Finite Group Databasebeta

← C109×#32699 #42150693

Subgroups — C109×#32699 #42150693

Order 27904.

538 conjugacy classes of subgroups.

The subgroup lattice has 498 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×#2070C211yesmaximal, characteristic, complemented
5C109×#2048C211yesmaximal, characteristic
6C109×#2003C211yesmaximal, characteristic
7C109×#1967C211yesmaximal, characteristic, complemented
8C109×#1932C211yesmaximal, characteristic, complemented
9C109×#1915C211yesmaximal, characteristic, complemented
10C109×#1912C211yesmaximal, characteristic, complemented
11C109×#1903C211yesmaximal, characteristic
12C109×#1888C211yesmaximal, characteristic, complemented
13C109×#1861C211yesmaximal, characteristic, complemented
14C109×#1841C211yesmaximal, characteristic, complemented
15C109×#1830C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)×C109C2211yescharacteristic
17#568×C109C2211yescharacteristic
18#536×C109C2211yescharacteristic
19C2×#124×C109C2211yescharacteristic
20#519×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×#101×C109C2211yescharacteristic
33C2×#101×C109C2211yescharacteristic
34C2×#101×C109C2211yescharacteristic
35C2×#101×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×Q8×C109C2311yescharacteristic
53C22×D4×C109C2311yescharacteristic
54AGL(1,ℤ/8)×C109—41no—
55AGL(1,ℤ/8)×C109—41no—
56C2×SD16×C109—21no—
57C2×D8×C109—21no—
58#131×C109—41no—
59#131×C109—41no—
60C2×M16×C109—41no—
61C2×M16×C109C2311yescharacteristic
62C2×M16×C109C2311yescharacteristic
63C2×M16×C109C2311yescharacteristic
64C2×M16×C109C2311yescharacteristic
65C2×M16×C109C2311yescharacteristic
66C2×M16×C109C2311yescharacteristic
67C2×M16×C109C2311yescharacteristic
68C2×M16×C109C2311yescharacteristic
69C4⋊Q8×C109—21no—
70#124×C109—21no—
71#124×C109—21no—
72#124×C109—21no—
73#124×C109—41no—
74#123×C109—22no—
75C22⋊Q8×C109—41no—
76C4×Q8×C109—41no—
77C4×Q8×C109—41no—
78#117×C109—22no—
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#101×C109—41no—
97#101×C109—41no—
98#101×C109—41no—
99#101×C109—41no—
100#100×C109—21no—
101#100×C109—41no—
102#100×C109—41no—
103#97×C109—21no—
104C4×C8×C109—21no—
105C24×C109—21no—
106#41×C109—21no—
107#41×C109—21no—
108#41×C109—21no—
109#41×C109—21no—
110C2×Q8×C109—21no—
111C2×Q8×C109—21no—
112C2×Q8×C109—21no—
113C2×Q8×C109—21no—
114C2×Q8×C109—21no—
115C2×Q8×C109—21no—
116C2×Q8×C109#4111yescharacteristic
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#4111yescharacteristic
125C22×C4×C109—21no—
126C22×C4×C109—21no—
127C22×C4×C109#4112yes—
128C22×C4×C109C2411yescharacteristic
129SD16×C109—42no—
130D8×C109—42no—
131M16×C109—21no—
132M16×C109—21no—
133M16×C109—21no—
134M16×C109—21no—
135M16×C109—21no—
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—42no—
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—42no—
157C4⋊C4×C109—21no—
158C4⋊C4×C109—21no—
159C4⋊C4×C109—21no—
160C4⋊C4×C109—42no—
161C4⋊C4×C109—42no—
162C22⋊C4×C109—21no—
163C22⋊C4×C109—21no—
164C22⋊C4×C109—21no—
165C22⋊C4×C109—42no—
166C22⋊C4×C109—42no—
167C22⋊C4×C109—42no—
168C42×C109—21no—
169C42×C109—21no—
170C42×C109—21no—
171C42×C109—21no—
172C42×C109—42no—
173C42×C109#4111yescharacteristic
174C42×C109#4111yescharacteristic
175C23×C109—21no—
176C23×C109—41no—
177C23×C109—41no—
178C23×C109—41no—
179C23×C109—82no—
180C23×C109C2×#4111yescharacteristic
181Q8×C109—21no—
182Q8×C109—21no—
183Q8×C109—41no—
184Q8×C109—41no—
185Q8×C109—41no—
186Q8×C109—41no—
187D4×C109—21no—
188D4×C109—21no—
189D4×C109—41no—
190D4×C109—41no—
191D4×C109—41no—
192D4×C109—41no—
193D4×C109—41no—
194D4×C109—41no—
195C2×C4×C109—22no—
196C2×C4×C109—21no—
197C2×C4×C109—21no—
198C2×C4×C109—22no—
199C2×C4×C109—22no—
200C2×C4×C109—21no—
201C2×C4×C109—41no—
202C2×C4×C109—41no—
203C2×C4×C109—41no—
204C2×C4×C109—41no—
205C2×C4×C109—41no—
206C2×C4×C109—81no—
207C2×C4×C109C2×#4111yescharacteristic
208C2×C4×C109C2×#4111yescharacteristic
209C2×C4×C109H(5,𝔽2)11yescharacteristic
210C2×C4×C109H(5,𝔽2)11yescharacteristic
211C2×C4×C109H(5,𝔽2)11yescharacteristic
212C2×C4×C109H(5,𝔽2)11yescharacteristic
213C8×C109—41no—
214C8×C109—41no—
215C8×C109—41no—
216C8×C109—41no—
217C8×C109—41no—
218C8×C109—41no—
219C8×C109—41no—
220C8×C109—41no—
221C22×C109—41no—
222C22×C109—41no—
223C22×C109—81no—
224C22×C109—82no—
225C22×C109—81no—
226C22×C109—161no—
227C22×C109—161no—
228C22×C109#53811yescharacteristic
229C22×C109#55111yescharacteristic
230C22×C109#55111yescharacteristic
231C4×C109—41no—
232C4×C109—41no—
233C4×C109—42no—
234C4×C109—41no—
235C4×C109—81no—
236C4×C109—81no—
237C4×C109—81no—
238C4×C109#53811yescharacteristic
239C4×C109#54011yescharacteristic
240C4×C109#55311yescharacteristic
241C4×C109#56011yescharacteristic
242#32699C10911yesmaximal, characteristic, direct product factor, Sylow
243C2×C109—21no—
244C2×C109—21no—
245C2×C109—21no—
246C2×C109—81no—
247C2×C109—161no—
248C2×C109#274911yescharacteristic
249#2102C2×C10911yescharacteristic, complemented
250#2081C2×C10911yescharacteristic, complemented
251#2081C2×C10911yescharacteristic, complemented
252#2070C2×C10911yescharacteristic, complemented
253#2048C2×C10911yescharacteristic
254#2003C2×C10911yescharacteristic
255#1967C2×C10911yescharacteristic, complemented
256#1932C2×C10911yescharacteristic, complemented
257#1915C2×C10911yescharacteristic, complemented
258#1912C2×C10911yescharacteristic, complemented
259#1903C2×C10911yescharacteristic
260#1888C2×C10911yescharacteristic, complemented
261#1861C2×C10911yescharacteristic, complemented
262#1841C2×C10911yescharacteristic, complemented
263#1830C2×C10911yescharacteristic, complemented
264C109#3269911yescharacteristic, direct product factor, Sylow
265C2×AGL(1,ℤ/8)C22×C10911yescharacteristic
266#568C22×C10911yescharacteristic
267#536C22×C10911yescharacteristic
268C2×#124C22×C10911yescharacteristic
269#519C22×C10911yescharacteristic
270#430C22×C10911yescharacteristic
271#430C22×C10911yescharacteristic
272#430C22×C10911yescharacteristic
273#423C22×C10911yescharacteristic
274#421C22×C10911yescharacteristic
275#421C22×C10911yescharacteristic
276C2×#104C22×C10911yescharacteristic
277#418C22×C10911yescharacteristic
278#418C22×C10911yescharacteristic
279#418C22×C10911yescharacteristic
280#413C22×C10911yescharacteristic
281C2×#101C22×C10911yescharacteristic
282C2×#101C22×C10911yescharacteristic
283C2×#101C22×C10911yescharacteristic
284C2×#101C22×C10911yescharacteristic
285C2×#100C22×C10911yescharacteristic
286C2×#100C22×C10911yescharacteristic
287C4×M16C22×C10911yescharacteristic
288#344C22×C10911yescharacteristic
289#344C22×C10911yescharacteristic
290#344C22×C10911yescharacteristic
291#344C22×C10911yescharacteristic
292#344C22×C10911yescharacteristic
293#344C22×C10911yescharacteristic
294#339C22×C10911yescharacteristic
295#339C22×C10911yescharacteristic
296#339C22×C10911yescharacteristic
297#338C22×C10911yescharacteristic
298H(3,ℤ/4)C22×C10911yescharacteristic
299H(3,ℤ/4)C22×C10911yescharacteristic
300C2×#41C23×C10911yescharacteristic
301C22×Q8C23×C10911yescharacteristic
302C22×D4C23×C10911yescharacteristic
303AGL(1,ℤ/8)—41no—
304AGL(1,ℤ/8)—41no—
305C2×SD16—21no—
306C2×D8—21no—
307#131—41no—
308#131—41no—
309C2×M16—41no—
310C2×M16C23×C10911yescharacteristic
311C2×M16C23×C10911yescharacteristic
312C2×M16C23×C10911yescharacteristic
313C2×M16C23×C10911yescharacteristic
314C2×M16C23×C10911yescharacteristic
315C2×M16C23×C10911yescharacteristic
316C2×M16C23×C10911yescharacteristic
317C2×M16C23×C10911yescharacteristic
318C4⋊Q8—21no—
319#124—21no—
320#124—21no—
321#124—21no—
322#124—41no—
323#123—22no—
324C22⋊Q8—41no—
325C4×Q8—41no—
326C4×Q8—41no—
327#117—22no—
328#117C23×C10911yescharacteristic
329#117C23×C10911yescharacteristic
330#117C23×C10911yescharacteristic
331C2×C22⋊C4—22no—
332C2×C42C23×C10911yescharacteristic
333#108—41no—
334#108—41no—
335#108—41no—
336C4⋊C8—41no—
337#104—21no—
338#104—21no—
339#104—41no—
340#104—41no—
341#102—41no—
342#102—41no—
343#102—41no—
344#101—21no—
345#101—41no—
346#101—41no—
347#101—41no—
348#101—41no—
349#100—21no—
350#100—41no—
351#100—41no—
352#97—21no—
353C4×C8—21no—
354C24—21no—
355#41—21no—
356#41—21no—
357#41—21no—
358#41—21no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8#41×C10911yescharacteristic
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4#41×C10911yescharacteristic
374C22×C4—21no—
375C22×C4—21no—
376C22×C4#41×C10912yes—
377C22×C4C24×C10911yescharacteristic
378SD16—42no—
379D8—42no—
380M16—21no—
381M16—21no—
382M16—21no—
383M16—21no—
384M16—21no—
385M16—21no—
386M16—21no—
387M16—21no—
388M16—21no—
389M16—21no—
390M16—21no—
391M16—21no—
392M16—21no—
393M16—21no—
394M16—21no—
395M16—21no—
396M16—42no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—42no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—21no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—42no—
422C42#41×C10911yescharacteristic
423C42#41×C10911yescharacteristic
424C23—21no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—82no—
429C23C2×#41×C10911yescharacteristic
430Q8—21no—
431Q8—21no—
432Q8—41no—
433Q8—41no—
434Q8—41no—
435Q8—41no—
436D4—21no—
437D4—21no—
438D4—41no—
439D4—41no—
440D4—41no—
441D4—41no—
442D4—41no—
443D4—41no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—81no—
456C2×C4C2×#41×C10911yescharacteristic
457C2×C4C2×#41×C10911yescharacteristic
458C2×C4H(5,𝔽2)×C10911yescharacteristic
459C2×C4H(5,𝔽2)×C10911yescharacteristic
460C2×C4H(5,𝔽2)×C10911yescharacteristic
461C2×C4H(5,𝔽2)×C10911yescharacteristic
462C8—41no—
463C8—41no—
464C8—41no—
465C8—41no—
466C8—41no—
467C8—41no—
468C8—41no—
469C8—41no—
470C22—41no—
471C22—41no—
472C22—81no—
473C22—82no—
474C22—81no—
475C22—161no—
476C22—161no—
477C22#538×C10911yescharacteristic
478C22#551×C10911yescharacteristic
479C22#551×C10911yescharacteristic
480C4—41no—
481C4—41no—
482C4—42no—
483C4—41no—
484C4—81no—
485C4—81no—
486C4—81no—
487C4#538×C10911yescharacteristic
488C4#540×C10911yescharacteristic
489C4#553×C10911yescharacteristic
490C4#560×C10911yescharacteristic
491C2—21no—
492C2—21no—
493C2—21no—
494C2—81no—
495C2—161no—
496C2C109×#274911yescharacteristic
subgroupG/H
center, socleC2×C109#2749
commutator, Frattini, cosocleC22×C4C24×C109
prime radicalC2×AGL(1,ℤ/8)×C109C22
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×#2070C2semidirect
7. C109×#1967C2semidirect
8. C109×#1932C2semidirect
9. C109×#1915C2semidirect
10. C109×#1912C2semidirect
12. C109×#1888C2semidirect
13. C109×#1861C2semidirect
14. C109×#1841C2semidirect
15. C109×#1830C2semidirect
242. #32699C109direct (N × H)
249. #2102C2×C109semidirect
250. #2081C2×C109semidirect
251. #2081C2×C109semidirect
252. #2070C2×C109semidirect
255. #1967C2×C109semidirect
256. #1932C2×C109semidirect
257. #1915C2×C109semidirect
258. #1912C2×C109semidirect
260. #1888C2×C109semidirect
261. #1861C2×C109semidirect
262. #1841C2×C109semidirect
263. #1830C2×C109semidirect
264. C109#32699direct (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×#20702
C109×#20482
C109×#20032
C109×#19672
C109×#19322
C109×#19152
C109×#19122
C109×#19032
C109×#18882
C109×#18612
C109×#18412
C109×#18302
#32699109