Subgroups — C₅₃×#14689 #27222201

Order 13568.

1422 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#382	C₂	1	2	yes	maximal
2	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×C₄⋊#95	C₂	1	4	yes	maximal
4	C₅₃×#1434	C₂	1	8	yes	maximal
5	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
6	C₂²×C₄²×C₅₃	C₂²	1	4	yes	—
7	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
8	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#382	—	2	8	no	—
10	C₂×#95×C₅₃	C₂²	1	16	yes	—
11	C₂×#95×C₅₃	C₂²	1	8	yes	—
12	C₄³×C₅₃	C₄	1	8	yes	complemented
13	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
14	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
15	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
16	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
18	C₂×C₄⋊C₄×C₅₃	—	2	16	no	—
19	C₂×C₄²×C₅₃	—	2	16	no	—
20	C₂×C₄²×C₅₃	—	2	8	no	—
21	C₂×C₄²×C₅₃	C₂×C₄	1	16	yes	—
22	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
23	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₅₃	D₄	1	4	yes	—
25	C₂×C₄²×C₅₃	Q₈	1	4	yes	—
26	#95×C₅₃	—	2	16	no	—
27	#95×C₅₃	—	2	32	no	—
28	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	—	2	8	no	—
30	C₂²×C₄×C₅₃	—	2	8	no	—
31	C₂²×C₄×C₅₃	—	2	8	no	—
32	C₂²×C₄×C₅₃	—	2	8	no	—
33	C₂²×C₄×C₅₃	—	2	32	no	—
34	C₂²×C₄×C₅₃	—	2	16	no	—
35	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
36	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
37	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
39	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
41	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
42	C₂²×C₄×C₅₃	#41	1	16	yes	—
43	C₂²×C₄×C₅₃	#41	1	8	yes	—
44	C₄⋊C₄×C₅₃	—	4	16	no	—
45	C₄²×C₅₃	—	2	32	no	—
46	C₄²×C₅₃	—	2	16	no	—
47	C₄²×C₅₃	C₄⋊C₄	1	16	yes	—
48	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
49	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
50	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
51	C₂³×C₅₃	C₂×#41	1	4	yes	—
52	C₂³×C₅₃	C₂×#41	1	8	yes	—
53	C₂×C₄×C₅₃	—	2	8	no	—
54	C₂×C₄×C₅₃	—	2	8	no	—
55	C₂×C₄×C₅₃	—	2	8	no	—
56	C₂×C₄×C₅₃	—	2	32	no	—
57	C₂×C₄×C₅₃	—	2	8	no	—
58	C₂×C₄×C₅₃	—	2	8	no	—
59	C₂×C₄×C₅₃	—	4	8	no	—
60	C₂×C₄×C₅₃	—	4	32	no	—
61	C₂×C₄×C₅₃	—	4	16	no	—
62	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
63	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
64	C₂×C₄×C₅₃	#117	1	16	yes	—
65	C₂×C₄×C₅₃	#117	1	8	yes	—
66	C₂×C₄×C₅₃	#124	1	8	yes	—
67	C₂×C₄×C₅₃	#125	1	8	yes	—
68	#14689	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₅₃	C₂×#117	1	2	yes	—
71	C₂²×C₅₃	C₂×#117	1	4	yes	—
72	C₂²×C₅₃	C₂×#124	1	2	yes	—
73	C₂²×C₅₃	C₂×#125	1	2	yes	—
74	C₂²×C₅₃	#529	1	16	yes	—
75	C₂²×C₅₃	#532	1	4	yes	—
76	C₂²×C₅₃	#533	1	4	yes	—
77	C₄×C₅₃	—	2	16	no	—
78	C₄×C₅₃	—	2	8	no	—
79	C₄×C₅₃	—	8	8	no	—
80	C₄×C₅₃	#382	1	8	yes	—
81	C₂×#382	C₂×C₅₃	1	2	yes	—
82	C₂×C₄³	C₂×C₅₃	1	1	yes	characteristic
83	C₄⋊#95	C₂×C₅₃	1	4	yes	—
84	#1434	C₂×C₅₃	1	8	yes	—
85	C₂×C₅₃	C₂×#382	1	1	yes	characteristic
86	C₂×C₅₃	#2283	1	4	yes	—
87	C₂×C₅₃	#2285	1	2	yes	—
88	C₂×C₅₃	#2861	1	4	yes	—
89	C₂×C₅₃	#2864	1	4	yes	—
90	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
91	C₂²×C₄²	C₂²×C₅₃	1	4	yes	—
92	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
93	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
94	#382	—	2	8	no	—
95	C₂×#95	C₂²×C₅₃	1	16	yes	—
96	C₂×#95	C₂²×C₅₃	1	8	yes	—
97	C₄³	C₄×C₅₃	1	8	yes	complemented
98	C₅₃	#14689	1	1	yes	characteristic, direct product factor, Sylow
99	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
100	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
101	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
102	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
103	C₂×C₄⋊C₄	—	2	8	no	—
104	C₂×C₄⋊C₄	—	2	16	no	—
105	C₂×C₄²	—	2	16	no	—
106	C₂×C₄²	—	2	8	no	—
107	C₂×C₄²	C₂×C₄×C₅₃	1	16	yes	—
108	C₂×C₄²	C₂×C₄×C₅₃	1	8	yes	—
109	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
110	C₂×C₄²	D₄×C₅₃	1	4	yes	—
111	C₂×C₄²	Q₈×C₅₃	1	4	yes	—
112	#95	—	2	16	no	—
113	#95	—	2	32	no	—
114	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
115	C₂²×C₄	—	2	8	no	—
116	C₂²×C₄	—	2	8	no	—
117	C₂²×C₄	—	2	8	no	—
118	C₂²×C₄	—	2	8	no	—
119	C₂²×C₄	—	2	32	no	—
120	C₂²×C₄	—	2	16	no	—
121	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
122	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
123	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
124	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
125	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
126	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
127	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
128	C₂²×C₄	#41×C₅₃	1	16	yes	—
129	C₂²×C₄	#41×C₅₃	1	8	yes	—
130	C₄⋊C₄	—	4	16	no	—
131	C₄²	—	2	32	no	—
132	C₄²	—	2	16	no	—
133	C₄²	C₄⋊C₄×C₅₃	1	16	yes	—
134	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
135	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
136	C₂³	C₂²×Q₈×C₅₃	1	1	yes	characteristic
137	C₂³	C₂×#41×C₅₃	1	4	yes	—
138	C₂³	C₂×#41×C₅₃	1	8	yes	—
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	2	8	no	—
141	C₂×C₄	—	2	8	no	—
142	C₂×C₄	—	2	32	no	—
143	C₂×C₄	—	2	8	no	—
144	C₂×C₄	—	2	8	no	—
145	C₂×C₄	—	4	8	no	—
146	C₂×C₄	—	4	32	no	—
147	C₂×C₄	—	4	16	no	—
148	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	8	yes	—
149	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
150	C₂×C₄	#117×C₅₃	1	16	yes	—
151	C₂×C₄	#117×C₅₃	1	8	yes	—
152	C₂×C₄	#124×C₅₃	1	8	yes	—
153	C₂×C₄	#125×C₅₃	1	8	yes	—
154	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
155	C₂²	C₂×#117×C₅₃	1	2	yes	—
156	C₂²	C₂×#117×C₅₃	1	4	yes	—
157	C₂²	C₂×#124×C₅₃	1	2	yes	—
158	C₂²	C₂×#125×C₅₃	1	2	yes	—
159	C₂²	C₅₃×#529	1	16	yes	—
160	C₂²	C₅₃×#532	1	4	yes	—
161	C₂²	C₅₃×#533	1	4	yes	—
162	C₄	—	2	16	no	—
163	C₄	—	2	8	no	—
164	C₄	—	8	8	no	—
165	C₄	C₅₃×#382	1	8	yes	—
166	C₂	C₂×C₅₃×#382	1	1	yes	characteristic
167	C₂	C₅₃×#2283	1	4	yes	—
168	C₂	C₅₃×#2285	1	2	yes	—
169	C₂	C₅₃×#2861	1	4	yes	—
170	C₂	C₅₃×#2864	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
68. #14689	C₅₃	direct (N × H)
97. C₄³	C₄×C₅₃	semidirect
98. C₅₃	#14689	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#382	2
C₂×C₄³×C₅₃	2
C₅₃×C₄⋊#95	2
C₅₃×#1434	2
#14689	53

Subgroups — C53×#14689 #27222201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#14689 #27222201