Finite Group Databasebeta

← C109×#15182 #42133176

Subgroups — C109×#15182 #42133176

Order 27904.

1990 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊Q8×C109C211yesmaximal, characteristic
2C2×#391×C109C211yesmaximal, characteristic, complemented
3C2×#391×C109C211yesmaximal, characteristic, complemented
4C2×#384×C109C211yesmaximal, characteristic, complemented
5C2×#380×C109C211yesmaximal, characteristic
6C2×C4×C22⋊C4×C109C211yesmaximal, characteristic
7C109×#1441C214yesmaximal, complemented
8C109×#1439C214yesmaximal
9C109×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×Q8×C109C2211yescharacteristic
11C24×C4×C109C2211yescharacteristic
12C2×C22⋊Q8×C109—24no—
13C2×C22⋊Q8×C109—24no—
14C2×C22⋊Q8×C109—24no—
15C22×C4⋊C4×C109C2211yescharacteristic
16C22×C4⋊C4×C109C2212yes—
17C22×C4⋊C4×C109C2211yescharacteristic
18C22×C4⋊C4×C109C2214yes—
19C22×C4⋊C4×C109C2211yescharacteristic
20C22×C22⋊C4×C109C2212yes—
21C22×C22⋊C4×C109C2212yes—
22C22×C22⋊C4×C109C2212yes—
23C22×C42×C109C2211yescharacteristic
24C22×C42×C109C2211yescharacteristic
25#391×C109—24no—
26#391×C109—24no—
27#384×C109—24no—
28#380×C109—24no—
29C4×C22⋊C4×C109—24no—
30C2×#95×C109C2214yes—
31C2×#95×C109C2214yes—
32C2×#95×C109C2212yes—
33C2×#95×C109C2212yes—
34C2×#95×C109C2214yes—
35C25×C109C2311yescharacteristic
36C22×Q8×C109—22no—
37C22×Q8×C109—22no—
38C22×Q8×C109—22no—
39C22×Q8×C109—22no—
40C22×Q8×C109—22no—
41C22×Q8×C109—22no—
42C22×Q8×C109D414yes—
43C23×C4×C109—22no—
44C23×C4×C109—22no—
45C23×C4×C109—22no—
46C23×C4×C109—22no—
47C23×C4×C109—22no—
48C23×C4×C109—22no—
49C23×C4×C109D414yes—
50C23×C4×C109C2312yes—
51C23×C4×C109C2312yes—
52C23×C4×C109C2312yes—
53C23×C4×C109C2312yes—
54C23×C4×C109C2311yescharacteristic
55C23×C4×C109C2312yes—
56C23×C4×C109C2312yes—
57C23×C4×C109C2311yescharacteristic
58C22⋊Q8×C109—232no—
59C2×C4⋊C4×C109—24no—
60C2×C4⋊C4×C109—24no—
61C2×C4⋊C4×C109—24no—
62C2×C4⋊C4×C109—22no—
63C2×C4⋊C4×C109—22no—
64C2×C4⋊C4×C109—22no—
65C2×C4⋊C4×C109—22no—
66C2×C4⋊C4×C109—28no—
67C2×C4⋊C4×C109—22no—
68C2×C4⋊C4×C109—22no—
69C2×C4⋊C4×C109—22no—
70C2×C4⋊C4×C109—28no—
71C2×C4⋊C4×C109—22no—
72C2×C4⋊C4×C109—44no—
73C2×C4⋊C4×C109D414yes—
74C2×C22⋊C4×C109—24no—
75C2×C22⋊C4×C109—24no—
76C2×C22⋊C4×C109—24no—
77C2×C22⋊C4×C109—24no—
78C2×C22⋊C4×C109—24no—
79C2×C22⋊C4×C109—24no—
80C2×C22⋊C4×C109—44no—
81C2×C22⋊C4×C109Q818yescomplemented
82C2×C42×C109—22no—
83C2×C42×C109—22no—
84C2×C42×C109—22no—
85C2×C42×C109—22no—
86C2×C42×C109—22no—
87C2×C42×C109D414yes—
88#95×C109—24no—
89#95×C109—24no—
90#95×C109—28no—
91#95×C109—28no—
92#95×C109—44no—
93C24×C109—21no—
94C24×C109—21no—
95C24×C109—21no—
96C24×C109—22no—
97C24×C109—21no—
98C24×C109—21no—
99C24×C109—21no—
100C24×C109—22no—
101C24×C109—22no—
102C24×C109C2×D412yes—
103C24×C109C2×Q814yes—
104C24×C109C2411yescharacteristic
105C2×Q8×C109—28no—
106C2×Q8×C109—28no—
107C2×Q8×C109—28no—
108C2×Q8×C109—44no—
109C2×Q8×C109—44no—
110C2×Q8×C109—44no—
111C2×Q8×C109—44no—
112C22×C4×C109—22no—
113C22×C4×C109—22no—
114C22×C4×C109—22no—
115C22×C4×C109—22no—
116C22×C4×C109—22no—
117C22×C4×C109—21no—
118C22×C4×C109—22no—
119C22×C4×C109—22no—
120C22×C4×C109—22no—
121C22×C4×C109—22no—
122C22×C4×C109—22no—
123C22×C4×C109—22no—
124C22×C4×C109—21no—
125C22×C4×C109—21no—
126C22×C4×C109—22no—
127C22×C4×C109—22no—
128C22×C4×C109—24no—
129C22×C4×C109—21no—
130C22×C4×C109—21no—
131C22×C4×C109—21no—
132C22×C4×C109—22no—
133C22×C4×C109—22no—
134C22×C4×C109—22no—
135C22×C4×C109—22no—
136C22×C4×C109—24no—
137C22×C4×C109—22no—
138C22×C4×C109—28no—
139C22×C4×C109—28no—
140C22×C4×C109—28no—
141C22×C4×C109—22no—
142C22×C4×C109—22no—
143C22×C4×C109—22no—
144C22×C4×C109—22no—
145C22×C4×C109—22no—
146C22×C4×C109—22no—
147C22×C4×C109—24no—
148C22×C4×C109—24no—
149C22×C4×C109—24no—
150C22×C4×C109—24no—
151C22×C4×C109—44no—
152C22×C4×C109—44no—
153C22×C4×C109—44no—
154C22×C4×C109—44no—
155C22×C4×C109C2×D414yes—
156C22×C4×C109C2×D412yes—
157C22×C4×C109C2×D414yes—
158C22×C4×C109C2×D414yes—
159C22×C4×C109C2×D412yes—
160C22×C4×C109C2×D412yes—
161C22×C4×C109C2×D412yes—
162C22×C4×C109C2×D412yes—
163C22×C4×C109C2×Q814yes—
164C22×C4×C109C2×Q814yes—
165C22×C4×C109#4114yes—
166C22×C4×C109#4112yes—
167C22×C4×C109#4112yes—
168C22×C4×C109#4112yes—
169C4⋊C4×C109—216no—
170C4⋊C4×C109—28no—
171C4⋊C4×C109—416no—
172C4⋊C4×C109—44no—
173C4⋊C4×C109—44no—
174C22⋊C4×C109—216no—
175C22⋊C4×C109—216no—
176C22⋊C4×C109—48no—
177C42×C109—28no—
178C42×C109—44no—
179C23×C109—22no—
180C23×C109—22no—
181C23×C109—24no—
182C23×C109—22no—
183C23×C109—22no—
184C23×C109—24no—
185C23×C109—22no—
186C23×C109—22no—
187C23×C109—24no—
188C23×C109—24no—
189C23×C109—24no—
190C23×C109—24no—
191C23×C109—41no—
192C23×C109—41no—
193C23×C109—41no—
194C23×C109—41no—
195C23×C109—42no—
196C23×C109—42no—
197C23×C109—42no—
198C23×C109—42no—
199C23×C109—42no—
200C23×C109—42no—
201C23×C109C4⋊Q814yes—
202C23×C109C22×D411yescharacteristic
203C23×C109C22×D411yescharacteristic
204C23×C109C22×D411yescharacteristic
205C23×C109C22×D411yescharacteristic
206C23×C109C22×Q812yes—
207C23×C109C2×#4111yescharacteristic
208C23×C109C2×#4111yescharacteristic
209C23×C109C2×#4111yescharacteristic
210C23×C109C2×#4112yes—
211C23×C109H(5,𝔽2)12yes—
212C23×C1092−1+412yes—
213Q8×C109—416no—
214C2×C4×C109—24no—
215C2×C4×C109—24no—
216C2×C4×C109—24no—
217C2×C4×C109—24no—
218C2×C4×C109—24no—
219C2×C4×C109—24no—
220C2×C4×C109—24no—
221C2×C4×C109—24no—
222C2×C4×C109—22no—
223C2×C4×C109—22no—
224C2×C4×C109—24no—
225C2×C4×C109—24no—
226C2×C4×C109—22no—
227C2×C4×C109—24no—
228C2×C4×C109—28no—
229C2×C4×C109—24no—
230C2×C4×C109—24no—
231C2×C4×C109—42no—
232C2×C4×C109—41no—
233C2×C4×C109—41no—
234C2×C4×C109—42no—
235C2×C4×C109—42no—
236C2×C4×C109—41no—
237C2×C4×C109—42no—
238C2×C4×C109—41no—
239C2×C4×C109—42no—
240C2×C4×C109—42no—
241C2×C4×C109—42no—
242C2×C4×C109—42no—
243C2×C4×C109—44no—
244C2×C4×C109—42no—
245C2×C4×C109—42no—
246C2×C4×C109—416no—
247C2×C4×C109—42no—
248C2×C4×C109—42no—
249C2×C4×C109—42no—
250C2×C4×C109—42no—
251C2×C4×C109—42no—
252C2×C4×C109—42no—
253C2×C4×C109—44no—
254C2×C4×C109—44no—
255C2×C4×C109—44no—
256C2×C4×C109—44no—
257C2×C4×C109#12014yes—
258C2×C4×C109#12114yes—
259C2×C4×C109#12114yes—
260C2×C4×C109#12114yes—
261C2×C4×C109C22⋊Q818yescomplemented
262C2×C4×C109#12314yes—
263C2×C4×C109#12314yes—
264C2×C4×C109#12414yes—
265C22×C109—24no—
266C22×C109—24no—
267C22×C109—24no—
268C22×C109—42no—
269C22×C109—42no—
270C22×C109—44no—
271C22×C109—44no—
272C22×C109—42no—
273C22×C109—44no—
274C22×C109—42no—
275C22×C109—44no—
276C22×C109C2×#12011yescharacteristic
277C22×C109C2×#12111yescharacteristic
278C22×C109C2×#12111yescharacteristic
279C22×C109C2×#12111yescharacteristic
280C22×C109C2×C22⋊Q812yes—
281C22×C109C2×#12311yescharacteristic
282C22×C109C2×#12311yescharacteristic
283C22×C109C2×#12411yescharacteristic
284C22×C109C2×C4⋊Q811yescharacteristic
285C22×C109#53211yescharacteristic
286C22×C109#53312yes—
287C22×C109#53511yescharacteristic
288C22×C109#53611yescharacteristic
289C22×C109#53712yes—
290C22×C109#54212yes—
291C22×C109#54612yes—
292C22×C109D4×Q812yes—
293C22×C109D4×Q812yes—
294C22×C109#55012yes—
295C22×C109#55212yes—
296C22×C109#55412yes—
297C22×C109#55412yes—
298C22×C109#55612yes—
299C4×C109—28no—
300C4×C109—42no—
301C4×C109—44no—
302C4×C109—44no—
303C4×C109—44no—
304C4×C109—82no—
305C4×C109—82no—
306C4×C109#39418yes—
307#15182C10911yesmaximal, characteristic, direct product factor, Sylow
308C2×C109—44no—
309C2×C109C2×#39411yescharacteristic
310C2×C109#241712yes—
311C2×C109#242312yes—
312C2×C109#243111yescharacteristic
313C2×C109#243411yescharacteristic
314C2×C109#249212yes—
315C2×C109#249911yescharacteristic
316C2×C109#250211yescharacteristic
317C2×C109#253812yes—
318C2×C109#262011yescharacteristic
319C2×C109#285711yescharacteristic
320C22×C22⋊Q8C2×C10911yescharacteristic
321C2×#391C2×C10911yescharacteristic, complemented
322C2×#391C2×C10911yescharacteristic, complemented
323C2×#384C2×C10911yescharacteristic, complemented
324C2×#380C2×C10911yescharacteristic
325C2×C4×C22⋊C4C2×C10911yescharacteristic
326#1441C2×C10914yescomplemented
327#1439C2×C10914yes—
328C4⋊#95C2×C10911yescharacteristic, complemented
329C109#1518211yescharacteristic, direct product factor, Sylow
330C23×Q8C22×C10911yescharacteristic
331C24×C4C22×C10911yescharacteristic
332C2×C22⋊Q8—24no—
333C2×C22⋊Q8—24no—
334C2×C22⋊Q8—24no—
335C22×C4⋊C4C22×C10912yes—
336C22×C4⋊C4C22×C10911yescharacteristic
337C22×C4⋊C4C22×C10911yescharacteristic
338C22×C4⋊C4C22×C10914yes—
339C22×C4⋊C4C22×C10911yescharacteristic
340C22×C22⋊C4C22×C10912yes—
341C22×C22⋊C4C22×C10912yes—
342C22×C22⋊C4C22×C10912yes—
343C22×C42C22×C10911yescharacteristic
344C22×C42C22×C10911yescharacteristic
345#391—24no—
346#391—24no—
347#384—24no—
348#380—24no—
349C4×C22⋊C4—24no—
350C2×#95C22×C10914yes—
351C2×#95C22×C10914yes—
352C2×#95C22×C10912yes—
353C2×#95C22×C10912yes—
354C2×#95C22×C10914yes—
355C25C23×C10911yescharacteristic
356C22×Q8—22no—
357C22×Q8—22no—
358C22×Q8—22no—
359C22×Q8—22no—
360C22×Q8—22no—
361C22×Q8—22no—
362C22×Q8D4×C10914yes—
363C23×C4—22no—
364C23×C4—22no—
365C23×C4—22no—
366C23×C4—22no—
367C23×C4—22no—
368C23×C4—22no—
369C23×C4D4×C10914yes—
370C23×C4C23×C10912yes—
371C23×C4C23×C10912yes—
372C23×C4C23×C10912yes—
373C23×C4C23×C10912yes—
374C23×C4C23×C10911yescharacteristic
375C23×C4C23×C10912yes—
376C23×C4C23×C10912yes—
377C23×C4C23×C10911yescharacteristic
378C22⋊Q8—232no—
379C2×C4⋊C4—24no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—28no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—28no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—44no—
393C2×C4⋊C4D4×C10914yes—
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—44no—
401C2×C22⋊C4Q8×C10918yescomplemented
402C2×C42—22no—
403C2×C42—22no—
404C2×C42—22no—
405C2×C42—22no—
406C2×C42—22no—
407C2×C42D4×C10914yes—
408#95—24no—
409#95—24no—
410#95—28no—
411#95—28no—
412#95—44no—
413C24—21no—
414C24—21no—
415C24—21no—
416C24—22no—
417C24—21no—
418C24—21no—
419C24—21no—
420C24—22no—
421C24—22no—
422C24C2×D4×C10912yes—
423C24C2×Q8×C10914yes—
424C24C24×C10911yescharacteristic
425C2×Q8—28no—
426C2×Q8—28no—
427C2×Q8—28no—
428C2×Q8—44no—
429C2×Q8—44no—
430C2×Q8—44no—
431C2×Q8—44no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—28no—
459C22×C4—28no—
460C22×C4—28no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4C2×D4×C10914yes—
476C22×C4C2×D4×C10912yes—
477C22×C4C2×D4×C10914yes—
478C22×C4C2×D4×C10914yes—
479C22×C4C2×D4×C10912yes—
480C22×C4C2×D4×C10912yes—
481C22×C4C2×D4×C10912yes—
482C22×C4C2×D4×C10912yes—
483C22×C4C2×Q8×C10914yes—
484C22×C4C2×Q8×C10914yes—
485C22×C4#41×C10914yes—
486C22×C4#41×C10912yes—
487C22×C4#41×C10912yes—
488C22×C4#41×C10912yes—
489C4⋊C4—216no—
490C4⋊C4—28no—
491C4⋊C4—416no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C22⋊C4—216no—
495C22⋊C4—216no—
496C22⋊C4—48no—
497C42—28no—
498C42—44no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23C4⋊Q8×C10914yes—
522C23C22×D4×C10911yescharacteristic
523C23C22×D4×C10911yescharacteristic
524C23C22×D4×C10911yescharacteristic
525C23C22×D4×C10911yescharacteristic
526C23C22×Q8×C10912yes—
527C23C2×#41×C10911yescharacteristic
528C23C2×#41×C10911yescharacteristic
529C23C2×#41×C10911yescharacteristic
530C23C2×#41×C10912yes—
531C23H(5,𝔽2)×C10912yes—
532C232−1+4×C10912yes—
533Q8—416no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—416no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4#120×C10914yes—
578C2×C4#121×C10914yes—
579C2×C4#121×C10914yes—
580C2×C4#121×C10914yes—
581C2×C4C22⋊Q8×C10918yescomplemented
582C2×C4#123×C10914yes—
583C2×C4#123×C10914yes—
584C2×C4#124×C10914yes—
585C22—24no—
586C22—24no—
587C22—24no—
588C22—42no—
589C22—44no—
590C22—42no—
591C22—44no—
592C22—42no—
593C22—42no—
594C22—44no—
595C22—44no—
596C22C2×#120×C10911yescharacteristic
597C22C2×#121×C10911yescharacteristic
598C22C2×#121×C10911yescharacteristic
599C22C2×#121×C10911yescharacteristic
600C22C2×C22⋊Q8×C10912yes—
601C22C2×#123×C10911yescharacteristic
602C22C2×#123×C10911yescharacteristic
603C22C2×#124×C10911yescharacteristic
604C22C2×C4⋊Q8×C10911yescharacteristic
605C22#532×C10911yescharacteristic
606C22#533×C10912yes—
607C22#535×C10911yescharacteristic
608C22#536×C10911yescharacteristic
609C22#537×C10912yes—
610C22#542×C10912yes—
611C22#546×C10912yes—
612C22D4×Q8×C10912yes—
613C22D4×Q8×C10912yes—
614C22#550×C10912yes—
615C22#552×C10912yes—
616C22#554×C10912yes—
617C22#554×C10912yes—
618C22#556×C10912yes—
619C4—28no—
620C4—44no—
621C4—44no—
622C4—42no—
623C4—44no—
624C4—82no—
625C4—82no—
626C4#394×C10918yes—
627C2—44no—
628C2C2×#394×C10911yescharacteristic
629C2C109×#241712yes—
630C2C109×#242312yes—
631C2C109×#243111yescharacteristic
632C2C109×#243411yescharacteristic
633C2C109×#249212yes—
634C2C109×#249911yescharacteristic
635C2C109×#250211yescharacteristic
636C2C109×#253812yes—
637C2C109×#262011yescharacteristic
638C2C109×#285711yescharacteristic
subgroupG/H
center, socleC24×C109C24
commutator, Frattini, cosocleC24C24×C109
prime radicalC25×C109C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#391×C109C2semidirect
3. C2×#391×C109C2semidirect
4. C2×#384×C109C2semidirect
7. C109×#1441C2semidirect
9. C109×C4⋊#95C2semidirect
81. C2×C22⋊C4×C109Q8semidirect
261. C2×C4×C109C22⋊Q8semidirect
307. #15182C109direct (N × H)
321. C2×#391C2×C109semidirect
322. C2×#391C2×C109semidirect
323. C2×#384C2×C109semidirect
326. #1441C2×C109semidirect
328. C4⋊#95C2×C109semidirect
329. C109#15182direct (N × H)
401. C2×C22⋊C4Q8×C109semidirect
581. C2×C4C22⋊Q8×C109semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Q8×C1092
C2×#391×C1092
C2×#384×C1092
C2×#380×C1092
C2×C4×C22⋊C4×C1092
C109×#14412
C109×#14392
C109×C4⋊#952
#15182109