Finite Group Databasebeta

← C109×#32219 #42150213

Subgroups — C109×#32219 #42150213

Order 27904.

636 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#2494C211yesmaximal, characteristic
2C109×#2489C211yesmaximal, characteristic
3C109×#2076C211yesmaximal, characteristic
4C109×#2074C211yesmaximal, characteristic, complemented
5C109×#2068C211yesmaximal, characteristic, complemented
6C109×#2068C211yesmaximal, characteristic, complemented
7C109×#2059C211yesmaximal, characteristic, complemented
8C109×#2035C211yesmaximal, characteristic
9C109×#1989C211yesmaximal, characteristic, complemented
10C109×#1949C211yesmaximal, characteristic, complemented
11C109×#1927C211yesmaximal, characteristic, complemented
12C109×#1919C211yesmaximal, characteristic, complemented
13C109×#1894C211yesmaximal, characteristic
14C109×#1851C211yesmaximal, characteristic
15C109×#1809C211yesmaximal, characteristic
16C22×M16×C109C2211yescharacteristic
17C2×#125×C109C2211yescharacteristic
18C2×C22⋊Q8×C109C2211yescharacteristic
19C2×#107×C109C2211yescharacteristic
20C2×#107×C109C2211yescharacteristic
21C2×#106×C109C2211yescharacteristic
22C2×#106×C109C2211yescharacteristic
23C2×C4⋊C8×C109C2211yescharacteristic
24C2×C4⋊C8×C109C2211yescharacteristic
25C2×#103×C109C2211yescharacteristic
26C2×#103×C109C2211yescharacteristic
27C2×#103×C109C2211yescharacteristic
28C2×#103×C109C2211yescharacteristic
29C2×C22⋊C8×C109C2211yescharacteristic
30#401×C109—21no—
31#400×C109—21no—
32#397×C109—21no—
33#397×C109—21no—
34#391×C109C2211yescharacteristic
35#391×C109C2211yescharacteristic
36#389×C109C2211yescharacteristic
37#389×C109C2211yescharacteristic
38#388×C109—21no—
39#388×C109—21no—
40#387×C109—21no—
41#382×C109—21no—
42#382×C109C2211yescharacteristic
43#380×C109C2211yescharacteristic
44#380×C109C2211yescharacteristic
45#380×C109C2211yescharacteristic
46C4×C4⋊C4×C109C2211yescharacteristic
47C4×C4⋊C4×C109C2211yescharacteristic
48C4×C22⋊C4×C109C2211yescharacteristic
49#340×C109C2211yescharacteristic
50#340×C109C2211yescharacteristic
51#340×C109C2211yescharacteristic
52#340×C109C2211yescharacteristic
53#340×C109C2211yescharacteristic
54#340×C109C2211yescharacteristic
55#340×C109C2211yescharacteristic
56#340×C109C2211yescharacteristic
57#336×C109C2211yescharacteristic
58#336×C109C2211yescharacteristic
59C22×Q8×C109C2311yescharacteristic
60C23×C4×C109C2311yescharacteristic
61C2×M16×C109—22no—
62C2×M16×C109—41no—
63C2×M16×C109—41no—
64C22×C8×C109C2311yescharacteristic
65C22×C8×C109C2311yescharacteristic
66C22×C8×C109C2311yescharacteristic
67C22×C8×C109C2311yescharacteristic
68#125×C109—22no—
69#125×C109—41no—
70C22⋊Q8×C109—22no—
71C22⋊Q8×C109—41no—
72C2×C4⋊C4×C109—21no—
73C2×C4⋊C4×C109—21no—
74C2×C4⋊C4×C109—21no—
75C2×C4⋊C4×C109—21no—
76C2×C4⋊C4×C109—21no—
77C2×C4⋊C4×C109C2311yescharacteristic
78C2×C4⋊C4×C109C2311yescharacteristic
79C2×C4⋊C4×C109C2311yescharacteristic
80C2×C4⋊C4×C109C2311yescharacteristic
81C2×C4⋊C4×C109C2311yescharacteristic
82C2×C4⋊C4×C109C2311yescharacteristic
83C2×C4⋊C4×C109C2311yescharacteristic
84C2×C22⋊C4×C109—21no—
85C2×C22⋊C4×C109—21no—
86C2×C22⋊C4×C109—21no—
87C2×C22⋊C4×C109—21no—
88C2×C22⋊C4×C109D411yescharacteristic
89C2×C22⋊C4×C109D411yescharacteristic
90C2×C42×C109—21no—
91C2×C42×C109—21no—
92C2×C42×C109C2311yescharacteristic
93C2×C42×C109C2311yescharacteristic
94#107×C109—41no—
95#107×C109—41no—
96#106×C109—22no—
97#106×C109—41no—
98C4⋊C8×C109—22no—
99C4⋊C8×C109—41no—
100#103×C109—22no—
101#103×C109—41no—
102#103×C109—41no—
103#103×C109—41no—
104C22⋊C8×C109—41no—
105#95×C109—21no—
106#95×C109—21no—
107#95×C109—21no—
108#95×C109—21no—
109#95×C109—21no—
110#95×C109—21no—
111#95×C109—21no—
112#95×C109—21no—
113#95×C109—21no—
114#95×C109—21no—
115#95×C109—21no—
116#95×C109—21no—
117#95×C109—21no—
118#95×C109D411yescharacteristic
119#95×C109D411yescharacteristic
120C24×C109C2×D411yescharacteristic
121C2×Q8×C109—21no—
122C2×Q8×C109—21no—
123C2×Q8×C109—21no—
124C2×Q8×C109—21no—
125C2×Q8×C109—21no—
126C2×Q8×C109—21no—
127C22×C4×C109—21no—
128C22×C4×C109—21no—
129C22×C4×C109—21no—
130C22×C4×C109—21no—
131C22×C4×C109—21no—
132C22×C4×C109—21no—
133C22×C4×C109—21no—
134C22×C4×C109—21no—
135C22×C4×C109—21no—
136C22×C4×C109—21no—
137C22×C4×C109—21no—
138C22×C4×C109—21no—
139C22×C4×C109—21no—
140C22×C4×C109—21no—
141C22×C4×C109C2×D411yescharacteristic
142C22×C4×C109C2×D411yescharacteristic
143C22×C4×C109C2×D411yescharacteristic
144C22×C4×C109C2×D411yescharacteristic
145C22×C4×C109C2×D411yescharacteristic
146C22×C4×C109C2411yescharacteristic
147M16×C109—44no—
148C2×C8×C109—21no—
149C2×C8×C109—21no—
150C2×C8×C109—21no—
151C2×C8×C109—21no—
152C2×C8×C109—21no—
153C2×C8×C109—21no—
154C2×C8×C109—21no—
155C2×C8×C109—21no—
156C2×C8×C109—21no—
157C2×C8×C109—21no—
158C2×C8×C109—21no—
159C2×C8×C109#4112yes—
160C4⋊C4×C109—21no—
161C4⋊C4×C109—21no—
162C4⋊C4×C109—21no—
163C4⋊C4×C109—21no—
164C4⋊C4×C109—21no—
165C4⋊C4×C109—21no—
166C4⋊C4×C109—21no—
167C4⋊C4×C109—21no—
168C4⋊C4×C109—21no—
169C4⋊C4×C109—21no—
170C4⋊C4×C109—21no—
171C4⋊C4×C109—21no—
172C4⋊C4×C109—21no—
173C4⋊C4×C109—42no—
174C4⋊C4×C109—42no—
175C4⋊C4×C109—42no—
176C4⋊C4×C109—42no—
177C4⋊C4×C109—81no—
178C4⋊C4×C109#4112yes—
179C22⋊C4×C109—41no—
180C22⋊C4×C109—41no—
181C22⋊C4×C109—42no—
182C22⋊C4×C109—42no—
183C22⋊C4×C109—42no—
184C22⋊C4×C109—81no—
185C42×C109—21no—
186C42×C109—21no—
187C42×C109—21no—
188C42×C109—21no—
189C42×C109—42no—
190C42×C109—81no—
191C23×C109—21no—
192C23×C109—21no—
193C23×C109—21no—
194C23×C109—21no—
195C23×C109—21no—
196C23×C109—21no—
197C23×C109—21no—
198C23×C109C22×D411yescharacteristic
199Q8×C109—41no—
200Q8×C109—41no—
201Q8×C109—41no—
202Q8×C109—41no—
203C2×C4×C109—21no—
204C2×C4×C109—21no—
205C2×C4×C109—21no—
206C2×C4×C109—22no—
207C2×C4×C109—21no—
208C2×C4×C109—21no—
209C2×C4×C109—21no—
210C2×C4×C109—21no—
211C2×C4×C109—21no—
212C2×C4×C109—21no—
213C2×C4×C109—21no—
214C2×C4×C109—21no—
215C2×C4×C109—21no—
216C2×C4×C109—21no—
217C2×C4×C109—21no—
218C2×C4×C109—21no—
219C2×C4×C109—41no—
220C2×C4×C109—41no—
221C2×C4×C109—41no—
222C2×C4×C109—41no—
223C2×C4×C109—41no—
224C2×C4×C109—41no—
225C2×C4×C109—41no—
226C2×C4×C109—41no—
227C2×C4×C109—41no—
228C2×C4×C109—41no—
229C2×C4×C109—41no—
230C2×C4×C109—41no—
231C2×C4×C109—41no—
232C2×C4×C109—41no—
233C2×C4×C109—41no—
234C2×C4×C109—41no—
235C2×C4×C109—41no—
236C2×C4×C109—41no—
237C2×C4×C109—41no—
238C2×C4×C109—41no—
239C2×C4×C109—41no—
240C2×C4×C109—41no—
241C2×C4×C109—41no—
242C2×C4×C109—41no—
243C2×C4×C109—41no—
244C2×C4×C109—41no—
245C2×C4×C109—41no—
246C2×C4×C109C2×#4111yescharacteristic
247C2×C4×C109C2×#4111yescharacteristic
248C2×C4×C109H(5,𝔽2)11yescharacteristic
249C2×C4×C109H(5,𝔽2)11yescharacteristic
250C2×C4×C1092−1+411yescharacteristic
251C2×C4×C1092−1+411yescharacteristic
252C8×C109—22no—
253C8×C109—41no—
254C8×C109—41no—
255C8×C109—41no—
256C22×C109—41no—
257C22×C109—41no—
258C22×C109—41no—
259C22×C109—41no—
260C22×C109—41no—
261C22×C109—41no—
262C22×C109—41no—
263C22×C109#53711yescharacteristic
264C22×C109#54611yescharacteristic
265C22×C109#54711yescharacteristic
266C22×C109#57511yescharacteristic
267C22×C109#57511yescharacteristic
268C22×C109#57711yescharacteristic
269C22×C109#57811yescharacteristic
270C4×C109—41no—
271C4×C109—41no—
272C4×C109—41no—
273C4×C109—41no—
274C4×C109—42no—
275C4×C109—41no—
276C4×C109—81no—
277C4×C109—81no—
278C4×C109—81no—
279C4×C109—81no—
280C4×C109—81no—
281C4×C109—81no—
282C4×C109—81no—
283C4×C109#53811yescharacteristic
284C4×C109#54211yescharacteristic
285C4×C109#55611yescharacteristic
286C4×C109#56311yescharacteristic
287#32219C10911yesmaximal, characteristic, direct product factor, Sylow
288C2×C109—81no—
289C2×C109#271511yescharacteristic
290C2×C109#325611yescharacteristic
291C2×C109#325811yescharacteristic
292C2×C109#331511yescharacteristic
293C2×C109#332611yescharacteristic
294C2×C109#334911yescharacteristic
295C2×C109#335111yescharacteristic
296#2494C2×C10911yescharacteristic
297#2489C2×C10911yescharacteristic
298#2076C2×C10911yescharacteristic
299#2074C2×C10911yescharacteristic, complemented
300#2068C2×C10911yescharacteristic, complemented
301#2068C2×C10911yescharacteristic, complemented
302#2059C2×C10911yescharacteristic, complemented
303#2035C2×C10911yescharacteristic
304#1989C2×C10911yescharacteristic, complemented
305#1949C2×C10911yescharacteristic, complemented
306#1927C2×C10911yescharacteristic, complemented
307#1919C2×C10911yescharacteristic, complemented
308#1894C2×C10911yescharacteristic
309#1851C2×C10911yescharacteristic
310#1809C2×C10911yescharacteristic
311C109#3221911yescharacteristic, direct product factor, Sylow
312C22×M16C22×C10911yescharacteristic
313C2×#125C22×C10911yescharacteristic
314C2×C22⋊Q8C22×C10911yescharacteristic
315C2×#107C22×C10911yescharacteristic
316C2×#107C22×C10911yescharacteristic
317C2×#106C22×C10911yescharacteristic
318C2×#106C22×C10911yescharacteristic
319C2×C4⋊C8C22×C10911yescharacteristic
320C2×C4⋊C8C22×C10911yescharacteristic
321C2×#103C22×C10911yescharacteristic
322C2×#103C22×C10911yescharacteristic
323C2×#103C22×C10911yescharacteristic
324C2×#103C22×C10911yescharacteristic
325C2×C22⋊C8C22×C10911yescharacteristic
326#401—21no—
327#400—21no—
328#397—21no—
329#397—21no—
330#391C22×C10911yescharacteristic
331#391C22×C10911yescharacteristic
332#389C22×C10911yescharacteristic
333#389C22×C10911yescharacteristic
334#388—21no—
335#388—21no—
336#387—21no—
337#382—21no—
338#382C22×C10911yescharacteristic
339#380C22×C10911yescharacteristic
340#380C22×C10911yescharacteristic
341#380C22×C10911yescharacteristic
342C4×C4⋊C4C22×C10911yescharacteristic
343C4×C4⋊C4C22×C10911yescharacteristic
344C4×C22⋊C4C22×C10911yescharacteristic
345#340C22×C10911yescharacteristic
346#340C22×C10911yescharacteristic
347#340C22×C10911yescharacteristic
348#340C22×C10911yescharacteristic
349#340C22×C10911yescharacteristic
350#340C22×C10911yescharacteristic
351#340C22×C10911yescharacteristic
352#340C22×C10911yescharacteristic
353#336C22×C10911yescharacteristic
354#336C22×C10911yescharacteristic
355C22×Q8C23×C10911yescharacteristic
356C23×C4C23×C10911yescharacteristic
357C2×M16—22no—
358C2×M16—41no—
359C2×M16—41no—
360C22×C8C23×C10911yescharacteristic
361C22×C8C23×C10911yescharacteristic
362C22×C8C23×C10911yescharacteristic
363C22×C8C23×C10911yescharacteristic
364#125—22no—
365#125—41no—
366C22⋊Q8—22no—
367C22⋊Q8—41no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4C23×C10911yescharacteristic
374C2×C4⋊C4C23×C10911yescharacteristic
375C2×C4⋊C4C23×C10911yescharacteristic
376C2×C4⋊C4C23×C10911yescharacteristic
377C2×C4⋊C4C23×C10911yescharacteristic
378C2×C4⋊C4C23×C10911yescharacteristic
379C2×C4⋊C4C23×C10911yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4D4×C10911yescharacteristic
385C2×C22⋊C4D4×C10911yescharacteristic
386C2×C42—21no—
387C2×C42—21no—
388C2×C42C23×C10911yescharacteristic
389C2×C42C23×C10911yescharacteristic
390#107—41no—
391#107—41no—
392#106—22no—
393#106—41no—
394C4⋊C8—22no—
395C4⋊C8—41no—
396#103—22no—
397#103—41no—
398#103—41no—
399#103—41no—
400C22⋊C8—41no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95D4×C10911yescharacteristic
415#95D4×C10911yescharacteristic
416C24C2×D4×C10911yescharacteristic
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4C2×D4×C10911yescharacteristic
438C22×C4C2×D4×C10911yescharacteristic
439C22×C4C2×D4×C10911yescharacteristic
440C22×C4C2×D4×C10911yescharacteristic
441C22×C4C2×D4×C10911yescharacteristic
442C22×C4C24×C10911yescharacteristic
443M16—44no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8#41×C10912yes—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—81no—
474C4⋊C4#41×C10912yes—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—81no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—42no—
486C42—81no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23C22×D4×C10911yescharacteristic
495Q8—41no—
496Q8—41no—
497Q8—41no—
498Q8—41no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4C2×#41×C10911yescharacteristic
543C2×C4C2×#41×C10911yescharacteristic
544C2×C4H(5,𝔽2)×C10911yescharacteristic
545C2×C4H(5,𝔽2)×C10911yescharacteristic
546C2×C42−1+4×C10911yescharacteristic
547C2×C42−1+4×C10911yescharacteristic
548C8—22no—
549C8—41no—
550C8—41no—
551C8—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22#537×C10911yescharacteristic
560C22#546×C10911yescharacteristic
561C22#547×C10911yescharacteristic
562C22#575×C10911yescharacteristic
563C22#575×C10911yescharacteristic
564C22#577×C10911yescharacteristic
565C22#578×C10911yescharacteristic
566C4—41no—
567C4—41no—
568C4—42no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—81no—
579C4#538×C10911yescharacteristic
580C4#542×C10911yescharacteristic
581C4#556×C10911yescharacteristic
582C4#563×C10911yescharacteristic
583C2—81no—
584C2C109×#271511yescharacteristic
585C2C109×#325611yescharacteristic
586C2C109×#325811yescharacteristic
587C2C109×#331511yescharacteristic
588C2C109×#332611yescharacteristic
589C2C109×#334911yescharacteristic
590C2C109×#335111yescharacteristic
subgroupG/H
center, socleC23×C109C22×D4
commutator, Frattini, cosocleC22×C4C24×C109
prime radicalC24×C109C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C109 ◁ C4×C22⋊C4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
4. C109×#2074C2semidirect
5. C109×#2068C2semidirect
6. C109×#2068C2semidirect
7. C109×#2059C2semidirect
9. C109×#1989C2semidirect
10. C109×#1949C2semidirect
11. C109×#1927C2semidirect
12. C109×#1919C2semidirect
287. #32219C109direct (N × H)
299. #2074C2×C109semidirect
300. #2068C2×C109semidirect
301. #2068C2×C109semidirect
302. #2059C2×C109semidirect
304. #1989C2×C109semidirect
305. #1949C2×C109semidirect
306. #1927C2×C109semidirect
307. #1919C2×C109semidirect
311. C109#32219direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#24942
C109×#24892
C109×#20762
C109×#20742
C109×#20682
C109×#20592
C109×#20352
C109×#19892
C109×#19492
C109×#19272
C109×#19192
C109×#18942
C109×#18512
C109×#18092
#32219109