Finite Group Databasebeta

← C109×#19315 #42137309

Subgroups — C109×#19315 #42137309

Order 27904.

598 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3165C211yesmaximal, characteristic, complemented
2C109×#3116C211yesmaximal, characteristic
3C109×#3079C211yesmaximal, characteristic
4C109×#2970C211yesmaximal, characteristic
5C109×#1696C212yesmaximal, complemented
6C109×#1692C212yesmaximal, complemented
7C109×#1675C212yesmaximal, complemented
8C109×#1671C212yesmaximal, complemented
9C109×#1569C211yesmaximal, characteristic, complemented
10C109×#1559C211yesmaximal, characteristic, complemented
11C109×#1558C211yesmaximal, characteristic, complemented
12#533×C109C2211yescharacteristic
13C2×C4⋊Q8×C109C2211yescharacteristic
14C4×#41×C109C2211yescharacteristic
15#501×C109—21no—
16#501×C109—21no—
17#500×C109C2212yes—
18#498×C109C2212yes—
19#485×C109—21no—
20#477×C109C2212yes—
21#477×C109C2211yescharacteristic
22#476×C109C2212yes—
23#476×C109C2211yescharacteristic
24#475×C109—21no—
25#474×C109—21no—
26#462×C109C2211yescharacteristic
27#461×C109C2211yescharacteristic
28#452×C109—21no—
29#446×C109C2212yes—
30#441×C109—21no—
31#440×C109—21no—
32#435×C109—22no—
33C8×D4×C109C2212yes—
34#432×C109—21no—
35#432×C109C2211yescharacteristic
36#432×C109C2211yescharacteristic
37#428×C109—22no—
38#428×C109—21no—
39#423×C109C2211yescharacteristic
40#417×C109—21no—
41C4×M16×C109C2211yescharacteristic
42C2×#97×C109—21no—
43#335×C109C2212yes—
44#334×C109C2212yes—
45#333×C109C2212yes—
46#333×C109C2212yes—
47#333×C109C2212yes—
48#332×C109C2212yes—
49C2×#41×C109D411yescharacteristic
50C22×Q8×C109—21no—
51C2×Q16×C109—21no—
52C2×SD16×C109—21no—
53C2×M16×C109—21no—
54C2×M16×C109D412yes—
55C22×C8×C109—22no—
56C4⋊Q8×C109—21no—
57C4⋊Q8×C109—21no—
58C4⋊Q8×C109—21no—
59C4⋊Q8×C109C2311yescharacteristic
60C4⋊Q8×C109C2312yes—
61C4⋊Q8×C109C2311yescharacteristic
62#125×C109—21no—
63C22⋊Q8×C109—21no—
64C22⋊Q8×C109—21no—
65C4×Q8×C109—21no—
66C4×Q8×C109—21no—
67C4×Q8×C109D411yescharacteristic
68C4×Q8×C109C2311yescharacteristic
69C4×D4×C109—21no—
70C4×D4×C109—21no—
71C4×D4×C109D411yescharacteristic
72C4×D4×C109C2311yescharacteristic
73#117×C109—21no—
74#117×C109—21no—
75#117×C109D411yescharacteristic
76C2×C4⋊C4×C109—21no—
77C2×C4⋊C4×C109—21no—
78C2×C42×C109—21no—
79C2×C42×C109C2311yescharacteristic
80#107×C109—22no—
81#107×C109—22no—
82#107×C109—22no—
83#107×C109—21no—
84#106×C109—22no—
85#106×C109—22no—
86#106×C109—22no—
87#106×C109—21no—
88C4⋊C8×C109—22no—
89C4⋊C8×C109—21no—
90C4⋊C8×C109C2312yes—
91C4⋊C8×C109C2312yes—
92C4⋊C8×C109C2311yescharacteristic
93C4⋊C8×C109C2311yescharacteristic
94#103×C109—22no—
95#103×C109—21no—
96#103×C109—21no—
97#103×C109—21no—
98#102×C109—22no—
99#102×C109—21no—
100C22⋊C8×C109—21no—
101C22⋊C8×C109—21no—
102C22⋊C8×C109—22no—
103#97×C109—22no—
104#97×C109—21no—
105#97×C109—21no—
106#97×C109D411yescharacteristic
107#97×C109D411yescharacteristic
108C4×C8×C109C2312yes—
109#41×C109—22no—
110#41×C109—41no—
111C2×Q8×C109—21no—
112C2×Q8×C109—21no—
113C2×Q8×C109—22no—
114C2×Q8×C109—41no—
115C2×Q8×C109—82no—
116C2×Q8×C109C2×D411yescharacteristic
117C2×D4×C109—21no—
118C2×D4×C109C2×D411yescharacteristic
119C22×C4×C109—21no—
120C22×C4×C109—21no—
121C22×C4×C109—21no—
122C22×C4×C109—41no—
123C22×C4×C109C2×D411yescharacteristic
124C22×C4×C109C2×D411yescharacteristic
125C22×C4×C109#4111yescharacteristic
126Q16×C109—42no—
127SD16×C109—42no—
128M16×C109—24no—
129M16×C109—42no—
130C2×C8×C109—22no—
131C2×C8×C109—21no—
132C2×C8×C109—21no—
133C2×C8×C109—22no—
134C2×C8×C109—44no—
135C2×C8×C109C2×D412yes—
136C2×C8×C109C2×D412yes—
137C4⋊C4×C109—21no—
138C4⋊C4×C109—21no—
139C4⋊C4×C109—22no—
140C4⋊C4×C109—21no—
141C4⋊C4×C109—21no—
142C4⋊C4×C109—22no—
143C4⋊C4×C109—21no—
144C4⋊C4×C109—21no—
145C4⋊C4×C109—41no—
146C4⋊C4×C109—41no—
147C4⋊C4×C109—41no—
148C4⋊C4×C109—41no—
149C4⋊C4×C109C2×D411yescharacteristic
150C4⋊C4×C109C2×D411yescharacteristic
151C22⋊C4×C109—21no—
152C22⋊C4×C109—21no—
153C22⋊C4×C109—21no—
154C22⋊C4×C109—41no—
155C42×C109—21no—
156C42×C109—21no—
157C42×C109—21no—
158C42×C109—41no—
159C42×C109C2×D411yescharacteristic
160C42×C109C2×D411yescharacteristic
161C42×C109#4111yescharacteristic
162C42×C109C2411yescharacteristic
163C23×C109—21no—
164C23×C109#12111yescharacteristic
165Q8×C109—21no—
166Q8×C109—21no—
167Q8×C109—41no—
168Q8×C109—41no—
169Q8×C109—81no—
170Q8×C109—82no—
171Q8×C109—82no—
172D4×C109—21no—
173D4×C109—42no—
174D4×C109#13712yes—
175C2×C4×C109—22no—
176C2×C4×C109—21no—
177C2×C4×C109—21no—
178C2×C4×C109—21no—
179C2×C4×C109—21no—
180C2×C4×C109—21no—
181C2×C4×C109—41no—
182C2×C4×C109—42no—
183C2×C4×C109—42no—
184C2×C4×C109—41no—
185C2×C4×C109—41no—
186C2×C4×C109—42no—
187C2×C4×C109—81no—
188C2×C4×C109—82no—
189C2×C4×C109#12111yescharacteristic
190C2×C4×C109#12111yescharacteristic
191C2×C4×C109#12111yescharacteristic
192C2×C4×C109C22×D411yescharacteristic
193C2×C4×C109C22×D411yescharacteristic
194C2×C4×C109C2×#4111yescharacteristic
195C8×C109—22no—
196C8×C109—42no—
197C8×C109—41no—
198C8×C109—42no—
199C8×C109—41no—
200C8×C109#13714yes—
201C22×C109—21no—
202C22×C109—21no—
203C22×C109—22no—
204C22×C109—21no—
205C22×C109—41no—
206C22×C109—41no—
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—81no—
216C4×C109—82no—
217C4×C109—81no—
218C4×C109C2×#13712yes—
219C4×C109C2×#13711yescharacteristic
220C4×C109#57511yescharacteristic
221#19315C10911yesmaximal, characteristic, direct product factor, Sylow
222C2×C109—41no—
223C2×C109—42no—
224C2×C109#303911yescharacteristic
225C2×C109#305611yescharacteristic
226C2×C109#306811yescharacteristic
227#3165C2×C10911yescharacteristic, complemented
228#3116C2×C10911yescharacteristic
229#3079C2×C10911yescharacteristic
230#2970C2×C10911yescharacteristic
231#1696C2×C10912yescomplemented
232#1692C2×C10912yescomplemented
233#1675C2×C10912yescomplemented
234#1671C2×C10912yescomplemented
235#1569C2×C10911yescharacteristic, complemented
236#1559C2×C10911yescharacteristic, complemented
237#1558C2×C10911yescharacteristic, complemented
238C109#1931511yescharacteristic, direct product factor, Sylow
239#533C22×C10911yescharacteristic
240C2×C4⋊Q8C22×C10911yescharacteristic
241C4×#41C22×C10911yescharacteristic
242#501—21no—
243#501—21no—
244#500C22×C10912yes—
245#498C22×C10912yes—
246#485—21no—
247#477C22×C10912yes—
248#477C22×C10911yescharacteristic
249#476C22×C10912yes—
250#476C22×C10911yescharacteristic
251#475—21no—
252#474—21no—
253#462C22×C10911yescharacteristic
254#461C22×C10911yescharacteristic
255#452—21no—
256#446C22×C10912yes—
257#441—21no—
258#440—21no—
259#435—22no—
260C8×D4C22×C10912yes—
261#432—21no—
262#432C22×C10911yescharacteristic
263#432C22×C10911yescharacteristic
264#428—22no—
265#428—21no—
266#423C22×C10911yescharacteristic
267#417—21no—
268C4×M16C22×C10911yescharacteristic
269C2×#97—21no—
270#335C22×C10912yes—
271#334C22×C10912yes—
272#333C22×C10912yes—
273#333C22×C10912yes—
274#333C22×C10912yes—
275#332C22×C10912yes—
276C2×#41D4×C10911yescharacteristic
277C22×Q8—21no—
278C2×Q16—21no—
279C2×SD16—21no—
280C2×M16—21no—
281C2×M16D4×C10912yes—
282C22×C8—22no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8C23×C10912yes—
287C4⋊Q8C23×C10911yescharacteristic
288C4⋊Q8C23×C10911yescharacteristic
289#125—21no—
290C22⋊Q8—21no—
291C22⋊Q8—21no—
292C4×Q8—21no—
293C4×Q8—21no—
294C4×Q8D4×C10911yescharacteristic
295C4×Q8C23×C10911yescharacteristic
296C4×D4—21no—
297C4×D4—21no—
298C4×D4D4×C10911yescharacteristic
299C4×D4C23×C10911yescharacteristic
300#117—21no—
301#117—21no—
302#117D4×C10911yescharacteristic
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C42—21no—
306C2×C42C23×C10911yescharacteristic
307#107—22no—
308#107—22no—
309#107—22no—
310#107—21no—
311#106—22no—
312#106—22no—
313#106—22no—
314#106—21no—
315C4⋊C8—22no—
316C4⋊C8—21no—
317C4⋊C8C23×C10912yes—
318C4⋊C8C23×C10912yes—
319C4⋊C8C23×C10911yescharacteristic
320C4⋊C8C23×C10911yescharacteristic
321#103—22no—
322#103—21no—
323#103—21no—
324#103—21no—
325#102—22no—
326#102—21no—
327C22⋊C8—21no—
328C22⋊C8—21no—
329C22⋊C8—22no—
330#97—22no—
331#97—21no—
332#97—21no—
333#97D4×C10911yescharacteristic
334#97D4×C10911yescharacteristic
335C4×C8C23×C10912yes—
336#41—22no—
337#41—41no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—22no—
341C2×Q8—41no—
342C2×Q8—82no—
343C2×Q8C2×D4×C10911yescharacteristic
344C2×D4—21no—
345C2×D4C2×D4×C10911yescharacteristic
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—41no—
350C22×C4C2×D4×C10911yescharacteristic
351C22×C4C2×D4×C10911yescharacteristic
352C22×C4#41×C10911yescharacteristic
353Q16—42no—
354SD16—42no—
355M16—24no—
356M16—42no—
357C2×C8—22no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—22no—
361C2×C8—44no—
362C2×C8C2×D4×C10912yes—
363C2×C8C2×D4×C10912yes—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4C2×D4×C10911yescharacteristic
377C4⋊C4C2×D4×C10911yescharacteristic
378C22⋊C4—21no—
379C22⋊C4—21no—
380C22⋊C4—21no—
381C22⋊C4—41no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—41no—
386C42C2×D4×C10911yescharacteristic
387C42C2×D4×C10911yescharacteristic
388C42#41×C10911yescharacteristic
389C42C24×C10911yescharacteristic
390C23—21no—
391C23#121×C10911yescharacteristic
392Q8—21no—
393Q8—21no—
394Q8—41no—
395Q8—41no—
396Q8—82no—
397Q8—81no—
398Q8—82no—
399D4—21no—
400D4—42no—
401D4#137×C10912yes—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—81no—
415C2×C4—82no—
416C2×C4#121×C10911yescharacteristic
417C2×C4#121×C10911yescharacteristic
418C2×C4#121×C10911yescharacteristic
419C2×C4C22×D4×C10911yescharacteristic
420C2×C4C22×D4×C10911yescharacteristic
421C2×C4C2×#41×C10911yescharacteristic
422C8—22no—
423C8—42no—
424C8—41no—
425C8—42no—
426C8—41no—
427C8#137×C10914yes—
428C22—21no—
429C22—21no—
430C22—22no—
431C22—21no—
432C22—41no—
433C22—41no—
434C22C2×#121×C10911yescharacteristic
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—41no—
440C4—41no—
441C4—41no—
442C4—81no—
443C4—82no—
444C4—81no—
445C4C2×#137×C10912yes—
446C4C2×#137×C10911yescharacteristic
447C4#575×C10911yescharacteristic
448C2—41no—
449C2—42no—
450C2C109×#303911yescharacteristic
451C2C109×#305611yescharacteristic
452C2C109×#306811yescharacteristic
subgroupG/H
center, socleC22×C109C2×#121
commutator, Frattini, cosocleC42C24×C109
prime radicalC2×#41×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
1. C109×#3165C2semidirect
5. C109×#1696C2semidirect
6. C109×#1692C2semidirect
7. C109×#1675C2semidirect
8. C109×#1671C2semidirect
9. C109×#1569C2semidirect
10. C109×#1559C2semidirect
11. C109×#1558C2semidirect
221. #19315C109direct (N × H)
227. #3165C2×C109semidirect
231. #1696C2×C109semidirect
232. #1692C2×C109semidirect
233. #1675C2×C109semidirect
234. #1671C2×C109semidirect
235. #1569C2×C109semidirect
236. #1559C2×C109semidirect
237. #1558C2×C109semidirect
238. C109#19315direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#31652
C109×#31162
C109×#30792
C109×#29702
C109×#16962
C109×#16922
C109×#16752
C109×#16712
C109×#15692
C109×#15592
C109×#15582
#19315109