Finite Group Databasebeta

← C107×#18176 #41044140

Subgroups — C107×#18176 #41044140

Order 27392.

1034 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C107×#3491C211yesmaximal, characteristic, complemented
2C107×#3114C212yesmaximal, complemented
3C107×#2392C211yesmaximal, characteristic, complemented
4C107×#1609C212yesmaximal, complemented
5C107×#1608C211yesmaximal, characteristic, complemented
6C107×#1607C211yesmaximal, characteristic, complemented
7C107×#1597C212yesmaximal, complemented
8C107×#1596C211yesmaximal, characteristic, complemented
9C107×#1595C211yesmaximal, characteristic
10C107×#1509C212yesmaximal, complemented
11C107×#1464C211yesmaximal, characteristic, complemented
12#552×C107—22no—
13#552×C107—22no—
14#551×C107—22no—
15#551×C107—22no—
16#543×C107C2211yescharacteristic
17#539×C107—21no—
18#539×C107C2212yes—
19#534×C107C2211yescharacteristic
20#529×C107—21no—
21#529×C107C2212yes—
22C2×#123×C107—21no—
23#518×C107—21no—
24C2×C4×D4×C107—21no—
25C2×C4×D4×C107C2211yescharacteristic, complemented
26C2×#117×C107C2211yescharacteristic
27#483×C107C2212yescomplemented
28#480×C107C2212yes—
29#478×C107—22no—
30#463×C107—22no—
31#450×C107C2212yescomplemented
32#447×C107C2212yes—
33#442×C107—22no—
34#440×C107—22no—
35#432×C107C2212yescomplemented
36#398×C107C2211yescharacteristic, complemented
37#397×C107C2211yescharacteristic
38#396×C107—21no—
39#394×C107—21no—
40#394×C107C2211yescharacteristic, complemented
41#392×C107C2211yescharacteristic
42C4×C4⋊C4×C107C2211yescharacteristic, complemented
43C4×C22⋊C4×C107—21no—
44#328×C107C2212yescomplemented
45#328×C107C2212yescomplemented
46#327×C107C2212yes—
47#327×C107C2212yes—
48#324×C107C2212yescomplemented
49#323×C107C2211yescharacteristic
50#323×C107C2211yescharacteristic
51C22×D4×C107—21no—
52C22×D4×C107—21no—
53C22×D4×C107C2311yescharacteristic
54C23×C4×C107—21no—
55C23×C4×C107D411yescharacteristic
56C2×SD16×C107—22no—
57C2×D8×C107—22no—
58#126×C107—22no—
59#126×C107—22no—
60#126×C107—22no—
61#126×C107—22no—
62#125×C107—21no—
63#125×C107—22no—
64#125×C107—22no—
65#125×C107D412yescomplemented
66#124×C107—21no—
67#124×C107D412yescomplemented
68#123×C107—22no—
69#123×C107—22no—
70#123×C107—22no—
71#123×C107—22no—
72#123×C107—22no—
73#123×C107—22no—
74#123×C107—22no—
75C22⋊Q8×C107—22no—
76C22⋊Q8×C107—22no—
77C22⋊Q8×C107—22no—
78C22⋊Q8×C107—21no—
79C22⋊Q8×C107—22no—
80C22⋊Q8×C107C2312yes—
81#121×C107—21no—
82#121×C107C2312yes—
83#120×C107—22no—
84C4×Q8×C107—21no—
85C4×Q8×C107D412yescomplemented
86C4×D4×C107—22no—
87C4×D4×C107—22no—
88C4×D4×C107—21no—
89C4×D4×C107—22no—
90C4×D4×C107—22no—
91C4×D4×C107—22no—
92C4×D4×C107—22no—
93C4×D4×C107—44no—
94C4×D4×C107D412yescomplemented
95#117×C107—22no—
96#117×C107—22no—
97#117×C107—22no—
98#117×C107—22no—
99#117×C107—22no—
100C2×C4⋊C4×C107—21no—
101C2×C4⋊C4×C107—21no—
102C2×C4⋊C4×C107—21no—
103C2×C4⋊C4×C107—21no—
104C2×C4⋊C4×C107D411yescharacteristic
105C2×C4⋊C4×C107C2311yescharacteristic
106C2×C4⋊C4×C107C2311yescharacteristic
107C2×C22⋊C4×C107—22no—
108C2×C22⋊C4×C107—21no—
109C2×C22⋊C4×C107—21no—
110C2×C22⋊C4×C107—21no—
111C2×C22⋊C4×C107—21no—
112C2×C22⋊C4×C107—21no—
113C2×C22⋊C4×C107—21no—
114C2×C22⋊C4×C107—21no—
115C2×C22⋊C4×C107—21no—
116C2×C22⋊C4×C107D411yescharacteristic
117C2×C22⋊C4×C107C2311yescharacteristic
118C2×C42×C107—21no—
119C2×C42×C107D411yescharacteristic
120C2×C42×C107C2311yescharacteristic
121#107×C107—22no—
122#106×C107—22no—
123C4⋊C8×C107—22no—
124#103×C107—22no—
125#102×C107—22no—
126#102×C107—22no—
127#102×C107—22no—
128C22⋊C8×C107C2312yes—
129C22⋊C8×C107C2312yes—
130#97×C107—22no—
131#95×C107—21no—
132#95×C107—21no—
133#95×C107C2311yescharacteristic
134#95×C107C2311yescharacteristic
135C24×C107—21no—
136C24×C107—21no—
137C24×C107C2×D411yescharacteristic
138C2×Q8×C107—21no—
139C2×Q8×C107C2×D412yes—
140C2×D4×C107—22no—
141C2×D4×C107—22no—
142C2×D4×C107—21no—
143C2×D4×C107—22no—
144C2×D4×C107—42no—
145C2×D4×C107—42no—
146C2×D4×C107—42no—
147C2×D4×C107—42no—
148C2×D4×C107—42no—
149C2×D4×C107—42no—
150C2×D4×C107C2×D412yes—
151C22×C4×C107—21no—
152C22×C4×C107—22no—
153C22×C4×C107—21no—
154C22×C4×C107—21no—
155C22×C4×C107—21no—
156C22×C4×C107—22no—
157C22×C4×C107—21no—
158C22×C4×C107—21no—
159C22×C4×C107—22no—
160C22×C4×C107—21no—
161C22×C4×C107—21no—
162C22×C4×C107—41no—
163C22×C4×C107—42no—
164C22×C4×C107—42no—
165C22×C4×C107—41no—
166C22×C4×C107—41no—
167C22×C4×C107C2×D411yescharacteristic
168C22×C4×C107C2×D411yescharacteristic
169C22×C4×C107C2×D411yescharacteristic
170C22×C4×C107C2×D411yescharacteristic
171C22×C4×C107C2×D411yescharacteristic
172C22×C4×C107C2411yescharacteristic
173SD16×C107—44no—
174D8×C107—44no—
175C2×C8×C107—22no—
176C2×C8×C107—22no—
177C4⋊C4×C107—22no—
178C4⋊C4×C107—22no—
179C4⋊C4×C107—22no—
180C4⋊C4×C107—22no—
181C4⋊C4×C107—21no—
182C4⋊C4×C107—21no—
183C4⋊C4×C107—22no—
184C4⋊C4×C107—22no—
185C4⋊C4×C107—22no—
186C4⋊C4×C107—22no—
187C4⋊C4×C107—22no—
188C4⋊C4×C107—22no—
189C4⋊C4×C107—22no—
190C4⋊C4×C107—42no—
191C4⋊C4×C107—42no—
192C4⋊C4×C107C2×D412yes—
193C4⋊C4×C107C2×D412yes—
194C22⋊C4×C107—22no—
195C22⋊C4×C107—22no—
196C22⋊C4×C107—22no—
197C22⋊C4×C107—22no—
198C22⋊C4×C107—22no—
199C22⋊C4×C107—22no—
200C22⋊C4×C107—22no—
201C22⋊C4×C107—22no—
202C22⋊C4×C107—22no—
203C22⋊C4×C107—22no—
204C22⋊C4×C107—22no—
205C22⋊C4×C107—22no—
206C22⋊C4×C107—42no—
207C22⋊C4×C107—42no—
208C22⋊C4×C107—42no—
209C22⋊C4×C107—42no—
210C22⋊C4×C107—42no—
211C42×C107—22no—
212C42×C107—21no—
213C42×C107—21no—
214C42×C107—22no—
215C42×C107—42no—
216C42×C107C2×D412yes—
217C42×C107C2×D412yes—
218C23×C107—21no—
219C23×C107—21no—
220C23×C107—22no—
221C23×C107—22no—
222C23×C107—41no—
223C23×C107—41no—
224C23×C107—41no—
225C23×C107—41no—
226C23×C107—41no—
227C23×C107—42no—
228C23×C107—41no—
229C23×C107—41no—
230C23×C107—41no—
231C23×C107—81no—
232C23×C107—81no—
233C23×C107AGL(1,ℤ/8)12yes—
234C23×C107C22×D411yescharacteristic
235Q8×C107—42no—
236Q8×C107—81no—
237D4×C107—24no—
238D4×C107—42no—
239D4×C107—42no—
240D4×C107—44no—
241D4×C107—48no—
242D4×C107—42no—
243D4×C107—42no—
244D4×C107—81no—
245C2×C4×C107—21no—
246C2×C4×C107—21no—
247C2×C4×C107—22no—
248C2×C4×C107—21no—
249C2×C4×C107—21no—
250C2×C4×C107—22no—
251C2×C4×C107—22no—
252C2×C4×C107—22no—
253C2×C4×C107—21no—
254C2×C4×C107—21no—
255C2×C4×C107—21no—
256C2×C4×C107—41no—
257C2×C4×C107—41no—
258C2×C4×C107—42no—
259C2×C4×C107—42no—
260C2×C4×C107—41no—
261C2×C4×C107—42no—
262C2×C4×C107—41no—
263C2×C4×C107—41no—
264C2×C4×C107—41no—
265C2×C4×C107—41no—
266C2×C4×C107—41no—
267C2×C4×C107—41no—
268C2×C4×C107—41no—
269C2×C4×C107—41no—
270C2×C4×C107—41no—
271C2×C4×C107—82no—
272C2×C4×C107—82no—
273C2×C4×C107—82no—
274C2×C4×C107—81no—
275C2×C4×C107—82no—
276C2×C4×C107—81no—
277C2×C4×C107#12012yes—
278C2×C4×C107#12011yescharacteristic
279C2×C4×C107#12011yescharacteristic
280C2×C4×C107C22×D412yes—
281C8×C107—42no—
282C8×C107—42no—
283C22×C107—21no—
284C22×C107—21no—
285C22×C107—22no—
286C22×C107—42no—
287C22×C107—42no—
288C22×C107—42no—
289C22×C107—41no—
290C22×C107—41no—
291C22×C107—42no—
292C22×C107—42no—
293C22×C107—42no—
294C22×C107—42no—
295C22×C107—81no—
296C22×C107—81no—
297C22×C107—81no—
298C22×C107—81no—
299C22×C107—81no—
300C22×C107—162no—
301C22×C107C2×#12011yescharacteristic
302C22×C107C2×AGL(1,ℤ/8)11yescharacteristic
303C22×C107#57511yescharacteristic
304C4×C107—21no—
305C4×C107—21no—
306C4×C107—22no—
307C4×C107—22no—
308C4×C107—42no—
309C4×C107—41no—
310C4×C107—42no—
311C4×C107—41no—
312C4×C107—42no—
313C4×C107—41no—
314C4×C107—42no—
315C4×C107—81no—
316C4×C107—81no—
317#18176C10711yesmaximal, characteristic, direct product factor, Sylow
318C2×C107—21no—
319C2×C107—21no—
320C2×C107—42no—
321C2×C107—82no—
322C2×C107—82no—
323C2×C107#300611yescharacteristic
324C2×C107#301811yescharacteristic
325C2×C107#302511yescharacteristic
326#3491C2×C10711yescharacteristic, complemented
327#3114C2×C10712yescomplemented
328#2392C2×C10711yescharacteristic, complemented
329#1609C2×C10712yescomplemented
330#1608C2×C10711yescharacteristic, complemented
331#1607C2×C10711yescharacteristic, complemented
332#1597C2×C10712yescomplemented
333#1596C2×C10711yescharacteristic, complemented
334#1595C2×C10711yescharacteristic
335#1509C2×C10712yescomplemented
336#1464C2×C10711yescharacteristic, complemented
337C107#1817611yescharacteristic, direct product factor, Sylow
338#552—22no—
339#552—22no—
340#551—22no—
341#551—22no—
342#543C22×C10711yescharacteristic
343#539—21no—
344#539C22×C10712yes—
345#534C22×C10711yescharacteristic
346#529—21no—
347#529C22×C10712yes—
348C2×#123—21no—
349#518—21no—
350C2×C4×D4—21no—
351C2×C4×D4C22×C10711yescharacteristic, complemented
352C2×#117C22×C10711yescharacteristic
353#483C22×C10712yescomplemented
354#480C22×C10712yes—
355#478—22no—
356#463—22no—
357#450C22×C10712yescomplemented
358#447C22×C10712yes—
359#442—22no—
360#440—22no—
361#432C22×C10712yescomplemented
362#398C22×C10711yescharacteristic, complemented
363#397C22×C10711yescharacteristic
364#396—21no—
365#394—21no—
366#394C22×C10711yescharacteristic, complemented
367#392C22×C10711yescharacteristic
368C4×C4⋊C4C22×C10711yescharacteristic, complemented
369C4×C22⋊C4—21no—
370#328C22×C10712yescomplemented
371#328C22×C10712yescomplemented
372#327C22×C10712yes—
373#327C22×C10712yes—
374#324C22×C10712yescomplemented
375#323C22×C10711yescharacteristic
376#323C22×C10711yescharacteristic
377C22×D4—21no—
378C22×D4—21no—
379C22×D4C23×C10711yescharacteristic
380C23×C4—21no—
381C23×C4D4×C10711yescharacteristic
382C2×SD16—22no—
383C2×D8—22no—
384#126—22no—
385#126—22no—
386#126—22no—
387#126—22no—
388#125—21no—
389#125—22no—
390#125—22no—
391#125D4×C10712yescomplemented
392#124—21no—
393#124D4×C10712yescomplemented
394#123—22no—
395#123—22no—
396#123—22no—
397#123—22no—
398#123—22no—
399#123—22no—
400#123—22no—
401C22⋊Q8—22no—
402C22⋊Q8—22no—
403C22⋊Q8—22no—
404C22⋊Q8—21no—
405C22⋊Q8—22no—
406C22⋊Q8C23×C10712yes—
407#121—21no—
408#121C23×C10712yes—
409#120—22no—
410C4×Q8—21no—
411C4×Q8D4×C10712yescomplemented
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—21no—
415C4×D4—22no—
416C4×D4—22no—
417C4×D4—22no—
418C4×D4—22no—
419C4×D4—44no—
420C4×D4D4×C10712yescomplemented
421#117—22no—
422#117—22no—
423#117—22no—
424#117—22no—
425#117—22no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4D4×C10711yescharacteristic
431C2×C4⋊C4C23×C10711yescharacteristic
432C2×C4⋊C4C23×C10711yescharacteristic
433C2×C22⋊C4—22no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4D4×C10711yescharacteristic
443C2×C22⋊C4C23×C10711yescharacteristic
444C2×C42—21no—
445C2×C42D4×C10711yescharacteristic
446C2×C42C23×C10711yescharacteristic
447#107—22no—
448#106—22no—
449C4⋊C8—22no—
450#103—22no—
451#102—22no—
452#102—22no—
453#102—22no—
454C22⋊C8C23×C10712yes—
455C22⋊C8C23×C10712yes—
456#97—22no—
457#95—21no—
458#95—21no—
459#95C23×C10711yescharacteristic
460#95C23×C10711yescharacteristic
461C24—21no—
462C24—21no—
463C24C2×D4×C10711yescharacteristic
464C2×Q8—21no—
465C2×Q8C2×D4×C10712yes—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—22no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4C2×D4×C10712yes—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—41no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4C2×D4×C10711yescharacteristic
494C22×C4C2×D4×C10711yescharacteristic
495C22×C4C2×D4×C10711yescharacteristic
496C22×C4C2×D4×C10711yescharacteristic
497C22×C4C2×D4×C10711yescharacteristic
498C22×C4C24×C10711yescharacteristic
499SD16—44no—
500D8—44no—
501C2×C8—22no—
502C2×C8—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4C2×D4×C10712yes—
519C4⋊C4C2×D4×C10712yes—
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—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C42—22no—
538C42—21no—
539C42—21no—
540C42—22no—
541C42—42no—
542C42C2×D4×C10712yes—
543C42C2×D4×C10712yes—
544C23—21no—
545C23—21no—
546C23—22no—
547C23—22no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—42no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—81no—
558C23—81no—
559C23AGL(1,ℤ/8)×C10712yes—
560C23C22×D4×C10711yescharacteristic
561Q8—42no—
562Q8—81no—
563D4—24no—
564D4—42no—
565D4—42no—
566D4—44no—
567D4—48no—
568D4—42no—
569D4—42no—
570D4—81no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—81no—
601C2×C4—82no—
602C2×C4—81no—
603C2×C4#120×C10712yes—
604C2×C4#120×C10711yescharacteristic
605C2×C4#120×C10711yescharacteristic
606C2×C4C22×D4×C10712yes—
607C8—42no—
608C8—42no—
609C22—21no—
610C22—21no—
611C22—22no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—41no—
616C22—41no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—162no—
627C22C2×#120×C10711yescharacteristic
628C22C2×AGL(1,ℤ/8)×C10711yescharacteristic
629C22#575×C10711yescharacteristic
630C4—21no—
631C4—21no—
632C4—22no—
633C4—22no—
634C4—42no—
635C4—41no—
636C4—42no—
637C4—41no—
638C4—42no—
639C4—41no—
640C4—42no—
641C4—81no—
642C4—81no—
643C2—21no—
644C2—21no—
645C2—42no—
646C2—82no—
647C2—82no—
648C2C107×#300611yescharacteristic
649C2C107×#301811yescharacteristic
650C2C107×#302511yescharacteristic
subgroupG/H
center, socleC22×C107C2×#120
commutator, Frattini, cosocleC22×C4C24×C107
prime radicalC107×#1595C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C107 ◁ C2×C42×C107 ◁ G

Complements

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

NH ≅ G/Ntype
1. C107×#3491C2semidirect
2. C107×#3114C2semidirect
3. C107×#2392C2semidirect
4. C107×#1609C2semidirect
5. C107×#1608C2semidirect
6. C107×#1607C2semidirect
7. C107×#1597C2semidirect
8. C107×#1596C2semidirect
10. C107×#1509C2semidirect
11. C107×#1464C2semidirect
25. C2×C4×D4×C107C22semidirect
27. #483×C107C22semidirect
31. #450×C107C22semidirect
35. #432×C107C22semidirect
36. #398×C107C22semidirect
40. #394×C107C22semidirect
42. C4×C4⋊C4×C107C22semidirect
44. #328×C107C22semidirect
45. #328×C107C22semidirect
48. #324×C107C22semidirect
65. #125×C107D4semidirect
67. #124×C107D4semidirect
85. C4×Q8×C107D4semidirect
94. C4×D4×C107D4semidirect
317. #18176C107direct (N × H)
326. #3491C2×C107semidirect
327. #3114C2×C107semidirect
328. #2392C2×C107semidirect
329. #1609C2×C107semidirect
330. #1608C2×C107semidirect
331. #1607C2×C107semidirect
332. #1597C2×C107semidirect
333. #1596C2×C107semidirect
335. #1509C2×C107semidirect
336. #1464C2×C107semidirect
337. C107#18176direct (N × H)
351. C2×C4×D4C22×C107semidirect
353. #483C22×C107semidirect
357. #450C22×C107semidirect
361. #432C22×C107semidirect
362. #398C22×C107semidirect
366. #394C22×C107semidirect
368. C4×C4⋊C4C22×C107semidirect
370. #328C22×C107semidirect
371. #328C22×C107semidirect
374. #324C22×C107semidirect
391. #125D4×C107semidirect
393. #124D4×C107semidirect
411. C4×Q8D4×C107semidirect
420. C4×D4D4×C107semidirect

Maximal subgroups

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

subgroupindex
C107×#34912
C107×#31142
C107×#23922
C107×#16092
C107×#16082
C107×#16072
C107×#15972
C107×#15962
C107×#15952
C107×#15092
C107×#14642
#18176107