Subgroups — C₄²×#11719910 #28310183

Order 13568.

558 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11719910	C₂	1	6	yes	maximal
2	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11719910	C₂²	1	4	yes	—
4	C₄×#11719910	C₄	1	24	yes	direct product factor
5	C₂×C₄×#88322	C₄	1	3	yes	—
6	C₂×C₄×#88322	C₂²	1	3	yes	—
7	C₄³×C₅₃	C₄	1	1	yes	characteristic
8	C₂×#11719910	C₂×C₄	1	24	yes	—
9	C₂²×#88322	C₂×C₄	1	3	yes	—
10	C₂²×#88322	C₂³	1	1	yes	characteristic
11	C₄×#88322	C₈	1	12	yes	—
12	C₄×#88322	C₂×C₄	1	12	yes	—
13	C₂×C₄²×C₅₃	C₈	1	4	yes	—
14	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
15	C₂²×C₄×C₅₃	C₂×C₈	1	6	yes	—
16	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
17	C₄²×C₅₃	C₁₆	1	16	yes	complemented
18	C₄²×C₅₃	C₄²	1	6	yes	—
19	C₄²×C₅₃	C₂×C₈	1	6	yes	—
20	C₂×#88322	C₄²	1	6	yes	—
21	C₂×#88322	C₂×C₈	1	12	yes	—
22	C₂×#88322	C₂²×C₄	1	6	yes	—
23	#11719910	C₄²	1	16	yes	direct product factor
24	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
25	C₂×C₄×C₅₃	C₄×C₈	1	12	yes	—
26	C₂×C₄×C₅₃	C₂×C₁₆	1	24	yes	—
27	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
28	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
29	#88322	C₄×C₈	1	12	yes	—
30	#88322	C₂×C₄²	1	4	yes	—
31	C₄²×C₁₆	—	53	1	no	maximal, Sylow
32	C₂²×C₅₃	C₂×C₄×C₈	1	3	yes	—
33	C₂²×C₅₃	C₂²×C₁₆	1	4	yes	—
34	C₄×C₅₃	C₄×C₁₆	1	24	yes	complemented
35	C₄×C₅₃	C₄³	1	1	yes	characteristic
36	C₄×C₅₃	C₂×C₄×C₈	1	3	yes	—
37	C₂×C₄×C₁₆	—	53	6	no	—
38	C₄²×C₈	—	53	1	no	—
39	C₂×C₅₃	C₄²×C₈	1	1	yes	characteristic
40	C₂×C₅₃	C₂×C₄×C₁₆	1	6	yes	—
41	C₂²×C₁₆	—	53	4	no	—
42	C₂×C₄×C₈	—	53	3	no	—
43	C₂×C₄×C₈	—	53	3	no	—
44	C₄³	#6157	1	1	yes	characteristic
45	C₄×C₁₆	—	53	24	no	—
46	C₅₃	C₄²×C₁₆	1	1	yes	characteristic, complemented, Sylow
47	C₂²×C₈	—	53	3	no	—
48	C₂²×C₈	—	53	1	no	—
49	C₂×C₄²	#88324	1	4	yes	—
50	C₂×C₄²	C₂×#6157	1	3	yes	—
51	C₂×C₁₆	—	53	24	no	—
52	C₄×C₈	—	53	12	no	—
53	C₄×C₈	—	53	12	no	—
54	C₂²×C₄	C₂×#88324	1	6	yes	—
55	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
56	C₂×C₈	—	53	6	no	—
57	C₂×C₈	—	53	6	no	—
58	C₂×C₈	—	53	12	no	—
59	C₄²	#11719910	1	16	yes	direct product factor
60	C₄²	D₅₃⋊C₈	1	6	yes	—
61	C₄²	C₄×#6157	1	6	yes	—
62	C₁₆	—	53	16	no	—
63	C₂³	C₂²×#88324	1	1	yes	characteristic
64	C₂×C₄	C₄×#88324	1	12	yes	—
65	C₂×C₄	C₂×#11719910	1	24	yes	—
66	C₂×C₄	C₂×C₄×#6157	1	3	yes	—
67	C₂×C₄	C₂×D₅₃⋊C₈	1	3	yes	—
68	C₈	—	53	4	no	—
69	C₈	—	53	12	no	—
70	C₂²	C₂×C₄×#88324	1	3	yes	—
71	C₂²	C₂²×#11719910	1	4	yes	—
72	C₄	C₄×#11719910	1	24	yes	direct product factor
73	C₄	C₄²×#6157	1	1	yes	characteristic
74	C₄	C₄×D₅₃⋊C₈	1	3	yes	—
75	C₂	C₄²×#88324	1	1	yes	characteristic
76	C₂	C₂×C₄×#11719910	1	6	yes	—

	subgroup	G/H
center, hypercenter	C₄³	#6157
commutator, hypocenter	C₅₃	C₄²×C₁₆
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₄³×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×#88322	C₂³
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₄³

Complements

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

N	H ≅ G/N	type
4. C₄×#11719910	C₄	direct (N × H)
17. C₄²×C₅₃	C₁₆	semidirect
23. #11719910	C₄²	direct (N × H)
34. C₄×C₅₃	C₄×C₁₆	semidirect
46. C₅₃	C₄²×C₁₆	semidirect
59. C₄²	#11719910	direct (N × H)
72. C₄	C₄×#11719910	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×#11719910	2
C₄²×#88322	2
C₄²×C₁₆	53

Subgroups — C42×#11719910 #28310183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×#11719910 #28310183