Finite Group Databasebeta

← C109×#13422 #42131416

Subgroups — C109×#13422 #42131416

Order 27904.

734 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#2669C211yesmaximal, characteristic
2C2×#353×C109C211yesmaximal, characteristic, complemented
3C2×AΣL(1,𝔽16)×C109C211yesmaximal, characteristic, complemented
4C109×#1889C211yesmaximal, characteristic, complemented
5C109×#1887C211yesmaximal, characteristic, complemented
6C109×#1867C211yesmaximal, characteristic, complemented
7C109×#1390C211yesmaximal, characteristic, complemented
8C2×#127×C109C2211yescharacteristic
9C2×#121×C109—21no—
10C2×#120×C109—21no—
11C2×#120×C109C2211yescharacteristic
12#518×C109C2211yescharacteristic
13#418×C109—21no—
14C2×#100×C109C2211yescharacteristic
15C2×#99×C109C2211yescharacteristic
16C2×#99×C109C2211yescharacteristic
17#394×C109—21no—
18#392×C109—21no—
19#392×C109—21no—
20#388×C109—21no—
21#386×C109C2211yescharacteristic
22#385×C109—21no—
23#379×C109—21no—
24#353×C109—22no—
25AΣL(1,𝔽16)×C109—22no—
26#342×C109—21no—
27#342×C109—21no—
28#339×C109—21no—
29C25×C109D411yescharacteristic
30C22×D4×C109—21no—
31C22×D4×C109—21no—
32C22×D4×C109—21no—
33C22×D4×C109—21no—
34C22×D4×C109—21no—
35C22×D4×C109D411yescharacteristic
36C22×D4×C109C2311yescharacteristic
37C23×C4×C109D411yescharacteristic
38C2×M16×C109—21no—
39#127×C109—21no—
40#127×C109—21no—
41#127×C109—21no—
42#127×C109—21no—
43#121×C109—44no—
44#120×C109—21no—
45#120×C109—21no—
46#120×C109—21no—
47#120×C109—21no—
48#120×C109—44no—
49C4×D4×C109—41no—
50C4×D4×C109—41no—
51#117×C109—21no—
52C2×C4⋊C4×C109—21no—
53C2×C4⋊C4×C109—41no—
54C2×C22⋊C4×C109—21no—
55C2×C22⋊C4×C109—21no—
56C2×C22⋊C4×C109—21no—
57C2×C22⋊C4×C109—21no—
58C2×C22⋊C4×C109—21no—
59C2×C22⋊C4×C109—21no—
60C2×C22⋊C4×C109—21no—
61C2×C22⋊C4×C109—21no—
62C2×C22⋊C4×C109—21no—
63C2×C22⋊C4×C109—21no—
64C2×C22⋊C4×C109—41no—
65C2×C22⋊C4×C109—41no—
66C2×C22⋊C4×C109D411yescharacteristic
67C2×C22⋊C4×C109D411yescharacteristic
68C2×C42×C109D411yescharacteristic
69#102×C109—41no—
70#102×C109—41no—
71#100×C109—21no—
72#100×C109—21no—
73#99×C109—21no—
74#99×C109—21no—
75#99×C109—21no—
76#99×C109D412yescomplemented
77#95×C109—21no—
78#95×C109—21no—
79#95×C109—41no—
80C24×C109—21no—
81C24×C109—21no—
82C24×C109—21no—
83C24×C109—21no—
84C24×C109—41no—
85C24×C109—42no—
86C24×C109—41no—
87C24×C109—81no—
88C24×C109C2×D411yescharacteristic
89C24×C109C2×D411yescharacteristic
90C2×D4×C109—21no—
91C2×D4×C109—21no—
92C2×D4×C109—41no—
93C2×D4×C109—41no—
94C2×D4×C109—41no—
95C2×D4×C109—41no—
96C2×D4×C109—42no—
97C2×D4×C109—42no—
98C2×D4×C109—42no—
99C2×D4×C109—42no—
100C2×D4×C109—42no—
101C2×D4×C109—42no—
102C2×D4×C109—41no—
103C2×D4×C109—41no—
104C2×D4×C109—41no—
105C2×D4×C109—41no—
106C2×D4×C109—41no—
107C2×D4×C109—41no—
108C2×D4×C109—81no—
109C2×D4×C109—81no—
110C2×D4×C109—81no—
111C2×D4×C109—81no—
112C2×D4×C109—81no—
113C2×D4×C109—81no—
114C2×D4×C109C2×D411yescharacteristic
115C2×D4×C109#4111yescharacteristic
116C2×D4×C109#4111yescharacteristic
117C2×D4×C109#4111yescharacteristic
118C22×C4×C109—21no—
119C22×C4×C109—21no—
120C22×C4×C109—21no—
121C22×C4×C109—21no—
122C22×C4×C109—21no—
123C22×C4×C109—21no—
124C22×C4×C109—21no—
125C22×C4×C109—21no—
126C22×C4×C109—42no—
127C22×C4×C109—41no—
128C22×C4×C109—41no—
129C22×C4×C109—41no—
130C22×C4×C109C2×D411yescharacteristic
131M16×C109—41no—
132M16×C109—41no—
133C2×C8×C109—41no—
134C4⋊C4×C109—21no—
135C4⋊C4×C109—21no—
136C4⋊C4×C109—42no—
137C4⋊C4×C109—82no—
138C22⋊C4×C109—24no—
139C22⋊C4×C109—22no—
140C22⋊C4×C109—21no—
141C22⋊C4×C109—21no—
142C22⋊C4×C109—42no—
143C22⋊C4×C109—42no—
144C22⋊C4×C109—42no—
145C22⋊C4×C109—41no—
146C22⋊C4×C109—41no—
147C22⋊C4×C109—41no—
148C22⋊C4×C109—41no—
149C22⋊C4×C109—41no—
150C22⋊C4×C109—41no—
151C22⋊C4×C109—41no—
152C22⋊C4×C109—41no—
153C22⋊C4×C109—41no—
154C22⋊C4×C109—81no—
155C22⋊C4×C109—82no—
156C22⋊C4×C109—82no—
157C42×C109—21no—
158C42×C109—21no—
159C42×C109—41no—
160C23×C109—21no—
161C23×C109—21no—
162C23×C109—21no—
163C23×C109—21no—
164C23×C109—22no—
165C23×C109—41no—
166C23×C109—41no—
167C23×C109—41no—
168C23×C109—41no—
169C23×C109—41no—
170C23×C109—41no—
171C23×C109—41no—
172C23×C109—42no—
173C23×C109—41no—
174C23×C109—41no—
175C23×C109—41no—
176C23×C109—41no—
177C23×C109—42no—
178C23×C109—42no—
179C23×C109—41no—
180C23×C109—41no—
181C23×C109—41no—
182C23×C109—41no—
183C23×C109—41no—
184C23×C109—41no—
185C23×C109—81no—
186C23×C109—82no—
187C23×C109—81no—
188C23×C109—82no—
189C23×C109—84no—
190C23×C109—81no—
191C23×C109#12011yescharacteristic
192C23×C109#12111yescharacteristic
193C23×C109#12111yescharacteristic
194C23×C109#12311yescharacteristic
195C23×C109#12411yescharacteristic
196D4×C109—41no—
197D4×C109—41no—
198D4×C109—41no—
199D4×C109—41no—
200D4×C109—42no—
201D4×C109—81no—
202D4×C109—82no—
203D4×C109—82no—
204D4×C109—82no—
205D4×C109—82no—
206D4×C109—84no—
207D4×C109—82no—
208D4×C109—82no—
209D4×C109—84no—
210C2×C4×C109—21no—
211C2×C4×C109—21no—
212C2×C4×C109—22no—
213C2×C4×C109—22no—
214C2×C4×C109—41no—
215C2×C4×C109—41no—
216C2×C4×C109—41no—
217C2×C4×C109—41no—
218C2×C4×C109—42no—
219C2×C4×C109—42no—
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—81no—
229C2×C4×C109—81no—
230C2×C4×C109—81no—
231C2×C4×C109—81no—
232C2×C4×C109—81no—
233C2×C4×C109—81no—
234C2×C4×C109—81no—
235C2×C4×C109—81no—
236C2×C4×C109—81no—
237C2×C4×C109—81no—
238C2×C4×C109#12111yescharacteristic
239C2×C4×C109#12311yescharacteristic
240C8×C109—81no—
241C22×C109—21no—
242C22×C109—21no—
243C22×C109—21no—
244C22×C109—21no—
245C22×C109—21no—
246C22×C109—21no—
247C22×C109—41no—
248C22×C109—41no—
249C22×C109—41no—
250C22×C109—41no—
251C22×C109—42no—
252C22×C109—42no—
253C22×C109—41no—
254C22×C109—41no—
255C22×C109—41no—
256C22×C109—41no—
257C22×C109—44no—
258C22×C109—42no—
259C22×C109—81no—
260C22×C109—81no—
261C22×C109—81no—
262C22×C109—81no—
263C22×C109—81no—
264C22×C109—81no—
265C22×C109—81no—
266C22×C109—81no—
267C22×C109—82no—
268C22×C109—82no—
269C22×C109—82no—
270C22×C109—81no—
271C22×C109—81no—
272C22×C109—81no—
273C22×C109—81no—
274C22×C109—81no—
275C22×C109#39411yescharacteristic
276C22×C109B(4,𝔽2)11yescharacteristic
277C22×C109#45811yescharacteristic
278C4×C109—21no—
279C4×C109—21no—
280C4×C109—42no—
281C4×C109—42no—
282C4×C109—81no—
283C4×C109—81no—
284C4×C109—82no—
285C4×C109—81no—
286C4×C109—81no—
287C4×C109—161no—
288#13422C10911yesmaximal, characteristic, direct product factor, Sylow
289C2×C109—21no—
290C2×C109—21no—
291C2×C109—41no—
292C2×C109—41no—
293C2×C109—41no—
294C2×C109—41no—
295C2×C109—82no—
296C2×C109—161no—
297C2×C109#204411yescharacteristic
298C2×C109#219611yescharacteristic
299C2×C109#220011yescharacteristic
300#2669C2×C10911yescharacteristic
301C2×#353C2×C10911yescharacteristic, complemented
302C2×AΣL(1,𝔽16)C2×C10911yescharacteristic, complemented
303#1889C2×C10911yescharacteristic, complemented
304#1887C2×C10911yescharacteristic, complemented
305#1867C2×C10911yescharacteristic, complemented
306#1390C2×C10911yescharacteristic, complemented
307C109#1342211yescharacteristic, direct product factor, Sylow
308C2×#127C22×C10911yescharacteristic
309C2×#121—21no—
310C2×#120—21no—
311C2×#120C22×C10911yescharacteristic
312#518C22×C10911yescharacteristic
313#418—21no—
314C2×#100C22×C10911yescharacteristic
315C2×#99C22×C10911yescharacteristic
316C2×#99C22×C10911yescharacteristic
317#394—21no—
318#392—21no—
319#392—21no—
320#388—21no—
321#386C22×C10911yescharacteristic
322#385—21no—
323#379—21no—
324#353—22no—
325AΣL(1,𝔽16)—22no—
326#342—21no—
327#342—21no—
328#339—21no—
329C25D4×C10911yescharacteristic
330C22×D4—21no—
331C22×D4—21no—
332C22×D4—21no—
333C22×D4—21no—
334C22×D4—21no—
335C22×D4D4×C10911yescharacteristic
336C22×D4C23×C10911yescharacteristic
337C23×C4D4×C10911yescharacteristic
338C2×M16—21no—
339#127—21no—
340#127—21no—
341#127—21no—
342#127—21no—
343#121—44no—
344#120—21no—
345#120—21no—
346#120—21no—
347#120—21no—
348#120—44no—
349C4×D4—41no—
350C4×D4—41no—
351#117—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—41no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—41no—
365C2×C22⋊C4—41no—
366C2×C22⋊C4D4×C10911yescharacteristic
367C2×C22⋊C4D4×C10911yescharacteristic
368C2×C42D4×C10911yescharacteristic
369#102—41no—
370#102—41no—
371#100—21no—
372#100—21no—
373#99—21no—
374#99—21no—
375#99—21no—
376#99D4×C10912yescomplemented
377#95—21no—
378#95—21no—
379#95—41no—
380C24—21no—
381C24—21no—
382C24—21no—
383C24—21no—
384C24—41no—
385C24—42no—
386C24—41no—
387C24—81no—
388C24C2×D4×C10911yescharacteristic
389C24C2×D4×C10911yescharacteristic
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—81no—
409C2×D4—81no—
410C2×D4—81no—
411C2×D4—81no—
412C2×D4—81no—
413C2×D4—81no—
414C2×D4C2×D4×C10911yescharacteristic
415C2×D4#41×C10911yescharacteristic
416C2×D4#41×C10911yescharacteristic
417C2×D4#41×C10911yescharacteristic
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—42no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4C2×D4×C10911yescharacteristic
431M16—41no—
432M16—41no—
433C2×C8—41no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—42no—
437C4⋊C4—82no—
438C22⋊C4—24no—
439C22⋊C4—22no—
440C22⋊C4—21no—
441C22⋊C4—21no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—41no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C22⋊C4—81no—
455C22⋊C4—82no—
456C22⋊C4—82no—
457C42—21no—
458C42—21no—
459C42—41no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—22no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—42no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—81no—
486C23—82no—
487C23—81no—
488C23—82no—
489C23—84no—
490C23—81no—
491C23#120×C10911yescharacteristic
492C23#121×C10911yescharacteristic
493C23#121×C10911yescharacteristic
494C23#123×C10911yescharacteristic
495C23#124×C10911yescharacteristic
496D4—41no—
497D4—41no—
498D4—41no—
499D4—41no—
500D4—42no—
501D4—81no—
502D4—82no—
503D4—82no—
504D4—82no—
505D4—82no—
506D4—84no—
507D4—82no—
508D4—82no—
509D4—84no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
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—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4#121×C10911yescharacteristic
539C2×C4#123×C10911yescharacteristic
540C8—81no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—42no—
552C22—42no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—44no—
558C22—42no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22#394×C10911yescharacteristic
576C22B(4,𝔽2)×C10911yescharacteristic
577C22#458×C10911yescharacteristic
578C4—21no—
579C4—21no—
580C4—42no—
581C4—42no—
582C4—81no—
583C4—81no—
584C4—82no—
585C4—81no—
586C4—81no—
587C4—161no—
588C2—21no—
589C2—21no—
590C2—41no—
591C2—41no—
592C2—41no—
593C2—41no—
594C2—82no—
595C2—161no—
596C2C109×#204411yescharacteristic
597C2C109×#219611yescharacteristic
598C2C109×#220011yescharacteristic
subgroupG/H
center, socleC22×C109B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C109
prime radicalC109×#2669C2
LCSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C109 ◁ C23×C109 ◁ C22×D4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#353×C109C2semidirect
3. C2×AΣL(1,𝔽16)×C109C2semidirect
4. C109×#1889C2semidirect
5. C109×#1887C2semidirect
6. C109×#1867C2semidirect
7. C109×#1390C2semidirect
76. #99×C109D4semidirect
288. #13422C109direct (N × H)
301. C2×#353C2×C109semidirect
302. C2×AΣL(1,𝔽16)C2×C109semidirect
303. #1889C2×C109semidirect
304. #1887C2×C109semidirect
305. #1867C2×C109semidirect
306. #1390C2×C109semidirect
307. C109#13422direct (N × H)
376. #99D4×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#26692
C2×#353×C1092
C2×AΣL(1,𝔽16)×C1092
C109×#18892
C109×#18872
C109×#18672
C109×#13902
#13422109