Finite Group Databasebeta

← C7×#8427 #13185643

Subgroups — C7×#8427 #13185643

Order 1792.

700 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
1C7×#3509C211yesmaximal, characteristic
2C7×#1988C211yesmaximal, characteristic, complemented
3C7×#1806C211yesmaximal, characteristic
4C7×#1550C211yesmaximal, characteristic, complemented
5C7×#1546C211yesmaximal, characteristic, complemented
6C7×#1503C211yesmaximal, characteristic, complemented
7C7×#1477C211yesmaximal, characteristic
8C7×#557—21no—
9C7×#557C412yescomplemented
10C7×#556—21no—
11C7×#556C412yescomplemented
12C7×#552—22no—
13C7×#552—22no—
14C7×#541—22no—
15C7×#536—21no—
16C7×#533—21no—
17C7×#533—21no—
18C7×#533C2211yescharacteristic
19C2×C7×C4⋊Q8C2211yescharacteristic
20C7×#519—21no—
21C4×C7×#41C2211yescharacteristic
22C7×#431C2211yescharacteristic
23C7×#423C2211yescharacteristic
24C2×C7×C4⋊C8C2211yescharacteristic
25C2×C7×#104—21no—
26C4×C7×C4⋊C4C2211yescharacteristic
27C7×#340—22no—
28C7×#339—21no—
29C7×#333—21no—
30C7×#332—22no—
31C7×#332—21no—
32C7×#328—22no—
33C7×Q8⋊C8—21no—
34C7×D4⋊C8—21no—
35#8427C711yesmaximal, characteristic, direct product factor, Sylow
36C2×C7×#41D411yescharacteristic
37C22×C7×Q8—21no—
38C2×C7×M16—21no—
39C22×C7×C8—21no—
40C7×C4⋊Q8—22no—
41C7×C4⋊Q8—21no—
42C7×C4⋊Q8—21no—
43C7×C4⋊Q8C2×C412yes—
44C7×C4⋊Q8D411yescharacteristic
45C7×C4⋊Q8D412yes—
46C7×C4⋊Q8D411yescharacteristic
47C7×#126—24no—
48C7×#126—24no—
49C7×#125—22no—
50C7×#125—24no—
51C7×#125—22no—
52C7×#125—24no—
53C7×#125—21no—
54C7×#124—22no—
55C7×#123—24no—
56C7×C22⋊Q8—22no—
57C7×C22⋊Q8—22no—
58C7×C22⋊Q8—22no—
59C7×C22⋊Q8—22no—
60C7×C22⋊Q8—21no—
61C7×C22⋊Q8D412yes—
62C4×C7×Q8—22no—
63C4×C7×Q8—22no—
64C4×C7×Q8—22no—
65C4×C7×Q8—22no—
66C4×C7×Q8—22no—
67C4×C7×Q8—22no—
68C4×C7×Q8—21no—
69C4×C7×Q8C2×C411yescharacteristic
70C4×C7×Q8C2×C412yes—
71C4×C7×Q8D411yescharacteristic
72C4×C7×D4—21no—
73C4×C7×D4—21no—
74C4×C7×D4C2×C411yescharacteristic
75C4×C7×D4D411yescharacteristic
76C7×#117—24no—
77C7×#117—22no—
78C7×#117—21no—
79C7×#117—22no—
80C7×#117—21no—
81C7×#117D411yescharacteristic
82C2×C7×C4⋊C4—21no—
83C2×C7×C4⋊C4—21no—
84C2×C7×C4⋊C4D412yes—
85C2×C42×C7—21no—
86C2×C42×C7—21no—
87C2×C42×C7C2311yescharacteristic
88C7×C4⋊C8—21no—
89C7×C4⋊C8—21no—
90C7×C4⋊C8—21no—
91C7×C4⋊C8—21no—
92C7×C4⋊C8—21no—
93C7×#104—42no—
94C7×C22⋊C8—21no—
95C4×C7×C8—21no—
96C7×#95—21no—
97#3509C2×C711yescharacteristic
98#1988C2×C711yescharacteristic, complemented
99#1806C2×C711yescharacteristic
100#1550C2×C711yescharacteristic, complemented
101#1546C2×C711yescharacteristic, complemented
102#1503C2×C711yescharacteristic, complemented
103#1477C2×C711yescharacteristic
104C7×#41—21no—
105C7×#41—21no—
106C7×#41—21no—
107C7×#41—21no—
108C2×C7×Q8—22no—
109C2×C7×Q8—21no—
110C2×C7×Q8—22no—
111C2×C7×Q8—41no—
112C2×C7×Q8—41no—
113C2×C7×Q8—41no—
114C2×C7×Q8—41no—
115C2×C7×Q8C22⋊C411yescharacteristic
116C2×C7×Q8C22⋊C412yes—
117C2×C7×D4—21no—
118C2×C7×D4C22⋊C411yescharacteristic
119C22×C4×C7—22no—
120C22×C4×C7—21no—
121C22×C4×C7—21no—
122C22×C4×C7—21no—
123C22×C4×C7—41no—
124C22×C4×C7C2×D411yescharacteristic
125C22×C4×C7C2×D411yescharacteristic
126C22×C4×C7C2×D411yescharacteristic
127C7×M16—41no—
128C7×M16—41no—
129C2×C7×C8—42no—
130C2×C7×C8—41no—
131C2×C7×C8—41no—
132C2×C7×C8—41no—
133C7×C4⋊C4—24no—
134C7×C4⋊C4—24no—
135C7×C4⋊C4—24no—
136C7×C4⋊C4—24no—
137C7×C4⋊C4—22no—
138C7×C4⋊C4—21no—
139C7×C4⋊C4—22no—
140C7×C4⋊C4—22no—
141C7×C4⋊C4—22no—
142C7×C4⋊C4—21no—
143C7×C4⋊C4—21no—
144C7×C4⋊C4—21no—
145C7×C4⋊C4—21no—
146C7×C4⋊C4—22no—
147C7×C4⋊C4—22no—
148C7×C4⋊C4—41no—
149C7×C4⋊C4C22⋊C414yes—
150C7×C4⋊C4C22⋊C412yes—
151C7×C4⋊C4C22⋊C411yescharacteristic
152C7×C4⋊C4C22⋊C411yescharacteristic
153C7×C22⋊C4—24no—
154C7×C22⋊C4—24no—
155C7×C22⋊C4—21no—
156C7×C22⋊C4—22no—
157C7×C22⋊C4—21no—
158C7×C22⋊C4—21no—
159C42×C7—24no—
160C42×C7—24no—
161C42×C7—21no—
162C42×C7—22no—
163C42×C7—21no—
164C42×C7—21no—
165C42×C7—41no—
166C42×C7—41no—
167C42×C7C22×C411yescharacteristic
168C42×C7C2×D411yescharacteristic
169C42×C7C2×D411yescharacteristic
170C42×C7C2×D411yescharacteristic
171#557—21no—
172#557C4×C712yescomplemented
173#556—21no—
174#556C4×C712yescomplemented
175#552—22no—
176#552—22no—
177#541—22no—
178#536—21no—
179#533—21no—
180#533—21no—
181#533C22×C711yescharacteristic
182C2×C4⋊Q8C22×C711yescharacteristic
183#519—21no—
184C4×#41C22×C711yescharacteristic
185#431C22×C711yescharacteristic
186#423C22×C711yescharacteristic
187C2×C4⋊C8C22×C711yescharacteristic
188C2×#104—21no—
189C4×C4⋊C4C22×C711yescharacteristic
190#340—22no—
191#339—21no—
192#333—21no—
193#332—22no—
194#332—21no—
195#328—22no—
196Q8⋊C8—21no—
197D4⋊C8—21no—
198C23×C7—21no—
199C23×C7#12011yescharacteristic
200C7×Q8—21no—
201C7×Q8—21no—
202C7×Q8—42no—
203C7×Q8—42no—
204C7×Q8—42no—
205C7×Q8—42no—
206C7×Q8—42no—
207C7×Q8—41no—
208C7×Q8—41no—
209C7×D4—21no—
210C7×D4—21no—
211C7×D4—41no—
212C7×D4—41no—
213C2×C4×C7—21no—
214C2×C4×C7—21no—
215C2×C4×C7—21no—
216C2×C4×C7—21no—
217C2×C4×C7—22no—
218C2×C4×C7—22no—
219C2×C4×C7—24no—
220C2×C4×C7—21no—
221C2×C4×C7—41no—
222C2×C4×C7—42no—
223C2×C4×C7—42no—
224C2×C4×C7—42no—
225C2×C4×C7—41no—
226C2×C4×C7—41no—
227C2×C4×C7—41no—
228C2×C4×C7—41no—
229C2×C4×C7—42no—
230C2×C4×C7—81no—
231C2×C4×C7—81no—
232C2×C4×C7C2×C22⋊C411yescharacteristic
233C2×C4×C7C2×C22⋊C411yescharacteristic
234C2×C4×C7C2×C22⋊C411yescharacteristic
235C2×C4×C7#12011yescharacteristic
236C2×C4×C7#12011yescharacteristic
237C2×C4×C7#12011yescharacteristic
238C2×C4×C7#13511yescharacteristic
239C2×C4×C7#13511yescharacteristic
240C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
241C2×C4×C7#13711yescharacteristic
242C7×C8—81no—
243C7×C8—81no—
244C7×C8—81no—
245C2×#41C7×D411yescharacteristic
246C22×Q8—21no—
247C2×M16—21no—
248C22×C8—21no—
249C4⋊Q8—22no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8C2×C4×C712yes—
253C4⋊Q8C7×D411yescharacteristic
254C4⋊Q8C7×D412yes—
255C4⋊Q8C7×D411yescharacteristic
256#126—24no—
257#126—24no—
258#125—22no—
259#125—24no—
260#125—22no—
261#125—24no—
262#125—21no—
263#124—22no—
264#123—24no—
265C22⋊Q8—22no—
266C22⋊Q8—22no—
267C22⋊Q8—22no—
268C22⋊Q8—22no—
269C22⋊Q8—21no—
270C22⋊Q8C7×D412yes—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×Q8—21no—
278C4×Q8C2×C4×C711yescharacteristic
279C4×Q8C2×C4×C712yes—
280C4×Q8C7×D411yescharacteristic
281C4×D4—21no—
282C4×D4—21no—
283C4×D4C2×C4×C711yescharacteristic
284C4×D4C7×D411yescharacteristic
285#117—24no—
286#117—22no—
287#117—21no—
288#117—22no—
289#117—21no—
290#117C7×D411yescharacteristic
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4C7×D412yes—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42C23×C711yescharacteristic
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8—21no—
301C4⋊C8—21no—
302#104—42no—
303C22⋊C8—21no—
304C4×C8—21no—
305#95—21no—
306C22×C7—21no—
307C22×C7—21no—
308C22×C7—41no—
309C22×C7—41no—
310C22×C7—41no—
311C22×C7#37911yescharacteristic
312C22×C7#44911yescharacteristic
313C22×C7#45211yescharacteristic
314C4×C7—21no—
315C4×C7—21no—
316C4×C7—21no—
317C4×C7—41no—
318C4×C7—42no—
319C4×C7—41no—
320C4×C7—42no—
321C4×C7—41no—
322C4×C7—44no—
323C4×C7—82no—
324C4×C7#41611yescharacteristic
325C4×C7#41711yescharacteristic
326C4×C7#44911yescharacteristic
327C4×C7#45211yescharacteristic
328C4×C7#45411yescharacteristic
329C4×C7#45611yescharacteristic
330#41—21no—
331#41—21no—
332#41—21no—
333#41—21no—
334C2×Q8—22no—
335C2×Q8—21no—
336C2×Q8—22no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8C7×C22⋊C411yescharacteristic
342C2×Q8C7×C22⋊C412yes—
343C2×D4—21no—
344C2×D4C7×C22⋊C411yescharacteristic
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—41no—
350C22×C4C2×C7×D411yescharacteristic
351C22×C4C2×C7×D411yescharacteristic
352C22×C4C2×C7×D411yescharacteristic
353M16—41no—
354M16—41no—
355C2×C8—42no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—21no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—41no—
375C4⋊C4C7×C22⋊C414yes—
376C4⋊C4C7×C22⋊C412yes—
377C4⋊C4C7×C22⋊C411yescharacteristic
378C4⋊C4C7×C22⋊C411yescharacteristic
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—21no—
382C22⋊C4—22no—
383C22⋊C4—21no—
384C22⋊C4—21no—
385C42—24no—
386C42—24no—
387C42—21no—
388C42—22no—
389C42—21no—
390C42—21no—
391C42—41no—
392C42—41no—
393C42C22×C4×C711yescharacteristic
394C42C2×C7×D411yescharacteristic
395C42C2×C7×D411yescharacteristic
396C42C2×C7×D411yescharacteristic
397C2×C7—21no—
398C2×C7—21no—
399C2×C7—81no—
400C2×C7#178811yescharacteristic
401C2×C7#179311yescharacteristic
402C2×C7#179611yescharacteristic
403C23—21no—
404C23C7×#12011yescharacteristic
405Q8—21no—
406Q8—21no—
407Q8—42no—
408Q8—42no—
409Q8—42no—
410Q8—42no—
411Q8—42no—
412Q8—41no—
413Q8—41no—
414D4—21no—
415D4—21no—
416D4—41no—
417D4—41no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—21no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4C2×C7×C22⋊C411yescharacteristic
438C2×C4C2×C7×C22⋊C411yescharacteristic
439C2×C4C2×C7×C22⋊C411yescharacteristic
440C2×C4C7×#12011yescharacteristic
441C2×C4C7×#12011yescharacteristic
442C2×C4C7×#12011yescharacteristic
443C2×C4C7×#13511yescharacteristic
444C2×C4C7×#13511yescharacteristic
445C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
446C2×C4C7×#13711yescharacteristic
447C8—81no—
448C8—81no—
449C8—81no—
450C7#842711yescharacteristic, direct product factor, Sylow
451C22—21no—
452C22—21no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22C7×#37911yescharacteristic
457C22C7×#44911yescharacteristic
458C22C7×#45211yescharacteristic
459C4—21no—
460C4—21no—
461C4—21no—
462C4—41no—
463C4—44no—
464C4—41no—
465C4—42no—
466C4—41no—
467C4—42no—
468C4—82no—
469C4C7×#41611yescharacteristic
470C4C7×#41711yescharacteristic
471C4C7×#44911yescharacteristic
472C4C7×#45211yescharacteristic
473C4C7×#45411yescharacteristic
474C4C7×#45611yescharacteristic
475C2—21no—
476C2—21no—
477C2—81no—
478C2C7×#178811yescharacteristic
479C2C7×#179311yescharacteristic
480C2C7×#179611yescharacteristic
subgroupG/H
center, socleC22×C7#379
commutatorC42C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC2×C7×#41D4
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×#1988C2semidirect
4. C7×#1550C2semidirect
5. C7×#1546C2semidirect
6. C7×#1503C2semidirect
9. C7×#557C4semidirect
11. C7×#556C4semidirect
35. #8427C7direct (N × H)
98. #1988C2×C7semidirect
100. #1550C2×C7semidirect
101. #1546C2×C7semidirect
102. #1503C2×C7semidirect
172. #557C4×C7semidirect
174. #556C4×C7semidirect
450. C7#8427direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35092
C7×#19882
C7×#18062
C7×#15502
C7×#15462
C7×#15032
C7×#14772
#84277