Finite Group Databasebeta

← C109×#24006 #42142000

Subgroups — C109×#24006 #42142000

Order 27904.

1500 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊Q8×C109C211yesmaximal, characteristic
2C2×#484×C109C212yesmaximal, complemented
3C2×#380×C109C211yesmaximal, characteristic
4C109×#2036C214yesmaximal, complemented
5C109×#1872C212yesmaximal, complemented
6C109×#1852C212yesmaximal, complemented
7C109×#1787C212yesmaximal, complemented
8C109×#1779C211yesmaximal, characteristic
9C23×Q8×C109C2211yescharacteristic
10C24×C4×C109C2211yescharacteristic
11C2×C22⋊Q8×C109—21no—
12C2×C22⋊Q8×C109—21no—
13C2×C22⋊Q8×C109—28no—
14C2×C22⋊Q8×C109C2212yes—
15C2×C22⋊Q8×C109C2212yes—
16C22×C4⋊C4×C109—21no—
17C22×C4⋊C4×C109C2211yescharacteristic
18C22×C4⋊C4×C109C2211yescharacteristic
19C22×C4⋊C4×C109C2211yescharacteristic
20C22×C22⋊C4×C109—21no—
21C22×C22⋊C4×C109—21no—
22#484×C109—28no—
23C2×#107×C109C2214yescomplemented
24C2×#103×C109C2214yescomplemented
25C2×#103×C109C2214yescomplemented
26C2×C22⋊C8×C109C2212yes—
27C2×C22⋊C8×C109C2212yes—
28C2×C22⋊C8×C109C2212yes—
29#380×C109—21no—
30#380×C109—21no—
31#380×C109C2212yes—
32#380×C109C2212yes—
33C2×#95×C109—21no—
34#340×C109C2214yescomplemented
35C25×C109D411yescharacteristic
36C22×Q8×C109—24no—
37C22×Q8×C109—21no—
38C22×Q8×C109—24no—
39C22×Q8×C109—42no—
40C22×Q8×C109C2312yes—
41C23×C4×C109—21no—
42C23×C4×C109—21no—
43C23×C4×C109—24no—
44C23×C4×C109—24no—
45C23×C4×C109—21no—
46C23×C4×C109—21no—
47C23×C4×C109—42no—
48C23×C4×C109D411yescharacteristic
49C23×C4×C109D411yescharacteristic
50C23×C4×C109D411yescharacteristic
51C23×C4×C109C2311yescharacteristic
52C23×C4×C109C2311yescharacteristic
53C23×C4×C109C2311yescharacteristic
54C22×C8×C109C2314yes—
55C22⋊Q8×C109—28no—
56C22⋊Q8×C109—28no—
57C22⋊Q8×C109—24no—
58C22⋊Q8×C109—24no—
59C22⋊Q8×C109—24no—
60C22⋊Q8×C109D418yes—
61C2×C4⋊C4×C109—28no—
62C2×C4⋊C4×C109—22no—
63C2×C4⋊C4×C109—22no—
64C2×C4⋊C4×C109—21no—
65C2×C4⋊C4×C109—24no—
66C2×C4⋊C4×C109—24no—
67C2×C4⋊C4×C109—21no—
68C2×C4⋊C4×C109—21no—
69C2×C4⋊C4×C109—42no—
70C2×C4⋊C4×C109C2312yes—
71C2×C4⋊C4×C109C2312yes—
72C2×C4⋊C4×C109C2312yes—
73C2×C22⋊C4×C109—28no—
74C2×C22⋊C4×C109—22no—
75C2×C22⋊C4×C109—22no—
76C2×C22⋊C4×C109—22no—
77C2×C22⋊C4×C109—22no—
78C2×C22⋊C4×C109—44no—
79#107×C109—28no—
80#103×C109—28no—
81#103×C109—44no—
82C22⋊C8×C109—24no—
83C22⋊C8×C109—24no—
84C22⋊C8×C109—42no—
85#95×C109—22no—
86#95×C109—22no—
87C24×C109—22no—
88C24×C109—22no—
89C24×C109—22no—
90C24×C109—22no—
91C24×C109—41no—
92C24×C109—41no—
93C24×C109Q1614yes—
94C24×C109C2×D411yescharacteristic
95C24×C109C2×D411yescharacteristic
96C24×C109C2×D411yescharacteristic
97C2×Q8×C109—24no—
98C2×Q8×C109—22no—
99C2×Q8×C109—24no—
100C2×Q8×C109—24no—
101C2×Q8×C109—416no—
102C2×Q8×C109—42no—
103C2×Q8×C109—42no—
104C2×Q8×C109C2×D414yes—
105C22×C4×C109—22no—
106C22×C4×C109—22no—
107C22×C4×C109—22no—
108C22×C4×C109—22no—
109C22×C4×C109—24no—
110C22×C4×C109—24no—
111C22×C4×C109—22no—
112C22×C4×C109—22no—
113C22×C4×C109—22no—
114C22×C4×C109—22no—
115C22×C4×C109—22no—
116C22×C4×C109—22no—
117C22×C4×C109—22no—
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—41no—
125C22×C4×C109—41no—
126C22×C4×C109—41no—
127C22×C4×C109—416no—
128C22×C4×C109—42no—
129C22×C4×C109—42no—
130C22×C4×C109—42no—
131C22×C4×C109—42no—
132C22×C4×C109—42no—
133C22×C4×C109—42no—
134C22×C4×C109—42no—
135C22×C4×C109—42no—
136C22×C4×C109—42no—
137C22×C4×C109—42no—
138C22×C4×C109C2×D411yescharacteristic
139C22×C4×C109C2×D411yescharacteristic
140C22×C4×C109C2×D414yes—
141C22×C4×C109C2×D411yescharacteristic
142C22×C4×C109#4114yes—
143C22×C4×C109C2411yescharacteristic
144C2×C8×C109—24no—
145C2×C8×C109—24no—
146C2×C8×C109—24no—
147C4⋊C4×C109—24no—
148C4⋊C4×C109—28no—
149C4⋊C4×C109—28no—
150C4⋊C4×C109—22no—
151C4⋊C4×C109—24no—
152C4⋊C4×C109—24no—
153C4⋊C4×C109—42no—
154C4⋊C4×C109—42no—
155C4⋊C4×C109C2×D414yes—
156C22⋊C4×C109—28no—
157C22⋊C4×C109—28no—
158C22⋊C4×C109—44no—
159C22⋊C4×C109—44no—
160C23×C109—21no—
161C23×C109—21no—
162C23×C109—21no—
163C23×C109—21no—
164C23×C109—21no—
165C23×C109—21no—
166C23×C109—21no—
167C23×C109—21no—
168C23×C109—21no—
169C23×C109—21no—
170C23×C109—21no—
171C23×C109—21no—
172C23×C109—21no—
173C23×C109—21no—
174C23×C109—28no—
175C23×C109—28no—
176C23×C109—44no—
177C23×C109—44no—
178C23×C109—44no—
179C23×C109—44no—
180C23×C109—44no—
181C23×C109#12012yes—
182C23×C109#12312yes—
183C23×C109C2×Q1612yes—
184C23×C109C2×Q1612yes—
185C23×C109C2×Q1612yes—
186C23×C109AGL(1,ℤ/8)12yes—
187C23×C109#13712yes—
188C23×C109C22×D411yescharacteristic
189Q8×C109—28no—
190Q8×C109—48no—
191Q8×C109—44no—
192C2×C4×C109—21no—
193C2×C4×C109—21no—
194C2×C4×C109—21no—
195C2×C4×C109—21no—
196C2×C4×C109—28no—
197C2×C4×C109—28no—
198C2×C4×C109—24no—
199C2×C4×C109—24no—
200C2×C4×C109—21no—
201C2×C4×C109—21no—
202C2×C4×C109—21no—
203C2×C4×C109—44no—
204C2×C4×C109—44no—
205C2×C4×C109—44no—
206C2×C4×C109—48no—
207C2×C4×C109—48no—
208C2×C4×C109—42no—
209C2×C4×C109—42no—
210C2×C4×C109—42no—
211C2×C4×C109—42no—
212C2×C4×C109—44no—
213C2×C4×C109—42no—
214C2×C4×C109—42no—
215C2×C4×C109—42no—
216C2×C4×C109—42no—
217C2×C4×C109—42no—
218C2×C4×C109—42no—
219C2×C4×C109—84no—
220C2×C4×C109—84no—
221C2×C4×C109#12012yes—
222C2×C4×C109#12312yes—
223C2×C4×C109C22×D412yes—
224C2×C4×C109C2×#4112yes—
225C2×C4×C109H(5,𝔽2)12yes—
226C8×C109—44no—
227C22×C109—22no—
228C22×C109—22no—
229C22×C109—22no—
230C22×C109—22no—
231C22×C109—22no—
232C22×C109—22no—
233C22×C109—22no—
234C22×C109—22no—
235C22×C109—22no—
236C22×C109—22no—
237C22×C109—22no—
238C22×C109—22no—
239C22×C109—22no—
240C22×C109—22no—
241C22×C109—22no—
242C22×C109—416no—
243C22×C109—41no—
244C22×C109—41no—
245C22×C109—41no—
246C22×C109—41no—
247C22×C109#45114yes—
248C22×C109#48414yescomplemented
249C22×C109C2×#12011yescharacteristic
250C22×C109C2×#12311yescharacteristic
251C22×C109#53411yescharacteristic
252C22×C109C22×Q1611yescharacteristic
253C22×C109C2×AGL(1,ℤ/8)11yescharacteristic
254C22×C109C2×#13711yescharacteristic
255C22×C109#57511yescharacteristic
256C4×C109—22no—
257C4×C109—22no—
258C4×C109—41no—
259C4×C109—44no—
260C4×C109—44no—
261C4×C109—82no—
262C4×C109—82no—
263C4×C109#54614yes—
264#24006C10911yesmaximal, characteristic, direct product factor, Sylow
265C2×C109—24no—
266C2×C109—24no—
267C2×C109—42no—
268C2×C109#240511yescharacteristic
269C2×C109C2×#45111yescharacteristic
270C2×C109#300611yescharacteristic
271C2×C109C2×#48411yescharacteristic
272C2×C109#309511yescharacteristic
273C2×C109#318911yescharacteristic
274C2×C109#319311yescharacteristic
275C22×C22⋊Q8C2×C10911yescharacteristic
276C2×#484C2×C10912yescomplemented
277C2×#380C2×C10911yescharacteristic
278#2036C2×C10914yescomplemented
279#1872C2×C10912yescomplemented
280#1852C2×C10912yescomplemented
281#1787C2×C10912yescomplemented
282#1779C2×C10911yescharacteristic
283C109#2400611yescharacteristic, direct product factor, Sylow
284C23×Q8C22×C10911yescharacteristic
285C24×C4C22×C10911yescharacteristic
286C2×C22⋊Q8—21no—
287C2×C22⋊Q8—21no—
288C2×C22⋊Q8—28no—
289C2×C22⋊Q8C22×C10912yes—
290C2×C22⋊Q8C22×C10912yes—
291C22×C4⋊C4—21no—
292C22×C4⋊C4C22×C10911yescharacteristic
293C22×C4⋊C4C22×C10911yescharacteristic
294C22×C4⋊C4C22×C10911yescharacteristic
295C22×C22⋊C4—21no—
296C22×C22⋊C4—21no—
297#484—28no—
298C2×#107C22×C10914yescomplemented
299C2×#103C22×C10914yescomplemented
300C2×#103C22×C10914yescomplemented
301C2×C22⋊C8C22×C10912yes—
302C2×C22⋊C8C22×C10912yes—
303C2×C22⋊C8C22×C10912yes—
304#380—21no—
305#380—21no—
306#380C22×C10912yes—
307#380C22×C10912yes—
308C2×#95—21no—
309#340C22×C10914yescomplemented
310C25D4×C10911yescharacteristic
311C22×Q8—24no—
312C22×Q8—21no—
313C22×Q8—24no—
314C22×Q8—42no—
315C22×Q8C23×C10912yes—
316C23×C4—21no—
317C23×C4—21no—
318C23×C4—24no—
319C23×C4—24no—
320C23×C4—21no—
321C23×C4—21no—
322C23×C4—42no—
323C23×C4D4×C10911yescharacteristic
324C23×C4D4×C10911yescharacteristic
325C23×C4D4×C10911yescharacteristic
326C23×C4C23×C10911yescharacteristic
327C23×C4C23×C10911yescharacteristic
328C23×C4C23×C10911yescharacteristic
329C22×C8C23×C10914yes—
330C22⋊Q8—28no—
331C22⋊Q8—28no—
332C22⋊Q8—24no—
333C22⋊Q8—24no—
334C22⋊Q8—24no—
335C22⋊Q8D4×C10918yes—
336C2×C4⋊C4—28no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—42no—
345C2×C4⋊C4C23×C10912yes—
346C2×C4⋊C4C23×C10912yes—
347C2×C4⋊C4C23×C10912yes—
348C2×C22⋊C4—28no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—44no—
354#107—28no—
355#103—28no—
356#103—44no—
357C22⋊C8—24no—
358C22⋊C8—24no—
359C22⋊C8—42no—
360#95—22no—
361#95—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—41no—
367C24—41no—
368C24Q16×C10914yes—
369C24C2×D4×C10911yescharacteristic
370C24C2×D4×C10911yescharacteristic
371C24C2×D4×C10911yescharacteristic
372C2×Q8—24no—
373C2×Q8—22no—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—416no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8C2×D4×C10914yes—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—416no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4C2×D4×C10911yescharacteristic
414C22×C4C2×D4×C10911yescharacteristic
415C22×C4C2×D4×C10914yes—
416C22×C4C2×D4×C10911yescharacteristic
417C22×C4#41×C10914yes—
418C22×C4C24×C10911yescharacteristic
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C4⋊C4—24no—
423C4⋊C4—28no—
424C4⋊C4—28no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4C2×D4×C10914yes—
431C22⋊C4—28no—
432C22⋊C4—28no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—28no—
450C23—28no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23#120×C10912yes—
457C23#123×C10912yes—
458C23C2×Q16×C10912yes—
459C23C2×Q16×C10912yes—
460C23C2×Q16×C10912yes—
461C23AGL(1,ℤ/8)×C10912yes—
462C23#137×C10912yes—
463C23C22×D4×C10911yescharacteristic
464Q8—28no—
465Q8—48no—
466Q8—44no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—28no—
472C2×C4—28no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—48no—
482C2×C4—48no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—84no—
495C2×C4—84no—
496C2×C4#120×C10912yes—
497C2×C4#123×C10912yes—
498C2×C4C22×D4×C10912yes—
499C2×C4C2×#41×C10912yes—
500C2×C4H(5,𝔽2)×C10912yes—
501C8—44no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—416no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22#451×C10914yes—
523C22#484×C10914yescomplemented
524C22C2×#120×C10911yescharacteristic
525C22C2×#123×C10911yescharacteristic
526C22#534×C10911yescharacteristic
527C22C22×Q16×C10911yescharacteristic
528C22C2×AGL(1,ℤ/8)×C10911yescharacteristic
529C22C2×#137×C10911yescharacteristic
530C22#575×C10911yescharacteristic
531C4—22no—
532C4—22no—
533C4—44no—
534C4—41no—
535C4—44no—
536C4—82no—
537C4—82no—
538C4#546×C10914yes—
539C2—24no—
540C2—24no—
541C2—42no—
542C2C109×#240511yescharacteristic
543C2C2×#451×C10911yescharacteristic
544C2C109×#300611yescharacteristic
545C2C2×#484×C10911yescharacteristic
546C2C109×#309511yescharacteristic
547C2C109×#318911yescharacteristic
548C2C109×#319311yescharacteristic
subgroupG/H
center, socleC23×C109C22×D4
commutator, Frattini, cosocleC22×C4C24×C109
prime radicalC25×C109D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C109 ◁ C24×C4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#484×C109C2semidirect
4. C109×#2036C2semidirect
5. C109×#1872C2semidirect
6. C109×#1852C2semidirect
7. C109×#1787C2semidirect
23. C2×#107×C109C22semidirect
24. C2×#103×C109C22semidirect
25. C2×#103×C109C22semidirect
34. #340×C109C22semidirect
248. C22×C109#484semidirect
264. #24006C109direct (N × H)
276. C2×#484C2×C109semidirect
278. #2036C2×C109semidirect
279. #1872C2×C109semidirect
280. #1852C2×C109semidirect
281. #1787C2×C109semidirect
283. C109#24006direct (N × H)
298. C2×#107C22×C109semidirect
299. C2×#103C22×C109semidirect
300. C2×#103C22×C109semidirect
309. #340C22×C109semidirect
523. C22#484×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×#484×C1092
C2×#380×C1092
C109×#20362
C109×#18722
C109×#18522
C109×#17872
C109×#17792
#24006109