Finite Group Databasebeta

← C109×#19365 #42137359

Subgroups — C109×#19365 #42137359

Order 27904.

584 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3239C211yesmaximal, characteristic
2C109×#3238C211yesmaximal, characteristic
3C2×#487×C109C211yesmaximal, characteristic, complemented
4C109×#2966C211yesmaximal, characteristic
5C109×#1704C212yesmaximal, complemented
6C109×#1703C212yesmaximal, complemented
7C109×#1684C212yesmaximal, complemented
8C109×#1683C212yesmaximal, complemented
9C109×#1568C211yesmaximal, characteristic, complemented
10C109×#1558C211yesmaximal, characteristic, complemented
11C109×#1556C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8×C109C2211yescharacteristic
13C2×C4⋊Q8×C109C2211yescharacteristic
14#529×C109C2211yescharacteristic
15#501×C109—21no—
16#501×C109—21no—
17#489×C109—21no—
18#489×C109—21no—
19#487×C109—21no—
20#487×C109—21no—
21#487×C109C2212yes—
22#487×C109C2212yes—
23#484×C109—21no—
24#483×C109—21no—
25#481×C109—21no—
26#480×C109—21no—
27#477×C109C2212yes—
28#476×C109C2212yes—
29#475×C109C2212yes—
30#474×C109C2212yes—
31#446×C109C2211yescharacteristic
32C8×Q8×C109C2211yescharacteristic
33#436×C109C2211yescharacteristic
34#435×C109—21no—
35C8×D4×C109—21no—
36C8×D4×C109C2211yescharacteristic
37#433×C109—21no—
38#432×C109C2211yescharacteristic
39#432×C109C2211yescharacteristic
40#432×C109C2211yescharacteristic
41C2×#103×C109—21no—
42C2×#103×C109—21no—
43#405×C109—21no—
44C2×C4×C8×C109C2211yescharacteristic
45#335×C109C2212yes—
46#334×C109C2212yes—
47#333×C109C2212yes—
48#333×C109C2212yes—
49#332×C109C2212yes—
50#332×C109C2212yes—
51C22×Q8×C109—21no—
52C22×Q8×C109—21no—
53C2×M16×C109—21no—
54C22×C8×C109—21no—
55C22×C8×C109D411yescharacteristic
56C22×C8×C109D411yescharacteristic
57C4⋊Q8×C109—21no—
58C4⋊Q8×C109—21no—
59C4⋊Q8×C109—21no—
60C4⋊Q8×C109—21no—
61C4⋊Q8×C109—21no—
62C4⋊Q8×C109—21no—
63C4⋊Q8×C109C2312yes—
64C4⋊Q8×C109C2312yes—
65#126×C109—21no—
66#125×C109D411yescharacteristic
67#124×C109D411yescharacteristic
68#123×C109—21no—
69C22⋊Q8×C109D411yescharacteristic
70#121×C109D411yescharacteristic
71C4×Q8×C109C2311yescharacteristic
72C4×D4×C109—21no—
73C4×D4×C109C2311yescharacteristic
74#117×C109—21no—
75C2×C4⋊C4×C109—21no—
76C2×C4⋊C4×C109—21no—
77C2×C4⋊C4×C109—21no—
78C2×C4⋊C4×C109—21no—
79C2×C42×C109C2311yescharacteristic
80#107×C109—22no—
81#107×C109—22no—
82#106×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#103×C109—22no—
92#103×C109—22no—
93#103×C109—22no—
94#103×C109—22no—
95#103×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
112C2×Q8×C109—22no—
113C2×Q8×C109—22no—
114C2×Q8×C109—42no—
115C2×Q8×C109—42no—
116C2×Q8×C109—41no—
117C2×Q8×C109—41no—
118C2×Q8×C109—82no—
119C2×Q8×C109C2×D411yescharacteristic
120C2×D4×C109—21no—
121C2×D4×C109C2×D411yescharacteristic
122C22×C4×C109—21no—
123C22×C4×C109—41no—
124C22×C4×C109—41no—
125C22×C4×C109C2×D411yescharacteristic
126C22×C4×C109C2×D411yescharacteristic
127C22×C4×C109#4111yescharacteristic
128M16×C109—42no—
129C2×C8×C109—21no—
130C2×C8×C109—21no—
131C2×C8×C109—22no—
132C2×C8×C109—22no—
133C2×C8×C109—41no—
134C2×C8×C109—41no—
135C2×C8×C109—42no—
136C2×C8×C109C2×D411yescharacteristic
137C2×C8×C109C2×D411yescharacteristic
138C2×C8×C109C2×D411yescharacteristic
139C2×C8×C109C2×D411yescharacteristic
140C4⋊C4×C109—21no—
141C4⋊C4×C109—21no—
142C4⋊C4×C109—21no—
143C4⋊C4×C109—21no—
144C4⋊C4×C109—22no—
145C4⋊C4×C109—22no—
146C4⋊C4×C109—22no—
147C4⋊C4×C109—22no—
148C4⋊C4×C109—41no—
149C4⋊C4×C109—41no—
150C4⋊C4×C109—41no—
151C4⋊C4×C109—41no—
152C4⋊C4×C109C2×D411yescharacteristic
153C4⋊C4×C109C2×D411yescharacteristic
154C22⋊C4×C109—21no—
155C22⋊C4×C109—21no—
156C22⋊C4×C109—21no—
157C22⋊C4×C109—21no—
158C22⋊C4×C109—21no—
159C42×C109—21no—
160C42×C109C2×D411yescharacteristic
161C42×C109C2×D411yescharacteristic
162C42×C109#4111yescharacteristic
163C42×C109C2411yescharacteristic
164C23×C109—21no—
165C23×C109#12111yescharacteristic
166Q8×C109—22no—
167Q8×C109—42no—
168Q8×C109—42no—
169Q8×C109—82no—
170Q8×C109—82no—
171Q8×C109—82no—
172D4×C109—22no—
173D4×C109—42no—
174C2×C4×C109—22no—
175C2×C4×C109—21no—
176C2×C4×C109—21no—
177C2×C4×C109—22no—
178C2×C4×C109—21no—
179C2×C4×C109—21no—
180C2×C4×C109—21no—
181C2×C4×C109—41no—
182C2×C4×C109—41no—
183C2×C4×C109—42no—
184C2×C4×C109—42no—
185C2×C4×C109—82no—
186C2×C4×C109—82no—
187C2×C4×C109#12111yescharacteristic
188C2×C4×C109#12111yescharacteristic
189C2×C4×C109#12111yescharacteristic
190C2×C4×C109C22×D411yescharacteristic
191C2×C4×C109C22×D411yescharacteristic
192C2×C4×C109C2×#4111yescharacteristic
193C8×C109—21no—
194C8×C109—21no—
195C8×C109—21no—
196C8×C109—21no—
197C8×C109—41no—
198C8×C109—41no—
199C8×C109—42no—
200C8×C109—42no—
201C22×C109—21no—
202C22×C109—21no—
203C22×C109—41no—
204C22×C109—41no—
205C22×C109—41no—
206C22×C109#47112yes—
207C22×C109C2×#12111yescharacteristic
208C4×C109—21no—
209C4×C109—21no—
210C4×C109—21no—
211C4×C109—21no—
212C4×C109—41no—
213C4×C109—41no—
214C4×C109—41no—
215C4×C109—82no—
216C4×C109—82no—
217C4×C109#57511yescharacteristic
218C4×C109#57511yescharacteristic
219C4×C109#57812yes—
220#19365C10911yesmaximal, characteristic, direct product factor, Sylow
221C2×C109—22no—
222C2×C109—81no—
223C2×C109#304011yescharacteristic
224C2×C109#305911yescharacteristic
225C2×C109C2×#47111yescharacteristic
226#3239C2×C10911yescharacteristic
227#3238C2×C10911yescharacteristic
228C2×#487C2×C10911yescharacteristic, complemented
229#2966C2×C10911yescharacteristic
230#1704C2×C10912yescomplemented
231#1703C2×C10912yescomplemented
232#1684C2×C10912yescomplemented
233#1683C2×C10912yescomplemented
234#1568C2×C10911yescharacteristic, complemented
235#1558C2×C10911yescharacteristic, complemented
236#1556C2×C10911yescharacteristic, complemented
237C109#1936511yescharacteristic, direct product factor, Sylow
238C2×C4⋊Q8C22×C10911yescharacteristic
239C2×C4⋊Q8C22×C10911yescharacteristic
240#529C22×C10911yescharacteristic
241#501—21no—
242#501—21no—
243#489—21no—
244#489—21no—
245#487—21no—
246#487—21no—
247#487C22×C10912yes—
248#487C22×C10912yes—
249#484—21no—
250#483—21no—
251#481—21no—
252#480—21no—
253#477C22×C10912yes—
254#476C22×C10912yes—
255#475C22×C10912yes—
256#474C22×C10912yes—
257#446C22×C10911yescharacteristic
258C8×Q8C22×C10911yescharacteristic
259#436C22×C10911yescharacteristic
260#435—21no—
261C8×D4—21no—
262C8×D4C22×C10911yescharacteristic
263#433—21no—
264#432C22×C10911yescharacteristic
265#432C22×C10911yescharacteristic
266#432C22×C10911yescharacteristic
267C2×#103—21no—
268C2×#103—21no—
269#405—21no—
270C2×C4×C8C22×C10911yescharacteristic
271#335C22×C10912yes—
272#334C22×C10912yes—
273#333C22×C10912yes—
274#333C22×C10912yes—
275#332C22×C10912yes—
276#332C22×C10912yes—
277C22×Q8—21no—
278C22×Q8—21no—
279C2×M16—21no—
280C22×C8—21no—
281C22×C8D4×C10911yescharacteristic
282C22×C8D4×C10911yescharacteristic
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8—21no—
287C4⋊Q8—21no—
288C4⋊Q8—21no—
289C4⋊Q8C23×C10912yes—
290C4⋊Q8C23×C10912yes—
291#126—21no—
292#125D4×C10911yescharacteristic
293#124D4×C10911yescharacteristic
294#123—21no—
295C22⋊Q8D4×C10911yescharacteristic
296#121D4×C10911yescharacteristic
297C4×Q8C23×C10911yescharacteristic
298C4×D4—21no—
299C4×D4C23×C10911yescharacteristic
300#117—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C42C23×C10911yescharacteristic
306#107—22no—
307#107—22no—
308#106—22no—
309#106—22no—
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8C23×C10911yescharacteristic
313C4⋊C8C23×C10911yescharacteristic
314C4⋊C8C23×C10912yes—
315C4⋊C8C23×C10912yes—
316#103—22no—
317#103—22no—
318#103—22no—
319#103—22no—
320#103—22no—
321#103—22no—
322#102—22no—
323#102—22no—
324C22⋊C8—21no—
325C22⋊C8—21no—
326C22⋊C8—21no—
327C22⋊C8—21no—
328C22⋊C8—21no—
329#97—21no—
330#97—21no—
331#97—21no—
332#97—21no—
333C4×C8—21no—
334C4×C8D4×C10911yescharacteristic
335C4×C8D4×C10911yescharacteristic
336C4×C8C23×C10911yescharacteristic
337C4×C8C23×C10911yescharacteristic
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—82no—
345C2×Q8C2×D4×C10911yescharacteristic
346C2×D4—21no—
347C2×D4C2×D4×C10911yescharacteristic
348C22×C4—21no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4C2×D4×C10911yescharacteristic
352C22×C4C2×D4×C10911yescharacteristic
353C22×C4#41×C10911yescharacteristic
354M16—42no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—42no—
362C2×C8C2×D4×C10911yescharacteristic
363C2×C8C2×D4×C10911yescharacteristic
364C2×C8C2×D4×C10911yescharacteristic
365C2×C8C2×D4×C10911yescharacteristic
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4C2×D4×C10911yescharacteristic
379C4⋊C4C2×D4×C10911yescharacteristic
380C22⋊C4—21no—
381C22⋊C4—21no—
382C22⋊C4—21no—
383C22⋊C4—21no—
384C22⋊C4—21no—
385C42—21no—
386C42C2×D4×C10911yescharacteristic
387C42C2×D4×C10911yescharacteristic
388C42#41×C10911yescharacteristic
389C42C24×C10911yescharacteristic
390C23—21no—
391C23#121×C10911yescharacteristic
392Q8—22no—
393Q8—42no—
394Q8—42no—
395Q8—82no—
396Q8—82no—
397Q8—82no—
398D4—22no—
399D4—42no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4#121×C10911yescharacteristic
414C2×C4#121×C10911yescharacteristic
415C2×C4#121×C10911yescharacteristic
416C2×C4C22×D4×C10911yescharacteristic
417C2×C4C22×D4×C10911yescharacteristic
418C2×C4C2×#41×C10911yescharacteristic
419C8—21no—
420C8—21no—
421C8—21no—
422C8—21no—
423C8—41no—
424C8—41no—
425C8—42no—
426C8—42no—
427C22—21no—
428C22—21no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22#471×C10912yes—
433C22C2×#121×C10911yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—41no—
439C4—41no—
440C4—41no—
441C4—82no—
442C4—82no—
443C4#575×C10911yescharacteristic
444C4#575×C10911yescharacteristic
445C4#578×C10912yes—
446C2—22no—
447C2—81no—
448C2C109×#304011yescharacteristic
449C2C109×#305911yescharacteristic
450C2C2×#471×C10911yescharacteristic
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
3. C2×#487×C109C2semidirect
5. C109×#1704C2semidirect
6. C109×#1703C2semidirect
7. C109×#1684C2semidirect
8. C109×#1683C2semidirect
9. C109×#1568C2semidirect
10. C109×#1558C2semidirect
11. C109×#1556C2semidirect
220. #19365C109direct (N × H)
228. C2×#487C2×C109semidirect
230. #1704C2×C109semidirect
231. #1703C2×C109semidirect
232. #1684C2×C109semidirect
233. #1683C2×C109semidirect
234. #1568C2×C109semidirect
235. #1558C2×C109semidirect
236. #1556C2×C109semidirect
237. C109#19365direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#32392
C109×#32382
C2×#487×C1092
C109×#29662
C109×#17042
C109×#17032
C109×#16842
C109×#16832
C109×#15682
C109×#15582
C109×#15562
#19365109