Finite Group Databasebeta

← C7×#19007 #13196223

Subgroups — C7×#19007 #13196223

Order 1792.

862 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#563C211yesmaximal, characteristic
2C7×#3236C211yesmaximal, characteristic, complemented
3C7×#3215C211yesmaximal, characteristic
4C7×#3166C211yesmaximal, characteristic, complemented
5C7×#1664C214yesmaximal, complemented
6C7×#1660C214yesmaximal, complemented
7C7×#1534C211yesmaximal, characteristic, complemented
8C7×#1528C211yesmaximal, characteristic, complemented
9C7×#1527C211yesmaximal, characteristic
10C7×#563—22no—
11C7×#563—22no—
12C7×#563—22no—
13C7×#563C2214yes—
14C7×#533C2211yescharacteristic
15C2×C7×C4⋊Q8—21no—
16C2×C7×#126—21no—
17C2×C7×#126—21no—
18C2×C7×#125C2211yescharacteristic
19C2×C7×#125C2211yescharacteristic
20C2×C7×#124C2211yescharacteristic
21C2×C7×#123—21no—
22C2×C7×C22⋊Q8—21no—
23C2×C7×C22⋊Q8—21no—
24C7×#501C2212yescomplemented
25C7×#501C2212yescomplemented
26C7×#501C2212yescomplemented
27C7×#499—22no—
28C7×#497C2212yes—
29C7×#491C2212yescomplemented
30C7×#489C2212yes—
31C7×#484—21no—
32C7×#483—21no—
33C7×#479—22no—
34C7×#464—22no—
35C7×#451—21no—
36C7×#450—21no—
37C7×#432C2211yescharacteristic
38C7×#432C2211yescharacteristic
39C7×#428—21no—
40C7×#428—21no—
41C4×C7×M16C2211yescharacteristic
42C7×#330C2214yescomplemented
43C7×#330C2214yescomplemented
44C7×#329C2214yes—
45#19007C711yesmaximal, characteristic, direct product factor, Sylow
46C22×C7×Q8—21no—
47C22×C7×Q8D411yescharacteristic
48C22×C7×D4D411yescharacteristic
49C23×C4×C7—21no—
50C2×C7×Q16—22no—
51C2×C7×SD16—22no—
52C2×C7×M16—21no—
53C7×C4⋊Q8—24no—
54C7×C4⋊Q8—24no—
55C7×C4⋊Q8C2311yescharacteristic
56C7×C4⋊Q8C2311yescharacteristic
57C7×#126—24no—
58C7×#126—24no—
59C7×#126—24no—
60C7×#126—24no—
61C7×#125—21no—
62C7×#125—21no—
63C7×#125—21no—
64C7×#125—21no—
65C7×#125—21no—
66C7×#125D412yes—
67C7×#125D412yes—
68C7×#125C2312yes—
69C7×#125C2312yes—
70C7×#124—21no—
71C7×#124—21no—
72C7×#124D412yes—
73C7×#124C2312yes—
74C7×#123—24no—
75C7×#123—24no—
76C7×C22⋊Q8—24no—
77C7×C22⋊Q8—24no—
78C7×C22⋊Q8—24no—
79C7×C22⋊Q8—24no—
80C7×C22⋊Q8—21no—
81C7×C22⋊Q8—21no—
82C4×C7×Q8—21no—
83C4×C7×Q8—21no—
84C7×#117—21no—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4—21no—
87C2×C7×C4⋊C4—21no—
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⋊C4D411yescharacteristic
94C2×C7×C4⋊C4D411yescharacteristic
95C2×C7×C22⋊C4—21no—
96C2×C7×C22⋊C4—21no—
97C2×C7×C22⋊C4—21no—
98C2×C7×C22⋊C4—21no—
99C2×C7×C22⋊C4—21no—
100C2×C42×C7—21no—
101C2×C42×C7C2311yescharacteristic
102C7×#107—22no—
103C7×#107—22no—
104C7×#107—22no—
105C7×#106—22no—
106C7×#106—22no—
107C7×#106—22no—
108C7×C4⋊C8—21no—
109C7×C4⋊C8—21no—
110C7×#103—22no—
111C7×#103—22no—
112C7×#103—22no—
113C7×#102—22no—
114C7×C22⋊C8—21no—
115C7×C22⋊C8—21no—
116C7×#97C2312yes—
117C7×#97C2312yes—
118C7×#97C2312yes—
119C4×C7×C8—21no—
120C2×#563C2×C711yescharacteristic
121#3236C2×C711yescharacteristic, complemented
122#3215C2×C711yescharacteristic
123#3166C2×C711yescharacteristic, complemented
124#1664C2×C714yescomplemented
125#1660C2×C714yescomplemented
126#1534C2×C711yescharacteristic, complemented
127#1528C2×C711yescharacteristic, complemented
128#1527C2×C711yescharacteristic
129C24×C7—21no—
130C2×C7×Q8—24no—
131C2×C7×Q8—21no—
132C2×C7×Q8—21no—
133C2×C7×Q8—21no—
134C2×C7×Q8—81no—
135C2×C7×Q8—82no—
136C2×C7×Q8C2×D412yes—
137C2×C7×D4—21no—
138C2×C7×D4—81no—
139C2×C7×D4C2×D412yes—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—24no—
146C22×C4×C7—82no—
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C7×Q16—82no—
151C7×SD16—82no—
152C7×M16—24no—
153C2×C7×C8—22no—
154C2×C7×C8—22no—
155C2×C7×C8—22no—
156C7×C4⋊C4—24no—
157C7×C4⋊C4—24no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4—22no—
162C7×C4⋊C4—21no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—41no—
175C7×C4⋊C4—41no—
176C7×C4⋊C4C2×D412yes—
177C7×C4⋊C4C2×D412yes—
178C7×C4⋊C4C2×D412yes—
179C7×C4⋊C4C2×D412yes—
180C7×C22⋊C4—24no—
181C7×C22⋊C4—24no—
182C7×C22⋊C4—24no—
183C7×C22⋊C4—22no—
184C7×C22⋊C4—22no—
185C7×C22⋊C4—22no—
186C7×C22⋊C4—22no—
187C7×C22⋊C4—41no—
188C42×C7—24no—
189C42×C7—41no—
190C42×C7C2×D411yescharacteristic
191C42×C7C2×D411yescharacteristic
192C42×C7C2×D411yescharacteristic
193C42×C7C2411yescharacteristic
194#563—22no—
195#563—22no—
196#563—22no—
197#563C22×C714yes—
198#533C22×C711yescharacteristic
199C2×C4⋊Q8—21no—
200C2×#126—21no—
201C2×#126—21no—
202C2×#125C22×C711yescharacteristic
203C2×#125C22×C711yescharacteristic
204C2×#124C22×C711yescharacteristic
205C2×#123—21no—
206C2×C22⋊Q8—21no—
207C2×C22⋊Q8—21no—
208#501C22×C712yescomplemented
209#501C22×C712yescomplemented
210#501C22×C712yescomplemented
211#499—22no—
212#497C22×C712yes—
213#491C22×C712yescomplemented
214#489C22×C712yes—
215#484—21no—
216#483—21no—
217#479—22no—
218#464—22no—
219#451—21no—
220#450—21no—
221#432C22×C711yescharacteristic
222#432C22×C711yescharacteristic
223#428—21no—
224#428—21no—
225C4×M16C22×C711yescharacteristic
226#330C22×C714yescomplemented
227#330C22×C714yescomplemented
228#329C22×C714yes—
229C23×C7—22no—
230C23×C7—42no—
231C23×C7—42no—
232C23×C7—42no—
233C23×C7#12011yescharacteristic
234C7×Q8—42no—
235C7×Q8—42no—
236C7×Q8—42no—
237C7×Q8—81no—
238C7×Q8—84no—
239C7×D4—42no—
240C7×D4—81no—
241C2×C4×C7—22no—
242C2×C4×C7—22no—
243C2×C4×C7—22no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—22no—
247C2×C4×C7—41no—
248C2×C4×C7—41no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7—42no—
254C2×C4×C7—41no—
255C2×C4×C7—41no—
256C2×C4×C7—84no—
257C2×C4×C7#12011yescharacteristic
258C2×C4×C7#12011yescharacteristic
259C2×C4×C7#12011yescharacteristic
260C2×C4×C7C22×D411yescharacteristic
261C2×C4×C7C22×D411yescharacteristic
262C2×C4×C7C22×D411yescharacteristic
263C7×C8—24no—
264C7×C8—42no—
265C7×C8—42no—
266C22×Q8—21no—
267C22×Q8C7×D411yescharacteristic
268C22×D4C7×D411yescharacteristic
269C23×C4—21no—
270C2×Q16—22no—
271C2×SD16—22no—
272C2×M16—21no—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8C23×C711yescharacteristic
276C4⋊Q8C23×C711yescharacteristic
277#126—24no—
278#126—24no—
279#126—24no—
280#126—24no—
281#125—21no—
282#125—21no—
283#125—21no—
284#125—21no—
285#125—21no—
286#125C7×D412yes—
287#125C7×D412yes—
288#125C23×C712yes—
289#125C23×C712yes—
290#124—21no—
291#124—21no—
292#124C7×D412yes—
293#124C23×C712yes—
294#123—24no—
295#123—24no—
296C22⋊Q8—24no—
297C22⋊Q8—24no—
298C22⋊Q8—24no—
299C22⋊Q8—24no—
300C22⋊Q8—21no—
301C22⋊Q8—21no—
302C4×Q8—21no—
303C4×Q8—21no—
304#117—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4C7×D411yescharacteristic
312C2×C4⋊C4C7×D411yescharacteristic
313C2×C4⋊C4C7×D411yescharacteristic
314C2×C4⋊C4C7×D411yescharacteristic
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C42—21no—
321C2×C42C23×C711yescharacteristic
322#107—22no—
323#107—22no—
324#107—22no—
325#106—22no—
326#106—22no—
327#106—22no—
328C4⋊C8—21no—
329C4⋊C8—21no—
330#103—22no—
331#103—22no—
332#103—22no—
333#102—22no—
334C22⋊C8—21no—
335C22⋊C8—21no—
336#97C23×C712yes—
337#97C23×C712yes—
338#97C23×C712yes—
339C4×C8—21no—
340C22×C7—21no—
341C22×C7—21no—
342C22×C7—42no—
343C22×C7—42no—
344C22×C7—42no—
345C22×C7—84no—
346C22×C7#45612yes—
347C22×C7C2×#12011yescharacteristic
348C4×C7—21no—
349C4×C7—21no—
350C4×C7—21no—
351C4×C7—21no—
352C4×C7—21no—
353C4×C7—21no—
354C4×C7—42no—
355C4×C7—42no—
356C4×C7—42no—
357C4×C7—42no—
358C4×C7—42no—
359C4×C7—81no—
360C4×C7—81no—
361C24—21no—
362C2×Q8—24no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—81no—
367C2×Q8—82no—
368C2×Q8C2×C7×D412yes—
369C2×D4—21no—
370C2×D4—81no—
371C2×D4C2×C7×D412yes—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—24no—
378C22×C4—82no—
379C22×C4C2×C7×D411yescharacteristic
380C22×C4C2×C7×D411yescharacteristic
381C22×C4C2×C7×D411yescharacteristic
382Q16—82no—
383SD16—82no—
384M16—24no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4C2×C7×D412yes—
409C4⋊C4C2×C7×D412yes—
410C4⋊C4C2×C7×D412yes—
411C4⋊C4C2×C7×D412yes—
412C22⋊C4—24no—
413C22⋊C4—24no—
414C22⋊C4—24no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—41no—
420C42—24no—
421C42—41no—
422C42C2×C7×D411yescharacteristic
423C42C2×C7×D411yescharacteristic
424C42C2×C7×D411yescharacteristic
425C42C24×C711yescharacteristic
426C2×C7—22no—
427C2×C7—82no—
428C2×C7C2×#45611yescharacteristic
429C2×C7#301911yescharacteristic
430C2×C7#302011yescharacteristic
431C23—22no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23C7×#12011yescharacteristic
436Q8—42no—
437Q8—42no—
438Q8—42no—
439Q8—81no—
440Q8—84no—
441D4—42no—
442D4—81no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—84no—
459C2×C4C7×#12011yescharacteristic
460C2×C4C7×#12011yescharacteristic
461C2×C4C7×#12011yescharacteristic
462C2×C4C22×C7×D411yescharacteristic
463C2×C4C22×C7×D411yescharacteristic
464C2×C4C22×C7×D411yescharacteristic
465C8—24no—
466C8—42no—
467C8—42no—
468C7#1900711yescharacteristic, direct product factor, Sylow
469C22—21no—
470C22—21no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—84no—
475C22C7×#45612yes—
476C22C2×C7×#12011yescharacteristic
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—42no—
487C4—42no—
488C4—81no—
489C4—81no—
490C2—22no—
491C2—82no—
492C2C2×C7×#45611yescharacteristic
493C2C7×#301911yescharacteristic
494C2C7×#302011yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC22×C7×D4D4
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
2. C7×#3236C2semidirect
4. C7×#3166C2semidirect
5. C7×#1664C2semidirect
6. C7×#1660C2semidirect
7. C7×#1534C2semidirect
8. C7×#1528C2semidirect
24. C7×#501C22semidirect
25. C7×#501C22semidirect
26. C7×#501C22semidirect
29. C7×#491C22semidirect
42. C7×#330C22semidirect
43. C7×#330C22semidirect
45. #19007C7direct (N × H)
121. #3236C2×C7semidirect
123. #3166C2×C7semidirect
124. #1664C2×C7semidirect
125. #1660C2×C7semidirect
126. #1534C2×C7semidirect
127. #1528C2×C7semidirect
208. #501C22×C7semidirect
209. #501C22×C7semidirect
210. #501C22×C7semidirect
213. #491C22×C7semidirect
226. #330C22×C7semidirect
227. #330C22×C7semidirect
468. C7#19007direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5632
C7×#32362
C7×#32152
C7×#31662
C7×#16642
C7×#16602
C7×#15342
C7×#15282
C7×#15272
#190077