Finite Group Databasebeta

← C7×#18339 #13195555

Subgroups — C7×#18339 #13195555

Order 1792.

1156 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3449C211yesmaximal, characteristic
2C7×#3234C212yesmaximal, complemented
3C7×#2611C211yesmaximal, characteristic, complemented
4C7×#1616C212yesmaximal, complemented
5C7×#1613C212yesmaximal, complemented
6C7×#1610C212yesmaximal, complemented
7C7×#1607C212yesmaximal, complemented
8C7×#1505C212yesmaximal, complemented
9C7×#1464C211yesmaximal, characteristic, complemented
10C22×C7×#41—21no—
11C7×#537—21no—
12C7×#537C2212yes—
13C7×#536—22no—
14C7×#535—22no—
15C7×#534—21no—
16C7×#534C2212yes—
17C7×#532—21no—
18C7×#532C2212yes—
19C2×C7×#125C2211yescharacteristic, complemented
20C7×#525—24no—
21C2×C7×#117C2211yescharacteristic
22C7×#501—22no—
23C7×#490—22no—
24C7×#481C2214yescomplemented
25C7×#480C2214yescomplemented
26C7×#478—24no—
27C7×#432C2212yescomplemented
28C7×#400C2211yescharacteristic, complemented
29C7×#399C2211yescharacteristic
30C7×#398C2211yescharacteristic, complemented
31C7×#397C2211yescharacteristic
32C7×#389—21no—
33C7×#388—21no—
34C7×#387—21no—
35C7×#385—21no—
36C7×#376C2211yescharacteristic, complemented
37C7×#327C2214yescomplemented
38C7×#327C2214yescomplemented
39C7×#324C2212yescomplemented
40C7×#323C2212yes—
41#18339C711yesmaximal, characteristic, direct product factor, Sylow
42C2×C7×#41—22no—
43C2×C7×#41—22no—
44C2×C7×#41—22no—
45C2×C7×#41—22no—
46C2×C7×#41—44no—
47C2×C7×#41—44no—
48C22×C7×Q8—21no—
49C22×C7×D4—21no—
50C22×C7×D4—22no—
51C23×C4×C7—22no—
52C23×C4×C7D411yescharacteristic
53C7×C4⋊Q8—21no—
54C7×C4⋊Q8D412yes—
55C7×#127—21no—
56C7×#127D412yes—
57C7×#125—22no—
58C7×#125—22no—
59C7×#124—22no—
60C7×#124—22no—
61C7×#123—24no—
62C7×#123—24no—
63C7×C22⋊Q8—24no—
64C7×C22⋊Q8—24no—
65C7×#121—24no—
66C7×#121—24no—
67C7×#121—22no—
68C7×#121—22no—
69C7×#121—22no—
70C7×#121C2314yes—
71C7×#120—24no—
72C7×#120—24no—
73C4×C7×D4—22no—
74C4×C7×D4—22no—
75C4×C7×D4—22no—
76C4×C7×D4D414yes—
77C7×#117—22no—
78C7×#117—22no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—21no—
82C2×C7×C4⋊C4—21no—
83C2×C7×C4⋊C4D411yescharacteristic
84C2×C7×C4⋊C4C2311yescharacteristic
85C2×C7×C4⋊C4C2311yescharacteristic
86C2×C7×C4⋊C4C2311yescharacteristic
87C2×C7×C22⋊C4—21no—
88C2×C7×C22⋊C4—21no—
89C2×C7×C22⋊C4D411yescharacteristic
90C2×C7×C22⋊C4C2311yescharacteristic
91C2×C42×C7—21no—
92C2×C42×C7D411yescharacteristic
93C2×C42×C7C2311yescharacteristic
94C7×#107—24no—
95C7×#106—24no—
96C7×C4⋊C8—22no—
97C7×#102—24no—
98C7×#102—24no—
99C7×C22⋊C8C2314yes—
100C7×#97—22no—
101C7×#95—21no—
102C7×#95—21no—
103C7×#95—21no—
104C7×#95—21no—
105C7×#95—21no—
106C7×#95C2311yescharacteristic
107C7×#95C2311yescharacteristic
108#3449C2×C711yescharacteristic
109#3234C2×C712yescomplemented
110#2611C2×C711yescharacteristic, complemented
111#1616C2×C712yescomplemented
112#1613C2×C712yescomplemented
113#1610C2×C712yescomplemented
114#1607C2×C712yescomplemented
115#1505C2×C712yescomplemented
116#1464C2×C711yescharacteristic, complemented
117C24×C7—22no—
118C24×C7C2×D411yescharacteristic
119C7×#41—48no—
120C7×#41—48no—
121C7×#41—44no—
122C7×#41—44no—
123C7×#41—44no—
124C7×#41—44no—
125C2×C7×Q8—22no—
126C2×C7×Q8—22no—
127C2×C7×Q8—44no—
128C2×C7×D4—24no—
129C2×C7×D4—22no—
130C2×C7×D4—24no—
131C2×C7×D4—22no—
132C2×C7×D4—22no—
133C2×C7×D4—22no—
134C2×C7×D4—22no—
135C2×C7×D4—44no—
136C2×C7×D4—44no—
137C2×C7×D4—44no—
138C2×C7×D4C2×D414yes—
139C22×C4×C7—22no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—22no—
143C22×C4×C7—24no—
144C22×C4×C7—22no—
145C22×C4×C7—21no—
146C22×C4×C7—24no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—48no—
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2411yescharacteristic
157C2×C7×C8—24no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4—22no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4—42no—
166C7×C4⋊C4—41no—
167C7×C4⋊C4—41no—
168C7×C4⋊C4—42no—
169C7×C4⋊C4—42no—
170C7×C4⋊C4C2×D414yes—
171C7×C22⋊C4—24no—
172C7×C22⋊C4—24no—
173C7×C22⋊C4—22no—
174C7×C22⋊C4—24no—
175C7×C22⋊C4—22no—
176C7×C22⋊C4—24no—
177C7×C22⋊C4—42no—
178C7×C22⋊C4—42no—
179C42×C7—21no—
180C42×C7—21no—
181C42×C7—42no—
182C42×C7C2×D412yes—
183C42×C7C2×D412yes—
184C22×#41—21no—
185#537—21no—
186#537C22×C712yes—
187#536—22no—
188#535—22no—
189#534—21no—
190#534C22×C712yes—
191#532—21no—
192#532C22×C712yes—
193C2×#125C22×C711yescharacteristic, complemented
194#525—24no—
195C2×#117C22×C711yescharacteristic
196#501—22no—
197#490—22no—
198#481C22×C714yescomplemented
199#480C22×C714yescomplemented
200#478—24no—
201#432C22×C712yescomplemented
202#400C22×C711yescharacteristic, complemented
203#399C22×C711yescharacteristic
204#398C22×C711yescharacteristic, complemented
205#397C22×C711yescharacteristic
206#389—21no—
207#388—21no—
208#387—21no—
209#385—21no—
210#376C22×C711yescharacteristic, complemented
211#327C22×C714yescomplemented
212#327C22×C714yescomplemented
213#324C22×C712yescomplemented
214#323C22×C712yes—
215C23×C7—21no—
216C23×C7—21no—
217C23×C7—24no—
218C23×C7—41no—
219C23×C7—41no—
220C23×C7—44no—
221C23×C7—82no—
222C23×C7—82no—
223C23×C7AGL(1,ℤ/8)12yes—
224C23×C7C22×D411yescharacteristic
225C7×Q8—44no—
226C7×Q8—44no—
227C7×D4—44no—
228C7×D4—44no—
229C7×D4—44no—
230C7×D4—48no—
231C7×D4—44no—
232C7×D4—44no—
233C7×D4—82no—
234C7×D4—82no—
235C7×D4—82no—
236C2×C4×C7—24no—
237C2×C4×C7—24no—
238C2×C4×C7—21no—
239C2×C4×C7—21no—
240C2×C4×C7—21no—
241C2×C4×C7—21no—
242C2×C4×C7—22no—
243C2×C4×C7—21no—
244C2×C4×C7—21no—
245C2×C4×C7—24no—
246C2×C4×C7—41no—
247C2×C4×C7—41no—
248C2×C4×C7—48no—
249C2×C4×C7—48no—
250C2×C4×C7—41no—
251C2×C4×C7—44no—
252C2×C4×C7—41no—
253C2×C4×C7—41no—
254C2×C4×C7—41no—
255C2×C4×C7—41no—
256C2×C4×C7—41no—
257C2×C4×C7—41no—
258C2×C4×C7—41no—
259C2×C4×C7—41no—
260C2×C4×C7—41no—
261C2×C4×C7—41no—
262C2×C4×C7—41no—
263C2×C4×C7—41no—
264C2×C4×C7#12011yescharacteristic
265C2×C4×C7#12011yescharacteristic
266C2×C4×C7#12012yes—
267C2×C4×C7AGL(1,ℤ/8)12yes—
268C2×C4×C7C22×D412yes—
269C7×C8—44no—
270C2×#41—22no—
271C2×#41—22no—
272C2×#41—22no—
273C2×#41—22no—
274C2×#41—44no—
275C2×#41—44no—
276C22×Q8—21no—
277C22×D4—21no—
278C22×D4—22no—
279C23×C4—22no—
280C23×C4C7×D411yescharacteristic
281C4⋊Q8—21no—
282C4⋊Q8C7×D412yes—
283#127—21no—
284#127C7×D412yes—
285#125—22no—
286#125—22no—
287#124—22no—
288#124—22no—
289#123—24no—
290#123—24no—
291C22⋊Q8—24no—
292C22⋊Q8—24no—
293#121—24no—
294#121—24no—
295#121—22no—
296#121—22no—
297#121—22no—
298#121C23×C714yes—
299#120—24no—
300#120—24no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—22no—
304C4×D4C7×D414yes—
305#117—22no—
306#117—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4C7×D411yescharacteristic
312C2×C4⋊C4C23×C711yescharacteristic
313C2×C4⋊C4C23×C711yescharacteristic
314C2×C4⋊C4C23×C711yescharacteristic
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4C7×D411yescharacteristic
318C2×C22⋊C4C23×C711yescharacteristic
319C2×C42—21no—
320C2×C42C7×D411yescharacteristic
321C2×C42C23×C711yescharacteristic
322#107—24no—
323#106—24no—
324C4⋊C8—22no—
325#102—24no—
326#102—24no—
327C22⋊C8C23×C714yes—
328#97—22no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95C23×C711yescharacteristic
335#95C23×C711yescharacteristic
336C22×C7—21no—
337C22×C7—21no—
338C22×C7—22no—
339C22×C7—42no—
340C22×C7—41no—
341C22×C7—41no—
342C22×C7—44no—
343C22×C7—44no—
344C22×C7—44no—
345C22×C7—81no—
346C22×C7—88no—
347C22×C7C2×#12011yescharacteristic
348C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
349C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
350C4×C7—21no—
351C4×C7—22no—
352C4×C7—21no—
353C4×C7—22no—
354C4×C7—22no—
355C4×C7—42no—
356C4×C7—44no—
357C4×C7—41no—
358C4×C7—41no—
359C4×C7—81no—
360C4×C7—81no—
361C4×C7—81no—
362C24—22no—
363C24C2×C7×D411yescharacteristic
364#41—48no—
365#41—48no—
366#41—44no—
367#41—44no—
368#41—44no—
369#41—44no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—44no—
373C2×D4—24no—
374C2×D4—22no—
375C2×D4—24no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4C2×C7×D414yes—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—48no—
396C22×C4C2×C7×D411yescharacteristic
397C22×C4C2×C7×D411yescharacteristic
398C22×C4C2×C7×D411yescharacteristic
399C22×C4C2×C7×D411yescharacteristic
400C22×C4C2×C7×D411yescharacteristic
401C22×C4C24×C711yescharacteristic
402C2×C8—24no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—42no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4C2×C7×D414yes—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—22no—
419C22⋊C4—24no—
420C22⋊C4—22no—
421C22⋊C4—24no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C42—21no—
425C42—21no—
426C42—42no—
427C42C2×C7×D412yes—
428C42C2×C7×D412yes—
429C2×C7—21no—
430C2×C7—21no—
431C2×C7—42no—
432C2×C7—84no—
433C2×C7#300311yescharacteristic
434C2×C7#301811yescharacteristic
435C2×C7#302811yescharacteristic
436C23—21no—
437C23—21no—
438C23—24no—
439C23—41no—
440C23—41no—
441C23—44no—
442C23—82no—
443C23—82no—
444C23C7×AGL(1,ℤ/8)12yes—
445C23C22×C7×D411yescharacteristic
446Q8—44no—
447Q8—44no—
448D4—44no—
449D4—44no—
450D4—44no—
451D4—48no—
452D4—44no—
453D4—44no—
454D4—82no—
455D4—82no—
456D4—82no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—24no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—48no—
470C2×C4—48no—
471C2×C4—41no—
472C2×C4—44no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4C7×#12011yescharacteristic
486C2×C4C7×#12011yescharacteristic
487C2×C4C7×#12012yes—
488C2×C4C7×AGL(1,ℤ/8)12yes—
489C2×C4C22×C7×D412yes—
490C8—44no—
491C7#1833911yescharacteristic, direct product factor, Sylow
492C22—21no—
493C22—21no—
494C22—22no—
495C22—42no—
496C22—41no—
497C22—41no—
498C22—44no—
499C22—44no—
500C22—44no—
501C22—81no—
502C22—88no—
503C22C2×C7×#12011yescharacteristic
504C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
505C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
506C4—21no—
507C4—22no—
508C4—21no—
509C4—22no—
510C4—22no—
511C4—42no—
512C4—44no—
513C4—41no—
514C4—41no—
515C4—81no—
516C4—81no—
517C4—81no—
518C2—21no—
519C2—21no—
520C2—42no—
521C2—84no—
522C2C7×#300311yescharacteristic
523C2C7×#301811yescharacteristic
524C2C7×#302811yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3449C2
LCSG ▷ C22×C4 ▷ 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×#3234C2semidirect
3. C7×#2611C2semidirect
4. C7×#1616C2semidirect
5. C7×#1613C2semidirect
6. C7×#1610C2semidirect
7. C7×#1607C2semidirect
8. C7×#1505C2semidirect
9. C7×#1464C2semidirect
19. C2×C7×#125C22semidirect
24. C7×#481C22semidirect
25. C7×#480C22semidirect
27. C7×#432C22semidirect
28. C7×#400C22semidirect
30. C7×#398C22semidirect
36. C7×#376C22semidirect
37. C7×#327C22semidirect
38. C7×#327C22semidirect
39. C7×#324C22semidirect
41. #18339C7direct (N × H)
109. #3234C2×C7semidirect
110. #2611C2×C7semidirect
111. #1616C2×C7semidirect
112. #1613C2×C7semidirect
113. #1610C2×C7semidirect
114. #1607C2×C7semidirect
115. #1505C2×C7semidirect
116. #1464C2×C7semidirect
193. C2×#125C22×C7semidirect
198. #481C22×C7semidirect
199. #480C22×C7semidirect
201. #432C22×C7semidirect
202. #400C22×C7semidirect
204. #398C22×C7semidirect
210. #376C22×C7semidirect
211. #327C22×C7semidirect
212. #327C22×C7semidirect
213. #324C22×C7semidirect
491. C7#18339direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34492
C7×#32342
C7×#26112
C7×#16162
C7×#16132
C7×#16102
C7×#16072
C7×#15052
C7×#14642
#183397