Subgroups — #41001178

Order 26816.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27211765	C₂	1	2	yes	maximal, complemented
2	#27211751	C₂	1	6	yes	maximal, complemented
3	#27211740	C₂	1	3	yes	maximal, complemented
4	C₂×Dic₁₆₇₆	C₂	1	3	yes	maximal, complemented
5	#131×C₄₁₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#20269411	C₂²	1	6	yes	—
7	#20269410	—	2	6	no	—
8	C₂×Dic₈₃₈	C₂²	1	6	yes	—
9	Q₈×D₄₁₉	—	2	2	no	—
10	Dic₁₆₇₆	C₂²	1	3	yes	—
11	Dic₁₆₇₆	C₂²	1	6	yes	—
12	#20269400	C₂²	1	6	yes	—
13	D₃₃₅₂	C₂²	1	1	yes	characteristic
14	#41×C₄₁₉	C₂²	1	1	yes	characteristic
15	M₁₆×C₄₁₉	C₂²	1	3	yes	—
16	C₂×C₈×C₄₁₉	C₂²	1	3	yes	—
17	Dic₈₃₈	—	2	6	no	—
18	Dic₈₃₈	C₂³	1	6	yes	—
19	D₁₆₇₆	C₂³	1	2	yes	—
20	C₂²⋊D₄₁₉	—	2	6	no	—
21	C₂×Dic₄₁₉	—	2	6	no	—
22	C₄×D₄₁₉	—	2	6	no	—
23	Q₈×C₄₁₉	D₄	1	1	yes	characteristic
24	D₄×C₄₁₉	D₄	1	3	yes	—
25	C₂×C₄×C₄₁₉	C₂³	1	3	yes	—
26	C₈×C₄₁₉	C₂³	1	1	yes	characteristic
27	C₈×C₄₁₉	C₂³	1	3	yes	—
28	C₂²×C₄₁₉	C₂×D₄	1	3	yes	—
29	C₂×D₄₁₉	—	2	2	no	—
30	C₄×C₄₁₉	C₂×D₄	1	3	yes	—
31	C₄×C₄₁₉	C₂⁴	1	1	yes	characteristic
32	Dic₄₁₉	—	2	6	no	—
33	C₂×C₄₁₉	—	2	3	no	—
34	C₂×C₄₁₉	C₂²×D₄	1	1	yes	characteristic
35	D₄₁₉	—	4	2	no	—
36	C₄₁₉	#578	1	1	yes	characteristic, complemented, Sylow
37	#578	—	419	1	no	maximal, Sylow
38	2₋¹⁺⁴	—	419	2	no	—
39	#137	—	419	6	no	—
40	#135	—	419	3	no	—
41	C₂×Q₁₆	—	419	3	no	—
42	#131	D₄₁₉	1	1	yes	characteristic, complemented
43	#41	—	838	6	no	—
44	#41	—	419	6	no	—
45	#41	C₂×D₄₁₉	1	1	yes	characteristic
46	C₂×Q₈	—	838	2	no	—
47	C₂×Q₈	—	419	6	no	—
48	Q₁₆	—	419	3	no	—
49	Q₁₆	—	419	6	no	—
50	SD₁₆	—	419	6	no	—
51	D₈	—	419	1	no	—
52	M₁₆	C₂×D₄₁₉	1	3	yes	—
53	C₂×C₈	C₂×D₄₁₉	1	3	yes	—
54	Q₈	—	838	6	no	—
55	Q₈	—	419	6	no	—
56	Q₈	D₁₆₇₆	1	1	yes	characteristic
57	D₄	—	838	6	no	—
58	D₄	—	419	2	no	—
59	D₄	D₁₆₇₆	1	3	yes	—
60	C₂×C₄	—	838	6	no	—
61	C₂×C₄	—	838	6	no	—
62	C₂×C₄	C₂²×D₄₁₉	1	3	yes	—
63	C₈	C₂²×D₄₁₉	1	1	yes	characteristic
64	C₈	C₂²×D₄₁₉	1	3	yes	—
65	C₂²	—	838	2	no	—
66	C₂²	C₂×D₁₆₇₆	1	3	yes	—
67	C₄	—	838	6	no	—
68	C₄	C₂×D₁₆₇₆	1	3	yes	—
69	C₄	C₂³×D₄₁₉	1	1	yes	characteristic
70	C₂	—	2	3	no	—
71	C₂	—	1676	2	no	—
72	C₂	C₂²×D₁₆₇₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₁₆₇₆
commutator, cosocle	C₄×C₄₁₉	C₂⁴
Frattini	C₄	C₂³×D₄₁₉
Fitting	#131×C₄₁₉	C₂
socle	C₂×C₄₁₉	C₂²×D₄
hypercenter	#131	D₄₁₉
hypocenter	C₄₁₉	#578
LCS	G ▷ C₄×C₄₁₉ ▷ C₂×C₄₁₉ ▷ C₄₁₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #27211765	C₂	semidirect
2. #27211751	C₂	semidirect
3. #27211740	C₂	semidirect
4. C₂×Dic₁₆₇₆	C₂	semidirect
5. #131×C₄₁₉	C₂	semidirect
36. C₄₁₉	#578	semidirect
42. #131	D₄₁₉	semidirect

Maximal subgroups

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

subgroup	index
#27211765	2
#27211751	2
#27211740	2
C₂×Dic₁₆₇₆	2
#131×C₄₁₉	2
#578	419