Subgroups — #39909302

Order 26176.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×D₄₀₉	C₂	1	1	yes	maximal, characteristic
2	#27189646	C₂	1	2	yes	maximal, complemented
3	#27189629	C₂	1	2	yes	maximal
4	#27189628	C₂	1	1	yes	maximal, characteristic, complemented
5	#97×C₄₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₄₀₉	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄₀₉	C₄	1	1	yes	characteristic
8	C₂×C₄×D₄₀₉	C₂²	1	1	yes	characteristic
9	C₂×#15692543	C₂²	1	2	yes	—
10	C₄×Dic₄₀₉	C₄	1	1	yes	characteristic
11	C₄×Dic₄₀₉	C₄	1	1	yes	characteristic
12	C₄×Dic₄₀₉	C₂²	1	1	yes	characteristic
13	C₂×C₈×C₄₀₉	C₂²	1	2	yes	—
14	C₄²×C₄₀₉	C₂²	1	1	yes	characteristic
15	C₂²×D₄₀₉	C₂×C₄	1	1	yes	characteristic
16	C₂×Dic₄₀₉	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₄₀₉	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₄₀₉	C₂×C₄	1	1	yes	characteristic
19	C₄×D₄₀₉	—	2	2	no	—
20	C₄×D₄₀₉	—	2	2	no	—
21	C₄×D₄₀₉	—	2	2	no	—
22	#15692543	—	2	2	no	—
23	C₂×C₄×C₄₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₄₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₄₀₉	C₂³	1	1	yes	characteristic
26	C₈×C₄₀₉	—	2	2	no	—
27	C₂²×C₄₀₉	C₂²×C₄	1	1	yes	characteristic
28	C₂×D₄₀₉	—	2	1	no	—
29	C₂×D₄₀₉	—	2	1	no	—
30	C₂×D₄₀₉	M₁₆	1	2	yes	—
31	C₄×C₄₀₉	—	2	1	no	—
32	C₄×C₄₀₉	—	2	1	no	—
33	C₄×C₄₀₉	#41	1	2	yes	—
34	Dic₄₀₉	—	2	1	no	—
35	Dic₄₀₉	—	2	1	no	—
36	Dic₄₀₉	M₁₆	1	2	yes	—
37	C₂×C₄₀₉	#117	1	1	yes	characteristic
38	C₂×C₄₀₉	C₂×M₁₆	1	1	yes	characteristic
39	C₂×C₄₀₉	C₂×M₁₆	1	1	yes	characteristic
40	D₄₀₉	—	2	2	no	—
41	C₄₀₉	#432	1	1	yes	characteristic, complemented, Sylow
42	#432	—	409	1	no	maximal, Sylow
43	C₂×C₄²	—	409	1	no	—
44	C₄⋊C₈	—	409	2	no	—
45	C₂²⋊C₈	—	409	2	no	—
46	#97	—	409	1	no	—
47	#97	D₄₀₉	1	1	yes	characteristic, complemented
48	C₂²×C₄	—	409	1	no	—
49	C₂²×C₄	—	409	1	no	—
50	C₂²×C₄	—	409	1	no	—
51	C₂×C₈	—	409	2	no	—
52	C₂×C₈	C₂×D₄₀₉	1	2	yes	—
53	C₄²	—	409	1	no	—
54	C₄²	—	409	1	no	—
55	C₄²	—	409	1	no	—
56	C₄²	C₂×D₄₀₉	1	1	yes	characteristic
57	C₂³	—	409	1	no	—
58	C₂×C₄	—	818	2	no	—
59	C₂×C₄	—	818	2	no	—
60	C₂×C₄	—	818	2	no	—
61	C₂×C₄	—	409	1	no	—
62	C₂×C₄	—	409	1	no	—
63	C₂×C₄	—	409	1	no	—
64	C₂×C₄	C₄×D₄₀₉	1	1	yes	characteristic
65	C₂×C₄	C₄×D₄₀₉	1	1	yes	characteristic
66	C₂×C₄	C₂²×D₄₀₉	1	1	yes	characteristic
67	C₈	—	2	2	no	—
68	C₈	—	818	2	no	—
69	C₂²	—	818	1	no	—
70	C₂²	—	818	1	no	—
71	C₂²	—	409	2	no	—
72	C₂²	C₂×C₄×D₄₀₉	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	2	1	no	—
75	C₄	—	818	1	no	—
76	C₄	—	818	1	no	—
77	C₄	—	409	2	no	—
78	C₄	#20266988	1	2	yes	—
79	C₂	—	818	2	no	—
80	C₂	#27189674	1	1	yes	characteristic
81	C₂	M₁₆×D₄₀₉	1	1	yes	characteristic
82	C₂	M₁₆×D₄₀₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₄₀₉
commutator, socle	C₂²×C₄₀₉	C₂²×C₄
Fitting	#97×C₄₀₉	C₂
prime radical	C₂²×D₄₀₉	C₂×C₄
cosocle	C₂×C₄×C₄₀₉	C₂³
hypercenter	#97	D₄₀₉
hypocenter	C₄₀₉	#432
LCS	G ▷ C₂²×C₄₀₉ ▷ C₄₀₉
UCS	C₁ ◁ C₂×C₄ ◁ #97

Complements

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

N	H ≅ G/N	type
2. #27189646	C₂	semidirect
4. #27189628	C₂	semidirect
5. #97×C₄₀₉	C₂	semidirect
41. C₄₀₉	#432	semidirect
47. #97	D₄₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₄²×D₄₀₉	2
#27189646	2
#27189629	2
#27189628	2
#97×C₄₀₉	2
#432	409