Finite Group Databasebeta

← C109×#12715 #42130709

Subgroups — C109×#12715 #42130709

Order 27904.

976 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3500C211yesmaximal, characteristic
2C109×#2132C211yesmaximal, characteristic, complemented
3C109×#2126C211yesmaximal, characteristic, complemented
4C109×#2122C212yesmaximal, complemented
5C109×#1796C212yesmaximal, complemented
6C2×2−1+4×C109C2211yescharacteristic
7C2×H(5,𝔽2)×C109C411yescharacteristic, complemented
8#548×C109—22no—
9#546×C109—22no—
10#546×C109—24no—
11#536×C109—21no—
12#536×C109C411yescharacteristic, complemented
13#535×C109—21no—
14#535×C109C2211yescharacteristic
15#532×C109—21no—
16#532×C109C2211yescharacteristic
17C2×#124×C109—21no—
18C2×#124×C109C411yescharacteristic, complemented
19C4×#41×C109C411yescharacteristic, complemented
20#421×C109—22no—
21#419×C109—22no—
22#413×C109C2212yescomplemented
23#410×C109C2212yescomplemented
24#356×C109—21no—
25#355×C109—21no—
26#354×C109—21no—
27#353×C109—21no—
28AΣL(1,𝔽16)×C109—24no—
292−1+4×C109—42no—
302−1+4×C109—42no—
31H(5,𝔽2)×C109—42no—
32H(5,𝔽2)×C109—42no—
33C2×#41×C109—21no—
34C2×#41×C109—21no—
35C2×#41×C109—21no—
36C2×#41×C109—22no—
37C2×#41×C109C2×C411yescharacteristic
38C2×#41×C109C2×C411yescharacteristic
39C2×#41×C109D412yes—
40C2×#41×C109C2311yescharacteristic
41C22×Q8×C109—21no—
42C22×Q8×C109C2×C411yescharacteristic
43C22×Q8×C109D412yes—
44C22×D4×C109—22no—
45C22×D4×C109—21no—
46C22×D4×C109—21no—
47C22×D4×C109C2×C411yescharacteristic
48C2×M16×C109—22no—
49C4⋊Q8×C109—21no—
50C4⋊Q8×C109D411yescharacteristic
51#127×C109—21no—
52#127×C109D411yescharacteristic
53#124×C109—24no—
54#124×C109—22no—
55#124×C109—21no—
56#124×C109—21no—
57#124×C109—21no—
58#124×C109—21no—
59#124×C109D411yescharacteristic
60#124×C109D411yescharacteristic
61#123×C109—24no—
62#123×C109—22no—
63C22⋊Q8×C109—24no—
64C22⋊Q8×C109—22no—
65#121×C109—24no—
66#121×C109—24no—
67#121×C109—22no—
68#121×C109—22no—
69#120×C109—24no—
70#120×C109D414yes—
71C4×Q8×C109—21no—
72C4×D4×C109—24no—
73C4×D4×C109—21no—
74C4×D4×C109—22no—
75C4×D4×C109—22no—
76#117×C109—21no—
77#117×C109—22no—
78#117×C109C2×C411yescharacteristic
79C2×C22⋊C4×C109—24no—
80C2×C22⋊C4×C109—22no—
81C2×C42×C109—21no—
82C2×C42×C109C2×C411yescharacteristic
83#104×C109—44no—
84#103×C109—44no—
85#101×C109—22no—
86#100×C109—22no—
87#99×C109—22no—
88#99×C109—22no—
89C24×C109—22no—
90C24×C109C22⋊C412yes—
91#41×C109—22no—
92#41×C109—22no—
93#41×C109—24no—
94#41×C109—41no—
95#41×C109—41no—
96#41×C109—42no—
97#41×C109—44no—
98#41×C109—44no—
99#41×C109—41no—
100#41×C109—41no—
101#41×C109—48no—
102#41×C109—42no—
103#41×C109—42no—
104C2×Q8×C109—21no—
105C2×Q8×C109—22no—
106C2×Q8×C109—24no—
107C2×Q8×C109—41no—
108C2×Q8×C109—41no—
109C2×Q8×C109—42no—
110C2×Q8×C109—44no—
111C2×Q8×C109C22⋊C411yescharacteristic
112C2×Q8×C109C22⋊C412yes—
113C2×Q8×C109C22⋊C411yescharacteristic
114C2×Q8×C109C2×D411yescharacteristic
115C2×D4×C109—21no—
116C2×D4×C109—22no—
117C2×D4×C109—22no—
118C2×D4×C109—21no—
119C2×D4×C109—22no—
120C2×D4×C109—22no—
121C2×D4×C109—24no—
122C2×D4×C109—21no—
123C2×D4×C109—21no—
124C2×D4×C109—44no—
125C2×D4×C109—41no—
126C2×D4×C109—41no—
127C2×D4×C109—48no—
128C2×D4×C109—42no—
129C2×D4×C109—42no—
130C2×D4×C109C22⋊C411yescharacteristic
131C2×D4×C109C22⋊C411yescharacteristic
132C2×D4×C109C2×D411yescharacteristic
133C2×D4×C109C2×D412yes—
134C22×C4×C109—21no—
135C22×C4×C109—22no—
136C22×C4×C109—21no—
137C22×C4×C109—22no—
138C22×C4×C109—22no—
139C22×C4×C109—21no—
140C22×C4×C109—21no—
141C22×C4×C109—22no—
142C22×C4×C109C22×C411yescharacteristic
143C22×C4×C109C2×D412yes—
144M16×C109—42no—
145M16×C109—42no—
146C2×C8×C109—42no—
147C4⋊C4×C109—22no—
148C4⋊C4×C109—24no—
149C4⋊C4×C109—21no—
150C4⋊C4×C109—22no—
151C4⋊C4×C109—42no—
152C22⋊C4×C109—24no—
153C22⋊C4×C109—24no—
154C22⋊C4×C109—24no—
155C22⋊C4×C109—22no—
156C22⋊C4×C109—22no—
157C22⋊C4×C109—22no—
158C22⋊C4×C109—24no—
159C22⋊C4×C109—42no—
160C22⋊C4×C109C22⋊C412yes—
161C42×C109—22no—
162C42×C109—21no—
163C42×C109—21no—
164C42×C109—42no—
165C42×C109C22⋊C411yescharacteristic
166C42×C109C22⋊C411yescharacteristic
167C23×C109—21no—
168C23×C109—22no—
169C23×C109—21no—
170C23×C109—44no—
171C23×C109—42no—
172C23×C109—82no—
173C23×C109—84no—
174C23×C109C2×C22⋊C411yescharacteristic
175C23×C109#12012yes—
176Q8×C109—21no—
177Q8×C109—22no—
178Q8×C109—22no—
179Q8×C109—41no—
180Q8×C109—44no—
181Q8×C109—41no—
182Q8×C109—42no—
183Q8×C109#9912yes—
184D4×C109—24no—
185D4×C109—21no—
186D4×C109—42no—
187D4×C109—41no—
188D4×C109—41no—
189D4×C109—42no—
190D4×C109—44no—
191D4×C109—44no—
192D4×C109—42no—
193D4×C109—44no—
194D4×C109—41no—
195D4×C109—41no—
196D4×C109—82no—
197D4×C109—84no—
198D4×C109#9912yes—
199C2×C4×C109—24no—
200C2×C4×C109—21no—
201C2×C4×C109—22no—
202C2×C4×C109—22no—
203C2×C4×C109—21no—
204C2×C4×C109—21no—
205C2×C4×C109—21no—
206C2×C4×C109—22no—
207C2×C4×C109—21no—
208C2×C4×C109—22no—
209C2×C4×C109—21no—
210C2×C4×C109—44no—
211C2×C4×C109—44no—
212C2×C4×C109—44no—
213C2×C4×C109—42no—
214C2×C4×C109—41no—
215C2×C4×C109—41no—
216C2×C4×C109—81no—
217C2×C4×C109—82no—
218C2×C4×C109—82no—
219C2×C4×C109C2×C22⋊C411yescharacteristic
220C2×C4×C109C2×C22⋊C411yescharacteristic
221C2×C4×C109#12012yes—
222C8×C109—82no—
223C22×C109—22no—
224C22×C109—21no—
225C22×C109—22no—
226C22×C109—41no—
227C22×C109—42no—
228C22×C109—42no—
229C22×C109—44no—
230C22×C109—42no—
231C22×C109—42no—
232C22×C109—41no—
233C22×C109—162no—
234C22×C109—164no—
235C22×C109#37911yescharacteristic
236C4×C109—22no—
237C4×C109—21no—
238C4×C109—22no—
239C4×C109—41no—
240C4×C109—42no—
241C4×C109—41no—
242C4×C109—41no—
243C4×C109—42no—
244C4×C109—162no—
245C4×C109C2×#9911yescharacteristic
246C4×C109C2×#9911yescharacteristic
247#12715C10911yesmaximal, characteristic, direct product factor, Sylow
248C2×C109—21no—
249C2×C109—42no—
250C2×C109—42no—
251C2×C109—41no—
252C2×C109—82no—
253C2×C109—81no—
254C2×C109#178111yescharacteristic
255#3500C2×C10911yescharacteristic
256#2132C2×C10911yescharacteristic, complemented
257#2126C2×C10911yescharacteristic, complemented
258#2122C2×C10912yescomplemented
259#1796C2×C10912yescomplemented
260C109#1271511yescharacteristic, direct product factor, Sylow
261C2×2−1+4C22×C10911yescharacteristic
262C2×H(5,𝔽2)C4×C10911yescharacteristic, complemented
263#548—22no—
264#546—22no—
265#546—24no—
266#536—21no—
267#536C4×C10911yescharacteristic, complemented
268#535—21no—
269#535C22×C10911yescharacteristic
270#532—21no—
271#532C22×C10911yescharacteristic
272C2×#124—21no—
273C2×#124C4×C10911yescharacteristic, complemented
274C4×#41C4×C10911yescharacteristic, complemented
275#421—22no—
276#419—22no—
277#413C22×C10912yescomplemented
278#410C22×C10912yescomplemented
279#356—21no—
280#355—21no—
281#354—21no—
282#353—21no—
283AΣL(1,𝔽16)—24no—
2842−1+4—42no—
2852−1+4—42no—
286H(5,𝔽2)—42no—
287H(5,𝔽2)—42no—
288C2×#41—21no—
289C2×#41—21no—
290C2×#41—21no—
291C2×#41—22no—
292C2×#41C2×C4×C10911yescharacteristic
293C2×#41C2×C4×C10911yescharacteristic
294C2×#41D4×C10912yes—
295C2×#41C23×C10911yescharacteristic
296C22×Q8—21no—
297C22×Q8C2×C4×C10911yescharacteristic
298C22×Q8D4×C10912yes—
299C22×D4—22no—
300C22×D4—21no—
301C22×D4—21no—
302C22×D4C2×C4×C10911yescharacteristic
303C2×M16—22no—
304C4⋊Q8—21no—
305C4⋊Q8D4×C10911yescharacteristic
306#127—21no—
307#127D4×C10911yescharacteristic
308#124—24no—
309#124—22no—
310#124—21no—
311#124—21no—
312#124—21no—
313#124—21no—
314#124D4×C10911yescharacteristic
315#124D4×C10911yescharacteristic
316#123—24no—
317#123—22no—
318C22⋊Q8—24no—
319C22⋊Q8—22no—
320#121—24no—
321#121—24no—
322#121—22no—
323#121—22no—
324#120—24no—
325#120D4×C10914yes—
326C4×Q8—21no—
327C4×D4—24no—
328C4×D4—21no—
329C4×D4—22no—
330C4×D4—22no—
331#117—21no—
332#117—22no—
333#117C2×C4×C10911yescharacteristic
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C42—21no—
337C2×C42C2×C4×C10911yescharacteristic
338#104—44no—
339#103—44no—
340#101—22no—
341#100—22no—
342#99—22no—
343#99—22no—
344C24—22no—
345C24C22⋊C4×C10912yes—
346#41—22no—
347#41—22no—
348#41—24no—
349#41—41no—
350#41—41no—
351#41—42no—
352#41—44no—
353#41—44no—
354#41—41no—
355#41—41no—
356#41—48no—
357#41—42no—
358#41—42no—
359C2×Q8—21no—
360C2×Q8—22no—
361C2×Q8—24no—
362C2×Q8—41no—
363C2×Q8—41no—
364C2×Q8—42no—
365C2×Q8—44no—
366C2×Q8C22⋊C4×C10912yes—
367C2×Q8C22⋊C4×C10911yescharacteristic
368C2×Q8C22⋊C4×C10911yescharacteristic
369C2×Q8C2×D4×C10911yescharacteristic
370C2×D4—21no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—21no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—24no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—44no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—48no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4C22⋊C4×C10911yescharacteristic
386C2×D4C22⋊C4×C10911yescharacteristic
387C2×D4C2×D4×C10911yescharacteristic
388C2×D4C2×D4×C10912yes—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4C22×C4×C10911yescharacteristic
398C22×C4C2×D4×C10912yes—
399M16—42no—
400M16—42no—
401C2×C8—42no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—42no—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—24no—
414C22⋊C4—42no—
415C22⋊C4C22⋊C4×C10912yes—
416C42—22no—
417C42—21no—
418C42—21no—
419C42—42no—
420C42C22⋊C4×C10911yescharacteristic
421C42C22⋊C4×C10911yescharacteristic
422C23—21no—
423C23—22no—
424C23—21no—
425C23—44no—
426C23—42no—
427C23—82no—
428C23—84no—
429C23C2×C22⋊C4×C10911yescharacteristic
430C23#120×C10912yes—
431Q8—21no—
432Q8—22no—
433Q8—22no—
434Q8—41no—
435Q8—44no—
436Q8—41no—
437Q8—42no—
438Q8#99×C10912yes—
439D4—24no—
440D4—21no—
441D4—42no—
442D4—41no—
443D4—41no—
444D4—42no—
445D4—44no—
446D4—44no—
447D4—42no—
448D4—44no—
449D4—41no—
450D4—41no—
451D4—82no—
452D4—84no—
453D4#99×C10912yes—
454C2×C4—24no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4C2×C22⋊C4×C10911yescharacteristic
475C2×C4C2×C22⋊C4×C10911yescharacteristic
476C2×C4#120×C10912yes—
477C8—82no—
478C22—22no—
479C22—21no—
480C22—22no—
481C22—41no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—44no—
486C22—42no—
487C22—41no—
488C22—162no—
489C22—164no—
490C22#379×C10911yescharacteristic
491C4—22no—
492C4—21no—
493C4—22no—
494C4—41no—
495C4—42no—
496C4—41no—
497C4—41no—
498C4—42no—
499C4—162no—
500C4C2×#99×C10911yescharacteristic
501C4C2×#99×C10911yescharacteristic
502C2—21no—
503C2—42no—
504C2—42no—
505C2—41no—
506C2—81no—
507C2—82no—
508C2C109×#178111yescharacteristic
subgroupG/H
center, socleC2×C109#1781
commutatorC22×C4C22×C4×C109
Frattini, cosocleC2×#41C23×C109
prime radicalC109×#3500C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C109 ◁ C2×C4×C109 ◁ C2×2−1+4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#2132C2semidirect
3. C109×#2126C2semidirect
4. C109×#2122C2semidirect
5. C109×#1796C2semidirect
7. C2×H(5,𝔽2)×C109C4semidirect
12. #536×C109C4semidirect
18. C2×#124×C109C4semidirect
19. C4×#41×C109C4semidirect
22. #413×C109C22semidirect
23. #410×C109C22semidirect
247. #12715C109direct (N × H)
256. #2132C2×C109semidirect
257. #2126C2×C109semidirect
258. #2122C2×C109semidirect
259. #1796C2×C109semidirect
260. C109#12715direct (N × H)
262. C2×H(5,𝔽2)C4×C109semidirect
267. #536C4×C109semidirect
273. C2×#124C4×C109semidirect
274. C4×#41C4×C109semidirect
277. #413C22×C109semidirect
278. #410C22×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#35002
C109×#21322
C109×#21262
C109×#21222
C109×#17962
#12715109