Subgroups — C₅₃×#14684 #27222196

Order 13568.

2286 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#381	C₂	1	2	yes	maximal
3	C₅₃×#1434	C₂	1	8	yes	maximal, complemented
4	C₅₃×C₂²⋊#95	C₂	1	4	yes	maximal
5	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
6	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
7	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
8	C₂²×C₄²×C₅₃	—	2	8	no	—
9	C₂²×C₄²×C₅₃	C₄	1	8	yes	complemented
10	C₂²×C₄²×C₅₃	C₂²	1	4	yes	—
11	C₅₃×#381	—	2	8	no	—
12	C₂×#95×C₅₃	C₂²	1	16	yes	—
13	C₂×#95×C₅₃	C₂²	1	8	yes	—
14	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
15	C₂³×C₄×C₅₃	—	2	8	no	—
16	C₂³×C₄×C₅₃	—	2	8	no	—
17	C₂³×C₄×C₅₃	—	2	16	no	—
18	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	—
19	C₂³×C₄×C₅₃	C₂×C₄	1	4	yes	—
20	C₂³×C₄×C₅₃	D₄	1	8	yes	—
21	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
22	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
23	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
24	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
25	C₂×C₂²⋊C₄×C₅₃	—	2	16	no	—
26	C₂×C₄²×C₅₃	—	2	32	no	—
27	C₂×C₄²×C₅₃	—	2	16	no	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	16	yes	—
29	#95×C₅₃	—	2	16	no	—
30	#95×C₅₃	—	2	32	no	—
31	C₂⁴×C₅₃	—	2	8	no	—
32	C₂⁴×C₅₃	C₂²×C₄	1	2	yes	—
33	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
34	C₂⁴×C₅₃	#41	1	8	yes	—
35	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	—	2	64	no	—
37	C₂²×C₄×C₅₃	—	2	16	no	—
38	C₂²×C₄×C₅₃	—	2	16	no	—
39	C₂²×C₄×C₅₃	—	2	32	no	—
40	C₂²×C₄×C₅₃	—	2	8	no	—
41	C₂²×C₄×C₅₃	—	2	8	no	—
42	C₂²×C₄×C₅₃	—	2	8	no	—
43	C₂²×C₄×C₅₃	—	2	32	no	—
44	C₂²×C₄×C₅₃	—	2	8	no	—
45	C₂²×C₄×C₅₃	—	2	32	no	—
46	C₂²×C₄×C₅₃	—	2	16	no	—
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	16	yes	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
49	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
50	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
51	C₂²×C₄×C₅₃	#41	1	16	yes	—
52	C₂²⋊C₄×C₅₃	—	4	16	no	—
53	C₄²×C₅₃	—	2	32	no	—
54	C₂³×C₅₃	—	2	8	no	—
55	C₂³×C₅₃	—	2	16	no	—
56	C₂³×C₅₃	—	2	32	no	—
57	C₂³×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
58	C₂³×C₅₃	#117	1	8	yes	—
59	C₂³×C₅₃	#123	1	16	yes	—
60	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
61	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
62	C₂³×C₅₃	C₂×#41	1	4	yes	—
63	C₂³×C₅₃	C₂×#41	1	8	yes	—
64	C₂×C₄×C₅₃	—	2	64	no	—
65	C₂×C₄×C₅₃	—	2	8	no	—
66	C₂×C₄×C₅₃	—	2	8	no	—
67	C₂×C₄×C₅₃	—	2	8	no	—
68	C₂×C₄×C₅₃	—	2	16	no	—
69	C₂×C₄×C₅₃	—	2	32	no	—
70	C₂×C₄×C₅₃	—	2	32	no	—
71	C₂×C₄×C₅₃	—	4	8	no	—
72	C₂×C₄×C₅₃	—	4	16	no	—
73	C₂×C₄×C₅₃	—	4	32	no	—
74	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	8	yes	—
75	C₂×C₄×C₅₃	#117	1	16	yes	—
76	#14684	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂²×C₅₃	—	2	32	no	—
78	C₂²×C₅₃	—	2	8	no	—
79	C₂²×C₅₃	—	2	16	no	—
80	C₂²×C₅₃	#381	1	8	yes	—
81	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₅₃	C₂×#117	1	2	yes	—
83	C₂²×C₅₃	C₂×#117	1	4	yes	—
84	C₂²×C₅₃	C₂×#123	1	4	yes	—
85	C₂²×C₅₃	#525	1	8	yes	—
86	C₂²×C₅₃	#529	1	16	yes	—
87	C₄×C₅₃	—	2	8	no	—
88	C₄×C₅₃	—	2	16	no	—
89	C₄×C₅₃	—	8	8	no	—
90	C₂³×C₄²	C₂×C₅₃	1	1	yes	characteristic
91	C₂×#381	C₂×C₅₃	1	2	yes	—
92	#1434	C₂×C₅₃	1	8	yes	complemented
93	C₂²⋊#95	C₂×C₅₃	1	4	yes	—
94	C₂×C₅₃	—	2	8	no	—
95	C₂×C₅₃	C₂×#381	1	1	yes	characteristic
96	C₂×C₅₃	#2280	1	2	yes	—
97	C₂×C₅₃	#2283	1	4	yes	—
98	C₂×C₅₃	#2853	1	8	yes	—
99	C₂⁴×C₄	C₂²×C₅₃	1	2	yes	—
100	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
101	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
102	C₂²×C₄²	—	2	8	no	—
103	C₂²×C₄²	C₄×C₅₃	1	8	yes	complemented
104	C₂²×C₄²	C₂²×C₅₃	1	4	yes	—
105	#381	—	2	8	no	—
106	C₂×#95	C₂²×C₅₃	1	16	yes	—
107	C₂×#95	C₂²×C₅₃	1	8	yes	—
108	C₅₃	#14684	1	1	yes	characteristic, direct product factor, Sylow
109	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
110	C₂³×C₄	—	2	8	no	—
111	C₂³×C₄	—	2	8	no	—
112	C₂³×C₄	—	2	16	no	—
113	C₂³×C₄	C₂×C₄×C₅₃	1	8	yes	—
114	C₂³×C₄	C₂×C₄×C₅₃	1	4	yes	—
115	C₂³×C₄	D₄×C₅₃	1	8	yes	—
116	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
117	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
118	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
119	C₂×C₂²⋊C₄	—	2	8	no	—
120	C₂×C₂²⋊C₄	—	2	16	no	—
121	C₂×C₄²	—	2	16	no	—
122	C₂×C₄²	—	2	32	no	—
123	C₂×C₄²	C₂×C₄×C₅₃	1	16	yes	—
124	#95	—	2	16	no	—
125	#95	—	2	32	no	—
126	C₂⁴	—	2	8	no	—
127	C₂⁴	C₂²×C₄×C₅₃	1	2	yes	—
128	C₂⁴	C₂×D₄×C₅₃	1	4	yes	—
129	C₂⁴	#41×C₅₃	1	8	yes	—
130	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
131	C₂²×C₄	—	2	64	no	—
132	C₂²×C₄	—	2	16	no	—
133	C₂²×C₄	—	2	16	no	—
134	C₂²×C₄	—	2	32	no	—
135	C₂²×C₄	—	2	8	no	—
136	C₂²×C₄	—	2	8	no	—
137	C₂²×C₄	—	2	8	no	—
138	C₂²×C₄	—	2	32	no	—
139	C₂²×C₄	—	2	8	no	—
140	C₂²×C₄	—	2	32	no	—
141	C₂²×C₄	—	2	16	no	—
142	C₂²×C₄	C₂²⋊C₄×C₅₃	1	16	yes	—
143	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
144	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
145	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
146	C₂²×C₄	#41×C₅₃	1	16	yes	—
147	C₂²⋊C₄	—	4	16	no	—
148	C₄²	—	2	32	no	—
149	C₂³	—	2	8	no	—
150	C₂³	—	2	16	no	—
151	C₂³	—	2	32	no	—
152	C₂³	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
153	C₂³	#117×C₅₃	1	8	yes	—
154	C₂³	#123×C₅₃	1	16	yes	—
155	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
156	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
157	C₂³	C₂×#41×C₅₃	1	4	yes	—
158	C₂³	C₂×#41×C₅₃	1	8	yes	—
159	C₂×C₄	—	2	64	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	2	8	no	—
162	C₂×C₄	—	2	8	no	—
163	C₂×C₄	—	2	16	no	—
164	C₂×C₄	—	2	32	no	—
165	C₂×C₄	—	2	32	no	—
166	C₂×C₄	—	4	8	no	—
167	C₂×C₄	—	4	16	no	—
168	C₂×C₄	—	4	32	no	—
169	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	8	yes	—
170	C₂×C₄	#117×C₅₃	1	16	yes	—
171	C₂²	—	2	8	no	—
172	C₂²	—	2	16	no	—
173	C₂²	—	2	32	no	—
174	C₂²	C₅₃×#381	1	8	yes	—
175	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
176	C₂²	C₂×#117×C₅₃	1	2	yes	—
177	C₂²	C₂×#117×C₅₃	1	4	yes	—
178	C₂²	C₂×#123×C₅₃	1	4	yes	—
179	C₂²	C₅₃×#525	1	8	yes	—
180	C₂²	C₅₃×#529	1	16	yes	—
181	C₄	—	2	8	no	—
182	C₄	—	2	16	no	—
183	C₄	—	8	8	no	—
184	C₂	—	2	8	no	—
185	C₂	C₂×C₅₃×#381	1	1	yes	characteristic
186	C₂	C₅₃×#2280	1	2	yes	—
187	C₂	C₅₃×#2283	1	4	yes	—
188	C₂	C₅₃×#2853	1	8	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₅₃
Frattini, cosocle	C₂⁴	C₂⁴×C₅₃
prime radical	C₂⁵×C₅₃	C₂³
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
3. C₅₃×#1434	C₂	semidirect
9. C₂²×C₄²×C₅₃	C₄	semidirect
76. #14684	C₅₃	direct (N × H)
92. #1434	C₂×C₅₃	semidirect
103. C₂²×C₄²	C₄×C₅₃	semidirect
108. C₅₃	#14684	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄²×C₅₃	2
C₂×C₅₃×#381	2
C₅₃×#1434	2
C₅₃×C₂²⋊#95	2
#14684	53

Subgroups — C53×#14684 #27222196

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#14684 #27222196