Subgroups — #14580810

Order 2624.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₆₄	C₂	1	1	yes	maximal, characteristic
2	C₄×D₁₆₄	C₂	1	4	yes	maximal, complemented
3	#13050126	C₂	1	4	yes	maximal, complemented
4	#13050123	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊C₄×D₄₁	C₂	1	4	yes	maximal, complemented
6	#117×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄₁	C₂²	1	2	yes	—
8	C₂²×C₄×C₄₁	C₂²	1	1	yes	characteristic
9	C₂×D₁₆₄	C₄	1	8	yes	complemented
10	C₂×D₁₆₄	C₂²	1	2	yes	—
11	C₂×D₁₆₄	C₂²	1	2	yes	—
12	C₂×C₄×D₄₁	C₂²	1	8	yes	complemented
13	C₄⋊C₄×C₄₁	C₂²	1	2	yes	complemented
14	C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
15	C₄²×C₄₁	C₂²	1	2	yes	complemented
16	C₂²⋊Dic₄₁	C₂²	1	2	yes	—
17	#10620281	C₂²	1	8	yes	complemented
18	C₄⋊Dic₄₁	C₂²	1	2	yes	complemented
19	C₄×Dic₄₁	C₂²	1	2	yes	complemented
20	C₂³×C₄₁	C₂³	1	1	yes	characteristic
21	C₂²×D₄₁	—	4	4	no	—
22	C₂²×D₄₁	C₂×C₄	1	8	yes	complemented
23	C₂²×D₄₁	C₂³	1	4	yes	—
24	C₂×C₄×C₄₁	C₂×C₄	1	4	yes	complemented
25	C₂×C₄×C₄₁	C₂³	1	4	yes	—
26	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
27	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
28	C₂×Dic₄₁	C₂³	1	4	yes	—
29	D₁₆₄	C₂×C₄	1	8	yes	complemented
30	D₁₆₄	C₂×C₄	1	8	yes	complemented
31	C₄×D₄₁	—	4	8	no	—
32	C₂²×C₄₁	—	2	2	no	—
33	C₂²×C₄₁	C₂²×C₄	1	2	yes	—
34	C₂²×C₄₁	C₂⁴	1	1	yes	characteristic
35	C₂×D₄₁	—	2	8	no	—
36	C₂×D₄₁	—	4	8	no	—
37	C₂×D₄₁	C₂²×C₄	1	8	yes	—
38	C₄×C₄₁	—	2	4	no	—
39	C₄×C₄₁	C₂²×C₄	1	2	yes	—
40	C₄×C₄₁	C₂²×C₄	1	2	yes	—
41	Dic₄₁	—	2	4	no	—
42	C₂×C₄₁	—	2	2	no	—
43	C₂×C₄₁	C₂³×C₄	1	1	yes	characteristic
44	C₂×C₄₁	H(5,𝔽₂)	1	2	yes	—
45	D₄₁	—	2	8	no	—
46	#518	—	41	1	no	maximal, Sylow
47	C₄₁	#518	1	1	yes	characteristic, complemented, Sylow
48	C₂²×D₄	—	41	1	no	—
49	C₄×D₄	—	41	4	no	—
50	C₄×D₄	—	41	4	no	—
51	#117	—	41	1	no	—
52	#117	D₄₁	1	1	yes	characteristic, complemented
53	C₂×C₂²⋊C₄	—	41	4	no	—
54	C₂⁴	—	41	2	no	—
55	C₂×D₄	—	41	8	no	—
56	C₂×D₄	—	41	2	no	—
57	C₂×D₄	—	41	2	no	—
58	C₂²×C₄	—	41	8	no	—
59	C₂²×C₄	C₂×D₄₁	1	1	yes	characteristic
60	C₄⋊C₄	—	41	2	no	—
61	C₄⋊C₄	C₂×D₄₁	1	2	yes	complemented
62	C₂²⋊C₄	—	41	2	no	—
63	C₂²⋊C₄	—	41	8	no	—
64	C₂²⋊C₄	C₂×D₄₁	1	2	yes	—
65	C₄²	—	41	2	no	—
66	C₄²	C₂×D₄₁	1	2	yes	complemented
67	C₂³	—	41	4	no	—
68	C₂³	—	41	8	no	—
69	C₂³	—	164	4	no	—
70	C₂³	C₂²×D₄₁	1	1	yes	characteristic
71	D₄	—	41	8	no	—
72	D₄	—	41	8	no	—
73	C₂×C₄	—	41	4	no	—
74	C₂×C₄	—	164	8	no	—
75	C₂×C₄	C₄×D₄₁	1	4	yes	complemented
76	C₂×C₄	C₂²×D₄₁	1	4	yes	—
77	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
78	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
79	C₂²	—	2	2	no	—
80	C₂²	—	41	8	no	—
81	C₂²	—	82	8	no	—
82	C₂²	—	164	8	no	—
83	C₂²	C₂×C₄×D₄₁	1	2	yes	—
84	C₂²	C₂³×D₄₁	1	1	yes	characteristic
85	C₄	—	2	4	no	—
86	C₄	—	82	4	no	—
87	C₄	C₂×C₄×D₄₁	1	2	yes	—
88	C₄	C₂×C₄×D₄₁	1	2	yes	—
89	C₂	—	2	2	no	—
90	C₂	—	82	8	no	—
91	C₂	C₂²×C₄×D₄₁	1	1	yes	characteristic
92	C₂	#13050210	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₁
commutator	C₂×C₄₁	C₂³×C₄
Fitting	#117×C₄₁	C₂
socle, cosocle	C₂²×C₄₁	C₂⁴
prime radical	C₂²×D₁₆₄	C₂
hypercenter	#117	D₄₁
hypocenter	C₄₁	#518
LCS	G ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₄×D₁₆₄	C₂	semidirect
3. #13050126	C₂	semidirect
4. #13050123	C₂	semidirect
5. C₂²⋊C₄×D₄₁	C₂	semidirect
6. #117×C₄₁	C₂	semidirect
9. C₂×D₁₆₄	C₄	semidirect
12. C₂×C₄×D₄₁	C₂²	semidirect
13. C₄⋊C₄×C₄₁	C₂²	semidirect
15. C₄²×C₄₁	C₂²	semidirect
17. #10620281	C₂²	semidirect
18. C₄⋊Dic₄₁	C₂²	semidirect
19. C₄×Dic₄₁	C₂²	semidirect
22. C₂²×D₄₁	C₂×C₄	semidirect
24. C₂×C₄×C₄₁	C₂×C₄	semidirect
29. D₁₆₄	C₂×C₄	semidirect
30. D₁₆₄	C₂×C₄	semidirect
47. C₄₁	#518	semidirect
52. #117	D₄₁	semidirect
61. C₄⋊C₄	C₂×D₄₁	semidirect
66. C₄²	C₂×D₄₁	semidirect
75. C₂×C₄	C₄×D₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₁₆₄	2
C₄×D₁₆₄	2
#13050126	2
#13050123	2
C₂²⋊C₄×D₄₁	2
#117×C₄₁	2
#518	41