Subgroups — #39910092

Order 26176.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊#15692550	C₂	1	2	yes	maximal
2	C₂×C₄⋊Dic₄₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#15692550	—	2	2	no	—
4	C₂²×Dic₄₀₉	C₄	1	1	yes	characteristic
5	C₂²×Dic₄₀₉	C₂²	1	1	yes	characteristic
6	C₄⋊Dic₄₀₉	—	2	2	no	—
7	C₂²×C₄×C₄₀₉	C₄	1	1	yes	characteristic
8	#15692550	—	4	2	no	—
9	C₂×Dic₄₀₉	—	2	1	no	—
10	C₂×Dic₄₀₉	—	4	1	no	—
11	C₂×Dic₄₀₉	—	4	1	no	—
12	C₂×Dic₄₀₉	D₄	1	2	yes	—
13	C₂³×C₄₀₉	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₄₀₉	—	2	1	no	—
15	C₂×C₄×C₄₀₉	—	2	1	no	—
16	C₂×C₄×C₄₀₉	C₈	1	2	yes	complemented
17	C₂²×C₄₀₉	—	2	1	no	—
18	C₂²×C₄₀₉	—	2	1	no	—
19	C₂²×C₄₀₉	C₂²⋊C₄	1	1	yes	characteristic
20	C₂²×C₄₀₉	C₂×C₈	1	1	yes	characteristic
21	C₂²×C₄₀₉	M₁₆	1	1	yes	characteristic
22	C₄×C₄₀₉	—	2	2	no	—
23	Dic₄₀₉	—	2	2	no	—
24	Dic₄₀₉	—	4	1	no	—
25	C₂×C₄₀₉	—	2	1	no	—
26	C₂×C₄₀₉	—	2	1	no	—
27	C₂×C₄₀₉	C₂²⋊C₈	1	1	yes	characteristic
28	C₂×C₄₀₉	#99	1	1	yes	characteristic
29	C₂×C₄₀₉	#101	1	1	yes	characteristic
30	C₄₀₉	#324	1	1	yes	characteristic, complemented, Sylow
31	#324	—	409	1	no	maximal, Sylow
32	C₂×C₄⋊C₄	—	409	1	no	—
33	C₂²⋊C₈	—	409	2	no	—
34	C₂²×C₄	—	409	1	no	—
35	C₂²×C₄	—	409	1	no	—
36	C₂²×C₄	#13109888	1	1	yes	characteristic
37	C₂×C₈	—	818	2	no	—
38	C₄⋊C₄	—	818	2	no	—
39	C₂³	C₂×#13109888	1	1	yes	characteristic
40	C₂×C₄	—	2	1	no	—
41	C₂×C₄	—	2	1	no	—
42	C₂×C₄	—	818	1	no	—
43	C₂×C₄	—	1636	1	no	—
44	C₂×C₄	—	1636	1	no	—
45	C₂×C₄	—	409	2	no	—
46	C₂×C₄	#15692550	1	2	yes	complemented
47	C₈	—	1636	2	no	—
48	C₂²	—	2	1	no	—
49	C₂²	—	2	1	no	—
50	C₂²	C₂²⋊#13109888	1	1	yes	characteristic
51	C₂²	C₂×#15692550	1	1	yes	characteristic
52	C₂²	#20266997	1	1	yes	characteristic
53	C₄	—	2	2	no	—
54	C₄	—	818	2	no	—
55	C₄	—	1636	1	no	—
56	C₂	—	2	1	no	—
57	C₂	—	2	1	no	—
58	C₂	C₂²⋊#15692550	1	1	yes	characteristic
59	C₂	#27189776	1	1	yes	characteristic
60	C₂	#27189778	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#13109888
commutator	C₂²×C₄₀₉	C₂×C₈
Frattini	C₂³	C₂×#13109888
Fitting	C₂²×C₄×C₄₀₉	C₄
socle	C₂²×C₄₀₉	C₂²⋊C₄
prime radical	C₂³×C₄₀₉	C₂×C₄
cosocle	C₂²×Dic₄₀₉	C₂²
hypercenter	C₂²×C₄	#13109888
hypocenter	C₄₀₉	#324
LCS	G ▷ C₂²×C₄₀₉ ▷ C₂×C₄₀₉ ▷ C₄₀₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
16. C₂×C₄×C₄₀₉	C₈	semidirect
30. C₄₀₉	#324	semidirect
46. C₂×C₄	#15692550	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊#15692550	2
C₂×C₄⋊Dic₄₀₉	2
#324	409