Finite Group Databasebeta

← C7×#24116 #13201332

Subgroups — C7×#24116 #13201332

Order 1792.

1434 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#547C211yesmaximal, characteristic, complemented
2C2×C7×#474C211yesmaximal, characteristic, complemented
3C2×C4×C7×D8C211yesmaximal, characteristic, complemented
4C7×#2537C211yesmaximal, characteristic, complemented
5C7×#2033C212yesmaximal, complemented
6C7×#1999C212yesmaximal, complemented
7C7×#1990C211yesmaximal, characteristic, complemented
8C7×#1954C211yesmaximal, characteristic, complemented
9C7×#1874C212yesmaximal, complemented
10C7×#1852C212yesmaximal, complemented
11C7×#1802C211yesmaximal, characteristic, complemented
12C22×C7×#41—21no—
13C22×C7×D8C2211yescharacteristic
14C7×#547—24no—
15C7×#547—24no—
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×#123—22no—
18C2×C7×C22⋊Q8—21no—
19C2×C7×#121—21no—
20C2×C7×#121C2212yes—
21C2×C7×#121C2212yes—
22C2×C4×C7×D4C2211yescharacteristic
23C2×C4×C7×D4C2211yescharacteristic
24C2×C4×C7×D4C2211yescharacteristic
25C22×C7×C4⋊C4C2212yes—
26C7×#474—24no—
27C4×C7×D8—24no—
28C2×C7×#107C2211yescharacteristic
29C2×C7×#107C2212yes—
30C2×C7×C4⋊C8C2211yescharacteristic
31C2×C7×#102C2212yes—
32C2×C7×#102C2211yescharacteristic
33C2×C7×#102C2211yescharacteristic
34C2×C7×#102C2212yes—
35C2×C7×C22⋊C8C2212yes—
36C2×C7×C22⋊C8C2212yes—
37C2×C4×C7×C8C2211yescharacteristic
38C7×#394—22no—
39C7×#384C2211yescharacteristic
40C7×#380—21no—
41C7×#380C2212yes—
42C7×#340C2212yes—
43C7×#340C2212yes—
44C7×#340C2212yes—
45#24116C711yesmaximal, characteristic, direct product factor, Sylow
46C2×C7×#41—28no—
47C2×C7×#41—44no—
48C2×C7×#41—44no—
49C22×C7×Q8—21no—
50C22×C7×D4—22no—
51C22×C7×D4—22no—
52C22×C7×D4D411yescharacteristic
53C22×C7×D4C2311yescharacteristic
54C22×C7×D4C2311yescharacteristic
55C23×C4×C7—22no—
56C23×C4×C7—21no—
57C23×C4×C7—21no—
58C23×C4×C7C2312yes—
59C2×C7×D8—22no—
60C2×C7×D8—22no—
61C2×C7×D8—22no—
62C22×C7×C8C2311yescharacteristic
63C22×C7×C8C2312yes—
64C22×C7×C8C2311yescharacteristic
65C7×C4⋊Q8—22no—
66C7×C4⋊Q8D414yescomplemented
67C7×#123—216no—
68C7×C22⋊Q8—28no—
69C7×#121—24no—
70C7×#121—24no—
71C7×#121—24no—
72C7×#121—42no—
73C7×#121—44no—
74C4×C7×D4—22no—
75C4×C7×D4—22no—
76C4×C7×D4—22no—
77C4×C7×D4—22no—
78C4×C7×D4—22no—
79C4×C7×D4D414yescomplemented
80C2×C7×C4⋊C4—22no—
81C2×C7×C4⋊C4—21no—
82C2×C7×C4⋊C4—24no—
83C2×C7×C4⋊C4—24no—
84C2×C7×C4⋊C4—21no—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4D411yescharacteristic
87C2×C7×C4⋊C4C2312yes—
88C2×C7×C4⋊C4C2311yescharacteristic
89C2×C7×C4⋊C4C2311yescharacteristic
90C2×C7×C4⋊C4C2312yes—
91C2×C7×C22⋊C4—21no—
92C2×C7×C22⋊C4—22no—
93C2×C7×C22⋊C4—22no—
94C2×C7×C22⋊C4—21no—
95C2×C7×C22⋊C4—22no—
96C2×C7×C22⋊C4D412yes—
97C2×C42×C7C2311yescharacteristic
98C7×#107—24no—
99C7×#107—22no—
100C7×C4⋊C8—22no—
101C7×#102—22no—
102C7×#102—24no—
103C7×#102—22no—
104C7×#102—42no—
105C7×C22⋊C8—24no—
106C7×C22⋊C8—42no—
107C4×C7×C8—22no—
108C7×#95—22no—
109C2×#547C2×C711yescharacteristic, complemented
110C2×#474C2×C711yescharacteristic, complemented
111C2×C4×D8C2×C711yescharacteristic, complemented
112#2537C2×C711yescharacteristic, complemented
113#2033C2×C712yescomplemented
114#1999C2×C712yescomplemented
115#1990C2×C711yescharacteristic, complemented
116#1954C2×C711yescharacteristic, complemented
117#1874C2×C712yescomplemented
118#1852C2×C712yescomplemented
119#1802C2×C711yescharacteristic, complemented
120C24×C7—21no—
121C24×C7—21no—
122C24×C7C2×D412yes—
123C7×#41—432no—
124C2×C7×Q8—24no—
125C2×C7×Q8—42no—
126C2×C7×Q8—42no—
127C2×C7×D4—28no—
128C2×C7×D4—22no—
129C2×C7×D4—22no—
130C2×C7×D4—21no—
131C2×C7×D4—21no—
132C2×C7×D4—21no—
133C2×C7×D4—21no—
134C2×C7×D4—21no—
135C2×C7×D4—21no—
136C2×C7×D4—21no—
137C2×C7×D4—21no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4—44no—
141C2×C7×D4—44no—
142C2×C7×D4—44no—
143C2×C7×D4—44no—
144C2×C7×D4—82no—
145C2×C7×D4D814yescomplemented
146C2×C7×D4C2×D412yes—
147C2×C7×D4#4112yescomplemented
148C22×C4×C7—28no—
149C22×C4×C7—21no—
150C22×C4×C7—22no—
151C22×C4×C7—24no—
152C22×C4×C7—22no—
153C22×C4×C7—22no—
154C22×C4×C7—22no—
155C22×C4×C7—22no—
156C22×C4×C7—22no—
157C22×C4×C7—22no—
158C22×C4×C7—21no—
159C22×C4×C7—22no—
160C22×C4×C7—44no—
161C22×C4×C7—44no—
162C22×C4×C7—42no—
163C22×C4×C7—42no—
164C22×C4×C7—42no—
165C22×C4×C7—42no—
166C22×C4×C7—42no—
167C22×C4×C7C2×D412yes—
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2411yescharacteristic
171C7×D8—28no—
172C2×C7×C8—21no—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C2×C7×C8—22no—
176C2×C7×C8—22no—
177C2×C7×C8—21no—
178C2×C7×C8—22no—
179C2×C7×C8—21no—
180C2×C7×C8#4112yescomplemented
181C7×C4⋊C4—28no—
182C7×C4⋊C4—24no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4—21no—
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⋊C4—81no—
192C7×C4⋊C4C2×D412yes—
193C7×C4⋊C4C2×D414yes—
194C7×C22⋊C4—28no—
195C7×C22⋊C4—24no—
196C7×C22⋊C4—24no—
197C7×C22⋊C4—44no—
198C7×C22⋊C4—44no—
199C7×C22⋊C4—42no—
200C42×C7C2×D412yes—
201C42×C7C2×D412yes—
202C22×#41—21no—
203C22×D8C22×C711yescharacteristic
204#547—24no—
205#547—24no—
206C2×C4⋊Q8C22×C711yescharacteristic
207C2×#123—22no—
208C2×C22⋊Q8—21no—
209C2×#121—21no—
210C2×#121C22×C712yes—
211C2×#121C22×C712yes—
212C2×C4×D4C22×C711yescharacteristic
213C2×C4×D4C22×C711yescharacteristic
214C2×C4×D4C22×C711yescharacteristic
215C22×C4⋊C4C22×C712yes—
216#474—24no—
217C4×D8—24no—
218C2×#107C22×C711yescharacteristic
219C2×#107C22×C712yes—
220C2×C4⋊C8C22×C711yescharacteristic
221C2×#102C22×C711yescharacteristic
222C2×#102C22×C712yes—
223C2×#102C22×C711yescharacteristic
224C2×#102C22×C712yes—
225C2×C22⋊C8C22×C712yes—
226C2×C22⋊C8C22×C712yes—
227C2×C4×C8C22×C711yescharacteristic
228#394—22no—
229#384C22×C711yescharacteristic
230#380—21no—
231#380C22×C712yes—
232#340C22×C712yes—
233#340C22×C712yes—
234#340C22×C712yes—
235C23×C7—22no—
236C23×C7—22no—
237C23×C7—22no—
238C23×C7—22no—
239C23×C7—22no—
240C23×C7—22no—
241C23×C7—22no—
242C23×C7—22no—
243C23×C7—41no—
244C23×C7—41no—
245C23×C7—41no—
246C23×C7—41no—
247C23×C7—41no—
248C23×C7—41no—
249C23×C7—41no—
250C23×C7—41no—
251C23×C7—41no—
252C23×C7—41no—
253C23×C7—41no—
254C23×C7—41no—
255C23×C7C2×D814yes—
256C23×C7C22×D411yescharacteristic
257C7×Q8—48no—
258C7×D4—22no—
259C7×D4—22no—
260C7×D4—22no—
261C7×D4—22no—
262C7×D4—28no—
263C7×D4—416no—
264C7×D4—41no—
265C7×D4—41no—
266C7×D4—41no—
267C7×D4—41no—
268C7×D4—88no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—22no—
272C2×C4×C7—22no—
273C2×C4×C7—24no—
274C2×C4×C7—22no—
275C2×C4×C7—22no—
276C2×C4×C7—416no—
277C2×C4×C7—41no—
278C2×C4×C7—42no—
279C2×C4×C7—41no—
280C2×C4×C7—42no—
281C2×C4×C7—42no—
282C2×C4×C7—42no—
283C2×C4×C7—42no—
284C2×C4×C7—42no—
285C2×C4×C7—48no—
286C2×C4×C7—41no—
287C2×C4×C7—42no—
288C2×C4×C7—41no—
289C2×C4×C7—42no—
290C2×C4×C7—41no—
291C2×C4×C7—42no—
292C2×C4×C7—84no—
293C2×C4×C7#12012yes—
294C2×C4×C7#12012yes—
295C2×C4×C7C2×D812yes—
296C2×C4×C7AGL(1,ℤ/8)12yes—
297C2×C4×C7C22×D411yescharacteristic
298C2×C4×C7C22×D411yescharacteristic
299C2×C4×C7C2×#4111yescharacteristic
300C2×C4×C7C2×#4111yescharacteristic
301C2×C4×C7H(5,𝔽2)11yescharacteristic
302C2×C4×C72−1+411yescharacteristic
303C7×C8—22no—
304C7×C8—41no—
305C7×C8—42no—
306C2×#41—28no—
307C2×#41—44no—
308C2×#41—44no—
309C22×Q8—21no—
310C22×D4—22no—
311C22×D4—22no—
312C22×D4C7×D411yescharacteristic
313C22×D4C23×C711yescharacteristic
314C22×D4C23×C711yescharacteristic
315C23×C4—22no—
316C23×C4—21no—
317C23×C4—21no—
318C23×C4C23×C712yes—
319C2×D8—22no—
320C2×D8—22no—
321C2×D8—22no—
322C22×C8C23×C711yescharacteristic
323C22×C8C23×C712yes—
324C22×C8C23×C711yescharacteristic
325C4⋊Q8—22no—
326C4⋊Q8C7×D414yescomplemented
327#123—216no—
328C22⋊Q8—28no—
329#121—24no—
330#121—24no—
331#121—24no—
332#121—42no—
333#121—44no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4—22no—
337C4×D4—22no—
338C4×D4—22no—
339C4×D4C7×D414yescomplemented
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4C7×D411yescharacteristic
347C2×C4⋊C4C23×C711yescharacteristic
348C2×C4⋊C4C23×C712yes—
349C2×C4⋊C4C23×C711yescharacteristic
350C2×C4⋊C4C23×C712yes—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4C7×D412yes—
357C2×C42C23×C711yescharacteristic
358#107—22no—
359#107—24no—
360C4⋊C8—22no—
361#102—22no—
362#102—22no—
363#102—24no—
364#102—42no—
365C22⋊C8—24no—
366C22⋊C8—42no—
367C4×C8—22no—
368#95—22no—
369C22×C7—24no—
370C22×C7—24no—
371C22×C7—24no—
372C22×C7—42no—
373C22×C7—42no—
374C22×C7—42no—
375C22×C7—42no—
376C22×C7—42no—
377C22×C7—42no—
378C22×C7—42no—
379C22×C7—42no—
380C22×C7—42no—
381C22×C7—42no—
382C22×C7—81no—
383C22×C7—81no—
384C22×C7—81no—
385C22×C7—81no—
386C22×C7—81no—
387C22×C7—81no—
388C22×C7—81no—
389C22×C7—81no—
390C22×C7C2×#12011yescharacteristic
391C22×C7#54611yescharacteristic
392C22×C7#54711yescharacteristic
393C22×C7C22×D811yescharacteristic
394C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
395C22×C7#57711yescharacteristic
396C22×C7#57811yescharacteristic
397C4×C7—22no—
398C4×C7—42no—
399C4×C7—42no—
400C4×C7—44no—
401C4×C7—81no—
402C4×C7—82no—
403C4×C7#44714yes—
404C4×C7#52511yescharacteristic
405C4×C7#53711yescharacteristic
406C4×C7#54811yescharacteristic
407C4×C7#55011yescharacteristic
408C24—21no—
409C24—21no—
410C24C2×C7×D412yes—
411#41—432no—
412C2×Q8—24no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×D4—28no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—82no—
433C2×D4C7×D814yescomplemented
434C2×D4C2×C7×D412yes—
435C2×D4C7×#4112yescomplemented
436C22×C4—28no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4C2×C7×D412yes—
456C22×C4C2×C7×D411yescharacteristic
457C22×C4C2×C7×D411yescharacteristic
458C22×C4C24×C711yescharacteristic
459D8—28no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—21no—
466C2×C8—22no—
467C2×C8—21no—
468C2×C8C7×#4112yescomplemented
469C4⋊C4—28no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—42no—
479C4⋊C4—81no—
480C4⋊C4C2×C7×D412yes—
481C4⋊C4C2×C7×D414yes—
482C22⋊C4—28no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—42no—
488C42C2×C7×D412yes—
489C42C2×C7×D412yes—
490C2×C7—44no—
491C2×C7—82no—
492C2×C7—82no—
493C2×C7#240711yescharacteristic
494C2×C7C2×#44711yescharacteristic
495C2×C7#301311yescharacteristic
496C2×C7#329411yescharacteristic
497C2×C7#333011yescharacteristic
498C2×C7#333411yescharacteristic
499C2×C7#335411yescharacteristic
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23C2×C7×D814yes—
521C23C22×C7×D411yescharacteristic
522Q8—48no—
523D4—22no—
524D4—22no—
525D4—22no—
526D4—22no—
527D4—28no—
528D4—416no—
529D4—41no—
530D4—41no—
531D4—41no—
532D4—41no—
533D4—88no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—416no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—48no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—84no—
558C2×C4C7×#12012yes—
559C2×C4C7×#12012yes—
560C2×C4C2×C7×D812yes—
561C2×C4C7×AGL(1,ℤ/8)12yes—
562C2×C4C22×C7×D411yescharacteristic
563C2×C4C22×C7×D411yescharacteristic
564C2×C4C2×C7×#4111yescharacteristic
565C2×C4C2×C7×#4111yescharacteristic
566C2×C4C7×H(5,𝔽2)11yescharacteristic
567C2×C4C7×2−1+411yescharacteristic
568C8—22no—
569C8—41no—
570C8—42no—
571C7#2411611yescharacteristic, direct product factor, Sylow
572C22—24no—
573C22—24no—
574C22—24no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22C2×C7×#12011yescharacteristic
594C22C7×#54611yescharacteristic
595C22C7×#54711yescharacteristic
596C22C22×C7×D811yescharacteristic
597C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
598C22C7×#57711yescharacteristic
599C22C7×#57811yescharacteristic
600C4—22no—
601C4—42no—
602C4—42no—
603C4—44no—
604C4—81no—
605C4—82no—
606C4C7×#44714yes—
607C4C7×#52511yescharacteristic
608C4C7×#53711yescharacteristic
609C4C7×#54811yescharacteristic
610C4C7×#55011yescharacteristic
611C2—44no—
612C2—82no—
613C2—82no—
614C2C7×#240711yescharacteristic
615C2C2×C7×#44711yescharacteristic
616C2C7×#301311yescharacteristic
617C2C7×#329411yescharacteristic
618C2C7×#333011yescharacteristic
619C2C7×#333411yescharacteristic
620C2C7×#335411yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#547C2semidirect
2. C2×C7×#474C2semidirect
3. C2×C4×C7×D8C2semidirect
4. C7×#2537C2semidirect
5. C7×#2033C2semidirect
6. C7×#1999C2semidirect
7. C7×#1990C2semidirect
8. C7×#1954C2semidirect
9. C7×#1874C2semidirect
10. C7×#1852C2semidirect
11. C7×#1802C2semidirect
45. #24116C7direct (N × H)
66. C7×C4⋊Q8D4semidirect
79. C4×C7×D4D4semidirect
109. C2×#547C2×C7semidirect
110. C2×#474C2×C7semidirect
111. C2×C4×D8C2×C7semidirect
112. #2537C2×C7semidirect
113. #2033C2×C7semidirect
114. #1999C2×C7semidirect
115. #1990C2×C7semidirect
116. #1954C2×C7semidirect
117. #1874C2×C7semidirect
118. #1852C2×C7semidirect
119. #1802C2×C7semidirect
145. C2×C7×D4D8semidirect
147. C2×C7×D4#41semidirect
180. C2×C7×C8#41semidirect
326. C4⋊Q8C7×D4semidirect
339. C4×D4C7×D4semidirect
433. C2×D4C7×D8semidirect
435. C2×D4C7×#41semidirect
468. C2×C8C7×#41semidirect
571. C7#24116direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5472
C2×C7×#4742
C2×C4×C7×D82
C7×#25372
C7×#20332
C7×#19992
C7×#19902
C7×#19542
C7×#18742
C7×#18522
C7×#18022
#241167