Subgroups — #39910228

Order 26176.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27189796	C₂	1	2	yes	maximal, complemented
2	#27189776	C₂	1	4	yes	maximal, complemented
3	C₂×#20266989	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#13109888	—	2	2	no	—
5	C₂²⋊#13109888	C₂²	1	4	yes	—
6	C₄×#13109888	—	2	2	no	—
7	#20266989	—	2	2	no	—
8	#20266989	—	2	2	no	—
9	C₂×D₁₆₃₆	C₄	1	1	yes	characteristic, complemented
10	C₂×D₁₆₃₆	C₂²	1	2	yes	—
11	C₂×C₄×D₄₀₉	C₄	1	2	yes	complemented
12	C₂×C₄×D₄₀₉	C₂²	1	1	yes	characteristic
13	C₂×Q₈×C₄₀₉	C₄	1	1	yes	characteristic, complemented
14	C₂×#13109888	—	2	4	no	—
15	C₂²×D₄₀₉	C₂×C₄	1	2	yes	—
16	C₂²×D₄₀₉	C₂³	1	1	yes	characteristic
17	D₁₆₃₆	—	2	4	no	—
18	D₁₆₃₆	—	2	2	no	—
19	C₂×Dic₄₀₉	C₂×C₄	1	1	yes	characteristic
20	C₄×D₄₀₉	—	2	2	no	—
21	C₄×D₄₀₉	—	2	2	no	—
22	C₄×D₄₀₉	D₄	1	2	yes	—
23	C₄×D₄₀₉	D₄	1	2	yes	—
24	Q₈×C₄₀₉	—	2	2	no	—
25	C₂×C₄×C₄₀₉	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₄₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄₀₉	C₂²×C₄	1	1	yes	characteristic
28	C₂×D₄₀₉	—	2	2	no	—
29	C₂×D₄₀₉	—	4	1	no	—
30	C₂×D₄₀₉	—	4	2	no	—
31	C₂×D₄₀₉	C₂×D₄	1	2	yes	—
32	#13109888	—	4	4	no	—
33	C₄×C₄₀₉	—	2	2	no	—
34	C₄×C₄₀₉	C₂²⋊C₄	1	2	yes	—
35	Dic₄₀₉	C₂²⋊C₄	1	1	yes	characteristic
36	Dic₄₀₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄₀₉	—	2	1	no	—
38	C₂×C₄₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
39	D₄₀₉	—	2	2	no	—
40	D₄₀₉	—	4	2	no	—
41	C₄₀₉	#410	1	1	yes	characteristic, complemented, Sylow
42	#410	—	409	1	no	maximal, Sylow
43	C₂×#41	—	409	1	no	—
44	#117	—	409	2	no	—
45	#99	—	409	4	no	—
46	#41	—	818	2	no	—
47	#41	—	818	2	no	—
48	C₂×Q₈	#13109888	1	1	yes	characteristic, complemented
49	C₂×D₄	—	409	2	no	—
50	C₂×D₄	—	409	1	no	—
51	C₂²×C₄	—	409	2	no	—
52	C₂²×C₄	—	409	1	no	—
53	C₄⋊C₄	—	818	2	no	—
54	C₂²⋊C₄	—	409	4	no	—
55	C₄²	—	818	2	no	—
56	C₂³	—	409	2	no	—
57	C₂³	—	409	1	no	—
58	Q₈	—	2	2	no	—
59	D₄	—	818	4	no	—
60	D₄	—	818	2	no	—
61	C₂×C₄	—	818	2	no	—
62	C₂×C₄	—	818	2	no	—
63	C₂×C₄	—	818	4	no	—
64	C₂×C₄	—	409	1	no	—
65	C₂×C₄	—	409	2	no	—
66	C₂×C₄	—	409	2	no	—
67	C₂×C₄	C₂×#13109888	1	2	yes	—
68	C₂×C₄	C₂×#13109888	1	1	yes	characteristic
69	C₂²	—	818	2	no	—
70	C₂²	—	1636	1	no	—
71	C₂²	—	1636	2	no	—
72	C₂²	—	409	2	no	—
73	C₂²	C₂²×#13109888	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	1636	4	no	—
76	C₄	—	409	1	no	—
77	C₄	—	409	1	no	—
78	C₄	C₂²⋊#13109888	1	2	yes	—
79	C₂	—	2	1	no	—
80	C₂	—	818	2	no	—
81	C₂	—	1636	2	no	—
82	C₂	C₂×C₂²⋊#13109888	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊#13109888
commutator	C₂²×C₄₀₉	C₂²×C₄
Frattini	C₂²	C₂²×#13109888
Fitting	C₂×Q₈×C₄₀₉	C₄
socle	C₂×C₄₀₉	C₂×C₂²⋊C₄
prime radical	C₂×#20266989	C₂
cosocle	C₂²×D₄₀₉	C₂³
hypercenter	C₂×Q₈	#13109888
hypocenter	C₄₀₉	#410
LCS	G ▷ C₂²×C₄₀₉ ▷ C₂×C₄₀₉ ▷ C₄₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. #27189796	C₂	semidirect
2. #27189776	C₂	semidirect
9. C₂×D₁₆₃₆	C₄	semidirect
11. C₂×C₄×D₄₀₉	C₄	semidirect
13. C₂×Q₈×C₄₀₉	C₄	semidirect
41. C₄₀₉	#410	semidirect
48. C₂×Q₈	#13109888	semidirect

Maximal subgroups

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

subgroup	index
#27189796	2
#27189776	2
C₂×#20266989	2
#410	409