Subgroups — C₂²×C₅₃×#500 #27268166

Order 13568.

2398 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₅₃	C₂	1	2	yes	maximal
2	C₂×C₅₃×#500	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#107×C₅₃	C₂	1	4	yes	maximal
4	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₅₃	—	2	2	no	—
6	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
7	C₂×C₄⋊Q₈×C₅₃	—	2	12	no	—
8	C₂×C₄⋊Q₈×C₅₃	C₂²	1	24	yes	—
9	C₂²×C₄⋊C₄×C₅₃	—	2	2	no	—
10	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
11	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#500	C₂²	1	64	yes	direct product factor
13	C₂×#107×C₅₃	C₂²	1	48	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	12	yes	—
15	C₂²×Q₈×C₅₃	—	2	24	no	—
16	C₂²×Q₈×C₅₃	—	4	16	no	—
17	C₂³×C₄×C₅₃	—	2	4	no	—
18	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
20	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	Q₈	1	16	yes	complemented
22	C₂²×C₈×C₅₃	C₂³	1	12	yes	—
23	C₄⋊Q₈×C₅₃	—	2	48	no	—
24	C₄⋊Q₈×C₅₃	C₂³	1	32	yes	—
25	C₂×C₄⋊C₄×C₅₃	—	2	24	no	—
26	C₂×C₄⋊C₄×C₅₃	—	2	12	no	—
27	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	—
28	C₂×C₄²×C₅₃	D₄	1	6	yes	—
29	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
30	#107×C₅₃	C₂³	1	64	yes	—
31	C₄×C₈×C₅₃	C₂³	1	16	yes	—
32	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂×Q₈×C₅₃	—	2	32	no	—
34	C₂×Q₈×C₅₃	—	4	96	no	—
35	C₂²×C₄×C₅₃	—	2	24	no	—
36	C₂²×C₄×C₅₃	—	4	16	no	—
37	C₂²×C₄×C₅₃	D₈	1	8	yes	—
38	C₂²×C₄×C₅₃	Q₁₆	1	8	yes	—
39	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
40	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
41	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
42	C₂²×C₄×C₅₃	C₂×Q₈	1	8	yes	—
43	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
44	C₂×C₈×C₅₃	C₂×Q₈	1	48	yes	complemented
45	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
46	C₄⋊C₄×C₅₃	—	2	32	no	—
47	C₄⋊C₄×C₅₃	—	2	24	no	—
48	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
49	C₄²×C₅₃	C₂×D₄	1	12	yes	—
50	C₄²×C₅₃	C₂⁴	1	4	yes	—
51	C₂³×C₅₃	C₄⋊Q₈	1	4	yes	—
52	C₂³×C₅₃	C₂×D₈	1	4	yes	—
53	C₂³×C₅₃	C₂×Q₁₆	1	4	yes	—
54	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
55	Q₈×C₅₃	—	4	64	no	—
56	C₂×C₄×C₅₃	—	2	16	no	—
57	C₂×C₄×C₅₃	—	4	48	no	—
58	C₂×C₄×C₅₃	C₄⋊Q₈	1	12	yes	—
59	C₂×C₄×C₅₃	C₂×D₈	1	24	yes	—
60	C₂×C₄×C₅₃	C₂×Q₁₆	1	24	yes	—
61	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
62	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
63	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
64	C₂×C₄×C₅₃	C₂²×Q₈	1	12	yes	—
65	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
66	C₈×C₅₃	C₂²×Q₈	1	16	yes	complemented
67	C₂²×#500	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂²×C₅₃	#500	1	16	yes	direct product factor
69	C₂²×C₅₃	C₂×C₄⋊Q₈	1	6	yes	—
70	C₂²×C₅₃	C₂²×D₈	1	6	yes	—
71	C₂²×C₅₃	C₂²×Q₁₆	1	6	yes	—
72	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
73	C₄×C₅₃	—	4	16	no	—
74	C₄×C₅₃	C₂×C₄⋊Q₈	1	6	yes	—
75	C₄×C₅₃	C₂²×D₈	1	8	yes	—
76	C₄×C₅₃	C₂²×Q₁₆	1	8	yes	—
77	C₄×C₅₃	C₂³×Q₈	1	2	yes	—
78	C₂²×C₄⋊Q₈	C₂×C₅₃	1	2	yes	—
79	C₂×#500	C₂×C₅₃	1	24	yes	direct product factor
80	C₂²×#107	C₂×C₅₃	1	4	yes	—
81	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
82	C₂×C₅₃	C₂×#500	1	12	yes	direct product factor
83	C₂×C₅₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
84	C₂×C₅₃	C₂³×D₈	1	1	yes	characteristic
85	C₂×C₅₃	C₂³×Q₁₆	1	1	yes	characteristic
86	C₂³×Q₈	—	2	2	no	—
87	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
88	C₂×C₄⋊Q₈	—	2	12	no	—
89	C₂×C₄⋊Q₈	C₂²×C₅₃	1	24	yes	—
90	C₂²×C₄⋊C₄	—	2	2	no	—
91	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
92	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
93	#500	C₂²×C₅₃	1	64	yes	direct product factor
94	C₂×#107	C₂²×C₅₃	1	48	yes	—
95	C₂×C₄×C₈	C₂²×C₅₃	1	12	yes	—
96	C₅₃	C₂²×#500	1	1	yes	characteristic, direct product factor, Sylow
97	C₂²×Q₈	—	2	24	no	—
98	C₂²×Q₈	—	4	16	no	—
99	C₂³×C₄	—	2	4	no	—
100	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
101	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
102	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
103	C₂²×C₈	Q₈×C₅₃	1	16	yes	complemented
104	C₂²×C₈	C₂³×C₅₃	1	12	yes	—
105	C₄⋊Q₈	—	2	48	no	—
106	C₄⋊Q₈	C₂³×C₅₃	1	32	yes	—
107	C₂×C₄⋊C₄	—	2	24	no	—
108	C₂×C₄⋊C₄	—	2	12	no	—
109	C₂×C₄⋊C₄	C₂³×C₅₃	1	24	yes	—
110	C₂×C₄²	D₄×C₅₃	1	6	yes	—
111	C₂×C₄²	C₂³×C₅₃	1	6	yes	—
112	#107	C₂³×C₅₃	1	64	yes	—
113	C₄×C₈	C₂³×C₅₃	1	16	yes	—
114	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
115	C₂×Q₈	—	2	32	no	—
116	C₂×Q₈	—	4	96	no	—
117	C₂²×C₄	—	2	24	no	—
118	C₂²×C₄	—	4	16	no	—
119	C₂²×C₄	D₈×C₅₃	1	8	yes	—
120	C₂²×C₄	Q₁₆×C₅₃	1	8	yes	—
121	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
122	C₂²×C₄	C₂×D₄×C₅₃	1	6	yes	—
123	C₂²×C₄	C₂×D₄×C₅₃	1	6	yes	—
124	C₂²×C₄	C₂×Q₈×C₅₃	1	8	yes	—
125	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
126	C₂×C₈	C₂×Q₈×C₅₃	1	48	yes	complemented
127	C₂×C₈	C₂⁴×C₅₃	1	8	yes	—
128	C₄⋊C₄	—	2	32	no	—
129	C₄⋊C₄	—	2	24	no	—
130	C₄⋊C₄	C₂⁴×C₅₃	1	16	yes	—
131	C₄²	C₂×D₄×C₅₃	1	12	yes	—
132	C₄²	C₂⁴×C₅₃	1	4	yes	—
133	C₂³	C₄⋊Q₈×C₅₃	1	4	yes	—
134	C₂³	C₂×D₈×C₅₃	1	4	yes	—
135	C₂³	C₂×Q₁₆×C₅₃	1	4	yes	—
136	C₂³	C₂²×D₄×C₅₃	1	3	yes	—
137	Q₈	—	4	64	no	—
138	C₂×C₄	—	2	16	no	—
139	C₂×C₄	—	4	48	no	—
140	C₂×C₄	C₄⋊Q₈×C₅₃	1	12	yes	—
141	C₂×C₄	C₂×D₈×C₅₃	1	24	yes	—
142	C₂×C₄	C₂×Q₁₆×C₅₃	1	24	yes	—
143	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
144	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
145	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
146	C₂×C₄	C₂²×Q₈×C₅₃	1	12	yes	—
147	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
148	C₈	C₂²×Q₈×C₅₃	1	16	yes	complemented
149	C₂²	C₅₃×#500	1	16	yes	direct product factor
150	C₂²	C₂×C₄⋊Q₈×C₅₃	1	6	yes	—
151	C₂²	C₂²×D₈×C₅₃	1	6	yes	—
152	C₂²	C₂²×Q₁₆×C₅₃	1	6	yes	—
153	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
154	C₄	—	4	16	no	—
155	C₄	C₂×C₄⋊Q₈×C₅₃	1	6	yes	—
156	C₄	C₂²×D₈×C₅₃	1	8	yes	—
157	C₄	C₂²×Q₁₆×C₅₃	1	8	yes	—
158	C₄	C₂³×Q₈×C₅₃	1	2	yes	—
159	C₂	C₂×C₅₃×#500	1	12	yes	direct product factor
160	C₂	C₂²×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
161	C₂	C₂³×D₈×C₅₃	1	1	yes	characteristic
162	C₂	C₂³×Q₁₆×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#500	C₂	direct (N × H)
12. C₅₃×#500	C₂²	direct (N × H)
21. C₂²×C₈×C₅₃	Q₈	semidirect
44. C₂×C₈×C₅₃	C₂×Q₈	semidirect
66. C₈×C₅₃	C₂²×Q₈	semidirect
67. C₂²×#500	C₅₃	direct (N × H)
68. C₂²×C₅₃	#500	direct (N × H)
79. C₂×#500	C₂×C₅₃	direct (N × H)
82. C₂×C₅₃	C₂×#500	direct (N × H)
93. #500	C₂²×C₅₃	direct (N × H)
96. C₅₃	C₂²×#500	direct (N × H)
103. C₂²×C₈	Q₈×C₅₃	semidirect
126. C₂×C₈	C₂×Q₈×C₅₃	semidirect
148. C₈	C₂²×Q₈×C₅₃	semidirect
149. C₂²	C₅₃×#500	direct (N × H)
159. C₂	C₂×C₅₃×#500	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄⋊Q₈×C₅₃	2
C₂×C₅₃×#500	2
C₂²×#107×C₅₃	2
C₂²×C₄×C₈×C₅₃	2
C₂²×#500	53

Subgroups — C22×C53×#500 #27268166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅₃×#500 #27268166