Finite Group Databasebeta

← C7×#8400 #13185616

Subgroups — C7×#8400 #13185616

Order 1792.

858 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3460C211yesmaximal, characteristic
2C7×#1871C212yesmaximal, complemented
3C7×#1870C211yesmaximal, characteristic, complemented
4C4×C7×#99C211yesmaximal, characteristic, complemented
5C7×#1501C212yesmaximal, complemented
6C22×C7×M16C2211yescharacteristic
7C7×#544C411yescharacteristic, complemented
8C7×#542—22no—
9C7×#540—22no—
10C7×#538—21no—
11C7×#538C412yescomplemented
12C7×#537C411yescharacteristic, complemented
13C7×#535—21no—
14C7×#532—22no—
15C7×#529—22no—
16C7×#525—21no—
17C7×#525C2212yes—
18C4×C7×#41—21no—
19C2×C4×C7×D4C2211yescharacteristic
20C2×C7×#104—22no—
21C2×C7×#99—21no—
22C2×C7×C22⋊C8—21no—
23C2×C7×C22⋊C8C2211yescharacteristic
24C4×C7×C22⋊C4C2212yes—
25C7×#342—21no—
26C7×#339—22no—
27C7×#336—21no—
28C7×#328—22no—
29C7×#327—22no—
30#8400C711yesmaximal, characteristic, direct product factor, Sylow
31C2×C7×#41—21no—
32C2×C7×#41—21no—
33C2×C7×#41D412yes—
34C22×C7×D4D411yescharacteristic
35C23×C4×C7D411yescharacteristic
36C23×C4×C7C2311yescharacteristic
37C2×C7×M16—22no—
38C2×C7×M16—42no—
39C22×C7×C8—21no—
40C22×C7×C8—21no—
41C7×C4⋊Q8—22no—
42C7×#127—22no—
43C7×#126—22no—
44C7×#126—22no—
45C7×#125—22no—
46C7×#124—22no—
47C7×#124—22no—
48C7×#124—22no—
49C7×#123—22no—
50C7×#123—22no—
51C7×#123—22no—
52C7×#123—22no—
53C7×C22⋊Q8—21no—
54C7×C22⋊Q8—21no—
55C7×C22⋊Q8—22no—
56C7×C22⋊Q8C2×C412yes—
57C7×C22⋊Q8D412yes—
58C7×#121—21no—
59C7×#121—22no—
60C7×#121—22no—
61C7×#121—22no—
62C7×#121—22no—
63C7×#121—22no—
64C7×#121—21no—
65C7×#121C2×C412yes—
66C7×#121D412yes—
67C7×#120—22no—
68C7×#120—22no—
69C4×C7×Q8—21no—
70C4×C7×Q8—21no—
71C4×C7×D4—21no—
72C4×C7×D4—21no—
73C4×C7×D4—22no—
74C4×C7×D4—22no—
75C4×C7×D4—22no—
76C4×C7×D4—22no—
77C4×C7×D4—22no—
78C4×C7×D4—22no—
79C7×#117—22no—
80C7×#117—21no—
81C7×#117—21no—
82C7×#117D412yes—
83C2×C7×C4⋊C4C2×C411yescharacteristic
84C2×C7×C22⋊C4—22no—
85C2×C7×C22⋊C4C2×C411yescharacteristic
86C2×C7×C22⋊C4D411yescharacteristic
87C2×C42×C7—22no—
88C2×C42×C7—22no—
89C2×C42×C7D411yescharacteristic
90C7×#104—44no—
91C7×#99—24no—
92C7×C22⋊C8—22no—
93C7×C22⋊C8—42no—
94C7×#95—22no—
95#3460C2×C711yescharacteristic
96#1871C2×C712yescomplemented
97#1870C2×C711yescharacteristic, complemented
98C4×#99C2×C711yescharacteristic, complemented
99#1501C2×C712yescomplemented
100C24×C7C22⋊C411yescharacteristic
101C24×C7C2×D411yescharacteristic
102C7×#41—42no—
103C7×#41—42no—
104C7×#41—42no—
105C7×#41—42no—
106C7×#41—81no—
107C7×#41—81no—
108C2×C7×Q8—21no—
109C2×C7×Q8—21no—
110C2×C7×Q8C22⋊C412yes—
111C2×C7×D4—21no—
112C2×C7×D4—22no—
113C2×C7×D4—22no—
114C2×C7×D4—22no—
115C2×C7×D4—22no—
116C2×C7×D4—21no—
117C2×C7×D4—22no—
118C2×C7×D4—22no—
119C2×C7×D4—22no—
120C2×C7×D4—41no—
121C2×C7×D4C22⋊C412yes—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—21no—
126C22×C4×C7—21no—
127C22×C4×C7—41no—
128C22×C4×C7—41no—
129C22×C4×C7—41no—
130C22×C4×C7—41no—
131C22×C4×C7—42no—
132C22×C4×C7C22⋊C411yescharacteristic
133C22×C4×C7C22⋊C411yescharacteristic
134C22×C4×C7C22⋊C411yescharacteristic
135C22×C4×C7C22×C411yescharacteristic
136C22×C4×C7C2×D412yes—
137C22×C4×C7C2×D411yescharacteristic
138C22×C4×C7C2×D412yes—
139C7×M16—24no—
140C7×M16—42no—
141C2×C7×C8—22no—
142C2×C7×C8—41no—
143C2×C7×C8—41no—
144C2×C7×C8—41no—
145C2×C7×C8—41no—
146C2×C7×C8—41no—
147C7×C4⋊C4—21no—
148C7×C4⋊C4—21no—
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—22no—
156C7×C4⋊C4C22⋊C412yes—
157C7×C4⋊C4C22⋊C412yes—
158C7×C22⋊C4—22no—
159C7×C22⋊C4—22no—
160C7×C22⋊C4—22no—
161C7×C22⋊C4—22no—
162C7×C22⋊C4—22no—
163C7×C22⋊C4—22no—
164C7×C22⋊C4—22no—
165C7×C22⋊C4—22no—
166C7×C22⋊C4—22no—
167C7×C22⋊C4—22no—
168C7×C22⋊C4—22no—
169C7×C22⋊C4—24no—
170C7×C22⋊C4—42no—
171C42×C7—22no—
172C42×C7—22no—
173C42×C7—21no—
174C42×C7—21no—
175C42×C7—22no—
176C42×C7—42no—
177C42×C7—42no—
178C22×M16C22×C711yescharacteristic
179#544C4×C711yescharacteristic, complemented
180#542—22no—
181#540—22no—
182#538—21no—
183#538C4×C712yescomplemented
184#537C4×C711yescharacteristic, complemented
185#535—21no—
186#532—22no—
187#529—22no—
188#525—21no—
189#525C22×C712yes—
190C4×#41—21no—
191C2×C4×D4C22×C711yescharacteristic
192C2×#104—22no—
193C2×#99—21no—
194C2×C22⋊C8—21no—
195C2×C22⋊C8C22×C711yescharacteristic
196C4×C22⋊C4C22×C712yes—
197#342—21no—
198#339—22no—
199#336—21no—
200#328—22no—
201#327—22no—
202C23×C7—21no—
203C23×C7—21no—
204C23×C7—21no—
205C23×C7—22no—
206C23×C7—21no—
207C23×C7—21no—
208C23×C7—41no—
209C23×C7—41no—
210C23×C7—41no—
211C23×C7—41no—
212C23×C7C2×C22⋊C411yescharacteristic
213C23×C7#12012yes—
214C7×Q8—42no—
215C7×Q8—42no—
216C7×Q8—81no—
217C7×D4—44no—
218C7×D4—42no—
219C7×D4—42no—
220C7×D4—42no—
221C7×D4—42no—
222C7×D4—82no—
223C7×D4—82no—
224C7×D4—82no—
225C7×D4—81no—
226C7×D4—82no—
227C2×C4×C7—21no—
228C2×C4×C7—21no—
229C2×C4×C7—22no—
230C2×C4×C7—21no—
231C2×C4×C7—22no—
232C2×C4×C7—21no—
233C2×C4×C7—21no—
234C2×C4×C7—22no—
235C2×C4×C7—22no—
236C2×C4×C7—22no—
237C2×C4×C7—21no—
238C2×C4×C7—41no—
239C2×C4×C7—42no—
240C2×C4×C7—44no—
241C2×C4×C7—41no—
242C2×C4×C7—42no—
243C2×C4×C7—41no—
244C2×C4×C7—42no—
245C2×C4×C7—41no—
246C2×C4×C7—41no—
247C2×C4×C7—44no—
248C2×C4×C7—81no—
249C2×C4×C7—82no—
250C2×C4×C7—82no—
251C2×C4×C7—82no—
252C2×C4×C7#10412yescomplemented
253C2×C4×C7C2×C22⋊C412yes—
254C2×C4×C7#12011yescharacteristic
255C2×C4×C7#12011yescharacteristic
256C7×C8—42no—
257C7×C8—81no—
258C2×#41—21no—
259C2×#41—21no—
260C2×#41C7×D412yes—
261C22×D4C7×D411yescharacteristic
262C23×C4C7×D411yescharacteristic
263C23×C4C23×C711yescharacteristic
264C2×M16—22no—
265C2×M16—42no—
266C22×C8—21no—
267C22×C8—21no—
268C4⋊Q8—22no—
269#127—22no—
270#126—22no—
271#126—22no—
272#125—22no—
273#124—22no—
274#124—22no—
275#124—22no—
276#123—22no—
277#123—22no—
278#123—22no—
279#123—22no—
280C22⋊Q8—21no—
281C22⋊Q8—21no—
282C22⋊Q8—22no—
283C22⋊Q8C2×C4×C712yes—
284C22⋊Q8C7×D412yes—
285#121—21no—
286#121—22no—
287#121—22no—
288#121—22no—
289#121—22no—
290#121—22no—
291#121—21no—
292#121C2×C4×C712yes—
293#121C7×D412yes—
294#120—22no—
295#120—22no—
296C4×Q8—21no—
297C4×Q8—21no—
298C4×D4—21no—
299C4×D4—21no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—22no—
304C4×D4—22no—
305C4×D4—22no—
306#117—22no—
307#117—21no—
308#117—21no—
309#117C7×D412yes—
310C2×C4⋊C4C2×C4×C711yescharacteristic
311C2×C22⋊C4—22no—
312C2×C22⋊C4C2×C4×C711yescharacteristic
313C2×C22⋊C4C7×D411yescharacteristic
314C2×C42—22no—
315C2×C42—22no—
316C2×C42C7×D411yescharacteristic
317#104—44no—
318#99—24no—
319C22⋊C8—22no—
320C22⋊C8—42no—
321#95—22no—
322C22×C7—21no—
323C22×C7—21no—
324C22×C7—22no—
325C22×C7—22no—
326C22×C7—22no—
327C22×C7—42no—
328C22×C7—42no—
329C22×C7—41no—
330C22×C7—42no—
331C22×C7—41no—
332C22×C7—41no—
333C22×C7—42no—
334C22×C7—81no—
335C22×C7—81no—
336C22×C7—81no—
337C22×C7#37911yescharacteristic
338C22×C7C2×#10411yescharacteristic
339C22×C7#42111yescharacteristic
340C4×C7—22no—
341C4×C7—21no—
342C4×C7—22no—
343C4×C7—41no—
344C4×C7—42no—
345C4×C7—42no—
346C4×C7—41no—
347C4×C7—42no—
348C4×C7—84no—
349C4×C7B(4,𝔽2)11yescharacteristic
350C4×C7#45811yescharacteristic
351C24C7×C22⋊C411yescharacteristic
352C24C2×C7×D411yescharacteristic
353#41—42no—
354#41—42no—
355#41—42no—
356#41—42no—
357#41—81no—
358#41—81no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8C7×C22⋊C412yes—
362C2×D4—21no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—21no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—41no—
372C2×D4C7×C22⋊C412yes—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4C7×C22⋊C411yescharacteristic
384C22×C4C7×C22⋊C411yescharacteristic
385C22×C4C7×C22⋊C411yescharacteristic
386C22×C4C22×C4×C711yescharacteristic
387C22×C4C2×C7×D412yes—
388C22×C4C2×C7×D411yescharacteristic
389C22×C4C2×C7×D412yes—
390M16—24no—
391M16—42no—
392C2×C8—22no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4C7×C22⋊C412yes—
408C4⋊C4C7×C22⋊C412yes—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—24no—
421C22⋊C4—42no—
422C42—22no—
423C42—22no—
424C42—21no—
425C42—22no—
426C42—21no—
427C42—42no—
428C42—42no—
429C2×C7—21no—
430C2×C7—21no—
431C2×C7—42no—
432C2×C7—82no—
433C2×C7—81no—
434C2×C7#178911yescharacteristic
435C2×C7#179011yescharacteristic
436C2×C7#179411yescharacteristic
437C23—21no—
438C23—21no—
439C23—21no—
440C23—22no—
441C23—21no—
442C23—21no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23C2×C7×C22⋊C411yescharacteristic
448C23C7×#12012yes—
449Q8—42no—
450Q8—42no—
451Q8—81no—
452D4—44no—
453D4—42no—
454D4—42no—
455D4—42no—
456D4—42no—
457D4—82no—
458D4—82no—
459D4—82no—
460D4—81no—
461D4—82no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—44no—
483C2×C4—81no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4C7×#10412yescomplemented
488C2×C4C2×C7×C22⋊C412yes—
489C2×C4C7×#12011yescharacteristic
490C2×C4C7×#12011yescharacteristic
491C8—42no—
492C8—81no—
493C7#840011yescharacteristic, direct product factor, Sylow
494C22—21no—
495C22—21no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—42no—
500C22—42no—
501C22—41no—
502C22—41no—
503C22—42no—
504C22—42no—
505C22—41no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22C7×#37911yescharacteristic
510C22C2×C7×#10411yescharacteristic
511C22C7×#42111yescharacteristic
512C4—22no—
513C4—21no—
514C4—22no—
515C4—42no—
516C4—41no—
517C4—42no—
518C4—41no—
519C4—42no—
520C4—84no—
521C4C7×B(4,𝔽2)11yescharacteristic
522C4C7×#45811yescharacteristic
523C2—21no—
524C2—21no—
525C2—42no—
526C2—81no—
527C2—82no—
528C2C7×#178911yescharacteristic
529C2C7×#179011yescharacteristic
530C2C7×#179411yescharacteristic
subgroupG/H
centerC2×C4×C7#120
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
socleC22×C7#379
prime radicalC7×#3460C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C7×#1871C2semidirect
3. C7×#1870C2semidirect
4. C4×C7×#99C2semidirect
5. C7×#1501C2semidirect
7. C7×#544C4semidirect
11. C7×#538C4semidirect
12. C7×#537C4semidirect
30. #8400C7direct (N × H)
96. #1871C2×C7semidirect
97. #1870C2×C7semidirect
98. C4×#99C2×C7semidirect
99. #1501C2×C7semidirect
179. #544C4×C7semidirect
183. #538C4×C7semidirect
184. #537C4×C7semidirect
252. C2×C4×C7#104semidirect
487. C2×C4C7×#104semidirect
493. C7#8400direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34602
C7×#18712
C7×#18702
C4×C7×#992
C7×#15012
#84007