Finite Group Databasebeta

← C109×#32697 #42150691

Subgroups — C109×#32697 #42150691

Order 27904.

528 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#2102C211yesmaximal, characteristic, complemented
2C109×#2082C211yesmaximal, characteristic, complemented
3C109×#2082C211yesmaximal, characteristic, complemented
4C109×#2070C211yesmaximal, characteristic, complemented
5C109×#2049C211yesmaximal, characteristic
6C109×#2005C211yesmaximal, characteristic
7C109×#1966C211yesmaximal, characteristic, complemented
8C109×#1932C211yesmaximal, characteristic
9C109×#1915C211yesmaximal, characteristic
10C109×#1912C211yesmaximal, characteristic
11C109×#1902C211yesmaximal, characteristic
12C109×#1886C211yesmaximal, characteristic, complemented
13C109×#1861C211yesmaximal, characteristic
14C109×#1841C211yesmaximal, characteristic, complemented
15C109×#1830C211yesmaximal, characteristic, complemented
16C2×#137×C109C2211yescharacteristic
17#568×C109C2211yescharacteristic
18#536×C109C2211yescharacteristic
19C2×C4⋊Q8×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#419×C109C2211yescharacteristic
29#419×C109C2211yescharacteristic
30#419×C109C2211yescharacteristic
31#413×C109C2211yescharacteristic
32C2×#101×C109C2211yescharacteristic
33C2×#101×C109C2211yescharacteristic
34C2×#101×C109C2211yescharacteristic
35C2×#101×C109C2211yescharacteristic
36C2×#101×C109C2211yescharacteristic
37C2×#101×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×Q8×C109C2311yescharacteristic
54#137×C109—41no—
55#137×C109—41no—
56C2×Q16×C109—21no—
57C2×SD16×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—
70C4⋊Q8×C109—21no—
71C4⋊Q8×C109—21no—
72C4⋊Q8×C109—41no—
73#124×C109—21no—
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×C4⋊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#103×C109—41no—
93#103×C109—41no—
94#103×C109—41no—
95#101×C109—21no—
96#101×C109—41no—
97#101×C109—41no—
98#101×C109—41no—
99#101×C109—41no—
100#101×C109—41no—
101#101×C109—41no—
102#100×C109—21no—
103#97×C109—21no—
104C4×C8×C109—21no—
105#41×C109—21no—
106#41×C109—21no—
107#41×C109—21no—
108#41×C109—21no—
109C2×Q8×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—21no—
117C2×Q8×C109—21no—
118C2×Q8×C109—21no—
119C2×Q8×C109—21no—
120C2×Q8×C109—21no—
121C2×Q8×C109#4111yescharacteristic
122C2×D4×C109—21no—
123C2×D4×C109#4111yescharacteristic
124C22×C4×C109—21no—
125C22×C4×C109—21no—
126C22×C4×C109—21no—
127C22×C4×C109#4112yes—
128C22×C4×C109C2411yescharacteristic
129Q16×C109—42no—
130SD16×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—
162C4⋊C4×C109—42no—
163C4⋊C4×C109—42no—
164C22⋊C4×C109—21no—
165C22⋊C4×C109—21no—
166C22⋊C4×C109—21no—
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×C109C2×#4111yescharacteristic
177Q8×C109—21no—
178Q8×C109—21no—
179Q8×C109—41no—
180Q8×C109—41no—
181Q8×C109—41no—
182Q8×C109—41no—
183Q8×C109—41no—
184Q8×C109—41no—
185Q8×C109—41no—
186Q8×C109—41no—
187D4×C109—21no—
188D4×C109—21no—
189D4×C109—41no—
190D4×C109—41no—
191C2×C4×C109—21no—
192C2×C4×C109—22no—
193C2×C4×C109—21no—
194C2×C4×C109—22no—
195C2×C4×C109—21no—
196C2×C4×C109—22no—
197C2×C4×C109—42no—
198C2×C4×C109—41no—
199C2×C4×C109—41no—
200C2×C4×C109—41no—
201C2×C4×C109—41no—
202C2×C4×C109—41no—
203C2×C4×C109—41no—
204C2×C4×C109—41no—
205C2×C4×C109—41no—
206C2×C4×C109C2×#4111yescharacteristic
207C2×C4×C109C2×#4111yescharacteristic
208C2×C4×C109H(5,𝔽2)11yescharacteristic
209C2×C4×C109H(5,𝔽2)11yescharacteristic
210C2×C4×C109H(5,𝔽2)11yescharacteristic
211C2×C4×C109H(5,𝔽2)11yescharacteristic
212C8×C109—41no—
213C8×C109—41no—
214C8×C109—41no—
215C8×C109—41no—
216C8×C109—41no—
217C8×C109—41no—
218C8×C109—41no—
219C8×C109—41no—
220C22×C109—41no—
221C22×C109—41no—
222C22×C109—81no—
223C22×C109#53811yescharacteristic
224C22×C109#55111yescharacteristic
225C22×C109#55111yescharacteristic
226C4×C109—22no—
227C4×C109—41no—
228C4×C109—42no—
229C4×C109—41no—
230C4×C109—81no—
231C4×C109—81no—
232C4×C109—81no—
233C4×C109—81no—
234C4×C109#53811yescharacteristic
235C4×C109#54011yescharacteristic
236C4×C109#55311yescharacteristic
237C4×C109#56011yescharacteristic
238#32697C10911yesmaximal, characteristic, direct product factor, Sylow
239C2×C109—21no—
240C2×C109—21no—
241C2×C109—21no—
242C2×C109—81no—
243C2×C109#274911yescharacteristic
244#2102C2×C10911yescharacteristic, complemented
245#2082C2×C10911yescharacteristic, complemented
246#2082C2×C10911yescharacteristic, complemented
247#2070C2×C10911yescharacteristic, complemented
248#2049C2×C10911yescharacteristic
249#2005C2×C10911yescharacteristic
250#1966C2×C10911yescharacteristic, complemented
251#1932C2×C10911yescharacteristic
252#1915C2×C10911yescharacteristic
253#1912C2×C10911yescharacteristic
254#1902C2×C10911yescharacteristic
255#1886C2×C10911yescharacteristic, complemented
256#1861C2×C10911yescharacteristic
257#1841C2×C10911yescharacteristic, complemented
258#1830C2×C10911yescharacteristic, complemented
259C109#3269711yescharacteristic, direct product factor, Sylow
260C2×#137C22×C10911yescharacteristic
261#568C22×C10911yescharacteristic
262#536C22×C10911yescharacteristic
263C2×C4⋊Q8C22×C10911yescharacteristic
264#519C22×C10911yescharacteristic
265#430C22×C10911yescharacteristic
266#430C22×C10911yescharacteristic
267#430C22×C10911yescharacteristic
268#423C22×C10911yescharacteristic
269#421C22×C10911yescharacteristic
270#421C22×C10911yescharacteristic
271C2×#104C22×C10911yescharacteristic
272#419C22×C10911yescharacteristic
273#419C22×C10911yescharacteristic
274#419C22×C10911yescharacteristic
275#413C22×C10911yescharacteristic
276C2×#101C22×C10911yescharacteristic
277C2×#101C22×C10911yescharacteristic
278C2×#101C22×C10911yescharacteristic
279C2×#101C22×C10911yescharacteristic
280C2×#101C22×C10911yescharacteristic
281C2×#101C22×C10911yescharacteristic
282C4×M16C22×C10911yescharacteristic
283#344C22×C10911yescharacteristic
284#344C22×C10911yescharacteristic
285#344C22×C10911yescharacteristic
286#344C22×C10911yescharacteristic
287#344C22×C10911yescharacteristic
288#344C22×C10911yescharacteristic
289#339C22×C10911yescharacteristic
290#339C22×C10911yescharacteristic
291#339C22×C10911yescharacteristic
292#338C22×C10911yescharacteristic
293H(3,ℤ/4)C22×C10911yescharacteristic
294H(3,ℤ/4)C22×C10911yescharacteristic
295C2×#41C23×C10911yescharacteristic
296C22×Q8C23×C10911yescharacteristic
297C22×Q8C23×C10911yescharacteristic
298#137—41no—
299#137—41no—
300C2×Q16—21no—
301C2×SD16—21no—
302#131—41no—
303#131—41no—
304C2×M16—41no—
305C2×M16C23×C10911yescharacteristic
306C2×M16C23×C10911yescharacteristic
307C2×M16C23×C10911yescharacteristic
308C2×M16C23×C10911yescharacteristic
309C2×M16C23×C10911yescharacteristic
310C2×M16C23×C10911yescharacteristic
311C2×M16C23×C10911yescharacteristic
312C2×M16C23×C10911yescharacteristic
313C4⋊Q8—21no—
314C4⋊Q8—21no—
315C4⋊Q8—21no—
316C4⋊Q8—41no—
317#124—21no—
318#123—22no—
319C22⋊Q8—41no—
320C4×Q8—41no—
321C4×Q8—41no—
322#117—22no—
323#117C23×C10911yescharacteristic
324#117C23×C10911yescharacteristic
325#117C23×C10911yescharacteristic
326C2×C4⋊C4—22no—
327C2×C42C23×C10911yescharacteristic
328#108—41no—
329#108—41no—
330#108—41no—
331C4⋊C8—41no—
332#104—21no—
333#104—21no—
334#104—41no—
335#104—41no—
336#103—41no—
337#103—41no—
338#103—41no—
339#101—21no—
340#101—41no—
341#101—41no—
342#101—41no—
343#101—41no—
344#101—41no—
345#101—41no—
346#100—21no—
347#97—21no—
348C4×C8—21no—
349#41—21no—
350#41—21no—
351#41—21no—
352#41—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—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#41×C10911yescharacteristic
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4#41×C10912yes—
372C22×C4C24×C10911yescharacteristic
373Q16—42no—
374SD16—42no—
375M16—21no—
376M16—21no—
377M16—21no—
378M16—21no—
379M16—21no—
380M16—21no—
381M16—21no—
382M16—21no—
383M16—21no—
384M16—21no—
385M16—21no—
386M16—21no—
387M16—21no—
388M16—21no—
389M16—21no—
390M16—21no—
391M16—42no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—42no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C22⋊C4—21no—
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—42no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—42no—
417C42#41×C10911yescharacteristic
418C42#41×C10911yescharacteristic
419C23—21no—
420C23C2×#41×C10911yescharacteristic
421Q8—21no—
422Q8—21no—
423Q8—41no—
424Q8—41no—
425Q8—41no—
426Q8—41no—
427Q8—41no—
428Q8—41no—
429Q8—41no—
430Q8—41no—
431D4—21no—
432D4—21no—
433D4—41no—
434D4—41no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4C2×#41×C10911yescharacteristic
451C2×C4C2×#41×C10911yescharacteristic
452C2×C4H(5,𝔽2)×C10911yescharacteristic
453C2×C4H(5,𝔽2)×C10911yescharacteristic
454C2×C4H(5,𝔽2)×C10911yescharacteristic
455C2×C4H(5,𝔽2)×C10911yescharacteristic
456C8—41no—
457C8—41no—
458C8—41no—
459C8—41no—
460C8—41no—
461C8—41no—
462C8—41no—
463C8—41no—
464C22—41no—
465C22—41no—
466C22—81no—
467C22#538×C10911yescharacteristic
468C22#551×C10911yescharacteristic
469C22#551×C10911yescharacteristic
470C4—22no—
471C4—41no—
472C4—42no—
473C4—41no—
474C4—81no—
475C4—81no—
476C4—81no—
477C4—81no—
478C4#538×C10911yescharacteristic
479C4#540×C10911yescharacteristic
480C4#553×C10911yescharacteristic
481C4#560×C10911yescharacteristic
482C2—21no—
483C2—21no—
484C2—21no—
485C2—81no—
486C2C109×#274911yescharacteristic
subgroupG/H
center, socleC2×C109#2749
commutator, Frattini, cosocleC22×C4C24×C109
prime radicalC2×#41×C109C23
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×#2082C2semidirect
3. C109×#2082C2semidirect
4. C109×#2070C2semidirect
7. C109×#1966C2semidirect
12. C109×#1886C2semidirect
14. C109×#1841C2semidirect
15. C109×#1830C2semidirect
238. #32697C109direct (N × H)
244. #2102C2×C109semidirect
245. #2082C2×C109semidirect
246. #2082C2×C109semidirect
247. #2070C2×C109semidirect
250. #1966C2×C109semidirect
255. #1886C2×C109semidirect
257. #1841C2×C109semidirect
258. #1830C2×C109semidirect
259. C109#32697direct (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×#20822
C109×#20702
C109×#20492
C109×#20052
C109×#19662
C109×#19322
C109×#19152
C109×#19122
C109×#19022
C109×#18862
C109×#18612
C109×#18412
C109×#18302
#32697109