Finite Group Databasebeta

← C7×#18763 #13195979

Subgroups — C7×#18763 #13195979

Order 1792.

910 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×#556C211yesmaximal, characteristic
2C7×#3230C211yesmaximal, characteristic, complemented
3C7×#3101C211yesmaximal, characteristic
4C7×#3078C211yesmaximal, characteristic, complemented
5C7×#1650C214yesmaximal, complemented
6C7×#1638C214yesmaximal, complemented
7C7×#1549C211yesmaximal, characteristic, complemented
8C7×#1486C211yesmaximal, characteristic
9C7×#1479C211yesmaximal, characteristic, complemented
10C7×#556—22no—
11C7×#556—22no—
12C7×#556—22no—
13C7×#556C2214yes—
14C7×#533C2211yescharacteristic
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#126—22no—
17C2×C7×#124—21no—
18C2×C7×C22⋊Q8—21no—
19C2×C4×C7×Q8—21no—
20C2×C4×C7×Q8C2211yescharacteristic
21C2×C4×C7×D4C2211yescharacteristic
22C2×C7×#117—21no—
23C7×#500C2212yescomplemented
24C7×#487C2212yescomplemented
25C7×#484—22no—
26C7×#483—21no—
27C7×#479—22no—
28C7×#477—22no—
29C7×#477C2212yescomplemented
30C7×#476C2212yes—
31C7×#475C2212yescomplemented
32C7×#461C2212yes—
33C7×#450—21no—
34C4×C7×SD16—22no—
35C7×#431C2211yescharacteristic
36C7×#431C2211yescharacteristic
37C7×#428—21no—
38C7×#423C2211yescharacteristic
39C7×#419—21no—
40C7×#332C2214yescomplemented
41C7×Q8⋊C8C2214yescomplemented
42C7×D4⋊C8C2214yes—
43#18763C711yesmaximal, characteristic, direct product factor, Sylow
44C22×C7×Q8—21no—
45C22×C7×Q8D411yescharacteristic
46C22×C7×D4D411yescharacteristic
47C23×C4×C7—21no—
48C2×C7×SD16—22no—
49C2×C7×M16—21no—
50C7×C4⋊Q8—21no—
51C7×C4⋊Q8—21no—
52C7×C4⋊Q8D412yes—
53C7×C4⋊Q8C2311yescharacteristic
54C7×C4⋊Q8C2311yescharacteristic
55C7×C4⋊Q8C2312yes—
56C7×#126—28no—
57C7×#126—28no—
58C7×#125—21no—
59C7×#124—24no—
60C7×#124—24no—
61C7×C22⋊Q8—24no—
62C7×C22⋊Q8—24no—
63C7×C22⋊Q8—21no—
64C7×C22⋊Q8—21no—
65C4×C7×Q8—24no—
66C4×C7×Q8—24no—
67C4×C7×Q8—21no—
68C4×C7×Q8—21no—
69C4×C7×Q8—21no—
70C4×C7×Q8—21no—
71C4×C7×Q8D412yes—
72C4×C7×Q8C2312yes—
73C4×C7×D4—21no—
74C4×C7×D4—21no—
75C4×C7×D4D412yes—
76C4×C7×D4C2312yes—
77C7×#117—24no—
78C7×#117—24no—
79C7×#117—21no—
80C2×C7×C4⋊C4—22no—
81C2×C7×C4⋊C4—21no—
82C2×C7×C4⋊C4—21no—
83C2×C7×C4⋊C4D411yescharacteristic
84C2×C7×C4⋊C4D412yes—
85C2×C7×C4⋊C4D411yescharacteristic
86C2×C7×C22⋊C4—22no—
87C2×C7×C22⋊C4—22no—
88C2×C7×C22⋊C4—21no—
89C2×C42×C7—22no—
90C2×C42×C7—21no—
91C2×C42×C7C2311yescharacteristic
92C7×#107—24no—
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—22no—
101C7×#103—22no—
102C7×#103—24no—
103C7×#103—22no—
104C7×#102—22no—
105C7×C22⋊C8—21no—
106C7×C22⋊C8—21no—
107C4×C7×C8—21no—
108C4×C7×C8C2312yes—
109C2×#556C2×C711yescharacteristic
110#3230C2×C711yescharacteristic, complemented
111#3101C2×C711yescharacteristic
112#3078C2×C711yescharacteristic, complemented
113#1650C2×C714yescomplemented
114#1638C2×C714yescomplemented
115#1549C2×C711yescharacteristic, complemented
116#1486C2×C711yescharacteristic
117#1479C2×C711yescharacteristic, complemented
118C24×C7—21no—
119C2×C7×Q8—21no—
120C2×C7×Q8—22no—
121C2×C7×Q8—22no—
122C2×C7×Q8—22no—
123C2×C7×Q8—22no—
124C2×C7×Q8—21no—
125C2×C7×Q8—21no—
126C2×C7×Q8—44no—
127C2×C7×Q8C2×D412yes—
128C2×C7×D4—21no—
129C2×C7×D4—42no—
130C2×C7×D4C2×D412yes—
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—21no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—42no—
138C22×C4×C7—42no—
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C7×SD16—44no—
143C7×M16—42no—
144C2×C7×C8—22no—
145C2×C7×C8—22no—
146C2×C7×C8—22no—
147C7×C4⋊C4—28no—
148C7×C4⋊C4—22no—
149C7×C4⋊C4—22no—
150C7×C4⋊C4—21no—
151C7×C4⋊C4—21no—
152C7×C4⋊C4—22no—
153C7×C4⋊C4—22no—
154C7×C4⋊C4—22no—
155C7×C4⋊C4—21no—
156C7×C4⋊C4—21no—
157C7×C4⋊C4—21no—
158C7×C4⋊C4—21no—
159C7×C4⋊C4—41no—
160C7×C4⋊C4—41no—
161C7×C4⋊C4C2×D412yes—
162C7×C4⋊C4C2×D414yes—
163C7×C4⋊C4C2×D412yes—
164C7×C22⋊C4—28no—
165C7×C22⋊C4—28no—
166C7×C22⋊C4—22no—
167C7×C22⋊C4—22no—
168C7×C22⋊C4—41no—
169C42×C7—28no—
170C42×C7—22no—
171C42×C7—22no—
172C42×C7—41no—
173C42×C7C2×D411yescharacteristic
174C42×C7C2×D411yescharacteristic
175C42×C7C2×D411yescharacteristic
176C42×C7C2411yescharacteristic
177#556—22no—
178#556—22no—
179#556—22no—
180#556C22×C714yes—
181#533C22×C711yescharacteristic
182C2×C4⋊Q8C22×C711yescharacteristic
183C2×#126—22no—
184C2×#124—21no—
185C2×C22⋊Q8—21no—
186C2×C4×Q8—21no—
187C2×C4×Q8C22×C711yescharacteristic
188C2×C4×D4C22×C711yescharacteristic
189C2×#117—21no—
190#500C22×C712yescomplemented
191#487C22×C712yescomplemented
192#484—22no—
193#483—21no—
194#479—22no—
195#477—22no—
196#477C22×C712yescomplemented
197#476C22×C712yes—
198#475C22×C712yescomplemented
199#461C22×C712yes—
200#450—21no—
201C4×SD16—22no—
202#431C22×C711yescharacteristic
203#431C22×C711yescharacteristic
204#428—21no—
205#423C22×C711yescharacteristic
206#419—21no—
207#332C22×C714yescomplemented
208Q8⋊C8C22×C714yescomplemented
209D4⋊C8C22×C714yes—
210C23×C7—22no—
211C23×C7—42no—
212C23×C7—42no—
213C23×C7—42no—
214C23×C7#12011yescharacteristic
215C7×Q8—22no—
216C7×Q8—22no—
217C7×Q8—44no—
218C7×Q8—42no—
219C7×Q8—44no—
220C7×Q8—42no—
221C7×Q8—42no—
222C7×D4—22no—
223C7×D4—22no—
224C7×D4—42no—
225C2×C4×C7—22no—
226C2×C4×C7—22no—
227C2×C4×C7—22no—
228C2×C4×C7—22no—
229C2×C4×C7—24no—
230C2×C4×C7—41no—
231C2×C4×C7—42no—
232C2×C4×C7—44no—
233C2×C4×C7—42no—
234C2×C4×C7—42no—
235C2×C4×C7—42no—
236C2×C4×C7—42no—
237C2×C4×C7—44no—
238C2×C4×C7—41no—
239C2×C4×C7—41no—
240C2×C4×C7#12011yescharacteristic
241C2×C4×C7#12011yescharacteristic
242C2×C4×C7#12011yescharacteristic
243C2×C4×C7#13512yes—
244C2×C4×C7#13712yes—
245C2×C4×C7C22×D411yescharacteristic
246C2×C4×C7C22×D411yescharacteristic
247C2×C4×C7C22×D411yescharacteristic
248C7×C8—42no—
249C7×C8—42no—
250C7×C8—42no—
251C22×Q8—21no—
252C22×Q8C7×D411yescharacteristic
253C22×D4C7×D411yescharacteristic
254C23×C4—21no—
255C2×SD16—22no—
256C2×M16—21no—
257C4⋊Q8—21no—
258C4⋊Q8—21no—
259C4⋊Q8C7×D412yes—
260C4⋊Q8C23×C711yescharacteristic
261C4⋊Q8C23×C711yescharacteristic
262C4⋊Q8C23×C712yes—
263#126—28no—
264#126—28no—
265#125—21no—
266#124—24no—
267#124—24no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—21no—
271C22⋊Q8—21no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—21no—
275C4×Q8—21no—
276C4×Q8—21no—
277C4×Q8—21no—
278C4×Q8C7×D412yes—
279C4×Q8C23×C712yes—
280C4×D4—21no—
281C4×D4—21no—
282C4×D4C7×D412yes—
283C4×D4C23×C712yes—
284#117—24no—
285#117—24no—
286#117—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4C7×D411yescharacteristic
291C2×C4⋊C4C7×D411yescharacteristic
292C2×C4⋊C4C7×D412yes—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—21no—
296C2×C42—22no—
297C2×C42—21no—
298C2×C42C23×C711yescharacteristic
299#107—22no—
300#107—24no—
301#106—22no—
302#106—22no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305C4⋊C8C23×C712yes—
306C4⋊C8C23×C712yes—
307#103—22no—
308#103—22no—
309#103—22no—
310#103—24no—
311#102—22no—
312C22⋊C8—21no—
313C22⋊C8—21no—
314C4×C8—21no—
315C4×C8C23×C712yes—
316C22×C7—21no—
317C22×C7—21no—
318C22×C7—42no—
319C22×C7—42no—
320C22×C7—42no—
321C22×C7—84no—
322C22×C7#45212yes—
323C22×C7C2×#12011yescharacteristic
324C4×C7—21no—
325C4×C7—21no—
326C4×C7—21no—
327C4×C7—42no—
328C4×C7—42no—
329C4×C7—44no—
330C4×C7—42no—
331C4×C7—81no—
332C4×C7—81no—
333C4×C7#45212yes—
334C4×C7C2×#13511yescharacteristic
335C4×C7C2×#13711yescharacteristic
336C4×C7#57512yes—
337C24—21no—
338C2×Q8—21no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—44no—
346C2×Q8C2×C7×D412yes—
347C2×D4—21no—
348C2×D4—42no—
349C2×D4C2×C7×D412yes—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4C2×C7×D411yescharacteristic
359C22×C4C2×C7×D411yescharacteristic
360C22×C4C2×C7×D411yescharacteristic
361SD16—44no—
362M16—42no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C4⋊C4—28no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4C2×C7×D412yes—
381C4⋊C4C2×C7×D414yes—
382C4⋊C4C2×C7×D412yes—
383C22⋊C4—28no—
384C22⋊C4—28no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—41no—
388C42—28no—
389C42—22no—
390C42—22no—
391C42—41no—
392C42C2×C7×D411yescharacteristic
393C42C2×C7×D411yescharacteristic
394C42C2×C7×D411yescharacteristic
395C42C24×C711yescharacteristic
396C2×C7—22no—
397C2×C7—82no—
398C2×C7C2×#45211yescharacteristic
399C2×C7#300511yescharacteristic
400C2×C7#302011yescharacteristic
401C23—22no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23C7×#12011yescharacteristic
406Q8—22no—
407Q8—22no—
408Q8—44no—
409Q8—42no—
410Q8—44no—
411Q8—42no—
412Q8—42no—
413D4—22no—
414D4—22no—
415D4—42no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4C7×#12011yescharacteristic
432C2×C4C7×#12011yescharacteristic
433C2×C4C7×#12011yescharacteristic
434C2×C4C7×#13512yes—
435C2×C4C7×#13712yes—
436C2×C4C22×C7×D411yescharacteristic
437C2×C4C22×C7×D411yescharacteristic
438C2×C4C22×C7×D411yescharacteristic
439C8—42no—
440C8—42no—
441C8—42no—
442C7#1876311yescharacteristic, direct product factor, Sylow
443C22—21no—
444C22—21no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—84no—
449C22C7×#45212yes—
450C22C2×C7×#12011yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—42no—
455C4—42no—
456C4—44no—
457C4—42no—
458C4—81no—
459C4—81no—
460C4C7×#45212yes—
461C4C2×C7×#13511yescharacteristic
462C4C2×C7×#13711yescharacteristic
463C4C7×#57512yes—
464C2—22no—
465C2—82no—
466C2C2×C7×#45211yescharacteristic
467C2C7×#300511yescharacteristic
468C2C7×#302011yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC22×C7×D4D4
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×#3230C2semidirect
4. C7×#3078C2semidirect
5. C7×#1650C2semidirect
6. C7×#1638C2semidirect
7. C7×#1549C2semidirect
9. C7×#1479C2semidirect
23. C7×#500C22semidirect
24. C7×#487C22semidirect
29. C7×#477C22semidirect
31. C7×#475C22semidirect
40. C7×#332C22semidirect
41. C7×Q8⋊C8C22semidirect
43. #18763C7direct (N × H)
110. #3230C2×C7semidirect
112. #3078C2×C7semidirect
113. #1650C2×C7semidirect
114. #1638C2×C7semidirect
115. #1549C2×C7semidirect
117. #1479C2×C7semidirect
190. #500C22×C7semidirect
191. #487C22×C7semidirect
196. #477C22×C7semidirect
198. #475C22×C7semidirect
207. #332C22×C7semidirect
208. Q8⋊C8C22×C7semidirect
442. C7#18763direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5562
C7×#32302
C7×#31012
C7×#30782
C7×#16502
C7×#16382
C7×#15492
C7×#14862
C7×#14792
#187637