Finite Group Databasebeta

← C7×#23634 #13200850

Subgroups — C7×#23634 #13200850

Order 1792.

1010 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#417C211yesmaximal, characteristic
2C2×C7×#416C211yesmaximal, characteristic
3C7×#2317C211yesmaximal, characteristic, complemented
4C7×#2311C211yesmaximal, characteristic
5C7×#2074C212yesmaximal, complemented
6C7×#2073C212yesmaximal, complemented
7C7×#2072C212yesmaximal, complemented
8C7×#2071C212yesmaximal, complemented
9C7×#1736C211yesmaximal, characteristic, complemented
10C2×C7×#340C211yesmaximal, characteristic, complemented
11C7×#1729C211yesmaximal, 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×#417—24no—
20C7×#416—24no—
21C2×C7×#103C2211yescharacteristic
22C2×C7×#103C2212yes—
23C2×C7×#103C2211yescharacteristic
24C2×C7×#102C2211yescharacteristic
25C2×C7×#102C2212yes—
26C2×C7×#102C2211yescharacteristic
27C7×#391C2212yes—
28C7×#390C2212yes—
29C7×#389—21no—
30C7×#389—21no—
31C7×#386—22no—
32C7×#385—22no—
33C7×#384C2212yes—
34C7×#382C2212yes—
35C7×#380—21no—
36C7×#380—21no—
37C7×#340C2212yes—
38C7×#340C2212yes—
39C7×#340C2212yes—
40C7×#340C2212yes—
41C7×#336C2212yes—
42C7×#336C2212yes—
43#23634C711yesmaximal, 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×#41D414yes—
50C22×C7×Q8C2311yescharacteristic
51C22×C7×D4—21no—
52C22×C7×D4C2311yescharacteristic
53C23×C4×C7—21no—
54C23×C4×C7—21no—
55C23×C4×C7—21no—
56C23×C4×C7C2311yescharacteristic
57C2×C7×M16—22no—
58C2×C7×M16—22no—
59C2×C7×M16—22no—
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⋊C4—22no—
77C2×C7×C4⋊C4—22no—
78C2×C7×C4⋊C4D411yescharacteristic
79C2×C7×C4⋊C4D411yescharacteristic
80C2×C7×C4⋊C4C2312yes—
81C2×C7×C4⋊C4C2312yes—
82C2×C7×C4⋊C4C2311yescharacteristic
83C2×C7×C4⋊C4C2311yescharacteristic
84C2×C7×C22⋊C4—21no—
85C2×C7×C22⋊C4—22no—
86C2×C7×C22⋊C4—22no—
87C2×C7×C22⋊C4D412yes—
88C2×C42×C7—21no—
89C2×C42×C7C2312yes—
90C7×#103—22no—
91C7×#103—22no—
92C7×#103—24no—
93C7×#102—22no—
94C7×#102—22no—
95C7×#102—24no—
96C7×#95—22no—
97C7×#95—22no—
98C7×#95—22no—
99C7×#95—22no—
100C2×#417C2×C711yescharacteristic
101C2×#416C2×C711yescharacteristic
102#2317C2×C711yescharacteristic, complemented
103#2311C2×C711yescharacteristic
104#2074C2×C712yescomplemented
105#2073C2×C712yescomplemented
106#2072C2×C712yescomplemented
107#2071C2×C712yescomplemented
108#1736C2×C711yescharacteristic, complemented
109C2×#340C2×C711yescharacteristic, complemented
110#1729C2×C711yescharacteristic, complemented
111C24×C7—21no—
112C24×C7C2×D411yescharacteristic
113C7×#41—28no—
114C7×#41—28no—
115C7×#41—44no—
116C7×#41—44no—
117C2×C7×Q8—21no—
118C2×C7×Q8—21no—
119C2×C7×Q8—21no—
120C2×C7×Q8—21no—
121C2×C7×Q8—21no—
122C2×C7×Q8C2×D412yes—
123C2×C7×D4—24no—
124C2×C7×D4—21no—
125C2×C7×D4—21no—
126C2×C7×D4—21no—
127C2×C7×D4—21no—
128C2×C7×D4—21no—
129C2×C7×D4—42no—
130C2×C7×D4—42no—
131C2×C7×D4C2×D412yes—
132C22×C4×C7—21no—
133C22×C4×C7—24no—
134C22×C4×C7—21no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—21no—
138C22×C4×C7—42no—
139C22×C4×C7—42no—
140C22×C4×C7—42no—
141C22×C4×C7—42no—
142C22×C4×C7—81no—
143C22×C4×C7—81no—
144C22×C4×C7—81no—
145C22×C4×C7—81no—
146C22×C4×C7C2×D412yes—
147C22×C4×C7C2×D412yes—
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D412yes—
150C22×C4×C7#4112yes—
151C22×C4×C7#4112yes—
152C22×C4×C7#4112yes—
153C22×C4×C7#4112yes—
154C22×C4×C7#4112yes—
155C22×C4×C7C2411yescharacteristic
156C7×M16—44no—
157C2×C7×C8—21no—
158C2×C7×C8—21no—
159C2×C7×C8—22no—
160C2×C7×C8—21no—
161C2×C7×C8—21no—
162C2×C7×C8—21no—
163C2×C7×C8—21no—
164C2×C7×C8—22no—
165C2×C7×C8—22no—
166C2×C7×C8—44no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—42no—
178C7×C4⋊C4—42no—
179C7×C4⋊C4—82no—
180C7×C4⋊C4—82no—
181C7×C22⋊C4—24no—
182C7×C22⋊C4—42no—
183C7×C22⋊C4—82no—
184C7×C22⋊C4—82no—
185C42×C7—22no—
186C42×C7—22no—
187C42×C7—42no—
188C22×#41C22×C711yescharacteristic
189C22×M16C22×C711yescharacteristic
190C23×C8C22×C711yescharacteristic
191C2×#117C22×C711yescharacteristic
192C2×#117C22×C711yescharacteristic
193C22×C4⋊C4C22×C711yescharacteristic
194C22×C4⋊C4C22×C711yescharacteristic
195#417—24no—
196#416—24no—
197C2×#103C22×C711yescharacteristic
198C2×#103C22×C712yes—
199C2×#103C22×C711yescharacteristic
200C2×#102C22×C711yescharacteristic
201C2×#102C22×C712yes—
202C2×#102C22×C711yescharacteristic
203#391C22×C712yes—
204#390C22×C712yes—
205#389—21no—
206#389—21no—
207#386—22no—
208#385—22no—
209#384C22×C712yes—
210#382C22×C712yes—
211#380—21no—
212#380—21no—
213#340C22×C712yes—
214#340C22×C712yes—
215#340C22×C712yes—
216#340C22×C712yes—
217#336C22×C712yes—
218#336C22×C712yes—
219C23×C7—21no—
220C23×C7—21no—
221C23×C7—21no—
222C23×C7—21no—
223C23×C7—22no—
224C23×C7—41no—
225C23×C7—41no—
226C23×C7—41no—
227C23×C7—41no—
228C23×C7—41no—
229C23×C7—41no—
230C23×C7#12112yes—
231C23×C7#12312yes—
232C23×C7#12412yes—
233C23×C7C22×D411yescharacteristic
234C7×Q8—22no—
235C7×Q8—22no—
236C7×Q8—41no—
237C7×Q8—41no—
238C7×D4—22no—
239C7×D4—22no—
240C7×D4—48no—
241C7×D4—41no—
242C7×D4—41no—
243C2×C4×C7—22no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—22no—
250C2×C4×C7—41no—
251C2×C4×C7—41no—
252C2×C4×C7—48no—
253C2×C4×C7—41no—
254C2×C4×C7—41no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—41no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—41no—
262C2×C4×C7—42no—
263C2×C4×C7—41no—
264C2×C4×C7—41no—
265C2×C4×C7—84no—
266C2×C4×C7—84no—
267C2×C4×C7#12112yes—
268C2×C4×C7#12314yes—
269C2×C4×C7#12314yes—
270C2×C4×C7#12312yes—
271C2×C4×C7#12412yes—
272C2×C4×C7#12614yes—
273C2×C4×C7#12614yes—
274C2×C4×C7C22×D411yescharacteristic
275C2×C4×C7C2×#4111yescharacteristic
276C2×C4×C7C2×#4111yescharacteristic
277C2×C4×C7C2×#4111yescharacteristic
278C2×C4×C7C2×#4111yescharacteristic
279C2×C4×C7C2×#4111yescharacteristic
280C7×C8—41no—
281C7×C8—41no—
282C7×C8—42no—
283C2×#41—22no—
284C2×#41—22no—
285C2×#41—22no—
286C2×#41—22no—
287C2×#41—22no—
288C2×#41C7×D414yes—
289C22×Q8C23×C711yescharacteristic
290C22×D4—21no—
291C22×D4C23×C711yescharacteristic
292C23×C4—21no—
293C23×C4—21no—
294C23×C4—21no—
295C23×C4C23×C711yescharacteristic
296C2×M16—22no—
297C2×M16—22no—
298C2×M16—22no—
299C22×C8—22no—
300C22×C8—22no—
301C22×C8—22no—
302C22×C8C23×C711yescharacteristic
303C22×C8C23×C711yescharacteristic
304C22×C8C23×C712yes—
305#117—22no—
306#117—22no—
307#117—22no—
308#117—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4C7×D411yescharacteristic
318C2×C4⋊C4C7×D411yescharacteristic
319C2×C4⋊C4C23×C712yes—
320C2×C4⋊C4C23×C712yes—
321C2×C4⋊C4C23×C711yescharacteristic
322C2×C4⋊C4C23×C711yescharacteristic
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4C7×D412yes—
327C2×C42—21no—
328C2×C42C23×C712yes—
329#103—22no—
330#103—22no—
331#103—24no—
332#102—22no—
333#102—22no—
334#102—24no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339C22×C7—22no—
340C22×C7—22no—
341C22×C7—22no—
342C22×C7—22no—
343C22×C7—22no—
344C22×C7—22no—
345C22×C7—42no—
346C22×C7—42no—
347C22×C7—42no—
348C22×C7—81no—
349C22×C7—81no—
350C22×C7—81no—
351C22×C7—81no—
352C22×C7#39714yes—
353C22×C7C2×#12111yescharacteristic
354C22×C7C2×#12311yescharacteristic
355C22×C7C2×#12411yescharacteristic
356C22×C7#57611yescharacteristic
357C22×C7#57711yescharacteristic
358C22×C7#57711yescharacteristic
359C22×C7#57811yescharacteristic
360C4×C7—42no—
361C4×C7—42no—
362C4×C7—42no—
363C4×C7—82no—
364C4×C7—81no—
365C4×C7—82no—
366C4×C7—81no—
367C4×C7#39714yes—
368C4×C7C2×#12311yescharacteristic
369C4×C7C2×#12311yescharacteristic
370C4×C7C2×#12611yescharacteristic
371C4×C7C2×#12611yescharacteristic
372C24—21no—
373C24C2×C7×D411yescharacteristic
374#41—28no—
375#41—28no—
376#41—44no—
377#41—44no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8C2×C7×D412yes—
384C2×D4—24no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4C2×C7×D412yes—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—81no—
404C22×C4—81no—
405C22×C4—81no—
406C22×C4—81no—
407C22×C4C2×C7×D412yes—
408C22×C4C2×C7×D412yes—
409C22×C4C2×C7×D411yescharacteristic
410C22×C4C2×C7×D412yes—
411C22×C4C7×#4112yes—
412C22×C4C7×#4112yes—
413C22×C4C7×#4112yes—
414C22×C4C7×#4112yes—
415C22×C4C7×#4112yes—
416C22×C4C24×C711yescharacteristic
417M16—44no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—22no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—44no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C22⋊C4—24no—
443C22⋊C4—42no—
444C22⋊C4—82no—
445C22⋊C4—82no—
446C42—22no—
447C42—22no—
448C42—42no—
449C2×C7—24no—
450C2×C7—82no—
451C2×C7C2×#39711yescharacteristic
452C2×C7#304511yescharacteristic
453C2×C7#304611yescharacteristic
454C2×C7#309811yescharacteristic
455C2×C7#309911yescharacteristic
456C2×C7#313611yescharacteristic
457C2×C7#313711yescharacteristic
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—22no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23C7×#12112yes—
470C23C7×#12312yes—
471C23C7×#12412yes—
472C23C22×C7×D411yescharacteristic
473Q8—22no—
474Q8—22no—
475Q8—41no—
476Q8—41no—
477D4—22no—
478D4—22no—
479D4—48no—
480D4—41no—
481D4—41no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—48no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—84no—
505C2×C4—84no—
506C2×C4C7×#12112yes—
507C2×C4C7×#12314yes—
508C2×C4C7×#12314yes—
509C2×C4C7×#12312yes—
510C2×C4C7×#12412yes—
511C2×C4C7×#12614yes—
512C2×C4C7×#12614yes—
513C2×C4C22×C7×D411yescharacteristic
514C2×C4C2×C7×#4111yescharacteristic
515C2×C4C2×C7×#4111yescharacteristic
516C2×C4C2×C7×#4111yescharacteristic
517C2×C4C2×C7×#4111yescharacteristic
518C2×C4C2×C7×#4111yescharacteristic
519C8—41no—
520C8—41no—
521C8—42no—
522C7#2363411yescharacteristic, direct product factor, Sylow
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22C7×#39714yes—
537C22C2×C7×#12111yescharacteristic
538C22C2×C7×#12311yescharacteristic
539C22C2×C7×#12411yescharacteristic
540C22C7×#57611yescharacteristic
541C22C7×#57711yescharacteristic
542C22C7×#57711yescharacteristic
543C22C7×#57811yescharacteristic
544C4—42no—
545C4—42no—
546C4—42no—
547C4—82no—
548C4—81no—
549C4—81no—
550C4—82no—
551C4C7×#39714yes—
552C4C2×C7×#12311yescharacteristic
553C4C2×C7×#12311yescharacteristic
554C4C2×C7×#12611yescharacteristic
555C4C2×C7×#12611yescharacteristic
556C2—24no—
557C2—82no—
558C2C2×C7×#39711yescharacteristic
559C2C7×#304511yescharacteristic
560C2C7×#304611yescharacteristic
561C2C7×#309811yescharacteristic
562C2C7×#309911yescharacteristic
563C2C7×#313611yescharacteristic
564C2C7×#313711yescharacteristic
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. C7×#2317C2semidirect
5. C7×#2074C2semidirect
6. C7×#2073C2semidirect
7. C7×#2072C2semidirect
8. C7×#2071C2semidirect
9. C7×#1736C2semidirect
10. C2×C7×#340C2semidirect
11. C7×#1729C2semidirect
43. #23634C7direct (N × H)
102. #2317C2×C7semidirect
104. #2074C2×C7semidirect
105. #2073C2×C7semidirect
106. #2072C2×C7semidirect
107. #2071C2×C7semidirect
108. #1736C2×C7semidirect
109. C2×#340C2×C7semidirect
110. #1729C2×C7semidirect
522. C7#23634direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4172
C2×C7×#4162
C7×#23172
C7×#23112
C7×#20742
C7×#20732
C7×#20722
C7×#20712
C7×#17362
C2×C7×#3402
C7×#17292
#236347