Finite Group Databasebeta

← C109×#12757 #42130751

Subgroups — C109×#12757 #42130751

Order 27904.

736 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3538C211yesmaximal, characteristic
2C109×#2135C211yesmaximal, characteristic, complemented
3C109×#2123C211yesmaximal, characteristic, complemented
4C109×#2120C211yesmaximal, characteristic
5C109×#1915C211yesmaximal, characteristic, complemented
6C109×#1909C211yesmaximal, characteristic, complemented
7C109×#1891C211yesmaximal, characteristic
8#568×C109C2211yescharacteristic
9#563×C109—21no—
10#563×C109C412yescomplemented
11#552×C109—21no—
12#552×C109—21no—
13#551×C109—21no—
14#551×C109C412yescomplemented
15D4×Q8×C109—21no—
16D4×Q8×C109—21no—
17#544×C109—22no—
18#541×C109—21no—
19#539×C109—22no—
20#536×C109C2211yescharacteristic
21#536×C109C2211yescharacteristic
22#525×C109C2211yescharacteristic
23C2×C22⋊Q8×C109—21no—
24#419×C109—21no—
25#413×C109C2211yescharacteristic
26#413×C109C2211yescharacteristic
27C2×#101×C109—21no—
28#410×C109C2211yescharacteristic
29#407×C109—21no—
30#356×C109—22no—
31#352×C109—21no—
32#352×C109—22no—
33AΣL(1,𝔽16)×C109—21no—
34#344×C109—21no—
35#344×C109—21no—
36H(3,ℤ/4)×C109—21no—
37C2×#41×C109C2311yescharacteristic
38C22×Q8×C109—21no—
39C22×Q8×C109D411yescharacteristic
40C22×Q8×C109D411yescharacteristic
41C23×C4×C109D411yescharacteristic
42#131×C109—42no—
43C2×M16×C109—21no—
44C2×M16×C109—21no—
45C2×M16×C109—21no—
46C2×M16×C109—21no—
47C2×M16×C109—21no—
48C4⋊Q8×C109—21no—
49C4⋊Q8×C109—22no—
50C4⋊Q8×C109—22no—
51C4⋊Q8×C109—21no—
52C4⋊Q8×C109D412yes—
53#126×C109—22no—
54#126×C109—22no—
55#126×C109—22no—
56#126×C109—22no—
57#125×C109—22no—
58#125×C109—22no—
59#125×C109—21no—
60#125×C109—22no—
61#125×C109—22no—
62#124×C109—21no—
63#124×C109C2×C412yes—
64#123×C109—22no—
65#123×C109—21no—
66#123×C109—21no—
67#123×C109—22no—
68#123×C109C2×C411yescharacteristic
69#123×C109C2×C412yes—
70#123×C109D411yescharacteristic
71#123×C109D412yes—
72C22⋊Q8×C109—22no—
73C22⋊Q8×C109—21no—
74C22⋊Q8×C109—21no—
75C22⋊Q8×C109—22no—
76C22⋊Q8×C109—22no—
77C22⋊Q8×C109—22no—
78C22⋊Q8×C109—21no—
79C22⋊Q8×C109—22no—
80C22⋊Q8×C109—22no—
81C22⋊Q8×C109—21no—
82C22⋊Q8×C109—21no—
83C22⋊Q8×C109—21no—
84C22⋊Q8×C109—21no—
85#121×C109—21no—
86#120×C109C2×C411yescharacteristic
87#120×C109D411yescharacteristic
88C4×Q8×C109—21no—
89C4×Q8×C109—21no—
90C4×D4×C109—22no—
91C4×D4×C109—22no—
92C4×D4×C109—21no—
93C4×D4×C109—21no—
94#117×C109—22no—
95#117×C109—22no—
96#117×C109—21no—
97#117×C109D411yescharacteristic
98#117×C109D411yescharacteristic
99#117×C109D411yescharacteristic
100C2×C4⋊C4×C109—22no—
101C2×C4⋊C4×C109—21no—
102C2×C22⋊C4×C109—22no—
103#103×C109—42no—
104#101×C109—21no—
105#101×C109—21no—
106#101×C109—42no—
107#100×C109—21no—
108#100×C109—21no—
109#99×C109—21no—
110#99×C109—21no—
111C22⋊C8×C109—42no—
112C24×C109C22⋊C411yescharacteristic
113#41×C109—41no—
114#41×C109—41no—
115C2×Q8×C109—22no—
116C2×Q8×C109—21no—
117C2×Q8×C109—21no—
118C2×Q8×C109—21no—
119C2×Q8×C109—41no—
120C2×Q8×C109—41no—
121C2×Q8×C109—41no—
122C2×Q8×C109—41no—
123C2×Q8×C109—81no—
124C2×Q8×C109—81no—
125C2×Q8×C109C22⋊C412yes—
126C2×Q8×C109C2×D411yescharacteristic
127C2×D4×C109—21no—
128C2×D4×C109—21no—
129C2×D4×C109C22×C411yescharacteristic
130C2×D4×C109C2×D411yescharacteristic
131C2×D4×C109C2×D411yescharacteristic
132C22×C4×C109—22no—
133C22×C4×C109—22no—
134C22×C4×C109—21no—
135C22×C4×C109—21no—
136C22×C4×C109—21no—
137C22×C4×C109—41no—
138C22×C4×C109—41no—
139C22×C4×C109C22⋊C412yes—
140C22×C4×C109C22⋊C411yescharacteristic
141C22×C4×C109C2×D411yescharacteristic
142C22×C4×C109C2×D411yescharacteristic
143C22×C4×C109C2×D411yescharacteristic
144M16×C109—22no—
145M16×C109—41no—
146M16×C109—41no—
147M16×C109—41no—
148M16×C109—41no—
149M16×C109—41no—
150M16×C109—41no—
151M16×C109—41no—
152C2×C8×C109—22no—
153C2×C8×C109—22no—
154C2×C8×C109—41no—
155C2×C8×C109—41no—
156C2×C8×C109—41no—
157C4⋊C4×C109—22no—
158C4⋊C4×C109—22no—
159C4⋊C4×C109—22no—
160C4⋊C4×C109—22no—
161C4⋊C4×C109—22no—
162C4⋊C4×C109—22no—
163C4⋊C4×C109—22no—
164C4⋊C4×C109—22no—
165C4⋊C4×C109—22no—
166C4⋊C4×C109—22no—
167C4⋊C4×C109—21no—
168C4⋊C4×C109—22no—
169C4⋊C4×C109—22no—
170C4⋊C4×C109—22no—
171C4⋊C4×C109—21no—
172C4⋊C4×C109—21no—
173C4⋊C4×C109—21no—
174C4⋊C4×C109—21no—
175C4⋊C4×C109—41no—
176C22⋊C4×C109—22no—
177C22⋊C4×C109—22no—
178C22⋊C4×C109—22no—
179C22⋊C4×C109—22no—
180C22⋊C4×C109—21no—
181C22⋊C4×C109—22no—
182C22⋊C4×C109—21no—
183C22⋊C4×C109—21no—
184C22⋊C4×C109—22no—
185C22⋊C4×C109—21no—
186C22⋊C4×C109—21no—
187C22⋊C4×C109—41no—
188C22⋊C4×C109C22⋊C411yescharacteristic
189C22⋊C4×C109C22⋊C411yescharacteristic
190C22⋊C4×C109C22⋊C412yes—
191C42×C109—22no—
192C42×C109—22no—
193C42×C109—21no—
194C42×C109—21no—
195C42×C109—41no—
196C42×C109C22⋊C412yes—
197C23×C109—21no—
198C23×C109—41no—
199C23×C109—42no—
200C23×C109C2×C22⋊C411yescharacteristic
201C23×C109C2×C22⋊C411yescharacteristic
202C23×C109#12011yescharacteristic
203Q8×C109—41no—
204Q8×C109—42no—
205Q8×C109—41no—
206Q8×C109—41no—
207Q8×C109—41no—
208Q8×C109—41no—
209Q8×C109—81no—
210Q8×C109—82no—
211D4×C109—41no—
212D4×C109—41no—
213D4×C109—41no—
214D4×C109—42no—
215D4×C109—42no—
216C2×C4×C109—21no—
217C2×C4×C109—21no—
218C2×C4×C109—21no—
219C2×C4×C109—21no—
220C2×C4×C109—21no—
221C2×C4×C109—21no—
222C2×C4×C109—21no—
223C2×C4×C109—21no—
224C2×C4×C109—22no—
225C2×C4×C109—22no—
226C2×C4×C109—22no—
227C2×C4×C109—22no—
228C2×C4×C109—21no—
229C2×C4×C109—44no—
230C2×C4×C109—41no—
231C2×C4×C109—41no—
232C2×C4×C109—41no—
233C2×C4×C109—42no—
234C2×C4×C109—81no—
235C2×C4×C109—81no—
236C2×C4×C109—82no—
237C2×C4×C109—81no—
238C2×C4×C109C2×C22⋊C411yescharacteristic
239C2×C4×C109#12011yescharacteristic
240C2×C4×C109#12011yescharacteristic
241C2×C4×C109#12011yescharacteristic
242C8×C109—42no—
243C8×C109—81no—
244C8×C109—81no—
245C22×C109—22no—
246C22×C109—21no—
247C22×C109—21no—
248C22×C109—21no—
249C22×C109—41no—
250C22×C109—41no—
251C22×C109—41no—
252C22×C109—41no—
253C22×C109—42no—
254C22×C109—41no—
255C22×C109—81no—
256C22×C109#37911yescharacteristic
257C4×C109—21no—
258C4×C109—21no—
259C4×C109—21no—
260C4×C109—42no—
261C4×C109—42no—
262C4×C109—41no—
263C4×C109—41no—
264C4×C109—41no—
265C4×C109—42no—
266C4×C109—42no—
267C4×C109—161no—
268C4×C109B(4,𝔽2)11yescharacteristic
269C4×C109#45811yescharacteristic
270#12757C10911yesmaximal, characteristic, direct product factor, Sylow
271C2×C109—21no—
272C2×C109—41no—
273C2×C109—41no—
274C2×C109—42no—
275C2×C109—41no—
276C2×C109#179011yescharacteristic
277#3538C2×C10911yescharacteristic
278#2135C2×C10911yescharacteristic, complemented
279#2123C2×C10911yescharacteristic, complemented
280#2120C2×C10911yescharacteristic
281#1915C2×C10911yescharacteristic, complemented
282#1909C2×C10911yescharacteristic, complemented
283#1891C2×C10911yescharacteristic
284C109#1275711yescharacteristic, direct product factor, Sylow
285#568C22×C10911yescharacteristic
286#563—21no—
287#563C4×C10912yescomplemented
288#552—21no—
289#552—21no—
290#551—21no—
291#551C4×C10912yescomplemented
292D4×Q8—21no—
293D4×Q8—21no—
294#544—22no—
295#541—21no—
296#539—22no—
297#536C22×C10911yescharacteristic
298#536C22×C10911yescharacteristic
299#525C22×C10911yescharacteristic
300C2×C22⋊Q8—21no—
301#419—21no—
302#413C22×C10911yescharacteristic
303#413C22×C10911yescharacteristic
304C2×#101—21no—
305#410C22×C10911yescharacteristic
306#407—21no—
307#356—22no—
308#352—21no—
309#352—22no—
310AΣL(1,𝔽16)—21no—
311#344—21no—
312#344—21no—
313H(3,ℤ/4)—21no—
314C2×#41C23×C10911yescharacteristic
315C22×Q8—21no—
316C22×Q8D4×C10911yescharacteristic
317C22×Q8D4×C10911yescharacteristic
318C23×C4D4×C10911yescharacteristic
319#131—42no—
320C2×M16—21no—
321C2×M16—21no—
322C2×M16—21no—
323C2×M16—21no—
324C2×M16—21no—
325C4⋊Q8—21no—
326C4⋊Q8—22no—
327C4⋊Q8—22no—
328C4⋊Q8—21no—
329C4⋊Q8D4×C10912yes—
330#126—22no—
331#126—22no—
332#126—22no—
333#126—22no—
334#125—22no—
335#125—22no—
336#125—21no—
337#125—22no—
338#125—22no—
339#124—21no—
340#124C2×C4×C10912yes—
341#123—22no—
342#123—21no—
343#123—21no—
344#123—22no—
345#123C2×C4×C10911yescharacteristic
346#123C2×C4×C10912yes—
347#123D4×C10911yescharacteristic
348#123D4×C10912yes—
349C22⋊Q8—22no—
350C22⋊Q8—21no—
351C22⋊Q8—21no—
352C22⋊Q8—22no—
353C22⋊Q8—22no—
354C22⋊Q8—22no—
355C22⋊Q8—21no—
356C22⋊Q8—22no—
357C22⋊Q8—22no—
358C22⋊Q8—21no—
359C22⋊Q8—21no—
360C22⋊Q8—21no—
361C22⋊Q8—21no—
362#121—21no—
363#120C2×C4×C10911yescharacteristic
364#120D4×C10911yescharacteristic
365C4×Q8—21no—
366C4×Q8—21no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—21no—
370C4×D4—21no—
371#117—22no—
372#117—22no—
373#117—21no—
374#117D4×C10911yescharacteristic
375#117D4×C10911yescharacteristic
376#117D4×C10911yescharacteristic
377C2×C4⋊C4—22no—
378C2×C4⋊C4—21no—
379C2×C22⋊C4—22no—
380#103—42no—
381#101—21no—
382#101—21no—
383#101—42no—
384#100—21no—
385#100—21no—
386#99—21no—
387#99—21no—
388C22⋊C8—42no—
389C24C22⋊C4×C10911yescharacteristic
390#41—41no—
391#41—41no—
392C2×Q8—22no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—41no—
397C2×Q8—41no—
398C2×Q8—41no—
399C2×Q8—41no—
400C2×Q8—81no—
401C2×Q8—81no—
402C2×Q8C22⋊C4×C10912yes—
403C2×Q8C2×D4×C10911yescharacteristic
404C2×D4—21no—
405C2×D4—21no—
406C2×D4C22×C4×C10911yescharacteristic
407C2×D4C2×D4×C10911yescharacteristic
408C2×D4C2×D4×C10911yescharacteristic
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4C22⋊C4×C10912yes—
417C22×C4C22⋊C4×C10911yescharacteristic
418C22×C4C2×D4×C10911yescharacteristic
419C22×C4C2×D4×C10911yescharacteristic
420C22×C4C2×D4×C10911yescharacteristic
421M16—22no—
422M16—41no—
423M16—41no—
424M16—41no—
425M16—41no—
426M16—41no—
427M16—41no—
428M16—41no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—41no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—21no—
458C22⋊C4—22no—
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—22no—
462C22⋊C4—21no—
463C22⋊C4—21no—
464C22⋊C4—41no—
465C22⋊C4C22⋊C4×C10911yescharacteristic
466C22⋊C4C22⋊C4×C10911yescharacteristic
467C22⋊C4C22⋊C4×C10912yes—
468C42—22no—
469C42—22no—
470C42—21no—
471C42—21no—
472C42—41no—
473C42C22⋊C4×C10912yes—
474C23—21no—
475C23—41no—
476C23—42no—
477C23C2×C22⋊C4×C10911yescharacteristic
478C23C2×C22⋊C4×C10911yescharacteristic
479C23#120×C10911yescharacteristic
480Q8—41no—
481Q8—42no—
482Q8—41no—
483Q8—41no—
484Q8—41no—
485Q8—41no—
486Q8—81no—
487Q8—82no—
488D4—41no—
489D4—41no—
490D4—41no—
491D4—42no—
492D4—42no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—44no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—82no—
514C2×C4—81no—
515C2×C4C2×C22⋊C4×C10911yescharacteristic
516C2×C4#120×C10911yescharacteristic
517C2×C4#120×C10911yescharacteristic
518C2×C4#120×C10911yescharacteristic
519C8—42no—
520C8—81no—
521C8—81no—
522C22—22no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—42no—
531C22—41no—
532C22—81no—
533C22#379×C10911yescharacteristic
534C4—21no—
535C4—21no—
536C4—21no—
537C4—42no—
538C4—42no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—42no—
543C4—42no—
544C4—161no—
545C4B(4,𝔽2)×C10911yescharacteristic
546C4#458×C10911yescharacteristic
547C2—21no—
548C2—41no—
549C2—42no—
550C2—41no—
551C2—41no—
552C2C109×#179011yescharacteristic
subgroupG/H
center, socleC2×C109#1790
commutatorC2×D4C22×C4×C109
Frattini, cosocleC2×#41C23×C109
prime radical#525×C109C22
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C109 ◁ C2×C4×C109 ◁ C2×#41×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#2135C2semidirect
3. C109×#2123C2semidirect
5. C109×#1915C2semidirect
6. C109×#1909C2semidirect
10. #563×C109C4semidirect
14. #551×C109C4semidirect
270. #12757C109direct (N × H)
278. #2135C2×C109semidirect
279. #2123C2×C109semidirect
281. #1915C2×C109semidirect
282. #1909C2×C109semidirect
284. C109#12757direct (N × H)
287. #563C4×C109semidirect
291. #551C4×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#35382
C109×#21352
C109×#21232
C109×#21202
C109×#19152
C109×#19092
C109×#18912
#12757109