Finite Group Databasebeta

← C7×#23643 #13200859

Subgroups — C7×#23643 #13200859

Order 1792.

1014 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×#428C211yesmaximal, characteristic, complemented
2C2×C7×#408C211yesmaximal, characteristic
3C7×#2311C211yesmaximal, characteristic
4C7×#2311C211yesmaximal, characteristic
5C7×#2077C212yesmaximal, complemented
6C7×#2077C212yesmaximal, complemented
7C7×#2076C212yesmaximal, complemented
8C7×#2075C212yesmaximal, complemented
9C7×#1736C211yesmaximal, characteristic, complemented
10C7×#1735C211yesmaximal, characteristic, complemented
11C2×C7×#340C211yesmaximal, characteristic, complemented
12C22×C7×#41C2211yescharacteristic
13C22×C7×M16C2211yescharacteristic
14C23×C7×C8C2211yescharacteristic
15C2×C7×#117C2211yescharacteristic
16C2×C7×#117C2211yescharacteristic
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×C7×C4⋊C4C2211yescharacteristic
19C7×#428—24no—
20C2×C7×#107C2212yes—
21C2×C7×#106C2212yes—
22C7×#408—24no—
23C2×C7×C22⋊C8C2212yes—
24C2×C7×C22⋊C8C2212yes—
25C7×#391—21no—
26C7×#391—21no—
27C7×#390—21no—
28C7×#390—21no—
29C7×#386—22no—
30C7×#386—22no—
31C7×#380C2212yes—
32C7×#380C2212yes—
33C7×#380C2212yes—
34C7×#380C2212yes—
35C7×#340C2212yes—
36C7×#340C2212yes—
37C7×#340C2211yescharacteristic
38C7×#340C2211yescharacteristic
39C7×#340C2211yescharacteristic
40C7×#340C2211yescharacteristic
41C7×#340C2212yes—
42C7×#340C2212yes—
43#23643C711yesmaximal, characteristic, direct product factor, Sylow
44C2×C7×#41—22no—
45C2×C7×#41—22no—
46C2×C7×#41—22no—
47C2×C7×#41—22no—
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C22×C7×Q8D411yescharacteristic
51C22×C7×D4D412yes—
52C22×C7×D4D411yescharacteristic
53C23×C4×C7—21no—
54C23×C4×C7—21no—
55C23×C4×C7C2312yes—
56C23×C4×C7C2311yescharacteristic
57C2×C7×M16—22no—
58C2×C7×M16—22no—
59C2×C7×M16D414yes—
60C22×C7×C8—22no—
61C22×C7×C8—22no—
62C22×C7×C8—22no—
63C22×C7×C8C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2312yes—
66C7×#117—22no—
67C7×#117—22no—
68C7×#117—22no—
69C7×#117—22no—
70C2×C7×C4⋊C4—22no—
71C2×C7×C4⋊C4—22no—
72C2×C7×C4⋊C4—22no—
73C2×C7×C4⋊C4—22no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4C2312yes—
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C4⋊C4C2312yes—
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C4⋊C4C2311yescharacteristic
82C2×C7×C22⋊C4—21no—
83C2×C7×C22⋊C4—21no—
84C2×C7×C22⋊C4—22no—
85C2×C7×C22⋊C4—22no—
86C2×C7×C22⋊C4—22no—
87C2×C7×C22⋊C4—22no—
88C2×C42×C7—21no—
89C2×C42×C7—21no—
90C7×#107—24no—
91C7×#106—24no—
92C7×C22⋊C8—24no—
93C7×C22⋊C8—24no—
94C7×#95—22no—
95C7×#95—22no—
96C7×#95—22no—
97C7×#95—22no—
98C2×#428C2×C711yescharacteristic, complemented
99C2×#408C2×C711yescharacteristic
100#2311C2×C711yescharacteristic
101#2311C2×C711yescharacteristic
102#2077C2×C712yescomplemented
103#2077C2×C712yescomplemented
104#2076C2×C712yescomplemented
105#2075C2×C712yescomplemented
106#1736C2×C711yescharacteristic, complemented
107#1735C2×C711yescharacteristic, complemented
108C2×#340C2×C711yescharacteristic, complemented
109C24×C7C2×D412yes—
110C24×C7C2×D411yescharacteristic
111C7×#41—44no—
112C7×#41—44no—
113C7×#41—44no—
114C7×#41—44no—
115C2×C7×Q8—22no—
116C2×C7×Q8—22no—
117C2×C7×Q8—22no—
118C2×C7×D4—24no—
119C2×C7×D4—24no—
120C2×C7×D4—24no—
121C2×C7×D4—22no—
122C2×C7×D4—22no—
123C2×C7×D4—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—21no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—42no—
137C22×C4×C7—42no—
138C22×C4×C7—81no—
139C22×C4×C7—81no—
140C22×C4×C7—81no—
141C22×C4×C7—81no—
142C22×C4×C7C2×D412yes—
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D412yes—
145C22×C4×C7#4112yes—
146C22×C4×C7#4112yes—
147C22×C4×C7#4112yes—
148C22×C4×C7#4112yes—
149C22×C4×C7#4112yes—
150C22×C4×C7C2411yescharacteristic
151C7×M16—28no—
152C2×C7×C8—22no—
153C2×C7×C8—22no—
154C2×C7×C8—21no—
155C2×C7×C8—21no—
156C2×C7×C8—21no—
157C2×C7×C8—21no—
158C2×C7×C8—21no—
159C2×C7×C8—21no—
160C2×C7×C8—44no—
161C2×C7×C8C2×D414yes—
162C7×C4⋊C4—21no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—22no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—42no—
175C7×C4⋊C4—42no—
176C7×C22⋊C4—42no—
177C7×C22⋊C4—42no—
178C7×C22⋊C4—82no—
179C7×C22⋊C4—82no—
180C7×C22⋊C4—82no—
181C7×C22⋊C4—82no—
182C42×C7—42no—
183C42×C7—42no—
184C22×#41C22×C711yescharacteristic
185C22×M16C22×C711yescharacteristic
186C23×C8C22×C711yescharacteristic
187C2×#117C22×C711yescharacteristic
188C2×#117C22×C711yescharacteristic
189C22×C4⋊C4C22×C711yescharacteristic
190C22×C4⋊C4C22×C711yescharacteristic
191#428—24no—
192C2×#107C22×C712yes—
193C2×#106C22×C712yes—
194#408—24no—
195C2×C22⋊C8C22×C712yes—
196C2×C22⋊C8C22×C712yes—
197#391—21no—
198#391—21no—
199#390—21no—
200#390—21no—
201#386—22no—
202#386—22no—
203#380C22×C712yes—
204#380C22×C712yes—
205#380C22×C712yes—
206#380C22×C712yes—
207#340C22×C711yescharacteristic
208#340C22×C712yes—
209#340C22×C712yes—
210#340C22×C711yescharacteristic
211#340C22×C712yes—
212#340C22×C712yes—
213#340C22×C711yescharacteristic
214#340C22×C711yescharacteristic
215C23×C7—21no—
216C23×C7—21no—
217C23×C7—21no—
218C23×C7—21no—
219C23×C7—22no—
220C23×C7—22no—
221C23×C7—22no—
222C23×C7—22no—
223C23×C7—22no—
224C23×C7—22no—
225C23×C7—22no—
226C23×C7#12112yes—
227C23×C7#12312yes—
228C23×C7#12312yes—
229C23×C7C22×D411yescharacteristic
230C7×Q8—44no—
231C7×D4—48no—
232C7×D4—44no—
233C2×C4×C7—22no—
234C2×C4×C7—22no—
235C2×C4×C7—22no—
236C2×C4×C7—42no—
237C2×C4×C7—42no—
238C2×C4×C7—42no—
239C2×C4×C7—42no—
240C2×C4×C7—41no—
241C2×C4×C7—41no—
242C2×C4×C7—42no—
243C2×C4×C7—41no—
244C2×C4×C7—41no—
245C2×C4×C7—42no—
246C2×C4×C7—41no—
247C2×C4×C7—41no—
248C2×C4×C7—42no—
249C2×C4×C7—42no—
250C2×C4×C7—41no—
251C2×C4×C7—41no—
252C2×C4×C7—42no—
253C2×C4×C7—42no—
254C2×C4×C7—41no—
255C2×C4×C7—41no—
256C2×C4×C7—41no—
257C2×C4×C7—41no—
258C2×C4×C7—84no—
259C2×C4×C7—84no—
260C2×C4×C7#12112yes—
261C2×C4×C7#12314yes—
262C2×C4×C7#12312yes—
263C2×C4×C7#12312yes—
264C2×C4×C7#12414yes—
265C2×C4×C7#12614yes—
266C2×C4×C7#12614yes—
267C2×C4×C7C22×D411yescharacteristic
268C2×C4×C7C2×#4111yescharacteristic
269C2×C4×C7C2×#4111yescharacteristic
270C2×C4×C7C2×#4111yescharacteristic
271C2×C4×C7C2×#4111yescharacteristic
272C2×C4×C7C2×#4111yescharacteristic
273C7×C8—24no—
274C7×C8—41no—
275C7×C8—41no—
276C2×#41—22no—
277C2×#41—22no—
278C2×#41—22no—
279C2×#41—22no—
280C2×#41—22no—
281C2×#41—22no—
282C22×Q8C7×D411yescharacteristic
283C22×D4C7×D412yes—
284C22×D4C7×D411yescharacteristic
285C23×C4—21no—
286C23×C4—21no—
287C23×C4C23×C712yes—
288C23×C4C23×C711yescharacteristic
289C2×M16—22no—
290C2×M16—22no—
291C2×M16C7×D414yes—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8—22no—
295C22×C8C23×C711yescharacteristic
296C22×C8C23×C711yescharacteristic
297C22×C8C23×C712yes—
298#117—22no—
299#117—22no—
300#117—22no—
301#117—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4C23×C712yes—
309C2×C4⋊C4C23×C712yes—
310C2×C4⋊C4C23×C711yescharacteristic
311C2×C4⋊C4C23×C711yescharacteristic
312C2×C4⋊C4C23×C711yescharacteristic
313C2×C4⋊C4C23×C711yescharacteristic
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C42—21no—
321C2×C42—21no—
322#107—24no—
323#106—24no—
324C22⋊C8—24no—
325C22⋊C8—24no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330C22×C7—22no—
331C22×C7—22no—
332C22×C7—22no—
333C22×C7—22no—
334C22×C7—22no—
335C22×C7—22no—
336C22×C7—42no—
337C22×C7—42no—
338C22×C7—42no—
339C22×C7—42no—
340C22×C7—42no—
341C22×C7—42no—
342C22×C7—42no—
343C22×C7#39714yes—
344C22×C7C2×#12111yescharacteristic
345C22×C7C2×#12311yescharacteristic
346C22×C7C2×#12311yescharacteristic
347C22×C7#57611yescharacteristic
348C22×C7#57711yescharacteristic
349C22×C7#57711yescharacteristic
350C22×C7#57811yescharacteristic
351C4×C7—42no—
352C4×C7—82no—
353C4×C7—81no—
354C4×C7—81no—
355C4×C7—81no—
356C4×C7—82no—
357C4×C7—81no—
358C4×C7#39714yes—
359C4×C7C2×#12311yescharacteristic
360C4×C7C2×#12411yescharacteristic
361C4×C7C2×#12611yescharacteristic
362C4×C7C2×#12611yescharacteristic
363C24C2×C7×D412yes—
364C24C2×C7×D411yescharacteristic
365#41—44no—
366#41—44no—
367#41—44no—
368#41—44no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—81no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4—81no—
396C22×C4C2×C7×D412yes—
397C22×C4C2×C7×D411yescharacteristic
398C22×C4C2×C7×D412yes—
399C22×C4C7×#4112yes—
400C22×C4C7×#4112yes—
401C22×C4C7×#4112yes—
402C22×C4C7×#4112yes—
403C22×C4C7×#4112yes—
404C22×C4C24×C711yescharacteristic
405M16—28no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—44no—
415C2×C8C2×C7×D414yes—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—82no—
433C22⋊C4—82no—
434C22⋊C4—82no—
435C22⋊C4—82no—
436C42—42no—
437C42—42no—
438C2×C7—24no—
439C2×C7—82no—
440C2×C7C2×#39711yescharacteristic
441C2×C7#306211yescharacteristic
442C2×C7#306311yescharacteristic
443C2×C7#309611yescharacteristic
444C2×C7#309711yescharacteristic
445C2×C7#309811yescharacteristic
446C2×C7#309911yescharacteristic
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23C7×#12112yes—
459C23C7×#12312yes—
460C23C7×#12312yes—
461C23C22×C7×D411yescharacteristic
462Q8—44no—
463D4—48no—
464D4—44no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—84no—
491C2×C4—84no—
492C2×C4C7×#12112yes—
493C2×C4C7×#12314yes—
494C2×C4C7×#12312yes—
495C2×C4C7×#12312yes—
496C2×C4C7×#12414yes—
497C2×C4C7×#12614yes—
498C2×C4C7×#12614yes—
499C2×C4C22×C7×D411yescharacteristic
500C2×C4C2×C7×#4111yescharacteristic
501C2×C4C2×C7×#4111yescharacteristic
502C2×C4C2×C7×#4111yescharacteristic
503C2×C4C2×C7×#4111yescharacteristic
504C2×C4C2×C7×#4111yescharacteristic
505C8—24no—
506C8—41no—
507C8—41no—
508C7#2364311yescharacteristic, direct product factor, Sylow
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22C7×#39714yes—
523C22C2×C7×#12111yescharacteristic
524C22C2×C7×#12311yescharacteristic
525C22C2×C7×#12311yescharacteristic
526C22C7×#57611yescharacteristic
527C22C7×#57711yescharacteristic
528C22C7×#57711yescharacteristic
529C22C7×#57811yescharacteristic
530C4—42no—
531C4—82no—
532C4—81no—
533C4—81no—
534C4—82no—
535C4—81no—
536C4—81no—
537C4C7×#39714yes—
538C4C2×C7×#12311yescharacteristic
539C4C2×C7×#12411yescharacteristic
540C4C2×C7×#12611yescharacteristic
541C4C2×C7×#12611yescharacteristic
542C2—24no—
543C2—82no—
544C2C2×C7×#39711yescharacteristic
545C2C7×#306211yescharacteristic
546C2C7×#306311yescharacteristic
547C2C7×#309611yescharacteristic
548C2C7×#309711yescharacteristic
549C2C7×#309811yescharacteristic
550C2C7×#309911yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC22×C7×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C22×C7×#41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#428C2semidirect
5. C7×#2077C2semidirect
6. C7×#2077C2semidirect
7. C7×#2076C2semidirect
8. C7×#2075C2semidirect
9. C7×#1736C2semidirect
10. C7×#1735C2semidirect
11. C2×C7×#340C2semidirect
43. #23643C7direct (N × H)
98. C2×#428C2×C7semidirect
102. #2077C2×C7semidirect
103. #2077C2×C7semidirect
104. #2076C2×C7semidirect
105. #2075C2×C7semidirect
106. #1736C2×C7semidirect
107. #1735C2×C7semidirect
108. C2×#340C2×C7semidirect
508. C7#23643direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4282
C2×C7×#4082
C7×#23112
C7×#20772
C7×#20762
C7×#20752
C7×#17362
C7×#17352
C2×C7×#3402
#236437