Subgroups — C₂×#125×D₅₃ #20287965

Order 6784.

1192 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133209	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13133208	C₂	1	2	yes	maximal, complemented
3	C₂×#13133207	C₂	1	4	yes	maximal, complemented
4	#125×D₅₃	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₄⋊C₄×D₅₃	C₂	1	4	yes	maximal
6	C₂×C₄⋊C₄×D₅₃	C₂	1	2	yes	maximal
7	C₂×C₄²×D₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#125×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₄×D₅₃	C₂²	1	2	yes	—
10	C₂²×C₄×D₅₃	C₂²	1	4	yes	—
11	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
12	#13133209	C₂²	1	8	yes	complemented
13	#13133208	C₂²	1	16	yes	complemented
14	#13133207	C₂²	1	32	yes	complemented
15	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
16	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
17	C₂×#11719919	C₂²	1	4	yes	—
18	C₂×#11719919	C₂²	1	4	yes	—
19	C₂×#11719919	C₂²	1	4	yes	—
20	C₄⋊C₄×D₅₃	C₂²	1	32	yes	—
21	C₄⋊C₄×D₅₃	C₂²	1	16	yes	—
22	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
23	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
24	C₄²×D₅₃	C₂²	1	8	yes	—
25	#125×C₅₃	C₂²	1	8	yes	complemented
26	C₂×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
27	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
29	C₂³×D₅₃	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
31	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
32	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
33	C₂²×Dic₅₃	C₂³	1	2	yes	—
34	C₂²×Dic₅₃	C₂³	1	4	yes	—
35	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
36	C₂×C₄×D₅₃	—	2	8	no	—
37	C₂×C₄×D₅₃	—	2	16	no	—
38	C₂×C₄×D₅₃	Q₈	1	8	yes	—
39	C₂×C₄×D₅₃	C₂³	1	8	yes	—
40	C₂×C₄×D₅₃	C₂³	1	16	yes	—
41	C₂×C₄×D₅₃	C₂³	1	4	yes	—
42	C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
43	C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
44	C₄²×C₅₃	C₂³	1	4	yes	—
45	C₄⋊Dic₅₃	C₂³	1	16	yes	—
46	C₄⋊Dic₅₃	C₂³	1	8	yes	—
47	#11719919	C₂³	1	16	yes	—
48	#11719919	C₂³	1	16	yes	—
49	#11719919	C₂³	1	16	yes	—
50	C₄×Dic₅₃	C₂³	1	8	yes	—
51	C₄×Dic₅₃	C₂³	1	4	yes	—
52	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
53	C₂²×D₅₃	C₂×Q₈	1	4	yes	—
54	C₂²×D₅₃	#41	1	8	yes	—
55	C₂²×D₅₃	C₂⁴	1	2	yes	—
56	C₂×C₄×C₅₃	—	2	4	no	—
57	C₂×C₄×C₅₃	—	2	8	no	—
58	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
59	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
60	C₂×C₄×C₅₃	C₂⁴	1	8	yes	—
61	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
62	C₂×Dic₅₃	—	2	4	no	—
63	C₂×Dic₅₃	—	2	8	no	—
64	C₂×Dic₅₃	C₂×Q₈	1	4	yes	—
65	C₂×Dic₅₃	C₂⁴	1	4	yes	—
66	C₂×Dic₅₃	C₂⁴	1	8	yes	—
67	C₂×Dic₅₃	C₂⁴	1	2	yes	—
68	C₄×D₅₃	—	2	16	no	—
69	C₄×D₅₃	—	2	32	no	—
70	C₄×D₅₃	C₂×Q₈	1	16	yes	—
71	C₂²×C₅₃	C₂²×Q₈	1	2	yes	—
72	C₂²×C₅₃	C₂×#41	1	4	yes	—
73	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
74	C₂×D₅₃	#125	1	16	yes	direct product factor
75	C₂×D₅₃	C₂²×Q₈	1	4	yes	—
76	C₂×D₅₃	C₂×#41	1	8	yes	—
77	C₄×C₅₃	—	2	4	no	—
78	C₄×C₅₃	—	2	8	no	—
79	C₄×C₅₃	C₂²×Q₈	1	4	yes	—
80	Dic₅₃	—	2	4	no	—
81	Dic₅₃	—	2	8	no	—
82	Dic₅₃	C₂²×Q₈	1	4	yes	—
83	C₂²×#125	—	53	1	no	maximal, Sylow
84	C₂×C₅₃	C₂×#125	1	4	yes	complemented
85	C₂×C₅₃	C₂³×Q₈	1	1	yes	characteristic
86	C₂×C₅₃	C₂²×#41	1	2	yes	—
87	D₅₃	C₂×#125	1	8	yes	direct product factor
88	C₂×#125	—	53	16	no	—
89	C₂×#125	—	53	4	no	—
90	C₂×#125	—	53	2	no	—
91	C₂×#125	—	53	1	no	—
92	C₂×#125	D₅₃	1	1	yes	characteristic, direct product factor
93	C₂²×C₄⋊C₄	—	53	2	no	—
94	C₂²×C₄⋊C₄	—	53	4	no	—
95	C₂²×C₄²	—	53	1	no	—
96	C₅₃	C₂²×#125	1	1	yes	characteristic, complemented, Sylow
97	C₂³×C₄	—	53	1	no	—
98	C₂³×C₄	—	53	4	no	—
99	C₂³×C₄	—	53	2	no	—
100	#125	—	53	32	no	—
101	#125	—	53	16	no	—
102	#125	—	53	8	no	—
103	#125	C₂×D₅₃	1	8	yes	direct product factor
104	C₂×C₄⋊C₄	—	53	4	no	—
105	C₂×C₄⋊C₄	—	53	4	no	—
106	C₂×C₄⋊C₄	—	53	4	no	—
107	C₂×C₄⋊C₄	—	53	4	no	—
108	C₂×C₄⋊C₄	—	53	2	no	—
109	C₂×C₄⋊C₄	—	53	16	no	—
110	C₂×C₄⋊C₄	—	53	32	no	—
111	C₂×C₄⋊C₄	C₂×D₅₃	1	4	yes	—
112	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
113	C₂×C₄²	—	53	2	no	—
114	C₂×C₄²	—	53	1	no	—
115	C₂×C₄²	—	53	8	no	—
116	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
117	C₂⁴	—	53	1	no	—
118	C₂²×C₄	—	53	1	no	—
119	C₂²×C₄	—	53	8	no	—
120	C₂²×C₄	—	53	4	no	—
121	C₂²×C₄	—	53	2	no	—
122	C₂²×C₄	—	53	4	no	—
123	C₂²×C₄	—	53	8	no	—
124	C₂²×C₄	—	53	16	no	—
125	C₂²×C₄	—	106	16	no	—
126	C₂²×C₄	—	106	8	no	—
127	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
128	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
129	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
130	C₄⋊C₄	—	53	16	no	—
131	C₄⋊C₄	—	53	16	no	—
132	C₄⋊C₄	—	53	16	no	—
133	C₄⋊C₄	—	53	16	no	—
134	C₄⋊C₄	—	53	8	no	—
135	C₄⋊C₄	C₂²×D₅₃	1	16	yes	—
136	C₄⋊C₄	C₂²×D₅₃	1	8	yes	—
137	C₄²	—	53	8	no	—
138	C₄²	—	53	4	no	—
139	C₄²	C₂²×D₅₃	1	4	yes	—
140	C₂³	—	53	4	no	—
141	C₂³	—	53	8	no	—
142	C₂³	—	53	2	no	—
143	C₂³	C₂³×D₅₃	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	8	no	—
146	C₂×C₄	—	53	4	no	—
147	C₂×C₄	—	53	2	no	—
148	C₂×C₄	—	53	16	no	—
149	C₂×C₄	—	53	4	no	—
150	C₂×C₄	—	53	8	no	—
151	C₂×C₄	—	106	16	no	—
152	C₂×C₄	—	106	32	no	—
153	C₂×C₄	—	106	4	no	—
154	C₂×C₄	—	106	8	no	—
155	C₂×C₄	Q₈×D₅₃	1	4	yes	—
156	C₂×C₄	C₂³×D₅₃	1	4	yes	—
157	C₂×C₄	C₂³×D₅₃	1	8	yes	—
158	C₂×C₄	C₂³×D₅₃	1	2	yes	—
159	C₂²	—	53	16	no	—
160	C₂²	—	53	4	no	—
161	C₂²	—	53	8	no	—
162	C₂²	C₂×Q₈×D₅₃	1	2	yes	—
163	C₂²	#41×D₅₃	1	4	yes	—
164	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
165	C₄	—	2	4	no	—
166	C₄	—	2	8	no	—
167	C₄	—	53	4	no	—
168	C₄	—	106	4	no	—
169	C₄	—	106	8	no	—
170	C₄	C₂×Q₈×D₅₃	1	4	yes	—
171	C₂	—	53	8	no	—
172	C₂	#125×D₅₃	1	4	yes	direct product factor
173	C₂	C₂²×Q₈×D₅₃	1	1	yes	characteristic
174	C₂	C₂×#41×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂⁵
Frattini	C₂²	C₂⁴×D₅₃
Fitting	C₂×#125×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂×#125	D₅₃
hypocenter	C₅₃	C₂²×#125
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
1. C₂×#13133209	C₂	semidirect
2. C₂×#13133208	C₂	semidirect
3. C₂×#13133207	C₂	semidirect
4. #125×D₅₃	C₂	direct (N × H)
8. C₂×#125×C₅₃	C₂	semidirect
12. #13133209	C₂²	semidirect
13. #13133208	C₂²	semidirect
14. #13133207	C₂²	semidirect
25. #125×C₅₃	C₂²	semidirect
74. C₂×D₅₃	#125	direct (N × H)
84. C₂×C₅₃	C₂×#125	semidirect
87. D₅₃	C₂×#125	direct (N × H)
92. C₂×#125	D₅₃	direct (N × H)
96. C₅₃	C₂²×#125	semidirect
103. #125	C₂×D₅₃	direct (N × H)
172. C₂	#125×D₅₃	direct (N × H)

Subgroups — C2×#125×D53 #20287965

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#125×D₅₃ #20287965