Subgroups — C₂×#16811681 #20291679

Order 6784.

368 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811681	C₂	1	8	yes	maximal, direct product factor
2	C₂×#13133291	C₂	1	3	yes	maximal, complemented
3	C₂²×#11719910	C₂	1	3	yes	maximal, complemented
4	C₂×#13133224	C₂	1	1	yes	maximal, characteristic
5	#13133291	C₂²	1	12	yes	complemented
6	C₂×#11719910	C₂²	1	1	yes	characteristic
7	C₂×#11719910	C₂²	1	3	yes	—
8	C₂×#11719910	C₂²	1	12	yes	complemented
9	#13133224	C₄	1	4	yes	—
10	#13133224	C₂²	1	4	yes	—
11	C₂×#11719917	C₄	1	3	yes	—
12	C₂²×#88322	C₂²	1	3	yes	—
13	C₂×#41×C₅₃	C₄	1	1	yes	characteristic
14	#41×C₅₃	C₈	1	4	yes	—
15	#41×C₅₃	C₂×C₄	1	4	yes	—
16	C₂×Q₈×C₅₃	C₈	1	1	yes	characteristic
17	C₂×D₄×C₅₃	C₈	1	3	yes	—
18	C₂²×C₄×C₅₃	C₂×C₄	1	3	yes	—
19	#11719917	C₂×C₄	1	12	yes	—
20	C₂×#88322	C₂×C₄	1	3	yes	—
21	C₂×#88322	C₂×C₄	1	6	yes	—
22	C₂×#88322	C₂³	1	1	yes	characteristic
23	C₂×#88322	C₂³	1	6	yes	—
24	#11719910	C₂³	1	2	yes	—
25	#11719910	C₂³	1	6	yes	—
26	C₂³×C₅₃	C₂×C₈	1	3	yes	—
27	Q₈×C₅₃	C₂×C₈	1	4	yes	—
28	D₄×C₅₃	C₂×C₈	1	12	yes	—
29	C₂×C₄×C₅₃	C₂×C₈	1	3	yes	—
30	C₂×C₄×C₅₃	C₂×C₈	1	6	yes	—
31	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
32	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	#88322	C₂²×C₄	1	1	yes	characteristic
34	#88322	C₂²×C₄	1	6	yes	—
35	#88322	C₂⁴	1	1	yes	characteristic
36	C₂²×C₅₃	—	2	6	no	—
37	C₂²×C₅₃	C₂²×C₈	1	6	yes	—
38	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
39	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
40	C₄×C₅₃	C₂²×C₈	1	6	yes	—
41	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
42	C₂×#504	—	53	1	no	maximal, Sylow
43	C₂×C₅₃	—	2	6	no	—
44	C₂×C₅₃	#504	1	2	yes	complemented
45	C₂×C₅₃	C₂³×C₈	1	1	yes	characteristic
46	C₂×#131	—	53	1	no	—
47	#504	—	53	8	no	—
48	C₂×M₃₂	—	53	3	no	—
49	C₂²×C₁₆	—	53	3	no	—
50	C₅₃	C₂×#504	1	1	yes	characteristic, complemented, Sylow
51	C₂×#41	#6157	1	1	yes	characteristic
52	#131	—	53	4	no	—
53	#131	—	53	4	no	—
54	C₂×M₁₆	—	53	3	no	—
55	C₂²×C₈	—	53	3	no	—
56	M₃₂	—	53	12	no	—
57	C₂×C₁₆	—	53	1	no	—
58	C₂×C₁₆	—	53	3	no	—
59	C₂×C₁₆	—	53	12	no	—
60	#41	#88324	1	4	yes	—
61	#41	C₂×#6157	1	4	yes	—
62	C₂×Q₈	#88324	1	1	yes	characteristic
63	C₂×D₄	#88324	1	3	yes	—
64	C₂²×C₄	C₂×#6157	1	3	yes	—
65	M₁₆	—	53	12	no	—
66	C₂×C₈	—	53	6	no	—
67	C₂×C₈	—	53	6	no	—
68	C₂×C₈	—	53	1	no	—
69	C₂×C₈	—	53	3	no	—
70	C₁₆	—	53	2	no	—
71	C₁₆	—	53	6	no	—
72	C₂³	C₂×#88324	1	3	yes	—
73	Q₈	C₂×#88324	1	4	yes	—
74	D₄	C₂×#88324	1	12	yes	—
75	C₂×C₄	C₂×#88324	1	3	yes	—
76	C₂×C₄	C₂×#88324	1	6	yes	—
77	C₂×C₄	C₂²×#6157	1	6	yes	—
78	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
79	C₈	—	53	1	no	—
80	C₈	—	53	6	no	—
81	C₈	—	53	1	no	—
82	C₂²	—	2	6	no	—
83	C₂²	C₂²×#88324	1	6	yes	—
84	C₂²	C₂²×#88324	1	1	yes	characteristic
85	C₄	C₂²×#88324	1	1	yes	characteristic
86	C₄	C₂²×#88324	1	6	yes	—
87	C₄	C₂³×#6157	1	1	yes	characteristic
88	C₂	—	2	6	no	—
89	C₂	#16811681	1	2	yes	direct product factor
90	C₂	C₂³×#88324	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#6157
commutator	C₂×C₅₃	C₂³×C₈
Frattini	C₄	C₂³×#6157
Fitting	C₂×#41×C₅₃	C₄
socle	C₂²×C₅₃	C₂²×C₈
hypercenter	C₂×#41	#6157
hypocenter	C₅₃	C₂×#504
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #16811681	C₂	direct (N × H)
2. C₂×#13133291	C₂	semidirect
3. C₂²×#11719910	C₂	semidirect
5. #13133291	C₂²	semidirect
8. C₂×#11719910	C₂²	semidirect
44. C₂×C₅₃	#504	semidirect
50. C₅₃	C₂×#504	semidirect
89. C₂	#16811681	direct (N × H)

Subgroups — C2×#16811681 #20291679

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811681 #20291679