Subgroups — #20288120

Order 6784.

996 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811254	C₂	1	4	yes	maximal, complemented
2	#16811251	C₂	1	8	yes	maximal, complemented
3	#16811249	C₂	1	4	yes	maximal
4	#16811180	C₂	1	4	yes	maximal, complemented
5	#16811176	C₂	1	8	yes	maximal, complemented
6	C₂×#13133176	C₂	1	2	yes	maximal
7	C₂×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
8	#13133217	C₂²	1	8	yes	complemented
9	#13133216	C₂²	1	16	yes	complemented
10	#13133209	C₂²	1	8	yes	complemented
11	#13133208	C₂²	1	16	yes	complemented
12	#13133196	C₂²	1	16	yes	—
13	#13133194	C₂²	1	32	yes	—
14	C₄×Dic₁₀₆	C₂²	1	16	yes	complemented
15	#13133176	C₂²	1	4	yes	—
16	#13133176	C₂²	1	4	yes	—
17	#13133176	C₂²	1	4	yes	—
18	#13133176	C₂²	1	4	yes	—
19	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
20	C₂×C₂²⋊Dic₅₃	C₂²	1	4	yes	—
21	C₂×C₄×Dic₅₃	C₂²	1	4	yes	—
22	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
23	#117×C₅₃	C₂²	1	4	yes	—
24	#117×C₅₃	C₂²	1	4	yes	—
25	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
28	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	Q₈	1	8	yes	complemented
30	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
31	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
34	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
35	C₂²×Dic₅₃	C₂³	1	8	yes	—
36	C₂×Dic₁₀₆	C₂³	1	16	yes	—
37	C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
38	C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
39	C₄²×C₅₃	C₂³	1	4	yes	—
40	C₄²×C₅₃	C₂³	1	4	yes	—
41	C₂²⋊Dic₅₃	C₂³	1	8	yes	—
42	C₂²⋊Dic₅₃	C₂³	1	8	yes	—
43	C₄⋊Dic₅₃	C₂³	1	8	yes	—
44	C₄⋊Dic₅₃	C₂³	1	8	yes	—
45	C₄⋊Dic₅₃	C₂³	1	16	yes	—
46	#11719919	C₂³	1	32	yes	—
47	C₄×Dic₅₃	C₂³	1	8	yes	—
48	C₄×Dic₅₃	C₂³	1	8	yes	—
49	C₂³×C₅₃	—	4	2	no	—
50	C₂³×C₅₃	C₂×Q₈	1	4	yes	—
51	C₂³×C₅₃	C₂⁴	1	2	yes	—
52	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
53	C₂×C₄×C₅₃	—	4	4	no	—
54	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
55	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
56	C₂×C₄×C₅₃	C₂×Q₈	1	8	yes	—
57	C₂×C₄×C₅₃	C₂×Q₈	1	8	yes	—
58	C₂×C₄×C₅₃	C₂⁴	1	8	yes	—
59	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
60	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
62	C₂×Dic₅₃	—	4	8	no	—
63	C₂×Dic₅₃	C₂⁴	1	16	yes	—
64	Dic₁₀₆	—	4	16	no	—
65	C₂²×C₅₃	—	2	4	no	—
66	C₂²×C₅₃	—	2	2	no	—
67	C₂²×C₅₃	—	4	4	no	—
68	C₂²×C₅₃	C₂²×Q₈	1	4	yes	—
69	C₂²×C₅₃	C₂²×Q₈	1	2	yes	—
70	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
71	C₄×C₅₃	—	2	8	no	—
72	C₄×C₅₃	C₂²×Q₈	1	4	yes	—
73	C₄×C₅₃	C₂²×Q₈	1	2	yes	—
74	C₄×C₅₃	C₂²×Q₈	1	2	yes	—
75	Dic₅₃	—	2	16	no	—
76	#3458	—	53	1	no	maximal, Sylow
77	C₂×C₅₃	—	2	4	no	—
78	C₂×C₅₃	—	2	2	no	—
79	C₂×C₅₃	C₂³×Q₈	1	1	yes	characteristic
80	C₂×C₅₃	#585	1	2	yes	—
81	#544	—	53	8	no	—
82	#544	—	53	4	no	—
83	#543	—	53	4	no	—
84	#533	—	53	8	no	—
85	#533	—	53	4	no	—
86	C₂×#117	—	53	2	no	—
87	C₂×#117	D₅₃	1	1	yes	characteristic
88	C₅₃	#3458	1	1	yes	characteristic, complemented, Sylow
89	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
90	C₄⋊Q₈	—	53	16	no	—
91	C₄⋊Q₈	—	53	8	no	—
92	#125	—	53	16	no	—
93	#125	—	53	8	no	—
94	C₂²⋊Q₈	—	53	16	no	—
95	C₂²⋊Q₈	—	53	32	no	—
96	C₄×Q₈	—	53	16	no	—
97	#117	—	53	4	no	—
98	#117	—	53	4	no	—
99	#117	—	53	4	no	—
100	#117	—	53	4	no	—
101	#117	C₂×D₅₃	1	4	yes	—
102	#117	C₂×D₅₃	1	4	yes	—
103	C₂×C₄⋊C₄	—	53	4	no	—
104	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
105	C₂×C₂²⋊C₄	—	53	4	no	—
106	C₂×C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
107	C₂×C₄²	—	53	4	no	—
108	C₂×C₄²	C₂×D₅₃	1	2	yes	—
109	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
110	C₂×Q₈	—	53	16	no	—
111	C₂²×C₄	—	53	8	no	—
112	C₂²×C₄	Dic₁₀₆	1	8	yes	complemented
113	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
114	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
115	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
117	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
118	C₄⋊C₄	—	53	8	no	—
119	C₄⋊C₄	—	53	8	no	—
120	C₄⋊C₄	—	53	32	no	—
121	C₄⋊C₄	—	53	16	no	—
122	C₄⋊C₄	C₂²×D₅₃	1	8	yes	—
123	C₂²⋊C₄	—	53	8	no	—
124	C₂²⋊C₄	—	53	8	no	—
125	C₂²⋊C₄	C₂²×D₅₃	1	8	yes	—
126	C₄²	—	53	8	no	—
127	C₄²	—	53	8	no	—
128	C₄²	C₂²×D₅₃	1	4	yes	—
129	C₄²	C₂²×D₅₃	1	4	yes	—
130	C₂³	—	4	2	no	—
131	C₂³	C₂×Dic₁₀₆	1	4	yes	—
132	C₂³	C₂³×D₅₃	1	1	yes	characteristic
133	C₂³	C₂³×D₅₃	1	2	yes	—
134	Q₈	—	212	16	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	53	16	no	—
137	C₂×C₄	—	212	8	no	—
138	C₂×C₄	C₂×Dic₁₀₆	1	8	yes	—
139	C₂×C₄	C₂×Dic₁₀₆	1	4	yes	—
140	C₂×C₄	C₂×Dic₁₀₆	1	8	yes	—
141	C₂×C₄	C₂×Dic₁₀₆	1	4	yes	—
142	C₂×C₄	C₂³×D₅₃	1	8	yes	—
143	C₂×C₄	C₂³×D₅₃	1	2	yes	—
144	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
145	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
146	C₂²	—	2	2	no	—
147	C₂²	—	2	4	no	—
148	C₂²	—	4	4	no	—
149	C₂²	C₂²×Dic₁₀₆	1	2	yes	—
150	C₂²	C₂²×Dic₁₀₆	1	4	yes	—
151	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
152	C₄	—	2	8	no	—
153	C₄	—	106	16	no	—
154	C₄	C₂²×Dic₁₀₆	1	4	yes	—
155	C₄	C₂²×Dic₁₀₆	1	2	yes	—
156	C₄	C₂²×Dic₁₀₆	1	2	yes	—
157	C₂	—	2	2	no	—
158	C₂	—	2	4	no	—
159	C₂	C₂³×Dic₁₀₆	1	1	yes	characteristic
160	C₂	#16811459	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Frattini	C₂²	C₂⁴×D₅₃
Fitting	C₂×#117×C₅₃	C₂
hypercenter	C₂×#117	D₅₃
hypocenter	C₅₃	#3458
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #16811254	C₂	semidirect
2. #16811251	C₂	semidirect
4. #16811180	C₂	semidirect
5. #16811176	C₂	semidirect
8. #13133217	C₂²	semidirect
9. #13133216	C₂²	semidirect
10. #13133209	C₂²	semidirect
11. #13133208	C₂²	semidirect
14. C₄×Dic₁₀₆	C₂²	semidirect
29. C₂²×C₄×C₅₃	Q₈	semidirect
88. C₅₃	#3458	semidirect
112. C₂²×C₄	Dic₁₀₆	semidirect