Subgroups — C₅₃×#14688 #27222200

Order 13568.

1422 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	C₂×C₅₃×#383	C₂	1	2	yes	maximal
2	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×C₄⋊#95	C₂	1	6	yes	maximal
4	C₅₃×#1434	C₂	1	6	yes	maximal
5	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
6	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
7	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
8	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#383	—	2	8	no	—
10	C₂×#95×C₅₃	C₂²	1	12	yes	—
11	C₂×#95×C₅₃	C₂²	1	12	yes	—
12	C₄³×C₅₃	C₄	1	8	yes	complemented
13	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
14	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
15	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
16	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₅₃	—	2	24	no	—
18	C₂×C₄²×C₅₃	—	2	12	no	—
19	C₂×C₄²×C₅₃	—	2	12	no	—
20	C₂×C₄²×C₅₃	C₂×C₄	1	12	yes	—
21	C₂×C₄²×C₅₃	C₂×C₄	1	12	yes	—
22	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₅₃	D₄	1	4	yes	—
24	C₂×C₄²×C₅₃	Q₈	1	4	yes	—
25	#95×C₅₃	—	2	24	no	—
26	#95×C₅₃	—	2	24	no	—
27	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	—	2	6	no	—
29	C₂²×C₄×C₅₃	—	2	6	no	—
30	C₂²×C₄×C₅₃	—	2	6	no	—
31	C₂²×C₄×C₅₃	—	2	6	no	—
32	C₂²×C₄×C₅₃	—	2	8	no	—
33	C₂²×C₄×C₅₃	—	2	24	no	—
34	C₂²×C₄×C₅₃	—	2	24	no	—
35	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
36	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
37	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
39	C₂²×C₄×C₅₃	C₂×Q₈	1	6	yes	—
40	C₂²×C₄×C₅₃	#41	1	12	yes	—
41	C₂²×C₄×C₅₃	#41	1	12	yes	—
42	C₄⋊C₄×C₅₃	—	4	16	no	—
43	C₄²×C₅₃	—	2	24	no	—
44	C₄²×C₅₃	—	2	24	no	—
45	C₄²×C₅₃	C₄⋊C₄	1	16	yes	complemented
46	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
47	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
48	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
49	C₂³×C₅₃	C₂×#41	1	6	yes	—
50	C₂³×C₅₃	C₂×#41	1	6	yes	—
51	C₂×C₄×C₅₃	—	2	6	no	—
52	C₂×C₄×C₅₃	—	2	6	no	—
53	C₂×C₄×C₅₃	—	2	6	no	—
54	C₂×C₄×C₅₃	—	2	6	no	—
55	C₂×C₄×C₅₃	—	2	24	no	—
56	C₂×C₄×C₅₃	—	2	12	no	—
57	C₂×C₄×C₅₃	—	2	12	no	—
58	C₂×C₄×C₅₃	—	4	8	no	—
59	C₂×C₄×C₅₃	—	4	24	no	—
60	C₂×C₄×C₅₃	—	4	24	no	—
61	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
62	C₂×C₄×C₅₃	#117	1	12	yes	—
63	C₂×C₄×C₅₃	#117	1	12	yes	—
64	C₂×C₄×C₅₃	#126	1	16	yes	—
65	#14688	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₅₃	C₂×#117	1	3	yes	—
68	C₂²×C₅₃	C₂×#117	1	3	yes	—
69	C₂²×C₅₃	C₂×#126	1	4	yes	—
70	C₂²×C₅₃	#529	1	12	yes	—
71	C₂²×C₅₃	#532	1	6	yes	—
72	C₂²×C₅₃	#533	1	6	yes	—
73	C₄×C₅₃	—	2	12	no	—
74	C₄×C₅₃	—	2	12	no	—
75	C₄×C₅₃	—	8	8	no	—
76	C₄×C₅₃	#383	1	8	yes	complemented
77	C₂×#383	C₂×C₅₃	1	2	yes	—
78	C₂×C₄³	C₂×C₅₃	1	1	yes	characteristic
79	C₄⋊#95	C₂×C₅₃	1	6	yes	—
80	#1434	C₂×C₅₃	1	6	yes	—
81	C₂×C₅₃	C₂×#383	1	1	yes	characteristic
82	C₂×C₅₃	#2283	1	3	yes	—
83	C₂×C₅₃	#2285	1	3	yes	—
84	C₂×C₅₃	#2860	1	4	yes	—
85	C₂×C₅₃	#2863	1	4	yes	—
86	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
87	C₂²×C₄²	C₂²×C₅₃	1	3	yes	—
88	C₂²×C₄²	C₂²×C₅₃	1	3	yes	—
89	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
90	#383	—	2	8	no	—
91	C₂×#95	C₂²×C₅₃	1	12	yes	—
92	C₂×#95	C₂²×C₅₃	1	12	yes	—
93	C₄³	C₄×C₅₃	1	8	yes	complemented
94	C₅₃	#14688	1	1	yes	characteristic, direct product factor, Sylow
95	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
96	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
97	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
98	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
99	C₂×C₄⋊C₄	—	2	24	no	—
100	C₂×C₄²	—	2	12	no	—
101	C₂×C₄²	—	2	12	no	—
102	C₂×C₄²	C₂×C₄×C₅₃	1	12	yes	—
103	C₂×C₄²	C₂×C₄×C₅₃	1	12	yes	—
104	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
105	C₂×C₄²	D₄×C₅₃	1	4	yes	—
106	C₂×C₄²	Q₈×C₅₃	1	4	yes	—
107	#95	—	2	24	no	—
108	#95	—	2	24	no	—
109	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
110	C₂²×C₄	—	2	6	no	—
111	C₂²×C₄	—	2	6	no	—
112	C₂²×C₄	—	2	6	no	—
113	C₂²×C₄	—	2	6	no	—
114	C₂²×C₄	—	2	8	no	—
115	C₂²×C₄	—	2	24	no	—
116	C₂²×C₄	—	2	24	no	—
117	C₂²×C₄	C₂²×C₄×C₅₃	1	6	yes	—
118	C₂²×C₄	C₂²×C₄×C₅₃	1	6	yes	—
119	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
120	C₂²×C₄	C₂×D₄×C₅₃	1	6	yes	—
121	C₂²×C₄	C₂×Q₈×C₅₃	1	6	yes	—
122	C₂²×C₄	#41×C₅₃	1	12	yes	—
123	C₂²×C₄	#41×C₅₃	1	12	yes	—
124	C₄⋊C₄	—	4	16	no	—
125	C₄²	—	2	24	no	—
126	C₄²	—	2	24	no	—
127	C₄²	C₄⋊C₄×C₅₃	1	16	yes	complemented
128	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
129	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
130	C₂³	C₂²×Q₈×C₅₃	1	1	yes	characteristic
131	C₂³	C₂×#41×C₅₃	1	6	yes	—
132	C₂³	C₂×#41×C₅₃	1	6	yes	—
133	C₂×C₄	—	2	6	no	—
134	C₂×C₄	—	2	6	no	—
135	C₂×C₄	—	2	6	no	—
136	C₂×C₄	—	2	6	no	—
137	C₂×C₄	—	2	24	no	—
138	C₂×C₄	—	2	12	no	—
139	C₂×C₄	—	2	12	no	—
140	C₂×C₄	—	4	8	no	—
141	C₂×C₄	—	4	24	no	—
142	C₂×C₄	—	4	24	no	—
143	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	12	yes	—
144	C₂×C₄	#117×C₅₃	1	12	yes	—
145	C₂×C₄	#117×C₅₃	1	12	yes	—
146	C₂×C₄	#126×C₅₃	1	16	yes	—
147	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
148	C₂²	C₂×#117×C₅₃	1	3	yes	—
149	C₂²	C₂×#117×C₅₃	1	3	yes	—
150	C₂²	C₂×#126×C₅₃	1	4	yes	—
151	C₂²	C₅₃×#529	1	12	yes	—
152	C₂²	C₅₃×#532	1	6	yes	—
153	C₂²	C₅₃×#533	1	6	yes	—
154	C₄	—	2	12	no	—
155	C₄	—	2	12	no	—
156	C₄	—	8	8	no	—
157	C₄	C₅₃×#383	1	8	yes	complemented
158	C₂	C₂×C₅₃×#383	1	1	yes	characteristic
159	C₂	C₅₃×#2283	1	3	yes	—
160	C₂	C₅₃×#2285	1	3	yes	—
161	C₂	C₅₃×#2860	1	4	yes	—
162	C₂	C₅₃×#2863	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
12. C₄³×C₅₃	C₄	semidirect
45. C₄²×C₅₃	C₄⋊C₄	semidirect
65. #14688	C₅₃	direct (N × H)
76. C₄×C₅₃	#383	semidirect
93. C₄³	C₄×C₅₃	semidirect
94. C₅₃	#14688	direct (N × H)
127. C₄²	C₄⋊C₄×C₅₃	semidirect
157. C₄	C₅₃×#383	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#383	2
C₂×C₄³×C₅₃	2
C₅₃×C₄⋊#95	2
C₅₃×#1434	2
#14688	53

Subgroups — C53×#14688 #27222200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#14688 #27222200