Finite Group Databasebeta

← C107×#14506 #41040470

Subgroups — C107×#14506 #41040470

Order 27392.

1962 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#123×C107C211yesmaximal, characteristic
2C2×#388×C107C212yesmaximal, complemented
3C107×#1445C212yesmaximal, complemented
4C107×#1439C212yesmaximal, complemented
5C107×#1438C211yesmaximal, characteristic
6C107×#1438C211yesmaximal, characteristic
7C107×#1437C212yesmaximal, complemented
8C107×C4⋊#95C212yesmaximal, complemented
9C107×C22⋊#95C212yesmaximal, complemented
10C23×D4×C107C2211yescharacteristic
11C24×C4×C107C2211yescharacteristic
12C2×#123×C107—28no—
13C2×#123×C107C418yescomplemented
14C22×C4⋊C4×C107C2212yes—
15C22×C4⋊C4×C107C2212yes—
16C22×C22⋊C4×C107C2212yes—
17C22×C22⋊C4×C107C2212yes—
18C22×C22⋊C4×C107C2211yescharacteristic
19C22×C22⋊C4×C107C2212yes—
20C22×C22⋊C4×C107C2212yes—
21C22×C22⋊C4×C107C2211yescharacteristic
22C22×C22⋊C4×C107C2211yescharacteristic
23C22×C42×C107C2212yes—
24#388×C107—28no—
25C2×#95×C107C2212yes—
26C2×#95×C107C2212yes—
27C2×#95×C107C2212yes—
28C2×#95×C107C2212yes—
29C2×#95×C107C2212yes—
30C2×#95×C107C2212yes—
31C2×#95×C107C2211yescharacteristic
32C2×#95×C107C2212yes—
33C2×#95×C107C2211yescharacteristic
34C25×C107C2311yescharacteristic
35C25×C107C2311yescharacteristic
36C22×D4×C107—24no—
37C22×D4×C107—42no—
38C22×D4×C107—41no—
39C22×D4×C107—41no—
40C22×D4×C107C2×C414yes—
41C23×C4×C107—24no—
42C23×C4×C107—42no—
43C23×C4×C107—41no—
44C23×C4×C107—41no—
45C23×C4×C107C2×C414yes—
46C23×C4×C107C2312yes—
47C23×C4×C107C2312yes—
48C23×C4×C107C2312yes—
49C23×C4×C107C2311yescharacteristic
50C23×C4×C107C2312yes—
51C23×C4×C107C2312yes—
52C23×C4×C107C2312yes—
53#123×C107—232no—
54C2×C4⋊C4×C107—24no—
55C2×C4⋊C4×C107—24no—
56C2×C4⋊C4×C107—24no—
57C2×C4⋊C4×C107—24no—
58C2×C4⋊C4×C107—42no—
59C2×C4⋊C4×C107C2×C418yes—
60C2×C22⋊C4×C107—24no—
61C2×C22⋊C4×C107—24no—
62C2×C22⋊C4×C107—24no—
63C2×C22⋊C4×C107—24no—
64C2×C22⋊C4×C107—24no—
65C2×C22⋊C4×C107—24no—
66C2×C22⋊C4×C107—24no—
67C2×C22⋊C4×C107—24no—
68C2×C22⋊C4×C107—41no—
69C2×C22⋊C4×C107—42no—
70C2×C22⋊C4×C107—42no—
71C2×C22⋊C4×C107—42no—
72C2×C22⋊C4×C107—42no—
73C2×C22⋊C4×C107—41no—
74C2×C22⋊C4×C107—42no—
75C2×C22⋊C4×C107C2×C418yes—
76C2×C22⋊C4×C107C2×C414yes—
77C2×C22⋊C4×C107D418yes—
78C2×C42×C107—24no—
79C2×C42×C107—24no—
80C2×C42×C107—42no—
81#95×C107—24no—
82#95×C107—24no—
83#95×C107—41no—
84#95×C107—42no—
85#95×C107—42no—
86#95×C107—42no—
87#95×C107—42no—
88#95×C107—42no—
89#95×C107—41no—
90C24×C107—22no—
91C24×C107—22no—
92C24×C107—22no—
93C24×C107—22no—
94C24×C107—21no—
95C24×C107—21no—
96C24×C107—22no—
97C24×C107—22no—
98C24×C107—22no—
99C24×C107—22no—
100C24×C107—22no—
101C24×C107—22no—
102C24×C107—21no—
103C24×C107—21no—
104C24×C107—22no—
105C24×C107C22×C412yes—
106C24×C107C22×C412yes—
107C24×C107C2×D414yes—
108C24×C107C2411yescharacteristic
109C2×D4×C107—28no—
110C2×D4×C107—44no—
111C2×D4×C107—44no—
112C2×D4×C107—48no—
113C2×D4×C107—48no—
114C22×C4×C107—22no—
115C22×C4×C107—22no—
116C22×C4×C107—22no—
117C22×C4×C107—22no—
118C22×C4×C107—28no—
119C22×C4×C107—22no—
120C22×C4×C107—22no—
121C22×C4×C107—22no—
122C22×C4×C107—22no—
123C22×C4×C107—22no—
124C22×C4×C107—22no—
125C22×C4×C107—22no—
126C22×C4×C107—22no—
127C22×C4×C107—22no—
128C22×C4×C107—22no—
129C22×C4×C107—22no—
130C22×C4×C107—22no—
131C22×C4×C107—22no—
132C22×C4×C107—22no—
133C22×C4×C107—22no—
134C22×C4×C107—22no—
135C22×C4×C107—21no—
136C22×C4×C107—21no—
137C22×C4×C107—22no—
138C22×C4×C107—22no—
139C22×C4×C107—22no—
140C22×C4×C107—22no—
141C22×C4×C107—22no—
142C22×C4×C107—22no—
143C22×C4×C107—22no—
144C22×C4×C107—22no—
145C22×C4×C107—22no—
146C22×C4×C107—22no—
147C22×C4×C107—22no—
148C22×C4×C107—22no—
149C22×C4×C107—22no—
150C22×C4×C107—22no—
151C22×C4×C107—22no—
152C22×C4×C107—22no—
153C22×C4×C107—22no—
154C22×C4×C107—22no—
155C22×C4×C107—44no—
156C22×C4×C107—44no—
157C22×C4×C107—48no—
158C22×C4×C107—48no—
159C22×C4×C107C22×C414yes—
160C22×C4×C107C22×C412yes—
161C22×C4×C107C22×C414yes—
162C22×C4×C107C2×D414yes—
163C22×C4×C107C2×D414yes—
164C22×C4×C107#4114yes—
165C22×C4×C107#4114yes—
166C4⋊C4×C107—216no—
167C4⋊C4×C107—48no—
168C22⋊C4×C107—216no—
169C22⋊C4×C107—44no—
170C22⋊C4×C107—48no—
171C22⋊C4×C107—48no—
172C22⋊C4×C107—48no—
173C22⋊C4×C107—44no—
174C22⋊C4×C107C22⋊C4116yescomplemented
175C42×C107—48no—
176C23×C107—24no—
177C23×C107—22no—
178C23×C107—22no—
179C23×C107—24no—
180C23×C107—24no—
181C23×C107—24no—
182C23×C107—24no—
183C23×C107—24no—
184C23×C107—24no—
185C23×C107—24no—
186C23×C107—22no—
187C23×C107—24no—
188C23×C107—22no—
189C23×C107—22no—
190C23×C107—24no—
191C23×C107—42no—
192C23×C107—42no—
193C23×C107—42no—
194C23×C107—42no—
195C23×C107—41no—
196C23×C107—42no—
197C23×C107—41no—
198C23×C107—41no—
199C23×C107—42no—
200C23×C107—41no—
201C23×C107—42no—
202C23×C107—42no—
203C23×C107—42no—
204C23×C107—42no—
205C23×C107—42no—
206C23×C107—42no—
207C23×C107—42no—
208C23×C107—42no—
209C23×C107—42no—
210C23×C107—42no—
211C23×C107—41no—
212C23×C107—42no—
213C23×C107—41no—
214C23×C107—41no—
215C23×C107—42no—
216C23×C107—41no—
217C23×C107C2×C22⋊C414yes—
218C23×C107C23×C411yescharacteristic
219C23×C107C22×D412yes—
220C23×C107C2×#4112yes—
221C23×C107C2×#4112yes—
222C23×C107H(5,𝔽2)12yes—
223C23×C107H(5,𝔽2)11yescharacteristic
224C23×C107H(5,𝔽2)11yescharacteristic
225C23×C107H(5,𝔽2)12yes—
226C23×C1072−1+412yes—
227D4×C107—416no—
228C2×C4×C107—24no—
229C2×C4×C107—24no—
230C2×C4×C107—24no—
231C2×C4×C107—24no—
232C2×C4×C107—24no—
233C2×C4×C107—24no—
234C2×C4×C107—24no—
235C2×C4×C107—22no—
236C2×C4×C107—24no—
237C2×C4×C107—24no—
238C2×C4×C107—22no—
239C2×C4×C107—22no—
240C2×C4×C107—24no—
241C2×C4×C107—24no—
242C2×C4×C107—24no—
243C2×C4×C107—24no—
244C2×C4×C107—24no—
245C2×C4×C107—24no—
246C2×C4×C107—416no—
247C2×C4×C107—42no—
248C2×C4×C107—42no—
249C2×C4×C107—42no—
250C2×C4×C107—42no—
251C2×C4×C107—42no—
252C2×C4×C107—42no—
253C2×C4×C107—42no—
254C2×C4×C107—42no—
255C2×C4×C107—42no—
256C2×C4×C107—42no—
257C2×C4×C107—42no—
258C2×C4×C107—42no—
259C2×C4×C107—42no—
260C2×C4×C107—42no—
261C2×C4×C107—42no—
262C2×C4×C107—42no—
263C2×C4×C107—42no—
264C2×C4×C107—42no—
265C2×C4×C107—42no—
266C2×C4×C107—42no—
267C2×C4×C107—42no—
268C2×C4×C107—42no—
269C2×C4×C107—42no—
270C2×C4×C107—42no—
271C2×C4×C107—42no—
272C2×C4×C107—42no—
273C2×C4×C107—42no—
274C2×C4×C107—42no—
275C2×C4×C107C2×C22⋊C418yes—
276C22×C107—24no—
277C22×C107—28no—
278C22×C107—44no—
279C22×C107—42no—
280C22×C107—42no—
281C22×C107—44no—
282C22×C107—44no—
283C22×C107—44no—
284C22×C107—44no—
285C22×C107—42no—
286C22×C107—44no—
287C22×C107—44no—
288C22×C107—42no—
289C22×C107—42no—
290C22×C107—82no—
291C22×C107—82no—
292C22×C107—82no—
293C22×C107—82no—
294C22×C107C22×C22⋊C411yescharacteristic
295C22×C107C4×#4112yes—
296C22×C107#51811yescharacteristic
297C22×C107#51811yescharacteristic
298C22×C107#52012yes—
299C22×C107#53512yes—
300C22×C107#53512yes—
301C22×C107#53812yes—
302C22×C107#53912yes—
303C22×C107#54012yes—
304C22×C107#54212yes—
305C22×C107#54612yes—
306C22×C107#54612yes—
307C22×C107#54712yes—
308C22×C107#54712yes—
309C22×C107#55111yescharacteristic
310C22×C107#55312yes—
311C22×C107#55911yescharacteristic
312C22×C107#56011yescharacteristic
313C22×C107#56212yes—
314C22×C107#56311yescharacteristic
315C4×C107—28no—
316C4×C107—44no—
317C4×C107—42no—
318C4×C107—44no—
319C4×C107—44no—
320C4×C107—82no—
321C4×C107—82no—
322#14506C10711yesmaximal, characteristic, direct product factor, Sylow
323C2×C107—44no—
324C2×C107—82no—
325C2×C107#230911yescharacteristic
326C2×C107#233512yes—
327C2×C107#233912yes—
328C2×C107#237411yescharacteristic
329C2×C107#237611yescharacteristic
330C2×C107#270312yes—
331C2×C107#270612yes—
332C2×C107#271012yes—
333C2×C107#271812yes—
334C22×#123C2×C10711yescharacteristic
335C2×#388C2×C10712yescomplemented
336#1445C2×C10712yescomplemented
337#1439C2×C10712yescomplemented
338#1438C2×C10711yescharacteristic
339#1438C2×C10711yescharacteristic
340#1437C2×C10712yescomplemented
341C4⋊#95C2×C10712yescomplemented
342C22⋊#95C2×C10712yescomplemented
343C107#1450611yescharacteristic, direct product factor, Sylow
344C23×D4C22×C10711yescharacteristic
345C24×C4C22×C10711yescharacteristic
346C2×#123—28no—
347C2×#123C4×C10718yescomplemented
348C22×C4⋊C4C22×C10712yes—
349C22×C4⋊C4C22×C10712yes—
350C22×C22⋊C4C22×C10712yes—
351C22×C22⋊C4C22×C10712yes—
352C22×C22⋊C4C22×C10711yescharacteristic
353C22×C22⋊C4C22×C10712yes—
354C22×C22⋊C4C22×C10712yes—
355C22×C22⋊C4C22×C10711yescharacteristic
356C22×C22⋊C4C22×C10711yescharacteristic
357C22×C42C22×C10712yes—
358#388—28no—
359C2×#95C22×C10712yes—
360C2×#95C22×C10712yes—
361C2×#95C22×C10712yes—
362C2×#95C22×C10712yes—
363C2×#95C22×C10712yes—
364C2×#95C22×C10712yes—
365C2×#95C22×C10711yescharacteristic
366C2×#95C22×C10712yes—
367C2×#95C22×C10711yescharacteristic
368C25C23×C10711yescharacteristic
369C25C23×C10711yescharacteristic
370C22×D4—24no—
371C22×D4—42no—
372C22×D4—41no—
373C22×D4—41no—
374C22×D4C2×C4×C10714yes—
375C23×C4—24no—
376C23×C4—42no—
377C23×C4—41no—
378C23×C4—41no—
379C23×C4C2×C4×C10714yes—
380C23×C4C23×C10712yes—
381C23×C4C23×C10712yes—
382C23×C4C23×C10712yes—
383C23×C4C23×C10711yescharacteristic
384C23×C4C23×C10712yes—
385C23×C4C23×C10712yes—
386C23×C4C23×C10712yes—
387#123—232no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—24no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—42no—
393C2×C4⋊C4C2×C4×C10718yes—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—41no—
403C2×C22⋊C4—42no—
404C2×C22⋊C4—42no—
405C2×C22⋊C4—42no—
406C2×C22⋊C4—42no—
407C2×C22⋊C4—41no—
408C2×C22⋊C4—42no—
409C2×C22⋊C4C2×C4×C10718yes—
410C2×C22⋊C4C2×C4×C10714yes—
411C2×C22⋊C4D4×C10718yes—
412C2×C42—24no—
413C2×C42—24no—
414C2×C42—42no—
415#95—24no—
416#95—24no—
417#95—41no—
418#95—42no—
419#95—42no—
420#95—42no—
421#95—42no—
422#95—42no—
423#95—41no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—21no—
429C24—21no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—21no—
437C24—21no—
438C24—22no—
439C24C22×C4×C10712yes—
440C24C22×C4×C10712yes—
441C24C2×D4×C10714yes—
442C24C24×C10711yescharacteristic
443C2×D4—28no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—48no—
447C2×D4—48no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—28no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4C22×C4×C10714yes—
494C22×C4C22×C4×C10712yes—
495C22×C4C22×C4×C10714yes—
496C22×C4C2×D4×C10714yes—
497C22×C4C2×D4×C10714yes—
498C22×C4#41×C10714yes—
499C22×C4#41×C10714yes—
500C4⋊C4—216no—
501C4⋊C4—48no—
502C22⋊C4—216no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4C22⋊C4×C107116yescomplemented
509C42—48no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—22no—
521C23—24no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—42no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23—42no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—41no—
551C23C2×C22⋊C4×C10714yes—
552C23C23×C4×C10711yescharacteristic
553C23C22×D4×C10712yes—
554C23C2×#41×C10712yes—
555C23C2×#41×C10712yes—
556C23H(5,𝔽2)×C10712yes—
557C23H(5,𝔽2)×C10711yescharacteristic
558C23H(5,𝔽2)×C10711yescharacteristic
559C23H(5,𝔽2)×C10712yes—
560C232−1+4×C10712yes—
561D4—416no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—416no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4C2×C22⋊C4×C10718yes—
610C22—24no—
611C22—28no—
612C22—42no—
613C22—42no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—44no—
623C22—44no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22C22×C22⋊C4×C10711yescharacteristic
629C22C4×#41×C10712yes—
630C22#518×C10711yescharacteristic
631C22#518×C10711yescharacteristic
632C22#520×C10712yes—
633C22#535×C10712yes—
634C22#535×C10712yes—
635C22#538×C10712yes—
636C22#539×C10712yes—
637C22#540×C10712yes—
638C22#542×C10712yes—
639C22#546×C10712yes—
640C22#546×C10712yes—
641C22#547×C10712yes—
642C22#547×C10712yes—
643C22#551×C10711yescharacteristic
644C22#553×C10712yes—
645C22#559×C10711yescharacteristic
646C22#560×C10711yescharacteristic
647C22#562×C10712yes—
648C22#563×C10711yescharacteristic
649C4—28no—
650C4—44no—
651C4—44no—
652C4—44no—
653C4—42no—
654C4—82no—
655C4—82no—
656C2—44no—
657C2—82no—
658C2C107×#230911yescharacteristic
659C2C107×#233512yes—
660C2C107×#233912yes—
661C2C107×#237411yescharacteristic
662C2C107×#237611yescharacteristic
663C2C107×#270312yes—
664C2C107×#270612yes—
665C2C107×#271012yes—
666C2C107×#271812yes—
subgroupG/H
center, socleC24×C107C24
commutatorC23C23×C4×C107
Frattini, cosocleC24C24×C107
prime radicalC23×D4×C107C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C107 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#388×C107C2semidirect
3. C107×#1445C2semidirect
4. C107×#1439C2semidirect
7. C107×#1437C2semidirect
8. C107×C4⋊#95C2semidirect
9. C107×C22⋊#95C2semidirect
13. C2×#123×C107C4semidirect
174. C22⋊C4×C107C22⋊C4semidirect
322. #14506C107direct (N × H)
335. C2×#388C2×C107semidirect
336. #1445C2×C107semidirect
337. #1439C2×C107semidirect
340. #1437C2×C107semidirect
341. C4⋊#95C2×C107semidirect
342. C22⋊#95C2×C107semidirect
343. C107#14506direct (N × H)
347. C2×#123C4×C107semidirect
508. C22⋊C4C22⋊C4×C107semidirect

Maximal subgroups

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

subgroupindex
C22×#123×C1072
C2×#388×C1072
C107×#14452
C107×#14392
C107×#14382
C107×#14372
C107×C4⋊#952
C107×C22⋊#952
#14506107