Finite Group Databasebeta

← C7×#23510 #13200726

Subgroups — C7×#23510 #13200726

Order 1792.

1414 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#536C211yesmaximal, characteristic
2C2×C7×#417C211yesmaximal, characteristic
3C2×C7×#419C211yesmaximal, characteristic, complemented
4C2×C7×#416C211yesmaximal, characteristic
5C22×C7×#103C211yesmaximal, characteristic, complemented
6C7×#2317C211yesmaximal, characteristic, complemented
7C7×#2043C214yesmaximal, complemented
8C7×#2042C214yesmaximal, complemented
9C7×#1729C211yesmaximal, characteristic, complemented
10C22×C7×#41C2211yescharacteristic
11C23×C7×Q8C2211yescharacteristic
12C22×C7×M16C2211yescharacteristic
13C23×C7×C8C2211yescharacteristic
14C7×#536—24no—
15C7×#536—24no—
16C2×C7×C4⋊Q8C2212yes—
17C2×C7×#124C2212yes—
18C2×C7×#123—22no—
19C2×C7×C22⋊Q8—21no—
20C2×C7×C22⋊Q8—21no—
21C2×C7×#117C2211yescharacteristic
22C2×C7×#117C2211yescharacteristic
23C22×C7×C4⋊C4C2211yescharacteristic
24C7×#419—24no—
25C7×#417—24no—
26C7×#416—24no—
27C2×C7×#103—24no—
28C2×C7×#103C2214yes—
29C2×C7×#103C2214yes—
30C2×C7×#103C2212yes—
31C2×C7×#103C2212yes—
32C2×C7×#102C2212yes—
33C2×C7×#102C2212yes—
34C7×#389C2214yes—
35C7×#385—22no—
36C7×#384—21no—
37C7×#382—21no—
38C7×#336C2214yes—
39#23510C711yesmaximal, characteristic, direct product factor, Sylow
40C2×C7×#41—22no—
41C2×C7×#41—22no—
42C2×C7×#41—22no—
43C2×C7×#41—22no—
44C2×C7×#41—22no—
45C2×C7×#41D414yes—
46C22×C7×Q8—21no—
47C22×C7×Q8—22no—
48C22×C7×Q8—22no—
49C22×C7×Q8—22no—
50C22×C7×Q8—22no—
51C22×C7×Q8—22no—
52C22×C7×Q8D414yes—
53C22×C7×Q8C2312yes—
54C22×C7×Q8C2311yescharacteristic
55C22×C7×D4—21no—
56C22×C7×D4C2311yescharacteristic
57C23×C4×C7—21no—
58C23×C4×C7—21no—
59C23×C4×C7—21no—
60C23×C4×C7C2311yescharacteristic
61C2×C7×M16—22no—
62C2×C7×M16—22no—
63C2×C7×M16—22no—
64C22×C7×C8—22no—
65C22×C7×C8—22no—
66C22×C7×C8—22no—
67C22×C7×C8C2312yes—
68C22×C7×C8C2312yes—
69C7×C4⋊Q8—24no—
70C7×C4⋊Q8—24no—
71C7×#124—24no—
72C7×#124—24no—
73C7×#123—216no—
74C7×C22⋊Q8—28no—
75C7×C22⋊Q8—28no—
76C7×#117—22no—
77C7×#117—22no—
78C7×#117—22no—
79C7×#117D414yes—
80C2×C7×C4⋊C4—22no—
81C2×C7×C4⋊C4—22no—
82C2×C7×C4⋊C4—22no—
83C2×C7×C4⋊C4—22no—
84C2×C7×C4⋊C4—21no—
85C2×C7×C4⋊C4—24no—
86C2×C7×C4⋊C4D411yescharacteristic
87C2×C7×C4⋊C4D411yescharacteristic
88C2×C7×C4⋊C4C2312yes—
89C2×C7×C4⋊C4C2312yes—
90C2×C7×C22⋊C4—21no—
91C2×C7×C22⋊C4—22no—
92C2×C7×C22⋊C4—22no—
93C2×C7×C22⋊C4D412yes—
94C2×C42×C7—21no—
95C2×C42×C7C2312yes—
96C7×#103—28no—
97C7×#103—24no—
98C7×#103—24no—
99C7×#103—28no—
100C7×#102—24no—
101C7×#102—24no—
102C7×#95—24no—
103C2×#536C2×C711yescharacteristic
104C2×#417C2×C711yescharacteristic
105C2×#419C2×C711yescharacteristic, complemented
106C2×#416C2×C711yescharacteristic
107C22×#103C2×C711yescharacteristic, complemented
108#2317C2×C711yescharacteristic, complemented
109#2043C2×C714yescomplemented
110#2042C2×C714yescomplemented
111#1729C2×C711yescharacteristic, complemented
112C24×C7—21no—
113C24×C7C2×D411yescharacteristic
114C7×#41—28no—
115C7×#41—28no—
116C7×#41—44no—
117C7×#41—44no—
118C2×C7×Q8—24no—
119C2×C7×Q8—22no—
120C2×C7×Q8—21no—
121C2×C7×Q8—28no—
122C2×C7×Q8—28no—
123C2×C7×Q8—22no—
124C2×C7×Q8—21no—
125C2×C7×Q8—22no—
126C2×C7×Q8—22no—
127C2×C7×Q8—22no—
128C2×C7×Q8—21no—
129C2×C7×Q8—21no—
130C2×C7×Q8—21no—
131C2×C7×Q8—42no—
132C2×C7×Q8—44no—
133C2×C7×Q8—44no—
134C2×C7×Q8—42no—
135C2×C7×Q8C2×D414yes—
136C2×C7×Q8C2×D412yes—
137C2×C7×D4—24no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4—21no—
141C2×C7×D4—21no—
142C2×C7×D4—21no—
143C2×C7×D4—42no—
144C2×C7×D4—42no—
145C2×C7×D4C2×D412yes—
146C22×C4×C7—24no—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7—21no—
150C22×C4×C7—24no—
151C22×C4×C7—22no—
152C22×C4×C7—42no—
153C22×C4×C7—42no—
154C22×C4×C7—42no—
155C22×C4×C7—42no—
156C22×C4×C7—42no—
157C22×C4×C7—82no—
158C22×C4×C7C2×D412yes—
159C22×C4×C7C2×D412yes—
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D412yes—
162C22×C4×C7C2×D412yes—
163C22×C4×C7C2×D412yes—
164C22×C4×C7#4112yes—
165C22×C4×C7#4112yes—
166C22×C4×C7#4112yes—
167C22×C4×C7C2411yescharacteristic
168C7×M16—44no—
169C2×C7×C8—22no—
170C2×C7×C8—22no—
171C2×C7×C8—22no—
172C2×C7×C8—22no—
173C2×C7×C8—22no—
174C2×C7×C8—22no—
175C2×C7×C8—44no—
176C7×C4⋊C4—28no—
177C7×C4⋊C4—28no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—22no—
181C7×C4⋊C4—22no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—42no—
184C7×C4⋊C4—84no—
185C7×C4⋊C4C2×D414yes—
186C7×C22⋊C4—28no—
187C7×C22⋊C4—28no—
188C7×C22⋊C4—24no—
189C7×C22⋊C4—24no—
190C42×C7—24no—
191C42×C7—22no—
192C42×C7—22no—
193C22×#41C22×C711yescharacteristic
194C23×Q8C22×C711yescharacteristic
195C22×M16C22×C711yescharacteristic
196C23×C8C22×C711yescharacteristic
197#536—24no—
198#536—24no—
199C2×C4⋊Q8C22×C712yes—
200C2×#124C22×C712yes—
201C2×#123—22no—
202C2×C22⋊Q8—21no—
203C2×C22⋊Q8—21no—
204C2×#117C22×C711yescharacteristic
205C2×#117C22×C711yescharacteristic
206C22×C4⋊C4C22×C711yescharacteristic
207#419—24no—
208#417—24no—
209#416—24no—
210C2×#103—24no—
211C2×#103C22×C714yes—
212C2×#103C22×C712yes—
213C2×#103C22×C714yes—
214C2×#103C22×C712yes—
215C2×#102C22×C712yes—
216C2×#102C22×C712yes—
217#389C22×C714yes—
218#385—22no—
219#384—21no—
220#382—21no—
221#336C22×C714yes—
222C23×C7—22no—
223C23×C7—22no—
224C23×C7—22no—
225C23×C7—41no—
226C23×C7—41no—
227C23×C7—42no—
228C23×C7—42no—
229C23×C7#12012yes—
230C23×C7#12112yes—
231C23×C7#12412yes—
232C23×C7C22×D411yescharacteristic
233C7×Q8—24no—
234C7×Q8—22no—
235C7×Q8—24no—
236C7×Q8—22no—
237C7×Q8—42no—
238C7×Q8—41no—
239C7×Q8—42no—
240C7×Q8—41no—
241C7×Q8—48no—
242C7×D4—22no—
243C7×D4—22no—
244C7×D4—48no—
245C7×D4—41no—
246C7×D4—41no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—24no—
250C2×C4×C7—22no—
251C2×C4×C7—24no—
252C2×C4×C7—24no—
253C2×C4×C7—24no—
254C2×C4×C7—41no—
255C2×C4×C7—42no—
256C2×C4×C7—41no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—48no—
261C2×C4×C7—48no—
262C2×C4×C7—42no—
263C2×C4×C7—44no—
264C2×C4×C7—84no—
265C2×C4×C7#12012yes—
266C2×C4×C7#12114yes—
267C2×C4×C7#12114yes—
268C2×C4×C7#12112yes—
269C2×C4×C7#12314yes—
270C2×C4×C7#12314yes—
271C2×C4×C7#12412yes—
272C2×C4×C7C22×D411yescharacteristic
273C2×C4×C7C22×D411yescharacteristic
274C2×C4×C7C22×D411yescharacteristic
275C2×C4×C7C2×#4111yescharacteristic
276C2×C4×C7C2×#4111yescharacteristic
277C2×C4×C7C2×#4111yescharacteristic
278C7×C8—42no—
279C7×C8—42no—
280C2×#41—22no—
281C2×#41—22no—
282C2×#41—22no—
283C2×#41—22no—
284C2×#41—22no—
285C2×#41C7×D414yes—
286C22×Q8—21no—
287C22×Q8—22no—
288C22×Q8—22no—
289C22×Q8—22no—
290C22×Q8—22no—
291C22×Q8—22no—
292C22×Q8C7×D414yes—
293C22×Q8C23×C712yes—
294C22×Q8C23×C711yescharacteristic
295C22×D4—21no—
296C22×D4C23×C711yescharacteristic
297C23×C4—21no—
298C23×C4—21no—
299C23×C4—21no—
300C23×C4C23×C711yescharacteristic
301C2×M16—22no—
302C2×M16—22no—
303C2×M16—22no—
304C22×C8—22no—
305C22×C8—22no—
306C22×C8—22no—
307C22×C8C23×C712yes—
308C22×C8C23×C712yes—
309C4⋊Q8—24no—
310C4⋊Q8—24no—
311#124—24no—
312#124—24no—
313#123—216no—
314C22⋊Q8—28no—
315C22⋊Q8—28no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117C7×D414yes—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4C7×D411yescharacteristic
327C2×C4⋊C4C7×D411yescharacteristic
328C2×C4⋊C4C23×C712yes—
329C2×C4⋊C4C23×C712yes—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4C7×D412yes—
334C2×C42—21no—
335C2×C42C23×C712yes—
336#103—28no—
337#103—24no—
338#103—24no—
339#103—28no—
340#102—24no—
341#102—24no—
342#95—24no—
343C22×C7—24no—
344C22×C7—22no—
345C22×C7—22no—
346C22×C7—24no—
347C22×C7—42no—
348C22×C7—42no—
349C22×C7—42no—
350C22×C7—82no—
351C22×C7—82no—
352C22×C7#39414yes—
353C22×C7C2×#12011yescharacteristic
354C22×C7C2×#12111yescharacteristic
355C22×C7C2×#12411yescharacteristic
356C22×C7#57712yes—
357C22×C7#57812yes—
358C4×C7—42no—
359C4×C7—44no—
360C4×C7—42no—
361C4×C7—42no—
362C4×C7—44no—
363C4×C7—82no—
364C4×C7#39414yes—
365C4×C7C2×#12111yescharacteristic
366C4×C7C2×#12111yescharacteristic
367C4×C7C2×#12311yescharacteristic
368C4×C7C2×#12311yescharacteristic
369C24—21no—
370C24C2×C7×D411yescharacteristic
371#41—28no—
372#41—28no—
373#41—44no—
374#41—44no—
375C2×Q8—24no—
376C2×Q8—22no—
377C2×Q8—21no—
378C2×Q8—28no—
379C2×Q8—28no—
380C2×Q8—22no—
381C2×Q8—21no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—42no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—42no—
392C2×Q8C2×C7×D414yes—
393C2×Q8C2×C7×D412yes—
394C2×D4—24no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4C2×C7×D412yes—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—82no—
415C22×C4C2×C7×D412yes—
416C22×C4C2×C7×D412yes—
417C22×C4C2×C7×D411yescharacteristic
418C22×C4C2×C7×D412yes—
419C22×C4C2×C7×D412yes—
420C22×C4C2×C7×D412yes—
421C22×C4C7×#4112yes—
422C22×C4C7×#4112yes—
423C22×C4C7×#4112yes—
424C22×C4C24×C711yescharacteristic
425M16—44no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—44no—
433C4⋊C4—28no—
434C4⋊C4—28no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—42no—
441C4⋊C4—84no—
442C4⋊C4C2×C7×D414yes—
443C22⋊C4—28no—
444C22⋊C4—28no—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C42—24no—
448C42—22no—
449C42—22no—
450C2×C7—24no—
451C2×C7—82no—
452C2×C7C2×#39411yescharacteristic
453C2×C7#301011yescharacteristic
454C2×C7#301111yescharacteristic
455C2×C7#304612yes—
456C2×C7#313712yes—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23—42no—
464C23C7×#12012yes—
465C23C7×#12112yes—
466C23C7×#12412yes—
467C23C22×C7×D411yescharacteristic
468Q8—24no—
469Q8—22no—
470Q8—24no—
471Q8—22no—
472Q8—42no—
473Q8—41no—
474Q8—42no—
475Q8—41no—
476Q8—48no—
477D4—22no—
478D4—22no—
479D4—48no—
480D4—41no—
481D4—41no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—48no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—48no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—84no—
500C2×C4C7×#12012yes—
501C2×C4C7×#12114yes—
502C2×C4C7×#12114yes—
503C2×C4C7×#12112yes—
504C2×C4C7×#12314yes—
505C2×C4C7×#12314yes—
506C2×C4C7×#12412yes—
507C2×C4C22×C7×D411yescharacteristic
508C2×C4C22×C7×D411yescharacteristic
509C2×C4C22×C7×D411yescharacteristic
510C2×C4C2×C7×#4111yescharacteristic
511C2×C4C2×C7×#4111yescharacteristic
512C2×C4C2×C7×#4111yescharacteristic
513C8—42no—
514C8—42no—
515C7#2351011yescharacteristic, direct product factor, Sylow
516C22—22no—
517C22—22no—
518C22—24no—
519C22—24no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—82no—
524C22—82no—
525C22C7×#39414yes—
526C22C2×C7×#12011yescharacteristic
527C22C2×C7×#12111yescharacteristic
528C22C2×C7×#12411yescharacteristic
529C22C7×#57712yes—
530C22C7×#57812yes—
531C4—42no—
532C4—42no—
533C4—44no—
534C4—42no—
535C4—44no—
536C4—82no—
537C4C7×#39414yes—
538C4C2×C7×#12111yescharacteristic
539C4C2×C7×#12111yescharacteristic
540C4C2×C7×#12311yescharacteristic
541C4C2×C7×#12311yescharacteristic
542C2—24no—
543C2—82no—
544C2C2×C7×#39411yescharacteristic
545C2C7×#301011yescharacteristic
546C2C7×#301111yescharacteristic
547C2C7×#304612yes—
548C2C7×#313712yes—
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC22×C7×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C7×#117 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C7×#419C2semidirect
5. C22×C7×#103C2semidirect
6. C7×#2317C2semidirect
7. C7×#2043C2semidirect
8. C7×#2042C2semidirect
9. C7×#1729C2semidirect
39. #23510C7direct (N × H)
105. C2×#419C2×C7semidirect
107. C22×#103C2×C7semidirect
108. #2317C2×C7semidirect
109. #2043C2×C7semidirect
110. #2042C2×C7semidirect
111. #1729C2×C7semidirect
515. C7#23510direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5362
C2×C7×#4172
C2×C7×#4192
C2×C7×#4162
C22×C7×#1032
C7×#23172
C7×#20432
C7×#20422
C7×#17292
#235107