Subgroups — #20283140

Order 6784.

556 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133223	C₂	1	2	yes	maximal
2	C₄×#11719945	C₂	1	1	yes	maximal, characteristic
3	#16810580	C₂	1	4	yes	maximal, complemented
4	#16810578	C₂	1	2	yes	maximal, complemented
5	C₄×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810575	C₂	1	2	yes	maximal, complemented
7	#97×D₅₃	C₂	1	2	yes	maximal, complemented
8	C₂×#97×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11719945	C₂²	1	1	yes	characteristic
10	#13133223	C₄	1	16	yes	complemented
11	C₂×#11719917	C₂²	1	2	yes	—
12	C₂×#11719912	C₂²	1	4	yes	—
13	C₂×C₈×D₅₃	C₂²	1	4	yes	—
14	C₄×Dic₁₀₆	C₄	1	2	yes	—
15	C₄×D₂₁₂	C₄	1	2	yes	—
16	C₄×C₂²⋊D₅₃	C₄	1	4	yes	—
17	#13133176	C₂²	1	1	yes	characteristic
18	#13133174	C₂²	1	2	yes	—
19	C₄²×D₅₃	C₂²	1	2	yes	—
20	#13133128	C₂²	1	2	yes	—
21	#13133127	C₂²	1	4	yes	—
22	C₄×#88322	C₂²	1	2	yes	—
23	C₈×Dic₅₃	C₂²	1	4	yes	—
24	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
25	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
26	#97×C₅₃	C₂²	1	1	yes	characteristic
27	#97×C₅₃	C₂²	1	1	yes	characteristic
28	#97×C₅₃	C₂²	1	2	yes	—
29	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
31	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
32	#11719945	C₂×C₄	1	8	yes	—
33	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₅₃	C₂×C₄	1	4	yes	—
35	C₂×C₄×D₅₃	C₂³	1	2	yes	—
36	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₂×C₄	1	8	yes	complemented
38	C₂×C₈×C₅₃	C₂³	1	2	yes	—
39	C₂×C₈×C₅₃	C₂³	1	2	yes	—
40	C₄²×C₅₃	C₂×C₄	1	2	yes	—
41	C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
44	#11719921	C₂×C₄	1	4	yes	—
45	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
46	#11719919	C₂×C₄	1	4	yes	—
47	C₄×Dic₅₃	C₂×C₄	1	4	yes	—
48	C₄×Dic₅₃	C₂³	1	2	yes	—
49	#11719917	C₂×C₄	1	8	yes	complemented
50	C₂×#88322	C₂³	1	4	yes	—
51	#11719912	C₂×C₄	1	16	yes	complemented
52	C₈×D₅₃	C₂×C₄	1	16	yes	complemented
53	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₅₃	—	4	2	no	—
56	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
61	C₂²⋊D₅₃	C₄²	1	8	yes	—
62	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
63	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
64	D₂₁₂	C₄²	1	4	yes	—
65	C₄×D₅₃	—	4	4	no	—
66	C₄×D₅₃	C₂²×C₄	1	8	yes	—
67	Dic₁₀₆	C₄²	1	4	yes	—
68	C₈×C₅₃	C₂²×C₄	1	4	yes	—
69	C₈×C₅₃	C₂²×C₄	1	4	yes	—
70	#88322	C₂²×C₄	1	8	yes	—
71	C₂²×C₅₃	—	2	2	no	—
72	C₂²×C₅₃	C₂×C₄²	1	2	yes	—
73	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
74	C₂×D₅₃	—	2	4	no	—
75	C₂×D₅₃	C₂×C₄²	1	4	yes	—
76	C₄×C₅₃	—	2	2	no	—
77	C₄×C₅₃	—	2	2	no	—
78	C₄×C₅₃	C₂×C₄²	1	2	yes	—
79	C₄×C₅₃	C₂³×C₄	1	2	yes	—
80	Dic₅₃	—	2	4	no	—
81	Dic₅₃	C₂×C₄²	1	4	yes	—
82	#2875	—	53	1	no	maximal, Sylow
83	C₂×C₅₃	—	2	2	no	—
84	C₂×C₅₃	C₂²×C₄²	1	1	yes	characteristic
85	C₂×C₅₃	#568	1	2	yes	—
86	D₅₃	—	2	4	no	—
87	C₂×#131	—	53	2	no	—
88	C₄×#41	—	53	1	no	—
89	#405	—	53	2	no	—
90	#405	—	53	4	no	—
91	C₄×M₁₆	—	53	1	no	—
92	C₄×M₁₆	—	53	2	no	—
93	C₂×#97	—	53	2	no	—
94	C₂×#97	D₅₃	1	1	yes	characteristic, complemented
95	C₅₃	#2875	1	1	yes	characteristic, complemented, Sylow
96	C₂×#41	—	53	1	no	—
97	#131	—	53	16	no	—
98	C₂×M₁₆	—	53	2	no	—
99	C₂×M₁₆	—	53	4	no	—
100	C₂²×C₈	—	53	4	no	—
101	C₂²×C₈	C₂×D₅₃	1	2	yes	—
102	C₄×Q₈	—	53	2	no	—
103	C₄×D₄	—	53	4	no	—
104	C₄×D₄	—	53	2	no	—
105	#117	—	53	1	no	—
106	#117	—	53	2	no	—
107	C₂×C₄²	—	53	2	no	—
108	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
109	#97	—	53	4	no	—
110	#97	—	53	2	no	—
111	#97	C₂×D₅₃	1	1	yes	characteristic
112	#97	C₂×D₅₃	1	1	yes	characteristic
113	#97	C₂×D₅₃	1	2	yes	—
114	C₄×C₈	—	53	4	no	—
115	C₄×C₈	—	53	2	no	—
116	#41	—	53	8	no	—
117	C₂×Q₈	—	53	1	no	—
118	C₂×D₄	—	53	2	no	—
119	C₂×D₄	—	53	1	no	—
120	C₂²×C₄	—	53	2	no	—
121	C₂²×C₄	—	53	4	no	—
122	C₂²×C₄	C₄×D₅₃	1	2	yes	—
123	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
124	M₁₆	—	53	8	no	—
125	M₁₆	—	53	16	no	—
126	C₂×C₈	—	53	4	no	—
127	C₂×C₈	—	53	16	no	—
128	C₂×C₈	C₄×D₅₃	1	8	yes	complemented
129	C₂×C₈	C₂²×D₅₃	1	2	yes	—
130	C₂×C₈	C₂²×D₅₃	1	2	yes	—
131	C₄⋊C₄	—	53	4	no	—
132	C₄⋊C₄	—	53	2	no	—
133	C₂²⋊C₄	—	53	2	no	—
134	C₂²⋊C₄	—	53	4	no	—
135	C₄²	—	53	4	no	—
136	C₄²	—	53	2	no	—
137	C₄²	C₄×D₅₃	1	2	yes	—
138	C₄²	C₂²×D₅₃	1	1	yes	characteristic
139	C₄²	C₂²×D₅₃	1	1	yes	characteristic
140	C₂³	—	53	2	no	—
141	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
142	Q₈	—	53	4	no	—
143	D₄	—	53	8	no	—
144	D₄	—	53	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	53	2	no	—
147	C₂×C₄	—	53	8	no	—
148	C₂×C₄	—	53	4	no	—
149	C₂×C₄	—	212	4	no	—
150	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
151	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
152	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₅₃	1	4	yes	—
154	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
155	C₈	—	53	8	no	—
156	C₈	C₂×C₄×D₅₃	1	4	yes	—
157	C₈	C₂×C₄×D₅₃	1	4	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	—	53	4	no	—
160	C₂²	—	106	4	no	—
161	C₂²	C₄²×D₅₃	1	2	yes	—
162	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	2	2	no	—
165	C₄	—	53	4	no	—
166	C₄	—	106	4	no	—
167	C₄	C₄²×D₅₃	1	2	yes	—
168	C₄	C₂²×C₄×D₅₃	1	2	yes	—
169	C₂	—	2	2	no	—
170	C₂	—	106	4	no	—
171	C₂	C₂×C₄²×D₅₃	1	1	yes	characteristic
172	C₂	#16811389	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator	C₂×C₅₃	C₂²×C₄²
Fitting	C₂×#97×C₅₃	C₂
socle	C₂²×C₅₃	C₂³×C₄
hypercenter	C₂×#97	D₅₃
hypocenter	C₅₃	#2875
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
3. #16810580	C₂	semidirect
4. #16810578	C₂	semidirect
5. C₄×#11719917	C₂	semidirect
6. #16810575	C₂	semidirect
7. #97×D₅₃	C₂	semidirect
8. C₂×#97×C₅₃	C₂	semidirect
10. #13133223	C₄	semidirect
37. C₂×C₈×C₅₃	C₂×C₄	semidirect
49. #11719917	C₂×C₄	semidirect
51. #11719912	C₂×C₄	semidirect
52. C₈×D₅₃	C₂×C₄	semidirect
94. C₂×#97	D₅₃	semidirect
95. C₅₃	#2875	semidirect
128. C₂×C₈	C₄×D₅₃	semidirect