Finite Group Databasebeta

← C7×#9045 #13186261

Subgroups — C7×#9045 #13186261

Order 1792.

678 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#481C211yesmaximal, characteristic
2C7×#2325C211yesmaximal, characteristic
3C7×#1824C211yesmaximal, characteristic, complemented
4C2×C7×#327C211yesmaximal, characteristic
5C7×#1445C211yesmaximal, characteristic, complemented
6C7×#1294C211yesmaximal, characteristic, complemented
7C7×#1280C211yesmaximal, characteristic, complemented
8C2×C7×#121C2211yescharacteristic
9C2×C4×C7×D4—21no—
10C22×C7×C4⋊C4C2211yescharacteristic
11C7×#481—22no—
12C7×#481C414yescomplemented
13C2×C7×#106—21no—
14C2×C7×#102—21no—
15C2×C7×C22⋊C8C2211yescharacteristic
16C2×C7×C22⋊C8C2211yescharacteristic
17C7×#388—21no—
18C7×#387—21no—
19C7×#386—21no—
20C7×#386—21no—
21C7×#385—21no—
22C7×#380C2211yescharacteristic
23C2×C7×#95—21no—
24C2×C7×#95—21no—
25C2×C7×#95C2211yescharacteristic
26C2×C7×#95C2211yescharacteristic
27C7×#340—21no—
28C7×#340—21no—
29C7×#327—22no—
30#9045C711yesmaximal, characteristic, direct product factor, Sylow
31C22×C7×D4—21no—
32C22×C7×D4D411yescharacteristic
33C23×C4×C7—21no—
34C23×C4×C7—21no—
35C23×C4×C7—21no—
36C23×C4×C7D411yescharacteristic
37C23×C4×C7D411yescharacteristic
38C23×C4×C7C2311yescharacteristic
39C22×C7×C8—21no—
40C22×C7×C8—21no—
41C7×#121—21no—
42C7×#121—21no—
43C7×#121C2×C412yes—
44C7×#121D412yes—
45C4×C7×D4—42no—
46C4×C7×D4—42no—
47C2×C7×C4⋊C4—21no—
48C2×C7×C4⋊C4—21no—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4C2×C412yescomplemented
54C2×C7×C4⋊C4D411yescharacteristic
55C2×C7×C4⋊C4D411yescharacteristic
56C2×C7×C4⋊C4D411yescharacteristic
57C2×C7×C22⋊C4—21no—
58C2×C7×C22⋊C4—21no—
59C2×C7×C22⋊C4—21no—
60C2×C7×C22⋊C4—21no—
61C2×C7×C22⋊C4—21no—
62C2×C42×C7—21no—
63C7×#106—24no—
64C7×#102—24no—
65C7×C22⋊C8—21no—
66C7×C22⋊C8—21no—
67C7×C22⋊C8—21no—
68C7×C22⋊C8C2×C412yescomplemented
69C7×#95—21no—
70C7×#95—21no—
71C7×#95—21no—
72C7×#95—21no—
73C7×#95—21no—
74C7×#95—21no—
75C7×#95—21no—
76C7×#95—42no—
77C7×#95—42no—
78C2×#481C2×C711yescharacteristic
79#2325C2×C711yescharacteristic
80#1824C2×C711yescharacteristic, complemented
81C2×#327C2×C711yescharacteristic
82#1445C2×C711yescharacteristic, complemented
83#1294C2×C711yescharacteristic, complemented
84#1280C2×C711yescharacteristic, complemented
85C24×C7—21no—
86C24×C7C2×D411yescharacteristic
87C2×C7×D4—22no—
88C2×C7×D4—22no—
89C2×C7×D4—21no—
90C2×C7×D4—41no—
91C2×C7×D4—41no—
92C2×C7×D4—41no—
93C2×C7×D4—41no—
94C2×C7×D4—41no—
95C2×C7×D4—41no—
96C2×C7×D4C22⋊C412yes—
97C22×C4×C7—21no—
98C22×C4×C7—21no—
99C22×C4×C7—21no—
100C22×C4×C7—22no—
101C22×C4×C7—22no—
102C22×C4×C7—21no—
103C22×C4×C7—21no—
104C22×C4×C7—21no—
105C22×C4×C7—21no—
106C22×C4×C7—21no—
107C22×C4×C7—41no—
108C22×C4×C7—41no—
109C22×C4×C7—42no—
110C22×C4×C7—41no—
111C22×C4×C7—41no—
112C22×C4×C7—41no—
113C22×C4×C7—41no—
114C22×C4×C7—41no—
115C22×C4×C7—41no—
116C22×C4×C7—41no—
117C22×C4×C7—41no—
118C22×C4×C7—41no—
119C22×C4×C7—41no—
120C22×C4×C7—41no—
121C22×C4×C7—41no—
122C22×C4×C7—41no—
123C22×C4×C7—41no—
124C22×C4×C7—41no—
125C22×C4×C7—41no—
126C22×C4×C7—81no—
127C22×C4×C7—81no—
128C22×C4×C7SD1612yes—
129C22×C4×C7C22×C411yescharacteristic
130C22×C4×C7C2×D411yescharacteristic
131C22×C4×C7C2×D411yescharacteristic
132C22×C4×C7C2×D411yescharacteristic
133C22×C4×C7#4111yescharacteristic
134C22×C4×C7#4111yescharacteristic
135C2×C7×C8—22no—
136C2×C7×C8—41no—
137C2×C7×C8—41no—
138C2×C7×C8—41no—
139C2×C7×C8—41no—
140C2×C7×C8—41no—
141C7×C4⋊C4—21no—
142C7×C4⋊C4—22no—
143C7×C4⋊C4—22no—
144C7×C4⋊C4—41no—
145C7×C4⋊C4—41no—
146C7×C4⋊C4—42no—
147C7×C4⋊C4—41no—
148C7×C4⋊C4—41no—
149C7×C4⋊C4C22⋊C412yes—
150C7×C22⋊C4—22no—
151C7×C22⋊C4—22no—
152C7×C22⋊C4—42no—
153C7×C22⋊C4—41no—
154C7×C22⋊C4—41no—
155C7×C22⋊C4—81no—
156C7×C22⋊C4—81no—
157C42×C7—42no—
158C2×#121C22×C711yescharacteristic
159C2×C4×D4—21no—
160C22×C4⋊C4C22×C711yescharacteristic
161#481—22no—
162#481C4×C714yescomplemented
163C2×#106—21no—
164C2×#102—21no—
165C2×C22⋊C8C22×C711yescharacteristic
166C2×C22⋊C8C22×C711yescharacteristic
167#388—21no—
168#387—21no—
169#386—21no—
170#386—21no—
171#385—21no—
172#380C22×C711yescharacteristic
173C2×#95—21no—
174C2×#95—21no—
175C2×#95C22×C711yescharacteristic
176C2×#95C22×C711yescharacteristic
177#340—21no—
178#340—21no—
179#327—22no—
180C23×C7—21no—
181C23×C7—21no—
182C23×C7—21no—
183C23×C7—21no—
184C23×C7—22no—
185C23×C7—41no—
186C23×C7—41no—
187C23×C7—41no—
188C23×C7—41no—
189C23×C7—41no—
190C23×C7—41no—
191C23×C7C4×D411yescharacteristic
192C23×C7#12011yescharacteristic
193C23×C7#12111yescharacteristic
194C23×C7C2×SD1611yescharacteristic
195C23×C7#13511yescharacteristic
196C23×C7AGL(1,ℤ/8)11yescharacteristic
197C23×C7#13711yescharacteristic
198C7×D4—42no—
199C7×D4—42no—
200C7×D4—42no—
201C7×D4—42no—
202C7×D4—42no—
203C7×D4—81no—
204C2×C4×C7—21no—
205C2×C4×C7—21no—
206C2×C4×C7—21no—
207C2×C4×C7—21no—
208C2×C4×C7—22no—
209C2×C4×C7—22no—
210C2×C4×C7—41no—
211C2×C4×C7—41no—
212C2×C4×C7—41no—
213C2×C4×C7—41no—
214C2×C4×C7—41no—
215C2×C4×C7—41no—
216C2×C4×C7—41no—
217C2×C4×C7—41no—
218C2×C4×C7—42no—
219C2×C4×C7—42no—
220C2×C4×C7—42no—
221C2×C4×C7—42no—
222C2×C4×C7—42no—
223C2×C4×C7—42no—
224C2×C4×C7—42no—
225C2×C4×C7—42no—
226C2×C4×C7—41no—
227C2×C4×C7—41no—
228C2×C4×C7—41no—
229C2×C4×C7—41no—
230C2×C4×C7—41no—
231C2×C4×C7—41no—
232C2×C4×C7—41no—
233C2×C4×C7—41no—
234C2×C4×C7—41no—
235C2×C4×C7—81no—
236C2×C4×C7—84no—
237C2×C4×C7—81no—
238C2×C4×C7—81no—
239C2×C4×C7—81no—
240C2×C4×C7—81no—
241C2×C4×C7—81no—
242C2×C4×C7—81no—
243C2×C4×C7—81no—
244C2×C4×C7—81no—
245C2×C4×C7—81no—
246C2×C4×C7C2×C22⋊C411yescharacteristic
247C2×C4×C7C4×D411yescharacteristic
248C2×C4×C7#12111yescharacteristic
249C2×C4×C7#12311yescharacteristic
250C7×C8—42no—
251C7×C8—81no—
252C22×D4—21no—
253C22×D4C7×D411yescharacteristic
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4C7×D411yescharacteristic
258C23×C4C7×D411yescharacteristic
259C23×C4C23×C711yescharacteristic
260C22×C8—21no—
261C22×C8—21no—
262#121—21no—
263#121—21no—
264#121C2×C4×C712yes—
265#121C7×D412yes—
266C4×D4—42no—
267C4×D4—42no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4C2×C4×C712yescomplemented
275C2×C4⋊C4C7×D411yescharacteristic
276C2×C4⋊C4C7×D411yescharacteristic
277C2×C4⋊C4C7×D411yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C42—21no—
284#106—24no—
285#102—24no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8C2×C4×C712yescomplemented
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—42no—
298#95—42no—
299C22×C7—21no—
300C22×C7—21no—
301C22×C7—21no—
302C22×C7—21no—
303C22×C7—21no—
304C22×C7—21no—
305C22×C7—21no—
306C22×C7—21no—
307C22×C7—21no—
308C22×C7—21no—
309C22×C7—21no—
310C22×C7—21no—
311C22×C7—42no—
312C22×C7—42no—
313C22×C7—42no—
314C22×C7—81no—
315C22×C7—81no—
316C22×C7—81no—
317C22×C7—81no—
318C22×C7#38611yescharacteristic
319C22×C7C4×SD1611yescharacteristic
320C22×C7#44011yescharacteristic
321C22×C7#45011yescharacteristic
322C22×C7#45211yescharacteristic
323C22×C7#45311yescharacteristic
324C22×C7#45411yescharacteristic
325C22×C7#45811yescharacteristic
326C22×C7#45811yescharacteristic
327C22×C7#46111yescharacteristic
328C22×C7#46311yescharacteristic
329C4×C7—21no—
330C4×C7—21no—
331C4×C7—21no—
332C4×C7—21no—
333C4×C7—42no—
334C4×C7—42no—
335C4×C7—82no—
336C4×C7—81no—
337C4×C7—81no—
338C4×C7—81no—
339C4×C7—82no—
340C24—21no—
341C24C2×C7×D411yescharacteristic
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—21no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4C7×C22⋊C412yes—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—81no—
382C22×C4—81no—
383C22×C4C7×SD1612yes—
384C22×C4C22×C4×C711yescharacteristic
385C22×C4C2×C7×D411yescharacteristic
386C22×C4C2×C7×D411yescharacteristic
387C22×C4C2×C7×D411yescharacteristic
388C22×C4C7×#4111yescharacteristic
389C22×C4C7×#4111yescharacteristic
390C2×C8—22no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4C7×C22⋊C412yes—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—42no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—81no—
411C22⋊C4—81no—
412C42—42no—
413C2×C7—21no—
414C2×C7—21no—
415C2×C7—21no—
416C2×C7—21no—
417C2×C7—82no—
418C2×C7#161311yescharacteristic
419C2×C7#161511yescharacteristic
420C2×C7#161611yescharacteristic
421C2×C7#161811yescharacteristic
422C2×C7#187711yescharacteristic
423C2×C7#188311yescharacteristic
424C2×C7#189011yescharacteristic
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—22no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23C4×C7×D411yescharacteristic
437C23C7×#12011yescharacteristic
438C23C7×#12111yescharacteristic
439C23C2×C7×SD1611yescharacteristic
440C23C7×#13511yescharacteristic
441C23C7×AGL(1,ℤ/8)11yescharacteristic
442C23C7×#13711yescharacteristic
443D4—42no—
444D4—42no—
445D4—42no—
446D4—42no—
447D4—42no—
448D4—81no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—81no—
481C2×C4—84no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4C2×C7×C22⋊C411yescharacteristic
492C2×C4C4×C7×D411yescharacteristic
493C2×C4C7×#12111yescharacteristic
494C2×C4C7×#12311yescharacteristic
495C8—42no—
496C8—81no—
497C7#904511yescharacteristic, direct product factor, Sylow
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22C7×#38611yescharacteristic
518C22C4×C7×SD1611yescharacteristic
519C22C7×#44011yescharacteristic
520C22C7×#45011yescharacteristic
521C22C7×#45211yescharacteristic
522C22C7×#45311yescharacteristic
523C22C7×#45411yescharacteristic
524C22C7×#45811yescharacteristic
525C22C7×#45811yescharacteristic
526C22C7×#46111yescharacteristic
527C22C7×#46311yescharacteristic
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—42no—
533C4—42no—
534C4—82no—
535C4—81no—
536C4—81no—
537C4—81no—
538C4—82no—
539C2—21no—
540C2—21no—
541C2—21no—
542C2—21no—
543C2—82no—
544C2C7×#161311yescharacteristic
545C2C7×#161511yescharacteristic
546C2C7×#161611yescharacteristic
547C2C7×#161811yescharacteristic
548C2C7×#187711yescharacteristic
549C2C7×#188311yescharacteristic
550C2C7×#189011yescharacteristic
subgroupG/H
center, socleC23×C7#120
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC2×C7×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#1824C2semidirect
5. C7×#1445C2semidirect
6. C7×#1294C2semidirect
7. C7×#1280C2semidirect
12. C7×#481C4semidirect
30. #9045C7direct (N × H)
53. C2×C7×C4⋊C4C2×C4semidirect
68. C7×C22⋊C8C2×C4semidirect
80. #1824C2×C7semidirect
82. #1445C2×C7semidirect
83. #1294C2×C7semidirect
84. #1280C2×C7semidirect
162. #481C4×C7semidirect
274. C2×C4⋊C4C2×C4×C7semidirect
289. C22⋊C8C2×C4×C7semidirect
497. C7#9045direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4812
C7×#23252
C7×#18242
C2×C7×#3272
C7×#14452
C7×#12942
C7×#12802
#90457