Finite Group Databasebeta

← C109×#12781 #42130775

Subgroups — C109×#12781 #42130775

Order 27904.

758 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
1C109×#3545C211yesmaximal, characteristic
2C109×#2133C212yesmaximal, complemented
3C2×#355×C109C211yesmaximal, characteristic, complemented
4C109×#1912C212yesmaximal, complemented
5C109×#1888C211yesmaximal, characteristic, complemented
6#563×C109C411yescharacteristic, complemented
7#562×C109—21no—
8H(3,𝔽4)×C109C411yescharacteristic, complemented
9#555×C109C412yescomplemented
10#551×C109—21no—
11#551×C109—22no—
12#546×C109—22no—
13#542×C109—22no—
14#538×C109—22no—
15#536×C109C2212yes—
16#534×C109—21no—
17#529×C109—21no—
18C2×#124×C109C2211yescharacteristic
19#519×C109C2211yescharacteristic, complemented
20#421×C109—22no—
21#418×C109—21no—
22C2×#101×C109C2212yescomplemented
23C2×#101×C109C2211yescharacteristic, complemented
24#356×C109—22no—
25#355×C109—21no—
26#355×C109—21no—
27#355×C109—22no—
28#339×C109—21no—
29H(3,ℤ/4)×C109—22no—
30C2×#41×C109D412yes—
31C22×Q8×C109C2311yescharacteristic
32C22×D4×C109—21no—
33C22×D4×C109D411yescharacteristic
34C2×M16×C109—22no—
35C2×M16×C109—21no—
36C4⋊Q8×C109C2×C412yes—
37C4⋊Q8×C109D412yes—
38#126×C109—22no—
39#126×C109—22no—
40#126×C109—22no—
41#125×C109—21no—
42#124×C109—22no—
43#124×C109—22no—
44#124×C109—21no—
45#124×C109—22no—
46#124×C109C2×C411yescharacteristic
47#124×C109C2×C411yescharacteristic
48#124×C109C2×C412yes—
49#124×C109D411yescharacteristic
50#124×C109D411yescharacteristic
51#124×C109D412yes—
52#123×C109—21no—
53#123×C109—22no—
54#123×C109—22no—
55#123×C109—22no—
56#123×C109—21no—
57#123×C109—22no—
58#123×C109—22no—
59C22⋊Q8×C109—21no—
60C22⋊Q8×C109—22no—
61C22⋊Q8×C109—22no—
62C22⋊Q8×C109—22no—
63C22⋊Q8×C109—22no—
64#121×C109—21no—
65#121×C109—22no—
66#121×C109—22no—
67#121×C109—22no—
68#121×C109—22no—
69#120×C109—21no—
70#120×C109—22no—
71#120×C109—21no—
72#120×C109—22no—
73C4×Q8×C109—21no—
74C4×Q8×C109—42no—
75C4×D4×C109—21no—
76C4×D4×C109—22no—
77C4×D4×C109—22no—
78C4×D4×C109—22no—
79#117×C109—22no—
80#117×C109—22no—
81#117×C109—22no—
82#117×C109—21no—
83#117×C109D412yes—
84C2×C22⋊C4×C109—21no—
85C2×C22⋊C4×C109—22no—
86C2×C22⋊C4×C109—22no—
87C2×C42×C109D411yescharacteristic
88#104×C109—44no—
89#102×C109—42no—
90#101×C109—22no—
91#101×C109—22no—
92#101×C109—22no—
93C24×C109—21no—
94C24×C109—21no—
95#41×C109—42no—
96#41×C109—42no—
97C2×Q8×C109—21no—
98C2×Q8×C109—41no—
99C2×Q8×C109—41no—
100C2×Q8×C109C22⋊C412yes—
101C2×Q8×C109C22×C411yescharacteristic
102C2×Q8×C109C2×D411yescharacteristic
103C2×Q8×C109C2×D412yes—
104C2×D4×C109—22no—
105C2×D4×C109—22no—
106C2×D4×C109—22no—
107C2×D4×C109—21no—
108C2×D4×C109—22no—
109C2×D4×C109—21no—
110C2×D4×C109—41no—
111C2×D4×C109—41no—
112C2×D4×C109—81no—
113C2×D4×C109—81no—
114C2×D4×C109C22⋊C412yes—
115C2×D4×C109C22⋊C411yescharacteristic
116C2×D4×C109C22⋊C411yescharacteristic
117C22×C4×C109—22no—
118C22×C4×C109—22no—
119C22×C4×C109—21no—
120C22×C4×C109—21no—
121C22×C4×C109—22no—
122C22×C4×C109C2×D411yescharacteristic
123C22×C4×C109C2×D412yes—
124M16×C109—42no—
125M16×C109—41no—
126M16×C109—41no—
127M16×C109—42no—
128C2×C8×C109—41no—
129C2×C8×C109—42no—
130C4⋊C4×C109—22no—
131C4⋊C4×C109—22no—
132C4⋊C4×C109—21no—
133C4⋊C4×C109—21no—
134C4⋊C4×C109—22no—
135C4⋊C4×C109—22no—
136C4⋊C4×C109—22no—
137C4⋊C4×C109—22no—
138C4⋊C4×C109—22no—
139C4⋊C4×C109—22no—
140C4⋊C4×C109—42no—
141C22⋊C4×C109—22no—
142C22⋊C4×C109—22no—
143C22⋊C4×C109—21no—
144C22⋊C4×C109—21no—
145C22⋊C4×C109—22no—
146C22⋊C4×C109—22no—
147C22⋊C4×C109—22no—
148C22⋊C4×C109—22no—
149C22⋊C4×C109—22no—
150C22⋊C4×C109—22no—
151C22⋊C4×C109—22no—
152C22⋊C4×C109—22no—
153C22⋊C4×C109—22no—
154C22⋊C4×C109—22no—
155C22⋊C4×C109—22no—
156C22⋊C4×C109—41no—
157C42×C109—22no—
158C42×C109—21no—
159C42×C109—24no—
160C42×C109—41no—
161C42×C109C22⋊C411yescharacteristic
162C42×C109C22⋊C411yescharacteristic
163C42×C109C22⋊C412yes—
164C42×C109C22⋊C412yes—
165C23×C109—21no—
166C23×C109—21no—
167C23×C109—22no—
168C23×C109—41no—
169C23×C109—41no—
170C23×C109—81no—
171C23×C109—82no—
172C23×C109—82no—
173C23×C109#12011yescharacteristic
174Q8×C109—41no—
175Q8×C109—41no—
176Q8×C109—42no—
177Q8×C109—42no—
178Q8×C109—81no—
179D4×C109—41no—
180D4×C109—41no—
181D4×C109—42no—
182D4×C109—42no—
183D4×C109—42no—
184D4×C109—81no—
185D4×C109—82no—
186D4×C109—82no—
187D4×C109—81no—
188D4×C109—82no—
189C2×C4×C109—22no—
190C2×C4×C109—21no—
191C2×C4×C109—22no—
192C2×C4×C109—21no—
193C2×C4×C109—21no—
194C2×C4×C109—22no—
195C2×C4×C109—21no—
196C2×C4×C109—22no—
197C2×C4×C109—22no—
198C2×C4×C109—42no—
199C2×C4×C109—82no—
200C2×C4×C109—82no—
201C2×C4×C109—81no—
202C2×C4×C109—82no—
203C2×C4×C109—81no—
204C2×C4×C109C2×C22⋊C412yes—
205C2×C4×C109C2×C22⋊C411yescharacteristic
206C2×C4×C109#12011yescharacteristic
207C2×C4×C109#12012yes—
208C8×C109—82no—
209C8×C109—81no—
210C22×C109—41no—
211C22×C109—42no—
212C22×C109—42no—
213C22×C109—42no—
214C22×C109—42no—
215C22×C109—42no—
216C22×C109—41no—
217C22×C109—41no—
218C22×C109—161no—
219C22×C109—161no—
220C22×C109—162no—
221C22×C109#37911yescharacteristic
222C22×C109B(4,𝔽2)11yescharacteristic
223C22×C109#45811yescharacteristic
224C4×C109—22no—
225C4×C109—21no—
226C4×C109—21no—
227C4×C109—22no—
228C4×C109—41no—
229C4×C109—42no—
230C4×C109—42no—
231C4×C109—41no—
232C4×C109—42no—
233C4×C109—82no—
234#12781C10911yesmaximal, characteristic, direct product factor, Sylow
235C2×C109—21no—
236C2×C109—21no—
237C2×C109—21no—
238C2×C109—81no—
239C2×C109—82no—
240C2×C109—81no—
241C2×C109#179011yescharacteristic
242#3545C2×C10911yescharacteristic
243#2133C2×C10912yescomplemented
244C2×#355C2×C10911yescharacteristic, complemented
245#1912C2×C10912yescomplemented
246#1888C2×C10911yescharacteristic, complemented
247C109#1278111yescharacteristic, direct product factor, Sylow
248#563C4×C10911yescharacteristic, complemented
249#562—21no—
250H(3,𝔽4)C4×C10911yescharacteristic, complemented
251#555C4×C10912yescomplemented
252#551—21no—
253#551—22no—
254#546—22no—
255#542—22no—
256#538—22no—
257#536C22×C10912yes—
258#534—21no—
259#529—21no—
260C2×#124C22×C10911yescharacteristic
261#519C22×C10911yescharacteristic, complemented
262#421—22no—
263#418—21no—
264C2×#101C22×C10912yescomplemented
265C2×#101C22×C10911yescharacteristic, complemented
266#356—22no—
267#355—21no—
268#355—21no—
269#355—22no—
270#339—21no—
271H(3,ℤ/4)—22no—
272C2×#41D4×C10912yes—
273C22×Q8C23×C10911yescharacteristic
274C22×D4—21no—
275C22×D4D4×C10911yescharacteristic
276C2×M16—22no—
277C2×M16—21no—
278C4⋊Q8C2×C4×C10912yes—
279C4⋊Q8D4×C10912yes—
280#126—22no—
281#126—22no—
282#126—22no—
283#125—21no—
284#124—22no—
285#124—22no—
286#124—21no—
287#124—22no—
288#124C2×C4×C10911yescharacteristic
289#124C2×C4×C10911yescharacteristic
290#124C2×C4×C10912yes—
291#124D4×C10911yescharacteristic
292#124D4×C10911yescharacteristic
293#124D4×C10912yes—
294#123—21no—
295#123—22no—
296#123—22no—
297#123—22no—
298#123—21no—
299#123—22no—
300#123—22no—
301C22⋊Q8—21no—
302C22⋊Q8—22no—
303C22⋊Q8—22no—
304C22⋊Q8—22no—
305C22⋊Q8—22no—
306#121—21no—
307#121—22no—
308#121—22no—
309#121—22no—
310#121—22no—
311#120—21no—
312#120—22no—
313#120—21no—
314#120—22no—
315C4×Q8—21no—
316C4×Q8—42no—
317C4×D4—21no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—21no—
325#117D4×C10912yes—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C42D4×C10911yescharacteristic
330#104—44no—
331#102—42no—
332#101—22no—
333#101—22no—
334#101—22no—
335C24—21no—
336C24—21no—
337#41—42no—
338#41—42no—
339C2×Q8—21no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8C22⋊C4×C10912yes—
343C2×Q8C22×C4×C10911yescharacteristic
344C2×Q8C2×D4×C10911yescharacteristic
345C2×Q8C2×D4×C10912yes—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—21no—
350C2×D4—22no—
351C2×D4—21no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—81no—
355C2×D4—81no—
356C2×D4C22⋊C4×C10912yes—
357C2×D4C22⋊C4×C10911yescharacteristic
358C2×D4C22⋊C4×C10911yescharacteristic
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4C2×D4×C10911yescharacteristic
365C22×C4C2×D4×C10912yes—
366M16—42no—
367M16—41no—
368M16—41no—
369M16—42no—
370C2×C8—41no—
371C2×C8—42no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—42no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—21no—
386C22⋊C4—21no—
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—41no—
399C42—22no—
400C42—21no—
401C42—24no—
402C42—41no—
403C42C22⋊C4×C10911yescharacteristic
404C42C22⋊C4×C10911yescharacteristic
405C42C22⋊C4×C10912yes—
406C42C22⋊C4×C10912yes—
407C23—21no—
408C23—21no—
409C23—22no—
410C23—41no—
411C23—41no—
412C23—81no—
413C23—82no—
414C23—82no—
415C23#120×C10911yescharacteristic
416Q8—41no—
417Q8—41no—
418Q8—42no—
419Q8—42no—
420Q8—81no—
421D4—41no—
422D4—41no—
423D4—42no—
424D4—42no—
425D4—42no—
426D4—81no—
427D4—82no—
428D4—82no—
429D4—82no—
430D4—81no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—42no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4C2×C22⋊C4×C10912yes—
447C2×C4C2×C22⋊C4×C10911yescharacteristic
448C2×C4#120×C10911yescharacteristic
449C2×C4#120×C10912yes—
450C8—82no—
451C8—81no—
452C22—41no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—41no—
459C22—41no—
460C22—161no—
461C22—161no—
462C22—162no—
463C22#379×C10911yescharacteristic
464C22B(4,𝔽2)×C10911yescharacteristic
465C22#458×C10911yescharacteristic
466C4—22no—
467C4—21no—
468C4—21no—
469C4—22no—
470C4—41no—
471C4—42no—
472C4—42no—
473C4—41no—
474C4—42no—
475C4—82no—
476C2—21no—
477C2—21no—
478C2—21no—
479C2—81no—
480C2—81no—
481C2—82no—
482C2C109×#179011yescharacteristic
subgroupG/H
center, socleC2×C109#1790
commutatorC2×Q8C22×C4×C109
Frattini, cosocleC22×Q8C23×C109
prime radicalC109×#3545C2
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C109 ◁ C23×C109 ◁ C22×Q8×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#2133C2semidirect
3. C2×#355×C109C2semidirect
4. C109×#1912C2semidirect
5. C109×#1888C2semidirect
6. #563×C109C4semidirect
8. H(3,𝔽4)×C109C4semidirect
9. #555×C109C4semidirect
19. #519×C109C22semidirect
22. C2×#101×C109C22semidirect
23. C2×#101×C109C22semidirect
234. #12781C109direct (N × H)
243. #2133C2×C109semidirect
244. C2×#355C2×C109semidirect
245. #1912C2×C109semidirect
246. #1888C2×C109semidirect
247. C109#12781direct (N × H)
248. #563C4×C109semidirect
250. H(3,𝔽4)C4×C109semidirect
251. #555C4×C109semidirect
261. #519C22×C109semidirect
264. C2×#101C22×C109semidirect
265. C2×#101C22×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#35452
C109×#21332
C2×#355×C1092
C109×#19122
C109×#18882
#12781109