Subgroups — #14581215

Order 2624.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄₁⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#4223	C₂	1	1	yes	maximal, characteristic
3	M₁₆×D₄₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×#4223	C₄	1	2	yes	—
5	C₂×C₄×D₄₁	C₂²	1	1	yes	characteristic
6	#10620301	C₄	1	2	yes	complemented
7	C₄⋊#4223	—	2	1	no	—
8	C₄⋊#4223	—	2	1	no	—
9	#10620297	—	2	1	no	—
10	#10620296	—	2	1	no	—
11	M₁₆×C₄₁	C₄	1	1	yes	characteristic, complemented
12	#10620277	C₄	1	1	yes	characteristic, complemented
13	#10620272	—	2	1	no	—
14	C₈×D₄₁	—	2	1	no	—
15	C₂²×D₄₁	C₂×C₄	1	1	yes	characteristic
16	C₂×#4223	—	2	2	no	—
17	C₂×#4223	—	4	2	no	—
18	C₂×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₄₁	C₂×C₄	1	1	yes	characteristic
20	C₄×D₄₁	D₄	1	1	yes	characteristic
21	C₄×D₄₁	D₄	1	1	yes	characteristic
22	C₄×D₄₁	D₄	1	1	yes	characteristic
23	C₄×D₄₁	Q₈	1	1	yes	characteristic
24	#66493	—	2	2	no	—
25	C₈×C₄₁	—	2	1	no	—
26	#66491	—	2	1	no	—
27	C₂²×C₄₁	C₄²	1	1	yes	characteristic
28	C₂×D₄₁	—	2	1	no	—
29	C₂×D₄₁	—	2	1	no	—
30	C₂×D₄₁	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×D₄₁	C₄⋊C₄	1	1	yes	characteristic
32	#4223	—	4	2	no	—
33	C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₄×C₄₁	C₄⋊C₄	1	1	yes	characteristic
35	Dic₄₁	C₂²⋊C₄	1	1	yes	characteristic
36	Dic₄₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄₁	—	2	1	no	—
38	C₂×C₄₁	#95	1	1	yes	characteristic
39	D₄₁	—	2	1	no	—
40	D₄₁	#100	1	1	yes	characteristic
41	D₄₁	#101	1	1	yes	characteristic
42	#343	—	41	1	no	maximal, Sylow
43	C₄₁	#343	1	1	yes	characteristic, complemented, Sylow
44	C₂×M₁₆	—	41	1	no	—
45	C₂×M₁₆	—	41	1	no	—
46	C₂×C₄⋊C₄	—	41	1	no	—
47	C₂²×C₄	—	41	1	no	—
48	C₂²×C₄	—	41	2	no	—
49	M₁₆	—	41	2	no	—
50	M₁₆	—	41	1	no	—
51	M₁₆	—	82	1	no	—
52	M₁₆	—	82	1	no	—
53	M₁₆	#4223	1	1	yes	characteristic, complemented
54	C₂×C₈	—	82	1	no	—
55	C₂×C₈	—	82	1	no	—
56	C₄⋊C₄	—	82	1	no	—
57	C₄⋊C₄	—	82	1	no	—
58	C₂³	—	41	1	no	—
59	C₂×C₄	—	41	1	no	—
60	C₂×C₄	—	41	1	no	—
61	C₂×C₄	—	41	1	no	—
62	C₂×C₄	—	41	1	no	—
63	C₂×C₄	—	41	1	no	—
64	C₂×C₄	—	82	2	no	—
65	C₂×C₄	—	164	2	no	—
66	C₂×C₄	C₂×#4223	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	—	82	2	no	—
69	C₈	—	82	1	no	—
70	C₂²	—	41	1	no	—
71	C₂²	—	41	1	no	—
72	C₂²	—	82	1	no	—
73	C₂²	—	82	1	no	—
74	C₂²	C₄×#4223	1	1	yes	characteristic
75	C₄	—	41	1	no	—
76	C₄	—	41	1	no	—
77	C₄	—	164	2	no	—
78	C₄	C₄⋊#4223	1	1	yes	characteristic
79	C₄	C₂²⋊#4223	1	1	yes	characteristic
80	C₂	—	2	1	no	—
81	C₂	—	41	1	no	—
82	C₂	—	41	1	no	—
83	C₂	—	82	1	no	—
84	C₂	(C₂×#4223)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	(C₂×#4223)⋊C₄
commutator	C₂²×C₄₁	C₄²
Frattini	C₂×C₄	C₂×#4223
Fitting	M₁₆×C₄₁	C₄
socle	C₂×C₄₁	#95
prime radical	C₂²×D₄₁	C₂×C₄
cosocle	C₂×C₄×D₄₁	C₂²
hypercenter	M₁₆	#4223
hypocenter	C₄₁	#343
LCS	G ▷ C₂²×C₄₁ ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
6. #10620301	C₄	semidirect
11. M₁₆×C₄₁	C₄	semidirect
12. #10620277	C₄	semidirect
43. C₄₁	#343	semidirect
53. M₁₆	#4223	semidirect

Maximal subgroups

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

subgroup	index
D₄₁⋊M₁₆	2
C₂×C₄⋊#4223	2
M₁₆×D₄₁	2
#343	41