Finite Group Databasebeta

← C7×#18654 #13195870

Subgroups — C7×#18654 #13195870

Order 1792.

904 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3509C211yesmaximal, characteristic
2C7×#3229C211yesmaximal, characteristic, complemented
3C2×C7×#500C211yesmaximal, characteristic, complemented
4C7×#3039C211yesmaximal, characteristic
5C7×#1650C214yesmaximal, complemented
6C7×#1648C214yesmaximal, complemented
7C7×#1546C211yesmaximal, characteristic, complemented
8C7×#1493C211yesmaximal, characteristic, complemented
9C7×#1476C211yesmaximal, characteristic
10C2×C7×#137—21no—
11C7×#557—21no—
12C7×#557C2212yes—
13C7×#556—21no—
14C7×#556C2212yes—
15C7×#552—22no—
16C7×#552—22no—
17C7×#541—22no—
18C7×#536—21no—
19C7×#533—21no—
20C7×#533—21no—
21C7×#533C2211yescharacteristic
22C2×C7×C4⋊Q8C2211yescharacteristic
23C2×C7×C4⋊Q8C2211yescharacteristic
24C7×#519—21no—
25C4×C7×#41C2211yescharacteristic
26C7×#500—22no—
27C7×#500C2214yes—
28C7×#498—21no—
29C7×#487—21no—
30C7×#483—22no—
31C7×#475C2214yes—
32C7×#462—21no—
33C7×#462C2212yes—
34C7×#461—21no—
35C7×#461C2212yes—
36C7×#431C2211yescharacteristic
37C2×C7×#107—22no—
38C7×#423C2211yescharacteristic
39C7×#419—21no—
40C2×C4×C7×C8C2211yescharacteristic
41C7×#332C2214yes—
42C7×Q8⋊C8C2212yes—
43C7×Q8⋊C8C2214yes—
44C7×D4⋊C8C2212yes—
45#18654C711yesmaximal, characteristic, direct product factor, Sylow
46C2×C7×#41D411yescharacteristic
47C22×C7×Q8—21no—
48C22×C7×Q8—21no—
49C7×#137—44no—
50C2×C7×Q16—22no—
51C2×C7×SD16—22no—
52C2×C7×M16—21no—
53C22×C7×C8—21no—
54C7×C4⋊Q8—22no—
55C7×C4⋊Q8—21no—
56C7×C4⋊Q8—21no—
57C7×C4⋊Q8—21no—
58C7×C4⋊Q8—21no—
59C7×C4⋊Q8—21no—
60C7×C4⋊Q8D411yescharacteristic
61C7×C4⋊Q8D411yescharacteristic
62C7×C4⋊Q8D412yes—
63C7×C4⋊Q8C2312yes—
64C7×C4⋊Q8C2312yes—
65C7×#126—24no—
66C7×#126—24no—
67C7×#125—22no—
68C7×#125—24no—
69C7×#125—22no—
70C7×#125—24no—
71C7×#125—21no—
72C7×#124—22no—
73C7×#123—24no—
74C7×C22⋊Q8—22no—
75C7×C22⋊Q8—22no—
76C7×C22⋊Q8—22no—
77C7×C22⋊Q8—22no—
78C7×C22⋊Q8—21no—
79C7×C22⋊Q8D412yes—
80C4×C7×Q8—22no—
81C4×C7×Q8—22no—
82C4×C7×Q8—22no—
83C4×C7×Q8—22no—
84C4×C7×Q8—22no—
85C4×C7×Q8—22no—
86C4×C7×Q8—21no—
87C4×C7×Q8D411yescharacteristic
88C4×C7×Q8C2312yes—
89C4×C7×Q8C2311yescharacteristic
90C4×C7×D4—21no—
91C4×C7×D4—21no—
92C4×C7×D4D411yescharacteristic
93C4×C7×D4C2311yescharacteristic
94C7×#117—24no—
95C7×#117—22no—
96C7×#117—21no—
97C7×#117—22no—
98C7×#117—21no—
99C7×#117D411yescharacteristic
100C2×C7×C4⋊C4—21no—
101C2×C7×C4⋊C4—21no—
102C2×C7×C4⋊C4—21no—
103C2×C7×C4⋊C4D412yes—
104C2×C42×C7—21no—
105C2×C42×C7C2311yescharacteristic
106C7×#107—28no—
107C7×#106—24no—
108C7×C4⋊C8—21no—
109C7×C4⋊C8C2312yes—
110C7×C4⋊C8C2312yes—
111C7×#103—24no—
112C7×#103—22no—
113C7×#103—22no—
114C7×C22⋊C8—21no—
115C4×C7×C8—21no—
116C4×C7×C8—21no—
117C4×C7×C8C2312yes—
118#3509C2×C711yescharacteristic
119#3229C2×C711yescharacteristic, complemented
120C2×#500C2×C711yescharacteristic, complemented
121#3039C2×C711yescharacteristic
122#1650C2×C714yescomplemented
123#1648C2×C714yescomplemented
124#1546C2×C711yescharacteristic, complemented
125#1493C2×C711yescharacteristic, complemented
126#1476C2×C711yescharacteristic
127C7×#41—22no—
128C7×#41—22no—
129C2×C7×Q8—21no—
130C2×C7×Q8—22no—
131C2×C7×Q8—22no—
132C2×C7×Q8—22no—
133C2×C7×Q8—41no—
134C2×C7×Q8—42no—
135C2×C7×Q8—42no—
136C2×C7×Q8—42no—
137C2×C7×Q8—42no—
138C2×C7×Q8C2×D412yes—
139C2×C7×Q8C2×D411yescharacteristic
140C2×C7×D4—21no—
141C2×C7×D4C2×D411yescharacteristic
142C22×C4×C7—22no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—41no—
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C7×Q16—44no—
151C7×SD16—44no—
152C7×M16—42no—
153C2×C7×C8—24no—
154C2×C7×C8—22no—
155C2×C7×C8—22no—
156C2×C7×C8—22no—
157C7×C4⋊C4—24no—
158C7×C4⋊C4—24no—
159C7×C4⋊C4—24no—
160C7×C4⋊C4—24no—
161C7×C4⋊C4—22no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—41no—
174C7×C4⋊C4—41no—
175C7×C4⋊C4C2×D411yescharacteristic
176C7×C4⋊C4C2×D411yescharacteristic
177C7×C4⋊C4C2×D412yes—
178C7×C4⋊C4C2×D414yes—
179C7×C22⋊C4—24no—
180C7×C22⋊C4—24no—
181C7×C22⋊C4—21no—
182C7×C22⋊C4—22no—
183C7×C22⋊C4—21no—
184C7×C22⋊C4—21no—
185C42×C7—24no—
186C42×C7—24no—
187C42×C7—21no—
188C42×C7—21no—
189C42×C7—21no—
190C42×C7—22no—
191C42×C7C2×D411yescharacteristic
192C42×C7C2×D411yescharacteristic
193C42×C7C2×D411yescharacteristic
194C42×C7C2411yescharacteristic
195C2×#137—21no—
196#557—21no—
197#557C22×C712yes—
198#556—21no—
199#556C22×C712yes—
200#552—22no—
201#552—22no—
202#541—22no—
203#536—21no—
204#533—21no—
205#533—21no—
206#533C22×C711yescharacteristic
207C2×C4⋊Q8C22×C711yescharacteristic
208C2×C4⋊Q8C22×C711yescharacteristic
209#519—21no—
210C4×#41C22×C711yescharacteristic
211#500—22no—
212#500C22×C714yes—
213#498—21no—
214#487—21no—
215#483—22no—
216#475C22×C714yes—
217#462—21no—
218#462C22×C712yes—
219#461—21no—
220#461C22×C712yes—
221#431C22×C711yescharacteristic
222C2×#107—22no—
223#423C22×C711yescharacteristic
224#419—21no—
225C2×C4×C8C22×C711yescharacteristic
226#332C22×C714yes—
227Q8⋊C8C22×C712yes—
228Q8⋊C8C22×C714yes—
229D4⋊C8C22×C712yes—
230C23×C7—21no—
231C23×C7#12011yescharacteristic
232C7×Q8—21no—
233C7×Q8—21no—
234C7×Q8—24no—
235C7×Q8—42no—
236C7×Q8—42no—
237C7×Q8—44no—
238C7×Q8—42no—
239C7×Q8—42no—
240C7×Q8—42no—
241C7×Q8—82no—
242C7×D4—21no—
243C7×D4—21no—
244C7×D4—42no—
245C2×C4×C7—22no—
246C2×C4×C7—22no—
247C2×C4×C7—21no—
248C2×C4×C7—21no—
249C2×C4×C7—22no—
250C2×C4×C7—22no—
251C2×C4×C7—24no—
252C2×C4×C7—21no—
253C2×C4×C7—44no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—82no—
259C2×C4×C7#12011yescharacteristic
260C2×C4×C7#12011yescharacteristic
261C2×C4×C7#12011yescharacteristic
262C2×C4×C7#13512yes—
263C2×C4×C7AGL(1,ℤ/8)12yes—
264C2×C4×C7C22×D411yescharacteristic
265C2×C4×C7C22×D411yescharacteristic
266C2×C4×C7C22×D411yescharacteristic
267C7×C8—24no—
268C7×C8—42no—
269C7×C8—42no—
270C2×#41C7×D411yescharacteristic
271C22×Q8—21no—
272C22×Q8—21no—
273#137—44no—
274C2×Q16—22no—
275C2×SD16—22no—
276C2×M16—21no—
277C22×C8—21no—
278C4⋊Q8—22no—
279C4⋊Q8—21no—
280C4⋊Q8—21no—
281C4⋊Q8—21no—
282C4⋊Q8—21no—
283C4⋊Q8—21no—
284C4⋊Q8C7×D411yescharacteristic
285C4⋊Q8C7×D412yes—
286C4⋊Q8C7×D411yescharacteristic
287C4⋊Q8C23×C712yes—
288C4⋊Q8C23×C712yes—
289#126—24no—
290#126—24no—
291#125—22no—
292#125—24no—
293#125—22no—
294#125—24no—
295#125—21no—
296#124—22no—
297#123—24no—
298C22⋊Q8—22no—
299C22⋊Q8—22no—
300C22⋊Q8—22no—
301C22⋊Q8—22no—
302C22⋊Q8—21no—
303C22⋊Q8C7×D412yes—
304C4×Q8—22no—
305C4×Q8—22no—
306C4×Q8—22no—
307C4×Q8—22no—
308C4×Q8—22no—
309C4×Q8—22no—
310C4×Q8—21no—
311C4×Q8C7×D411yescharacteristic
312C4×Q8C23×C711yescharacteristic
313C4×Q8C23×C712yes—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4C7×D411yescharacteristic
317C4×D4C23×C711yescharacteristic
318#117—24no—
319#117—22no—
320#117—21no—
321#117—22no—
322#117—21no—
323#117C7×D411yescharacteristic
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4C7×D412yes—
328C2×C42—21no—
329C2×C42C23×C711yescharacteristic
330#107—28no—
331#106—24no—
332C4⋊C8—21no—
333C4⋊C8C23×C712yes—
334C4⋊C8C23×C712yes—
335#103—24no—
336#103—22no—
337#103—22no—
338C22⋊C8—21no—
339C4×C8—21no—
340C4×C8—21no—
341C4×C8C23×C712yes—
342C22×C7—21no—
343C22×C7—21no—
344C22×C7—41no—
345C22×C7—41no—
346C22×C7—41no—
347C22×C7#44912yes—
348C22×C7C2×#12011yescharacteristic
349C4×C7—21no—
350C4×C7—21no—
351C4×C7—41no—
352C4×C7—44no—
353C4×C7—41no—
354C4×C7—42no—
355C4×C7—41no—
356C4×C7—42no—
357C4×C7—82no—
358C4×C7#44912yes—
359C4×C7#45612yes—
360C4×C7C2×#13511yescharacteristic
361C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
362C4×C7#57812yes—
363#41—22no—
364#41—22no—
365C2×Q8—21no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—41no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8C2×C7×D412yes—
375C2×Q8C2×C7×D411yescharacteristic
376C2×D4—21no—
377C2×D4C2×C7×D411yescharacteristic
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—41no—
383C22×C4C2×C7×D411yescharacteristic
384C22×C4C2×C7×D411yescharacteristic
385C22×C4C2×C7×D411yescharacteristic
386Q16—44no—
387SD16—44no—
388M16—42no—
389C2×C8—24no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—21no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4C2×C7×D411yescharacteristic
412C4⋊C4C2×C7×D411yescharacteristic
413C4⋊C4C2×C7×D412yes—
414C4⋊C4C2×C7×D414yes—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—21no—
418C22⋊C4—22no—
419C22⋊C4—21no—
420C22⋊C4—21no—
421C42—24no—
422C42—24no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—22no—
427C42C2×C7×D411yescharacteristic
428C42C2×C7×D411yescharacteristic
429C42C2×C7×D411yescharacteristic
430C42C24×C711yescharacteristic
431C2×C7—22no—
432C2×C7—81no—
433C2×C7C2×#44911yescharacteristic
434C2×C7#301111yescharacteristic
435C2×C7C2×#45611yescharacteristic
436C23—21no—
437C23C7×#12011yescharacteristic
438Q8—21no—
439Q8—21no—
440Q8—24no—
441Q8—42no—
442Q8—42no—
443Q8—44no—
444Q8—42no—
445Q8—42no—
446Q8—42no—
447Q8—82no—
448D4—21no—
449D4—21no—
450D4—42no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—21no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—82no—
465C2×C4C7×#12011yescharacteristic
466C2×C4C7×#12011yescharacteristic
467C2×C4C7×#12011yescharacteristic
468C2×C4C7×#13512yes—
469C2×C4C7×AGL(1,ℤ/8)12yes—
470C2×C4C22×C7×D411yescharacteristic
471C2×C4C22×C7×D411yescharacteristic
472C2×C4C22×C7×D411yescharacteristic
473C8—24no—
474C8—42no—
475C8—42no—
476C7#1865411yescharacteristic, direct product factor, Sylow
477C22—21no—
478C22—21no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22C7×#44912yes—
483C22C2×C7×#12011yescharacteristic
484C4—21no—
485C4—21no—
486C4—41no—
487C4—44no—
488C4—41no—
489C4—42no—
490C4—41no—
491C4—42no—
492C4—82no—
493C4C7×#44912yes—
494C4C7×#45612yes—
495C4C2×C7×#13511yescharacteristic
496C4C2×C7×AGL(1,ℤ/8)11yescharacteristic
497C4C7×#57812yes—
498C2—22no—
499C2—81no—
500C2C2×C7×#44911yescharacteristic
501C2C7×#301111yescharacteristic
502C2C2×C7×#45611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC2×C7×#41D4
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×#3229C2semidirect
3. C2×C7×#500C2semidirect
5. C7×#1650C2semidirect
6. C7×#1648C2semidirect
7. C7×#1546C2semidirect
8. C7×#1493C2semidirect
45. #18654C7direct (N × H)
119. #3229C2×C7semidirect
120. C2×#500C2×C7semidirect
122. #1650C2×C7semidirect
123. #1648C2×C7semidirect
124. #1546C2×C7semidirect
125. #1493C2×C7semidirect
476. C7#18654direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35092
C7×#32292
C2×C7×#5002
C7×#30392
C7×#16502
C7×#16482
C7×#15462
C7×#14932
C7×#14762
#186547