Finite Group Databasebeta

← C7×#18318 #13195534

Subgroups — C7×#18318 #13195534

Order 1792.

1172 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#543C211yesmaximal, characteristic, complemented
2C7×#3193C212yesmaximal, complemented
3C7×#2539C211yesmaximal, characteristic
4C7×#1614C214yesmaximal, complemented
5C7×#1596C214yesmaximal, complemented
6C7×#1508C212yesmaximal, complemented
7C7×#1462C211yesmaximal, characteristic, complemented
8C7×#543—24no—
9C7×#543—22no—
10C7×#543C2214yes—
11C2×C7×#123—21no—
12C2×C7×C22⋊Q8—22no—
13C2×C7×C22⋊Q8—21no—
14C2×C7×C22⋊Q8—22no—
15C2×C7×C22⋊Q8C2212yes—
16C2×C7×#121C2211yescharacteristic
17C2×C7×#120—21no—
18C22×C7×C22⋊C4—21no—
19C22×C7×C22⋊C4C2211yescharacteristic
20C7×#484C2214yescomplemented
21C7×#480—22no—
22C7×#468—24no—
23C7×#450C2214yescomplemented
24C7×#448—22no—
25C7×#407C2212yes—
26C7×#399C2212yes—
27C7×#392C2212yes—
28C7×#390—22no—
29C4×C7×C22⋊C4C2211yescharacteristic
30C7×#328C2218yescomplemented
31C7×#323C2214yes—
32#18318C711yesmaximal, characteristic, direct product factor, Sylow
33C25×C7D411yescharacteristic
34C22×C7×Q8—21no—
35C22×C7×Q8D412yescomplemented
36C22×C7×D4—21no—
37C22×C7×D4—22no—
38C22×C7×D4C2311yescharacteristic
39C23×C4×C7—21no—
40C23×C4×C7—21no—
41C23×C4×C7D411yescharacteristic
42C23×C4×C7C2311yescharacteristic
43C2×C7×SD16—24no—
44C2×C7×M16—22no—
45C7×#123—44no—
46C7×C22⋊Q8—28no—
47C7×C22⋊Q8—28no—
48C7×C22⋊Q8—24no—
49C7×C22⋊Q8—28no—
50C7×C22⋊Q8—24no—
51C7×C22⋊Q8—22no—
52C7×C22⋊Q8—22no—
53C7×C22⋊Q8—28no—
54C7×C22⋊Q8D414yescomplemented
55C7×C22⋊Q8C2314yes—
56C7×#121—22no—
57C7×#121—22no—
58C7×#120—44no—
59C2×C7×C4⋊C4—21no—
60C2×C7×C4⋊C4—22no—
61C2×C7×C4⋊C4—22no—
62C2×C7×C4⋊C4—22no—
63C2×C7×C4⋊C4D412yescomplemented
64C2×C7×C4⋊C4C2311yescharacteristic
65C2×C7×C22⋊C4—22no—
66C2×C7×C22⋊C4—24no—
67C2×C7×C22⋊C4—22no—
68C2×C7×C22⋊C4—24no—
69C2×C7×C22⋊C4—21no—
70C2×C7×C22⋊C4—22no—
71C2×C7×C22⋊C4—22no—
72C2×C7×C22⋊C4—22no—
73C2×C7×C22⋊C4—21no—
74C2×C7×C22⋊C4—22no—
75C2×C7×C22⋊C4D412yes—
76C2×C7×C22⋊C4C2312yes—
77C2×C42×C7—21no—
78C7×#107—24no—
79C7×#103—24no—
80C7×#102—24no—
81C7×C22⋊C8—22no—
82C7×C22⋊C8C2314yes—
83C7×#95C2312yes—
84C2×#543C2×C711yescharacteristic, complemented
85#3193C2×C712yescomplemented
86#2539C2×C711yescharacteristic
87#1614C2×C714yescomplemented
88#1596C2×C714yescomplemented
89#1508C2×C712yescomplemented
90#1462C2×C711yescharacteristic, complemented
91C24×C7—21no—
92C24×C7—21no—
93C24×C7—24no—
94C24×C7—21no—
95C24×C7—42no—
96C24×C7—81no—
97C24×C7C2×D411yescharacteristic
98C24×C7C2×D412yes—
99C2×C7×Q8—24no—
100C2×C7×Q8—22no—
101C2×C7×Q8—82no—
102C2×C7×Q8—82no—
103C2×C7×Q8C2×D414yes—
104C2×C7×D4—22no—
105C2×C7×D4—44no—
106C2×C7×D4—42no—
107C2×C7×D4—42no—
108C2×C7×D4—41no—
109C2×C7×D4—41no—
110C2×C7×D4—44no—
111C2×C7×D4—81no—
112C2×C7×D4—82no—
113C22×C4×C7—22no—
114C22×C4×C7—21no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—24no—
118C22×C4×C7—24no—
119C22×C4×C7—21no—
120C22×C4×C7—21no—
121C22×C4×C7—42no—
122C22×C4×C7—81no—
123C22×C4×C7—82no—
124C22×C4×C7—82no—
125C22×C4×C7C2×D412yes—
126C22×C4×C7C2×D411yescharacteristic
127C22×C4×C7C2×D412yes—
128C22×C4×C7C2×D412yes—
129C22×C4×C7C2411yescharacteristic
130C7×SD16—84no—
131C7×M16—82no—
132C2×C7×C8—24no—
133C7×C4⋊C4—24no—
134C7×C4⋊C4—28no—
135C7×C4⋊C4—24no—
136C7×C4⋊C4—22no—
137C7×C4⋊C4—24no—
138C7×C4⋊C4—22no—
139C7×C4⋊C4—44no—
140C7×C4⋊C4C2×D414yes—
141C7×C22⋊C4—28no—
142C7×C22⋊C4—24no—
143C7×C22⋊C4—24no—
144C7×C22⋊C4—24no—
145C7×C22⋊C4—24no—
146C7×C22⋊C4—24no—
147C7×C22⋊C4—44no—
148C7×C22⋊C4—44no—
149C7×C22⋊C4—42no—
150C42×C7—81no—
151#543—24no—
152#543—22no—
153#543C22×C714yes—
154C2×#123—21no—
155C2×C22⋊Q8—22no—
156C2×C22⋊Q8—21no—
157C2×C22⋊Q8—22no—
158C2×C22⋊Q8C22×C712yes—
159C2×#121C22×C711yescharacteristic
160C2×#120—21no—
161C22×C22⋊C4—21no—
162C22×C22⋊C4C22×C711yescharacteristic
163#484C22×C714yescomplemented
164#480—22no—
165#468—24no—
166#450C22×C714yescomplemented
167#448—22no—
168#407C22×C712yes—
169#399C22×C712yes—
170#392C22×C712yes—
171#390—22no—
172C4×C22⋊C4C22×C711yescharacteristic
173#328C22×C718yescomplemented
174#323C22×C714yes—
175C23×C7—24no—
176C23×C7—24no—
177C23×C7—22no—
178C23×C7—22no—
179C23×C7—22no—
180C23×C7—22no—
181C23×C7—41no—
182C23×C7—42no—
183C23×C7—42no—
184C23×C7—42no—
185C23×C7—41no—
186C23×C7—41no—
187C23×C7—42no—
188C23×C7—42no—
189C23×C7—41no—
190C23×C7—42no—
191C23×C7—81no—
192C23×C7—81no—
193C23×C7—88no—
194C23×C7#12011yescharacteristic
195C23×C7#12011yescharacteristic
196C23×C7#13714yes—
197C23×C7C22×D411yescharacteristic
198C7×Q8—44no—
199C7×Q8—82no—
200C7×Q8—84no—
201C7×D4—44no—
202C7×D4—88no—
203C7×D4—82no—
204C7×D4—82no—
205C2×C4×C7—24no—
206C2×C4×C7—24no—
207C2×C4×C7—22no—
208C2×C4×C7—24no—
209C2×C4×C7—41no—
210C2×C4×C7—42no—
211C2×C4×C7—42no—
212C2×C4×C7—41no—
213C2×C4×C7—42no—
214C2×C4×C7—44no—
215C2×C4×C7—42no—
216C2×C4×C7—42no—
217C2×C4×C7—44no—
218C2×C4×C7—42no—
219C2×C4×C7—41no—
220C2×C4×C7—41no—
221C2×C4×C7—41no—
222C2×C4×C7—81no—
223C2×C4×C7—84no—
224C2×C4×C7—84no—
225C2×C4×C7#12012yes—
226C2×C4×C7C22×D412yes—
227C7×C8—44no—
228C25C7×D411yescharacteristic
229C22×Q8—21no—
230C22×Q8C7×D412yescomplemented
231C22×D4—21no—
232C22×D4—22no—
233C22×D4C23×C711yescharacteristic
234C23×C4—21no—
235C23×C4—21no—
236C23×C4C7×D411yescharacteristic
237C23×C4C23×C711yescharacteristic
238C2×SD16—24no—
239C2×M16—22no—
240#123—44no—
241C22⋊Q8—28no—
242C22⋊Q8—28no—
243C22⋊Q8—24no—
244C22⋊Q8—28no—
245C22⋊Q8—24no—
246C22⋊Q8—22no—
247C22⋊Q8—22no—
248C22⋊Q8—28no—
249C22⋊Q8C7×D414yescomplemented
250C22⋊Q8C23×C714yes—
251#121—22no—
252#121—22no—
253#120—44no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4C7×D412yescomplemented
259C2×C4⋊C4C23×C711yescharacteristic
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4C7×D412yes—
271C2×C22⋊C4C23×C712yes—
272C2×C42—21no—
273#107—24no—
274#103—24no—
275#102—24no—
276C22⋊C8—22no—
277C22⋊C8C23×C714yes—
278#95C23×C712yes—
279C22×C7—21no—
280C22×C7—21no—
281C22×C7—21no—
282C22×C7—21no—
283C22×C7—21no—
284C22×C7—24no—
285C22×C7—21no—
286C22×C7—21no—
287C22×C7—44no—
288C22×C7—44no—
289C22×C7—48no—
290C22×C7—42no—
291C22×C7—42no—
292C22×C7—82no—
293C22×C7—82no—
294C22×C7—82no—
295C22×C7—84no—
296C22×C7—81no—
297C22×C7—81no—
298C22×C7—81no—
299C22×C7#45312yes—
300C22×C7C2×#12011yescharacteristic
301C22×C7C2×#13712yes—
302C4×C7—22no—
303C4×C7—22no—
304C4×C7—44no—
305C4×C7—42no—
306C4×C7—44no—
307C4×C7—44no—
308C4×C7—81no—
309C24—21no—
310C24—21no—
311C24—24no—
312C24—21no—
313C24—42no—
314C24—81no—
315C24C2×C7×D411yescharacteristic
316C24C2×C7×D412yes—
317C2×Q8—24no—
318C2×Q8—22no—
319C2×Q8—82no—
320C2×Q8—82no—
321C2×Q8C2×C7×D414yes—
322C2×D4—22no—
323C2×D4—44no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—44no—
329C2×D4—81no—
330C2×D4—82no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—42no—
340C22×C4—81no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4C2×C7×D412yes—
344C22×C4C2×C7×D411yescharacteristic
345C22×C4C2×C7×D412yes—
346C22×C4C2×C7×D412yes—
347C22×C4C24×C711yescharacteristic
348SD16—84no—
349M16—82no—
350C2×C8—24no—
351C4⋊C4—24no—
352C4⋊C4—28no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—22no—
357C4⋊C4—44no—
358C4⋊C4C2×C7×D414yes—
359C22⋊C4—28no—
360C22⋊C4—24no—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—44no—
366C22⋊C4—44no—
367C22⋊C4—42no—
368C42—81no—
369C2×C7—22no—
370C2×C7—22no—
371C2×C7—41no—
372C2×C7—44no—
373C2×C7—161no—
374C2×C7#300411yescharacteristic
375C2×C7C2×#45311yescharacteristic
376C2×C7#302611yescharacteristic
377C23—24no—
378C23—24no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—41no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—41no—
388C23—41no—
389C23—42no—
390C23—42no—
391C23—41no—
392C23—42no—
393C23—81no—
394C23—81no—
395C23—88no—
396C23C7×#12011yescharacteristic
397C23C7×#12011yescharacteristic
398C23C7×#13714yes—
399C23C22×C7×D411yescharacteristic
400Q8—44no—
401Q8—82no—
402Q8—84no—
403D4—44no—
404D4—88no—
405D4—82no—
406D4—82no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—81no—
425C2×C4—84no—
426C2×C4—84no—
427C2×C4C7×#12012yes—
428C2×C4C22×C7×D412yes—
429C8—44no—
430C7#1831811yescharacteristic, direct product factor, Sylow
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—24no—
437C22—21no—
438C22—21no—
439C22—44no—
440C22—44no—
441C22—48no—
442C22—42no—
443C22—42no—
444C22—82no—
445C22—82no—
446C22—82no—
447C22—84no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22C7×#45312yes—
452C22C2×C7×#12011yescharacteristic
453C22C2×C7×#13712yes—
454C4—22no—
455C4—22no—
456C4—44no—
457C4—42no—
458C4—44no—
459C4—44no—
460C4—81no—
461C2—22no—
462C2—22no—
463C2—41no—
464C2—44no—
465C2—161no—
466C2C7×#300411yescharacteristic
467C2C2×C7×#45311yescharacteristic
468C2C7×#302611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#2539C2
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
1. C2×C7×#543C2semidirect
2. C7×#3193C2semidirect
4. C7×#1614C2semidirect
5. C7×#1596C2semidirect
6. C7×#1508C2semidirect
7. C7×#1462C2semidirect
20. C7×#484C22semidirect
23. C7×#450C22semidirect
30. C7×#328C22semidirect
32. #18318C7direct (N × H)
35. C22×C7×Q8D4semidirect
54. C7×C22⋊Q8D4semidirect
63. C2×C7×C4⋊C4D4semidirect
84. C2×#543C2×C7semidirect
85. #3193C2×C7semidirect
87. #1614C2×C7semidirect
88. #1596C2×C7semidirect
89. #1508C2×C7semidirect
90. #1462C2×C7semidirect
163. #484C22×C7semidirect
166. #450C22×C7semidirect
173. #328C22×C7semidirect
230. C22×Q8C7×D4semidirect
249. C22⋊Q8C7×D4semidirect
258. C2×C4⋊C4C7×D4semidirect
430. C7#18318direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5432
C7×#31932
C7×#25392
C7×#16142
C7×#15962
C7×#15082
C7×#14622
#183187