Finite Group Databasebeta

← C109×#12726 #42130720

Subgroups — C109×#12726 #42130720

Order 27904.

708 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3491C211yesmaximal, characteristic
2C109×#2121C212yesmaximal
3C2×#352×C109C212yesmaximal, complemented
4C109×#1829C211yesmaximal, characteristic, complemented
5C109×#1828C211yesmaximal, characteristic, complemented
6#552×C109—22no—
7#552×C109—22no—
8#551×C109—22no—
9#551×C109—22no—
10#543×C109C411yescharacteristic, complemented
11#539×C109—22no—
12#534×C109C2211yescharacteristic
13#529×C109—22no—
14C2×#123×C109C411yescharacteristic, complemented
15C2×#123×C109C2211yescharacteristic
16#518×C109C411yescharacteristic, complemented
17#518×C109C2211yescharacteristic
18C2×#117×C109C411yescharacteristic, complemented
19C2×#100×C109C2212yes—
20C2×#99×C109C2212yes—
21#352×C109—22no—
22#352×C109—22no—
23#352×C109—22no—
24AΣL(1,𝔽16)×C109—22no—
25#344×C109—22no—
26#342×C109—22no—
27C22×D4×C109C2×C411yescharacteristic
28C22×D4×C109C2311yescharacteristic
29C23×C4×C109C2×C411yescharacteristic
30C2×M16×C109—22no—
31#126×C109—22no—
32#126×C109—22no—
33#126×C109—22no—
34#126×C109—22no—
35#125×C109—22no—
36#125×C109—22no—
37#125×C109—22no—
38#124×C109—22no—
39#123×C109—22no—
40#123×C109—22no—
41#123×C109—22no—
42#123×C109—21no—
43#123×C109—21no—
44#123×C109—21no—
45#123×C109—22no—
46#123×C109D412yes—
47#123×C109D412yes—
48#123×C109D412yes—
49C22⋊Q8×C109—22no—
50C22⋊Q8×C109—22no—
51C22⋊Q8×C109—22no—
52C22⋊Q8×C109—22no—
53C22⋊Q8×C109—22no—
54#121×C109—22no—
55#120×C109—21no—
56#120×C109D412yes—
57C4×Q8×C109—22no—
58C4×D4×C109—22no—
59C4×D4×C109—22no—
60C4×D4×C109—22no—
61C4×D4×C109—22no—
62C4×D4×C109—22no—
63#117×C109—22no—
64#117×C109—22no—
65#117×C109—21no—
66#117×C109—22no—
67#117×C109—22no—
68#117×C109D412yes—
69C2×C4⋊C4×C109—21no—
70C2×C4⋊C4×C109—21no—
71C2×C4⋊C4×C109—21no—
72C2×C22⋊C4×C109—21no—
73C2×C22⋊C4×C109—21no—
74C2×C22⋊C4×C109—21no—
75C2×C22⋊C4×C109—22no—
76C2×C22⋊C4×C109C2×C411yescharacteristic
77C2×C22⋊C4×C109C2×C411yescharacteristic
78C2×C22⋊C4×C109C2×C411yescharacteristic
79C2×C22⋊C4×C109C2×C411yescharacteristic
80C2×C22⋊C4×C109D412yes—
81C2×C42×C109—21no—
82#100×C109—22no—
83#100×C109—22no—
84#99×C109—22no—
85#99×C109—22no—
86C24×C109C22⋊C411yescharacteristic
87C24×C109C22×C411yescharacteristic
88C24×C109C2×D411yescharacteristic
89C2×Q8×C109—22no—
90C2×D4×C109—21no—
91C2×D4×C109—21no—
92C2×D4×C109—22no—
93C2×D4×C109—42no—
94C2×D4×C109—42no—
95C2×D4×C109C2×D412yes—
96C2×D4×C109C2×D412yes—
97C22×C4×C109—21no—
98C22×C4×C109—21no—
99C22×C4×C109—22no—
100C22×C4×C109—21no—
101C22×C4×C109—21no—
102C22×C4×C109—21no—
103C22×C4×C109—21no—
104C22×C4×C109—22no—
105C22×C4×C109—42no—
106C22×C4×C109—42no—
107C22×C4×C109C22⋊C411yescharacteristic
108C22×C4×C109C22⋊C411yescharacteristic
109C22×C4×C109C22⋊C411yescharacteristic
110C22×C4×C109C2×D411yescharacteristic
111M16×C109—42no—
112M16×C109—42no—
113C2×C8×C109—42no—
114C4⋊C4×C109—22no—
115C4⋊C4×C109—22no—
116C4⋊C4×C109—22no—
117C4⋊C4×C109—22no—
118C4⋊C4×C109—22no—
119C4⋊C4×C109—22no—
120C4⋊C4×C109—22no—
121C4⋊C4×C109—22no—
122C4⋊C4×C109—22no—
123C4⋊C4×C109—22no—
124C4⋊C4×C109—22no—
125C4⋊C4×C109—22no—
126C22⋊C4×C109—22no—
127C22⋊C4×C109—22no—
128C22⋊C4×C109—22no—
129C22⋊C4×C109—21no—
130C22⋊C4×C109—21no—
131C22⋊C4×C109—21no—
132C22⋊C4×C109—21no—
133C22⋊C4×C109—22no—
134C22⋊C4×C109—22no—
135C22⋊C4×C109—22no—
136C22⋊C4×C109—22no—
137C22⋊C4×C109—22no—
138C22⋊C4×C109—42no—
139C22⋊C4×C109—42no—
140C22⋊C4×C109C22⋊C412yes—
141C22⋊C4×C109C22⋊C412yes—
142C22⋊C4×C109C22⋊C412yes—
143C22⋊C4×C109C22⋊C412yes—
144C42×C109—22no—
145C42×C109—22no—
146C42×C109—22no—
147C42×C109—22no—
148C23×C109—22no—
149C23×C109—22no—
150C23×C109—21no—
151C23×C109—41no—
152C23×C109—81no—
153C23×C109—81no—
154C23×C109—81no—
155C23×C109C2×C22⋊C411yescharacteristic
156C23×C109C2×C22⋊C411yescharacteristic
157C23×C109C2×C22⋊C411yescharacteristic
158C23×C109#12012yes—
159Q8×C109—82no—
160D4×C109—42no—
161D4×C109—42no—
162D4×C109—42no—
163D4×C109—42no—
164D4×C109—82no—
165C2×C4×C109—22no—
166C2×C4×C109—21no—
167C2×C4×C109—22no—
168C2×C4×C109—22no—
169C2×C4×C109—21no—
170C2×C4×C109—22no—
171C2×C4×C109—22no—
172C2×C4×C109—21no—
173C2×C4×C109—42no—
174C2×C4×C109—42no—
175C2×C4×C109—41no—
176C2×C4×C109—41no—
177C2×C4×C109—41no—
178C2×C4×C109—81no—
179C2×C4×C109—82no—
180C2×C4×C109—82no—
181C2×C4×C109—81no—
182C2×C4×C109—81no—
183C2×C4×C109—81no—
184C2×C4×C109—82no—
185C2×C4×C109—162no—
186C2×C4×C109#12012yes—
187C8×C109—82no—
188C22×C109—21no—
189C22×C109—21no—
190C22×C109—22no—
191C22×C109—41no—
192C22×C109—41no—
193C22×C109—42no—
194C22×C109—41no—
195C22×C109—42no—
196C22×C109—41no—
197C22×C109—82no—
198C22×C109—82no—
199C22×C109—161no—
200C22×C109#37911yescharacteristic
201C22×C109#41012yes—
202C4×C109—22no—
203C4×C109—42no—
204C4×C109—42no—
205C4×C109—42no—
206C4×C109—42no—
207C4×C109—41no—
208C4×C109—41no—
209C4×C109—41no—
210C4×C109—162no—
211#12726C10911yesmaximal, characteristic, direct product factor, Sylow
212C2×C109—21no—
213C2×C109—22no—
214C2×C109—42no—
215C2×C109—41no—
216C2×C109—41no—
217C2×C109—81no—
218C2×C109#178211yescharacteristic
219#3491C2×C10911yescharacteristic
220#2121C2×C10912yes—
221C2×#352C2×C10912yescomplemented
222#1829C2×C10911yescharacteristic, complemented
223#1828C2×C10911yescharacteristic, complemented
224C109#1272611yescharacteristic, direct product factor, Sylow
225#552—22no—
226#552—22no—
227#551—22no—
228#551—22no—
229#543C4×C10911yescharacteristic, complemented
230#539—22no—
231#534C22×C10911yescharacteristic
232#529—22no—
233C2×#123C4×C10911yescharacteristic, complemented
234C2×#123C22×C10911yescharacteristic
235#518C4×C10911yescharacteristic, complemented
236#518C22×C10911yescharacteristic
237C2×#117C4×C10911yescharacteristic, complemented
238C2×#100C22×C10912yes—
239C2×#99C22×C10912yes—
240#352—22no—
241#352—22no—
242#352—22no—
243AΣL(1,𝔽16)—22no—
244#344—22no—
245#342—22no—
246C22×D4C2×C4×C10911yescharacteristic
247C22×D4C23×C10911yescharacteristic
248C23×C4C2×C4×C10911yescharacteristic
249C2×M16—22no—
250#126—22no—
251#126—22no—
252#126—22no—
253#126—22no—
254#125—22no—
255#125—22no—
256#125—22no—
257#124—22no—
258#123—22no—
259#123—22no—
260#123—22no—
261#123—21no—
262#123—21no—
263#123—21no—
264#123—22no—
265#123D4×C10912yes—
266#123D4×C10912yes—
267#123D4×C10912yes—
268C22⋊Q8—22no—
269C22⋊Q8—22no—
270C22⋊Q8—22no—
271C22⋊Q8—22no—
272C22⋊Q8—22no—
273#121—22no—
274#120—21no—
275#120D4×C10912yes—
276C4×Q8—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—22no—
282#117—22no—
283#117—22no—
284#117—21no—
285#117—22no—
286#117—22no—
287#117D4×C10912yes—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4C2×C4×C10911yescharacteristic
296C2×C22⋊C4C2×C4×C10911yescharacteristic
297C2×C22⋊C4C2×C4×C10911yescharacteristic
298C2×C22⋊C4C2×C4×C10911yescharacteristic
299C2×C22⋊C4D4×C10912yes—
300C2×C42—21no—
301#100—22no—
302#100—22no—
303#99—22no—
304#99—22no—
305C24C22⋊C4×C10911yescharacteristic
306C24C22×C4×C10911yescharacteristic
307C24C2×D4×C10911yescharacteristic
308C2×Q8—22no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—22no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4C2×D4×C10912yes—
315C2×D4C2×D4×C10912yes—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4C22⋊C4×C10911yescharacteristic
327C22×C4C22⋊C4×C10911yescharacteristic
328C22×C4C22⋊C4×C10911yescharacteristic
329C22×C4C2×D4×C10911yescharacteristic
330M16—42no—
331M16—42no—
332C2×C8—42no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—22no—
347C22⋊C4—22no—
348C22⋊C4—21no—
349C22⋊C4—21no—
350C22⋊C4—21no—
351C22⋊C4—21no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4C22⋊C4×C10912yes—
360C22⋊C4C22⋊C4×C10912yes—
361C22⋊C4C22⋊C4×C10912yes—
362C22⋊C4C22⋊C4×C10912yes—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C23—22no—
368C23—22no—
369C23—21no—
370C23—41no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23C2×C22⋊C4×C10911yescharacteristic
375C23C2×C22⋊C4×C10911yescharacteristic
376C23C2×C22⋊C4×C10911yescharacteristic
377C23#120×C10912yes—
378Q8—82no—
379D4—42no—
380D4—42no—
381D4—42no—
382D4—42no—
383D4—82no—
384C2×C4—22no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—81no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—82no—
404C2×C4—162no—
405C2×C4#120×C10912yes—
406C8—82no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—42no—
413C22—41no—
414C22—42no—
415C22—41no—
416C22—82no—
417C22—82no—
418C22—161no—
419C22#379×C10911yescharacteristic
420C22#410×C10912yes—
421C4—22no—
422C4—42no—
423C4—42no—
424C4—42no—
425C4—42no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—162no—
430C2—21no—
431C2—22no—
432C2—42no—
433C2—41no—
434C2—41no—
435C2—81no—
436C2C109×#178211yescharacteristic
subgroupG/H
center, socleC2×C109#1782
commutatorC24C22×C4×C109
Frattini, cosocleC22×D4C23×C109
prime radical#534×C109C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C109 ◁ C23×C109 ◁ #518×C109 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×#352×C109C2semidirect
4. C109×#1829C2semidirect
5. C109×#1828C2semidirect
10. #543×C109C4semidirect
14. C2×#123×C109C4semidirect
16. #518×C109C4semidirect
18. C2×#117×C109C4semidirect
211. #12726C109direct (N × H)
221. C2×#352C2×C109semidirect
222. #1829C2×C109semidirect
223. #1828C2×C109semidirect
224. C109#12726direct (N × H)
229. #543C4×C109semidirect
233. C2×#123C4×C109semidirect
235. #518C4×C109semidirect
237. C2×#117C4×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#34912
C109×#21212
C2×#352×C1092
C109×#18292
C109×#18282
#12726109