Finite Group Databasebeta

← C7×#18751 #13195967

Subgroups — C7×#18751 #13195967

Order 1792.

986 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#559C211yesmaximal, characteristic
2C7×#3208C211yesmaximal, characteristic
3C7×#3104C211yesmaximal, characteristic, complemented
4C7×#3078C211yesmaximal, characteristic, complemented
5C7×#1640C214yesmaximal, complemented
6C7×#1630C214yesmaximal, complemented
7C7×#1545C211yesmaximal, characteristic
8C7×#1487C211yesmaximal, characteristic, complemented
9C7×#1479C211yesmaximal, characteristic, complemented
10C7×#559—22no—
11C7×#559—22no—
12C7×#559—22no—
13C7×#559C2214yes—
14C7×#533C2211yescharacteristic
15C2×C7×#127C2211yescharacteristic
16C2×C7×#124—21no—
17C2×C7×#124—21no—
18C2×C7×#123—22no—
19C2×C4×C7×Q8C2211yescharacteristic
20C2×C4×C7×Q8C2211yescharacteristic
21C2×C4×C7×D4—21no—
22C2×C4×C7×D4—21no—
23C7×#492C2212yes—
24C7×#486C2212yes—
25C7×#484—21no—
26C7×#479—22no—
27C7×#477C2212yescomplemented
28C7×#477C2212yescomplemented
29C7×#475C2212yescomplemented
30C7×#462C2212yescomplemented
31C7×#460—22no—
32C7×#451—21no—
33C7×#450—22no—
34C4×C7×Q16—22no—
35C7×#431C2211yescharacteristic
36C7×#431C2211yescharacteristic
37C7×#428—21no—
38C7×#423C2211yescharacteristic
39C7×#419—21no—
40C7×#331C2214yes—
41C7×Q8⋊C8C2214yescomplemented
42C7×Q8⋊C8C2214yescomplemented
43#18751C711yesmaximal, characteristic, direct product factor, Sylow
44C22×C7×Q8D411yescharacteristic
45C22×C7×Q8D411yescharacteristic
46C22×C7×D4—21no—
47C22×C7×D4—21no—
48C22×C7×D4D412yes—
49C23×C4×C7—22no—
50C2×C7×Q16—22no—
51C2×C7×SD16—24no—
52C2×C7×M16—21no—
53C7×C4⋊Q8C2311yescharacteristic
54C7×C4⋊Q8C2311yescharacteristic
55C7×#127—21no—
56C7×#127—21no—
57C7×#127D412yes—
58C7×#127C2312yes—
59C7×#125—21no—
60C7×#124—24no—
61C7×#124—24no—
62C7×#124—24no—
63C7×#124—24no—
64C7×#123—28no—
65C7×#123—28no—
66C7×C22⋊Q8—21no—
67C7×C22⋊Q8—21no—
68C4×C7×Q8—21no—
69C4×C7×Q8—21no—
70C4×C7×Q8—21no—
71C4×C7×Q8—21no—
72C4×C7×Q8—21no—
73C4×C7×Q8—21no—
74C4×C7×Q8D412yes—
75C4×C7×Q8D412yes—
76C4×C7×Q8C2312yes—
77C4×C7×Q8C2312yes—
78C4×C7×D4—24no—
79C4×C7×D4—24no—
80C4×C7×D4—24no—
81C4×C7×D4—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—22no—
90C2×C42×C7—21no—
91C2×C42×C7—21no—
92C2×C42×C7C2311yescharacteristic
93C7×#107—22no—
94C7×#107—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—22no—
103C7×#103—22no—
104C7×#102—24no—
105C7×C22⋊C8—21no—
106C7×C22⋊C8—21no—
107C4×C7×C8—21no—
108C4×C7×C8C2312yes—
109C2×#559C2×C711yescharacteristic
110#3208C2×C711yescharacteristic
111#3104C2×C711yescharacteristic, complemented
112#3078C2×C711yescharacteristic, complemented
113#1640C2×C714yescomplemented
114#1630C2×C714yescomplemented
115#1545C2×C711yescharacteristic
116#1487C2×C711yescharacteristic, complemented
117#1479C2×C711yescharacteristic, complemented
118C24×C7—22no—
119C2×C7×Q8—21no—
120C2×C7×Q8—21no—
121C2×C7×Q8—22no—
122C2×C7×Q8—22no—
123C2×C7×Q8—21no—
124C2×C7×Q8—21no—
125C2×C7×Q8—42no—
126C2×C7×Q8C2×D412yes—
127C2×C7×Q8C2×D412yes—
128C2×C7×D4—22no—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4—22no—
132C2×C7×D4—22no—
133C2×C7×D4—44no—
134C2×C7×D4—42no—
135C2×C7×D4—42no—
136C2×C7×D4—82no—
137C2×C7×D4C2×D414yes—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—28no—
143C22×C4×C7—84no—
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C7×Q16—44no—
148C7×SD16—84no—
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—28no—
170C7×C22⋊C4—41no—
171C42×C7—22no—
172C42×C7—22no—
173C42×C7—22no—
174C42×C7—22no—
175C42×C7—41no—
176C42×C7C2×D411yescharacteristic
177C42×C7C2×D411yescharacteristic
178C42×C7C2×D411yescharacteristic
179C42×C7C2411yescharacteristic
180#559—22no—
181#559—22no—
182#559—22no—
183#559C22×C714yes—
184#533C22×C711yescharacteristic
185C2×#127C22×C711yescharacteristic
186C2×#124—21no—
187C2×#124—21no—
188C2×#123—22no—
189C2×C4×Q8C22×C711yescharacteristic
190C2×C4×Q8C22×C711yescharacteristic
191C2×C4×D4—21no—
192C2×C4×D4—21no—
193#492C22×C712yes—
194#486C22×C712yes—
195#484—21no—
196#479—22no—
197#477C22×C712yescomplemented
198#477C22×C712yescomplemented
199#475C22×C712yescomplemented
200#462C22×C712yescomplemented
201#460—22no—
202#451—21no—
203#450—22no—
204C4×Q16—22no—
205#431C22×C711yescharacteristic
206#431C22×C711yescharacteristic
207#428—21no—
208#423C22×C711yescharacteristic
209#419—21no—
210#331C22×C714yes—
211Q8⋊C8C22×C714yescomplemented
212Q8⋊C8C22×C714yescomplemented
213C23×C7—24no—
214C23×C7—44no—
215C23×C7—44no—
216C23×C7—44no—
217C23×C7#12011yescharacteristic
218C7×Q8—22no—
219C7×Q8—22no—
220C7×Q8—22no—
221C7×Q8—22no—
222C7×Q8—42no—
223C7×Q8—42no—
224C7×Q8—42no—
225C7×Q8—42no—
226C7×D4—44no—
227C7×D4—44no—
228C7×D4—42no—
229C7×D4—42no—
230C7×D4—44no—
231C7×D4—44no—
232C7×D4—82no—
233C2×C4×C7—22no—
234C2×C4×C7—22no—
235C2×C4×C7—22no—
236C2×C4×C7—22no—
237C2×C4×C7—22no—
238C2×C4×C7—41no—
239C2×C4×C7—42no—
240C2×C4×C7—42no—
241C2×C4×C7—42no—
242C2×C4×C7—42no—
243C2×C4×C7—41no—
244C2×C4×C7—41no—
245C2×C4×C7—88no—
246C2×C4×C7#12011yescharacteristic
247C2×C4×C7#12011yescharacteristic
248C2×C4×C7#12011yescharacteristic
249C2×C4×C7#13512yes—
250C2×C4×C7#13712yes—
251C2×C4×C7C22×D411yescharacteristic
252C2×C4×C7C22×D411yescharacteristic
253C2×C4×C7C22×D411yescharacteristic
254C7×C8—42no—
255C7×C8—42no—
256C7×C8—42no—
257C22×Q8C7×D411yescharacteristic
258C22×Q8C7×D411yescharacteristic
259C22×D4—21no—
260C22×D4—21no—
261C22×D4C7×D412yes—
262C23×C4—22no—
263C2×Q16—22no—
264C2×SD16—24no—
265C2×M16—21no—
266C4⋊Q8C23×C711yescharacteristic
267C4⋊Q8C23×C711yescharacteristic
268#127—21no—
269#127—21no—
270#127C7×D412yes—
271#127C23×C712yes—
272#125—21no—
273#124—24no—
274#124—24no—
275#124—24no—
276#124—24no—
277#123—28no—
278#123—28no—
279C22⋊Q8—21no—
280C22⋊Q8—21no—
281C4×Q8—21no—
282C4×Q8—21no—
283C4×Q8—21no—
284C4×Q8—21no—
285C4×Q8—21no—
286C4×Q8—21no—
287C4×Q8C7×D412yes—
288C4×Q8C7×D412yes—
289C4×Q8C23×C712yes—
290C4×Q8C23×C712yes—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—24no—
295#117—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4C7×D411yescharacteristic
299C2×C4⋊C4C7×D411yescharacteristic
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42C23×C711yescharacteristic
306#107—22no—
307#107—22no—
308#106—22no—
309C4⋊C8—21no—
310C4⋊C8—21no—
311C4⋊C8C23×C712yes—
312C4⋊C8C23×C712yes—
313#103—22no—
314#103—22no—
315#103—22no—
316#103—22no—
317#102—24no—
318C22⋊C8—21no—
319C22⋊C8—21no—
320C4×C8—21no—
321C4×C8C23×C712yes—
322C22×C7—21no—
323C22×C7—21no—
324C22×C7—44no—
325C22×C7—44no—
326C22×C7—44no—
327C22×C7—88no—
328C22×C7#45212yes—
329C22×C7C2×#12011yescharacteristic
330C4×C7—21no—
331C4×C7—21no—
332C4×C7—21no—
333C4×C7—42no—
334C4×C7—42no—
335C4×C7—42no—
336C4×C7—42no—
337C4×C7—81no—
338C4×C7—81no—
339C4×C7#45212yes—
340C4×C7C2×#13511yescharacteristic
341C4×C7C2×#13711yescharacteristic
342C4×C7#57512yes—
343C24—22no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—42no—
351C2×Q8C2×C7×D412yes—
352C2×Q8C2×C7×D412yes—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—44no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—82no—
362C2×D4C2×C7×D414yes—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—28no—
368C22×C4—84no—
369C22×C4C2×C7×D411yescharacteristic
370C22×C4C2×C7×D411yescharacteristic
371C22×C4C2×C7×D411yescharacteristic
372Q16—44no—
373SD16—84no—
374M16—42no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4C2×C7×D412yes—
391C4⋊C4C2×C7×D412yes—
392C22⋊C4—28no—
393C22⋊C4—28no—
394C22⋊C4—28no—
395C22⋊C4—41no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—41no—
401C42C2×C7×D411yescharacteristic
402C42C2×C7×D411yescharacteristic
403C42C2×C7×D411yescharacteristic
404C42C24×C711yescharacteristic
405C2×C7—22no—
406C2×C7—84no—
407C2×C7C2×#45211yescharacteristic
408C2×C7#300511yescharacteristic
409C2×C7#301911yescharacteristic
410C23—24no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23C7×#12011yescharacteristic
415Q8—22no—
416Q8—22no—
417Q8—22no—
418Q8—22no—
419Q8—42no—
420Q8—42no—
421Q8—42no—
422Q8—42no—
423D4—44no—
424D4—44no—
425D4—42no—
426D4—42no—
427D4—44no—
428D4—44no—
429D4—82no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—88no—
443C2×C4C7×#12011yescharacteristic
444C2×C4C7×#12011yescharacteristic
445C2×C4C7×#12011yescharacteristic
446C2×C4C7×#13512yes—
447C2×C4C7×#13712yes—
448C2×C4C22×C7×D411yescharacteristic
449C2×C4C22×C7×D411yescharacteristic
450C2×C4C22×C7×D411yescharacteristic
451C8—42no—
452C8—42no—
453C8—42no—
454C7#1875111yescharacteristic, direct product factor, Sylow
455C22—21no—
456C22—21no—
457C22—44no—
458C22—44no—
459C22—44no—
460C22—88no—
461C22C7×#45212yes—
462C22C2×C7×#12011yescharacteristic
463C4—21no—
464C4—21no—
465C4—21no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—42no—
470C4—81no—
471C4—81no—
472C4C7×#45212yes—
473C4C2×C7×#13511yescharacteristic
474C4C2×C7×#13711yescharacteristic
475C4C7×#57512yes—
476C2—22no—
477C2—84no—
478C2C2×C7×#45211yescharacteristic
479C2C7×#300511yescharacteristic
480C2C7×#301911yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC2×C7×#127C22
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
3. C7×#3104C2semidirect
4. C7×#3078C2semidirect
5. C7×#1640C2semidirect
6. C7×#1630C2semidirect
8. C7×#1487C2semidirect
9. C7×#1479C2semidirect
27. C7×#477C22semidirect
28. C7×#477C22semidirect
29. C7×#475C22semidirect
30. C7×#462C22semidirect
41. C7×Q8⋊C8C22semidirect
42. C7×Q8⋊C8C22semidirect
43. #18751C7direct (N × H)
111. #3104C2×C7semidirect
112. #3078C2×C7semidirect
113. #1640C2×C7semidirect
114. #1630C2×C7semidirect
116. #1487C2×C7semidirect
117. #1479C2×C7semidirect
197. #477C22×C7semidirect
198. #477C22×C7semidirect
199. #475C22×C7semidirect
200. #462C22×C7semidirect
211. Q8⋊C8C22×C7semidirect
212. Q8⋊C8C22×C7semidirect
454. C7#18751direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5592
C7×#32082
C7×#31042
C7×#30782
C7×#16402
C7×#16302
C7×#15452
C7×#14872
C7×#14792
#187517