Subgroups — #41000175

Order 26816.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₃₅₂	C₂	1	1	yes	maximal, characteristic
2	#27211654	C₂	1	1	yes	maximal, characteristic, complemented
3	M₃₂×C₄₁₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₆₇₆	—	2	1	no	—
5	D₃₃₅₂	—	2	1	no	—
6	D₃₃₅₂	C₄	1	2	yes	—
7	#20269395	—	2	1	no	—
8	C₂×C₈×C₄₁₉	C₂²	1	1	yes	characteristic
9	C₁₆×C₄₁₉	—	2	1	no	—
10	C₂²×D₄₁₉	—	4	1	no	—
11	D₁₆₇₆	—	2	2	no	—
12	D₁₆₇₆	—	4	1	no	—
13	C₄₁₉⋊C₈	—	4	1	no	—
14	C₂×C₄×C₄₁₉	D₄	1	1	yes	characteristic
15	C₈×C₄₁₉	C₂×C₄	1	1	yes	characteristic
16	C₈×C₄₁₉	D₄	1	1	yes	characteristic
17	C₂²×C₄₁₉	SD₁₆	1	1	yes	characteristic
18	C₂×D₄₁₉	—	4	2	no	—
19	C₂×D₄₁₉	—	8	2	no	—
20	C₄×C₄₁₉	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₄₁₉	D₈	1	1	yes	characteristic
22	C₂×C₄₁₉	—	2	1	no	—
23	C₂×C₄₁₉	#102	1	1	yes	characteristic
24	D₄₁₉	—	8	2	no	—
25	C₄₁₉	#361	1	1	yes	characteristic, complemented, Sylow
26	#361	—	419	1	no	maximal, Sylow
27	C₂×D₈	—	419	1	no	—
28	M₃₂	D₄₁₉	1	1	yes	characteristic, complemented
29	#108	—	419	1	no	—
30	C₂×D₄	—	838	1	no	—
31	D₈	—	838	1	no	—
32	D₈	—	419	2	no	—
33	M₁₆	—	838	1	no	—
34	C₂×C₈	C₂×D₄₁₉	1	1	yes	characteristic
35	C₁₆	—	2	1	no	—
36	C₂³	—	1676	1	no	—
37	D₄	—	838	2	no	—
38	D₄	—	1676	1	no	—
39	C₂×C₄	D₁₆₇₆	1	1	yes	characteristic
40	C₈	—	1676	1	no	—
41	C₈	C₄×D₄₁₉	1	1	yes	characteristic
42	C₈	C₂²⋊D₄₁₉	1	1	yes	characteristic
43	C₂²	—	3352	2	no	—
44	C₂²	—	1676	2	no	—
45	C₂²	#20269400	1	1	yes	characteristic
46	C₄	(C₂×D₄₁₉)⋊C₄	1	1	yes	characteristic
47	C₄	D₃₃₅₂	1	1	yes	characteristic
48	C₂	—	2	1	no	—
49	C₂	—	3352	2	no	—
50	C₂	#27211640	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27211640
commutator	C₈×C₄₁₉	C₂×C₄
Frattini	C₂×C₈	C₂×D₄₁₉
Fitting	M₃₂×C₄₁₉	C₂
socle	C₂×C₄₁₉	#102
prime radical	C₂×D₃₃₅₂	C₂
cosocle	C₂×C₈×C₄₁₉	C₂²
hypercenter	M₃₂	D₄₁₉
hypocenter	C₄₁₉	#361
LCS	G ▷ C₈×C₄₁₉ ▷ C₄×C₄₁₉ ▷ C₂×C₄₁₉ ▷ C₄₁₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. #27211654	C₂	semidirect
3. M₃₂×C₄₁₉	C₂	semidirect
25. C₄₁₉	#361	semidirect
28. M₃₂	D₄₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃₃₅₂	2
#27211654	2
M₃₂×C₄₁₉	2
#361	419