Subgroups — C₂×C₄×#11719910 #20290911

Order 6784.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₂×C₄²	#6157
commutator, hypocenter	C₅₃	C₂×C₄×C₁₆
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₂×C₄²×C₅₃	C₄
socle	C₂³×C₅₃	C₂×C₈
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×#11719910	C₂	direct (N × H)
4. C₂×#11719910	C₄	direct (N × H)
20. #11719910	C₂×C₄	direct (N × H)
22. C₂×C₄×C₅₃	C₁₆	semidirect
35. C₄×C₅₃	C₂×C₁₆	semidirect
39. C₂×C₅₃	C₄×C₁₆	semidirect
45. C₅₃	C₂×C₄×C₁₆	semidirect
64. C₂×C₄	#11719910	direct (N × H)
77. C₄	C₂×#11719910	direct (N × H)
80. C₂	C₄×#11719910	direct (N × H)

Subgroups — C2×C4×#11719910 #20290911

Conjugacy classes of subgroups

Complements

Subgroups — C₂×C₄×#11719910 #20290911