Subgroups — C₂×C₅₃×#2247 #27241222

Order 13568.

814 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₃₂×C₅₃	C₂	1	4	yes	maximal
2	C₂×C₅₃×#494	C₂	1	2	yes	maximal
3	C₅₃×#2247	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×C₅₃	—	2	2	no	—
6	C₂²×Q₁₆×C₅₃	C₂²	1	4	yes	—
7	C₂×C₄⋊Q₈×C₅₃	—	2	2	no	—
8	C₂×Q₃₂×C₅₃	—	2	16	no	—
9	C₂×Q₃₂×C₅₃	C₂²	1	16	yes	—
10	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#494	—	2	4	no	—
12	C₅₃×#494	C₂²	1	8	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₅₃	C₂²	1	4	yes	—
15	C₂²×Q₈×C₅₃	—	2	4	no	—
16	C₂×Q₁₆×C₅₃	—	2	4	no	—
17	C₂×Q₁₆×C₅₃	—	2	8	no	—
18	C₂×Q₁₆×C₅₃	—	2	16	no	—
19	C₂×Q₁₆×C₅₃	—	4	8	no	—
20	C₂×Q₁₆×C₅₃	C₂³	1	8	yes	—
21	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
23	C₄⋊Q₈×C₅₃	—	2	8	no	—
24	C₄⋊Q₈×C₅₃	—	4	4	no	—
25	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
26	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
27	Q₃₂×C₅₃	—	2	32	no	—
28	C₂×C₁₆×C₅₃	D₄	1	8	yes	—
29	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
30	C₄×C₈×C₅₃	D₄	1	2	yes	—
31	C₄×C₈×C₅₃	C₂³	1	2	yes	—
32	C₂×Q₈×C₅₃	—	2	8	no	—
33	C₂×Q₈×C₅₃	—	4	4	no	—
34	C₂×Q₈×C₅₃	—	4	8	no	—
35	C₂×Q₈×C₅₃	—	4	8	no	—
36	C₂²×C₄×C₅₃	—	4	4	no	—
37	C₂²×C₄×C₅₃	D₈	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	Q₁₆×C₅₃	—	2	16	no	—
40	Q₁₆×C₅₃	—	4	8	no	—
41	Q₁₆×C₅₃	—	4	16	no	—
42	C₂×C₈×C₅₃	D₈	1	4	yes	—
43	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
45	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
47	C₄⋊C₄×C₅₃	—	4	16	no	—
48	C₄²×C₅₃	D₈	1	2	yes	—
49	C₄²×C₅₃	C₂×D₄	1	2	yes	—
50	C₁₆×C₅₃	C₂×D₄	1	8	yes	—
51	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
52	Q₈×C₅₃	—	4	8	no	—
53	Q₈×C₅₃	—	4	8	no	—
54	Q₈×C₅₃	—	8	8	no	—
55	C₂×C₄×C₅₃	—	4	8	no	—
56	C₂×C₄×C₅₃	—	8	8	no	—
57	C₂×C₄×C₅₃	Q₃₂	1	8	yes	complemented
58	C₂×C₄×C₅₃	#127	1	2	yes	—
59	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
60	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
62	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
63	C₈×C₅₃	#127	1	2	yes	—
64	C₈×C₅₃	C₂×D₈	1	4	yes	—
65	C₈×C₅₃	C₂²×D₄	1	2	yes	—
66	C₂×#2247	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂²×C₅₃	#493	1	2	yes	—
68	C₂²×C₅₃	C₂×Q₃₂	1	4	yes	—
69	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
70	C₄×C₅₃	—	8	8	no	—
71	C₄×C₅₃	#493	1	2	yes	—
72	C₄×C₅₃	C₂×Q₃₂	1	8	yes	complemented
73	C₄×C₅₃	C₂×#127	1	1	yes	characteristic
74	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
75	C₂²×Q₃₂	C₂×C₅₃	1	4	yes	—
76	C₂×#494	C₂×C₅₃	1	2	yes	—
77	#2247	C₂×C₅₃	1	8	yes	direct product factor
78	C₂×C₄×C₁₆	C₂×C₅₃	1	1	yes	characteristic
79	C₂×C₅₃	#2247	1	4	yes	direct product factor
80	C₂×C₅₃	C₂×#493	1	1	yes	characteristic
81	C₂×C₅₃	C₂²×Q₃₂	1	2	yes	—
82	C₂²×Q₁₆	—	2	2	no	—
83	C₂²×Q₁₆	C₂²×C₅₃	1	4	yes	—
84	C₂×C₄⋊Q₈	—	2	2	no	—
85	C₂×Q₃₂	—	2	16	no	—
86	C₂×Q₃₂	C₂²×C₅₃	1	16	yes	—
87	C₂²×C₁₆	C₂²×C₅₃	1	2	yes	—
88	#494	—	2	4	no	—
89	#494	C₂²×C₅₃	1	8	yes	—
90	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
91	C₄×C₁₆	C₂²×C₅₃	1	4	yes	—
92	C₅₃	C₂×#2247	1	1	yes	characteristic, direct product factor, Sylow
93	C₂²×Q₈	—	2	4	no	—
94	C₂×Q₁₆	—	2	16	no	—
95	C₂×Q₁₆	—	2	8	no	—
96	C₂×Q₁₆	—	2	4	no	—
97	C₂×Q₁₆	—	4	8	no	—
98	C₂×Q₁₆	C₂³×C₅₃	1	8	yes	—
99	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
100	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
101	C₄⋊Q₈	—	2	8	no	—
102	C₄⋊Q₈	—	4	4	no	—
103	C₂×C₄⋊C₄	—	4	4	no	—
104	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
105	Q₃₂	—	2	32	no	—
106	C₂×C₁₆	D₄×C₅₃	1	8	yes	—
107	C₂×C₁₆	C₂³×C₅₃	1	4	yes	—
108	C₄×C₈	D₄×C₅₃	1	2	yes	—
109	C₄×C₈	C₂³×C₅₃	1	2	yes	—
110	C₂×Q₈	—	2	8	no	—
111	C₂×Q₈	—	4	4	no	—
112	C₂×Q₈	—	4	8	no	—
113	C₂×Q₈	—	4	8	no	—
114	C₂²×C₄	—	4	4	no	—
115	C₂²×C₄	D₈×C₅₃	1	2	yes	—
116	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
117	Q₁₆	—	2	16	no	—
118	Q₁₆	—	4	8	no	—
119	Q₁₆	—	4	16	no	—
120	C₂×C₈	D₈×C₅₃	1	4	yes	—
121	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
122	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
123	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
124	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
125	C₄⋊C₄	—	4	16	no	—
126	C₄²	D₈×C₅₃	1	2	yes	—
127	C₄²	C₂×D₄×C₅₃	1	2	yes	—
128	C₁₆	C₂×D₄×C₅₃	1	8	yes	—
129	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
130	Q₈	—	4	8	no	—
131	Q₈	—	4	8	no	—
132	Q₈	—	8	8	no	—
133	C₂×C₄	—	4	8	no	—
134	C₂×C₄	—	8	8	no	—
135	C₂×C₄	Q₃₂×C₅₃	1	8	yes	complemented
136	C₂×C₄	#127×C₅₃	1	2	yes	—
137	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
138	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
139	C₂×C₄	C₂×D₈×C₅₃	1	4	yes	—
140	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
141	C₈	#127×C₅₃	1	2	yes	—
142	C₈	C₂×D₈×C₅₃	1	4	yes	—
143	C₈	C₂²×D₄×C₅₃	1	2	yes	—
144	C₂²	C₅₃×#493	1	2	yes	—
145	C₂²	C₂×Q₃₂×C₅₃	1	4	yes	—
146	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
147	C₄	—	8	8	no	—
148	C₄	C₅₃×#493	1	2	yes	—
149	C₄	C₂×Q₃₂×C₅₃	1	8	yes	complemented
150	C₄	C₂×#127×C₅₃	1	1	yes	characteristic
151	C₄	C₂²×D₈×C₅₃	1	1	yes	characteristic
152	C₂	C₅₃×#2247	1	4	yes	direct product factor
153	C₂	C₂×C₅₃×#493	1	1	yes	characteristic
154	C₂	C₂²×Q₃₂×C₅₃	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#2247	C₂	direct (N × H)
57. C₂×C₄×C₅₃	Q₃₂	semidirect
66. C₂×#2247	C₅₃	direct (N × H)
72. C₄×C₅₃	C₂×Q₃₂	semidirect
77. #2247	C₂×C₅₃	direct (N × H)
79. C₂×C₅₃	#2247	direct (N × H)
92. C₅₃	C₂×#2247	direct (N × H)
135. C₂×C₄	Q₃₂×C₅₃	semidirect
149. C₄	C₂×Q₃₂×C₅₃	semidirect
152. C₂	C₅₃×#2247	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×Q₃₂×C₅₃	2
C₂×C₅₃×#494	2
C₅₃×#2247	2
C₂×C₄×C₁₆×C₅₃	2
C₂×#2247	53

Subgroups — C2×C53×#2247 #27241222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2247 #27241222