Subgroups — #39909455

Order 26176.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20266978	C₂	1	1	yes	maximal, characteristic
2	#27189715	C₂	1	1	yes	maximal, characteristic
3	Q₈×Dic₄₀₉	C₂	1	1	yes	maximal, characteristic
4	#27189646	C₂	1	1	yes	maximal, characteristic
5	#27189642	C₂	1	1	yes	maximal, characteristic
6	#27189641	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₁₆×C₄₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₈₁₈	C₂²	1	1	yes	characteristic
9	#20266978	—	2	2	no	—
10	C₂×#15692543	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₄₀₉	—	2	1	no	—
12	C₄⋊Dic₄₀₉	C₂²	1	1	yes	characteristic
13	#20266967	—	2	1	no	—
14	#20266967	—	2	1	no	—
15	C₄×Dic₄₀₉	—	2	1	no	—
16	C₄×Dic₄₀₉	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₄₀₉	C₂²	1	1	yes	characteristic
18	C₂×Q₈×C₄₀₉	C₂²	1	1	yes	characteristic
19	Q₁₆×C₄₀₉	—	2	2	no	—
20	C₂×C₈×C₄₀₉	C₂²	1	1	yes	characteristic
21	Dic₈₁₈	—	2	1	no	—
22	Dic₈₁₈	—	2	1	no	—
23	C₂×Dic₄₀₉	—	2	1	no	—
24	C₂×Dic₄₀₉	—	2	1	no	—
25	C₂×Dic₄₀₉	D₄	1	1	yes	characteristic
26	C₂×Dic₄₀₉	D₄	1	1	yes	characteristic
27	#15692543	—	2	1	no	—
28	Q₈×C₄₀₉	—	2	1	no	—
29	Q₈×C₄₀₉	—	2	1	no	—
30	Q₈×C₄₀₉	—	2	1	no	—
31	Q₈×C₄₀₉	D₄	1	2	yes	—
32	C₂×C₄×C₄₀₉	—	2	1	no	—
33	C₂×C₄×C₄₀₉	—	2	1	no	—
34	C₂×C₄×C₄₀₉	C₂³	1	1	yes	characteristic
35	C₈×C₄₀₉	—	2	1	no	—
36	C₂²×C₄₀₉	C₂×D₄	1	1	yes	characteristic
37	C₄×C₄₀₉	—	2	2	no	—
38	C₄×C₄₀₉	—	4	1	no	—
39	C₄×C₄₀₉	C₂×D₄	1	1	yes	characteristic
40	C₄×C₄₀₉	#41	1	1	yes	characteristic
41	Dic₄₀₉	—	2	1	no	—
42	Dic₄₀₉	—	4	1	no	—
43	Dic₄₀₉	—	4	1	no	—
44	Dic₄₀₉	Q₁₆	1	2	yes	complemented
45	C₂×C₄₀₉	#121	1	1	yes	characteristic
46	C₂×C₄₀₉	C₂×Q₁₆	1	1	yes	characteristic
47	C₂×C₄₀₉	#137	1	1	yes	characteristic
48	C₄₀₉	#462	1	1	yes	characteristic, complemented, Sylow
49	#462	—	409	1	no	maximal, Sylow
50	C₂×Q₁₆	—	409	1	no	—
51	C₂×Q₁₆	D₄₀₉	1	1	yes	characteristic
52	C₄⋊Q₈	—	409	1	no	—
53	C₄×Q₈	—	409	1	no	—
54	C₄⋊C₈	—	409	1	no	—
55	#103	—	409	1	no	—
56	#103	—	409	1	no	—
57	C₂×Q₈	—	409	1	no	—
58	C₂×Q₈	C₂×D₄₀₉	1	1	yes	characteristic
59	C₂×Q₈	C₂×D₄₀₉	1	1	yes	characteristic
60	Q₁₆	—	2	2	no	—
61	Q₁₆	—	818	2	no	—
62	C₂×C₈	—	409	1	no	—
63	C₂×C₈	C₂×D₄₀₉	1	1	yes	characteristic
64	C₄⋊C₄	—	818	1	no	—
65	C₄⋊C₄	—	818	1	no	—
66	C₄⋊C₄	—	818	1	no	—
67	C₄⋊C₄	—	409	1	no	—
68	C₄²	—	818	1	no	—
69	C₄²	—	409	1	no	—
70	Q₈	—	2	1	no	—
71	Q₈	—	2	1	no	—
72	Q₈	—	2	1	no	—
73	Q₈	—	818	1	no	—
74	Q₈	—	818	1	no	—
75	Q₈	C₂²⋊D₄₀₉	1	2	yes	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	818	1	no	—
79	C₂×C₄	—	818	1	no	—
80	C₂×C₄	—	409	1	no	—
81	C₂×C₄	—	409	1	no	—
82	C₂×C₄	C₂²×D₄₀₉	1	1	yes	characteristic
83	C₈	—	2	1	no	—
84	C₈	—	818	1	no	—
85	C₂²	D₄×D₄₀₉	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	4	1	no	—
88	C₄	—	818	1	no	—
89	C₄	—	1636	1	no	—
90	C₄	—	1636	1	no	—
91	C₄	—	409	2	no	—
92	C₄	C₂×C₂²⋊D₄₀₉	1	1	yes	characteristic
93	C₄	#20266987	1	1	yes	characteristic
94	C₂	#27189689	1	1	yes	characteristic
95	C₂	Q₁₆×D₄₀₉	1	1	yes	characteristic
96	C₂	#27189749	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₄₀₉
commutator, cosocle	C₂×C₄×C₄₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₄₀₉
Fitting	C₂×Q₁₆×C₄₀₉	C₂
socle, prime radical	C₂²×C₄₀₉	C₂×D₄
hypercenter	C₂×Q₁₆	D₄₀₉
hypocenter	C₄₀₉	#462
LCS	G ▷ C₂×C₄×C₄₀₉ ▷ C₂×C₄₀₉ ▷ C₄₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
44. Dic₄₀₉	Q₁₆	semidirect
48. C₄₀₉	#462	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20266978	2
#27189715	2
Q₈×Dic₄₀₉	2
#27189646	2
#27189642	2
#27189641	2
C₂×Q₁₆×C₄₀₉	2
#462	409