Finite Group Databasebeta

← C4×AΣL(1,𝔽16)×C109 #42130613

Subgroups — C4×AΣL(1,𝔽16)×C109 #42130613

Order 27904.

844 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#120×C109C211yesmaximal, characteristic
2C2×AΣL(1,𝔽16)×C109C212yesmaximal
3C4×#100×C109C211yesmaximal, characteristic, complemented
4C4×#99×C109C211yesmaximal, characteristic, complemented
5C109×#1390C212yesmaximal, complemented
6C24×C4×C109C411yescharacteristic, complemented
7C2×#120×C109C2211yescharacteristic
8C2×C4×D4×C109—21no—
9C2×C4×D4×C109C2211yescharacteristic
10C2×#100×C109C2212yes—
11C2×#99×C109C2212yes—
12C4×M16×C109—21no—
13#386×C109—21no—
14#386×C109C2211yescharacteristic
15#385×C109—21no—
16#385×C109C411yescharacteristic, complemented
17#379×C109C411yescharacteristic, complemented
18C4×C22⋊C4×C109—21no—
19C4×C22⋊C4×C109—21no—
20C4×C22⋊C4×C109C411yescharacteristic, complemented
21AΣL(1,𝔽16)×C109C418yesdirect product factor
22#343×C109—21no—
23#342×C109—21no—
24C25×C109C2×C411yescharacteristic
25C22×D4×C109—21no—
26C22×D4×C109C2311yescharacteristic
27C23×C4×C109—21no—
28C23×C4×C109—21no—
29C23×C4×C109—44no—
30C23×C4×C109—41no—
31C23×C4×C109—41no—
32C23×C4×C109C2×C411yescharacteristic
33C23×C4×C109C2×C411yescharacteristic
34C23×C4×C109D411yescharacteristic
35C2×M16×C109—22no—
36#120×C109—22no—
37#120×C109C2×C414yes—
38C4×D4×C109—41no—
39C4×D4×C109—41no—
40C4×D4×C109—44no—
41C2×C4⋊C4×C109—21no—
42C2×C4⋊C4×C109D411yescharacteristic
43C2×C22⋊C4×C109—21no—
44C2×C22⋊C4×C109—22no—
45C2×C22⋊C4×C109—21no—
46C2×C22⋊C4×C109—21no—
47C2×C22⋊C4×C109—21no—
48C2×C22⋊C4×C109C2×C411yescharacteristic
49C2×C22⋊C4×C109C2×C411yescharacteristic
50C2×C22⋊C4×C109C2×C411yescharacteristic
51C2×C22⋊C4×C109D411yescharacteristic
52C2×C42×C109—21no—
53C2×C42×C109—41no—
54C2×C42×C109D411yescharacteristic
55#100×C109—22no—
56#100×C109C2×C414yescomplemented
57#99×C109—22no—
58#99×C109C2×C414yescomplemented
59#97×C109—41no—
60C4×C8×C109—41no—
61#95×C109—21no—
62#95×C109—21no—
63#95×C109—21no—
64#95×C109—41no—
65C24×C109—21no—
66C24×C109—21no—
67C24×C109—42no—
68C24×C109—41no—
69C24×C109—42no—
70C24×C109—41no—
71C24×C109C4212yescomplemented
72C24×C109C22×C411yescharacteristic
73C24×C109C2×D411yescharacteristic
74C2×D4×C109—24no—
75C2×D4×C109—41no—
76C2×D4×C109—41no—
77C2×D4×C109—42no—
78C2×D4×C109—42no—
79C2×D4×C109C22×C412yes—
80C2×D4×C109#4112yes—
81C22×C4×C109—24no—
82C22×C4×C109—21no—
83C22×C4×C109—21no—
84C22×C4×C109—21no—
85C22×C4×C109—21no—
86C22×C4×C109—21no—
87C22×C4×C109—21no—
88C22×C4×C109—21no—
89C22×C4×C109—42no—
90C22×C4×C109—42no—
91C22×C4×C109—42no—
92C22×C4×C109—42no—
93C22×C4×C109—42no—
94C22×C4×C109—42no—
95C22×C4×C109—42no—
96C22×C4×C109—44no—
97C22×C4×C109—44no—
98C22×C4×C109—44no—
99C22×C4×C109—41no—
100C22×C4×C109—41no—
101C22×C4×C109—41no—
102C22×C4×C109—44no—
103C22×C4×C109—42no—
104C22×C4×C109C22⋊C411yescharacteristic
105C22×C4×C109C22⋊C411yescharacteristic
106C22×C4×C109C22⋊C411yescharacteristic
107C22×C4×C109C22⋊C411yescharacteristic
108C22×C4×C109C2×D411yescharacteristic
109M16×C109—24no—
110M16×C109—42no—
111C2×C8×C109—42no—
112C4⋊C4×C109—41no—
113C4⋊C4×C109—42no—
114C22⋊C4×C109—21no—
115C22⋊C4×C109—24no—
116C22⋊C4×C109—24no—
117C22⋊C4×C109—41no—
118C22⋊C4×C109—41no—
119C22⋊C4×C109—42no—
120C22⋊C4×C109—42no—
121C22⋊C4×C109—42no—
122C22⋊C4×C109—42no—
123C22⋊C4×C109—41no—
124C22⋊C4×C109C4212yescomplemented
125C42×C109—21no—
126C42×C109—21no—
127C42×C109—42no—
128C42×C109—82no—
129C23×C109—22no—
130C23×C109—22no—
131C23×C109—22no—
132C23×C109—21no—
133C23×C109—21no—
134C23×C109—21no—
135C23×C109—21no—
136C23×C109—41no—
137C23×C109—44no—
138C23×C109—42no—
139C23×C109—41no—
140C23×C109—41no—
141C23×C109—41no—
142C23×C109—44no—
143C23×C109—42no—
144C23×C109—41no—
145C23×C109—41no—
146C23×C109—42no—
147C23×C109—42no—
148C23×C109—42no—
149C23×C109—48no—
150C23×C109—42no—
151C23×C109—41no—
152C23×C109—41no—
153C23×C109C2×C4211yescharacteristic
154C23×C109C2×C22⋊C411yescharacteristic
155C23×C109#11711yescharacteristic
156C23×C109C4×D412yes—
157D4×C109—42no—
158D4×C109—42no—
159D4×C109—48no—
160C2×C4×C109—21no—
161C2×C4×C109—21no—
162C2×C4×C109—22no—
163C2×C4×C109—22no—
164C2×C4×C109—22no—
165C2×C4×C109—21no—
166C2×C4×C109—21no—
167C2×C4×C109—21no—
168C2×C4×C109—44no—
169C2×C4×C109—42no—
170C2×C4×C109—41no—
171C2×C4×C109—41no—
172C2×C4×C109—41no—
173C2×C4×C109—48no—
174C2×C4×C109—42no—
175C2×C4×C109—41no—
176C2×C4×C109—42no—
177C2×C4×C109—44no—
178C2×C4×C109—42no—
179C2×C4×C109—41no—
180C2×C4×C109—41no—
181C2×C4×C109—41no—
182C2×C4×C109—42no—
183C2×C4×C109—41no—
184C2×C4×C109—44no—
185C2×C4×C109—41no—
186C2×C4×C109—41no—
187C2×C4×C109—81no—
188C2×C4×C109—82no—
189C2×C4×C109—82no—
190C2×C4×C109#9911yescharacteristic
191C2×C4×C109#9911yescharacteristic
192C2×C4×C109C4×D412yes—
193C8×C109—44no—
194C22×C109—21no—
195C22×C109—21no—
196C22×C109—24no—
197C22×C109—22no—
198C22×C109—21no—
199C22×C109—21no—
200C22×C109—42no—
201C22×C109—41no—
202C22×C109—41no—
203C22×C109—42no—
204C22×C109—42no—
205C22×C109—42no—
206C22×C109—42no—
207C22×C109—44no—
208C22×C109—44no—
209C22×C109—44no—
210C22×C109—44no—
211C22×C109—42no—
212C22×C109—41no—
213C22×C109—41no—
214C22×C109—41no—
215C22×C109—42no—
216C22×C109—41no—
217C22×C109—42no—
218C22×C109—81no—
219C22×C109C4×C22⋊C411yescharacteristic
220C22×C109C2×#9911yescharacteristic
221C22×C109#41011yescharacteristic
222C4×C109—21no—
223C4×C109—22no—
224C4×C109—42no—
225C4×C109—42no—
226C4×C109—41no—
227C4×C109—41no—
228C4×C109—41no—
229C4×C109—81no—
230C4×C109—84no—
231C4×C109AΣL(1,𝔽16)12yesdirect product factor
232C4×AΣL(1,𝔽16)C10911yesmaximal, characteristic, direct product factor, Sylow
233C2×C109—21no—
234C2×C109—21no—
235C2×C109—42no—
236C2×C109—44no—
237C2×C109—41no—
238C2×C109—41no—
239C2×C109C4×#9911yescharacteristic
240C2×C109C2×AΣL(1,𝔽16)11yescharacteristic
241C2×C109#212011yescharacteristic
242C4×#120C2×C10911yescharacteristic
243C2×AΣL(1,𝔽16)C2×C10912yes—
244C4×#100C2×C10911yescharacteristic, complemented
245C4×#99C2×C10911yescharacteristic, complemented
246#1390C2×C10912yescomplemented
247C109C4×AΣL(1,𝔽16)11yescharacteristic, direct product factor, Sylow
248C24×C4C4×C10911yescharacteristic, complemented
249C2×#120C22×C10911yescharacteristic
250C2×C4×D4—21no—
251C2×C4×D4C22×C10911yescharacteristic
252C2×#100C22×C10912yes—
253C2×#99C22×C10912yes—
254C4×M16—21no—
255#386—21no—
256#386C22×C10911yescharacteristic
257#385—21no—
258#385C4×C10911yescharacteristic, complemented
259#379C4×C10911yescharacteristic, complemented
260C4×C22⋊C4—21no—
261C4×C22⋊C4—21no—
262C4×C22⋊C4C4×C10911yescharacteristic, complemented
263AΣL(1,𝔽16)C4×C10918yesdirect product factor
264#343—21no—
265#342—21no—
266C25C2×C4×C10911yescharacteristic
267C22×D4—21no—
268C22×D4C23×C10911yescharacteristic
269C23×C4—21no—
270C23×C4—21no—
271C23×C4—44no—
272C23×C4—41no—
273C23×C4—41no—
274C23×C4C2×C4×C10911yescharacteristic
275C23×C4C2×C4×C10911yescharacteristic
276C23×C4D4×C10911yescharacteristic
277C2×M16—22no—
278#120—22no—
279#120C2×C4×C10914yes—
280C4×D4—41no—
281C4×D4—41no—
282C4×D4—44no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4D4×C10911yescharacteristic
285C2×C22⋊C4—21no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4C2×C4×C10911yescharacteristic
291C2×C22⋊C4C2×C4×C10911yescharacteristic
292C2×C22⋊C4C2×C4×C10911yescharacteristic
293C2×C22⋊C4D4×C10911yescharacteristic
294C2×C42—21no—
295C2×C42—41no—
296C2×C42D4×C10911yescharacteristic
297#100—22no—
298#100C2×C4×C10914yescomplemented
299#99—22no—
300#99C2×C4×C10914yescomplemented
301#97—41no—
302C4×C8—41no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—41no—
307C24—21no—
308C24—21no—
309C24—42no—
310C24—41no—
311C24—42no—
312C24—41no—
313C24C42×C10912yescomplemented
314C24C22×C4×C10911yescharacteristic
315C24C2×D4×C10911yescharacteristic
316C2×D4—24no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4C22×C4×C10912yes—
322C2×D4#41×C10912yes—
323C22×C4—24no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—44no—
345C22×C4—42no—
346C22×C4C22⋊C4×C10911yescharacteristic
347C22×C4C22⋊C4×C10911yescharacteristic
348C22×C4C22⋊C4×C10911yescharacteristic
349C22×C4C22⋊C4×C10911yescharacteristic
350C22×C4C2×D4×C10911yescharacteristic
351M16—24no—
352M16—42no—
353C2×C8—42no—
354C4⋊C4—41no—
355C4⋊C4—42no—
356C22⋊C4—21no—
357C22⋊C4—24no—
358C22⋊C4—24no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—41no—
366C22⋊C4C42×C10912yescomplemented
367C42—21no—
368C42—21no—
369C42—42no—
370C42—82no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—41no—
379C23—44no—
380C23—42no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—44no—
385C23—42no—
386C23—41no—
387C23—41no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—48no—
392C23—42no—
393C23—41no—
394C23—41no—
395C23C2×C42×C10911yescharacteristic
396C23C2×C22⋊C4×C10911yescharacteristic
397C23#117×C10911yescharacteristic
398C23C4×D4×C10912yes—
399D4—42no—
400D4—42no—
401D4—48no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—48no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—44no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—81no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4#99×C10911yescharacteristic
433C2×C4#99×C10911yescharacteristic
434C2×C4C4×D4×C10912yes—
435C8—44no—
436C22—21no—
437C22—21no—
438C22—24no—
439C22—22no—
440C22—21no—
441C22—21no—
442C22—42no—
443C22—41no—
444C22—41no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—44no—
450C22—44no—
451C22—44no—
452C22—44no—
453C22—42no—
454C22—42no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—42no—
459C22—41no—
460C22—81no—
461C22C4×C22⋊C4×C10911yescharacteristic
462C22C2×#99×C10911yescharacteristic
463C22#410×C10911yescharacteristic
464C4—22no—
465C4—21no—
466C4—42no—
467C4—42no—
468C4—41no—
469C4—41no—
470C4—41no—
471C4—81no—
472C4—84no—
473C4AΣL(1,𝔽16)×C10912yesdirect product factor
474C2—21no—
475C2—21no—
476C2—42no—
477C2—44no—
478C2—41no—
479C2—41no—
480C2C4×#99×C10911yescharacteristic
481C2C2×AΣL(1,𝔽16)×C10911yescharacteristic
482C2C109×#212011yescharacteristic
subgroupG/H
centerC2×C4×C109#99
commutatorC23C2×C42×C109
Frattini, cosocleC22×D4C23×C109
socleC22×C109C2×#99
prime radicalC2×#120×C109C22
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C109 ◁ C22×C4×C109 ◁ C2×C4×D4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
3. C4×#100×C109C2semidirect
4. C4×#99×C109C2semidirect
5. C109×#1390C2semidirect
6. C24×C4×C109C4semidirect
16. #385×C109C4semidirect
17. #379×C109C4semidirect
20. C4×C22⋊C4×C109C4semidirect
21. AΣL(1,𝔽16)×C109C4direct (N × H)
56. #100×C109C2×C4semidirect
58. #99×C109C2×C4semidirect
71. C24×C109C42semidirect
124. C22⋊C4×C109C42semidirect
231. C4×C109AΣL(1,𝔽16)direct (N × H)
232. C4×AΣL(1,𝔽16)C109direct (N × H)
244. C4×#100C2×C109semidirect
245. C4×#99C2×C109semidirect
246. #1390C2×C109semidirect
247. C109C4×AΣL(1,𝔽16)direct (N × H)
248. C24×C4C4×C109semidirect
258. #385C4×C109semidirect
259. #379C4×C109semidirect
262. C4×C22⋊C4C4×C109semidirect
263. AΣL(1,𝔽16)C4×C109direct (N × H)
298. #100C2×C4×C109semidirect
300. #99C2×C4×C109semidirect
313. C24C42×C109semidirect
366. C22⋊C4C42×C109semidirect
473. C4AΣL(1,𝔽16)×C109direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#120×C1092
C2×AΣL(1,𝔽16)×C1092
C4×#100×C1092
C4×#99×C1092
C109×#13902
C4×AΣL(1,𝔽16)109