Finite Group Databasebeta

← C7×#24112 #13201328

Subgroups — C7×#24112 #13201328

Order 1792.

1518 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3490C211yesmaximal, characteristic
2C7×#2849C212yesmaximal, complemented
3C2×C7×#385C211yesmaximal, characteristic, complemented
4C7×#2062C212yesmaximal, complemented
5C7×#2044C212yesmaximal, complemented
6C7×#1898C212yesmaximal, complemented
7C7×#1867C214yesmaximal, complemented
8C7×#1819C211yesmaximal, characteristic, complemented
9C24×C4×C7C2211yescharacteristic
10C7×#551—22no—
11C7×#551—22no—
12C7×#546—22no—
13C7×#546—22no—
14C7×#543C2211yescharacteristic
15C7×#538—21no—
16C7×#538—21no—
17C7×#534C2211yescharacteristic
18C7×#525—21no—
19C7×#525—21no—
20C2×C7×#123—22no—
21C2×C7×#120C2212yes—
22C7×#518C2212yes—
23C22×C7×C4⋊C4—21no—
24C22×C7×C22⋊C4C2211yescharacteristic
25C22×C7×C22⋊C4C2211yescharacteristic
26C2×C7×#99C2214yes—
27C7×#386—22no—
28C7×#386C2214yes—
29C7×#385—21no—
30C7×#385—21no—
31C7×#385—22no—
32C7×#385C2212yes—
33C7×#385C2214yes—
34C7×#385C2212yes—
35C7×#381—22no—
36C7×#379C2212yes—
37C2×C7×#95—21no—
38C7×#342C2212yes—
39C7×#342C2214yes—
40C7×#342C2212yes—
41#24112C711yesmaximal, characteristic, direct product factor, Sylow
42C25×C7C2311yescharacteristic
43C2×C7×#41—21no—
44C2×C7×#41—21no—
45C22×C7×D4D412yes—
46C22×C7×D4D412yes—
47C22×C7×D4C2312yes—
48C23×C4×C7—21no—
49C23×C4×C7—21no—
50C23×C4×C7—22no—
51C23×C4×C7—21no—
52C23×C4×C7—22no—
53C23×C4×C7—44no—
54C23×C4×C7D411yescharacteristic
55C23×C4×C7D411yescharacteristic
56C23×C4×C7C2312yes—
57C7×#126—22no—
58C7×#126—22no—
59C7×#124—22no—
60C7×#124—22no—
61C7×#123—22no—
62C7×#123—22no—
63C7×#123—22no—
64C7×#123—22no—
65C7×#123—22no—
66C7×#123—48no—
67C7×C22⋊Q8—21no—
68C7×C22⋊Q8—21no—
69C7×C22⋊Q8—21no—
70C7×C22⋊Q8—21no—
71C7×C22⋊Q8—22no—
72C7×C22⋊Q8—22no—
73C7×#121—21no—
74C7×#121—21no—
75C7×#121—22no—
76C7×#121—22no—
77C7×#121—21no—
78C7×#121—21no—
79C7×#120—22no—
80C7×#120—22no—
81C7×#120—22no—
82C7×#120—22no—
83C7×#120—22no—
84C4×C7×D4—22no—
85C4×C7×D4—22no—
86C4×C7×D4—22no—
87C4×C7×D4—22no—
88C4×C7×D4—22no—
89C4×C7×D4—22no—
90C7×#117—21no—
91C7×#117—21no—
92C7×#117—22no—
93C2×C7×C4⋊C4—22no—
94C2×C7×C4⋊C4—22no—
95C2×C7×C4⋊C4—24no—
96C2×C7×C4⋊C4—44no—
97C2×C7×C22⋊C4—21no—
98C2×C7×C22⋊C4—22no—
99C2×C7×C22⋊C4—22no—
100C2×C7×C22⋊C4—22no—
101C2×C7×C22⋊C4—24no—
102C2×C7×C22⋊C4—22no—
103C2×C7×C22⋊C4—22no—
104C2×C7×C22⋊C4—24no—
105C2×C7×C22⋊C4D411yescharacteristic
106C2×C7×C22⋊C4D411yescharacteristic
107C2×C7×C22⋊C4D412yes—
108C2×C7×C22⋊C4D412yes—
109C2×C7×C22⋊C4C2314yes—
110C2×C7×C22⋊C4C2311yescharacteristic
111C2×C7×C22⋊C4C2311yescharacteristic
112C2×C7×C22⋊C4C2312yes—
113C2×C7×C22⋊C4C2312yes—
114C7×#99—28no—
115C7×#95—22no—
116C7×#95—22no—
117C7×#95—24no—
118C7×#95—24no—
119#3490C2×C711yescharacteristic
120#2849C2×C712yescomplemented
121C2×#385C2×C711yescharacteristic, complemented
122#2062C2×C712yescomplemented
123#2044C2×C712yescomplemented
124#1898C2×C712yescomplemented
125#1867C2×C714yescomplemented
126#1819C2×C711yescharacteristic, complemented
127C24×C7—22no—
128C24×C7—22no—
129C24×C7—42no—
130C24×C7—42no—
131C24×C7C2×D411yescharacteristic
132C24×C7C2×D412yes—
133C24×C7C2×D412yes—
134C24×C7C2×D412yes—
135C24×C7C2×D411yescharacteristic
136C24×C7C2411yescharacteristic
137C7×#41—82no—
138C7×#41—82no—
139C2×C7×Q8—21no—
140C2×C7×Q8—21no—
141C2×C7×D4—22no—
142C2×C7×D4—22no—
143C2×C7×D4—22no—
144C2×C7×D4—22no—
145C2×C7×D4—24no—
146C2×C7×D4—24no—
147C2×C7×D4—21no—
148C2×C7×D4—21no—
149C2×C7×D4—22no—
150C2×C7×D4—22no—
151C2×C7×D4—44no—
152C2×C7×D4—44no—
153C22×C4×C7—24no—
154C22×C4×C7—24no—
155C22×C4×C7—28no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7—22no—
159C22×C4×C7—24no—
160C22×C4×C7—22no—
161C22×C4×C7—22no—
162C22×C4×C7—22no—
163C22×C4×C7—21no—
164C22×C4×C7—21no—
165C22×C4×C7—24no—
166C22×C4×C7—44no—
167C22×C4×C7—48no—
168C22×C4×C7—44no—
169C22×C4×C7—42no—
170C22×C4×C7—42no—
171C22×C4×C7—42no—
172C22×C4×C7—44no—
173C22×C4×C7—42no—
174C22×C4×C7—41no—
175C22×C4×C7—41no—
176C22×C4×C7—42no—
177C22×C4×C7—42no—
178C22×C4×C7C2×D412yes—
179C22×C4×C7C2×D411yescharacteristic
180C22×C4×C7C2×D412yes—
181C22×C4×C7C2×D411yescharacteristic
182C22×C4×C7C2×D411yescharacteristic
183C22×C4×C7C2×D412yes—
184C22×C4×C7C2×D411yescharacteristic
185C7×C4⋊C4—22no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—22no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—44no—
194C7×C4⋊C4—44no—
195C7×C4⋊C4—48no—
196C7×C22⋊C4—24no—
197C7×C22⋊C4—22no—
198C7×C22⋊C4—22no—
199C7×C22⋊C4—22no—
200C7×C22⋊C4—22no—
201C7×C22⋊C4—22no—
202C7×C22⋊C4—22no—
203C7×C22⋊C4—22no—
204C7×C22⋊C4—22no—
205C7×C22⋊C4—22no—
206C7×C22⋊C4—22no—
207C7×C22⋊C4—22no—
208C7×C22⋊C4—22no—
209C7×C22⋊C4—22no—
210C7×C22⋊C4—22no—
211C7×C22⋊C4—22no—
212C7×C22⋊C4—24no—
213C7×C22⋊C4—24no—
214C7×C22⋊C4—48no—
215C7×C22⋊C4—44no—
216C7×C22⋊C4—44no—
217C7×C22⋊C4#4114yescomplemented
218C42×C7—22no—
219C42×C7—22no—
220C24×C4C22×C711yescharacteristic
221#551—22no—
222#551—22no—
223#546—22no—
224#546—22no—
225#543C22×C711yescharacteristic
226#538—21no—
227#538—21no—
228#534C22×C711yescharacteristic
229#525—21no—
230#525—21no—
231C2×#123—22no—
232C2×#120C22×C712yes—
233#518C22×C712yes—
234C22×C4⋊C4—21no—
235C22×C22⋊C4C22×C711yescharacteristic
236C22×C22⋊C4C22×C711yescharacteristic
237C2×#99C22×C714yes—
238#386—22no—
239#386C22×C714yes—
240#385—22no—
241#385—21no—
242#385—21no—
243#385C22×C714yes—
244#385C22×C712yes—
245#385C22×C712yes—
246#381—22no—
247#379C22×C712yes—
248C2×#95—21no—
249#342C22×C712yes—
250#342C22×C714yes—
251#342C22×C712yes—
252C23×C7—21no—
253C23×C7—21no—
254C23×C7—22no—
255C23×C7—22no—
256C23×C7—21no—
257C23×C7—21no—
258C23×C7—24no—
259C23×C7—24no—
260C23×C7—24no—
261C23×C7—24no—
262C23×C7—22no—
263C23×C7—24no—
264C23×C7—41no—
265C23×C7—42no—
266C23×C7—41no—
267C23×C7—41no—
268C23×C7—42no—
269C23×C7—41no—
270C23×C7—48no—
271C23×C7—48no—
272C23×C7—42no—
273C23×C7—42no—
274C23×C7#12012yes—
275C23×C7#12011yescharacteristic
276C23×C7#12011yescharacteristic
277C23×C7C22×D411yescharacteristic
278C23×C7C22×D412yes—
279C23×C7C2×#4112yes—
280C23×C72−1+412yes—
281C7×Q8—81no—
282C7×Q8—81no—
283C7×D4—42no—
284C7×D4—42no—
285C7×D4—44no—
286C7×D4—44no—
287C7×D4—44no—
288C7×D4—82no—
289C7×D4—82no—
290C7×D4—81no—
291C7×D4—81no—
292C2×C4×C7—28no—
293C2×C4×C7—24no—
294C2×C4×C7—22no—
295C2×C4×C7—21no—
296C2×C4×C7—21no—
297C2×C4×C7—22no—
298C2×C4×C7—22no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—24no—
302C2×C4×C7—24no—
303C2×C4×C7—42no—
304C2×C4×C7—48no—
305C2×C4×C7—42no—
306C2×C4×C7—42no—
307C2×C4×C7—44no—
308C2×C4×C7—44no—
309C2×C4×C7—48no—
310C2×C4×C7—42no—
311C2×C4×C7—44no—
312C2×C4×C7—42no—
313C2×C4×C7—42no—
314C2×C4×C7—41no—
315C2×C4×C7—41no—
316C2×C4×C7—42no—
317C2×C4×C7—41no—
318C2×C4×C7—41no—
319C2×C4×C7—42no—
320C2×C4×C7—44no—
321C2×C4×C7—44no—
322C2×C4×C7—44no—
323C2×C4×C7—81no—
324C2×C4×C7—81no—
325C2×C4×C7—82no—
326C2×C4×C7—82no—
327C25C23×C711yescharacteristic
328C2×#41—21no—
329C2×#41—21no—
330C22×D4C7×D412yes—
331C22×D4C7×D412yes—
332C22×D4C23×C712yes—
333C23×C4—21no—
334C23×C4—21no—
335C23×C4—22no—
336C23×C4—21no—
337C23×C4—22no—
338C23×C4—44no—
339C23×C4C7×D411yescharacteristic
340C23×C4C7×D411yescharacteristic
341C23×C4C23×C712yes—
342#126—22no—
343#126—22no—
344#124—22no—
345#124—22no—
346#123—22no—
347#123—22no—
348#123—22no—
349#123—22no—
350#123—22no—
351#123—48no—
352C22⋊Q8—21no—
353C22⋊Q8—21no—
354C22⋊Q8—21no—
355C22⋊Q8—21no—
356C22⋊Q8—22no—
357C22⋊Q8—22no—
358#121—21no—
359#121—21no—
360#121—22no—
361#121—22no—
362#121—21no—
363#121—21no—
364#120—22no—
365#120—22no—
366#120—22no—
367#120—22no—
368#120—22no—
369C4×D4—22no—
370C4×D4—22no—
371C4×D4—22no—
372C4×D4—22no—
373C4×D4—22no—
374C4×D4—22no—
375#117—21no—
376#117—21no—
377#117—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—44no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—24no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4C7×D411yescharacteristic
391C2×C22⋊C4C7×D411yescharacteristic
392C2×C22⋊C4C7×D412yes—
393C2×C22⋊C4C7×D412yes—
394C2×C22⋊C4C23×C714yes—
395C2×C22⋊C4C23×C711yescharacteristic
396C2×C22⋊C4C23×C711yescharacteristic
397C2×C22⋊C4C23×C712yes—
398C2×C22⋊C4C23×C712yes—
399#99—28no—
400#95—22no—
401#95—22no—
402#95—24no—
403#95—24no—
404C22×C7—24no—
405C22×C7—24no—
406C22×C7—21no—
407C22×C7—21no—
408C22×C7—21no—
409C22×C7—22no—
410C22×C7—21no—
411C22×C7—21no—
412C22×C7—22no—
413C22×C7—21no—
414C22×C7—22no—
415C22×C7—22no—
416C22×C7—22no—
417C22×C7—42no—
418C22×C7—42no—
419C22×C7—48no—
420C22×C7—48no—
421C22×C7—42no—
422C22×C7—42no—
423C22×C7—42no—
424C22×C7—42no—
425C22×C7—42no—
426C22×C7—84no—
427C22×C7B(4,𝔽2)12yes—
428C22×C7#45812yes—
429C22×C7C2×#12011yescharacteristic
430C22×C7#52511yescharacteristic
431C22×C7#53611yescharacteristic
432C22×C7#54714yes—
433C4×C7—44no—
434C4×C7—42no—
435C4×C7—44no—
436C4×C7—41no—
437C4×C7—41no—
438C4×C7—42no—
439C4×C7—41no—
440C4×C7—41no—
441C4×C7—42no—
442C4×C7—84no—
443C24—22no—
444C24—22no—
445C24—42no—
446C24—42no—
447C24C2×C7×D411yescharacteristic
448C24C2×C7×D412yes—
449C24C2×C7×D412yes—
450C24C2×C7×D412yes—
451C24C2×C7×D411yescharacteristic
452C24C24×C711yescharacteristic
453#41—82no—
454#41—82no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—24no—
462C2×D4—24no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—44no—
468C2×D4—44no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—28no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—24no—
482C22×C4—44no—
483C22×C4—48no—
484C22×C4—44no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—44no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4C2×C7×D412yes—
495C22×C4C2×C7×D411yescharacteristic
496C22×C4C2×C7×D412yes—
497C22×C4C2×C7×D411yescharacteristic
498C22×C4C2×C7×D411yescharacteristic
499C22×C4C2×C7×D412yes—
500C22×C4C2×C7×D411yescharacteristic
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—48no—
512C22⋊C4—24no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—48no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4C7×#4114yescomplemented
534C42—22no—
535C42—22no—
536C2×C7—21no—
537C2×C7—24no—
538C2×C7—21no—
539C2×C7—22no—
540C2×C7—22no—
541C2×C7—42no—
542C2×C7—82no—
543C2×C7#240611yescharacteristic
544C2×C7C2×B(4,𝔽2)11yescharacteristic
545C2×C7C2×#45811yescharacteristic
546C23—21no—
547C23—21no—
548C23—22no—
549C23—22no—
550C23—21no—
551C23—21no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—22no—
557C23—24no—
558C23—41no—
559C23—42no—
560C23—41no—
561C23—41no—
562C23—42no—
563C23—41no—
564C23—48no—
565C23—48no—
566C23—42no—
567C23—42no—
568C23C7×#12012yes—
569C23C7×#12011yescharacteristic
570C23C7×#12011yescharacteristic
571C23C22×C7×D411yescharacteristic
572C23C22×C7×D412yes—
573C23C2×C7×#4112yes—
574C23C7×2−1+412yes—
575Q8—81no—
576Q8—81no—
577D4—42no—
578D4—42no—
579D4—44no—
580D4—44no—
581D4—44no—
582D4—82no—
583D4—82no—
584D4—81no—
585D4—81no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—42no—
598C2×C4—48no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—48no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—82no—
620C2×C4—82no—
621C7#2411211yescharacteristic, direct product factor, Sylow
622C22—24no—
623C22—24no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—22no—
628C22—21no—
629C22—21no—
630C22—22no—
631C22—21no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—42no—
636C22—42no—
637C22—48no—
638C22—48no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—84no—
645C22C7×B(4,𝔽2)12yes—
646C22C7×#45812yes—
647C22C2×C7×#12011yescharacteristic
648C22C7×#52511yescharacteristic
649C22C7×#53611yescharacteristic
650C22C7×#54714yes—
651C4—44no—
652C4—42no—
653C4—44no—
654C4—41no—
655C4—41no—
656C4—42no—
657C4—41no—
658C4—41no—
659C4—42no—
660C4—84no—
661C2—21no—
662C2—24no—
663C2—21no—
664C2—22no—
665C2—22no—
666C2—42no—
667C2—82no—
668C2C7×#240611yescharacteristic
669C2C2×C7×B(4,𝔽2)11yescharacteristic
670C2C2×C7×#45811yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC24C24×C7
prime radicalC7×#3490C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C25×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2849C2semidirect
3. C2×C7×#385C2semidirect
4. C7×#2062C2semidirect
5. C7×#2044C2semidirect
6. C7×#1898C2semidirect
7. C7×#1867C2semidirect
8. C7×#1819C2semidirect
41. #24112C7direct (N × H)
120. #2849C2×C7semidirect
121. C2×#385C2×C7semidirect
122. #2062C2×C7semidirect
123. #2044C2×C7semidirect
124. #1898C2×C7semidirect
125. #1867C2×C7semidirect
126. #1819C2×C7semidirect
217. C7×C22⋊C4#41semidirect
533. C22⋊C4C7×#41semidirect
621. C7#24112direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34902
C7×#28492
C2×C7×#3852
C7×#20622
C7×#20442
C7×#18982
C7×#18672
C7×#18192
#241127