Subgroups — C₂²×#20266997 #39910319

Order 26176.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20266997	C₂	1	12	yes	maximal, direct product factor
2	C₂²×#15692550	C₂	1	2	yes	maximal, complemented
3	C₂³×Dic₄₀₉	C₂	1	1	yes	maximal, characteristic
4	#20266997	C₂²	1	16	yes	direct product factor
5	C₂×#15692550	C₂²	1	12	yes	complemented
6	C₂²×Dic₄₀₉	C₄	1	6	yes	—
7	C₂²×Dic₄₀₉	C₄	1	1	yes	characteristic
8	C₂²×Dic₄₀₉	C₂²	1	1	yes	characteristic
9	C₂²×Dic₄₀₉	C₂²	1	6	yes	—
10	C₂⁴×C₄₀₉	C₄	1	1	yes	characteristic
11	#15692550	C₂³	1	8	yes	complemented
12	C₂×Dic₄₀₉	C₂×C₄	1	12	yes	—
13	C₂×Dic₄₀₉	C₂×C₄	1	3	yes	—
14	C₂×Dic₄₀₉	C₂×C₄	1	6	yes	—
15	C₂×Dic₄₀₉	C₂³	1	3	yes	—
16	C₂×Dic₄₀₉	C₂³	1	4	yes	—
17	C₂³×C₄₀₉	—	2	4	no	—
18	C₂³×C₄₀₉	C₂×C₄	1	6	yes	—
19	C₂³×C₄₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄₀₉	—	2	12	no	—
21	C₂²×C₄₀₉	M₁₆	1	4	yes	complemented
22	C₂²×C₄₀₉	C₂²×C₄	1	4	yes	—
23	C₂²×C₄₀₉	C₂²×C₄	1	3	yes	—
24	Dic₄₀₉	C₂²×C₄	1	3	yes	—
25	Dic₄₀₉	C₂²×C₄	1	4	yes	—
26	Dic₄₀₉	C₂⁴	1	1	yes	characteristic
27	C₂×C₄₀₉	—	2	4	no	—
28	C₂×C₄₀₉	C₂×M₁₆	1	6	yes	complemented
29	C₂×C₄₀₉	C₂³×C₄	1	1	yes	characteristic
30	C₄₀₉	C₂²×M₁₆	1	1	yes	characteristic, complemented, Sylow
31	C₂²×M₁₆	—	409	1	no	maximal, Sylow
32	C₂³×C₄	—	409	1	no	—
33	C₂×M₁₆	—	409	12	no	—
34	C₂²×C₈	—	409	2	no	—
35	C₂⁴	#13109888	1	1	yes	characteristic
36	C₂²×C₄	—	409	6	no	—
37	C₂²×C₄	—	409	6	no	—
38	C₂²×C₄	—	409	1	no	—
39	C₂²×C₄	—	409	1	no	—
40	M₁₆	—	409	16	no	—
41	C₂×C₈	—	409	12	no	—
42	C₂³	—	2	4	no	—
43	C₂³	C₂×#13109888	1	6	yes	—
44	C₂³	C₂×#13109888	1	1	yes	characteristic
45	C₂×C₄	—	409	12	no	—
46	C₂×C₄	—	409	6	no	—
47	C₂×C₄	—	409	3	no	—
48	C₂×C₄	—	409	3	no	—
49	C₂×C₄	—	409	4	no	—
50	C₈	—	409	8	no	—
51	C₂²	—	2	12	no	—
52	C₂²	#20266997	1	4	yes	direct product factor
53	C₂²	C₂²×#13109888	1	4	yes	—
54	C₂²	C₂²×#13109888	1	3	yes	—
55	C₄	—	409	1	no	—
56	C₄	—	409	3	no	—
57	C₄	—	409	4	no	—
58	C₂	—	2	4	no	—
59	C₂	C₂×#20266997	1	6	yes	direct product factor
60	C₂	C₂³×#13109888	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13109888
commutator	C₂×C₄₀₉	C₂³×C₄
Frattini	C₂	C₂³×#13109888
Fitting, prime radical	C₂⁴×C₄₀₉	C₄
socle	C₂³×C₄₀₉	C₂×C₄
cosocle	Dic₄₀₉	C₂⁴
hypercenter	C₂⁴	#13109888
hypocenter	C₄₀₉	C₂²×M₁₆
LCS	G ▷ C₂×C₄₀₉ ▷ C₄₀₉
UCS	C₁ ◁ C₂³ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂×#20266997	C₂	direct (N × H)
2. C₂²×#15692550	C₂	semidirect
4. #20266997	C₂²	direct (N × H)
5. C₂×#15692550	C₂²	semidirect
11. #15692550	C₂³	semidirect
21. C₂²×C₄₀₉	M₁₆	semidirect
28. C₂×C₄₀₉	C₂×M₁₆	semidirect
30. C₄₀₉	C₂²×M₁₆	semidirect
52. C₂²	#20266997	direct (N × H)
59. C₂	C₂×#20266997	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#20266997	2
C₂²×#15692550	2
C₂³×Dic₄₀₉	2
C₂²×M₁₆	409

Subgroups — C22×#20266997 #39910319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#20266997 #39910319