Subgroups — C₂×C₄×#11719917 #20283124

Order 6784.

656 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11719917	C₂	1	2	yes	maximal
2	C₄×#11719917	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13133128	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×#88322	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₄	1	16	yes	direct product factor
7	C₂×#11719917	C₂²	1	8	yes	—
8	C₂²×#88322	C₂²	1	4	yes	—
9	#13133128	C₂²	1	8	yes	complemented
10	C₄×#88322	C₂²	1	8	yes	complemented
11	C₂³×C₄×C₅₃	C₄	1	2	yes	—
12	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₅₃	C₄	1	4	yes	—
14	C₂×C₄²×C₅₃	C₄	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₅₃	C₂²	1	4	yes	—
17	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	—	2	8	no	—
20	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₅₃	C₂×C₄	1	8	yes	—
22	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂×C₄	1	8	yes	—
25	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
28	C₄²×C₅₃	C₂×C₄	1	2	yes	—
29	C₄²×C₅₃	C₂×C₄	1	8	yes	—
30	C₄²×C₅₃	C₂×C₄	1	2	yes	—
31	C₄²×C₅₃	C₂³	1	2	yes	—
32	C₄²×C₅₃	C₂³	1	2	yes	—
33	#11719917	C₂×C₄	1	32	yes	direct product factor
34	C₂×#88322	C₂×C₄	1	16	yes	complemented
35	C₂×#88322	C₂³	1	8	yes	—
36	C₂³×C₅₃	—	2	4	no	—
37	C₂³×C₅₃	C₄²	1	4	yes	—
38	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
39	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	—	2	4	no	—
41	C₂×C₄×C₅₃	—	2	16	no	—
42	C₂×C₄×C₅₃	C₄²	1	8	yes	—
43	C₂×C₄×C₅₃	C₄²	1	4	yes	—
44	C₂×C₄×C₅₃	M₁₆	1	8	yes	complemented
45	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
46	C₂×C₄×C₅₃	C₂²×C₄	1	8	yes	—
47	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
51	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
52	#88322	C₂²×C₄	1	16	yes	complemented
53	C₂²×C₅₃	—	2	4	no	—
54	C₂²×C₅₃	—	2	8	no	—
55	C₂²×C₅₃	C₂×C₄²	1	4	yes	—
56	C₂²×C₅₃	C₂×C₄²	1	2	yes	—
57	C₂²×C₅₃	C₂×M₁₆	1	4	yes	—
58	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
59	C₄×C₅₃	—	2	4	no	—
60	C₄×C₅₃	C₂×C₄²	1	2	yes	—
61	C₄×C₅₃	C₂×C₄²	1	4	yes	—
62	C₄×C₅₃	C₂×M₁₆	1	8	yes	complemented
63	C₄×C₅₃	C₂³×C₄	1	2	yes	—
64	C₂×C₄×M₁₆	—	53	1	no	maximal, Sylow
65	C₂×C₅₃	—	2	4	no	—
66	C₂×C₅₃	C₄×M₁₆	1	4	yes	complemented
67	C₂×C₅₃	C₂²×C₄²	1	1	yes	characteristic
68	C₂×C₅₃	C₂²×M₁₆	1	2	yes	—
69	C₂²×M₁₆	—	53	2	no	—
70	C₂²×C₄²	D₅₃	1	1	yes	characteristic
71	C₄×M₁₆	—	53	8	no	—
72	C₂×#97	—	53	2	no	—
73	C₂×C₄×C₈	—	53	2	no	—
74	C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
75	C₂³×C₄	Dic₅₃	1	2	yes	—
76	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
77	C₂×M₁₆	—	53	16	no	—
78	C₂×M₁₆	—	53	8	no	—
79	C₂²×C₈	—	53	4	no	—
80	C₂×C₄²	Dic₅₃	1	2	yes	—
81	C₂×C₄²	Dic₅₃	1	4	yes	—
82	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
83	C₂×C₄²	C₂×D₅₃	1	4	yes	—
84	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
85	#97	—	53	8	no	—
86	C₄×C₈	—	53	8	no	—
87	C₂⁴	C₂×Dic₅₃	1	1	yes	characteristic
88	C₂²×C₄	—	2	8	no	—
89	C₂²×C₄	C₄×D₅₃	1	8	yes	—
90	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
91	C₂²×C₄	C₂×Dic₅₃	1	8	yes	—
92	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
93	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
94	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
96	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
97	M₁₆	—	53	32	no	—
98	C₂×C₈	—	53	16	no	—
99	C₂×C₈	—	53	8	no	—
100	C₄²	C₂×Dic₅₃	1	2	yes	—
101	C₄²	C₂×Dic₅₃	1	8	yes	—
102	C₄²	C₂×Dic₅₃	1	2	yes	—
103	C₄²	C₂²×D₅₃	1	2	yes	—
104	C₄²	C₂²×D₅₃	1	2	yes	—
105	C₂³	—	2	4	no	—
106	C₂³	C₄×Dic₅₃	1	4	yes	—
107	C₂³	C₂²×Dic₅₃	1	2	yes	—
108	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	2	16	no	—
111	C₂×C₄	#11719917	1	8	yes	direct product factor
112	C₂×C₄	C₄×Dic₅₃	1	8	yes	—
113	C₂×C₄	C₄×Dic₅₃	1	4	yes	—
114	C₂×C₄	C₂×C₄×D₅₃	1	4	yes	—
115	C₂×C₄	C₂×C₄×D₅₃	1	8	yes	—
116	C₂×C₄	C₂²×Dic₅₃	1	4	yes	—
117	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
118	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
119	C₂×C₄	C₂²×Dic₅₃	1	4	yes	—
120	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
121	C₈	—	53	16	no	—
122	C₂²	—	2	8	no	—
123	C₂²	—	2	4	no	—
124	C₂²	C₂×C₄×Dic₅₃	1	4	yes	—
125	C₂²	C₂×C₄×Dic₅₃	1	2	yes	—
126	C₂²	C₂×#11719917	1	4	yes	—
127	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
128	C₄	—	2	4	no	—
129	C₄	C₂×C₄×Dic₅₃	1	2	yes	—
130	C₄	C₂×C₄×Dic₅₃	1	4	yes	—
131	C₄	C₂×#11719917	1	8	yes	direct product factor
132	C₄	C₂²×C₄×D₅₃	1	2	yes	—
133	C₂	—	2	4	no	—
134	C₂	C₄×#11719917	1	4	yes	direct product factor
135	C₂	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
136	C₂	C₂²×#11719917	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×D₅₃
commutator	C₂×C₅₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂²×C₄²×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₂²×C₄²	D₅₃
hypocenter	C₅₃	C₂×C₄×M₁₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×#11719917	C₂	direct (N × H)
3. C₂×#13133128	C₂	semidirect
4. C₂×C₄×#88322	C₂	semidirect
6. C₂×#11719917	C₄	direct (N × H)
9. #13133128	C₂²	semidirect
10. C₄×#88322	C₂²	semidirect
33. #11719917	C₂×C₄	direct (N × H)
34. C₂×#88322	C₂×C₄	semidirect
44. C₂×C₄×C₅₃	M₁₆	semidirect
52. #88322	C₂²×C₄	semidirect
62. C₄×C₅₃	C₂×M₁₆	semidirect
66. C₂×C₅₃	C₄×M₁₆	semidirect
74. C₅₃	C₂×C₄×M₁₆	semidirect
111. C₂×C₄	#11719917	direct (N × H)
131. C₄	C₂×#11719917	direct (N × H)
134. C₂	C₄×#11719917	direct (N × H)

Subgroups — C2×C4×#11719917 #20283124

Conjugacy classes of subgroups

Complements

Subgroups — C₂×C₄×#11719917 #20283124