Subgroups — #41000409

Order 26816.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20269395	C₂	1	1	yes	maximal, characteristic
2	C₂×#20269395	C₂	1	1	yes	maximal, characteristic
3	#27211653	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₄₁₉	C₂	1	1	yes	maximal, characteristic
5	#20269395	—	2	1	no	—
6	#20269395	C₄	1	2	yes	—
7	#20269395	C₄	1	2	yes	—
8	#20269395	C₂²	1	2	yes	—
9	C₂×C₄₁₉⋊C₈	—	2	1	no	—
10	C₂×C₄₁₉⋊C₈	C₂²	1	2	yes	—
11	C₂²×C₄×C₄₁₉	C₂²	1	1	yes	characteristic
12	M₁₆×C₄₁₉	—	2	1	no	—
13	M₁₆×C₄₁₉	C₂²	1	2	yes	—
14	C₂×C₈×C₄₁₉	—	2	1	no	—
15	C₄₁₉⋊C₈	—	2	2	no	—
16	C₄₁₉⋊C₈	C₂×C₄	1	4	yes	—
17	C₂³×C₄₁₉	Q₈	1	1	yes	characteristic
18	C₂×C₄×C₄₁₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₄₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₄₁₉	D₄	1	1	yes	characteristic
21	C₂×C₄×C₄₁₉	D₄	1	1	yes	characteristic
22	C₂×C₄×C₄₁₉	Q₈	1	1	yes	characteristic
23	C₂×C₄×C₄₁₉	C₂³	1	1	yes	characteristic
24	C₈×C₄₁₉	—	2	2	no	—
25	C₂²×C₄₁₉	—	4	1	no	—
26	C₂²×C₄₁₉	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₄₁₉	C₄⋊C₄	1	1	yes	characteristic
28	C₂²×C₄₁₉	C₂×Q₈	1	1	yes	characteristic
29	C₄×C₄₁₉	C₄⋊C₄	1	1	yes	characteristic
30	C₄×C₄₁₉	C₄⋊C₄	1	1	yes	characteristic
31	C₄×C₄₁₉	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₄₁₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄₁₉	—	2	1	no	—
34	C₂×C₄₁₉	—	2	1	no	—
35	C₂×C₄₁₉	—	2	1	no	—
36	C₂×C₄₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
37	C₄₁₉	#430	1	1	yes	characteristic, complemented, Sylow
38	#430	—	419	1	no	maximal, Sylow
39	C₂×M₁₆	—	419	1	no	—
40	C₂×M₁₆	—	419	1	no	—
41	C₂×M₁₆	D₄₁₉	1	1	yes	characteristic
42	#108	—	419	4	no	—
43	C₂²×C₄	C₂×D₄₁₉	1	1	yes	characteristic
44	M₁₆	—	2	1	no	—
45	M₁₆	—	838	1	no	—
46	M₁₆	—	419	2	no	—
47	M₁₆	—	419	2	no	—
48	M₁₆	—	419	2	no	—
49	M₁₆	C₂×D₄₁₉	1	2	yes	—
50	C₂×C₈	—	2	1	no	—
51	C₂×C₈	—	838	1	no	—
52	C₂×C₈	—	419	2	no	—
53	C₂³	Dic₈₃₈	1	1	yes	characteristic
54	C₂×C₄	C₄×D₄₁₉	1	1	yes	characteristic
55	C₂×C₄	C₄×D₄₁₉	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₄₁₉	1	1	yes	characteristic
57	C₂×C₄	C₂²⋊D₄₁₉	1	1	yes	characteristic
58	C₂×C₄	Dic₈₃₈	1	1	yes	characteristic
59	C₂×C₄	C₂²×D₄₁₉	1	1	yes	characteristic
60	C₈	—	2	2	no	—
61	C₈	—	838	2	no	—
62	C₈	—	419	4	no	—
63	C₂²	—	4	1	no	—
64	C₂²	Dic₄₁₉⋊C₄	1	1	yes	characteristic
65	C₂²	Dic₄₁₉⋊C₄	1	1	yes	characteristic
66	C₂²	C₂×Dic₈₃₈	1	1	yes	characteristic
67	C₄	Dic₄₁₉⋊C₄	1	1	yes	characteristic
68	C₄	Dic₄₁₉⋊C₄	1	1	yes	characteristic
69	C₄	C₂×C₄×D₄₁₉	1	1	yes	characteristic
70	C₄	C₂×C₂²⋊D₄₁₉	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	—	2	1	no	—
73	C₂	—	2	1	no	—
74	C₂	C₂×Dic₄₁₉⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₂²⋊D₄₁₉
commutator	C₄×C₄₁₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₄₁₉
Fitting	C₂×M₁₆×C₄₁₉	C₂
socle	C₂×C₄₁₉	C₂×C₄⋊C₄
prime radical	C₂³×C₄₁₉	Q₈
cosocle	C₂×C₄×C₄₁₉	C₂³
hypercenter	C₂×M₁₆	D₄₁₉
hypocenter	C₄₁₉	#430
LCS	G ▷ C₄×C₄₁₉ ▷ C₂×C₄₁₉ ▷ C₄₁₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #27211653	C₂	semidirect
37. C₄₁₉	#430	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20269395	2
#27211653	2
C₂×M₁₆×C₄₁₉	2
#430	419