Subgroups — #41003715

Order 26944.

90 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×D₄₂₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	M₁₆×D₄₂₁	C₂	1	1	yes	maximal, characteristic
4	#20269781	C₄	1	2	yes	—
5	#20269780	—	2	1	no	—
6	D₄₂₁⋊C₈	—	2	1	no	—
7	D₄₂₁⋊C₈	—	2	1	no	—
8	D₄₂₁⋊C₈	C₄	1	2	yes	—
9	C₂×#16809897	—	2	1	no	—
10	C₂×C₄×D₄₂₁	C₂²	1	1	yes	characteristic
11	#20269755	C₄	1	1	yes	characteristic
12	#20269754	—	2	1	no	—
13	C₈×D₄₂₁	—	2	1	no	—
14	M₁₆×C₄₂₁	C₄	1	1	yes	characteristic
15	#16809897	—	2	2	no	—
16	#16809897	—	2	2	no	—
17	C₂²×D₄₂₁	Q₈	1	1	yes	characteristic
18	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
19	C₄×D₄₂₁	C₂×C₄	1	1	yes	characteristic
20	C₄×D₄₂₁	C₂×C₄	1	1	yes	characteristic
21	C₄×D₄₂₁	D₄	1	1	yes	characteristic
22	C₄×D₄₂₁	D₄	1	1	yes	characteristic
23	#16809890	—	2	1	no	—
24	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
25	C₈×C₄₂₁	—	2	1	no	—
26	C₂²×C₄₂₁	C₄⋊C₄	1	1	yes	characteristic
27	C₂×D₄₂₁	—	2	2	no	—
28	C₂×D₄₂₁	C₄⋊C₄	1	1	yes	characteristic
29	C₂×D₄₂₁	C₄⋊C₄	1	1	yes	characteristic
30	C₄×C₄₂₁	C₄²	1	1	yes	characteristic
31	C₄×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
32	Dic₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
33	Dic₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄₂₁	—	2	1	no	—
35	C₂×C₄₂₁	#95	1	1	yes	characteristic
36	D₄₂₁	—	2	1	no	—
37	D₄₂₁	#108	1	2	yes	—
38	C₄₂₁	#341	1	1	yes	characteristic, complemented, Sylow
39	#341	—	421	1	no	maximal, Sylow
40	C₂×M₁₆	—	421	1	no	—
41	C₂×M₁₆	—	421	1	no	—
42	C₂²×C₈	—	421	1	no	—
43	C₂²×C₄	—	421	1	no	—
44	M₁₆	—	842	1	no	—
45	M₁₆	—	842	1	no	—
46	M₁₆	—	421	2	no	—
47	M₁₆	—	421	1	no	—
48	M₁₆	#13132997	1	1	yes	characteristic
49	C₂×C₈	—	842	1	no	—
50	C₂×C₈	—	842	1	no	—
51	C₂×C₈	—	842	1	no	—
52	C₂×C₈	—	842	1	no	—
53	C₂×C₈	—	421	2	no	—
54	C₂³	—	421	1	no	—
55	C₂×C₄	—	421	1	no	—
56	C₂×C₄	—	421	1	no	—
57	C₂×C₄	—	421	1	no	—
58	C₂×C₄	—	421	1	no	—
59	C₂×C₄	—	421	1	no	—
60	C₂×C₄	C₂×#13132997	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	—	842	2	no	—
63	C₈	—	842	2	no	—
64	C₈	—	842	1	no	—
65	C₂²	—	842	2	no	—
66	C₂²	—	421	1	no	—
67	C₂²	—	421	1	no	—
68	C₂²	C₄⋊#13132997	1	1	yes	characteristic
69	C₄	—	421	1	no	—
70	C₄	—	421	1	no	—
71	C₄	C₄×#13132997	1	1	yes	characteristic
72	C₄	C₂²⋊#13132997	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	—	842	1	no	—
75	C₂	—	421	2	no	—
76	C₂	(C₂×#13132997)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊#13132997
commutator	C₄×C₄₂₁	C₄²
Frattini	C₂×C₄	C₂×#13132997
Fitting	M₁₆×C₄₂₁	C₄
socle	C₂×C₄₂₁	#95
prime radical	C₂²×D₄₂₁	Q₈
cosocle	C₂×C₄×D₄₂₁	C₂²
hypercenter	M₁₆	#13132997
hypocenter	C₄₂₁	#341
LCS	G ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
38. C₄₂₁	#341	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×D₄₂₁⋊C₈	2
M₁₆×D₄₂₁	2
#341	421