Finite Group Databasebeta

← C7×#18133 #13195349

Subgroups — C7×#18133 #13195349

Order 1792.

1310 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#543C211yesmaximal, characteristic, complemented
2C7×#3007C212yesmaximal, complemented
3C7×#2404C211yesmaximal, characteristic
4C7×#1602C214yesmaximal, complemented
5C7×#1596C214yesmaximal, complemented
6C7×#1511C212yesmaximal, complemented
7C7×#1462C211yesmaximal, characteristic, complemented
8C22×C7×#41C2211yescharacteristic
9C2×C7×#137—22no—
10C7×#543—24no—
11C7×#543—22no—
12C7×#543C2214yes—
13C2×C7×C22⋊Q8—22no—
14C2×C7×C22⋊Q8—21no—
15C2×C7×C22⋊Q8—22no—
16C2×C7×C22⋊Q8C2212yes—
17C2×C7×#120—21no—
18C22×C7×C22⋊C4—21no—
19C22×C7×C22⋊C4C2211yescharacteristic
20C7×#452—24no—
21C7×#451C2214yescomplemented
22C7×#450C2214yescomplemented
23C7×#417—22no—
24C7×#407C2212yes—
25C7×#393C2212yes—
26C7×#392C2212yes—
27C7×#386—21no—
28C7×#386—22no—
29C7×#381C2211yescharacteristic
30C7×#328C2218yescomplemented
31C7×#323C2214yes—
32#18133C711yesmaximal, characteristic, direct product factor, Sylow
33C25×C7D411yescharacteristic
34C2×C7×#41—22no—
35C2×C7×#41—22no—
36C2×C7×#41—44no—
37C22×C7×Q8—21no—
38C22×C7×Q8D412yescomplemented
39C22×C7×Q8C2311yescharacteristic
40C22×C7×D4—21no—
41C22×C7×D4—22no—
42C22×C7×D4C2311yescharacteristic
43C23×C4×C7—21no—
44C23×C4×C7—21no—
45C23×C4×C7—21no—
46C23×C4×C7D411yescharacteristic
47C23×C4×C7C2311yescharacteristic
48C7×#137—48no—
49C2×C7×Q16—24no—
50C2×C7×SD16—24no—
51C2×C7×M16—22no—
52C7×C22⋊Q8—28no—
53C7×C22⋊Q8—28no—
54C7×C22⋊Q8—24no—
55C7×C22⋊Q8—28no—
56C7×C22⋊Q8—24no—
57C7×C22⋊Q8—22no—
58C7×C22⋊Q8—22no—
59C7×C22⋊Q8—28no—
60C7×C22⋊Q8D414yescomplemented
61C7×C22⋊Q8C2314yes—
62C7×#120—44no—
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4—22no—
65C2×C7×C4⋊C4—22no—
66C2×C7×C4⋊C4D412yescomplemented
67C2×C7×C22⋊C4—22no—
68C2×C7×C22⋊C4—24no—
69C2×C7×C22⋊C4—22no—
70C2×C7×C22⋊C4—24no—
71C2×C7×C22⋊C4—22no—
72C2×C7×C22⋊C4—22no—
73C2×C7×C22⋊C4—21no—
74C2×C7×C22⋊C4—22no—
75C2×C7×C22⋊C4—22no—
76C2×C7×C22⋊C4D412yes—
77C2×C7×C22⋊C4C2312yes—
78C7×#103—24no—
79C7×#102—24no—
80C7×C22⋊C8—22no—
81C7×C22⋊C8C2314yes—
82C7×#95—21no—
83C7×#95—22no—
84C7×#95C2312yes—
85C2×#543C2×C711yescharacteristic, complemented
86#3007C2×C712yescomplemented
87#2404C2×C711yescharacteristic
88#1602C2×C714yescomplemented
89#1596C2×C714yescomplemented
90#1511C2×C712yescomplemented
91#1462C2×C711yescharacteristic, complemented
92C24×C7—21no—
93C24×C7—21no—
94C24×C7—24no—
95C24×C7—21no—
96C24×C7—42no—
97C24×C7—81no—
98C24×C7C2×D411yescharacteristic
99C24×C7C2×D412yes—
100C7×#41—28no—
101C7×#41—44no—
102C7×#41—44no—
103C7×#41—44no—
104C2×C7×Q8—24no—
105C2×C7×Q8—22no—
106C2×C7×Q8—22no—
107C2×C7×Q8—42no—
108C2×C7×Q8—44no—
109C2×C7×Q8—82no—
110C2×C7×Q8C2×D414yes—
111C2×C7×D4—24no—
112C2×C7×D4—22no—
113C2×C7×D4—42no—
114C2×C7×D4—42no—
115C2×C7×D4—41no—
116C2×C7×D4—41no—
117C2×C7×D4—44no—
118C2×C7×D4—44no—
119C2×C7×D4—82no—
120C22×C4×C7—22no—
121C22×C4×C7—21no—
122C22×C4×C7—22no—
123C22×C4×C7—21no—
124C22×C4×C7—24no—
125C22×C4×C7—24no—
126C22×C4×C7—21no—
127C22×C4×C7—22no—
128C22×C4×C7—42no—
129C22×C4×C7—42no—
130C22×C4×C7—42no—
131C22×C4×C7—42no—
132C22×C4×C7—81no—
133C22×C4×C7—82no—
134C22×C4×C7—82no—
135C22×C4×C7C2×D412yes—
136C22×C4×C7C2×D412yes—
137C22×C4×C7C2×D411yescharacteristic
138C22×C4×C7C2×D412yes—
139C22×C4×C7C2411yescharacteristic
140C7×Q16—48no—
141C7×SD16—48no—
142C7×M16—44no—
143C2×C7×C8—24no—
144C7×C4⋊C4—24no—
145C7×C4⋊C4—28no—
146C7×C4⋊C4—24no—
147C7×C4⋊C4—22no—
148C7×C4⋊C4—24no—
149C7×C4⋊C4C2×D414yes—
150C7×C22⋊C4—28no—
151C7×C22⋊C4—24no—
152C7×C22⋊C4—24no—
153C7×C22⋊C4—24no—
154C7×C22⋊C4—24no—
155C7×C22⋊C4—44no—
156C7×C22⋊C4—42no—
157C7×C22⋊C4—82no—
158C22×#41C22×C711yescharacteristic
159C2×#137—22no—
160#543—24no—
161#543—22no—
162#543C22×C714yes—
163C2×C22⋊Q8—22no—
164C2×C22⋊Q8—21no—
165C2×C22⋊Q8—22no—
166C2×C22⋊Q8C22×C712yes—
167C2×#120—21no—
168C22×C22⋊C4—21no—
169C22×C22⋊C4C22×C711yescharacteristic
170#452—24no—
171#451C22×C714yescomplemented
172#450C22×C714yescomplemented
173#417—22no—
174#407C22×C712yes—
175#393C22×C712yes—
176#392C22×C712yes—
177#386—21no—
178#386—22no—
179#381C22×C711yescharacteristic
180#328C22×C718yescomplemented
181#323C22×C714yes—
182C23×C7—24no—
183C23×C7—24no—
184C23×C7—22no—
185C23×C7—22no—
186C23×C7—22no—
187C23×C7—22no—
188C23×C7—22no—
189C23×C7—41no—
190C23×C7—42no—
191C23×C7—42no—
192C23×C7—42no—
193C23×C7—42no—
194C23×C7—41no—
195C23×C7—42no—
196C23×C7—41no—
197C23×C7—42no—
198C23×C7—81no—
199C23×C7—81no—
200C23×C7—88no—
201C23×C7#12011yescharacteristic
202C23×C7#12011yescharacteristic
203C23×C7#13714yescomplemented
204C23×C7C22×D411yescharacteristic
205C7×Q8—24no—
206C7×Q8—44no—
207C7×Q8—42no—
208C7×Q8—82no—
209C7×Q8—84no—
210C7×D4—24no—
211C7×D4—42no—
212C7×D4—44no—
213C7×D4—44no—
214C7×D4—88no—
215C2×C4×C7—24no—
216C2×C4×C7—22no—
217C2×C4×C7—24no—
218C2×C4×C7—22no—
219C2×C4×C7—24no—
220C2×C4×C7—41no—
221C2×C4×C7—42no—
222C2×C4×C7—41no—
223C2×C4×C7—48no—
224C2×C4×C7—41no—
225C2×C4×C7—42no—
226C2×C4×C7—44no—
227C2×C4×C7—41no—
228C2×C4×C7—44no—
229C2×C4×C7—42no—
230C2×C4×C7—42no—
231C2×C4×C7—42no—
232C2×C4×C7—42no—
233C2×C4×C7—81no—
234C2×C4×C7—84no—
235C2×C4×C7—84no—
236C2×C4×C7#12012yes—
237C2×C4×C7C22×D412yes—
238C7×C8—44no—
239C25C7×D411yescharacteristic
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—44no—
243C22×Q8—21no—
244C22×Q8C7×D412yescomplemented
245C22×Q8C23×C711yescharacteristic
246C22×D4—21no—
247C22×D4—22no—
248C22×D4C23×C711yescharacteristic
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4C7×D411yescharacteristic
253C23×C4C23×C711yescharacteristic
254#137—48no—
255C2×Q16—24no—
256C2×SD16—24no—
257C2×M16—22no—
258C22⋊Q8—28no—
259C22⋊Q8—28no—
260C22⋊Q8—24no—
261C22⋊Q8—28no—
262C22⋊Q8—24no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265C22⋊Q8—28no—
266C22⋊Q8C7×D414yescomplemented
267C22⋊Q8C23×C714yes—
268#120—44no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4C7×D412yescomplemented
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4C7×D412yes—
283C2×C22⋊C4C23×C712yes—
284#103—24no—
285#102—24no—
286C22⋊C8—22no—
287C22⋊C8C23×C714yes—
288#95—21no—
289#95—22no—
290#95C23×C712yes—
291C22×C7—21no—
292C22×C7—21no—
293C22×C7—21no—
294C22×C7—21no—
295C22×C7—21no—
296C22×C7—24no—
297C22×C7—21no—
298C22×C7—21no—
299C22×C7—44no—
300C22×C7—44no—
301C22×C7—48no—
302C22×C7—44no—
303C22×C7—42no—
304C22×C7—42no—
305C22×C7—42no—
306C22×C7—82no—
307C22×C7—82no—
308C22×C7—84no—
309C22×C7—81no—
310C22×C7—81no—
311C22×C7B(4,𝔽2)12yes—
312C22×C7C2×#12011yescharacteristic
313C22×C7C2×#13712yes—
314C4×C7—22no—
315C4×C7—22no—
316C4×C7—42no—
317C4×C7—44no—
318C4×C7—42no—
319C4×C7—44no—
320C4×C7—82no—
321C24—21no—
322C24—21no—
323C24—24no—
324C24—21no—
325C24—42no—
326C24—81no—
327C24C2×C7×D411yescharacteristic
328C24C2×C7×D412yes—
329#41—28no—
330#41—44no—
331#41—44no—
332#41—44no—
333C2×Q8—24no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—42no—
337C2×Q8—44no—
338C2×Q8—82no—
339C2×Q8C2×C7×D414yes—
340C2×D4—24no—
341C2×D4—22no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—82no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—81no—
362C22×C4—82no—
363C22×C4—82no—
364C22×C4C2×C7×D412yes—
365C22×C4C2×C7×D412yes—
366C22×C4C2×C7×D411yescharacteristic
367C22×C4C2×C7×D412yes—
368C22×C4C24×C711yescharacteristic
369Q16—48no—
370SD16—48no—
371M16—44no—
372C2×C8—24no—
373C4⋊C4—24no—
374C4⋊C4—28no—
375C4⋊C4—24no—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4C2×C7×D414yes—
379C22⋊C4—28no—
380C22⋊C4—24no—
381C22⋊C4—24no—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—44no—
385C22⋊C4—42no—
386C22⋊C4—82no—
387C2×C7—22no—
388C2×C7—22no—
389C2×C7—41no—
390C2×C7—44no—
391C2×C7—82no—
392C2×C7#300411yescharacteristic
393C2×C7#301911yescharacteristic
394C2×C7C2×B(4,𝔽2)11yescharacteristic
395C23—24no—
396C23—24no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—41no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—41no—
408C23—42no—
409C23—41no—
410C23—42no—
411C23—81no—
412C23—81no—
413C23—88no—
414C23C7×#12011yescharacteristic
415C23C7×#12011yescharacteristic
416C23C7×#13714yescomplemented
417C23C22×C7×D411yescharacteristic
418Q8—24no—
419Q8—44no—
420Q8—42no—
421Q8—82no—
422Q8—84no—
423D4—24no—
424D4—42no—
425D4—44no—
426D4—44no—
427D4—88no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—48no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—41no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—81no—
447C2×C4—84no—
448C2×C4—84no—
449C2×C4C7×#12012yes—
450C2×C4C22×C7×D412yes—
451C8—44no—
452C7#1813311yescharacteristic, direct product factor, Sylow
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—24no—
459C22—21no—
460C22—21no—
461C22—44no—
462C22—44no—
463C22—48no—
464C22—44no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—82no—
469C22—82no—
470C22—84no—
471C22—81no—
472C22—81no—
473C22C7×B(4,𝔽2)12yes—
474C22C2×C7×#12011yescharacteristic
475C22C2×C7×#13712yes—
476C4—22no—
477C4—22no—
478C4—42no—
479C4—44no—
480C4—42no—
481C4—44no—
482C4—82no—
483C2—22no—
484C2—22no—
485C2—41no—
486C2—44no—
487C2—82no—
488C2C7×#300411yescharacteristic
489C2C7×#301911yescharacteristic
490C2C2×C7×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#2404C2
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×#3007C2semidirect
4. C7×#1602C2semidirect
5. C7×#1596C2semidirect
6. C7×#1511C2semidirect
7. C7×#1462C2semidirect
21. C7×#451C22semidirect
22. C7×#450C22semidirect
30. C7×#328C22semidirect
32. #18133C7direct (N × H)
38. C22×C7×Q8D4semidirect
60. C7×C22⋊Q8D4semidirect
66. C2×C7×C4⋊C4D4semidirect
85. C2×#543C2×C7semidirect
86. #3007C2×C7semidirect
88. #1602C2×C7semidirect
89. #1596C2×C7semidirect
90. #1511C2×C7semidirect
91. #1462C2×C7semidirect
171. #451C22×C7semidirect
172. #450C22×C7semidirect
180. #328C22×C7semidirect
203. C23×C7#137semidirect
244. C22×Q8C7×D4semidirect
266. C22⋊Q8C7×D4semidirect
272. C2×C4⋊C4C7×D4semidirect
416. C23C7×#137semidirect
452. C7#18133direct (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×#30072
C7×#24042
C7×#16022
C7×#15962
C7×#15112
C7×#14622
#181337