Finite Group Databasebeta

← C109×#12720 #42130714

Subgroups — C109×#12720 #42130714

Order 27904.

694 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3455C211yesmaximal, characteristic
2C109×#2132C211yesmaximal, characteristic, complemented
3C109×#2126C211yesmaximal, characteristic, complemented
4C109×#2120C212yesmaximal, complemented
5C109×#1827C212yesmaximal, complemented
6#542×C109—21no—
7#542×C109—21no—
8#541×C109—21no—
9#540×C109—21no—
10#539×C109—22no—
11#538×C109—22no—
12#533×C109C411yescharacteristic, complemented
13#532×C109C2211yescharacteristic
14#529×C109—22no—
15#529×C109—22no—
16#525×C109C411yescharacteristic, complemented
17#525×C109C2211yescharacteristic
18C4×#41×C109C411yescharacteristic, complemented
19C4×#41×C109C2211yescharacteristic
20C2×#117×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×C109C2×C411yescharacteristic
32C2×#41×C109C2311yescharacteristic
33C23×C4×C109C2×C411yescharacteristic
34C2×M16×C109—22no—
35C4⋊Q8×C109—21no—
36C4⋊Q8×C109D411yescharacteristic
37#127×C109D411yescharacteristic
38#126×C109—22no—
39#126×C109—22no—
40#126×C109—22no—
41#126×C109—22no—
42#125×C109—22no—
43#125×C109—22no—
44#125×C109—21no—
45#124×C109—22no—
46#124×C109—22no—
47#124×C109D411yescharacteristic
48#124×C109D411yescharacteristic
49#123×C109—22no—
50#123×C109—22no—
51#123×C109—22no—
52#123×C109—22no—
53#123×C109—21no—
54#123×C109D412yes—
55C22⋊Q8×C109—22no—
56C22⋊Q8×C109—22no—
57C22⋊Q8×C109—21no—
58C22⋊Q8×C109—21no—
59C22⋊Q8×C109—22no—
60#121×C109—22no—
61#121×C109—22no—
62#121×C109—21no—
63#121×C109—21no—
64#121×C109—22no—
65#120×C109—21no—
66#120×C109D412yes—
67C4×Q8×C109—21no—
68C4×Q8×C109—21no—
69C4×Q8×C109—22no—
70C4×D4×C109—21no—
71C4×D4×C109—22no—
72C4×D4×C109—21no—
73C4×D4×C109—22no—
74C4×D4×C109—22no—
75C4×D4×C109—22no—
76C4×D4×C109—22no—
77#117×C109—21no—
78#117×C109—21no—
79#117×C109—21no—
80#117×C109—21no—
81#117×C109—21no—
82#117×C109—22no—
83#117×C109C2×C411yescharacteristic
84#117×C109C2×C411yescharacteristic
85#117×C109D412yes—
86C2×C4⋊C4×C109—21no—
87C2×C22⋊C4×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—41no—
100#41×C109—41no—
101C2×Q8×C109—22no—
102C2×Q8×C109C22⋊C411yescharacteristic
103C2×Q8×C109C2×D411yescharacteristic
104C2×D4×C109—21no—
105C2×D4×C109—22no—
106C2×D4×C109—22no—
107C2×D4×C109—22no—
108C2×D4×C109C22⋊C411yescharacteristic
109C2×D4×C109C2×D411yescharacteristic
110C2×D4×C109C2×D412yes—
111C22×C4×C109—22no—
112C22×C4×C109—21no—
113C22×C4×C109—21no—
114C22×C4×C109—22no—
115C22×C4×C109—21no—
116C22×C4×C109—21no—
117C22×C4×C109—21no—
118C22×C4×C109—41no—
119C22×C4×C109—41no—
120C22×C4×C109C22⋊C411yescharacteristic
121C22×C4×C109C22×C411yescharacteristic
122C22×C4×C109C2×D412yes—
123M16×C109—42no—
124M16×C109—42no—
125C2×C8×C109—42no—
126C4⋊C4×C109—21no—
127C4⋊C4×C109—21no—
128C4⋊C4×C109—21no—
129C4⋊C4×C109—21no—
130C4⋊C4×C109—22no—
131C4⋊C4×C109—22no—
132C4⋊C4×C109—22no—
133C4⋊C4×C109—22no—
134C4⋊C4×C109—22no—
135C4⋊C4×C109—21no—
136C4⋊C4×C109—21no—
137C4⋊C4×C109—22no—
138C4⋊C4×C109—22no—
139C4⋊C4×C109—22no—
140C4⋊C4×C109—42no—
141C22⋊C4×C109—21no—
142C22⋊C4×C109—21no—
143C22⋊C4×C109—21no—
144C22⋊C4×C109—21no—
145C22⋊C4×C109—22no—
146C22⋊C4×C109—22no—
147C22⋊C4×C109—22no—
148C22⋊C4×C109—21no—
149C22⋊C4×C109—22no—
150C22⋊C4×C109—22no—
151C22⋊C4×C109—22no—
152C22⋊C4×C109—22no—
153C22⋊C4×C109—22no—
154C22⋊C4×C109—42no—
155C22⋊C4×C109C22⋊C412yes—
156C42×C109—22no—
157C42×C109—21no—
158C42×C109—21no—
159C42×C109—21no—
160C42×C109—22no—
161C42×C109—42no—
162C42×C109C22⋊C411yescharacteristic
163C42×C109C22⋊C411yescharacteristic
164C42×C109C22⋊C411yescharacteristic
165C42×C109C22⋊C412yes—
166C42×C109C22⋊C411yescharacteristic
167C23×C109—21no—
168C23×C109—21no—
169C23×C109—41no—
170C23×C109—81no—
171C23×C109C2×C22⋊C411yescharacteristic
172C23×C109#12012yes—
173Q8×C109—41no—
174Q8×C109—41no—
175Q8×C109—82no—
176D4×C109—42no—
177D4×C109—41no—
178D4×C109—41no—
179D4×C109—42no—
180D4×C109—82no—
181D4×C109—82no—
182D4×C109—82no—
183C2×C4×C109—22no—
184C2×C4×C109—21no—
185C2×C4×C109—22no—
186C2×C4×C109—21no—
187C2×C4×C109—21no—
188C2×C4×C109—21no—
189C2×C4×C109—21no—
190C2×C4×C109—22no—
191C2×C4×C109—21no—
192C2×C4×C109—22no—
193C2×C4×C109—21no—
194C2×C4×C109—21no—
195C2×C4×C109—42no—
196C2×C4×C109—41no—
197C2×C4×C109—41no—
198C2×C4×C109—41no—
199C2×C4×C109—41no—
200C2×C4×C109—41no—
201C2×C4×C109—81no—
202C2×C4×C109—82no—
203C2×C4×C109—82no—
204C2×C4×C109—81no—
205C2×C4×C109—82no—
206C2×C4×C109C2×C22⋊C411yescharacteristic
207C2×C4×C109C2×C22⋊C411yescharacteristic
208C2×C4×C109#12012yes—
209C8×C109—82no—
210C22×C109—21no—
211C22×C109—22no—
212C22×C109—41no—
213C22×C109—42no—
214C22×C109—42no—
215C22×C109—41no—
216C22×C109—41no—
217C22×C109—41no—
218C22×C109—41no—
219C22×C109—82no—
220C22×C109#37911yescharacteristic
221C4×C109—22no—
222C4×C109—21no—
223C4×C109—41no—
224C4×C109—41no—
225C4×C109—41no—
226C4×C109—42no—
227C4×C109—42no—
228C4×C109—41no—
229C4×C109—41no—
230C4×C109—41no—
231C4×C109—162no—
232C4×C109#41011yescharacteristic
233C4×C109#41011yescharacteristic
234#12720C10911yesmaximal, characteristic, direct product factor, Sylow
235C2×C109—21no—
236C2×C109—42no—
237C2×C109—41no—
238C2×C109—81no—
239C2×C109—81no—
240C2×C109#178211yescharacteristic
241#3455C2×C10911yescharacteristic
242#2132C2×C10911yescharacteristic, complemented
243#2126C2×C10911yescharacteristic, complemented
244#2120C2×C10912yescomplemented
245#1827C2×C10912yescomplemented
246C109#1272011yescharacteristic, direct product factor, Sylow
247#542—21no—
248#542—21no—
249#541—21no—
250#540—21no—
251#539—22no—
252#538—22no—
253#533C4×C10911yescharacteristic, complemented
254#532C22×C10911yescharacteristic
255#529—22no—
256#529—22no—
257#525C4×C10911yescharacteristic, complemented
258#525C22×C10911yescharacteristic
259C4×#41C4×C10911yescharacteristic, complemented
260C4×#41C22×C10911yescharacteristic
261C2×#117C4×C10911yescharacteristic, complemented
262#413C22×C10912yes—
263#410C22×C10912yes—
264#356—21no—
265#355—21no—
266#354—21no—
267#353—21no—
268#352—22no—
269AΣL(1,𝔽16)—22no—
270#339—22no—
271H(3,ℤ/4)—22no—
272C2×#41C2×C4×C10911yescharacteristic
273C2×#41C23×C10911yescharacteristic
274C23×C4C2×C4×C10911yescharacteristic
275C2×M16—22no—
276C4⋊Q8—21no—
277C4⋊Q8D4×C10911yescharacteristic
278#127D4×C10911yescharacteristic
279#126—22no—
280#126—22no—
281#126—22no—
282#126—22no—
283#125—22no—
284#125—22no—
285#125—21no—
286#124—22no—
287#124—22no—
288#124D4×C10911yescharacteristic
289#124D4×C10911yescharacteristic
290#123—22no—
291#123—22no—
292#123—22no—
293#123—22no—
294#123—21no—
295#123D4×C10912yes—
296C22⋊Q8—22no—
297C22⋊Q8—22no—
298C22⋊Q8—21no—
299C22⋊Q8—21no—
300C22⋊Q8—22no—
301#121—22no—
302#121—22no—
303#121—21no—
304#121—21no—
305#121—22no—
306#120—21no—
307#120D4×C10912yes—
308C4×Q8—21no—
309C4×Q8—21no—
310C4×Q8—22no—
311C4×D4—21no—
312C4×D4—22no—
313C4×D4—21no—
314C4×D4—22no—
315C4×D4—22no—
316C4×D4—22no—
317C4×D4—22no—
318#117—21no—
319#117—21no—
320#117—21no—
321#117—21no—
322#117—21no—
323#117—22no—
324#117C2×C4×C10911yescharacteristic
325#117C2×C4×C10911yescharacteristic
326#117D4×C10912yes—
327C2×C4⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C42—21no—
330C2×C42C2×C4×C10911yescharacteristic
331C2×C42C2×C4×C10911yescharacteristic
332C2×C42D4×C10912yes—
333#101—22no—
334#100—22no—
335#99—22no—
336#99—22no—
337C24C22⋊C4×C10911yescharacteristic
338#41—41no—
339#41—41no—
340#41—41no—
341#41—41no—
342C2×Q8—22no—
343C2×Q8C22⋊C4×C10911yescharacteristic
344C2×Q8C2×D4×C10911yescharacteristic
345C2×D4—21no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4C22⋊C4×C10911yescharacteristic
350C2×D4C2×D4×C10911yescharacteristic
351C2×D4C2×D4×C10912yes—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4C22⋊C4×C10911yescharacteristic
362C22×C4C22×C4×C10911yescharacteristic
363C22×C4C2×D4×C10912yes—
364M16—42no—
365M16—42no—
366C2×C8—42no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—42no—
382C22⋊C4—21no—
383C22⋊C4—21no—
384C22⋊C4—21no—
385C22⋊C4—21no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—21no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—42no—
396C22⋊C4C22⋊C4×C10912yes—
397C42—22no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—22no—
402C42—42no—
403C42C22⋊C4×C10911yescharacteristic
404C42C22⋊C4×C10911yescharacteristic
405C42C22⋊C4×C10911yescharacteristic
406C42C22⋊C4×C10912yes—
407C42C22⋊C4×C10911yescharacteristic
408C23—21no—
409C23—21no—
410C23—41no—
411C23—81no—
412C23C2×C22⋊C4×C10911yescharacteristic
413C23#120×C10912yes—
414Q8—41no—
415Q8—41no—
416Q8—82no—
417D4—42no—
418D4—41no—
419D4—41no—
420D4—42no—
421D4—82no—
422D4—82no—
423D4—82no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4C2×C22⋊C4×C10911yescharacteristic
448C2×C4C2×C22⋊C4×C10911yescharacteristic
449C2×C4#120×C10912yes—
450C8—82no—
451C22—21no—
452C22—22no—
453C22—41no—
454C22—42no—
455C22—42no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—82no—
461C22#379×C10911yescharacteristic
462C4—22no—
463C4—21no—
464C4—41no—
465C4—41no—
466C4—41no—
467C4—42no—
468C4—42no—
469C4—41no—
470C4—41no—
471C4—41no—
472C4—162no—
473C4#410×C10911yescharacteristic
474C4#410×C10911yescharacteristic
475C2—21no—
476C2—42no—
477C2—41no—
478C2—81no—
479C2—81no—
480C2C109×#178211yescharacteristic
subgroupG/H
centerC4×C109#410
commutatorC22×C4C22×C4×C109
Frattini, cosocleC2×#41C23×C109
socleC2×C109#1782
prime radicalC109×#3455C2
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
12. #533×C109C4semidirect
16. #525×C109C4semidirect
18. C4×#41×C109C4semidirect
20. C2×#117×C109C4semidirect
234. #12720C109direct (N × H)
242. #2132C2×C109semidirect
243. #2126C2×C109semidirect
244. #2120C2×C109semidirect
245. #1827C2×C109semidirect
246. C109#12720direct (N × H)
253. #533C4×C109semidirect
257. #525C4×C109semidirect
259. C4×#41C4×C109semidirect
261. C2×#117C4×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#34552
C109×#21322
C109×#21262
C109×#21202
C109×#18272
#12720109