Subgroups — #41000090

Order 26816.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₁₆₇₆	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₄₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₈×C₄₁₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₆₇₆	C₄	1	1	yes	characteristic
5	C₂×C₄×D₄₁₉	—	2	1	no	—
6	C₂×C₄₁₉⋊C₈	—	2	1	no	—
7	C₄⋊Dic₄₁₉	C₄	1	1	yes	characteristic
8	(C₂×D₄₁₉)⋊C₄	—	2	1	no	—
9	C₂×C₈×C₄₁₉	—	2	1	no	—
10	C₄²×C₄₁₉	C₂²	1	1	yes	characteristic
11	C₂²×D₄₁₉	—	2	1	no	—
12	D₁₆₇₆	—	2	1	no	—
13	D₁₆₇₆	C₈	1	2	yes	complemented
14	C₂×Dic₄₁₉	—	2	1	no	—
15	C₄×D₄₁₉	—	4	1	no	—
16	C₄×D₄₁₉	—	4	1	no	—
17	C₄₁₉⋊C₈	—	4	1	no	—
18	C₂×C₄×C₄₁₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₄₁₉	D₄	1	1	yes	characteristic
20	C₂×C₄×C₄₁₉	D₄	1	1	yes	characteristic
21	C₈×C₄₁₉	—	2	2	no	—
22	C₂²×C₄₁₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×D₄₁₉	—	2	2	no	—
24	C₂×D₄₁₉	—	4	1	no	—
25	C₂×D₄₁₉	—	4	1	no	—
26	C₄×C₄₁₉	—	2	1	no	—
27	C₄×C₄₁₉	C₂×C₈	1	1	yes	characteristic
28	C₄×C₄₁₉	M₁₆	1	1	yes	characteristic
29	C₄×C₄₁₉	D₈	1	1	yes	characteristic
30	C₄×C₄₁₉	SD₁₆	1	1	yes	characteristic
31	Dic₄₁₉	—	4	1	no	—
32	C₂×C₄₁₉	C₂²⋊C₈	1	1	yes	characteristic
33	C₂×C₄₁₉	#102	1	1	yes	characteristic
34	C₂×C₄₁₉	#104	1	1	yes	characteristic
35	D₄₁₉	—	4	2	no	—
36	C₄₁₉	D₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
37	D₄⋊C₈	—	419	1	no	maximal, Sylow
38	C₄×D₄	—	419	1	no	—
39	C₄⋊C₈	—	419	1	no	—
40	C₄×C₈	D₄₁₉	1	1	yes	characteristic, complemented
41	C₂×D₄	—	419	1	no	—
42	C₂²×C₄	—	838	1	no	—
43	C₂×C₈	—	2	1	no	—
44	C₂×C₈	—	838	1	no	—
45	C₄⋊C₄	—	419	1	no	—
46	C₂²⋊C₄	—	838	1	no	—
47	C₄²	C₂×D₄₁₉	1	1	yes	characteristic
48	C₂³	—	838	1	no	—
49	D₄	—	838	1	no	—
50	D₄	—	419	2	no	—
51	C₂×C₄	—	838	1	no	—
52	C₂×C₄	—	1676	1	no	—
53	C₂×C₄	—	1676	1	no	—
54	C₂×C₄	C₄×D₄₁₉	1	1	yes	characteristic
55	C₂×C₄	C₂²⋊D₄₁₉	1	1	yes	characteristic
56	C₂×C₄	D₁₆₇₆	1	1	yes	characteristic
57	C₈	—	2	2	no	—
58	C₈	—	1676	1	no	—
59	C₂²	—	838	2	no	—
60	C₂²	—	1676	1	no	—
61	C₂²	—	1676	1	no	—
62	C₂²	(C₂×D₄₁₉)⋊C₄	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	1676	1	no	—
65	C₄	C₈×D₄₁₉	1	1	yes	characteristic
66	C₄	#20269394	1	1	yes	characteristic
67	C₄	D₃₃₅₂	1	1	yes	characteristic
68	C₄	#20269400	1	1	yes	characteristic
69	C₂	—	1676	2	no	—
70	C₂	(C₂×D₄₁₉)⋊C₈	1	1	yes	characteristic
71	C₂	#27211640	1	1	yes	characteristic
72	C₂	#27211646	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₆₇₆
commutator	C₄×C₄₁₉	C₂×C₈
Frattini	C₄²	C₂×D₄₁₉
Fitting	C₄×C₈×C₄₁₉	C₂
socle	C₂²×C₄₁₉	C₂²⋊C₄
prime radical	C₂×D₁₆₇₆	C₄
cosocle	C₄²×C₄₁₉	C₂²
hypercenter	C₄×C₈	D₄₁₉
hypocenter	C₄₁₉	D₄⋊C₈
LCS	G ▷ C₄×C₄₁₉ ▷ C₂×C₄₁₉ ▷ C₄₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₄⋊(C₄₁₉⋊C₈)	C₂	semidirect
3. C₄×C₈×C₄₁₉	C₂	semidirect
13. D₁₆₇₆	C₈	semidirect
36. C₄₁₉	D₄⋊C₈	semidirect
40. C₄×C₈	D₄₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₁₆₇₆	2
C₄⋊(C₄₁₉⋊C₈)	2
C₄×C₈×C₄₁₉	2
D₄⋊C₈	419