Finite Group Databasebeta

← C7×#18137 #13195353

Subgroups — C7×#18137 #13195353

Order 1792.

1398 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#534C211yesmaximal, characteristic
2C7×#3094C212yesmaximal, complemented
3C7×#2482C211yesmaximal, characteristic, complemented
4C7×#1613C214yesmaximal, complemented
5C7×#1607C214yesmaximal, complemented
6C7×#1510C212yesmaximal, complemented
7C7×#1462C211yesmaximal, characteristic, complemented
8C23×C7×D4—21no—
9C7×#534—24no—
10C7×#534—22no—
11C7×#534C2214yes—
12C2×C7×#123—22no—
13C2×C7×#121—21no—
14C2×C7×#121C2212yes—
15C2×C7×#120—22no—
16C2×C7×#117C2211yescharacteristic, complemented
17C22×C7×C22⋊C4C2211yescharacteristic
18C7×#482—24no—
19C7×#481C2214yescomplemented
20C7×#480C2214yescomplemented
21C7×#428—22no—
22C7×#418—22no—
23C7×#407C2212yescomplemented
24C7×#399C2212yescomplemented
25C7×#397C2212yescomplemented
26C7×#388—22no—
27C7×#385—21no—
28C7×#381C2211yescharacteristic, complemented
29C7×#379—21no—
30C7×#327C2218yescomplemented
31C7×#323C2214yescomplemented
32#18137C711yesmaximal, characteristic, direct product factor, Sylow
33C25×C7—21no—
34C25×C7D411yescharacteristic
35C22×C7×D4—22no—
36C22×C7×D4—24no—
37C22×C7×D4—22no—
38C22×C7×D4—24no—
39C22×C7×D4—21no—
40C22×C7×D4—48no—
41C22×C7×D4D412yes—
42C23×C4×C7—21no—
43C23×C4×C7D411yescharacteristic
44C23×C4×C7C2311yescharacteristic
45C2×C7×M16—22no—
46C7×#123—28no—
47C7×#123—28no—
48C7×#121—24no—
49C7×#121—24no—
50C7×#121—22no—
51C7×#121—22no—
52C7×#121D414yes—
53C7×#121C2314yes—
54C7×#120—28no—
55C7×#120—28no—
56C7×#117—22no—
57C7×#117—22no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4—22no—
60C2×C7×C4⋊C4D412yes—
61C2×C7×C4⋊C4C2311yescharacteristic, complemented
62C2×C7×C4⋊C4C2311yescharacteristic, complemented
63C2×C7×C22⋊C4—22no—
64C2×C7×C22⋊C4—22no—
65C2×C7×C22⋊C4—22no—
66C2×C7×C22⋊C4—22no—
67C2×C7×C22⋊C4—21no—
68C2×C7×C22⋊C4—21no—
69C2×C7×C22⋊C4—22no—
70C2×C7×C22⋊C4—22no—
71C2×C7×C22⋊C4D412yes—
72C2×C7×C22⋊C4C2312yes—
73C2×C42×C7—21no—
74C7×#107—24no—
75C7×#106—24no—
76C7×#102—24no—
77C7×#102—24no—
78C7×C22⋊C8—22no—
79C7×C22⋊C8C2314yescomplemented
80C7×#95—21no—
81C7×#95—22no—
82C7×#95C2312yescomplemented
83C2×#534C2×C711yescharacteristic
84#3094C2×C712yescomplemented
85#2482C2×C711yescharacteristic, complemented
86#1613C2×C714yescomplemented
87#1607C2×C714yescomplemented
88#1510C2×C712yescomplemented
89#1462C2×C711yescharacteristic, complemented
90C24×C7—21no—
91C24×C7—21no—
92C24×C7—24no—
93C24×C7—24no—
94C24×C7—22no—
95C24×C7—44no—
96C24×C7—42no—
97C24×C7—81no—
98C24×C7—82no—
99C24×C7—82no—
100C24×C7C2×D411yescharacteristic
101C24×C7C2×D412yes—
102C2×C7×D4—28no—
103C2×C7×D4—24no—
104C2×C7×D4—24no—
105C2×C7×D4—24no—
106C2×C7×D4—22no—
107C2×C7×D4—44no—
108C2×C7×D4—48no—
109C2×C7×D4—416no—
110C2×C7×D4—48no—
111C2×C7×D4—48no—
112C2×C7×D4—48no—
113C2×C7×D4—82no—
114C2×C7×D4C2×D414yes—
115C22×C4×C7—24no—
116C22×C4×C7—21no—
117C22×C4×C7—22no—
118C22×C4×C7—21no—
119C22×C4×C7—22no—
120C22×C4×C7—21no—
121C22×C4×C7—21no—
122C22×C4×C7—42no—
123C22×C4×C7—44no—
124C22×C4×C7—81no—
125C22×C4×C7C2×D412yes—
126C22×C4×C7C2×D412yes—
127C22×C4×C7C2×D411yescharacteristic
128C22×C4×C7C2×D412yes—
129C22×C4×C7C2411yescharacteristic
130C7×M16—44no—
131C2×C7×C8—24no—
132C7×C4⋊C4—24no—
133C7×C4⋊C4—22no—
134C7×C4⋊C4—22no—
135C7×C4⋊C4—22no—
136C7×C4⋊C4—82no—
137C7×C4⋊C4C2×D414yes—
138C7×C22⋊C4—28no—
139C7×C22⋊C4—24no—
140C7×C22⋊C4—24no—
141C7×C22⋊C4—24no—
142C7×C22⋊C4—24no—
143C7×C22⋊C4—42no—
144C7×C22⋊C4—41no—
145C7×C22⋊C4—41no—
146C7×C22⋊C4—44no—
147C7×C22⋊C4—82no—
148C42×C7—42no—
149C23×D4—21no—
150#534—24no—
151#534—22no—
152#534C22×C714yes—
153C2×#123—22no—
154C2×#121—21no—
155C2×#121C22×C712yes—
156C2×#120—22no—
157C2×#117C22×C711yescharacteristic, complemented
158C22×C22⋊C4C22×C711yescharacteristic
159#482—24no—
160#481C22×C714yescomplemented
161#480C22×C714yescomplemented
162#428—22no—
163#418—22no—
164#407C22×C712yescomplemented
165#399C22×C712yescomplemented
166#397C22×C712yescomplemented
167#388—22no—
168#385—21no—
169#381C22×C711yescharacteristic, complemented
170#379—21no—
171#327C22×C718yescomplemented
172#323C22×C714yescomplemented
173C23×C7—24no—
174C23×C7—24no—
175C23×C7—22no—
176C23×C7—22no—
177C23×C7—22no—
178C23×C7—22no—
179C23×C7—24no—
180C23×C7—41no—
181C23×C7—42no—
182C23×C7—42no—
183C23×C7—42no—
184C23×C7—44no—
185C23×C7—48no—
186C23×C7—44no—
187C23×C7—42no—
188C23×C7—44no—
189C23×C7—81no—
190C23×C7—81no—
191C23×C7—88no—
192C23×C7—84no—
193C23×C7—84no—
194C23×C7—816no—
195C23×C7#12011yescharacteristic
196C23×C7#12011yescharacteristic
197C23×C7AGL(1,ℤ/8)14yes—
198C23×C7C22×D411yescharacteristic
199C7×D4—44no—
200C7×D4—416no—
201C7×D4—48no—
202C7×D4—48no—
203C7×D4—82no—
204C7×D4—84no—
205C2×C4×C7—24no—
206C2×C4×C7—22no—
207C2×C4×C7—24no—
208C2×C4×C7—42no—
209C2×C4×C7—41no—
210C2×C4×C7—41no—
211C2×C4×C7—44no—
212C2×C4×C7—48no—
213C2×C4×C7—42no—
214C2×C4×C7—42no—
215C2×C4×C7—42no—
216C2×C4×C7—42no—
217C2×C4×C7—42no—
218C2×C4×C7—41no—
219C2×C4×C7—41no—
220C2×C4×C7—42no—
221C2×C4×C7—42no—
222C2×C4×C7—81no—
223C2×C4×C7#12012yes—
224C2×C4×C7C22×D412yes—
225C7×C8—44no—
226C25—21no—
227C25C7×D411yescharacteristic
228C22×D4—22no—
229C22×D4—24no—
230C22×D4—22no—
231C22×D4—24no—
232C22×D4—21no—
233C22×D4—48no—
234C22×D4C7×D412yes—
235C23×C4—21no—
236C23×C4C7×D411yescharacteristic
237C23×C4C23×C711yescharacteristic
238C2×M16—22no—
239#123—28no—
240#123—28no—
241#121—24no—
242#121—24no—
243#121—22no—
244#121—22no—
245#121C7×D414yes—
246#121C23×C714yes—
247#120—28no—
248#120—28no—
249#117—22no—
250#117—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4C7×D412yes—
254C2×C4⋊C4C23×C711yescharacteristic, complemented
255C2×C4⋊C4C23×C711yescharacteristic, complemented
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4C7×D412yes—
265C2×C22⋊C4C23×C712yes—
266C2×C42—21no—
267#107—24no—
268#106—24no—
269#102—24no—
270#102—24no—
271C22⋊C8—22no—
272C22⋊C8C23×C714yescomplemented
273#95—21no—
274#95—22no—
275#95C23×C712yescomplemented
276C22×C7—21no—
277C22×C7—21no—
278C22×C7—21no—
279C22×C7—21no—
280C22×C7—24no—
281C22×C7—21no—
282C22×C7—21no—
283C22×C7—21no—
284C22×C7—44no—
285C22×C7—44no—
286C22×C7—48no—
287C22×C7—42no—
288C22×C7—42no—
289C22×C7—44no—
290C22×C7—44no—
291C22×C7—44no—
292C22×C7—82no—
293C22×C7—88no—
294C22×C7—84no—
295C22×C7—816no—
296C22×C7#45812yes—
297C22×C7C2×#12011yescharacteristic
298C22×C7C2×AGL(1,ℤ/8)12yes—
299C4×C7—22no—
300C4×C7—22no—
301C4×C7—44no—
302C4×C7—42no—
303C4×C7—82no—
304C4×C7—81no—
305C4×C7—81no—
306C24—21no—
307C24—21no—
308C24—24no—
309C24—24no—
310C24—22no—
311C24—44no—
312C24—42no—
313C24—81no—
314C24—82no—
315C24—82no—
316C24C2×C7×D411yescharacteristic
317C24C2×C7×D412yes—
318C2×D4—28no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—22no—
323C2×D4—44no—
324C2×D4—48no—
325C2×D4—416no—
326C2×D4—48no—
327C2×D4—48no—
328C2×D4—48no—
329C2×D4—82no—
330C2×D4C2×C7×D414yes—
331C22×C4—24no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—81no—
341C22×C4C2×C7×D412yes—
342C22×C4C2×C7×D412yes—
343C22×C4C2×C7×D411yescharacteristic
344C22×C4C2×C7×D412yes—
345C22×C4C24×C711yescharacteristic
346M16—44no—
347C2×C8—24no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—82no—
353C4⋊C4C2×C7×D414yes—
354C22⋊C4—28no—
355C22⋊C4—24no—
356C22⋊C4—24no—
357C22⋊C4—24no—
358C22⋊C4—24no—
359C22⋊C4—42no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—44no—
363C22⋊C4—82no—
364C42—42no—
365C2×C7—22no—
366C2×C7—22no—
367C2×C7—41no—
368C2×C7—44no—
369C2×C7—84no—
370C2×C7#300311yescharacteristic
371C2×C7#301911yescharacteristic
372C2×C7C2×#45811yescharacteristic
373C23—24no—
374C23—24no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—24no—
380C23—41no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—44no—
385C23—48no—
386C23—44no—
387C23—42no—
388C23—44no—
389C23—81no—
390C23—81no—
391C23—88no—
392C23—84no—
393C23—84no—
394C23—816no—
395C23C7×#12011yescharacteristic
396C23C7×#12011yescharacteristic
397C23C7×AGL(1,ℤ/8)14yes—
398C23C22×C7×D411yescharacteristic
399D4—44no—
400D4—416no—
401D4—48no—
402D4—48no—
403D4—82no—
404D4—84no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—44no—
412C2×C4—48no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—81no—
423C2×C4C7×#12012yes—
424C2×C4C22×C7×D412yes—
425C8—44no—
426C7#1813711yescharacteristic, direct product factor, Sylow
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—24no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—44no—
436C22—44no—
437C22—48no—
438C22—42no—
439C22—42no—
440C22—44no—
441C22—44no—
442C22—44no—
443C22—82no—
444C22—88no—
445C22—84no—
446C22—816no—
447C22C7×#45812yes—
448C22C2×C7×#12011yescharacteristic
449C22C2×C7×AGL(1,ℤ/8)12yes—
450C4—22no—
451C4—22no—
452C4—44no—
453C4—42no—
454C4—82no—
455C4—81no—
456C4—81no—
457C2—22no—
458C2—22no—
459C2—41no—
460C2—44no—
461C2—84no—
462C2C7×#300311yescharacteristic
463C2C7×#301911yescharacteristic
464C2C2×C7×#45811yescharacteristic
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×#3094C2semidirect
3. C7×#2482C2semidirect
4. C7×#1613C2semidirect
5. C7×#1607C2semidirect
6. C7×#1510C2semidirect
7. C7×#1462C2semidirect
16. C2×C7×#117C22semidirect
19. C7×#481C22semidirect
20. C7×#480C22semidirect
23. C7×#407C22semidirect
24. C7×#399C22semidirect
25. C7×#397C22semidirect
28. C7×#381C22semidirect
30. C7×#327C22semidirect
31. C7×#323C22semidirect
32. #18137C7direct (N × H)
61. C2×C7×C4⋊C4C23semidirect
62. C2×C7×C4⋊C4C23semidirect
79. C7×C22⋊C8C23semidirect
82. C7×#95C23semidirect
84. #3094C2×C7semidirect
85. #2482C2×C7semidirect
86. #1613C2×C7semidirect
87. #1607C2×C7semidirect
88. #1510C2×C7semidirect
89. #1462C2×C7semidirect
157. C2×#117C22×C7semidirect
160. #481C22×C7semidirect
161. #480C22×C7semidirect
164. #407C22×C7semidirect
165. #399C22×C7semidirect
166. #397C22×C7semidirect
169. #381C22×C7semidirect
171. #327C22×C7semidirect
172. #323C22×C7semidirect
254. C2×C4⋊C4C23×C7semidirect
255. C2×C4⋊C4C23×C7semidirect
272. C22⋊C8C23×C7semidirect
275. #95C23×C7semidirect
426. C7#18137direct (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×#30942
C7×#24822
C7×#16132
C7×#16072
C7×#15102
C7×#14622
#181377