Finite Group Databasebeta

← C109×#15174 #42133168

Subgroups — C109×#15174 #42133168

Order 27904.

1418 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
1C2×#398×C109C211yesmaximal, characteristic, complemented
2C2×#397×C109C211yesmaximal, characteristic
3C2×#388×C109C211yesmaximal, characteristic
4C2×C4×C4⋊C4×C109C211yesmaximal, characteristic, complemented
5C109×#1443C212yesmaximal, complemented
6C109×#1441C211yesmaximal, characteristic, complemented
7C109×#1439C212yesmaximal
8C109×#1439C211yesmaximal, characteristic
9C109×#1437C212yesmaximal
10C109×#1436C212yesmaximal, complemented
11C109×#1434C211yesmaximal, characteristic, complemented
12C22×C4⋊C4×C109C2211yescharacteristic
13C22×C4⋊C4×C109C2211yescharacteristic
14C22×C4⋊C4×C109C2212yes—
15C22×C4⋊C4×C109C2211yescharacteristic
16C22×C22⋊C4×C109C2211yescharacteristic
17C22×C22⋊C4×C109C2211yescharacteristic
18C22×C22⋊C4×C109C2211yescharacteristic
19C22×C22⋊C4×C109C2212yes—
20C22×C22⋊C4×C109C2212yes—
21C22×C42×C109C2211yescharacteristic
22C22×C42×C109C2212yes—
23#398×C109—24no—
24#397×C109—24no—
25#388×C109—24no—
26C4×C4⋊C4×C109—24no—
27C2×#95×C109C2212yes—
28C2×#95×C109C2212yes—
29C2×#95×C109C2211yescharacteristic
30C2×#95×C109C2211yescharacteristic
31C2×#95×C109C2211yescharacteristic
32C2×#95×C109C2212yes—
33C2×#95×C109C2212yes—
34C2×#95×C109C2212yes—
35C2×#95×C109C2212yes—
36C2×#95×C109C2212yes—
37C2×#95×C109C2211yescharacteristic
38C2×#95×C109C2212yes—
39C25×C109C2311yescharacteristic
40C23×C4×C109C2311yescharacteristic
41C23×C4×C109C2311yescharacteristic
42C23×C4×C109C2312yes—
43C23×C4×C109C2312yes—
44C23×C4×C109C2311yescharacteristic
45C23×C4×C109C2312yes—
46C23×C4×C109C2311yescharacteristic
47C23×C4×C109C2312yes—
48C23×C4×C109C2311yescharacteristic
49C23×C4×C109C2311yescharacteristic
50C2×C4⋊C4×C109—22no—
51C2×C4⋊C4×C109—22no—
52C2×C4⋊C4×C109—24no—
53C2×C4⋊C4×C109—22no—
54C2×C4⋊C4×C109—22no—
55C2×C4⋊C4×C109—24no—
56C2×C4⋊C4×C109—22no—
57C2×C4⋊C4×C109—24no—
58C2×C4⋊C4×C109—22no—
59C2×C4⋊C4×C109—42no—
60C2×C4⋊C4×C109D414yes—
61C2×C22⋊C4×C109—22no—
62C2×C22⋊C4×C109—22no—
63C2×C22⋊C4×C109—22no—
64C2×C22⋊C4×C109—24no—
65C2×C22⋊C4×C109—22no—
66C2×C22⋊C4×C109—22no—
67C2×C22⋊C4×C109—24no—
68C2×C22⋊C4×C109—22no—
69C2×C22⋊C4×C109—24no—
70C2×C22⋊C4×C109—24no—
71C2×C22⋊C4×C109—22no—
72C2×C22⋊C4×C109—24no—
73C2×C22⋊C4×C109—42no—
74C2×C22⋊C4×C109—42no—
75C2×C42×C109—24no—
76C2×C42×C109—24no—
77C2×C42×C109—42no—
78C2×C42×C109—42no—
79C2×C42×C109D414yes—
80#95×C109—24no—
81#95×C109—22no—
82#95×C109—22no—
83#95×C109—24no—
84#95×C109—22no—
85#95×C109—22no—
86#95×C109—24no—
87#95×C109—24no—
88#95×C109—24no—
89#95×C109—24no—
90#95×C109—24no—
91#95×C109—42no—
92C24×C109—21no—
93C24×C109—21no—
94C24×C109—21no—
95C24×C109—22no—
96C24×C109—22no—
97C24×C109—22no—
98C24×C109—21no—
99C24×C109—21no—
100C24×C109—21no—
101C24×C109—22no—
102C24×C109#4112yes—
103C24×C109C2411yescharacteristic
104C22×C4×C109—22no—
105C22×C4×C109—22no—
106C22×C4×C109—21no—
107C22×C4×C109—21no—
108C22×C4×C109—22no—
109C22×C4×C109—22no—
110C22×C4×C109—22no—
111C22×C4×C109—22no—
112C22×C4×C109—21no—
113C22×C4×C109—21no—
114C22×C4×C109—22no—
115C22×C4×C109—22no—
116C22×C4×C109—22no—
117C22×C4×C109—22no—
118C22×C4×C109—21no—
119C22×C4×C109—21no—
120C22×C4×C109—21no—
121C22×C4×C109—22no—
122C22×C4×C109—22no—
123C22×C4×C109—22no—
124C22×C4×C109—22no—
125C22×C4×C109—22no—
126C22×C4×C109—22no—
127C22×C4×C109—22no—
128C22×C4×C109—21no—
129C22×C4×C109—21no—
130C22×C4×C109—22no—
131C22×C4×C109—22no—
132C22×C4×C109—22no—
133C22×C4×C109—22no—
134C22×C4×C109—22no—
135C22×C4×C109—21no—
136C22×C4×C109—22no—
137C22×C4×C109—21no—
138C22×C4×C109—21no—
139C22×C4×C109—22no—
140C22×C4×C109—21no—
141C22×C4×C109—21no—
142C22×C4×C109—22no—
143C22×C4×C109—22no—
144C22×C4×C109—22no—
145C22×C4×C109—22no—
146C22×C4×C109—22no—
147C22×C4×C109—21no—
148C22×C4×C109—22no—
149C22×C4×C109—22no—
150C22×C4×C109—21no—
151C22×C4×C109—22no—
152C22×C4×C109—22no—
153C22×C4×C109—22no—
154C22×C4×C109C2×D412yes—
155C22×C4×C109C2×D414yes—
156C22×C4×C109C2×D412yes—
157C22×C4×C109C2×D412yes—
158C22×C4×C109C2×D412yes—
159C22×C4×C109#4114yes—
160C22×C4×C109#4114yes—
161C22×C4×C109#4112yes—
162C22×C4×C109#4112yes—
163C22×C4×C109#4114yes—
164C4⋊C4×C109—28no—
165C4⋊C4×C109—28no—
166C4⋊C4×C109—48no—
167C4⋊C4×C109—44no—
168C22⋊C4×C109—28no—
169C22⋊C4×C109—48no—
170C22⋊C4×C109—48no—
171C22⋊C4×C109—44no—
172C22⋊C4×C109—44no—
173C42×C109—28no—
174C42×C109—48no—
175C23×C109—22no—
176C23×C109—24no—
177C23×C109—22no—
178C23×C109—24no—
179C23×C109—22no—
180C23×C109—41no—
181C23×C109—41no—
182C23×C109—41no—
183C23×C109—42no—
184C23×C109—42no—
185C23×C109—41no—
186C23×C109—42no—
187C23×C109—41no—
188C23×C109—41no—
189C23×C109—42no—
190C23×C109—42no—
191C23×C109—41no—
192C23×C109—42no—
193C23×C109—42no—
194C23×C109—41no—
195C23×C109—42no—
196C23×C109—42no—
197C23×C109—42no—
198C23×C109C22×D411yescharacteristic
199C23×C109C22×D411yescharacteristic
200C23×C109C2×#4111yescharacteristic
201C23×C109C2×#4112yes—
202C23×C109C2×#4111yescharacteristic
203C23×C109C2×#4111yescharacteristic
204C23×C109C2×#4112yes—
205C23×C109C2×#4112yes—
206C23×C109H(5,𝔽2)11yescharacteristic
207C23×C109H(5,𝔽2)11yescharacteristic
208C23×C1092−1+412yes—
209C2×C4×C109—22no—
210C2×C4×C109—22no—
211C2×C4×C109—24no—
212C2×C4×C109—24no—
213C2×C4×C109—22no—
214C2×C4×C109—22no—
215C2×C4×C109—24no—
216C2×C4×C109—24no—
217C2×C4×C109—24no—
218C2×C4×C109—22no—
219C2×C4×C109—22no—
220C2×C4×C109—22no—
221C2×C4×C109—24no—
222C2×C4×C109—24no—
223C2×C4×C109—22no—
224C2×C4×C109—24no—
225C2×C4×C109—22no—
226C2×C4×C109—24no—
227C2×C4×C109—24no—
228C2×C4×C109—24no—
229C2×C4×C109—24no—
230C2×C4×C109—24no—
231C2×C4×C109—22no—
232C2×C4×C109—41no—
233C2×C4×C109—42no—
234C2×C4×C109—42no—
235C2×C4×C109—42no—
236C2×C4×C109—42no—
237C2×C4×C109—42no—
238C2×C4×C109—42no—
239C2×C4×C109—42no—
240C2×C4×C109—41no—
241C2×C4×C109—42no—
242C2×C4×C109—42no—
243C2×C4×C109—41no—
244C2×C4×C109—42no—
245C2×C4×C109—42no—
246C2×C4×C109—42no—
247C2×C4×C109—41no—
248C2×C4×C109—42no—
249C2×C4×C109—42no—
250C2×C4×C109—41no—
251C2×C4×C109—41no—
252C2×C4×C109—41no—
253C2×C4×C109—41no—
254C2×C4×C109—42no—
255C2×C4×C109—42no—
256C2×C4×C109—42no—
257C2×C4×C109—41no—
258C2×C4×C109—42no—
259C2×C4×C109—41no—
260C2×C4×C109—42no—
261C2×C4×C109—42no—
262C2×C4×C109—42no—
263C2×C4×C109—42no—
264C2×C4×C109—42no—
265C2×C4×C109—42no—
266C2×C4×C109—42no—
267C2×C4×C109#12114yes—
268C2×C4×C109#12314yes—
269C22×C109—44no—
270C22×C109—42no—
271C22×C109—42no—
272C22×C109—42no—
273C22×C109—44no—
274C22×C109—82no—
275C22×C109—82no—
276C22×C109—81no—
277C22×C109—81no—
278C22×C109—81no—
279C22×C109—81no—
280C22×C109C2×#12111yescharacteristic
281C22×C109C2×#12311yescharacteristic
282C22×C109#52511yescharacteristic
283C22×C109#52511yescharacteristic
284C22×C109#52912yes—
285C22×C109#52912yes—
286C22×C109#52912yes—
287C22×C109#53212yes—
288C22×C109#53211yescharacteristic
289C22×C109#53511yescharacteristic
290C22×C109#53611yescharacteristic
291C22×C109#53912yes—
292C22×C109#53912yes—
293C22×C109#54611yescharacteristic
294C22×C109#54611yescharacteristic
295C22×C109#54812yes—
296C22×C109#55112yes—
297C22×C109#55212yes—
298C22×C109#55311yescharacteristic
299C22×C109#55311yescharacteristic
300C22×C109#55312yes—
301C22×C109#55612yes—
302C22×C109#55612yes—
303C4×C109—24no—
304C4×C109—24no—
305C4×C109—44no—
306C4×C109—44no—
307C4×C109—42no—
308C4×C109—44no—
309C4×C109—42no—
310C4×C109—81no—
311C4×C109—81no—
312C4×C109—82no—
313#15174C10911yesmaximal, characteristic, direct product factor, Sylow
314C2×C109—82no—
315C2×C109#239511yescharacteristic
316C2×C109#253511yescharacteristic
317C2×C109#254511yescharacteristic
318C2×C109#255812yes—
319C2×C109#256012yes—
320C2×C109#257712yes—
321C2×C109#259412yes—
322C2×C109#259911yescharacteristic
323C2×C109#260311yescharacteristic
324C2×C109#262311yescharacteristic
325C2×C109#285411yescharacteristic
326C2×#398C2×C10911yescharacteristic, complemented
327C2×#397C2×C10911yescharacteristic
328C2×#388C2×C10911yescharacteristic
329C2×C4×C4⋊C4C2×C10911yescharacteristic, complemented
330#1443C2×C10912yescomplemented
331#1441C2×C10911yescharacteristic, complemented
332#1439C2×C10911yescharacteristic
333#1439C2×C10912yes—
334#1437C2×C10912yes—
335#1436C2×C10912yescomplemented
336#1434C2×C10911yescharacteristic, complemented
337C109#1517411yescharacteristic, direct product factor, Sylow
338C22×C4⋊C4C22×C10911yescharacteristic
339C22×C4⋊C4C22×C10911yescharacteristic
340C22×C4⋊C4C22×C10912yes—
341C22×C4⋊C4C22×C10911yescharacteristic
342C22×C22⋊C4C22×C10911yescharacteristic
343C22×C22⋊C4C22×C10911yescharacteristic
344C22×C22⋊C4C22×C10911yescharacteristic
345C22×C22⋊C4C22×C10912yes—
346C22×C22⋊C4C22×C10912yes—
347C22×C42C22×C10911yescharacteristic
348C22×C42C22×C10912yes—
349#398—24no—
350#397—24no—
351#388—24no—
352C4×C4⋊C4—24no—
353C2×#95C22×C10911yescharacteristic
354C2×#95C22×C10912yes—
355C2×#95C22×C10911yescharacteristic
356C2×#95C22×C10912yes—
357C2×#95C22×C10912yes—
358C2×#95C22×C10912yes—
359C2×#95C22×C10911yescharacteristic
360C2×#95C22×C10912yes—
361C2×#95C22×C10912yes—
362C2×#95C22×C10912yes—
363C2×#95C22×C10911yescharacteristic
364C2×#95C22×C10912yes—
365C25C23×C10911yescharacteristic
366C23×C4C23×C10911yescharacteristic
367C23×C4C23×C10911yescharacteristic
368C23×C4C23×C10912yes—
369C23×C4C23×C10912yes—
370C23×C4C23×C10911yescharacteristic
371C23×C4C23×C10911yescharacteristic
372C23×C4C23×C10912yes—
373C23×C4C23×C10912yes—
374C23×C4C23×C10911yescharacteristic
375C23×C4C23×C10911yescharacteristic
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—24no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—24no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—42no—
386C2×C4⋊C4D4×C10914yes—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—24no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—42no—
400C2×C22⋊C4—42no—
401C2×C42—24no—
402C2×C42—24no—
403C2×C42—42no—
404C2×C42—42no—
405C2×C42D4×C10914yes—
406#95—24no—
407#95—22no—
408#95—22no—
409#95—24no—
410#95—22no—
411#95—22no—
412#95—24no—
413#95—24no—
414#95—24no—
415#95—24no—
416#95—24no—
417#95—42no—
418C24—21no—
419C24—21no—
420C24—21no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—21no—
425C24—21no—
426C24—21no—
427C24—22no—
428C24#41×C10912yes—
429C24C24×C10911yescharacteristic
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4C2×D4×C10912yes—
481C22×C4C2×D4×C10914yes—
482C22×C4C2×D4×C10912yes—
483C22×C4C2×D4×C10912yes—
484C22×C4C2×D4×C10912yes—
485C22×C4#41×C10914yes—
486C22×C4#41×C10914yes—
487C22×C4#41×C10912yes—
488C22×C4#41×C10912yes—
489C22×C4#41×C10914yes—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—48no—
493C4⋊C4—44no—
494C22⋊C4—28no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C42—28no—
500C42—48no—
501C23—22no—
502C23—24no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—41no—
518C23—42no—
519C23—42no—
520C23—41no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23C22×D4×C10911yescharacteristic
525C23C22×D4×C10911yescharacteristic
526C23C2×#41×C10911yescharacteristic
527C23C2×#41×C10912yes—
528C23C2×#41×C10911yescharacteristic
529C23C2×#41×C10911yescharacteristic
530C23C2×#41×C10912yes—
531C23C2×#41×C10912yes—
532C23H(5,𝔽2)×C10911yescharacteristic
533C23H(5,𝔽2)×C10911yescharacteristic
534C232−1+4×C10912yes—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4#121×C10914yes—
594C2×C4#123×C10914yes—
595C22—42no—
596C22—44no—
597C22—44no—
598C22—42no—
599C22—42no—
600C22—81no—
601C22—82no—
602C22—82no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22C2×#121×C10911yescharacteristic
607C22C2×#123×C10911yescharacteristic
608C22#525×C10911yescharacteristic
609C22#525×C10911yescharacteristic
610C22#529×C10912yes—
611C22#529×C10912yes—
612C22#529×C10912yes—
613C22#532×C10911yescharacteristic
614C22#532×C10912yes—
615C22#535×C10911yescharacteristic
616C22#536×C10911yescharacteristic
617C22#539×C10912yes—
618C22#539×C10912yes—
619C22#546×C10911yescharacteristic
620C22#546×C10911yescharacteristic
621C22#548×C10912yes—
622C22#551×C10912yes—
623C22#552×C10912yes—
624C22#553×C10911yescharacteristic
625C22#553×C10911yescharacteristic
626C22#553×C10912yes—
627C22#556×C10912yes—
628C22#556×C10912yes—
629C4—24no—
630C4—24no—
631C4—44no—
632C4—42no—
633C4—44no—
634C4—44no—
635C4—42no—
636C4—81no—
637C4—82no—
638C4—81no—
639C2—82no—
640C2C109×#239511yescharacteristic
641C2C109×#253511yescharacteristic
642C2C109×#254511yescharacteristic
643C2C109×#255812yes—
644C2C109×#256012yes—
645C2C109×#257712yes—
646C2C109×#259412yes—
647C2C109×#259911yescharacteristic
648C2C109×#260311yescharacteristic
649C2C109×#262311yescharacteristic
650C2C109×#285411yescharacteristic
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
1. C2×#398×C109C2semidirect
4. C2×C4×C4⋊C4×C109C2semidirect
5. C109×#1443C2semidirect
6. C109×#1441C2semidirect
10. C109×#1436C2semidirect
11. C109×#1434C2semidirect
313. #15174C109direct (N × H)
326. C2×#398C2×C109semidirect
329. C2×C4×C4⋊C4C2×C109semidirect
330. #1443C2×C109semidirect
331. #1441C2×C109semidirect
335. #1436C2×C109semidirect
336. #1434C2×C109semidirect
337. C109#15174direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#398×C1092
C2×#397×C1092
C2×#388×C1092
C2×C4×C4⋊C4×C1092
C109×#14432
C109×#14412
C109×#14392
C109×#14372
C109×#14362
C109×#14342
#15174109