Finite Group Databasebeta

← C109×#32698 #42150692

Subgroups — C109×#32698 #42150692

Order 27904.

594 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#2102C211yesmaximal, characteristic, complemented
2C109×#2081C211yesmaximal, characteristic, complemented
3C109×#2081C211yesmaximal, characteristic, complemented
4C109×#2069C211yesmaximal, characteristic, complemented
5C109×#2048C211yesmaximal, characteristic, complemented
6C109×#2002C211yesmaximal, characteristic
7C109×#1965C211yesmaximal, characteristic, complemented
8C109×#1932C211yesmaximal, characteristic, complemented
9C109×#1914C211yesmaximal, characteristic, complemented
10C109×#1911C211yesmaximal, characteristic, complemented
11C109×#1903C211yesmaximal, characteristic, complemented
12C109×#1887C211yesmaximal, characteristic, complemented
13C109×#1861C211yesmaximal, characteristic, complemented
14C109×#1841C211yesmaximal, characteristic, complemented
15C109×#1829C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)×C109C2211yescharacteristic
17#568×C109C2211yescharacteristic
18#535×C109C2211yescharacteristic
19C2×#127×C109C2211yescharacteristic
20#518×C109C2211yescharacteristic
21#430×C109C2211yescharacteristic
22#430×C109C2211yescharacteristic
23#430×C109C2211yescharacteristic
24#423×C109C2211yescharacteristic
25#421×C109C2211yescharacteristic
26#421×C109C2211yescharacteristic
27C2×#104×C109C2211yescharacteristic
28#418×C109C2211yescharacteristic
29#418×C109C2211yescharacteristic
30#418×C109C2211yescharacteristic
31#413×C109C2211yescharacteristic
32C2×#100×C109C2211yescharacteristic
33C2×#100×C109C2211yescharacteristic
34C2×#100×C109C2211yescharacteristic
35C2×#100×C109C2211yescharacteristic
36C2×#100×C109C2211yescharacteristic
37C2×#100×C109C2211yescharacteristic
38C4×M16×C109C2211yescharacteristic
39#344×C109C2211yescharacteristic
40#344×C109C2211yescharacteristic
41#344×C109C2211yescharacteristic
42#344×C109C2211yescharacteristic
43#344×C109C2211yescharacteristic
44#344×C109C2211yescharacteristic
45#339×C109C2211yescharacteristic
46#339×C109C2211yescharacteristic
47#339×C109C2211yescharacteristic
48#338×C109C2211yescharacteristic
49H(3,ℤ/4)×C109C2211yescharacteristic
50H(3,ℤ/4)×C109C2211yescharacteristic
51C2×#41×C109C2311yescharacteristic
52C22×D4×C109—22no—
53C22×D4×C109C2311yescharacteristic
54C22×D4×C109C2311yescharacteristic
55AGL(1,ℤ/8)×C109—41no—
56AGL(1,ℤ/8)×C109—41no—
57C2×SD16×C109—21no—
58C2×D8×C109—21no—
59#131×C109—41no—
60#131×C109—41no—
61C2×M16×C109—41no—
62C2×M16×C109C2311yescharacteristic
63C2×M16×C109C2311yescharacteristic
64C2×M16×C109C2311yescharacteristic
65C2×M16×C109C2311yescharacteristic
66C2×M16×C109C2311yescharacteristic
67C2×M16×C109C2311yescharacteristic
68C2×M16×C109C2311yescharacteristic
69C2×M16×C109C2311yescharacteristic
70#127×C109—21no—
71#127×C109—21no—
72#127×C109—21no—
73#127×C109—41no—
74#124×C109—21no—
75#121×C109—41no—
76#120×C109—22no—
77C4×D4×C109—41no—
78C4×D4×C109—41no—
79#117×C109C2311yescharacteristic
80#117×C109C2311yescharacteristic
81#117×C109C2311yescharacteristic
82C2×C22⋊C4×C109—22no—
83C2×C42×C109C2311yescharacteristic
84#108×C109—41no—
85#108×C109—41no—
86#108×C109—41no—
87C4⋊C8×C109—41no—
88#104×C109—21no—
89#104×C109—21no—
90#104×C109—41no—
91#104×C109—41no—
92#102×C109—41no—
93#102×C109—41no—
94#102×C109—41no—
95#101×C109—21no—
96#100×C109—21no—
97#100×C109—41no—
98#100×C109—41no—
99#100×C109—41no—
100#100×C109—41no—
101#100×C109—41no—
102#100×C109—41no—
103#97×C109—21no—
104C4×C8×C109—21no—
105C24×C109—21no—
106C24×C109#4112yes—
107#41×C109—21no—
108#41×C109—21no—
109#41×C109—21no—
110#41×C109—21no—
111C2×Q8×C109#4111yescharacteristic
112C2×D4×C109—21no—
113C2×D4×C109—21no—
114C2×D4×C109—21no—
115C2×D4×C109—21no—
116C2×D4×C109—21no—
117C2×D4×C109—21no—
118C2×D4×C109—21no—
119C2×D4×C109—21no—
120C2×D4×C109—21no—
121C2×D4×C109—21no—
122C2×D4×C109—21no—
123C2×D4×C109—21no—
124C2×D4×C109—21no—
125C2×D4×C109—42no—
126C2×D4×C109—42no—
127C2×D4×C109—42no—
128C2×D4×C109—84no—
129C2×D4×C109#4111yescharacteristic
130C22×C4×C109—21no—
131C22×C4×C109—21no—
132C22×C4×C109—42no—
133C22×C4×C109C2411yescharacteristic
134SD16×C109—42no—
135D8×C109—42no—
136M16×C109—21no—
137M16×C109—21no—
138M16×C109—21no—
139M16×C109—21no—
140M16×C109—21no—
141M16×C109—21no—
142M16×C109—21no—
143M16×C109—21no—
144M16×C109—21no—
145M16×C109—21no—
146M16×C109—21no—
147M16×C109—21no—
148M16×C109—21no—
149M16×C109—21no—
150M16×C109—21no—
151M16×C109—21no—
152M16×C109—42no—
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—21no—
160C2×C8×C109—21no—
161C2×C8×C109—42no—
162C4⋊C4×C109—21no—
163C4⋊C4×C109—21no—
164C4⋊C4×C109—21no—
165C22⋊C4×C109—21no—
166C22⋊C4×C109—21no—
167C22⋊C4×C109—21no—
168C22⋊C4×C109—42no—
169C22⋊C4×C109—42no—
170C42×C109—21no—
171C42×C109—21no—
172C42×C109—21no—
173C42×C109—21no—
174C42×C109#4111yescharacteristic
175C42×C109#4111yescharacteristic
176C23×C109—22no—
177C23×C109—22no—
178C23×C109—21no—
179C23×C109—22no—
180C23×C109—41no—
181C23×C109—41no—
182C23×C109—41no—
183C23×C109—82no—
184C23×C109—82no—
185C23×C109C2×#4111yescharacteristic
186Q8×C109—21no—
187Q8×C109—21no—
188D4×C109—21no—
189D4×C109—21no—
190D4×C109—41no—
191D4×C109—41no—
192D4×C109—41no—
193D4×C109—41no—
194D4×C109—41no—
195D4×C109—41no—
196D4×C109—41no—
197D4×C109—41no—
198D4×C109—41no—
199D4×C109—41no—
200D4×C109—84no—
201D4×C109—84no—
202D4×C109—162no—
203C2×C4×C109—21no—
204C2×C4×C109—21no—
205C2×C4×C109—21no—
206C2×C4×C109—42no—
207C2×C4×C109—41no—
208C2×C4×C109—41no—
209C2×C4×C109—41no—
210C2×C4×C109—41no—
211C2×C4×C109—41no—
212C2×C4×C109—162no—
213C2×C4×C109C2×#4111yescharacteristic
214C2×C4×C109C2×#4111yescharacteristic
215C2×C4×C109H(5,𝔽2)11yescharacteristic
216C2×C4×C109H(5,𝔽2)11yescharacteristic
217C2×C4×C109H(5,𝔽2)11yescharacteristic
218C2×C4×C109H(5,𝔽2)11yescharacteristic
219C8×C109—41no—
220C8×C109—41no—
221C8×C109—41no—
222C8×C109—41no—
223C8×C109—41no—
224C8×C109—41no—
225C8×C109—41no—
226C8×C109—41no—
227C22×C109—41no—
228C22×C109—41no—
229C22×C109—42no—
230C22×C109—82no—
231C22×C109—82no—
232C22×C109—81no—
233C22×C109—82no—
234C22×C109—82no—
235C22×C109—81no—
236C22×C109—161no—
237C22×C109—161no—
238C22×C109#53811yescharacteristic
239C22×C109#55111yescharacteristic
240C22×C109#55111yescharacteristic
241C4×C109—22no—
242C4×C109—41no—
243C4×C109—41no—
244C4×C109—81no—
245C4×C109—81no—
246C4×C109—81no—
247C4×C109#53811yescharacteristic
248C4×C109#54011yescharacteristic
249C4×C109#55311yescharacteristic
250C4×C109#56011yescharacteristic
251#32698C10911yesmaximal, characteristic, direct product factor, Sylow
252C2×C109—21no—
253C2×C109—21no—
254C2×C109—21no—
255C2×C109—81no—
256C2×C109—82no—
257C2×C109—161no—
258C2×C109#274911yescharacteristic
259#2102C2×C10911yescharacteristic, complemented
260#2081C2×C10911yescharacteristic, complemented
261#2081C2×C10911yescharacteristic, complemented
262#2069C2×C10911yescharacteristic, complemented
263#2048C2×C10911yescharacteristic, complemented
264#2002C2×C10911yescharacteristic
265#1965C2×C10911yescharacteristic, complemented
266#1932C2×C10911yescharacteristic, complemented
267#1914C2×C10911yescharacteristic, complemented
268#1911C2×C10911yescharacteristic, complemented
269#1903C2×C10911yescharacteristic, complemented
270#1887C2×C10911yescharacteristic, complemented
271#1861C2×C10911yescharacteristic, complemented
272#1841C2×C10911yescharacteristic, complemented
273#1829C2×C10911yescharacteristic, complemented
274C109#3269811yescharacteristic, direct product factor, Sylow
275C2×AGL(1,ℤ/8)C22×C10911yescharacteristic
276#568C22×C10911yescharacteristic
277#535C22×C10911yescharacteristic
278C2×#127C22×C10911yescharacteristic
279#518C22×C10911yescharacteristic
280#430C22×C10911yescharacteristic
281#430C22×C10911yescharacteristic
282#430C22×C10911yescharacteristic
283#423C22×C10911yescharacteristic
284#421C22×C10911yescharacteristic
285#421C22×C10911yescharacteristic
286C2×#104C22×C10911yescharacteristic
287#418C22×C10911yescharacteristic
288#418C22×C10911yescharacteristic
289#418C22×C10911yescharacteristic
290#413C22×C10911yescharacteristic
291C2×#100C22×C10911yescharacteristic
292C2×#100C22×C10911yescharacteristic
293C2×#100C22×C10911yescharacteristic
294C2×#100C22×C10911yescharacteristic
295C2×#100C22×C10911yescharacteristic
296C2×#100C22×C10911yescharacteristic
297C4×M16C22×C10911yescharacteristic
298#344C22×C10911yescharacteristic
299#344C22×C10911yescharacteristic
300#344C22×C10911yescharacteristic
301#344C22×C10911yescharacteristic
302#344C22×C10911yescharacteristic
303#344C22×C10911yescharacteristic
304#339C22×C10911yescharacteristic
305#339C22×C10911yescharacteristic
306#339C22×C10911yescharacteristic
307#338C22×C10911yescharacteristic
308H(3,ℤ/4)C22×C10911yescharacteristic
309H(3,ℤ/4)C22×C10911yescharacteristic
310C2×#41C23×C10911yescharacteristic
311C22×D4—22no—
312C22×D4C23×C10911yescharacteristic
313C22×D4C23×C10911yescharacteristic
314AGL(1,ℤ/8)—41no—
315AGL(1,ℤ/8)—41no—
316C2×SD16—21no—
317C2×D8—21no—
318#131—41no—
319#131—41no—
320C2×M16—41no—
321C2×M16C23×C10911yescharacteristic
322C2×M16C23×C10911yescharacteristic
323C2×M16C23×C10911yescharacteristic
324C2×M16C23×C10911yescharacteristic
325C2×M16C23×C10911yescharacteristic
326C2×M16C23×C10911yescharacteristic
327C2×M16C23×C10911yescharacteristic
328C2×M16C23×C10911yescharacteristic
329#127—21no—
330#127—21no—
331#127—21no—
332#127—41no—
333#124—21no—
334#121—41no—
335#120—22no—
336C4×D4—41no—
337C4×D4—41no—
338#117C23×C10911yescharacteristic
339#117C23×C10911yescharacteristic
340#117C23×C10911yescharacteristic
341C2×C22⋊C4—22no—
342C2×C42C23×C10911yescharacteristic
343#108—41no—
344#108—41no—
345#108—41no—
346C4⋊C8—41no—
347#104—21no—
348#104—21no—
349#104—41no—
350#104—41no—
351#102—41no—
352#102—41no—
353#102—41no—
354#101—21no—
355#100—21no—
356#100—41no—
357#100—41no—
358#100—41no—
359#100—41no—
360#100—41no—
361#100—41no—
362#97—21no—
363C4×C8—21no—
364C24—21no—
365C24#41×C10912yes—
366#41—21no—
367#41—21no—
368#41—21no—
369#41—21no—
370C2×Q8#41×C10911yescharacteristic
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—84no—
388C2×D4#41×C10911yescharacteristic
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—42no—
392C22×C4C24×C10911yescharacteristic
393SD16—42no—
394D8—42no—
395M16—21no—
396M16—21no—
397M16—21no—
398M16—21no—
399M16—21no—
400M16—21no—
401M16—21no—
402M16—21no—
403M16—21no—
404M16—21no—
405M16—21no—
406M16—21no—
407M16—21no—
408M16—21no—
409M16—21no—
410M16—21no—
411M16—42no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—42no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C22⋊C4—21no—
425C22⋊C4—21no—
426C22⋊C4—21no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42#41×C10911yescharacteristic
434C42#41×C10911yescharacteristic
435C23—22no—
436C23—22no—
437C23—21no—
438C23—22no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—82no—
443C23—82no—
444C23C2×#41×C10911yescharacteristic
445Q8—21no—
446Q8—21no—
447D4—21no—
448D4—21no—
449D4—41no—
450D4—41no—
451D4—41no—
452D4—41no—
453D4—41no—
454D4—41no—
455D4—41no—
456D4—41no—
457D4—41no—
458D4—41no—
459D4—84no—
460D4—84no—
461D4—162no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—162no—
472C2×C4C2×#41×C10911yescharacteristic
473C2×C4C2×#41×C10911yescharacteristic
474C2×C4H(5,𝔽2)×C10911yescharacteristic
475C2×C4H(5,𝔽2)×C10911yescharacteristic
476C2×C4H(5,𝔽2)×C10911yescharacteristic
477C2×C4H(5,𝔽2)×C10911yescharacteristic
478C8—41no—
479C8—41no—
480C8—41no—
481C8—41no—
482C8—41no—
483C8—41no—
484C8—41no—
485C8—41no—
486C22—41no—
487C22—41no—
488C22—42no—
489C22—82no—
490C22—82no—
491C22—81no—
492C22—82no—
493C22—82no—
494C22—81no—
495C22—161no—
496C22—161no—
497C22#538×C10911yescharacteristic
498C22#551×C10911yescharacteristic
499C22#551×C10911yescharacteristic
500C4—22no—
501C4—41no—
502C4—41no—
503C4—81no—
504C4—81no—
505C4—81no—
506C4#538×C10911yescharacteristic
507C4#540×C10911yescharacteristic
508C4#553×C10911yescharacteristic
509C4#560×C10911yescharacteristic
510C2—21no—
511C2—21no—
512C2—21no—
513C2—82no—
514C2—81no—
515C2—161no—
516C2C109×#274911yescharacteristic
subgroupG/H
center, socleC2×C109#2749
commutator, Frattini, cosocleC22×C4C24×C109
prime radicalC109×#2002C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C109 ◁ C22×C4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
1. C109×#2102C2semidirect
2. C109×#2081C2semidirect
3. C109×#2081C2semidirect
4. C109×#2069C2semidirect
5. C109×#2048C2semidirect
7. C109×#1965C2semidirect
8. C109×#1932C2semidirect
9. C109×#1914C2semidirect
10. C109×#1911C2semidirect
11. C109×#1903C2semidirect
12. C109×#1887C2semidirect
13. C109×#1861C2semidirect
14. C109×#1841C2semidirect
15. C109×#1829C2semidirect
251. #32698C109direct (N × H)
259. #2102C2×C109semidirect
260. #2081C2×C109semidirect
261. #2081C2×C109semidirect
262. #2069C2×C109semidirect
263. #2048C2×C109semidirect
265. #1965C2×C109semidirect
266. #1932C2×C109semidirect
267. #1914C2×C109semidirect
268. #1911C2×C109semidirect
269. #1903C2×C109semidirect
270. #1887C2×C109semidirect
271. #1861C2×C109semidirect
272. #1841C2×C109semidirect
273. #1829C2×C109semidirect
274. C109#32698direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#21022
C109×#20812
C109×#20692
C109×#20482
C109×#20022
C109×#19652
C109×#19322
C109×#19142
C109×#19112
C109×#19032
C109×#18872
C109×#18612
C109×#18412
C109×#18292
#32698109