Subgroups — #14581347

Order 2624.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13050243	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#4223	C₂	1	1	yes	maximal, characteristic
3	#13050233	C₂	1	2	yes	maximal, complemented
4	#13050232	C₂	1	2	yes	maximal, complemented
5	M₁₆×D₄₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#4223	—	2	1	no	—
7	C₂×C₄×D₄₁	C₂²	1	1	yes	characteristic
8	C₂²⋊#4223	—	2	1	no	—
9	C₄⋊#4223	—	2	1	no	—
10	C₄⋊#4223	C₂²	1	1	yes	characteristic
11	C₄⋊#4223	C₂²	1	2	yes	—
12	C₄⋊#4223	C₂²	1	1	yes	characteristic
13	C₄×#4223	—	2	1	no	—
14	M₁₆×C₄₁	C₄	1	1	yes	characteristic, complemented
15	#10620277	C₄	1	1	yes	characteristic, complemented
16	#10620272	C₄	1	2	yes	complemented
17	C₈×D₄₁	C₂²	1	2	yes	—
18	C₂²×D₄₁	D₄	1	1	yes	characteristic
19	C₂×#4223	—	2	1	no	—
20	C₂×#4223	—	2	1	no	—
21	C₂×#4223	—	2	2	no	—
22	C₂×#4223	—	4	2	no	—
23	C₂×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₄₁	Q₈	1	1	yes	characteristic
25	C₄×D₄₁	C₂×C₄	1	1	yes	characteristic
26	C₄×D₄₁	D₄	1	1	yes	characteristic
27	C₄×D₄₁	Q₈	1	1	yes	characteristic
28	C₄×D₄₁	C₂³	1	1	yes	characteristic
29	C₈×C₄₁	C₂×C₄	1	2	yes	complemented
30	#66491	C₂×C₄	1	2	yes	complemented
31	C₂²×C₄₁	C₄⋊C₄	1	1	yes	characteristic
32	C₂×D₄₁	—	2	1	no	—
33	C₂×D₄₁	—	2	1	no	—
34	C₂×D₄₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂×D₄₁	C₂×D₄	1	1	yes	characteristic
36	#4223	—	4	1	no	—
37	#4223	—	4	2	no	—
38	#4223	—	4	1	no	—
39	C₄×C₄₁	C₄⋊C₄	1	1	yes	characteristic
40	C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
41	Dic₄₁	C₄⋊C₄	1	1	yes	characteristic
42	Dic₄₁	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄₁	—	2	1	no	—
44	C₂×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
45	D₄₁	—	2	1	no	—
46	D₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
47	D₄₁	#137	1	1	yes	characteristic
48	#428	—	41	1	no	maximal, Sylow
49	C₄₁	#428	1	1	yes	characteristic, complemented, Sylow
50	C₂×M₁₆	—	41	1	no	—
51	#117	—	41	1	no	—
52	C₂×C₄⋊C₄	—	41	1	no	—
53	#107	—	41	2	no	—
54	#106	—	41	2	no	—
55	C₂²×C₄	—	41	1	no	—
56	C₂²×C₄	—	82	1	no	—
57	M₁₆	—	41	1	no	—
58	M₁₆	—	41	2	no	—
59	M₁₆	#4223	1	1	yes	characteristic, complemented
60	C₂×C₈	—	41	2	no	—
61	C₄⋊C₄	—	41	1	no	—
62	C₄⋊C₄	—	41	2	no	—
63	C₄⋊C₄	—	41	1	no	—
64	C₄⋊C₄	—	82	1	no	—
65	C₂²⋊C₄	—	82	1	no	—
66	C₄²	—	82	1	no	—
67	C₂³	—	41	1	no	—
68	C₂×C₄	—	41	1	no	—
69	C₂×C₄	—	41	1	no	—
70	C₂×C₄	—	41	1	no	—
71	C₂×C₄	—	41	1	no	—
72	C₂×C₄	—	41	1	no	—
73	C₂×C₄	—	82	1	no	—
74	C₂×C₄	—	82	1	no	—
75	C₂×C₄	—	82	2	no	—
76	C₂×C₄	—	164	2	no	—
77	C₂×C₄	C₂×#4223	1	1	yes	characteristic
78	C₈	—	41	2	no	—
79	C₈	C₂×#4223	1	2	yes	complemented
80	C₂²	—	41	1	no	—
81	C₂²	—	41	1	no	—
82	C₂²	—	82	1	no	—
83	C₂²	—	82	1	no	—
84	C₂²	C₄⋊#4223	1	1	yes	characteristic
85	C₄	—	41	1	no	—
86	C₄	—	41	1	no	—
87	C₄	—	164	2	no	—
88	C₄	—	164	1	no	—
89	C₄	—	164	1	no	—
90	C₄	C₄⋊#4223	1	1	yes	characteristic
91	C₄	C₂²×#4223	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	41	1	no	—
94	C₂	—	41	1	no	—
95	C₂	—	82	1	no	—
96	C₂	C₂×C₄⋊#4223	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₄⋊#4223
commutator	C₄×C₄₁	C₂²×C₄
Frattini	C₄	C₂²×#4223
Fitting	M₁₆×C₄₁	C₄
socle	C₂×C₄₁	C₂×C₄⋊C₄
prime radical	C₂²×D₄₁	D₄
cosocle	C₄×D₄₁	C₂³
hypercenter	M₁₆	#4223
hypocenter	C₄₁	#428
LCS	G ▷ C₄×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
3. #13050233	C₂	semidirect
4. #13050232	C₂	semidirect
14. M₁₆×C₄₁	C₄	semidirect
15. #10620277	C₄	semidirect
16. #10620272	C₄	semidirect
29. C₈×C₄₁	C₂×C₄	semidirect
30. #66491	C₂×C₄	semidirect
49. C₄₁	#428	semidirect
59. M₁₆	#4223	semidirect
79. C₈	C₂×#4223	semidirect

Maximal subgroups

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

subgroup	index
#13050243	2
C₂×C₄⋊#4223	2
#13050233	2
#13050232	2
M₁₆×D₄₁	2
#428	41