Finite Group Databasebeta

← C7×#23458 #13200674

Subgroups — C7×#23458 #13200674

Order 1792.

1250 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#544C211yesmaximal, characteristic, complemented
2C2×C7×#417C211yesmaximal, characteristic
3C2×C7×#416C211yesmaximal, characteristic
4C7×#2311C211yesmaximal, characteristic
5C7×#2026C212yesmaximal, complemented
6C7×#2025C212yesmaximal, complemented
7C7×#2024C212yesmaximal, complemented
8C7×#2023C212yesmaximal, complemented
9C7×#1736C211yesmaximal, characteristic, complemented
10C2×C7×#340C211yesmaximal, characteristic, complemented
11C7×#1729C211yesmaximal, characteristic, complemented
12C22×C7×#41C2211yescharacteristic
13C22×C7×M16C2211yescharacteristic
14C23×C7×C8C2211yescharacteristic
15C7×#544—24no—
16C7×#544—24no—
17C2×C7×C4⋊Q8—21no—
18C2×C7×C4⋊Q8C2212yes—
19C2×C7×#125—21no—
20C2×C7×#125C2212yes—
21C2×C7×C22⋊Q8—22no—
22C2×C7×#117C2211yescharacteristic
23C2×C7×#117C2211yescharacteristic
24C22×C7×C4⋊C4C2211yescharacteristic
25C22×C7×C4⋊C4C2211yescharacteristic
26C7×#417—24no—
27C7×#416—24no—
28C2×C7×#103C2211yescharacteristic
29C2×C7×#103C2212yes—
30C2×C7×#103C2211yescharacteristic
31C2×C7×#102C2211yescharacteristic
32C2×C7×#102C2212yes—
33C2×C7×#102C2211yescharacteristic
34C7×#391C2212yes—
35C7×#390C2212yes—
36C7×#386—22no—
37C7×#380—21no—
38C7×#380—21no—
39C7×#340C2212yes—
40C7×#340C2212yes—
41C7×#340C2212yes—
42C7×#340C2212yes—
43C7×#336C2212yes—
44C7×#336C2212yes—
45#23458C711yesmaximal, characteristic, direct product factor, Sylow
46C2×C7×#41—22no—
47C2×C7×#41—22no—
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41—22no—
51C2×C7×#41Q814yescomplemented
52C22×C7×Q8—22no—
53C22×C7×Q8C2311yescharacteristic
54C22×C7×D4—21no—
55C22×C7×D4C2311yescharacteristic
56C23×C4×C7—21no—
57C23×C4×C7—21no—
58C23×C4×C7—21no—
59C23×C4×C7C2311yescharacteristic
60C2×C7×M16—22no—
61C2×C7×M16—22no—
62C2×C7×M16—22no—
63C22×C7×C8—22no—
64C22×C7×C8—22no—
65C22×C7×C8—22no—
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C22×C7×C8C2312yes—
69C7×C4⋊Q8—28no—
70C7×C4⋊Q8—24no—
71C7×C4⋊Q8—24no—
72C7×#125—28no—
73C7×#125—24no—
74C7×#125—24no—
75C7×C22⋊Q8—216no—
76C7×#117—22no—
77C7×#117—22no—
78C7×#117—22no—
79C7×#117D414yes—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—22no—
82C2×C7×C4⋊C4—22no—
83C2×C7×C4⋊C4—22no—
84C2×C7×C4⋊C4—22no—
85C2×C7×C4⋊C4—22no—
86C2×C7×C4⋊C4—22no—
87C2×C7×C4⋊C4—21no—
88C2×C7×C4⋊C4D411yescharacteristic
89C2×C7×C4⋊C4D411yescharacteristic
90C2×C7×C4⋊C4D414yes—
91C2×C7×C4⋊C4C2312yes—
92C2×C7×C4⋊C4C2312yes—
93C2×C7×C4⋊C4C2311yescharacteristic
94C2×C7×C4⋊C4C2311yescharacteristic
95C2×C7×C22⋊C4—21no—
96C2×C7×C22⋊C4—22no—
97C2×C7×C22⋊C4—22no—
98C2×C7×C22⋊C4D412yes—
99C2×C42×C7—21no—
100C2×C42×C7C2312yes—
101C7×#103—22no—
102C7×#103—22no—
103C7×#103—24no—
104C7×#102—22no—
105C7×#102—22no—
106C7×#102—24no—
107C7×#95—22no—
108C7×#95—22no—
109C2×#544C2×C711yescharacteristic, complemented
110C2×#417C2×C711yescharacteristic
111C2×#416C2×C711yescharacteristic
112#2311C2×C711yescharacteristic
113#2026C2×C712yescomplemented
114#2025C2×C712yescomplemented
115#2024C2×C712yescomplemented
116#2023C2×C712yescomplemented
117#1736C2×C711yescharacteristic, complemented
118C2×#340C2×C711yescharacteristic, complemented
119#1729C2×C711yescharacteristic, complemented
120C24×C7—21no—
121C24×C7C2×D411yescharacteristic
122C7×#41—28no—
123C7×#41—28no—
124C7×#41—44no—
125C7×#41—44no—
126C2×C7×Q8—28no—
127C2×C7×Q8—21no—
128C2×C7×Q8—21no—
129C2×C7×Q8—21no—
130C2×C7×Q8—21no—
131C2×C7×Q8—21no—
132C2×C7×Q8—82no—
133C2×C7×Q8—82no—
134C2×C7×Q8C2×Q812yes—
135C2×C7×D4—24no—
136C2×C7×D4—21no—
137C2×C7×D4—21no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4—21no—
141C2×C7×D4—42no—
142C2×C7×D4—42no—
143C2×C7×D4C2×Q812yes—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—22no—
148C22×C4×C7—24no—
149C22×C4×C7—24no—
150C22×C4×C7—22no—
151C22×C4×C7—42no—
152C22×C4×C7—42no—
153C22×C4×C7—42no—
154C22×C4×C7—81no—
155C22×C4×C7—81no—
156C22×C4×C7—81no—
157C22×C4×C7—81no—
158C22×C4×C7C2×D412yes—
159C22×C4×C7C2×D412yes—
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D412yes—
162C22×C4×C7C2×D412yes—
163C22×C4×C7C2×Q812yes—
164C22×C4×C7#4112yes—
165C22×C4×C7#4112yes—
166C22×C4×C7#4112yes—
167C22×C4×C7C2411yescharacteristic
168C7×M16—44no—
169C2×C7×C8—21no—
170C2×C7×C8—21no—
171C2×C7×C8—22no—
172C2×C7×C8—21no—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C2×C7×C8—21no—
176C2×C7×C8—22no—
177C2×C7×C8—22no—
178C2×C7×C8—44no—
179C7×C4⋊C4—28no—
180C7×C4⋊C4—28no—
181C7×C4⋊C4—28no—
182C7×C4⋊C4—24no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—42no—
191C7×C4⋊C4C2×D414yes—
192C7×C4⋊C4C2×D412yes—
193C7×C4⋊C4C2×D412yes—
194C7×C22⋊C4—24no—
195C7×C22⋊C4—24no—
196C7×C22⋊C4—82no—
197C7×C22⋊C4—82no—
198C42×C7—24no—
199C42×C7—22no—
200C42×C7—22no—
201C22×#41C22×C711yescharacteristic
202C22×M16C22×C711yescharacteristic
203C23×C8C22×C711yescharacteristic
204#544—24no—
205#544—24no—
206C2×C4⋊Q8—21no—
207C2×C4⋊Q8C22×C712yes—
208C2×#125—21no—
209C2×#125C22×C712yes—
210C2×C22⋊Q8—22no—
211C2×#117C22×C711yescharacteristic
212C2×#117C22×C711yescharacteristic
213C22×C4⋊C4C22×C711yescharacteristic
214C22×C4⋊C4C22×C711yescharacteristic
215#417—24no—
216#416—24no—
217C2×#103C22×C711yescharacteristic
218C2×#103C22×C711yescharacteristic
219C2×#103C22×C712yes—
220C2×#102C22×C711yescharacteristic
221C2×#102C22×C711yescharacteristic
222C2×#102C22×C712yes—
223#391C22×C712yes—
224#390C22×C712yes—
225#386—22no—
226#380—21no—
227#380—21no—
228#340C22×C712yes—
229#340C22×C712yes—
230#340C22×C712yes—
231#340C22×C712yes—
232#336C22×C712yes—
233#336C22×C712yes—
234C23×C7—21no—
235C23×C7—21no—
236C23×C7—21no—
237C23×C7—21no—
238C23×C7—22no—
239C23×C7—41no—
240C23×C7—41no—
241C23×C7—41no—
242C23×C7—41no—
243C23×C7—41no—
244C23×C7—41no—
245C23×C7#12012yes—
246C23×C7C22⋊Q812yes—
247C23×C7#12412yes—
248C23×C7C22×D411yescharacteristic
249C7×Q8—22no—
250C7×Q8—22no—
251C7×Q8—41no—
252C7×Q8—41no—
253C7×Q8—88no—
254C7×D4—22no—
255C7×D4—22no—
256C7×D4—48no—
257C7×D4—41no—
258C7×D4—41no—
259C2×C4×C7—22no—
260C2×C4×C7—22no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—24no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—42no—
272C2×C4×C7—41no—
273C2×C4×C7—41no—
274C2×C4×C7—42no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—48no—
278C2×C4×C7—42no—
279C2×C4×C7—42no—
280C2×C4×C7—42no—
281C2×C4×C7—84no—
282C2×C4×C7—84no—
283C2×C4×C7#12012yes—
284C2×C4×C7C22⋊Q814yes—
285C2×C4×C7C22⋊Q814yes—
286C2×C4×C7C22⋊Q812yes—
287C2×C4×C7#12414yes—
288C2×C4×C7#12414yes—
289C2×C4×C7#12412yes—
290C2×C4×C7C22×D411yescharacteristic
291C2×C4×C7C22×D411yescharacteristic
292C2×C4×C7C22×Q811yescharacteristic
293C2×C4×C7C2×#4111yescharacteristic
294C2×C4×C7C2×#4111yescharacteristic
295C2×C4×C7C2×#4111yescharacteristic
296C7×C8—41no—
297C7×C8—41no—
298C7×C8—42no—
299C2×#41—22no—
300C2×#41—22no—
301C2×#41—22no—
302C2×#41—22no—
303C2×#41—22no—
304C2×#41C7×Q814yescomplemented
305C22×Q8—22no—
306C22×Q8C23×C711yescharacteristic
307C22×D4—21no—
308C22×D4C23×C711yescharacteristic
309C23×C4—21no—
310C23×C4—21no—
311C23×C4—21no—
312C23×C4C23×C711yescharacteristic
313C2×M16—22no—
314C2×M16—22no—
315C2×M16—22no—
316C22×C8—22no—
317C22×C8—22no—
318C22×C8—22no—
319C22×C8C23×C711yescharacteristic
320C22×C8C23×C711yescharacteristic
321C22×C8C23×C712yes—
322C4⋊Q8—28no—
323C4⋊Q8—24no—
324C4⋊Q8—24no—
325#125—28no—
326#125—24no—
327#125—24no—
328C22⋊Q8—216no—
329#117—22no—
330#117—22no—
331#117—22no—
332#117C7×D414yes—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4C7×D411yescharacteristic
342C2×C4⋊C4C7×D411yescharacteristic
343C2×C4⋊C4C7×D414yes—
344C2×C4⋊C4C23×C712yes—
345C2×C4⋊C4C23×C712yes—
346C2×C4⋊C4C23×C711yescharacteristic
347C2×C4⋊C4C23×C711yescharacteristic
348C2×C22⋊C4—21no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4C7×D412yes—
352C2×C42—21no—
353C2×C42C23×C712yes—
354#103—22no—
355#103—22no—
356#103—24no—
357#102—22no—
358#102—22no—
359#102—24no—
360#95—22no—
361#95—22no—
362C22×C7—22no—
363C22×C7—22no—
364C22×C7—22no—
365C22×C7—22no—
366C22×C7—22no—
367C22×C7—22no—
368C22×C7—42no—
369C22×C7—42no—
370C22×C7—42no—
371C22×C7—81no—
372C22×C7—81no—
373C22×C7—81no—
374C22×C7—81no—
375C22×C7#39314yes—
376C22×C7C2×#12011yescharacteristic
377C22×C7C2×C22⋊Q811yescharacteristic
378C22×C7C2×#12411yescharacteristic
379C22×C7#57611yescharacteristic
380C22×C7#57711yescharacteristic
381C22×C7#57711yescharacteristic
382C22×C7#57811yescharacteristic
383C4×C7—42no—
384C4×C7—42no—
385C4×C7—44no—
386C4×C7—42no—
387C4×C7—42no—
388C4×C7—42no—
389C4×C7—82no—
390C4×C7#39314yes—
391C4×C7C2×C22⋊Q811yescharacteristic
392C4×C7C2×C22⋊Q811yescharacteristic
393C4×C7C2×#12411yescharacteristic
394C4×C7C2×#12411yescharacteristic
395C24—21no—
396C24C2×C7×D411yescharacteristic
397#41—28no—
398#41—28no—
399#41—44no—
400#41—44no—
401C2×Q8—28no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—82no—
408C2×Q8—82no—
409C2×Q8C2×C7×Q812yes—
410C2×D4—24no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4C2×C7×Q812yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—81no—
430C22×C4—81no—
431C22×C4—81no—
432C22×C4—81no—
433C22×C4C2×C7×D412yes—
434C22×C4C2×C7×D412yes—
435C22×C4C2×C7×D411yescharacteristic
436C22×C4C2×C7×D412yes—
437C22×C4C2×C7×D412yes—
438C22×C4C2×C7×Q812yes—
439C22×C4C7×#4112yes—
440C22×C4C7×#4112yes—
441C22×C4C7×#4112yes—
442C22×C4C24×C711yescharacteristic
443M16—44no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—22no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—44no—
454C4⋊C4—28no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—42no—
466C4⋊C4C2×C7×D414yes—
467C4⋊C4C2×C7×D412yes—
468C4⋊C4C2×C7×D412yes—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—82no—
472C22⋊C4—82no—
473C42—24no—
474C42—22no—
475C42—22no—
476C2×C7—24no—
477C2×C7—82no—
478C2×C7C2×#39311yescharacteristic
479C2×C7#301211yescharacteristic
480C2×C7#301311yescharacteristic
481C2×C7#308311yescharacteristic
482C2×C7#308411yescharacteristic
483C2×C7#313611yescharacteristic
484C2×C7#313711yescharacteristic
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—22no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23C7×#12012yes—
497C23C7×C22⋊Q812yes—
498C23C7×#12412yes—
499C23C22×C7×D411yescharacteristic
500Q8—22no—
501Q8—22no—
502Q8—41no—
503Q8—41no—
504Q8—88no—
505D4—22no—
506D4—22no—
507D4—48no—
508D4—41no—
509D4—41no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—48no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—84no—
533C2×C4—84no—
534C2×C4C7×#12012yes—
535C2×C4C7×C22⋊Q814yes—
536C2×C4C7×C22⋊Q814yes—
537C2×C4C7×C22⋊Q812yes—
538C2×C4C7×#12414yes—
539C2×C4C7×#12414yes—
540C2×C4C7×#12412yes—
541C2×C4C22×C7×D411yescharacteristic
542C2×C4C22×C7×D411yescharacteristic
543C2×C4C22×C7×Q811yescharacteristic
544C2×C4C2×C7×#4111yescharacteristic
545C2×C4C2×C7×#4111yescharacteristic
546C2×C4C2×C7×#4111yescharacteristic
547C8—41no—
548C8—41no—
549C8—42no—
550C7#2345811yescharacteristic, direct product factor, Sylow
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22C7×#39314yes—
565C22C2×C7×#12011yescharacteristic
566C22C2×C7×C22⋊Q811yescharacteristic
567C22C2×C7×#12411yescharacteristic
568C22C7×#57611yescharacteristic
569C22C7×#57711yescharacteristic
570C22C7×#57711yescharacteristic
571C22C7×#57811yescharacteristic
572C4—42no—
573C4—44no—
574C4—42no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—82no—
579C4C7×#39314yes—
580C4C2×C7×C22⋊Q811yescharacteristic
581C4C2×C7×C22⋊Q811yescharacteristic
582C4C2×C7×#12411yescharacteristic
583C4C2×C7×#12411yescharacteristic
584C2—24no—
585C2—82no—
586C2C2×C7×#39311yescharacteristic
587C2C7×#301211yescharacteristic
588C2C7×#301311yescharacteristic
589C2C7×#308311yescharacteristic
590C2C7×#308411yescharacteristic
591C2C7×#313611yescharacteristic
592C2C7×#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
1. C2×C7×#544C2semidirect
5. C7×#2026C2semidirect
6. C7×#2025C2semidirect
7. C7×#2024C2semidirect
8. C7×#2023C2semidirect
9. C7×#1736C2semidirect
10. C2×C7×#340C2semidirect
11. C7×#1729C2semidirect
45. #23458C7direct (N × H)
51. C2×C7×#41Q8semidirect
109. C2×#544C2×C7semidirect
113. #2026C2×C7semidirect
114. #2025C2×C7semidirect
115. #2024C2×C7semidirect
116. #2023C2×C7semidirect
117. #1736C2×C7semidirect
118. C2×#340C2×C7semidirect
119. #1729C2×C7semidirect
304. C2×#41C7×Q8semidirect
550. C7#23458direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5442
C2×C7×#4172
C2×C7×#4162
C7×#23112
C7×#20262
C7×#20252
C7×#20242
C7×#20232
C7×#17362
C2×C7×#3402
C7×#17292
#234587