Finite Group Databasebeta

← C7×#18773 #13195989

Subgroups — C7×#18773 #13195989

Order 1792.

986 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#555C211yesmaximal, characteristic
2C7×#3209C211yesmaximal, characteristic, complemented
3C7×#3101C211yesmaximal, characteristic, complemented
4C7×#3077C211yesmaximal, characteristic, complemented
5C7×#1646C214yesmaximal, complemented
6C7×#1634C214yesmaximal, complemented
7C7×#1548C211yesmaximal, characteristic, complemented
8C7×#1486C211yesmaximal, characteristic, complemented
9C7×#1478C211yesmaximal, characteristic, complemented
10C7×#555—22no—
11C7×#555—22no—
12C7×#555—22no—
13C7×#555C2214yes—
14C7×#533C2211yescharacteristic
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#124—22no—
17C2×C7×#121—21no—
18C2×C7×#121—21no—
19C2×C4×C7×D4C2211yescharacteristic
20C2×C4×C7×D4C2211yescharacteristic
21C2×C7×#117—21no—
22C2×C7×#117—21no—
23C7×#494C2212yescomplemented
24C7×#487C2212yescomplemented
25C7×#483—21no—
26C7×#478—22no—
27C7×#476C2212yescomplemented
28C7×#476C2212yescomplemented
29C7×#474C2212yescomplemented
30C7×#462—22no—
31C7×#461C2212yescomplemented
32C7×#451—22no—
33C7×#450—21no—
34C4×C7×SD16—22no—
35C7×#431C2211yescharacteristic
36C7×#431C2211yescharacteristic
37C7×#428—21no—
38C7×#423C2211yescharacteristic
39C7×#418—21no—
40C7×#333C2214yescomplemented
41C7×D4⋊C8C2214yescomplemented
42C7×D4⋊C8C2214yescomplemented
43#18773C711yesmaximal, characteristic, direct product factor, Sylow
44C22×C7×Q8D412yes—
45C22×C7×D4—22no—
46C22×C7×D4D411yescharacteristic
47C22×C7×D4D411yescharacteristic
48C23×C4×C7—21no—
49C23×C4×C7—21no—
50C2×C7×Q16—24no—
51C2×C7×SD16—22no—
52C2×C7×M16—21no—
53C7×C4⋊Q8—21no—
54C7×C4⋊Q8—21no—
55C7×C4⋊Q8D412yes—
56C7×C4⋊Q8C2311yescharacteristic
57C7×C4⋊Q8C2311yescharacteristic
58C7×C4⋊Q8C2312yes—
59C7×#125—21no—
60C7×#124—28no—
61C7×#124—28no—
62C7×C22⋊Q8—21no—
63C7×C22⋊Q8—21no—
64C7×#121—24no—
65C7×#121—24no—
66C7×#121—24no—
67C7×#121—24no—
68C4×C7×Q8—21no—
69C4×C7×Q8—21no—
70C4×C7×D4—21no—
71C4×C7×D4—21no—
72C4×C7×D4—21no—
73C4×C7×D4—21no—
74C4×C7×D4D412yes—
75C4×C7×D4D412yes—
76C4×C7×D4C2312yes—
77C4×C7×D4C2312yes—
78C7×#117—24no—
79C7×#117—24no—
80C7×#117—24no—
81C7×#117—24no—
82C7×#117—21no—
83C2×C7×C4⋊C4—21no—
84C2×C7×C4⋊C4—21no—
85C2×C7×C4⋊C4D411yescharacteristic
86C2×C7×C4⋊C4D411yescharacteristic
87C2×C7×C22⋊C4—22no—
88C2×C7×C22⋊C4—22no—
89C2×C7×C22⋊C4—21no—
90C2×C7×C22⋊C4—21no—
91C2×C42×C7—22no—
92C2×C42×C7C2311yescharacteristic
93C7×#107—22no—
94C7×#106—22no—
95C7×#106—22no—
96C7×C4⋊C8—21no—
97C7×C4⋊C8—21no—
98C7×C4⋊C8C2312yes—
99C7×C4⋊C8C2312yes—
100C7×#103—24no—
101C7×#103—22no—
102C7×#102—22no—
103C7×#102—22no—
104C7×#102—22no—
105C7×C22⋊C8—21no—
106C7×C22⋊C8—21no—
107C4×C7×C8—21no—
108C4×C7×C8C2312yes—
109C2×#555C2×C711yescharacteristic
110#3209C2×C711yescharacteristic, complemented
111#3101C2×C711yescharacteristic, complemented
112#3077C2×C711yescharacteristic, complemented
113#1646C2×C714yescomplemented
114#1634C2×C714yescomplemented
115#1548C2×C711yescharacteristic, complemented
116#1486C2×C711yescharacteristic, complemented
117#1478C2×C711yescharacteristic, complemented
118C24×C7—21no—
119C24×C7—21no—
120C2×C7×Q8—22no—
121C2×C7×Q8—21no—
122C2×C7×Q8—21no—
123C2×C7×Q8—82no—
124C2×C7×Q8C2×D414yes—
125C2×C7×D4—28no—
126C2×C7×D4—21no—
127C2×C7×D4—21no—
128C2×C7×D4—22no—
129C2×C7×D4—22no—
130C2×C7×D4—42no—
131C2×C7×D4—84no—
132C2×C7×D4C2×D412yes—
133C2×C7×D4C2×D412yes—
134C22×C4×C7—21no—
135C22×C4×C7—21no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—44no—
142C22×C4×C7—42no—
143C22×C4×C7—42no—
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C7×Q16—84no—
148C7×SD16—44no—
149C7×M16—42no—
150C2×C7×C8—22no—
151C2×C7×C8—22no—
152C2×C7×C8—22no—
153C7×C4⋊C4—22no—
154C7×C4⋊C4—22no—
155C7×C4⋊C4—21no—
156C7×C4⋊C4—21no—
157C7×C4⋊C4—22no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4—21no—
160C7×C4⋊C4—21no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4—21no—
163C7×C4⋊C4—41no—
164C7×C4⋊C4—41no—
165C7×C4⋊C4C2×D412yes—
166C7×C4⋊C4C2×D412yes—
167C7×C22⋊C4—28no—
168C7×C22⋊C4—28no—
169C7×C22⋊C4—22no—
170C7×C22⋊C4—22no—
171C7×C22⋊C4—22no—
172C7×C22⋊C4—22no—
173C7×C22⋊C4—41no—
174C42×C7—28no—
175C42×C7—41no—
176C42×C7C2×D411yescharacteristic
177C42×C7C2×D411yescharacteristic
178C42×C7C2×D411yescharacteristic
179C42×C7C2411yescharacteristic
180#555—22no—
181#555—22no—
182#555—22no—
183#555C22×C714yes—
184#533C22×C711yescharacteristic
185C2×C4⋊Q8C22×C711yescharacteristic
186C2×#124—22no—
187C2×#121—21no—
188C2×#121—21no—
189C2×C4×D4C22×C711yescharacteristic
190C2×C4×D4C22×C711yescharacteristic
191C2×#117—21no—
192C2×#117—21no—
193#494C22×C712yescomplemented
194#487C22×C712yescomplemented
195#483—21no—
196#478—22no—
197#476C22×C712yescomplemented
198#476C22×C712yescomplemented
199#474C22×C712yescomplemented
200#462—22no—
201#461C22×C712yescomplemented
202#451—22no—
203#450—21no—
204C4×SD16—22no—
205#431C22×C711yescharacteristic
206#431C22×C711yescharacteristic
207#428—21no—
208#423C22×C711yescharacteristic
209#418—21no—
210#333C22×C714yescomplemented
211D4⋊C8C22×C714yescomplemented
212D4⋊C8C22×C714yescomplemented
213C23×C7—22no—
214C23×C7—22no—
215C23×C7—42no—
216C23×C7—42no—
217C23×C7—42no—
218C23×C7—42no—
219C23×C7—42no—
220C23×C7—42no—
221C23×C7#12011yescharacteristic
222C7×Q8—44no—
223C7×Q8—42no—
224C7×Q8—42no—
225C7×Q8—82no—
226C7×D4—22no—
227C7×D4—22no—
228C7×D4—22no—
229C7×D4—22no—
230C7×D4—42no—
231C7×D4—42no—
232C7×D4—88no—
233C2×C4×C7—22no—
234C2×C4×C7—22no—
235C2×C4×C7—22no—
236C2×C4×C7—24no—
237C2×C4×C7—41no—
238C2×C4×C7—44no—
239C2×C4×C7—42no—
240C2×C4×C7—44no—
241C2×C4×C7—42no—
242C2×C4×C7—42no—
243C2×C4×C7—42no—
244C2×C4×C7—44no—
245C2×C4×C7—44no—
246C2×C4×C7—41no—
247C2×C4×C7—41no—
248C2×C4×C7#12011yescharacteristic
249C2×C4×C7#12011yescharacteristic
250C2×C4×C7#12011yescharacteristic
251C2×C4×C7#13512yes—
252C2×C4×C7#13712yes—
253C2×C4×C7C22×D411yescharacteristic
254C2×C4×C7C22×D411yescharacteristic
255C2×C4×C7C22×D411yescharacteristic
256C7×C8—42no—
257C7×C8—42no—
258C7×C8—42no—
259C22×Q8C7×D412yes—
260C22×D4—22no—
261C22×D4C7×D411yescharacteristic
262C22×D4C7×D411yescharacteristic
263C23×C4—21no—
264C23×C4—21no—
265C2×Q16—24no—
266C2×SD16—22no—
267C2×M16—21no—
268C4⋊Q8—21no—
269C4⋊Q8—21no—
270C4⋊Q8C7×D412yes—
271C4⋊Q8C23×C711yescharacteristic
272C4⋊Q8C23×C711yescharacteristic
273C4⋊Q8C23×C712yes—
274#125—21no—
275#124—28no—
276#124—28no—
277C22⋊Q8—21no—
278C22⋊Q8—21no—
279#121—24no—
280#121—24no—
281#121—24no—
282#121—24no—
283C4×Q8—21no—
284C4×Q8—21no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4C7×D412yes—
290C4×D4C7×D412yes—
291C4×D4C23×C712yes—
292C4×D4C23×C712yes—
293#117—24no—
294#117—24no—
295#117—24no—
296#117—24no—
297#117—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4C7×D411yescharacteristic
301C2×C4⋊C4C7×D411yescharacteristic
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C42—22no—
307C2×C42C23×C711yescharacteristic
308#107—22no—
309#106—22no—
310#106—22no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8C23×C712yes—
314C4⋊C8C23×C712yes—
315#103—24no—
316#103—22no—
317#102—22no—
318#102—22no—
319#102—22no—
320C22⋊C8—21no—
321C22⋊C8—21no—
322C4×C8—21no—
323C4×C8C23×C712yes—
324C22×C7—21no—
325C22×C7—21no—
326C22×C7—42no—
327C22×C7—42no—
328C22×C7—42no—
329C22×C7—42no—
330C22×C7—42no—
331C22×C7—42no—
332C22×C7—84no—
333C22×C7—84no—
334C22×C7#45212yes—
335C22×C7C2×#12011yescharacteristic
336C4×C7—21no—
337C4×C7—21no—
338C4×C7—21no—
339C4×C7—42no—
340C4×C7—42no—
341C4×C7—44no—
342C4×C7—81no—
343C4×C7—81no—
344C4×C7#45212yes—
345C4×C7C2×#13511yescharacteristic
346C4×C7C2×#13711yescharacteristic
347C4×C7#57512yes—
348C24—21no—
349C24—21no—
350C2×Q8—22no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—82no—
354C2×Q8C2×C7×D414yes—
355C2×D4—28no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—42no—
361C2×D4—84no—
362C2×D4C2×C7×D412yes—
363C2×D4C2×C7×D412yes—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—44no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4C2×C7×D411yescharacteristic
375C22×C4C2×C7×D411yescharacteristic
376C22×C4C2×C7×D411yescharacteristic
377Q16—84no—
378SD16—44no—
379M16—42no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4C2×C7×D412yes—
396C4⋊C4C2×C7×D412yes—
397C22⋊C4—28no—
398C22⋊C4—28no—
399C22⋊C4—22no—
400C22⋊C4—22no—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C22⋊C4—41no—
404C42—28no—
405C42—41no—
406C42C2×C7×D411yescharacteristic
407C42C2×C7×D411yescharacteristic
408C42C2×C7×D411yescharacteristic
409C42C24×C711yescharacteristic
410C2×C7—22no—
411C2×C7—82no—
412C2×C7—82no—
413C2×C7C2×#45211yescharacteristic
414C2×C7#300511yescharacteristic
415C2×C7#301911yescharacteristic
416C23—22no—
417C23—22no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23C7×#12011yescharacteristic
425Q8—44no—
426Q8—42no—
427Q8—42no—
428Q8—82no—
429D4—22no—
430D4—22no—
431D4—22no—
432D4—22no—
433D4—42no—
434D4—42no—
435D4—88no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—41no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4C7×#12011yescharacteristic
452C2×C4C7×#12011yescharacteristic
453C2×C4C7×#12011yescharacteristic
454C2×C4C7×#13512yes—
455C2×C4C7×#13712yes—
456C2×C4C22×C7×D411yescharacteristic
457C2×C4C22×C7×D411yescharacteristic
458C2×C4C22×C7×D411yescharacteristic
459C8—42no—
460C8—42no—
461C8—42no—
462C7#1877311yescharacteristic, direct product factor, Sylow
463C22—21no—
464C22—21no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—84no—
472C22—84no—
473C22C7×#45212yes—
474C22C2×C7×#12011yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—42no—
479C4—42no—
480C4—44no—
481C4—81no—
482C4—81no—
483C4C7×#45212yes—
484C4C2×C7×#13511yescharacteristic
485C4C2×C7×#13711yescharacteristic
486C4C7×#57512yes—
487C2—22no—
488C2—82no—
489C2—82no—
490C2C2×C7×#45211yescharacteristic
491C2C7×#300511yescharacteristic
492C2C7×#301911yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC2×C7×#555C2
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×#3209C2semidirect
3. C7×#3101C2semidirect
4. C7×#3077C2semidirect
5. C7×#1646C2semidirect
6. C7×#1634C2semidirect
7. C7×#1548C2semidirect
8. C7×#1486C2semidirect
9. C7×#1478C2semidirect
23. C7×#494C22semidirect
24. C7×#487C22semidirect
27. C7×#476C22semidirect
28. C7×#476C22semidirect
29. C7×#474C22semidirect
31. C7×#461C22semidirect
40. C7×#333C22semidirect
41. C7×D4⋊C8C22semidirect
42. C7×D4⋊C8C22semidirect
43. #18773C7direct (N × H)
110. #3209C2×C7semidirect
111. #3101C2×C7semidirect
112. #3077C2×C7semidirect
113. #1646C2×C7semidirect
114. #1634C2×C7semidirect
115. #1548C2×C7semidirect
116. #1486C2×C7semidirect
117. #1478C2×C7semidirect
193. #494C22×C7semidirect
194. #487C22×C7semidirect
197. #476C22×C7semidirect
198. #476C22×C7semidirect
199. #474C22×C7semidirect
201. #461C22×C7semidirect
210. #333C22×C7semidirect
211. D4⋊C8C22×C7semidirect
212. D4⋊C8C22×C7semidirect
462. C7#18773direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5552
C7×#32092
C7×#31012
C7×#30772
C7×#16462
C7×#16342
C7×#15482
C7×#14862
C7×#14782
#187737