×#103—22no—
102C7×#102—22no—
103C7×#102—24no—
104C7×#102—22no—
105C7×#102—22no—
106C7×C22⋊C8—21no—
107C7×C22⋊C8—21no—
108C4×C7×C8—21no—
109C4×C7×C8C2312yes—
110C2×#556C2×C711yescharacteristic, complemented
111#3227C2×C711yescharacteristic, complemented
112#3100C2×C711yescharacteristic
113#3037C2×C711yescharacteristic, complemented
114#1652C2×C714yescomplemented
115#1628C2×C714yescomplemented
116#1549C2×C711yescharacteristic, complemented
117#1486C2×C711yescharacteristic, complemented
118#1479C2×C711yescharacteristic, complemented
119C24×C7—21no—
120C7×#41—42no—
121C7×#41—42no—
122C2×C7×Q8—21no—
123C2×C7×Q8—22no—
124C2×C7×Q8—22no—
125C2×C7×Q8—22no—
126C2×C7×Q8—22no—
127C2×C7×Q8—21no—
128C2×C7×Q8—44no—
129C2×C7×Q8C2×D412yes—
130C2×C7×D4—21no—
131C2×C7×D4—21no—
132C2×C7×D4—21no—
133C2×C7×D4—21no—
134C2×C7×D4—42no—
135C2×C7×D4—41no—
136C2×C7×D4C2×D412yes—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—42no—
144C22×C4×C7—42no—
145C22×C4×C7—41no—
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C7×Q16—44no—
150C7×SD16—44no—
151C7×D8—44no—
152C7×M16—42no—
153C2×C7×C8—22no—
154C2×C7×C8—22no—
155C2×C7×C8—22no—
156C7×C4⋊C4—28no—
157C7×C4⋊C4—22no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4—21no—
160C7×C4⋊C4—21no—
161C7×C4⋊C4—22no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4C2×D412yes—
167C7×C4⋊C4C2×D414yes—
168C7×C4⋊C4C2×D412yes—
169C7×C22⋊C4—28no—
170C7×C22⋊C4—28no—
171C7×C22⋊C4—22no—
172C7×C22⋊C4—22no—
173C7×C22⋊C4—41no—
174C7×C22⋊C4—41no—
175C42×C7—28no—
176C42×C7—22no—
177C42×C7—22no—
178C42×C7C2×D411yescharacteristic
179C42×C7C2×D411yescharacteristic
180C42×C7C2×D411yescharacteristic
181C42×C7C2411yescharacteristic
182C2×#137—21no—
183#556—22no—
184#556—22no—
185#556—22no—
186#556C22×C714yes—
187#532C22×C711yescharacteristic
188C2×C4⋊Q8C22×C711yescharacteristic
189C2×#126—22no—
190C2×#124—21no—
191C2×C22⋊Q8—21no—
192C2×C4×Q8—21no—
193C2×C4×Q8C22×C711yescharacteristic
194C2×C4×D4C22×C711yescharacteristic
195C2×#117—21no—
196#498C22×C712yescomplemented
197#486C22×C712yescomplemented
198#481—22no—
199#480—21no—
200#476—22no—
201#474C22×C712yes—
202#464—22no—
203#462C22×C712yescomplemented
204#460C22×C712yescomplemented
205#459C22×C712yes—
206#447—21no—
207C4×D8—22no—
208#431C22×C711yescharacteristic
209#431C22×C711yescharacteristic
210#423C22×C711yescharacteristic
211#417—21no—
212#332C22×C714yescomplemented
213Q8⋊C8C22×C714yescomplemented
214D4⋊C8C22×C714yes—
215C23×C7—22no—
216C23×C7—42no—
217C23×C7—42no—
218C23×C7—42no—
219C23×C7—41no—
220C23×C7#12011yescharacteristic
221C7×Q8—22no—
222C7×Q8—22no—
223C7×Q8—44no—
224C7×Q8—44no—
225C7×Q8—42no—
226C7×Q8—41no—
227C7×Q8—41no—
228C7×D4—22no—
229C7×D4—22no—
230C7×D4—42no—
231C7×D4—42no—
232C7×D4—41no—
233C7×D4—41no—
234C7×D4—42no—
235C7×D4—82no—
236C2×C4×C7—22no—
237C2×C4×C7—22no—
238C2×C4×C7—22no—
239C2×C4×C7—22no—
240C2×C4×C7—24no—
241C2×C4×C7—41no—
242C2×C4×C7—42no—
243C2×C4×C7—44no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7—42no—
247C2×C4×C7—42no—
248C2×C4×C7—44no—
249C2×C4×C7—41no—
250C2×C4×C7—82no—
251C2×C4×C7#12011yescharacteristic
252C2×C4×C7#12011yescharacteristic
253C2×C4×C7#12011yescharacteristic
254C2×C4×C7#13512yes—
255C2×C4×C7AGL(1,ℤ/8)12yes—
256C2×C4×C7C22×D411yescharacteristic
257C2×C4×C7C22×D411yescharacteristic
258C2×C4×C7C22×D411yescharacteristic
259C7×C8—42no—
260C7×C8—42no—
261C7×C8—42no—
262C2×#41—21no—
263C22×Q8—21no—
264C22×Q8C7×D411yescharacteristic
265C22×D4C7×D411yescharacteristic
266C23×C4—21no—
267#137—44no—
268C2×Q16—22no—
269C2×SD16—22no—
270C2×D8—22no—
271C2×M16—21no—
272C4⋊Q8—21no—
273C4⋊Q8—21no—
274C4⋊Q8C7×D412yes—
275C4⋊Q8C23×C711yescharacteristic
276C4⋊Q8C23×C712yes—
277#127C23×C711yescharacteristic
278#126—28no—
279#126—28no—
280#124—24no—
281#124—24no—
282#124—21no—
283C22⋊Q8—24no—
284C22⋊Q8—24no—
285#121—21no—
286#121—21no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—21no—
290C4×Q8—21no—
291C4×Q8C7×D412yes—
292C4×Q8C23×C712yes—
293C4×D4—21no—
294C4×D4—21no—
295C4×D4—21no—
296C4×D4—21no—
297C4×D4C7×D412yes—
298C4×D4C23×C712yes—
299#117—24no—
300#117—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4C7×D411yescharacteristic
305C2×C4⋊C4C7×D411yescharacteristic
306C2×C4⋊C4C7×D412yes—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—21no—
310C2×C42—22no—
311C2×C42—21no—
312C2×C42C23×C711yescharacteristic
313#107—22no—
314#106—24no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8C23×C712yes—
318C4⋊C8C23×C712yes—
319#103—22no—
320#102—22no—
321#102—22no—
322#102—22no—
323#102—24no—
324C22⋊C8—21no—
325C22⋊C8—21no—
326C4×C8—21no—
327C4×C8C23×C712yes—
328C22×C7—21no—
329C22×C7—21no—
330C22×C7—42no—
331C22×C7—42no—
332C22×C7—42no—
333C22×C7—84no—
334C22×C7—81no—
335C22×C7—81no—
336C22×C7—81no—
337C22×C7#44912yes—
338C22×C7C2×#12011yescharacteristic
339C4×C7—21no—
340C4×C7—21no—
341C4×C7—21no—
342C4×C7—42no—
343C4×C7—42no—
344C4×C7—44no—
345C4×C7—42no—
346C4×C7—81no—
347C4×C7#44912yes—
348C4×C7C2×#13511yescharacteristic
349C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
350C4×C7#57512yes—
351C24—21no—
352#41—42no—
353#41—42no—
354C2×Q8—21no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—21no—
360C2×Q8—44no—
361C2×Q8C2×C7×D412yes—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—42no—
367C2×D4—41no—
368C2×D4C2×C7×D412yes—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4C2×C7×D411yescharacteristic
379C22×C4C2×C7×D411yescharacteristic
380C22×C4C2×C7×D411yescharacteristic
381Q16—44no—
382SD16—44no—
383D8—44no—
384M16—42no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C4⋊C4—28no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4C2×C7×D412yes—
399C4⋊C4C2×C7×D414yes—
400C4⋊C4C2×C7×D412yes—
401C22⋊C4—28no—
402C22⋊C4—28no—
403C22⋊C4—22no—
404C22⋊C4—22no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C42—28no—
408C42—22no—
409C42—22no—
410C42C2×C7×D411yescharacteristic
411C42C2×C7×D411yescharacteristic
412C42C2×C7×D411yescharacteristic
413C42C24×C711yescharacteristic
414C2×C7—22no—
415C2×C7—82no—
416C2×C7—161no—
417C2×C7C2×#44911yescharacteristic
418C2×C7#300511yescharacteristic
419C2×C7#302011yescharacteristic
420C23—22no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—41no—
425C23C7×#12011yescharacteristic
426Q8—22no—
427Q8—22no—
428Q8—44no—
429Q8—44no—
430Q8—42no—
431Q8—41no—
432Q8—41no—
433D4—22no—
434D4—22no—
435D4—42no—
436D4—42no—
437D4—41no—
438D4—41no—
439D4—42no—
440D4—82no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—41no—
455C2×C4—82no—
456C2×C4C7×#12011yescharacteristic
457C2×C4C7×#12011yescharacteristic
458C2×C4C7×#12011yescharacteristic
459C2×C4C7×#13512yes—
460C2×C4C7×AGL(1,ℤ/8)12yes—
461C2×C4C22×C7×D411yescharacteristic
462C2×C4C22×C7×D411yescharacteristic
463C2×C4C22×C7×D411yescharacteristic
464C8—42no—
465C8—42no—
466C8—42no—
467C7#1876211yescharacteristic, direct product factor, Sylow
468C22—21no—
469C22—21no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—84no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22C7×#44912yes—
478C22C2×C7×#12011yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—42no—
483C4—42no—
484C4—44no—
485C4—42no—
486C4—81no—
487C4C7×#44912yes—
488C4C2×C7×#13511yescharacteristic
489C4C2×C7×AGL(1,ℤ/8)11yescharacteristic
490C4C7×#57512yes—
491C2—22no—
492C2—82no—
493C2—161no—
494C2C2×C7×#44911yescharacteristic
495C2C7×#300511yescharacteristic
496C2C7×#302011yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3100C2
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
1. C2×C7×#556C2semidirect
2. C7×#3227C2semidirect
4. C7×#3037C2semidirect
5. C7×#1652C2semidirect
6. C7×#1628C2semidirect
7. C7×#1549C2semidirect
8. C7×#1486C2semidirect
9. C7×#1479C2semidirect
24. C7×#498C22semidirect
25. C7×#486C22semidirect
31. C7×#462C22semidirect
32. C7×#460C22semidirect
40. C7×#332C22semidirect
41. C7×Q8⋊C8C22semidirect
43. #18762C7direct (N × H)
110. C2×#556C2×C7semidirect
111. #3227C2×C7semidirect
113. #3037C2×C7semidirect
114. #1652C2×C7semidirect
115. #1628C2×C7semidirect
116. #1549C2×C7semidirect
117. #1486C2×C7semidirect
118. #1479C2×C7semidirect
196. #498C22×C7semidirect
197. #486C22×C7semidirect
203. #462C22×C7semidirect
204. #460C22×C7semidirect
212. #332C22×C7semidirect
213. Q8⋊C8C22×C7semidirect
467. C7#18762direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5562
C7×#32272
C7×#31002
C7×#30372
C7×#16522
C7×#16282
C7×#15492
C7×#14862
C7×#14792
#187627