Subgroups — C₅₃×#12596 #27220108

Order 13568.

310 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#347	C₂	1	2	yes	maximal
2	C₅₃×#1726	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1377	C₂	1	2	yes	maximal
4	C₅₃×#1376	C₂	1	2	yes	maximal
5	C₂×M₃₂×C₅₃	C₂²	1	4	yes	—
6	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
7	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#376	C₄	1	1	yes	characteristic
13	C₅₃×#347	C₄	1	8	yes	complemented
14	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
21	M₃₂×C₅₃	—	2	4	no	—
22	M₃₂×C₅₃	C₂×C₄	1	8	yes	—
23	C₂×C₁₆×C₅₃	—	2	4	no	—
24	#97×C₅₃	—	2	2	no	—
25	#97×C₅₃	—	2	2	no	—
26	#97×C₅₃	C₂×C₄	1	4	yes	—
27	C₄×C₈×C₅₃	—	2	2	no	—
28	C₄×C₈×C₅₃	—	2	2	no	—
29	C₄×C₈×C₅₃	—	2	2	no	—
30	#95×C₅₃	—	2	2	no	—
31	C₂²×C₄×C₅₃	—	2	1	no	—
32	C₂²×C₄×C₅₃	—	2	1	no	—
33	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
34	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₅₃	—	2	1	no	—
36	C₂×C₈×C₅₃	—	2	1	no	—
37	C₂×C₈×C₅₃	—	2	1	no	—
38	C₂×C₈×C₅₃	—	2	1	no	—
39	C₂×C₈×C₅₃	—	2	1	no	—
40	C₂×C₈×C₅₃	—	2	1	no	—
41	C₂×C₈×C₅₃	C₄²	1	2	yes	—
42	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
43	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
44	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
45	C₂×C₈×C₅₃	#41	1	2	yes	—
46	C₂×C₈×C₅₃	#41	1	2	yes	—
47	C₄²×C₅₃	—	2	1	no	—
48	C₄²×C₅₃	—	2	1	no	—
49	C₄²×C₅₃	—	2	1	no	—
50	C₄²×C₅₃	—	2	1	no	—
51	C₄²×C₅₃	C₄²	1	2	yes	—
52	C₄²×C₅₃	M₁₆	1	2	yes	—
53	C₁₆×C₅₃	—	2	8	no	—
54	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	2	2	no	—
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	4	2	no	—
59	C₂×C₄×C₅₃	—	4	2	no	—
60	C₂×C₄×C₅₃	—	4	2	no	—
61	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
67	C₈×C₅₃	—	2	2	no	—
68	C₈×C₅₃	—	2	2	no	—
69	C₈×C₅₃	—	2	2	no	—
70	C₈×C₅₃	#117	1	4	yes	—
71	#12596	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂²×C₅₃	—	2	2	no	—
73	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
74	C₂²×C₅₃	#432	1	1	yes	characteristic
75	C₂²×C₅₃	#432	1	1	yes	characteristic
76	C₄×C₅₃	—	4	2	no	—
77	C₄×C₅₃	—	4	2	no	—
78	C₄×C₅₃	—	4	2	no	—
79	C₄×C₅₃	#376	1	1	yes	characteristic
80	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
81	C₄×C₅₃	#432	1	1	yes	characteristic
82	C₄×C₅₃	#432	1	1	yes	characteristic
83	C₂×#347	C₂×C₅₃	1	2	yes	—
84	#1726	C₂×C₅₃	1	1	yes	characteristic
85	#1377	C₂×C₅₃	1	2	yes	—
86	#1376	C₂×C₅₃	1	2	yes	—
87	C₂×C₅₃	—	2	1	no	—
88	C₂×C₅₃	—	2	1	no	—
89	C₂×C₅₃	#1745	1	1	yes	characteristic
90	C₂×C₅₃	#2166	1	2	yes	—
91	C₂×M₃₂	C₂²×C₅₃	1	4	yes	—
92	C₂×#97	C₄×C₅₃	1	1	yes	characteristic
93	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
94	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
95	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
96	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
98	#376	C₄×C₅₃	1	1	yes	characteristic
99	#347	C₄×C₅₃	1	8	yes	complemented
100	C₅₃	#12596	1	1	yes	characteristic, direct product factor, Sylow
101	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
102	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
103	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
104	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
105	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
106	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
107	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
108	M₃₂	—	2	4	no	—
109	M₃₂	C₂×C₄×C₅₃	1	8	yes	—
110	C₂×C₁₆	—	2	4	no	—
111	#97	—	2	2	no	—
112	#97	—	2	2	no	—
113	#97	C₂×C₄×C₅₃	1	4	yes	—
114	C₄×C₈	—	2	2	no	—
115	C₄×C₈	—	2	2	no	—
116	C₄×C₈	—	2	2	no	—
117	#95	—	2	2	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	M₁₆×C₅₃	1	2	yes	—
121	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	C₄²×C₅₃	1	2	yes	—
129	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
130	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
131	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
132	C₂×C₈	#41×C₅₃	1	2	yes	—
133	C₂×C₈	#41×C₅₃	1	2	yes	—
134	C₄²	—	2	1	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	2	1	no	—
138	C₄²	C₄²×C₅₃	1	2	yes	—
139	C₄²	M₁₆×C₅₃	1	2	yes	—
140	C₁₆	—	2	8	no	—
141	C₂³	C₂×M₁₆×C₅₃	1	1	yes	characteristic
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
149	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
150	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
151	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
152	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
153	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	—	2	2	no	—
156	C₈	—	2	2	no	—
157	C₈	#117×C₅₃	1	4	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	C₄×M₁₆×C₅₃	1	1	yes	characteristic
160	C₂²	C₅₃×#432	1	1	yes	characteristic
161	C₂²	C₅₃×#432	1	1	yes	characteristic
162	C₄	—	4	2	no	—
163	C₄	—	4	2	no	—
164	C₄	—	4	2	no	—
165	C₄	C₅₃×#376	1	1	yes	characteristic
166	C₄	C₄×M₁₆×C₅₃	1	1	yes	characteristic
167	C₄	C₅₃×#432	1	1	yes	characteristic
168	C₄	C₅₃×#432	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	C₅₃×#1745	1	1	yes	characteristic
172	C₂	C₅₃×#2166	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	#117
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	#432
prime radical	C₂³×C₅₃	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×#97×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₅₃×#347	C₄	semidirect
71. #12596	C₅₃	direct (N × H)
99. #347	C₄×C₅₃	semidirect
100. C₅₃	#12596	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×#347	2
C₅₃×#1726	2
C₅₃×#1377	2
C₅₃×#1376	2
#12596	53

Subgroups — C53×#12596 #27220108

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12596 #27220108