Subgroups — #14580898

Order 2624.

346 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊D₄₁	C₂	1	2	yes	maximal, complemented
2	#13050148	C₂	1	4	yes	maximal, complemented
3	#13050138	C₂	1	4	yes	maximal, complemented
4	#13050134	C₂	1	4	yes	maximal, complemented
5	C₂×C₂²⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×C₄₁	C₂²	1	1	yes	characteristic
7	C₂³×D₄₁	C₂²	1	2	yes	complemented
8	C₂²×C₄×C₄₁	C₂²	1	2	yes	complemented
9	C₂×C₂²⋊D₄₁	—	2	8	no	—
10	C₂×C₂²⋊D₄₁	C₂²	1	4	yes	—
11	C₂×C₂²⋊D₄₁	C₂²	1	4	yes	complemented
12	C₂²×Dic₄₁	C₂²	1	2	yes	complemented
13	C₂×D₁₆₄	C₂²	1	4	yes	complemented
14	C₂²⋊C₄×C₄₁	C₂²	1	4	yes	complemented
15	#10620281	C₂²	1	8	yes	complemented
16	C₄⋊Dic₄₁	C₂²	1	4	yes	complemented
17	C₂³×C₄₁	—	4	2	no	—
18	C₂³×C₄₁	D₄	1	4	yes	complemented
19	C₂³×C₄₁	C₂³	1	2	yes	—
20	C₂³×C₄₁	C₂³	1	1	yes	characteristic
21	C₂²×D₄₁	—	2	4	no	—
22	C₂²×D₄₁	—	4	4	no	—
23	C₂²×D₄₁	C₂³	1	4	yes	—
24	C₂×C₄×C₄₁	—	4	2	no	—
25	C₂×C₄×C₄₁	C₂³	1	4	yes	—
26	C₂²⋊D₄₁	—	2	8	no	—
27	C₂²⋊D₄₁	—	2	8	no	—
28	C₂×Dic₄₁	—	2	4	no	—
29	C₂×Dic₄₁	C₂³	1	4	yes	—
30	D₁₆₄	—	4	4	no	—
31	C₂²×C₄₁	—	2	2	no	—
32	C₂²×C₄₁	—	2	4	no	—
33	C₂²×C₄₁	—	4	4	no	—
34	C₂²×C₄₁	C₂×D₄	1	4	yes	—
35	C₂²×C₄₁	C₂×D₄	1	2	yes	—
36	C₂²×C₄₁	C₂⁴	1	1	yes	characteristic
37	C₂×D₄₁	—	2	4	no	—
38	C₂×D₄₁	—	2	8	no	—
39	C₂×D₄₁	—	4	8	no	—
40	C₄×C₄₁	—	2	4	no	—
41	Dic₄₁	—	2	4	no	—
42	C₂×C₄₁	—	2	4	no	—
43	C₂×C₄₁	—	2	2	no	—
44	C₂×C₄₁	C₂²×D₄	1	1	yes	characteristic
45	C₂×C₄₁	H(5,𝔽₂)	1	2	yes	—
46	D₄₁	—	4	4	no	—
47	#534	—	41	1	no	maximal, Sylow
48	C₄₁	#534	1	1	yes	characteristic, complemented, Sylow
49	C₂²×D₄	—	41	2	no	—
50	#123	—	41	4	no	—
51	#121	—	41	4	no	—
52	#120	—	41	4	no	—
53	C₂×C₂²⋊C₄	D₄₁	1	1	yes	characteristic, complemented
54	C₂⁴	—	41	2	no	—
55	C₂⁴	C₂×D₄₁	1	1	yes	characteristic
56	C₂×D₄	—	41	4	no	—
57	C₂×D₄	—	41	4	no	—
58	C₂×D₄	—	41	4	no	—
59	C₂×D₄	—	82	8	no	—
60	C₂²×C₄	—	41	2	no	—
61	C₂²×C₄	C₂×D₄₁	1	2	yes	complemented
62	C₄⋊C₄	—	41	4	no	—
63	C₂²⋊C₄	—	41	8	no	—
64	C₂²⋊C₄	C₂×D₄₁	1	4	yes	complemented
65	C₂³	—	4	2	no	—
66	C₂³	—	41	4	no	—
67	C₂³	—	82	4	no	—
68	C₂³	—	164	4	no	—
69	C₂³	D₁₆₄	1	4	yes	complemented
70	C₂³	C₂²×D₄₁	1	2	yes	—
71	C₂³	C₂²×D₄₁	1	1	yes	characteristic
72	D₄	—	82	8	no	—
73	D₄	—	82	8	no	—
74	D₄	—	164	4	no	—
75	C₂×C₄	—	4	2	no	—
76	C₂×C₄	—	41	4	no	—
77	C₂×C₄	—	82	4	no	—
78	C₂×C₄	C₂²×D₄₁	1	4	yes	—
79	C₂²	—	2	2	no	—
80	C₂²	—	2	4	no	—
81	C₂²	—	4	4	no	—
82	C₂²	—	82	4	no	—
83	C₂²	—	82	8	no	—
84	C₂²	—	164	8	no	—
85	C₂²	C₂×D₁₆₄	1	4	yes	—
86	C₂²	C₂×D₁₆₄	1	2	yes	—
87	C₂²	C₂³×D₄₁	1	1	yes	characteristic
88	C₄	—	2	4	no	—
89	C₄	—	82	4	no	—
90	C₂	—	2	4	no	—
91	C₂	—	2	2	no	—
92	C₂	—	164	4	no	—
93	C₂	C₂²×D₁₆₄	1	1	yes	characteristic
94	C₂	#13050209	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₁
commutator, socle, cosocle	C₂²×C₄₁	C₂⁴
Fitting	C₂×C₂²⋊C₄×C₄₁	C₂
hypercenter	C₂×C₂²⋊C₄	D₄₁
hypocenter	C₄₁	#534
LCS	G ▷ C₂²×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂²⋊D₄₁	C₂	semidirect
2. #13050148	C₂	semidirect
3. #13050138	C₂	semidirect
4. #13050134	C₂	semidirect
5. C₂×C₂²⋊C₄×C₄₁	C₂	semidirect
7. C₂³×D₄₁	C₂²	semidirect
8. C₂²×C₄×C₄₁	C₂²	semidirect
11. C₂×C₂²⋊D₄₁	C₂²	semidirect
12. C₂²×Dic₄₁	C₂²	semidirect
13. C₂×D₁₆₄	C₂²	semidirect
14. C₂²⋊C₄×C₄₁	C₂²	semidirect
15. #10620281	C₂²	semidirect
16. C₄⋊Dic₄₁	C₂²	semidirect
18. C₂³×C₄₁	D₄	semidirect
48. C₄₁	#534	semidirect
53. C₂×C₂²⋊C₄	D₄₁	semidirect
61. C₂²×C₄	C₂×D₄₁	semidirect
64. C₂²⋊C₄	C₂×D₄₁	semidirect
69. C₂³	D₁₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂²⋊D₄₁	2
#13050148	2
#13050138	2
#13050134	2
C₂×C₂²⋊C₄×C₄₁	2
#534	41