Finite Group Databasebeta

← C109×#19362 #42137356

Subgroups — C109×#19362 #42137356

Order 27904.

648 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×#3238C211yesmaximal, characteristic, complemented
2C109×#3217C211yesmaximal, characteristic
3C2×#486×C109C211yesmaximal, characteristic, complemented
4C109×#2966C211yesmaximal, characteristic, complemented
5C109×#1698C212yesmaximal, complemented
6C109×#1697C212yesmaximal, complemented
7C109×#1678C212yesmaximal, complemented
8C109×#1677C212yesmaximal, complemented
9C109×#1568C211yesmaximal, characteristic, complemented
10C109×#1558C211yesmaximal, characteristic, complemented
11C109×#1555C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8×C109C2211yescharacteristic
13C2×#127×C109C2211yescharacteristic
14#529×C109C2211yescharacteristic
15#501×C109—21no—
16#496×C109—21no—
17#490×C109—21no—
18#489×C109—21no—
19#486×C109—21no—
20#486×C109—21no—
21#486×C109C2212yes—
22#486×C109C2212yes—
23#483×C109—21no—
24#480×C109—21no—
25#477×C109C2212yescomplemented
26#476×C109C2212yescomplemented
27#460×C109C2212yes—
28#459×C109C2212yes—
29#450×C109—21no—
30#447×C109—21no—
31#446×C109C2211yescharacteristic, complemented
32C8×Q8×C109C2211yescharacteristic, complemented
33#436×C109C2211yescharacteristic, complemented
34#435×C109—21no—
35C8×D4×C109—21no—
36C8×D4×C109C2211yescharacteristic, complemented
37#433×C109—21no—
38#432×C109C2211yescharacteristic
39#432×C109C2211yescharacteristic
40#432×C109C2211yescharacteristic
41C2×#102×C109—21no—
42C2×#102×C109—21no—
43#405×C109—21no—
44C2×C4×C8×C109C2211yescharacteristic
45#335×C109C2212yescomplemented
46#334×C109C2212yescomplemented
47#333×C109C2212yescomplemented
48#333×C109C2212yescomplemented
49#331×C109C2212yes—
50#331×C109C2212yes—
51C22×Q8×C109—21no—
52C22×D4×C109—21no—
53C22×D4×C109—21no—
54C22×D4×C109—21no—
55C2×SD16×C109—22no—
56C2×D8×C109—22no—
57C2×M16×C109—21no—
58C22×C8×C109—21no—
59C22×C8×C109D411yescharacteristic
60C22×C8×C109D411yescharacteristic
61C4⋊Q8×C109—21no—
62C4⋊Q8×C109—21no—
63C4⋊Q8×C109—21no—
64C4⋊Q8×C109C2312yes—
65#127×C109—21no—
66#127×C109—21no—
67#127×C109—21no—
68#127×C109C2312yes—
69#126×C109—21no—
70#125×C109D411yescharacteristic
71#124×C109D411yescharacteristic
72#123×C109—21no—
73C22⋊Q8×C109D411yescharacteristic
74#121×C109D411yescharacteristic
75C4×Q8×C109C2311yescharacteristic
76C4×D4×C109—21no—
77C4×D4×C109C2311yescharacteristic
78#117×C109—21no—
79C2×C4⋊C4×C109—21no—
80C2×C4⋊C4×C109—21no—
81C2×C42×C109C2311yescharacteristic
82#107×C109—22no—
83#106×C109—22no—
84C4⋊C8×C109—21no—
85C4⋊C8×C109—21no—
86C4⋊C8×C109C2311yescharacteristic
87C4⋊C8×C109C2311yescharacteristic
88C4⋊C8×C109C2312yes—
89C4⋊C8×C109C2312yes—
90#103×C109—22no—
91#102×C109—22no—
92#102×C109—22no—
93#102×C109—22no—
94#102×C109—22no—
95#102×C109—22no—
96#102×C109—22no—
97#102×C109—22no—
98C22⋊C8×C109—21no—
99C22⋊C8×C109—21no—
100C22⋊C8×C109—21no—
101C22⋊C8×C109—21no—
102C22⋊C8×C109—21no—
103#97×C109—21no—
104#97×C109—21no—
105#97×C109—21no—
106#97×C109—21no—
107C4×C8×C109—21no—
108C4×C8×C109D411yescharacteristic
109C4×C8×C109D411yescharacteristic
110C4×C8×C109C2311yescharacteristic
111C4×C8×C109C2311yescharacteristic
112C24×C109—41no—
113C2×Q8×C109—22no—
114C2×Q8×C109—41no—
115C2×Q8×C109—82no—
116C2×Q8×C109C2×D411yescharacteristic
117C2×D4×C109—21no—
118C2×D4×C109—22no—
119C2×D4×C109—22no—
120C2×D4×C109—22no—
121C2×D4×C109—42no—
122C2×D4×C109—42no—
123C2×D4×C109—41no—
124C2×D4×C109—41no—
125C2×D4×C109—41no—
126C2×D4×C109—82no—
127C2×D4×C109—82no—
128C2×D4×C109C2×D411yescharacteristic
129C22×C4×C109—21no—
130C22×C4×C109—41no—
131C22×C4×C109C2×D411yescharacteristic
132C22×C4×C109C2×D411yescharacteristic
133C22×C4×C109#4111yescharacteristic
134SD16×C109—82no—
135D8×C109—82no—
136M16×C109—42no—
137C2×C8×C109—21no—
138C2×C8×C109—21no—
139C2×C8×C109—22no—
140C2×C8×C109—22no—
141C2×C8×C109—41no—
142C2×C8×C109—41no—
143C2×C8×C109—42no—
144C2×C8×C109C2×D411yescharacteristic
145C2×C8×C109C2×D411yescharacteristic
146C2×C8×C109C2×D411yescharacteristic
147C2×C8×C109C2×D411yescharacteristic
148C4⋊C4×C109—21no—
149C4⋊C4×C109—21no—
150C4⋊C4×C109—21no—
151C4⋊C4×C109—21no—
152C4⋊C4×C109—22no—
153C4⋊C4×C109—22no—
154C4⋊C4×C109—41no—
155C4⋊C4×C109—41no—
156C4⋊C4×C109C2×D411yescharacteristic
157C4⋊C4×C109C2×D411yescharacteristic
158C22⋊C4×C109—21no—
159C22⋊C4×C109—21no—
160C22⋊C4×C109—21no—
161C22⋊C4×C109—21no—
162C22⋊C4×C109—21no—
163C42×C109—21no—
164C42×C109C2×D411yescharacteristic
165C42×C109C2×D411yescharacteristic
166C42×C109#4111yescharacteristic
167C42×C109C2411yescharacteristic
168C23×C109—21no—
169C23×C109—42no—
170C23×C109—82no—
171C23×C109—82no—
172C23×C109—82no—
173C23×C109#12111yescharacteristic
174Q8×C109—22no—
175Q8×C109—82no—
176Q8×C109—82no—
177D4×C109—22no—
178D4×C109—42no—
179D4×C109—42no—
180D4×C109—42no—
181D4×C109—42no—
182D4×C109—42no—
183D4×C109—82no—
184D4×C109—82no—
185D4×C109—82no—
186D4×C109—82no—
187C2×C4×C109—22no—
188C2×C4×C109—21no—
189C2×C4×C109—21no—
190C2×C4×C109—22no—
191C2×C4×C109—21no—
192C2×C4×C109—21no—
193C2×C4×C109—21no—
194C2×C4×C109—41no—
195C2×C4×C109—41no—
196C2×C4×C109—42no—
197C2×C4×C109—82no—
198C2×C4×C109#12111yescharacteristic
199C2×C4×C109#12111yescharacteristic
200C2×C4×C109#12111yescharacteristic
201C2×C4×C109C22×D411yescharacteristic
202C2×C4×C109C22×D411yescharacteristic
203C2×C4×C109C2×#4111yescharacteristic
204C8×C109—21no—
205C8×C109—21no—
206C8×C109—21no—
207C8×C109—21no—
208C8×C109—41no—
209C8×C109—41no—
210C8×C109—42no—
211C8×C109—42no—
212C22×C109—21no—
213C22×C109—21no—
214C22×C109—41no—
215C22×C109—41no—
216C22×C109—41no—
217C22×C109—82no—
218C22×C109—82no—
219C22×C109—82no—
220C22×C109—164no—
221C22×C109#47212yes—
222C22×C109C2×#12111yescharacteristic
223C4×C109—21no—
224C4×C109—21no—
225C4×C109—21no—
226C4×C109—21no—
227C4×C109—41no—
228C4×C109—41no—
229C4×C109—41no—
230C4×C109—82no—
231C4×C109#57511yescharacteristic
232C4×C109#57511yescharacteristic
233C4×C109#57712yes—
234#19362C10911yesmaximal, characteristic, direct product factor, Sylow
235C2×C109—22no—
236C2×C109—81no—
237C2×C109—162no—
238C2×C109#304011yescharacteristic
239C2×C109#305711yescharacteristic
240C2×C109C2×#47211yescharacteristic
241#3238C2×C10911yescharacteristic, complemented
242#3217C2×C10911yescharacteristic
243C2×#486C2×C10911yescharacteristic, complemented
244#2966C2×C10911yescharacteristic, complemented
245#1698C2×C10912yescomplemented
246#1697C2×C10912yescomplemented
247#1678C2×C10912yescomplemented
248#1677C2×C10912yescomplemented
249#1568C2×C10911yescharacteristic, complemented
250#1558C2×C10911yescharacteristic, complemented
251#1555C2×C10911yescharacteristic, complemented
252C109#1936211yescharacteristic, direct product factor, Sylow
253C2×C4⋊Q8C22×C10911yescharacteristic
254C2×#127C22×C10911yescharacteristic
255#529C22×C10911yescharacteristic
256#501—21no—
257#496—21no—
258#490—21no—
259#489—21no—
260#486—21no—
261#486—21no—
262#486C22×C10912yes—
263#486C22×C10912yes—
264#483—21no—
265#480—21no—
266#477C22×C10912yescomplemented
267#476C22×C10912yescomplemented
268#460C22×C10912yes—
269#459C22×C10912yes—
270#450—21no—
271#447—21no—
272#446C22×C10911yescharacteristic, complemented
273C8×Q8C22×C10911yescharacteristic, complemented
274#436C22×C10911yescharacteristic, complemented
275#435—21no—
276C8×D4—21no—
277C8×D4C22×C10911yescharacteristic, complemented
278#433—21no—
279#432C22×C10911yescharacteristic
280#432C22×C10911yescharacteristic
281#432C22×C10911yescharacteristic
282C2×#102—21no—
283C2×#102—21no—
284#405—21no—
285C2×C4×C8C22×C10911yescharacteristic
286#335C22×C10912yescomplemented
287#334C22×C10912yescomplemented
288#333C22×C10912yescomplemented
289#333C22×C10912yescomplemented
290#331C22×C10912yes—
291#331C22×C10912yes—
292C22×Q8—21no—
293C22×D4—21no—
294C22×D4—21no—
295C22×D4—21no—
296C2×SD16—22no—
297C2×D8—22no—
298C2×M16—21no—
299C22×C8—21no—
300C22×C8D4×C10911yescharacteristic
301C22×C8D4×C10911yescharacteristic
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—21no—
305C4⋊Q8C23×C10912yes—
306#127—21no—
307#127—21no—
308#127—21no—
309#127C23×C10912yes—
310#126—21no—
311#125D4×C10911yescharacteristic
312#124D4×C10911yescharacteristic
313#123—21no—
314C22⋊Q8D4×C10911yescharacteristic
315#121D4×C10911yescharacteristic
316C4×Q8C23×C10911yescharacteristic
317C4×D4—21no—
318C4×D4C23×C10911yescharacteristic
319#117—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C42C23×C10911yescharacteristic
323#107—22no—
324#106—22no—
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8C23×C10911yescharacteristic
328C4⋊C8C23×C10911yescharacteristic
329C4⋊C8C23×C10912yes—
330C4⋊C8C23×C10912yes—
331#103—22no—
332#102—22no—
333#102—22no—
334#102—22no—
335#102—22no—
336#102—22no—
337#102—22no—
338#102—22no—
339C22⋊C8—21no—
340C22⋊C8—21no—
341C22⋊C8—21no—
342C22⋊C8—21no—
343C22⋊C8—21no—
344#97—21no—
345#97—21no—
346#97—21no—
347#97—21no—
348C4×C8—21no—
349C4×C8D4×C10911yescharacteristic
350C4×C8D4×C10911yescharacteristic
351C4×C8C23×C10911yescharacteristic
352C4×C8C23×C10911yescharacteristic
353C24—41no—
354C2×Q8—22no—
355C2×Q8—41no—
356C2×Q8—82no—
357C2×Q8C2×D4×C10911yescharacteristic
358C2×D4—21no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—82no—
368C2×D4—82no—
369C2×D4C2×D4×C10911yescharacteristic
370C22×C4—21no—
371C22×C4—41no—
372C22×C4C2×D4×C10911yescharacteristic
373C22×C4C2×D4×C10911yescharacteristic
374C22×C4#41×C10911yescharacteristic
375SD16—82no—
376D8—82no—
377M16—42no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—42no—
385C2×C8C2×D4×C10911yescharacteristic
386C2×C8C2×D4×C10911yescharacteristic
387C2×C8C2×D4×C10911yescharacteristic
388C2×C8C2×D4×C10911yescharacteristic
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4C2×D4×C10911yescharacteristic
398C4⋊C4C2×D4×C10911yescharacteristic
399C22⋊C4—21no—
400C22⋊C4—21no—
401C22⋊C4—21no—
402C22⋊C4—21no—
403C22⋊C4—21no—
404C42—21no—
405C42C2×D4×C10911yescharacteristic
406C42C2×D4×C10911yescharacteristic
407C42#41×C10911yescharacteristic
408C42C24×C10911yescharacteristic
409C23—21no—
410C23—42no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23#121×C10911yescharacteristic
415Q8—22no—
416Q8—82no—
417Q8—82no—
418D4—22no—
419D4—42no—
420D4—42no—
421D4—42no—
422D4—42no—
423D4—42no—
424D4—82no—
425D4—82no—
426D4—82no—
427D4—82no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—82no—
439C2×C4#121×C10911yescharacteristic
440C2×C4#121×C10911yescharacteristic
441C2×C4#121×C10911yescharacteristic
442C2×C4C22×D4×C10911yescharacteristic
443C2×C4C22×D4×C10911yescharacteristic
444C2×C4C2×#41×C10911yescharacteristic
445C8—21no—
446C8—21no—
447C8—21no—
448C8—21no—
449C8—41no—
450C8—41no—
451C8—42no—
452C8—42no—
453C22—21no—
454C22—21no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22—164no—
462C22#472×C10912yes—
463C22C2×#121×C10911yescharacteristic
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—41no—
469C4—41no—
470C4—41no—
471C4—82no—
472C4#575×C10911yescharacteristic
473C4#575×C10911yescharacteristic
474C4#577×C10912yes—
475C2—22no—
476C2—81no—
477C2—162no—
478C2C109×#304011yescharacteristic
479C2C109×#305711yescharacteristic
480C2C2×#472×C10911yescharacteristic
subgroupG/H
center, socleC22×C109C2×#121
commutator, Frattini, cosocleC42C24×C109
prime radicalC109×#3217C2
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
1. C109×#3238C2semidirect
3. C2×#486×C109C2semidirect
4. C109×#2966C2semidirect
5. C109×#1698C2semidirect
6. C109×#1697C2semidirect
7. C109×#1678C2semidirect
8. C109×#1677C2semidirect
9. C109×#1568C2semidirect
10. C109×#1558C2semidirect
11. C109×#1555C2semidirect
25. #477×C109C22semidirect
26. #476×C109C22semidirect
31. #446×C109C22semidirect
32. C8×Q8×C109C22semidirect
33. #436×C109C22semidirect
36. C8×D4×C109C22semidirect
45. #335×C109C22semidirect
46. #334×C109C22semidirect
47. #333×C109C22semidirect
48. #333×C109C22semidirect
234. #19362C109direct (N × H)
241. #3238C2×C109semidirect
243. C2×#486C2×C109semidirect
244. #2966C2×C109semidirect
245. #1698C2×C109semidirect
246. #1697C2×C109semidirect
247. #1678C2×C109semidirect
248. #1677C2×C109semidirect
249. #1568C2×C109semidirect
250. #1558C2×C109semidirect
251. #1555C2×C109semidirect
252. C109#19362direct (N × H)
266. #477C22×C109semidirect
267. #476C22×C109semidirect
272. #446C22×C109semidirect
273. C8×Q8C22×C109semidirect
274. #436C22×C109semidirect
277. C8×D4C22×C109semidirect
286. #335C22×C109semidirect
287. #334C22×C109semidirect
288. #333C22×C109semidirect
289. #333C22×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#32382
C109×#32172
C2×#486×C1092
C109×#29662
C109×#16982
C109×#16972
C109×#16782
C109×#16772
C109×#15682
C109×#15582
C109×#15552
#19362109