Subgroups — #39910149

Order 26176.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27189777	C₂	1	1	yes	maximal, characteristic
2	#27189775	C₂	1	1	yes	maximal, characteristic
3	#27189700	C₂	1	1	yes	maximal, characteristic
4	#20266997	—	2	1	no	—
5	C₂²⋊#13109888	—	2	1	no	—
6	C₂×C₂²⋊D₄₀₉	C₂²	1	1	yes	characteristic
7	#20266967	—	2	1	no	—
8	C₂²⋊Dic₄₀₉	C₄	1	1	yes	characteristic, complemented
9	#20266965	—	2	1	no	—
10	C₂²×C₄×C₄₀₉	C₄	1	1	yes	characteristic, complemented
11	C₂×#13109888	—	4	1	no	—
12	#15692550	—	4	1	no	—
13	C₂²×D₄₀₉	D₄	1	1	yes	characteristic
14	C₂²⋊D₄₀₉	—	4	1	no	—
15	C₂×Dic₄₀₉	—	2	1	no	—
16	C₂×Dic₄₀₉	D₄	1	1	yes	characteristic
17	C₂³×C₄₀₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₄₀₉	—	2	1	no	—
19	C₂×C₄×C₄₀₉	—	4	1	no	—
20	C₂²×C₄₀₉	—	2	1	no	—
21	C₂²×C₄₀₉	—	4	1	no	—
22	C₂²×C₄₀₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×D₄₀₉	—	2	1	no	—
24	C₂×D₄₀₉	—	4	1	no	—
25	#13109888	—	8	1	no	—
26	C₄×C₄₀₉	—	4	1	no	—
27	Dic₄₀₉	—	2	1	no	—
28	Dic₄₀₉	—	4	1	no	—
29	C₂×C₄₀₉	—	2	1	no	—
30	C₂×C₄₀₉	—	4	1	no	—
31	C₂×C₄₀₉	#99	1	1	yes	characteristic
32	D₄₀₉	—	4	1	no	—
33	C₄₀₉	#352	1	1	yes	characteristic, complemented, Sylow
34	#352	—	409	1	no	maximal, Sylow
35	#123	—	409	1	no	—
36	#100	—	409	1	no	—
37	#99	—	409	1	no	—
38	C₂×D₄	—	409	1	no	—
39	C₂²×C₄	#13109888	1	1	yes	characteristic, complemented
40	M₁₆	—	818	1	no	—
41	C₄⋊C₄	—	818	1	no	—
42	C₂²⋊C₄	—	818	1	no	—
43	C₂²⋊C₄	—	818	1	no	—
44	C₂²⋊C₄	—	409	1	no	—
45	C₂³	—	409	1	no	—
46	C₂³	C₂×#13109888	1	1	yes	characteristic
47	D₄	—	1636	1	no	—
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	—	4	1	no	—
50	C₂×C₄	—	818	1	no	—
51	C₂×C₄	—	1636	1	no	—
52	C₂×C₄	—	409	1	no	—
53	C₈	—	1636	1	no	—
54	C₂²	—	2	1	no	—
55	C₂²	—	4	1	no	—
56	C₂²	—	818	1	no	—
57	C₂²	—	1636	1	no	—
58	C₂²	C₂²⋊#13109888	1	1	yes	characteristic
59	C₄	—	4	1	no	—
60	C₄	—	818	1	no	—
61	C₄	—	1636	1	no	—
62	C₄	—	3272	1	no	—
63	C₂	—	2	1	no	—
64	C₂	—	4	1	no	—
65	C₂	—	1636	1	no	—
66	C₂	#27189775	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27189775
commutator	C₂³×C₄₀₉	C₂×C₄
Frattini	C₂³	C₂×#13109888
Fitting	C₂²×C₄×C₄₀₉	C₄
socle	C₂×C₄₀₉	#99
prime radical, cosocle	C₂×C₂²⋊D₄₀₉	C₂²
hypercenter	C₂²×C₄	#13109888
hypocenter	C₄₀₉	#352
LCS	G ▷ C₂³×C₄₀₉ ▷ C₂²×C₄₀₉ ▷ C₂×C₄₀₉ ▷ C₄₀₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
8. C₂²⋊Dic₄₀₉	C₄	semidirect
10. C₂²×C₄×C₄₀₉	C₄	semidirect
33. C₄₀₉	#352	semidirect
39. C₂²×C₄	#13109888	semidirect

Maximal subgroups

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

subgroup	index
#27189777	2
#27189775	2
#27189700	2
#352	409