Finite Group Databasebeta

← C109×#19361 #42137355

Subgroups — C109×#19361 #42137355

Order 27904.

570 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3252C212yesmaximal
2C109×#3160C211yesmaximal, characteristic, complemented
3C109×#2966C211yesmaximal, characteristic
4C109×#1692C212yesmaximal, complemented
5C109×#1691C212yesmaximal, complemented
6C109×#1676C211yesmaximal, characteristic, complemented
7C109×#1675C211yesmaximal, characteristic, complemented
8C109×#1674C211yesmaximal, characteristic, complemented
9C109×#1673C211yesmaximal, characteristic, complemented
10C109×#1567C211yesmaximal, characteristic, complemented
11C109×#1559C212yesmaximal, complemented
12#533×C109C2212yes—
13#529×C109C2211yescharacteristic
14#500×C109C2211yescharacteristic
15#500×C109C2211yescharacteristic
16#499×C109—21no—
17#499×C109—21no—
18#498×C109C2212yes—
19#479×C109—22no—
20#477×C109C2212yes—
21#476×C109C2212yes—
22#470×C109—22no—
23#468×C109—22no—
24#464×C109—22no—
25#462×C109C2212yes—
26#461×C109C2212yes—
27#446×C109C2211yescharacteristic
28C8×Q8×C109C2211yescharacteristic
29#436×C109C2211yescharacteristic
30#435×C109—21no—
31C8×D4×C109—21no—
32C8×D4×C109C2211yescharacteristic
33#433×C109—21no—
34#432×C109C2211yescharacteristic
35#427×C109—21no—
36#427×C109—21no—
37#423×C109C2212yes—
38#405×C109—21no—
39C2×C4×C8×C109C2211yescharacteristic
40#335×C109C2211yescharacteristic
41#335×C109C2211yescharacteristic
42#335×C109C2212yes—
43#333×C109C2212yes—
44#333×C109C2212yes—
45#332×C109C2212yes—
46#332×C109C2212yes—
47C2×Q16×C109—22no—
48C2×SD16×C109—22no—
49C2×M16×C109—22no—
50C2×M16×C109—21no—
51C22×C8×C109—21no—
52C22×C8×C109D411yescharacteristic
53C22×C8×C109D411yescharacteristic
54C4⋊Q8×C109C2312yes—
55C4⋊Q8×C109C2312yes—
56#126×C109—21no—
57#125×C109—22no—
58#125×C109D411yescharacteristic
59#124×C109D411yescharacteristic
60#123×C109—21no—
61C22⋊Q8×C109—22no—
62C22⋊Q8×C109—22no—
63C22⋊Q8×C109D411yescharacteristic
64#121×C109D411yescharacteristic
65C4×Q8×C109—22no—
66C4×Q8×C109—22no—
67C4×Q8×C109C2311yescharacteristic
68C4×D4×C109—21no—
69C4×D4×C109C2311yescharacteristic
70#117×C109—21no—
71#117×C109—22no—
72C2×C42×C109C2311yescharacteristic
73#107×C109—21no—
74#107×C109—21no—
75#107×C109—21no—
76#107×C109—21no—
77#107×C109—22no—
78#106×C109—22no—
79#106×C109—22no—
80#106×C109—22no—
81C4⋊C8×C109—21no—
82C4⋊C8×C109—21no—
83C4⋊C8×C109—22no—
84C4⋊C8×C109C2311yescharacteristic
85C4⋊C8×C109C2311yescharacteristic
86C4⋊C8×C109C2312yes—
87C4⋊C8×C109C2312yes—
88#103×C109—22no—
89#103×C109—22no—
90#103×C109—22no—
91#102×C109—22no—
92C22⋊C8×C109—21no—
93C22⋊C8×C109—21no—
94C22⋊C8×C109—21no—
95#97×C109—21no—
96#97×C109—21no—
97C4×C8×C109—21no—
98C4×C8×C109D411yescharacteristic
99C4×C8×C109D411yescharacteristic
100C4×C8×C109C2311yescharacteristic
101C4×C8×C109C2311yescharacteristic
102C2×Q8×C109—22no—
103C2×Q8×C109—22no—
104C2×Q8×C109C2×D411yescharacteristic
105C2×D4×C109—21no—
106C2×D4×C109C2×D411yescharacteristic
107C22×C4×C109—21no—
108C22×C4×C109C2×D411yescharacteristic
109C22×C4×C109C2×D411yescharacteristic
110C22×C4×C109#4111yescharacteristic
111Q16×C109—82no—
112SD16×C109—82no—
113M16×C109—42no—
114M16×C109—82no—
115C2×C8×C109—22no—
116C2×C8×C109—22no—
117C2×C8×C109—21no—
118C2×C8×C109—21no—
119C2×C8×C109—22no—
120C2×C8×C109—22no—
121C2×C8×C109—42no—
122C2×C8×C109C2×D411yescharacteristic
123C2×C8×C109C2×D411yescharacteristic
124C2×C8×C109C2×D411yescharacteristic
125C2×C8×C109C2×D411yescharacteristic
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—42no—
135C4⋊C4×C109—42no—
136C4⋊C4×C109C2×D411yescharacteristic
137C4⋊C4×C109C2×D411yescharacteristic
138C22⋊C4×C109—21no—
139C22⋊C4×C109—21no—
140C22⋊C4×C109—21no—
141C22⋊C4×C109—21no—
142C22⋊C4×C109—21no—
143C22⋊C4×C109—42no—
144C42×C109—21no—
145C42×C109—42no—
146C42×C109C2×D411yescharacteristic
147C42×C109C2×D411yescharacteristic
148C42×C109#4111yescharacteristic
149C42×C109C2411yescharacteristic
150C23×C109—21no—
151C23×C109#12111yescharacteristic
152Q8×C109—21no—
153Q8×C109—21no—
154Q8×C109—42no—
155Q8×C109—42no—
156Q8×C109—82no—
157D4×C109—21no—
158D4×C109—21no—
159D4×C109—42no—
160C2×C4×C109—21no—
161C2×C4×C109—21no—
162C2×C4×C109—21no—
163C2×C4×C109—21no—
164C2×C4×C109—21no—
165C2×C4×C109—21no—
166C2×C4×C109—21no—
167C2×C4×C109—41no—
168C2×C4×C109—41no—
169C2×C4×C109—41no—
170C2×C4×C109—42no—
171C2×C4×C109—42no—
172C2×C4×C109#12111yescharacteristic
173C2×C4×C109#12111yescharacteristic
174C2×C4×C109#12111yescharacteristic
175C2×C4×C109C22×D411yescharacteristic
176C2×C4×C109C22×D411yescharacteristic
177C2×C4×C109C2×#4111yescharacteristic
178C8×C109—21no—
179C8×C109—21no—
180C8×C109—41no—
181C8×C109—41no—
182C8×C109—42no—
183C8×C109—42no—
184C8×C109#13712yes—
185C8×C109#13712yes—
186C22×C109—21no—
187C22×C109—21no—
188C22×C109—21no—
189C22×C109—41no—
190C22×C109—41no—
191C22×C109—41no—
192C22×C109C2×#12111yescharacteristic
193C4×C109—21no—
194C4×C109—21no—
195C4×C109—21no—
196C4×C109—21no—
197C4×C109—41no—
198C4×C109—41no—
199C4×C109—41no—
200C4×C109—82no—
201C4×C109—82no—
202C4×C109C2×#13711yescharacteristic
203C4×C109C2×#13711yescharacteristic
204C4×C109#57611yescharacteristic
205C4×C109#57811yescharacteristic
206#19361C10911yesmaximal, characteristic, direct product factor, Sylow
207C2×C109—41no—
208C2×C109—81no—
209C2×C109#303911yescharacteristic
210C2×C109#306111yescharacteristic
211C2×C109#306711yescharacteristic
212#3252C2×C10912yes—
213#3160C2×C10911yescharacteristic, complemented
214#2966C2×C10911yescharacteristic
215#1692C2×C10912yescomplemented
216#1691C2×C10912yescomplemented
217#1676C2×C10911yescharacteristic, complemented
218#1675C2×C10911yescharacteristic, complemented
219#1674C2×C10911yescharacteristic, complemented
220#1673C2×C10911yescharacteristic, complemented
221#1567C2×C10911yescharacteristic, complemented
222#1559C2×C10912yescomplemented
223C109#1936111yescharacteristic, direct product factor, Sylow
224#533C22×C10912yes—
225#529C22×C10911yescharacteristic
226#500C22×C10911yescharacteristic
227#500C22×C10911yescharacteristic
228#499—21no—
229#499—21no—
230#498C22×C10912yes—
231#479—22no—
232#477C22×C10912yes—
233#476C22×C10912yes—
234#470—22no—
235#468—22no—
236#464—22no—
237#462C22×C10912yes—
238#461C22×C10912yes—
239#446C22×C10911yescharacteristic
240C8×Q8C22×C10911yescharacteristic
241#436C22×C10911yescharacteristic
242#435—21no—
243C8×D4—21no—
244C8×D4C22×C10911yescharacteristic
245#433—21no—
246#432C22×C10911yescharacteristic
247#427—21no—
248#427—21no—
249#423C22×C10912yes—
250#405—21no—
251C2×C4×C8C22×C10911yescharacteristic
252#335C22×C10911yescharacteristic
253#335C22×C10911yescharacteristic
254#335C22×C10912yes—
255#333C22×C10912yes—
256#333C22×C10912yes—
257#332C22×C10912yes—
258#332C22×C10912yes—
259C2×Q16—22no—
260C2×SD16—22no—
261C2×M16—22no—
262C2×M16—21no—
263C22×C8—21no—
264C22×C8D4×C10911yescharacteristic
265C22×C8D4×C10911yescharacteristic
266C4⋊Q8C23×C10912yes—
267C4⋊Q8C23×C10912yes—
268#126—21no—
269#125—22no—
270#125D4×C10911yescharacteristic
271#124D4×C10911yescharacteristic
272#123—21no—
273C22⋊Q8—22no—
274C22⋊Q8—22no—
275C22⋊Q8D4×C10911yescharacteristic
276#121D4×C10911yescharacteristic
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8C23×C10911yescharacteristic
280C4×D4—21no—
281C4×D4C23×C10911yescharacteristic
282#117—21no—
283#117—22no—
284C2×C42C23×C10911yescharacteristic
285#107—21no—
286#107—21no—
287#107—21no—
288#107—21no—
289#107—22no—
290#106—22no—
291#106—22no—
292#106—22no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—22no—
296C4⋊C8C23×C10911yescharacteristic
297C4⋊C8C23×C10911yescharacteristic
298C4⋊C8C23×C10912yes—
299C4⋊C8C23×C10912yes—
300#103—22no—
301#103—22no—
302#103—22no—
303#102—22no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306C22⋊C8—21no—
307#97—21no—
308#97—21no—
309C4×C8—21no—
310C4×C8D4×C10911yescharacteristic
311C4×C8D4×C10911yescharacteristic
312C4×C8C23×C10911yescharacteristic
313C4×C8C23×C10911yescharacteristic
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8C2×D4×C10911yescharacteristic
317C2×D4—21no—
318C2×D4C2×D4×C10911yescharacteristic
319C22×C4—21no—
320C22×C4C2×D4×C10911yescharacteristic
321C22×C4C2×D4×C10911yescharacteristic
322C22×C4#41×C10911yescharacteristic
323Q16—82no—
324SD16—82no—
325M16—42no—
326M16—82no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—42no—
334C2×C8C2×D4×C10911yescharacteristic
335C2×C8C2×D4×C10911yescharacteristic
336C2×C8C2×D4×C10911yescharacteristic
337C2×C8C2×D4×C10911yescharacteristic
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4C2×D4×C10911yescharacteristic
349C4⋊C4C2×D4×C10911yescharacteristic
350C22⋊C4—21no—
351C22⋊C4—21no—
352C22⋊C4—21no—
353C22⋊C4—21no—
354C22⋊C4—21no—
355C22⋊C4—42no—
356C42—21no—
357C42—42no—
358C42C2×D4×C10911yescharacteristic
359C42C2×D4×C10911yescharacteristic
360C42#41×C10911yescharacteristic
361C42C24×C10911yescharacteristic
362C23—21no—
363C23#121×C10911yescharacteristic
364Q8—21no—
365Q8—21no—
366Q8—42no—
367Q8—42no—
368Q8—82no—
369D4—21no—
370D4—21no—
371D4—42no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4#121×C10911yescharacteristic
385C2×C4#121×C10911yescharacteristic
386C2×C4#121×C10911yescharacteristic
387C2×C4C22×D4×C10911yescharacteristic
388C2×C4C22×D4×C10911yescharacteristic
389C2×C4C2×#41×C10911yescharacteristic
390C8—21no—
391C8—21no—
392C8—41no—
393C8—41no—
394C8—42no—
395C8—42no—
396C8#137×C10912yes—
397C8#137×C10912yes—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22C2×#121×C10911yescharacteristic
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—82no—
413C4—82no—
414C4C2×#137×C10911yescharacteristic
415C4C2×#137×C10911yescharacteristic
416C4#576×C10911yescharacteristic
417C4#578×C10911yescharacteristic
418C2—41no—
419C2—81no—
420C2C109×#303911yescharacteristic
421C2C109×#306111yescharacteristic
422C2C109×#306711yescharacteristic
subgroupG/H
center, socleC22×C109C2×#121
commutator, Frattini, cosocleC42C24×C109
prime radical#121×C109D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C109 ◁ C2×C42×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#3160C2semidirect
4. C109×#1692C2semidirect
5. C109×#1691C2semidirect
6. C109×#1676C2semidirect
7. C109×#1675C2semidirect
8. C109×#1674C2semidirect
9. C109×#1673C2semidirect
10. C109×#1567C2semidirect
11. C109×#1559C2semidirect
206. #19361C109direct (N × H)
213. #3160C2×C109semidirect
215. #1692C2×C109semidirect
216. #1691C2×C109semidirect
217. #1676C2×C109semidirect
218. #1675C2×C109semidirect
219. #1674C2×C109semidirect
220. #1673C2×C109semidirect
221. #1567C2×C109semidirect
222. #1559C2×C109semidirect
223. C109#19361direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#32522
C109×#31602
C109×#29662
C109×#16922
C109×#16912
C109×#16762
C109×#16752
C109×#16742
C109×#16732
C109×#15672
C109×#15592
#19361109