Subgroups — #41003719

Order 26944.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄₂₁⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#13132997	C₂	1	1	yes	maximal, characteristic
3	M₁₆×D₄₂₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13132997	C₄	1	2	yes	—
5	#20269781	—	2	1	no	—
6	#20269780	C₄	1	2	yes	complemented
7	D₄₂₁⋊C₈	—	2	1	no	—
8	C₄⋊#13132997	—	2	1	no	—
9	C₄⋊#13132997	—	2	1	no	—
10	C₂×C₄×D₄₂₁	C₂²	1	1	yes	characteristic
11	#20269755	C₄	1	1	yes	characteristic, complemented
12	#20269754	—	2	1	no	—
13	C₈×D₄₂₁	—	2	1	no	—
14	M₁₆×C₄₂₁	C₄	1	1	yes	characteristic, complemented
15	C₂×#13132997	—	2	2	no	—
16	C₂×#13132997	—	4	2	no	—
17	#16809897	—	2	2	no	—
18	C₂²×D₄₂₁	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₄₂₁	C₂×C₄	1	1	yes	characteristic
20	C₄×D₄₂₁	D₄	1	1	yes	characteristic
21	C₄×D₄₂₁	D₄	1	1	yes	characteristic
22	C₄×D₄₂₁	D₄	1	1	yes	characteristic
23	C₄×D₄₂₁	Q₈	1	1	yes	characteristic
24	#16809890	—	2	1	no	—
25	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
26	C₈×C₄₂₁	—	2	1	no	—
27	C₂²×C₄₂₁	C₄²	1	1	yes	characteristic
28	C₂×D₄₂₁	—	2	1	no	—
29	C₂×D₄₂₁	—	2	1	no	—
30	C₂×D₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×D₄₂₁	C₄⋊C₄	1	1	yes	characteristic
32	#13132997	—	4	2	no	—
33	C₄×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₄×C₄₂₁	C₄⋊C₄	1	1	yes	characteristic
35	Dic₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
36	Dic₄₂₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄₂₁	—	2	1	no	—
38	C₂×C₄₂₁	#95	1	1	yes	characteristic
39	D₄₂₁	—	2	1	no	—
40	D₄₂₁	#100	1	1	yes	characteristic
41	D₄₂₁	#101	1	1	yes	characteristic
42	C₄₂₁	#343	1	1	yes	characteristic, complemented, Sylow
43	#343	—	421	1	no	maximal, Sylow
44	C₂×M₁₆	—	421	1	no	—
45	C₂×M₁₆	—	421	1	no	—
46	C₂×C₄⋊C₄	—	421	1	no	—
47	C₂²×C₄	—	421	2	no	—
48	C₂²×C₄	—	421	1	no	—
49	M₁₆	—	842	1	no	—
50	M₁₆	—	842	1	no	—
51	M₁₆	—	421	2	no	—
52	M₁₆	—	421	1	no	—
53	M₁₆	#13132997	1	1	yes	characteristic, complemented
54	C₂×C₈	—	842	1	no	—
55	C₂×C₈	—	842	1	no	—
56	C₄⋊C₄	—	842	1	no	—
57	C₄⋊C₄	—	842	1	no	—
58	C₂³	—	421	1	no	—
59	C₂×C₄	—	842	2	no	—
60	C₂×C₄	—	1684	2	no	—
61	C₂×C₄	—	421	1	no	—
62	C₂×C₄	—	421	1	no	—
63	C₂×C₄	—	421	1	no	—
64	C₂×C₄	—	421	1	no	—
65	C₂×C₄	—	421	1	no	—
66	C₂×C₄	C₂×#13132997	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	—	842	2	no	—
69	C₈	—	842	1	no	—
70	C₂²	—	842	1	no	—
71	C₂²	—	842	1	no	—
72	C₂²	—	421	1	no	—
73	C₂²	—	421	1	no	—
74	C₂²	C₄×#13132997	1	1	yes	characteristic
75	C₄	—	1684	2	no	—
76	C₄	—	421	1	no	—
77	C₄	—	421	1	no	—
78	C₄	C₂²⋊#13132997	1	1	yes	characteristic
79	C₄	C₄⋊#13132997	1	1	yes	characteristic
80	C₂	—	2	1	no	—
81	C₂	—	842	1	no	—
82	C₂	—	421	1	no	—
83	C₂	—	421	1	no	—
84	C₂	(C₂×#13132997)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	(C₂×#13132997)⋊C₄
commutator	C₂²×C₄₂₁	C₄²
Frattini	C₂×C₄	C₂×#13132997
Fitting	M₁₆×C₄₂₁	C₄
socle	C₂×C₄₂₁	#95
prime radical	C₂²×D₄₂₁	C₂×C₄
cosocle	C₂×C₄×D₄₂₁	C₂²
hypercenter	M₁₆	#13132997
hypocenter	C₄₂₁	#343
LCS	G ▷ C₂²×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
6. #20269780	C₄	semidirect
11. #20269755	C₄	semidirect
14. M₁₆×C₄₂₁	C₄	semidirect
42. C₄₂₁	#343	semidirect
53. M₁₆	#13132997	semidirect

Maximal subgroups

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

subgroup	index
D₄₂₁⋊M₁₆	2
C₂×C₄⋊#13132997	2
M₁₆×D₄₂₁	2
#343	421