Finite Group Databasebeta

← C109×#12689 #42130683

Subgroups — C109×#12689 #42130683

Order 27904.

1160 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#525×C109C211yesmaximal, characteristic
2C2×#352×C109C212yesmaximal, complemented
3C2×AΣL(1,𝔽16)×C109C212yesmaximal, complemented
4C109×#1785C211yesmaximal, characteristic, complemented
5C109×#1782C211yesmaximal, characteristic, complemented
6C22×#41×C109C2211yescharacteristic
7C24×C4×C109C411yescharacteristic, complemented
8#525×C109—24no—
9#525×C109—24no—
10C2×#123×C109C411yescharacteristic, complemented
11C2×#123×C109C2211yescharacteristic
12C2×C22⋊Q8×C109—21no—
13C2×#121×C109—21no—
14C2×#120×C109C411yescharacteristic, complemented
15C2×#120×C109C2211yescharacteristic
16C2×C4×D4×C109—21no—
17C2×C4×D4×C109—21no—
18C2×#117×C109C411yescharacteristic, complemented
19C2×#101×C109—21no—
20C2×#100×C109C2212yes—
21C2×#99×C109—21no—
22C2×#99×C109C2212yes—
23#407×C109—21no—
24#381×C109—21no—
25#352×C109—24no—
26AΣL(1,𝔽16)×C109—24no—
27C25×C109C2×C411yescharacteristic
28C2×#41×C109—22no—
29C2×#41×C109—22no—
30C2×#41×C109—42no—
31C2×#41×C109—42no—
32C22×Q8×C109D411yescharacteristic
33C22×D4×C109—21no—
34C22×D4×C109—21no—
35C22×D4×C109C2×C411yescharacteristic
36C22×D4×C109D411yescharacteristic
37C22×D4×C109C2311yescharacteristic
38C23×C4×C109—22no—
39C23×C4×C109—21no—
40C23×C4×C109—22no—
41C23×C4×C109—21no—
42C23×C4×C109—42no—
43C23×C4×C109—42no—
44C23×C4×C109C2×C411yescharacteristic
45C23×C4×C109C2×C411yescharacteristic
46C23×C4×C109D411yescharacteristic
47C23×C4×C109D411yescharacteristic
48C2×M16×C109—22no—
49#123×C109—22no—
50#123×C109—22no—
51#123×C109—22no—
52#123×C109D414yes—
53C22⋊Q8×C109—28no—
54#121×C109—28no—
55#120×C109—22no—
56#120×C109—22no—
57#120×C109—22no—
58#120×C109D414yes—
59C4×D4×C109—28no—
60C4×D4×C109—28no—
61#117×C109—22no—
62#117×C109—22no—
63C2×C4⋊C4×C109—21no—
64C2×C4⋊C4×C109—21no—
65C2×C4⋊C4×C109—21no—
66C2×C22⋊C4×C109—21no—
67C2×C22⋊C4×C109—21no—
68C2×C22⋊C4×C109—21no—
69C2×C22⋊C4×C109—21no—
70C2×C22⋊C4×C109—22no—
71C2×C22⋊C4×C109C2×C411yescharacteristic
72C2×C22⋊C4×C109C2×C411yescharacteristic
73C2×C42×C109—21no—
74#101×C109—42no—
75#100×C109—22no—
76#100×C109—22no—
77#99×C109—22no—
78#99×C109—22no—
79#99×C109—42no—
80C22⋊C8×C109—42no—
81#95×C109—42no—
82C24×C109—21no—
83C24×C109—21no—
84C24×C109—41no—
85C24×C109—41no—
86C24×C109—44no—
87C24×C109C22⋊C411yescharacteristic
88C24×C109C22⋊C412yes—
89C24×C109C22×C411yescharacteristic
90C24×C109C2×D411yescharacteristic
91#41×C109—48no—
92#41×C109—48no—
93C2×Q8×C109—22no—
94C2×Q8×C109—22no—
95C2×Q8×C109—41no—
96C2×D4×C109—24no—
97C2×D4×C109—21no—
98C2×D4×C109—21no—
99C2×D4×C109—24no—
100C2×D4×C109—22no—
101C2×D4×C109—22no—
102C2×D4×C109—41no—
103C2×D4×C109—41no—
104C2×D4×C109—41no—
105C2×D4×C109—41no—
106C2×D4×C109—41no—
107C2×D4×C109—44no—
108C2×D4×C109—44no—
109C2×D4×C109C2×D412yes—
110C2×D4×C109C2×D412yes—
111C22×C4×C109—21no—
112C22×C4×C109—21no—
113C22×C4×C109—22no—
114C22×C4×C109—22no—
115C22×C4×C109—24no—
116C22×C4×C109—24no—
117C22×C4×C109—21no—
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—41no—
126C22×C4×C109—41no—
127C22×C4×C109—48no—
128C22×C4×C109—48no—
129C22×C4×C109—42no—
130C22×C4×C109—42no—
131C22×C4×C109—42no—
132C22×C4×C109—42no—
133C22×C4×C109—41no—
134C22×C4×C109—41no—
135C22×C4×C109—42no—
136C22×C4×C109C22⋊C411yescharacteristic
137C22×C4×C109C22⋊C411yescharacteristic
138C22×C4×C109C22⋊C412yes—
139C22×C4×C109C22⋊C411yescharacteristic
140C22×C4×C109C2×D411yescharacteristic
141M16×C109—42no—
142M16×C109—42no—
143C2×C8×C109—42no—
144C4⋊C4×C109—24no—
145C4⋊C4×C109—24no—
146C4⋊C4×C109—24no—
147C22⋊C4×C109—24no—
148C22⋊C4×C109—24no—
149C22⋊C4×C109—21no—
150C22⋊C4×C109—21no—
151C22⋊C4×C109—24no—
152C22⋊C4×C109—24no—
153C22⋊C4×C109—42no—
154C22⋊C4×C109—42no—
155C22⋊C4×C109C22⋊C412yes—
156C22⋊C4×C109C22⋊C412yes—
157C42×C109—24no—
158C23×C109—24no—
159C23×C109—22no—
160C23×C109—21no—
161C23×C109—21no—
162C23×C109—21no—
163C23×C109—21no—
164C23×C109—21no—
165C23×C109—21no—
166C23×C109—21no—
167C23×C109—42no—
168C23×C109—44no—
169C23×C109—42no—
170C23×C109—42no—
171C23×C109—42no—
172C23×C109—44no—
173C23×C109—42no—
174C23×C109—42no—
175C23×C109—44no—
176C23×C109—48no—
177C23×C109—44no—
178C23×C109—41no—
179C23×C109—41no—
180C23×C109C2×C22⋊C411yescharacteristic
181C23×C109C2×C22⋊C411yescharacteristic
182C23×C109C2×C22⋊C411yescharacteristic
183C23×C109#12012yes—
184Q8×C109—44no—
185D4×C109—42no—
186D4×C109—42no—
187D4×C109—44no—
188D4×C109—48no—
189D4×C109—48no—
190D4×C109—42no—
191D4×C109—42no—
192C2×C4×C109—24no—
193C2×C4×C109—24no—
194C2×C4×C109—21no—
195C2×C4×C109—21no—
196C2×C4×C109—21no—
197C2×C4×C109—21no—
198C2×C4×C109—21no—
199C2×C4×C109—21no—
200C2×C4×C109—22no—
201C2×C4×C109—22no—
202C2×C4×C109—22no—
203C2×C4×C109—21no—
204C2×C4×C109—21no—
205C2×C4×C109—21no—
206C2×C4×C109—44no—
207C2×C4×C109—44no—
208C2×C4×C109—48no—
209C2×C4×C109—48no—
210C2×C4×C109—42no—
211C2×C4×C109—42no—
212C2×C4×C109—42no—
213C2×C4×C109—41no—
214C2×C4×C109—41no—
215C2×C4×C109—42no—
216C2×C4×C109—82no—
217C2×C4×C109—84no—
218C2×C4×C109#9912yes—
219C2×C4×C109#9912yes—
220C2×C4×C109#12012yes—
221C8×C109—82no—
222C22×C109—22no—
223C22×C109—24no—
224C22×C109—21no—
225C22×C109—21no—
226C22×C109—22no—
227C22×C109—42no—
228C22×C109—41no—
229C22×C109—41no—
230C22×C109—42no—
231C22×C109—44no—
232C22×C109—42no—
233C22×C109—42no—
234C22×C109—44no—
235C22×C109—44no—
236C22×C109—48no—
237C22×C109—42no—
238C22×C109—44no—
239C22×C109—42no—
240C22×C109—42no—
241C22×C109—42no—
242C22×C109—41no—
243C22×C109—81no—
244C22×C109—81no—
245C22×C109#37911yescharacteristic
246C22×C109C2×#9911yescharacteristic
247C22×C109C2×#9911yescharacteristic
248C4×C109—22no—
249C4×C109—22no—
250C4×C109—22no—
251C4×C109—42no—
252C4×C109—42no—
253C4×C109—41no—
254C4×C109—42no—
255C4×C109—162no—
256#12689C10911yesmaximal, characteristic, direct product factor, Sylow
257C2×C109—21no—
258C2×C109—21no—
259C2×C109—42no—
260C2×C109—44no—
261C2×C109—41no—
262C2×C109—41no—
263C2×C109—81no—
264C2×C109#178111yescharacteristic
265C2×C109#212012yes—
266C2×#525C2×C10911yescharacteristic
267C2×#352C2×C10912yescomplemented
268C2×AΣL(1,𝔽16)C2×C10912yescomplemented
269#1785C2×C10911yescharacteristic, complemented
270#1782C2×C10911yescharacteristic, complemented
271C109#1268911yescharacteristic, direct product factor, Sylow
272C22×#41C22×C10911yescharacteristic
273C24×C4C4×C10911yescharacteristic, complemented
274#525—24no—
275#525—24no—
276C2×#123C4×C10911yescharacteristic, complemented
277C2×#123C22×C10911yescharacteristic
278C2×C22⋊Q8—21no—
279C2×#121—21no—
280C2×#120C4×C10911yescharacteristic, complemented
281C2×#120C22×C10911yescharacteristic
282C2×C4×D4—21no—
283C2×C4×D4—21no—
284C2×#117C4×C10911yescharacteristic, complemented
285C2×#101—21no—
286C2×#100C22×C10912yes—
287C2×#99—21no—
288C2×#99C22×C10912yes—
289#407—21no—
290#381—21no—
291#352—24no—
292AΣL(1,𝔽16)—24no—
293C25C2×C4×C10911yescharacteristic
294C2×#41—22no—
295C2×#41—22no—
296C2×#41—42no—
297C2×#41—42no—
298C22×Q8D4×C10911yescharacteristic
299C22×D4—21no—
300C22×D4—21no—
301C22×D4C2×C4×C10911yescharacteristic
302C22×D4D4×C10911yescharacteristic
303C22×D4C23×C10911yescharacteristic
304C23×C4—22no—
305C23×C4—21no—
306C23×C4—21no—
307C23×C4—22no—
308C23×C4—42no—
309C23×C4—42no—
310C23×C4C2×C4×C10911yescharacteristic
311C23×C4C2×C4×C10911yescharacteristic
312C23×C4D4×C10911yescharacteristic
313C23×C4D4×C10911yescharacteristic
314C2×M16—22no—
315#123—22no—
316#123—22no—
317#123—22no—
318#123D4×C10914yes—
319C22⋊Q8—28no—
320#121—28no—
321#120—22no—
322#120—22no—
323#120—22no—
324#120D4×C10914yes—
325C4×D4—28no—
326C4×D4—28no—
327#117—22no—
328#117—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4C2×C4×C10911yescharacteristic
338C2×C22⋊C4C2×C4×C10911yescharacteristic
339C2×C42—21no—
340#101—42no—
341#100—22no—
342#100—22no—
343#99—22no—
344#99—22no—
345#99—42no—
346C22⋊C8—42no—
347#95—42no—
348C24—21no—
349C24—21no—
350C24—41no—
351C24—41no—
352C24—44no—
353C24C22⋊C4×C10911yescharacteristic
354C24C22⋊C4×C10912yes—
355C24C22×C4×C10911yescharacteristic
356C24C2×D4×C10911yescharacteristic
357#41—48no—
358#41—48no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—41no—
362C2×D4—24no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—24no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4C2×D4×C10912yes—
376C2×D4C2×D4×C10912yes—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4C22⋊C4×C10911yescharacteristic
403C22×C4C22⋊C4×C10911yescharacteristic
404C22×C4C22⋊C4×C10911yescharacteristic
405C22×C4C22⋊C4×C10912yes—
406C22×C4C2×D4×C10911yescharacteristic
407M16—42no—
408M16—42no—
409C2×C8—42no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C22⋊C4—24no—
414C22⋊C4—24no—
415C22⋊C4—21no—
416C22⋊C4—21no—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—42no—
420C22⋊C4—42no—
421C22⋊C4C22⋊C4×C10912yes—
422C22⋊C4C22⋊C4×C10912yes—
423C42—24no—
424C23—24no—
425C23—22no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—42no—
434C23—44no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—44no—
439C23—42no—
440C23—42no—
441C23—44no—
442C23—48no—
443C23—44no—
444C23—41no—
445C23—41no—
446C23C2×C22⋊C4×C10911yescharacteristic
447C23C2×C22⋊C4×C10911yescharacteristic
448C23C2×C22⋊C4×C10911yescharacteristic
449C23#120×C10912yes—
450Q8—44no—
451D4—42no—
452D4—42no—
453D4—44no—
454D4—42no—
455D4—42no—
456D4—48no—
457D4—48no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—48no—
475C2×C4—48no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—82no—
483C2×C4—84no—
484C2×C4#99×C10912yes—
485C2×C4#99×C10912yes—
486C2×C4#120×C10912yes—
487C8—82no—
488C22—22no—
489C22—24no—
490C22—21no—
491C22—21no—
492C22—22no—
493C22—42no—
494C22—41no—
495C22—41no—
496C22—42no—
497C22—44no—
498C22—42no—
499C22—42no—
500C22—44no—
501C22—44no—
502C22—48no—
503C22—42no—
504C22—42no—
505C22—44no—
506C22—42no—
507C22—41no—
508C22—42no—
509C22—81no—
510C22—81no—
511C22#379×C10911yescharacteristic
512C22C2×#99×C10911yescharacteristic
513C22C2×#99×C10911yescharacteristic
514C4—22no—
515C4—22no—
516C4—22no—
517C4—42no—
518C4—42no—
519C4—41no—
520C4—42no—
521C4—162no—
522C2—21no—
523C2—21no—
524C2—42no—
525C2—44no—
526C2—41no—
527C2—41no—
528C2—81no—
529C2C109×#178111yescharacteristic
530C2C109×#212012yes—
subgroupG/H
center, socleC22×C109C2×#99
commutatorC24C22×C4×C109
Frattini, cosocleC22×D4C23×C109
prime radicalC2×#525×C109C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C109 ◁ C22×C4×C109 ◁ C22×#41×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#352×C109C2semidirect
3. C2×AΣL(1,𝔽16)×C109C2semidirect
4. C109×#1785C2semidirect
5. C109×#1782C2semidirect
7. C24×C4×C109C4semidirect
10. C2×#123×C109C4semidirect
14. C2×#120×C109C4semidirect
18. C2×#117×C109C4semidirect
256. #12689C109direct (N × H)
267. C2×#352C2×C109semidirect
268. C2×AΣL(1,𝔽16)C2×C109semidirect
269. #1785C2×C109semidirect
270. #1782C2×C109semidirect
271. C109#12689direct (N × H)
273. C24×C4C4×C109semidirect
276. C2×#123C4×C109semidirect
280. C2×#120C4×C109semidirect
284. C2×#117C4×C109semidirect

Maximal subgroups

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

subgroupindex
C2×#525×C1092
C2×#352×C1092
C2×AΣL(1,𝔽16)×C1092
C109×#17852
C109×#17822
#12689109