Finite Group Databasebeta

← C109×#12719 #42130713

Subgroups — C109×#12719 #42130713

Order 27904.

780 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3444C211yesmaximal, characteristic
2C109×#2132C211yesmaximal, characteristic, complemented
3C109×#2126C211yesmaximal, characteristic, complemented
4C109×#2120C212yesmaximal, complemented
5C109×#1827C212yesmaximal, complemented
6#533×C109—21no—
7#533×C109C411yescharacteristic, complemented
8#532×C109—21no—
9#532×C109C2211yescharacteristic
10#529×C109—21no—
11#529×C109—21no—
12#529×C109—22no—
13#529×C109—22no—
14#525×C109—22no—
15#525×C109C411yescharacteristic, complemented
16#525×C109C2211yescharacteristic
17C4×#41×C109—22no—
18C4×#41×C109C411yescharacteristic, complemented
19C4×#41×C109C2211yescharacteristic
20C22×C42×C109C411yescharacteristic, complemented
21#413×C109C2212yes—
22#410×C109C2212yes—
23#356×C109—21no—
24#355×C109—21no—
25#354×C109—21no—
26#353×C109—21no—
27#352×C109—22no—
28AΣL(1,𝔽16)×C109—22no—
29#339×C109—22no—
30H(3,ℤ/4)×C109—22no—
31C2×#41×C109—22no—
32C2×#41×C109C2×C411yescharacteristic
33C2×#41×C109C2311yescharacteristic
34C23×C4×C109—21no—
35C23×C4×C109C2×C411yescharacteristic
36C2×M16×C109—22no—
37C4⋊Q8×C109—21no—
38C4⋊Q8×C109D411yescharacteristic
39#127×C109D411yescharacteristic
40#126×C109—22no—
41#126×C109—22no—
42#125×C109—22no—
43#125×C109—21no—
44#124×C109—22no—
45#124×C109D411yescharacteristic
46#124×C109D411yescharacteristic
47#123×C109—22no—
48#123×C109—22no—
49#123×C109—21no—
50#123×C109D412yes—
51C22⋊Q8×C109—22no—
52C22⋊Q8×C109—21no—
53C22⋊Q8×C109—21no—
54C22⋊Q8×C109—22no—
55#121×C109—22no—
56#121×C109—21no—
57#121×C109—22no—
58#121×C109—21no—
59#120×C109—21no—
60#120×C109D412yes—
61C4×Q8×C109—22no—
62C4×Q8×C109—21no—
63C4×Q8×C109—21no—
64C4×Q8×C109—22no—
65C4×D4×C109—22no—
66C4×D4×C109—22no—
67C4×D4×C109—22no—
68C4×D4×C109—21no—
69C4×D4×C109—22no—
70C4×D4×C109—21no—
71C4×D4×C109—22no—
72C4×D4×C109—22no—
73C4×D4×C109—22no—
74C4×D4×C109—22no—
75#117×C109—22no—
76#117×C109—22no—
77#117×C109—22no—
78#117×C109—21no—
79#117×C109—22no—
80#117×C109C2×C411yescharacteristic
81#117×C109C2×C411yescharacteristic
82#117×C109D412yes—
83C2×C42×C109—21no—
84C2×C42×C109—21no—
85C2×C42×C109—21no—
86C2×C42×C109—21no—
87C2×C42×C109—21no—
88C2×C42×C109—21no—
89C2×C42×C109C2×C411yescharacteristic
90C2×C42×C109C2×C411yescharacteristic
91C2×C42×C109D412yes—
92#101×C109—22no—
93#100×C109—22no—
94#99×C109—22no—
95#99×C109—22no—
96C24×C109C22⋊C411yescharacteristic
97#41×C109—41no—
98#41×C109—41no—
99#41×C109—48no—
100#41×C109—41no—
101#41×C109—41no—
102C2×Q8×C109—22no—
103C2×Q8×C109C22⋊C411yescharacteristic
104C2×Q8×C109C2×D411yescharacteristic
105C2×D4×C109—22no—
106C2×D4×C109—21no—
107C2×D4×C109—22no—
108C2×D4×C109—22no—
109C2×D4×C109C22⋊C411yescharacteristic
110C2×D4×C109C2×D411yescharacteristic
111C2×D4×C109C2×D412yes—
112C22×C4×C109—21no—
113C22×C4×C109—21no—
114C22×C4×C109—21no—
115C22×C4×C109—21no—
116C22×C4×C109—22no—
117C22×C4×C109—21no—
118C22×C4×C109—21no—
119C22×C4×C109—22no—
120C22×C4×C109—21no—
121C22×C4×C109—21no—
122C22×C4×C109—21no—
123C22×C4×C109—44no—
124C22×C4×C109—41no—
125C22×C4×C109—41no—
126C22×C4×C109C22⋊C411yescharacteristic
127C22×C4×C109C22×C411yescharacteristic
128C22×C4×C109C2×D412yes—
129M16×C109—42no—
130M16×C109—42no—
131C2×C8×C109—42no—
132C4⋊C4×C109—22no—
133C4⋊C4×C109—22no—
134C4⋊C4×C109—22no—
135C4⋊C4×C109—22no—
136C4⋊C4×C109—22no—
137C4⋊C4×C109—21no—
138C4⋊C4×C109—21no—
139C4⋊C4×C109—22no—
140C4⋊C4×C109—22no—
141C4⋊C4×C109—22no—
142C4⋊C4×C109—42no—
143C22⋊C4×C109—22no—
144C22⋊C4×C109—22no—
145C22⋊C4×C109—22no—
146C22⋊C4×C109—21no—
147C22⋊C4×C109—22no—
148C22⋊C4×C109—22no—
149C22⋊C4×C109—22no—
150C22⋊C4×C109—22no—
151C22⋊C4×C109—22no—
152C22⋊C4×C109—42no—
153C22⋊C4×C109C22⋊C412yes—
154C42×C109—21no—
155C42×C109—21no—
156C42×C109—21no—
157C42×C109—21no—
158C42×C109—22no—
159C42×C109—21no—
160C42×C109—21no—
161C42×C109—21no—
162C42×C109—22no—
163C42×C109—42no—
164C42×C109C22⋊C411yescharacteristic
165C42×C109C22⋊C411yescharacteristic
166C42×C109C22⋊C411yescharacteristic
167C42×C109C22⋊C412yes—
168C42×C109C22⋊C411yescharacteristic
169C23×C109—21no—
170C23×C109—21no—
171C23×C109—42no—
172C23×C109—41no—
173C23×C109C2×C22⋊C411yescharacteristic
174C23×C109#12012yes—
175Q8×C109—41no—
176Q8×C109—41no—
177Q8×C109—44no—
178D4×C109—42no—
179D4×C109—41no—
180D4×C109—41no—
181D4×C109—44no—
182D4×C109—44no—
183D4×C109—44no—
184D4×C109—42no—
185C2×C4×C109—24no—
186C2×C4×C109—21no—
187C2×C4×C109—21no—
188C2×C4×C109—22no—
189C2×C4×C109—21no—
190C2×C4×C109—22no—
191C2×C4×C109—21no—
192C2×C4×C109—21no—
193C2×C4×C109—21no—
194C2×C4×C109—21no—
195C2×C4×C109—21no—
196C2×C4×C109—22no—
197C2×C4×C109—22no—
198C2×C4×C109—42no—
199C2×C4×C109—42no—
200C2×C4×C109—42no—
201C2×C4×C109—42no—
202C2×C4×C109—42no—
203C2×C4×C109—44no—
204C2×C4×C109—44no—
205C2×C4×C109—42no—
206C2×C4×C109—41no—
207C2×C4×C109—41no—
208C2×C4×C109—41no—
209C2×C4×C109—41no—
210C2×C4×C109—41no—
211C2×C4×C109—82no—
212C2×C4×C109C2×C22⋊C411yescharacteristic
213C2×C4×C109C2×C22⋊C411yescharacteristic
214C2×C4×C109#12012yes—
215C8×C109—82no—
216C22×C109—22no—
217C22×C109—21no—
218C22×C109—22no—
219C22×C109—41no—
220C22×C109—42no—
221C22×C109—41no—
222C22×C109—42no—
223C22×C109—42no—
224C22×C109—42no—
225C22×C109—41no—
226C22×C109—41no—
227C22×C109#37911yescharacteristic
228C4×C109—22no—
229C4×C109—22no—
230C4×C109—22no—
231C4×C109—21no—
232C4×C109—41no—
233C4×C109—41no—
234C4×C109—42no—
235C4×C109—42no—
236C4×C109—41no—
237C4×C109—41no—
238C4×C109—162no—
239C4×C109#41011yescharacteristic
240C4×C109#41011yescharacteristic
241#12719C10911yesmaximal, characteristic, direct product factor, Sylow
242C2×C109—21no—
243C2×C109—42no—
244C2×C109—42no—
245C2×C109—41no—
246C2×C109—81no—
247C2×C109#178211yescharacteristic
248#3444C2×C10911yescharacteristic
249#2132C2×C10911yescharacteristic, complemented
250#2126C2×C10911yescharacteristic, complemented
251#2120C2×C10912yescomplemented
252#1827C2×C10912yescomplemented
253C109#1271911yescharacteristic, direct product factor, Sylow
254#533—21no—
255#533C4×C10911yescharacteristic, complemented
256#532—21no—
257#532C22×C10911yescharacteristic
258#529—21no—
259#529—21no—
260#529—22no—
261#529—22no—
262#525—22no—
263#525C4×C10911yescharacteristic, complemented
264#525C22×C10911yescharacteristic
265C4×#41—22no—
266C4×#41C4×C10911yescharacteristic, complemented
267C4×#41C22×C10911yescharacteristic
268C22×C42C4×C10911yescharacteristic, complemented
269#413C22×C10912yes—
270#410C22×C10912yes—
271#356—21no—
272#355—21no—
273#354—21no—
274#353—21no—
275#352—22no—
276AΣL(1,𝔽16)—22no—
277#339—22no—
278H(3,ℤ/4)—22no—
279C2×#41—22no—
280C2×#41C2×C4×C10911yescharacteristic
281C2×#41C23×C10911yescharacteristic
282C23×C4—21no—
283C23×C4C2×C4×C10911yescharacteristic
284C2×M16—22no—
285C4⋊Q8—21no—
286C4⋊Q8D4×C10911yescharacteristic
287#127D4×C10911yescharacteristic
288#126—22no—
289#126—22no—
290#125—22no—
291#125—21no—
292#124—22no—
293#124D4×C10911yescharacteristic
294#124D4×C10911yescharacteristic
295#123—22no—
296#123—22no—
297#123—21no—
298#123D4×C10912yes—
299C22⋊Q8—22no—
300C22⋊Q8—21no—
301C22⋊Q8—21no—
302C22⋊Q8—22no—
303#121—22no—
304#121—21no—
305#121—22no—
306#121—21no—
307#120—21no—
308#120D4×C10912yes—
309C4×Q8—22no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×Q8—22no—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4—22no—
316C4×D4—21no—
317C4×D4—22no—
318C4×D4—21no—
319C4×D4—22no—
320C4×D4—22no—
321C4×D4—22no—
322C4×D4—22no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—21no—
327#117—22no—
328#117C2×C4×C10911yescharacteristic
329#117C2×C4×C10911yescharacteristic
330#117D4×C10912yes—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42C2×C4×C10911yescharacteristic
338C2×C42C2×C4×C10911yescharacteristic
339C2×C42D4×C10912yes—
340#101—22no—
341#100—22no—
342#99—22no—
343#99—22no—
344C24C22⋊C4×C10911yescharacteristic
345#41—41no—
346#41—41no—
347#41—48no—
348#41—41no—
349#41—41no—
350C2×Q8—22no—
351C2×Q8C22⋊C4×C10911yescharacteristic
352C2×Q8C2×D4×C10911yescharacteristic
353C2×D4—22no—
354C2×D4—21no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4C22⋊C4×C10911yescharacteristic
358C2×D4C2×D4×C10911yescharacteristic
359C2×D4C2×D4×C10912yes—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—44no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4C22⋊C4×C10911yescharacteristic
375C22×C4C22×C4×C10911yescharacteristic
376C22×C4C2×D4×C10912yes—
377M16—42no—
378M16—42no—
379C2×C8—42no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—42no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—21no—
395C22⋊C4—22no—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—42no—
401C22⋊C4C22⋊C4×C10912yes—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—22no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—22no—
411C42—42no—
412C42C22⋊C4×C10911yescharacteristic
413C42C22⋊C4×C10911yescharacteristic
414C42C22⋊C4×C10911yescharacteristic
415C42C22⋊C4×C10912yes—
416C42C22⋊C4×C10911yescharacteristic
417C23—21no—
418C23—21no—
419C23—42no—
420C23—41no—
421C23C2×C22⋊C4×C10911yescharacteristic
422C23#120×C10912yes—
423Q8—41no—
424Q8—41no—
425Q8—44no—
426D4—42no—
427D4—41no—
428D4—41no—
429D4—44no—
430D4—44no—
431D4—44no—
432D4—42no—
433C2×C4—24no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—82no—
460C2×C4C2×C22⋊C4×C10911yescharacteristic
461C2×C4C2×C22⋊C4×C10911yescharacteristic
462C2×C4#120×C10912yes—
463C8—82no—
464C22—22no—
465C22—21no—
466C22—22no—
467C22—41no—
468C22—42no—
469C22—41no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—41no—
474C22—41no—
475C22#379×C10911yescharacteristic
476C4—22no—
477C4—22no—
478C4—22no—
479C4—21no—
480C4—41no—
481C4—41no—
482C4—42no—
483C4—42no—
484C4—41no—
485C4—41no—
486C4—162no—
487C4#410×C10911yescharacteristic
488C4#410×C10911yescharacteristic
489C2—21no—
490C2—42no—
491C2—42no—
492C2—41no—
493C2—81no—
494C2C109×#178211yescharacteristic
subgroupG/H
centerC4×C109#410
commutatorC22×C4C22×C4×C109
Frattini, cosocleC2×#41C23×C109
socleC2×C109#1782
prime radicalC109×#3444C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C109 ◁ C42×C109 ◁ C4×#41×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#2132C2semidirect
3. C109×#2126C2semidirect
4. C109×#2120C2semidirect
5. C109×#1827C2semidirect
7. #533×C109C4semidirect
15. #525×C109C4semidirect
18. C4×#41×C109C4semidirect
20. C22×C42×C109C4semidirect
241. #12719C109direct (N × H)
249. #2132C2×C109semidirect
250. #2126C2×C109semidirect
251. #2120C2×C109semidirect
252. #1827C2×C109semidirect
253. C109#12719direct (N × H)
255. #533C4×C109semidirect
263. #525C4×C109semidirect
266. C4×#41C4×C109semidirect
268. C22×C42C4×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#34442
C109×#21322
C109×#21262
C109×#21202
C109×#18272
#12719109