Subgroups — C₅₃×#7209 #27214721

Order 13568.

222 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2162	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₆×C₅₃	C₄	1	2	yes	—
5	C₅₃×#431	C₄	1	1	yes	characteristic
6	C₄×M₁₆×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×C₄⋊C₁₆	—	2	1	no	—
9	C₅₃×C₄⋊C₁₆	—	2	1	no	—
10	C₅₃×C₄⋊C₁₆	—	2	1	no	—
11	C₅₃×C₂²⋊C₁₆	C₄	1	2	yes	—
12	C₄×C₁₆×C₅₃	—	2	1	no	—
13	C₅₃×#322	—	2	1	no	—
14	C₈²×C₅₃	—	2	1	no	—
15	C₂×M₁₆×C₅₃	C₈	1	2	yes	—
16	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₅₃	—	2	2	no	—
20	C₂×C₁₆×C₅₃	—	2	2	no	—
21	C₂×C₁₆×C₅₃	—	4	2	no	—
22	C₄⋊C₈×C₅₃	—	2	1	no	—
23	C₂²⋊C₈×C₅₃	C₈	1	2	yes	—
24	#97×C₅₃	—	2	1	no	—
25	C₄×C₈×C₅₃	—	2	1	no	—
26	C₄×C₈×C₅₃	—	2	1	no	—
27	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
34	M₁₆×C₅₃	—	2	2	no	—
35	M₁₆×C₅₃	C₁₆	1	4	yes	complemented
36	C₂×C₈×C₅₃	—	2	2	no	—
37	C₂×C₈×C₅₃	—	2	1	no	—
38	C₂×C₈×C₅₃	—	2	1	no	—
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	—	4	1	no	—
41	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
48	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
49	C₁₆×C₅₃	—	4	2	no	—
50	C₁₆×C₅₃	—	4	2	no	—
51	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	—	2	1	no	—
53	C₂×C₄×C₅₃	—	2	1	no	—
54	C₂×C₄×C₅₃	—	4	1	no	—
55	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
62	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
63	C₈×C₅₃	—	2	1	no	—
64	C₈×C₅₃	—	2	1	no	—
65	C₈×C₅₃	—	2	1	no	—
66	C₈×C₅₃	—	2	4	no	—
67	C₈×C₅₃	—	4	2	no	—
68	C₈×C₅₃	#100	1	1	yes	characteristic
69	C₈×C₅₃	#101	1	1	yes	characteristic
70	#7209	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂²×C₅₃	—	2	1	no	—
72	C₂²×C₅₃	—	2	1	no	—
73	C₂²×C₅₃	#336	1	1	yes	characteristic
74	C₂²×C₅₃	C₄×C₁₆	1	1	yes	characteristic
75	C₂²×C₅₃	#346	1	1	yes	characteristic
76	C₄×C₅₃	—	2	1	no	—
77	C₄×C₅₃	—	2	1	no	—
78	C₄×C₅₃	—	2	1	no	—
79	C₄×C₅₃	#343	1	1	yes	characteristic
80	C₄×C₅₃	#344	1	1	yes	characteristic
81	C₄×C₅₃	C₂²⋊C₁₆	1	2	yes	—
82	C₄×C₅₃	C₄⋊C₁₆	1	2	yes	—
83	#2162	C₂×C₅₃	1	1	yes	characteristic
84	C₂×C₄⋊C₁₆	C₂×C₅₃	1	1	yes	characteristic
85	C₈×M₁₆	C₂×C₅₃	1	1	yes	characteristic
86	C₂×C₅₃	—	2	1	no	—
87	C₂×C₅₃	—	2	1	no	—
88	C₂×C₅₃	#1283	1	1	yes	characteristic
89	C₂×C₅₃	#1374	1	1	yes	characteristic
90	C₂×C₅₃	#1378	1	1	yes	characteristic
91	C₂²×C₁₆	C₄×C₅₃	1	2	yes	—
92	#431	C₄×C₅₃	1	1	yes	characteristic
93	C₄×M₁₆	C₄×C₅₃	1	1	yes	characteristic
94	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
95	C₄⋊C₁₆	—	2	1	no	—
96	C₄⋊C₁₆	—	2	1	no	—
97	C₄⋊C₁₆	—	2	1	no	—
98	C₂²⋊C₁₆	C₄×C₅₃	1	2	yes	—
99	C₄×C₁₆	—	2	1	no	—
100	#322	—	2	1	no	—
101	C₈²	—	2	1	no	—
102	C₅₃	#7209	1	1	yes	characteristic, direct product factor, Sylow
103	C₂×M₁₆	C₈×C₅₃	1	2	yes	—
104	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
105	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
106	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
107	C₂×C₁₆	—	2	2	no	—
108	C₂×C₁₆	—	2	2	no	—
109	C₂×C₁₆	—	4	2	no	—
110	C₄⋊C₈	—	2	1	no	—
111	C₂²⋊C₈	C₈×C₅₃	1	2	yes	—
112	#97	—	2	1	no	—
113	C₄×C₈	—	2	1	no	—
114	C₄×C₈	—	2	1	no	—
115	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
116	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
117	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
118	C₄×C₈	Q₈×C₅₃	1	1	yes	characteristic
119	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
120	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
121	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
122	M₁₆	—	2	2	no	—
123	M₁₆	C₁₆×C₅₃	1	4	yes	complemented
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	4	1	no	—
129	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
130	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
131	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
132	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
133	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
134	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
135	C₄²	M₁₆×C₅₃	1	1	yes	characteristic
136	C₄²	M₁₆×C₅₃	1	1	yes	characteristic
137	C₁₆	—	4	2	no	—
138	C₁₆	—	4	2	no	—
139	C₂³	C₄×C₈×C₅₃	1	1	yes	characteristic
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
144	C₂×C₄	#97×C₅₃	1	1	yes	characteristic
145	C₂×C₄	C₂²⋊C₈×C₅₃	1	1	yes	characteristic
146	C₂×C₄	C₂²⋊C₈×C₅₃	1	1	yes	characteristic
147	C₂×C₄	C₄⋊C₈×C₅₃	1	1	yes	characteristic
148	C₂×C₄	C₄⋊C₈×C₅₃	1	1	yes	characteristic
149	C₂×C₄	C₂×C₁₆×C₅₃	1	2	yes	—
150	C₂×C₄	M₃₂×C₅₃	1	2	yes	—
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	2	4	no	—
155	C₈	—	4	2	no	—
156	C₈	#100×C₅₃	1	1	yes	characteristic
157	C₈	#101×C₅₃	1	1	yes	characteristic
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	C₅₃×#336	1	1	yes	characteristic
161	C₂²	C₄×C₁₆×C₅₃	1	1	yes	characteristic
162	C₂²	C₅₃×#346	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	C₅₃×#343	1	1	yes	characteristic
167	C₄	C₅₃×#344	1	1	yes	characteristic
168	C₄	C₅₃×C₂²⋊C₁₆	1	2	yes	—
169	C₄	C₅₃×C₄⋊C₁₆	1	2	yes	—
170	C₂	—	2	1	no	—
171	C₂	—	2	1	no	—
172	C₂	C₅₃×#1283	1	1	yes	characteristic
173	C₂	C₅₃×#1374	1	1	yes	characteristic
174	C₂	C₅₃×#1378	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₅₃	C₂²⋊C₄
commutator	C₂²	C₄×C₁₆×C₅₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅₃
socle	C₂²×C₅₃	#336
prime radical	C₂³×C₅₃	C₄×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
35. M₁₆×C₅₃	C₁₆	semidirect
70. #7209	C₅₃	direct (N × H)
102. C₅₃	#7209	direct (N × H)
123. M₁₆	C₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2162	2
C₂×C₅₃×C₄⋊C₁₆	2
C₈×M₁₆×C₅₃	2
#7209	53

Subgroups — C53×#7209 #27214721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7209 #27214721